Subgroups — C₆₇×#60939 #30740190

Order 17152.

1758 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3495	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3263	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3261	C₂	1	2	yes	maximal, complemented
4	C₆₇×#3258	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3256	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3255	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3252	C₂	1	4	yes	maximal, complemented
8	C₆₇×#3084	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3082	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3056	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3054	C₂	1	1	yes	maximal, characteristic
12	C₂×#470×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#3046	C₂	1	2	yes	maximal, complemented
14	C₆₇×#3044	C₂	1	2	yes	maximal, complemented
15	C₆₇×#2905	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#137×C₆₇	C₂²	1	2	yes	—
17	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	#556×C₆₇	—	2	2	no	—
22	#554×C₆₇	—	2	2	no	—
23	#552×C₆₇	—	2	4	no	—
24	#552×C₆₇	—	2	2	no	—
25	#552×C₆₇	—	2	2	no	—
26	#551×C₆₇	—	2	4	no	—
27	#544×C₆₇	C₂²	1	2	yes	—
28	#542×C₆₇	C₂²	1	4	yes	—
29	#541×C₆₇	C₂²	1	4	yes	—
30	#536×C₆₇	C₂²	1	2	yes	—
31	#533×C₆₇	C₂²	1	4	yes	—
32	#529×C₆₇	C₂²	1	4	yes	—
33	#525×C₆₇	C₂²	1	2	yes	—
34	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
35	#520×C₆₇	C₂²	1	2	yes	—
36	#519×C₆₇	C₂²	1	2	yes	—
37	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
38	#479×C₆₇	C₂²	1	4	yes	complemented
39	#479×C₆₇	C₂²	1	4	yes	complemented
40	#479×C₆₇	C₂²	1	4	yes	complemented
41	#478×C₆₇	C₂²	1	4	yes	complemented
42	#477×C₆₇	C₂²	1	4	yes	complemented
43	#476×C₆₇	C₂²	1	4	yes	complemented
44	#475×C₆₇	C₂²	1	4	yes	complemented
45	#475×C₆₇	C₂²	1	4	yes	complemented
46	#470×C₆₇	C₂²	1	2	yes	—
47	#470×C₆₇	C₂²	1	2	yes	—
48	#470×C₆₇	C₂²	1	2	yes	—
49	#470×C₆₇	C₂²	1	2	yes	—
50	#470×C₆₇	C₂²	1	1	yes	characteristic
51	#470×C₆₇	C₂²	1	1	yes	characteristic
52	#470×C₆₇	C₂²	1	4	yes	—
53	#469×C₆₇	C₂²	1	1	yes	characteristic
54	#469×C₆₇	C₂²	1	1	yes	characteristic
55	#468×C₆₇	C₂²	1	2	yes	—
56	#468×C₆₇	C₂²	1	2	yes	—
57	#468×C₆₇	C₂²	1	4	yes	—
58	#467×C₆₇	C₂²	1	4	yes	—
59	#465×C₆₇	C₂²	1	4	yes	—
60	#464×C₆₇	C₂²	1	4	yes	complemented
61	#464×C₆₇	C₂²	1	4	yes	complemented
62	#464×C₆₇	C₂²	1	4	yes	complemented
63	#463×C₆₇	C₂²	1	4	yes	complemented
64	#462×C₆₇	C₂²	1	4	yes	complemented
65	#462×C₆₇	C₂²	1	4	yes	complemented
66	#462×C₆₇	C₂²	1	4	yes	complemented
67	#461×C₆₇	C₂²	1	4	yes	complemented
68	#428×C₆₇	C₂²	1	2	yes	—
69	#427×C₆₇	C₂²	1	1	yes	characteristic
70	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
71	#424×C₆₇	C₂²	1	2	yes	—
72	#424×C₆₇	C₂²	1	2	yes	—
73	#424×C₆₇	C₂²	1	2	yes	—
74	#424×C₆₇	C₂²	1	2	yes	—
75	#423×C₆₇	C₂²	1	2	yes	—
76	#423×C₆₇	C₂²	1	2	yes	—
77	#419×C₆₇	C₂²	1	2	yes	—
78	#417×C₆₇	C₂²	1	2	yes	—
79	#416×C₆₇	C₂²	1	2	yes	—
80	C₂×#103×C₆₇	C₂²	1	2	yes	—
81	C₂×#41×C₆₇	C₂³	1	2	yes	—
82	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
83	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
84	#137×C₆₇	—	4	4	no	—
85	#135×C₆₇	—	4	2	no	—
86	C₂×Q₁₆×C₆₇	—	4	2	no	—
87	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
88	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
89	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
90	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
91	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
92	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
93	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×M₁₆×C₆₇	—	4	2	no	—
95	C₂×M₁₆×C₆₇	—	4	2	no	—
96	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
97	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
98	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
99	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
100	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
102	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
103	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
104	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
105	C₄⋊Q₈×C₆₇	—	2	2	no	—
106	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
107	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
108	#126×C₆₇	—	2	4	no	—
109	#126×C₆₇	—	2	4	no	—
110	#126×C₆₇	—	2	4	no	—
111	#126×C₆₇	—	2	4	no	—
112	#125×C₆₇	—	2	4	no	—
113	#125×C₆₇	—	2	4	no	—
114	#125×C₆₇	—	2	2	no	—
115	#125×C₆₇	—	2	4	no	—
116