Subgroups — C₆₇×#61214 #30740465

Order 17152.

2026 conjugacy classes of subgroups.

The subgroup lattice has 818 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3507	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3497	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3333	C₂	1	2	yes	maximal, complemented
4	C₆₇×#3328	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3327	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3304	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3300	C₂	1	4	yes	maximal, complemented
8	C₆₇×#3299	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3131	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#3098	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3078	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#3063	C₂	1	2	yes	maximal, complemented
13	C₆₇×#3037	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#3011	C₂	1	2	yes	maximal, complemented
15	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
17	C₆₇×#2898	C₂	1	2	yes	maximal, complemented
18	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	2	yes	—
19	C₂×#137×C₆₇	C₂²	1	2	yes	—
20	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#131×C₆₇	C₂²	1	2	yes	—
22	#559×C₆₇	—	2	1	no	—
23	#557×C₆₇	—	2	1	no	—
24	#556×C₆₇	—	2	2	no	—
25	#556×C₆₇	C₂²	1	2	yes	—
26	#554×C₆₇	C₂²	1	2	yes	—
27	#552×C₆₇	C₂²	1	4	yes	—
28	#550×C₆₇	—	2	1	no	—
29	#550×C₆₇	C₂²	1	2	yes	—
30	D₄×Q₈×C₆₇	—	2	1	no	—
31	D₄×Q₈×C₆₇	C₂²	1	2	yes	—
32	#548×C₆₇	—	2	2	no	—
33	#548×C₆₇	C₂²	1	4	yes	—
34	#539×C₆₇	—	2	2	no	—
35	#537×C₆₇	—	2	2	no	—
36	#537×C₆₇	C₂²	1	2	yes	—
37	#536×C₆₇	C₂²	1	2	yes	—
38	#533×C₆₇	C₂²	1	1	yes	characteristic
39	#532×C₆₇	—	2	2	no	—
40	#532×C₆₇	C₂²	1	1	yes	characteristic
41	#532×C₆₇	C₂²	1	1	yes	characteristic
42	#529×C₆₇	—	2	2	no	—
43	#520×C₆₇	C₂²	1	2	yes	—
44	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
46	#488×C₆₇	C₂²	1	2	yes	—
47	#488×C₆₇	C₂²	1	2	yes	—
48	#487×C₆₇	C₂²	1	1	yes	characteristic
49	#487×C₆₇	C₂²	1	1	yes	characteristic
50	#486×C₆₇	C₂²	1	1	yes	characteristic
51	#486×C₆₇	C₂²	1	1	yes	characteristic
52	#485×C₆₇	C₂²	1	4	yes	—
53	#484×C₆₇	C₂²	1	4	yes	—
54	#482×C₆₇	C₂²	1	4	yes	—
55	#480×C₆₇	C₂²	1	4	yes	—
56	#479×C₆₇	C₂²	1	4	yes	—
57	#477×C₆₇	C₂²	1	2	yes	—
58	#475×C₆₇	C₂²	1	2	yes	—
59	#470×C₆₇	C₂²	1	4	yes	—
60	#468×C₆₇	C₂²	1	4	yes	—
61	#467×C₆₇	C₂²	1	4	yes	—
62	#465×C₆₇	C₂²	1	4	yes	—
63	#464×C₆₇	C₂²	1	4	yes	—
64	#462×C₆₇	C₂²	1	2	yes	—
65	#460×C₆₇	C₂²	1	2	yes	—
66	#452×C₆₇	C₂²	1	4	yes	—
67	#452×C₆₇	C₂²	1	4	yes	—
68	#451×C₆₇	C₂²	1	4	yes	—
69	#451×C₆₇	C₂²	1	4	yes	—
70	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
71	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
72	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
73	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
74	#436×C₆₇	C₂²	1	2	yes	—
75	#436×C₆₇	C₂²	1	2	yes	—
76	C₈×D₄×C₆₇	C₂²	1	2	yes	—
77	C₈×D₄×C₆₇	C₂²	1	2	yes	—
78	#428×C₆₇	C₂²	1	2	yes	—
79	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
80	#423×C₆₇	C₂²	1	1	yes	characteristic
81	#419×C₆₇	C₂²	1	2	yes	—
82	#419×C₆₇	C₂²	1	4	yes	—
83	#417×C₆₇	C₂²	1	2	yes	—
84	#417×C₆₇	C₂²	1	4	yes	—
85	#416×C₆₇	C₂²	1	2	yes	—
86	#416×C₆₇	C₂²	1	4	yes	—
87	#416×C₆₇	C₂²	1	2	yes	—
88	C₂×#103×C₆₇	C₂²	1	4	yes	—
89	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
90	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
91	#408×C₆₇	C₂²	1	2	yes	—
92	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
93	2₋¹⁺⁴×C₆₇	—	2	8	no	—
94	2₋¹⁺⁴×C₆₇	—	4	4	no	—
95	C₂×#41×C₆₇	—	2	4	no	—
96	C₂×#41×C₆₇	—	2	2	no	—
97	C₂×#41×C₆₇	C₂³	1	2	yes	—
98	C₂×#41×C₆₇	C₂³	1	4	yes	—
99	C₂×#41×C₆₇	C₂³	1	2	yes	—
100	C₂²×Q₈×C₆₇	—	2	2	no	—
101	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
102	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
103	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
104	#137×C₆₇	—	4	4	no	—
105	C₂×Q₁₆×C₆₇	—	2	4	no	—
106	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
107	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
108	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
109	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
110	#131×C₆₇	—	4	4	no	—
111	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
112	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
113	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
114	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
115	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
116	C₄⋊Q₈×C₆₇	—	2	2	no	—
117	C₄⋊Q₈×C₆₇	—	2	2	no	—
118	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
119	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
120	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
121	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
122	#127×C₆₇	—	2	2	no	—
123	#127×C₆₇	D₄	1	1	yes	characteristic
124	#127×C₆₇	C₂³	1	1	yes	characteristic
125	#126×C₆₇	—	2	4	no	—
126	#126×C₆₇	—	2	4	no	—
127	#125×C₆₇	—	2	2	no	—
128	#125×C₆₇	—	2	4	no	—
129	#125×C₆₇	C₂³	1	2	yes	—
130	#124×C₆₇	—	2	2	no	—
131	#124×C₆₇	—	2	2	no	—
132	#124×C₆₇	—	2	4	no	—
133	#124×C₆₇	D₄	1	2	yes	—
134	#124×C₆₇	C₂³	1	2	yes	—
135	#123×C₆₇	—	2	4	no	—
136	#123×C₆₇	—	2	4	no	—
137	C₂²⋊Q₈×C₆₇	—	2	2	no	—
138	C₂²⋊Q₈×C₆₇	—	2	1	no	—
139	C₂²⋊Q₈×C₆₇	—	2	1	no	—
140	C₂²⋊Q₈×C₆₇	—	2	4	no	—
141	C₂²⋊Q₈×C₆₇	—	2	4	no	—
142	C₂²⋊Q₈×C₆₇	—	2	4	no	—
143	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
144	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
145	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
146	#121×C₆₇	—	2	4	no	—
147	#121×C₆₇	—	2	4	no	—
148	#121×C₆₇	—	2	2	no	—
149	#121×C₆₇	—	2	1	no	—
150	#121×C₆₇	—	2	1	no	—
151	#121×C₆₇	—	2	4	no	—
152	#121×C₆₇	D₄	1	2	yes	—
153	#121×C₆₇	D₄	1	2	yes	—
154	#121×C₆₇	C₂³	1	4	yes	—
155	C₄×Q₈×C₆₇	—	2	2	no	—
156	C₄×Q₈×C₆₇	—	2	2	no	—
157	C₄×Q₈×C₆₇	—	2	1	no	—
158	C₄×Q₈×C₆₇	—	2	1	no	—
159	C₄×Q₈×C₆₇	—	2	1	no	—
160	C₄×Q₈×C₆₇	—	2	1	no	—
161	C₄×Q₈×C₆₇	—	2	1	no	—
162	C₄×Q₈×C₆₇	—	2	1	no	—
163	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
164	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
165	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
166	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
167	C₄×D₄×C₆₇	—	2	4	no	—
168	C₄×D₄×C₆₇	—	2	2	no	—
169	C₄×D₄×C₆₇	—	2	2	no	—
170	C₄×D₄×C₆₇	—	2	2	no	—
171	C₄×D₄×C₆₇	—	2	2	no	—
172	C₄×D₄×C₆₇	—	2	4	no	—
173	C₄×D₄×C₆₇	—	2	1	no	—
174	C₄×D₄×C₆₇	—	2	1	no	—
175	C₄×D₄×C₆₇	C₂³	1	2	yes	—
176	C₄×D₄×C₆₇	C₂³	1	2	yes	—
177	#117×C₆₇	—	2	4	no	—
178	#117×C₆₇	C₂³	1	4	yes	—
179	#117×C₆₇	C₂³	1	2	yes	—
180	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
181	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
182	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
183	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
184	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
185	C₂×C₄²×C₆₇	—	2	1	no	—
186	C₂×C₄²×C₆₇	—	2	1	no	—
187	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
188	#107×C₆₇	C₂³	1	4	yes	—
189	#107×C₆₇	C₂³	1	2	yes	—
190	#107×C₆₇	C₂³	1	2	yes	—
191	#106×C₆₇	C₂³	1	4	yes	—
192	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
193	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
194	#103×C₆₇	C₂³	1	2	yes	—
195	#103×C₆₇	C₂³	1	2	yes	—
196	#103×C₆₇	C₂³	1	4	yes	—
197	#103×C₆₇	C₂³	1	4	yes	—
198	#103×C₆₇	C₂³	1	4	yes	—
199	#103×C₆₇	C₂³	1	2	yes	—
200	#103×C₆₇	C₂³	1	2	yes	—
201	#103×C₆₇	C₂³	1	2	yes	—
202	#103×C₆₇	C₂³	1	2	yes	—
203	#103×C₆₇	C₂³	1	2	yes	—
204	#103×C₆₇	C₂³	1	2	yes	—
205	#102×C₆₇	C₂³	1	2	yes	—
206	#102×C₆₇	C₂³	1	2	yes	—
207	#102×C₆₇	C₂³	1	4	yes	—
208	#102×C₆₇	C₂³	1	2	yes	—
209	#102×C₆₇	C₂³	1	2	yes	—
210	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
211	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
212	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
213	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
214	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
215	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
216	#41×C₆₇	—	2	4	no	—
217	#41×C₆₇	—	2	4	no	—
218	#41×C₆₇	—	2	8	no	—
219	#41×C₆₇	—	2	4	no	—
220	#41×C₆₇	—	2	4	no	—
221	#41×C₆₇	—	4	4	no	—
222	#41×C₆₇	—	4	16	no	—
223	#41×C₆₇	—	4	8	no	—
224	C₂×Q₈×C₆₇	—	2	1	no	—
225	C₂×Q₈×C₆₇	—	2	1	no	—
226	C₂×Q₈×C₆₇	—	2	1	no	—
227	C₂×Q₈×C₆₇	—	2	4	no	—
228	C₂×Q₈×C₆₇	—	2	1	no	—
229	C₂×Q₈×C₆₇	—	2	2	no	—
230	C₂×Q₈×C₆₇	—	2	8	no	—
231	C₂×Q₈×C₆₇	—	2	2	no	—
232	C₂×Q₈×C₆₇	—	2	2	no	—
233	C₂×Q₈×C₆₇	—	4	4	no	—
234	C₂×Q₈×C₆₇	—	4	8	no	—
235	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
236	C₂×Q₈×C₆₇	#41	1	4	yes	—
237	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
238	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
239	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
240	C₂×D₄×C₆₇	—	2	2	no	—
241	C₂×D₄×C₆₇	—	2	4	no	—
242	C₂×D₄×C₆₇	—	2	4	no	—
243	C₂×D₄×C₆₇	—	2	1	no	—
244	C₂×D₄×C₆₇	—	2	1	no	—
245	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
246	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
247	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
248	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
249	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
250	C₂²×C₄×C₆₇	—	2	1	no	—
251	C₂²×C₄×C₆₇	—	2	1	no	—
252	C₂²×C₄×C₆₇	—	2	4	no	—
253	C₂²×C₄×C₆₇	—	2	1	no	—
254	C₂²×C₄×C₆₇	—	2	1	no	—
255	C₂²×C₄×C₆₇	—	2	4	no	—
256	C₂²×C₄×C₆₇	—	2	4	no	—
257	C₂²×C₄×C₆₇	—	2	2	no	—
258	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
259	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
260	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
261	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
262	Q₁₆×C₆₇	—	2	4	no	—
263	Q₁₆×C₆₇	—	2	4	no	—
264	Q₁₆×C₆₇	—	4	4	no	—
265	SD₁₆×C₆₇	—	4	4	no	—
266	M₁₆×C₆₇	—	2	4	no	—
267	M₁₆×C₆₇	—	4	4	no	—
268	C₂×C₈×C₆₇	—	2	2	no	—
269	C₂×C₈×C₆₇	—	2	2	no	—
270	C₂×C₈×C₆₇	—	4	4	no	—
271	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
272	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
273	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
274	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
275	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
276	C₄⋊C₄×C₆₇	—	2	2	no	—
277	C₄⋊C₄×C₆₇	—	2	2	no	—
278	C₄⋊C₄×C₆₇	—	2	2	no	—
279	C₄⋊C₄×C₆₇	—	2	2	no	—
280	C₄⋊C₄×C₆₇	—	2	1	no	—
281	C₄⋊C₄×C₆₇	—	2	4	no	—
282	C₄⋊C₄×C₆₇	—	2	1	no	—
283	C₄⋊C₄×C₆₇	—	2	2	no	—
284	C₄⋊C₄×C₆₇	—	2	4	no	—
285	C₄⋊C₄×C₆₇	—	2	1	no	—
286	C₄⋊C₄×C₆₇	—	2	1	no	—
287	C₄⋊C₄×C₆₇	—	2	1	no	—
288	C₄⋊C₄×C₆₇	—	2	4	no	—
289	C₄⋊C₄×C₆₇	—	2	2	no	—
290	C₄⋊C₄×C₆₇	—	2	1	no	—
291	C₄⋊C₄×C₆₇	—	2	4	no	—
292	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
293	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
294	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
295	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
296	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
297	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
298	C₂²⋊C₄×C₆₇	—	2	4	no	—
299	C₂²⋊C₄×C₆₇	—	2	2	no	—
300	C₂²⋊C₄×C₆₇	—	2	4	no	—
301	C₂²⋊C₄×C₆₇	—	2	4	no	—
302	C₂²⋊C₄×C₆₇	—	2	4	no	—
303	C₂²⋊C₄×C₆₇	—	2	2	no	—
304	C₂²⋊C₄×C₆₇	—	2	2	no	—
305	C₂²⋊C₄×C₆₇	—	2	4	no	—
306	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
307	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
308	C₄²×C₆₇	—	2	2	no	—
309	C₄²×C₆₇	—	2	2	no	—
310	C₄²×C₆₇	—	2	2	no	—
311	C₄²×C₆₇	—	2	2	no	—
312	C₄²×C₆₇	—	2	4	no	—
313	C₄²×C₆₇	—	2	2	no	—
314	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
315	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
316	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
317	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
318	C₂³×C₆₇	—	2	2	no	—
319	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
320	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
321	Q₈×C₆₇	—	2	4	no	—
322	Q₈×C₆₇	—	2	4	no	—
323	Q₈×C₆₇	—	2	2	no	—
324	Q₈×C₆₇	—	2	2	no	—
325	Q₈×C₆₇	—	2	2	no	—
326	Q₈×C₆₇	—	2	2	no	—
327	Q₈×C₆₇	—	2	2	no	—
328	Q₈×C₆₇	—	4	8	no	—
329	Q₈×C₆₇	—	4	2	no	—
330	Q₈×C₆₇	—	4	2	no	—
331	Q₈×C₆₇	—	8	1	no	—
332	Q₈×C₆₇	C₂×#41	1	4	yes	—
333	D₄×C₆₇	—	2	8	no	—
334	D₄×C₆₇	—	2	4	no	—
335	D₄×C₆₇	—	2	4	no	—
336	D₄×C₆₇	—	2	2	no	—
337	D₄×C₆₇	—	2	2	no	—
338	D₄×C₆₇	—	4	2	no	—
339	D₄×C₆₇	—	4	8	no	—
340	D₄×C₆₇	—	4	4	no	—
341	D₄×C₆₇	—	4	2	no	—
342	D₄×C₆₇	—	8	1	no	—
343	D₄×C₆₇	—	8	4	no	—
344	C₂×C₄×C₆₇	—	2	4	no	—
345	C₂×C₄×C₆₇	—	2	2	no	—
346	C₂×C₄×C₆₇	—	2	2	no	—
347	C₂×C₄×C₆₇	—	2	4	no	—
348	C₂×C₄×C₆₇	—	2	4	no	—
349	C₂×C₄×C₆₇	—	2	2	no	—
350	C₂×C₄×C₆₇	—	2	2	no	—
351	C₂×C₄×C₆₇	—	2	2	no	—
352	C₂×C₄×C₆₇	—	2	2	no	—
353	C₂×C₄×C₆₇	—	2	2	no	—
354	C₂×C₄×C₆₇	—	2	2	no	—
355	C₂×C₄×C₆₇	—	4	1	no	—
356	C₂×C₄×C₆₇	—	4	8	no	—
357	C₂×C₄×C₆₇	—	4	8	no	—
358	C₂×C₄×C₆₇	—	4	2	no	—
359	C₂×C₄×C₆₇	—	4	4	no	—
360	C₂×C₄×C₆₇	—	8	1	no	—
361	C₂×C₄×C₆₇	—	8	4	no	—
362	C₂×C₄×C₆₇	—	8	1	no	—
363	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
364	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
365	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
366	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
367	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
368	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
369	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
370	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
371	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
372	C₈×C₆₇	—	2	1	no	—
373	C₈×C₆₇	—	2	1	no	—
374	C₈×C₆₇	—	2	1	no	—
375	C₈×C₆₇	—	2	1	no	—
376	C₈×C₆₇	—	2	4	no	—
377	C₂²×C₆₇	—	2	1	no	—
378	C₂²×C₆₇	—	2	1	no	—
379	C₂²×C₆₇	—	2	4	no	—
380	C₂²×C₆₇	—	2	4	no	—
381	C₂²×C₆₇	—	2	4	no	—
382	C₂²×C₆₇	—	4	2	no	—
383	C₂²×C₆₇	—	4	2	no	—
384	C₂²×C₆₇	—	4	2	no	—
385	C₂²×C₆₇	#546	1	2	yes	—
386	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
387	C₄×C₆₇	—	2	1	no	—
388	C₄×C₆₇	—	2	1	no	—
389	C₄×C₆₇	—	2	4	no	—
390	C₄×C₆₇	—	2	4	no	—
391	C₄×C₆₇	—	2	1	no	—
392	C₄×C₆₇	—	4	2	no	—
393	C₄×C₆₇	—	4	2	no	—
394	C₄×C₆₇	—	4	2	no	—
395	C₄×C₆₇	—	4	2	no	—
396	C₄×C₆₇	—	4	2	no	—
397	C₄×C₆₇	—	4	2	no	—
398	C₄×C₆₇	#546	1	2	yes	—
399	C₄×C₆₇	#578	1	2	yes	—
400	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
401	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
402	#61214	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
403	C₂×C₆₇	—	2	2	no	—
404	C₂×C₆₇	—	4	4	no	—
405	C₂×C₆₇	—	8	2	no	—
406	C₂×C₆₇	C₂×#546	1	1	yes	characteristic
407	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
408	C₂×C₆₇	#3584	1	1	yes	characteristic
409	#3507	C₂×C₆₇	1	1	yes	characteristic, complemented
410	#3497	C₂×C₆₇	1	1	yes	characteristic
411	#3333	C₂×C₆₇	1	2	yes	complemented
412	#3328	C₂×C₆₇	1	2	yes	complemented
413	#3327	C₂×C₆₇	1	4	yes	complemented
414	#3304	C₂×C₆₇	1	2	yes	complemented
415	#3300	C₂×C₆₇	1	4	yes	complemented
416	#3299	C₂×C₆₇	1	2	yes	complemented
417	#3131	C₂×C₆₇	1	1	yes	characteristic, complemented
418	#3098	C₂×C₆₇	1	2	yes	complemented
419	#3078	C₂×C₆₇	1	1	yes	characteristic, complemented
420	#3063	C₂×C₆₇	1	2	yes	complemented
421	#3037	C₂×C₆₇	1	1	yes	characteristic, complemented
422	#3011	C₂×C₆₇	1	2	yes	complemented
423	C₂×C₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
424	#2931	C₂×C₆₇	1	1	yes	characteristic, complemented
425	#2898	C₂×C₆₇	1	2	yes	complemented
426	C₆₇	#61214	1	1	yes	characteristic, direct product factor, Sylow
427	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	2	yes	—
428	C₂×#137	C₂²×C₆₇	1	2	yes	—
429	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
430	C₂×#131	C₂²×C₆₇	1	2	yes	—
431	#559	—	2	1	no	—
432	#557	—	2	1	no	—
433	#556	—	2	2	no	—
434	#556	C₂²×C₆₇	1	2	yes	—
435	#554	C₂²×C₆₇	1	2	yes	—
436	#552	C₂²×C₆₇	1	4	yes	—
437	#550	—	2	1	no	—
438	#550	C₂²×C₆₇	1	2	yes	—
439	D₄×Q₈	—	2	1	no	—
440	D₄×Q₈	C₂²×C₆₇	1	2	yes	—
441	#548	—	2	2	no	—
442	#548	C₂²×C₆₇	1	4	yes	—
443	#539	—	2	2	no	—
444	#537	—	2	2	no	—
445	#537	C₂²×C₆₇	1	2	yes	—
446	#536	C₂²×C₆₇	1	2	yes	—
447	#533	C₂²×C₆₇	1	1	yes	characteristic
448	#532	—	2	2	no	—
449	#532	C₂²×C₆₇	1	1	yes	characteristic
450	#532	C₂²×C₆₇	1	1	yes	characteristic
451	#529	—	2	2	no	—
452	#520	C₂²×C₆₇	1	2	yes	—
453	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
454	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
455	#488	C₂²×C₆₇	1	2	yes	—
456	#488	C₂²×C₆₇	1	2	yes	—
457	#487	C₂²×C₆₇	1	1	yes	characteristic
458	#487	C₂²×C₆₇	1	1	yes	characteristic
459	#486	C₂²×C₆₇	1	1	yes	characteristic
460	#486	C₂²×C₆₇	1	1	yes	characteristic
461	#485	C₂²×C₆₇	1	4	yes	—
462	#484	C₂²×C₆₇	1	4	yes	—
463	#482	C₂²×C₆₇	1	4	yes	—
464	#480	C₂²×C₆₇	1	4	yes	—
465	#479	C₂²×C₆₇	1	4	yes	—
466	#477	C₂²×C₆₇	1	2	yes	—
467	#475	C₂²×C₆₇	1	2	yes	—
468	#470	C₂²×C₆₇	1	4	yes	—
469	#468	C₂²×C₆₇	1	4	yes	—
470	#467	C₂²×C₆₇	1	4	yes	—
471	#465	C₂²×C₆₇	1	4	yes	—
472	#464	C₂²×C₆₇	1	4	yes	—
473	#462	C₂²×C₆₇	1	2	yes	—
474	#460	C₂²×C₆₇	1	2	yes	—
475	#452	C₂²×C₆₇	1	4	yes	—
476	#452	C₂²×C₆₇	1	4	yes	—
477	#451	C₂²×C₆₇	1	4	yes	—
478	#451	C₂²×C₆₇	1	4	yes	—
479	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
480	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
481	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
482	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
483	#436	C₂²×C₆₇	1	2	yes	—
484	#436	C₂²×C₆₇	1	2	yes	—
485	C₈×D₄	C₂²×C₆₇	1	2	yes	—
486	C₈×D₄	C₂²×C₆₇	1	2	yes	—
487	#428	C₂²×C₆₇	1	2	yes	—
488	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
489	#423	C₂²×C₆₇	1	1	yes	characteristic
490	#419	C₂²×C₆₇	1	4	yes	—
491	#419	C₂²×C₆₇	1	2	yes	—
492	#417	C₂²×C₆₇	1	4	yes	—
493	#417	C₂²×C₆₇	1	2	yes	—
494	#416	C₂²×C₆₇	1	2	yes	—
495	#416	C₂²×C₆₇	1	2	yes	—
496	#416	C₂²×C₆₇	1	4	yes	—
497	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
498	C₂×#103	C₂²×C₆₇	1	4	yes	—
499	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
500	#408	C₂²×C₆₇	1	2	yes	—
501	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
502	2₋¹⁺⁴	—	2	8	no	—
503	2₋¹⁺⁴	—	4	4	no	—
504	C₂×#41	—	2	4	no	—
505	C₂×#41	—	2	2	no	—
506	C₂×#41	C₂³×C₆₇	1	2	yes	—
507	C₂×#41	C₂³×C₆₇	1	4	yes	—
508	C₂×#41	C₂³×C₆₇	1	2	yes	—
509	C₂²×Q₈	—	2	2	no	—
510	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
511	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
512	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
513	#137	—	4	4	no	—
514	C₂×Q₁₆	—	2	4	no	—
515	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
516	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
517	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
518	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
519	#131	—	4	4	no	—
520	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
521	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
522	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
523	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
524	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
525	C₄⋊Q₈	—	2	2	no	—
526	C₄⋊Q₈	—	2	2	no	—
527	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
528	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
529	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
530	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
531	#127	—	2	2	no	—
532	#127	D₄×C₆₇	1	1	yes	characteristic
533	#127	C₂³×C₆₇	1	1	yes	characteristic
534	#126	—	2	4	no	—
535	#126	—	2	4	no	—
536	#125	—	2	2	no	—
537	#125	—	2	4	no	—
538	#125	C₂³×C₆₇	1	2	yes	—
539	#124	—	2	2	no	—
540	#124	—	2	2	no	—
541	#124	—	2	4	no	—
542	#124	D₄×C₆₇	1	2	yes	—
543	#124	C₂³×C₆₇	1	2	yes	—
544	#123	—	2	4	no	—
545	#123	—	2	4	no	—
546	C₂²⋊Q₈	—	2	2	no	—
547	C₂²⋊Q₈	—	2	1	no	—
548	C₂²⋊Q₈	—	2	1	no	—
549	C₂²⋊Q₈	—	2	4	no	—
550	C₂²⋊Q₈	—	2	4	no	—
551	C₂²⋊Q₈	—	2	4	no	—
552	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
553	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
554	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
555	#121	—	2	4	no	—
556	#121	—	2	4	no	—
557	#121	—	2	2	no	—
558	#121	—	2	1	no	—
559	#121	—	2	1	no	—
560	#121	—	2	4	no	—
561	#121	D₄×C₆₇	1	2	yes	—
562	#121	D₄×C₆₇	1	2	yes	—
563	#121	C₂³×C₆₇	1	4	yes	—
564	C₄×Q₈	—	2	2	no	—
565	C₄×Q₈	—	2	2	no	—
566	C₄×Q₈	—	2	1	no	—
567	C₄×Q₈	—	2	1	no	—
568	C₄×Q₈	—	2	1	no	—
569	C₄×Q₈	—	2	1	no	—
570	C₄×Q₈	—	2	1	no	—
571	C₄×Q₈	—	2	1	no	—
572	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
573	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
574	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
575	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
576	C₄×D₄	—	2	4	no	—
577	C₄×D₄	—	2	2	no	—
578	C₄×D₄	—	2	2	no	—
579	C₄×D₄	—	2	2	no	—
580	C₄×D₄	—	2	2	no	—
581	C₄×D₄	—	2	4	no	—
582	C₄×D₄	—	2	1	no	—
583	C₄×D₄	—	2	1	no	—
584	C₄×D₄	C₂³×C₆₇	1	2	yes	—
585	C₄×D₄	C₂³×C₆₇	1	2	yes	—
586	#117	—	2	4	no	—
587	#117	C₂³×C₆₇	1	2	yes	—
588	#117	C₂³×C₆₇	1	4	yes	—
589	C₂×C₄⋊C₄	—	2	1	no	—
590	C₂×C₄⋊C₄	—	2	1	no	—
591	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
592	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
593	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
594	C₂×C₄²	—	2	1	no	—
595	C₂×C₄²	—	2	1	no	—
596	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
597	#107	C₂³×C₆₇	1	2	yes	—
598	#107	C₂³×C₆₇	1	2	yes	—
599	#107	C₂³×C₆₇	1	4	yes	—
600	#106	C₂³×C₆₇	1	4	yes	—
601	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
602	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
603	#103	C₂³×C₆₇	1	2	yes	—
604	#103	C₂³×C₆₇	1	2	yes	—
605	#103	C₂³×C₆₇	1	2	yes	—
606	#103	C₂³×C₆₇	1	4	yes	—
607	#103	C₂³×C₆₇	1	4	yes	—
608	#103	C₂³×C₆₇	1	2	yes	—
609	#103	C₂³×C₆₇	1	2	yes	—
610	#103	C₂³×C₆₇	1	4	yes	—
611	#103	C₂³×C₆₇	1	2	yes	—
612	#103	C₂³×C₆₇	1	2	yes	—
613	#103	C₂³×C₆₇	1	2	yes	—
614	#102	C₂³×C₆₇	1	2	yes	—
615	#102	C₂³×C₆₇	1	4	yes	—
616	#102	C₂³×C₆₇	1	2	yes	—
617	#102	C₂³×C₆₇	1	2	yes	—
618	#102	C₂³×C₆₇	1	2	yes	—
619	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
620	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
621	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
622	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
623	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
624	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
625	#41	—	2	4	no	—
626	#41	—	2	4	no	—
627	#41	—	2	8	no	—
628	#41	—	2	4	no	—
629	#41	—	2	4	no	—
630	#41	—	4	4	no	—
631	#41	—	4	16	no	—
632	#41	—	4	8	no	—
633	C₂×Q₈	—	2	1	no	—
634	C₂×Q₈	—	2	1	no	—
635	C₂×Q₈	—	2	1	no	—
636	C₂×Q₈	—	2	4	no	—
637	C₂×Q₈	—	2	1	no	—
638	C₂×Q₈	—	2	2	no	—
639	C₂×Q₈	—	2	8	no	—
640	C₂×Q₈	—	2	2	no	—
641	C₂×Q₈	—	2	2	no	—
642	C₂×Q₈	—	4	4	no	—
643	C₂×Q₈	—	4	8	no	—
644	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
645	C₂×Q₈	#41×C₆₇	1	4	yes	—
646	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
647	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
648	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
649	C₂×D₄	—	2	2	no	—
650	C₂×D₄	—	2	4	no	—
651	C₂×D₄	—	2	4	no	—
652	C₂×D₄	—	2	1	no	—
653	C₂×D₄	—	2	1	no	—
654	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
655	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
656	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
657	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
658	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
659	C₂²×C₄	—	2	1	no	—
660	C₂²×C₄	—	2	1	no	—
661	C₂²×C₄	—	2	4	no	—
662	C₂²×C₄	—	2	1	no	—
663	C₂²×C₄	—	2	1	no	—
664	C₂²×C₄	—	2	4	no	—
665	C₂²×C₄	—	2	4	no	—
666	C₂²×C₄	—	2	2	no	—
667	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
668	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
669	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
670	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
671	Q₁₆	—	2	4	no	—
672	Q₁₆	—	2	4	no	—
673	Q₁₆	—	4	4	no	—
674	SD₁₆	—	4	4	no	—
675	M₁₆	—	2	4	no	—
676	M₁₆	—	4	4	no	—
677	C₂×C₈	—	2	2	no	—
678	C₂×C₈	—	2	2	no	—
679	C₂×C₈	—	4	4	no	—
680	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
681	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
682	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
683	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
684	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
685	C₄⋊C₄	—	2	2	no	—
686	C₄⋊C₄	—	2	2	no	—
687	C₄⋊C₄	—	2	2	no	—
688	C₄⋊C₄	—	2	2	no	—
689	C₄⋊C₄	—	2	1	no	—
690	C₄⋊C₄	—	2	4	no	—
691	C₄⋊C₄	—	2	1	no	—
692	C₄⋊C₄	—	2	2	no	—
693	C₄⋊C₄	—	2	4	no	—
694	C₄⋊C₄	—	2	1	no	—
695	C₄⋊C₄	—	2	1	no	—
696	C₄⋊C₄	—	2	1	no	—
697	C₄⋊C₄	—	2	4	no	—
698	C₄⋊C₄	—	2	2	no	—
699	C₄⋊C₄	—	2	1	no	—
700	C₄⋊C₄	—	2	4	no	—
701	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
702	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
703	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
704	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
705	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
706	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
707	C₂²⋊C₄	—	2	4	no	—
708	C₂²⋊C₄	—	2	2	no	—
709	C₂²⋊C₄	—	2	4	no	—
710	C₂²⋊C₄	—	2	4	no	—
711	C₂²⋊C₄	—	2	4	no	—
712	C₂²⋊C₄	—	2	2	no	—
713	C₂²⋊C₄	—	2	2	no	—
714	C₂²⋊C₄	—	2	4	no	—
715	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
716	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
717	C₄²	—	2	2	no	—
718	C₄²	—	2	2	no	—
719	C₄²	—	2	2	no	—
720	C₄²	—	2	2	no	—
721	C₄²	—	2	4	no	—
722	C₄²	—	2	2	no	—
723	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
724	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
725	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
726	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
727	C₂³	—	2	2	no	—
728	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
729	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
730	Q₈	—	2	4	no	—
731	Q₈	—	2	4	no	—
732	Q₈	—	2	2	no	—
733	Q₈	—	2	2	no	—
734	Q₈	—	2	2	no	—
735	Q₈	—	2	2	no	—
736	Q₈	—	2	2	no	—
737	Q₈	—	4	8	no	—
738	Q₈	—	4	2	no	—
739	Q₈	—	4	2	no	—
740	Q₈	—	8	1	no	—
741	Q₈	C₂×#41×C₆₇	1	4	yes	—
742	D₄	—	2	8	no	—
743	D₄	—	2	4	no	—
744	D₄	—	2	4	no	—
745	D₄	—	2	2	no	—
746	D₄	—	2	2	no	—
747	D₄	—	4	2	no	—
748	D₄	—	4	8	no	—
749	D₄	—	4	4	no	—
750	D₄	—	4	2	no	—
751	D₄	—	8	1	no	—
752	D₄	—	8	4	no	—
753	C₂×C₄	—	2	4	no	—
754	C₂×C₄	—	2	2	no	—
755	C₂×C₄	—	2	2	no	—
756	C₂×C₄	—	2	4	no	—
757	C₂×C₄	—	2	4	no	—
758	C₂×C₄	—	2	2	no	—
759	C₂×C₄	—	2	2	no	—
760	C₂×C₄	—	2	2	no	—
761	C₂×C₄	—	2	2	no	—
762	C₂×C₄	—	2	2	no	—
763	C₂×C₄	—	2	2	no	—
764	C₂×C₄	—	4	1	no	—
765	C₂×C₄	—	4	8	no	—
766	C₂×C₄	—	4	8	no	—
767	C₂×C₄	—	4	2	no	—
768	C₂×C₄	—	4	4	no	—
769	C₂×C₄	—	8	1	no	—
770	C₂×C₄	—	8	4	no	—
771	C₂×C₄	—	8	1	no	—
772	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
773	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
774	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
775	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
776	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
777	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
778	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
779	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
780	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
781	C₈	—	2	1	no	—
782	C₈	—	2	1	no	—
783	C₈	—	2	1	no	—
784	C₈	—	2	1	no	—
785	C₈	—	2	4	no	—
786	C₂²	—	2	1	no	—
787	C₂²	—	2	1	no	—
788	C₂²	—	2	4	no	—
789	C₂²	—	2	4	no	—
790	C₂²	—	2	4	no	—
791	C₂²	—	4	2	no	—
792	C₂²	—	4	2	no	—
793	C₂²	—	4	2	no	—
794	C₂²	#546×C₆₇	1	2	yes	—
795	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
796	C₄	—	2	1	no	—
797	C₄	—	2	1	no	—
798	C₄	—	2	1	no	—
799	C₄	—	2	4	no	—
800	C₄	—	2	4	no	—
801	C₄	—	4	2	no	—
802	C₄	—	4	2	no	—
803	C₄	—	4	2	no	—
804	C₄	—	4	2	no	—
805	C₄	—	4	2	no	—
806	C₄	—	4	2	no	—
807	C₄	#546×C₆₇	1	2	yes	—
808	C₄	#578×C₆₇	1	2	yes	—
809	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
810	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
811	C₂	—	2	2	no	—
812	C₂	—	4	4	no	—
813	C₂	—	8	2	no	—
814	C₂	C₂×#546×C₆₇	1	1	yes	characteristic
815	C₂	C₂×#578×C₆₇	1	1	yes	characteristic
816	C₂	C₆₇×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₆₇×#3497	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #532×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₆₇×#3507	C₂	semidirect
3. C₆₇×#3333	C₂	semidirect
4. C₆₇×#3328	C₂	semidirect
5. C₆₇×#3327	C₂	semidirect
6. C₆₇×#3304	C₂	semidirect
7. C₆₇×#3300	C₂	semidirect
8. C₆₇×#3299	C₂	semidirect
9. C₆₇×#3131	C₂	semidirect
10. C₆₇×#3098	C₂	semidirect
11. C₆₇×#3078	C₂	semidirect
12. C₆₇×#3063	C₂	semidirect
13. C₆₇×#3037	C₂	semidirect
14. C₆₇×#3011	C₂	semidirect
15. C₂×C₄×Q₁₆×C₆₇	C₂	semidirect
16. C₆₇×#2931	C₂	semidirect
17. C₆₇×#2898	C₂	semidirect
402. #61214	C₆₇	direct (N × H)
409. #3507	C₂×C₆₇	semidirect
411. #3333	C₂×C₆₇	semidirect
412. #3328	C₂×C₆₇	semidirect
413. #3327	C₂×C₆₇	semidirect
414. #3304	C₂×C₆₇	semidirect
415. #3300	C₂×C₆₇	semidirect
416. #3299	C₂×C₆₇	semidirect
417. #3131	C₂×C₆₇	semidirect
418. #3098	C₂×C₆₇	semidirect
419. #3078	C₂×C₆₇	semidirect
420. #3063	C₂×C₆₇	semidirect
421. #3037	C₂×C₆₇	semidirect
422. #3011	C₂×C₆₇	semidirect
423. C₂×C₄×Q₁₆	C₂×C₆₇	semidirect
424. #2931	C₂×C₆₇	semidirect
425. #2898	C₂×C₆₇	semidirect
426. C₆₇	#61214	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#3507	2
C₆₇×#3497	2
C₆₇×#3333	2
C₆₇×#3328	2
C₆₇×#3327	2
C₆₇×#3304	2
C₆₇×#3300	2
C₆₇×#3299	2
C₆₇×#3131	2
C₆₇×#3098	2
C₆₇×#3078	2
C₆₇×#3063	2
C₆₇×#3037	2
C₆₇×#3011	2
C₂×C₄×Q₁₆×C₆₇	2
C₆₇×#2931	2
C₆₇×#2898	2
#61214	67

Subgroups — C67×#61214 #30740465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61214 #30740465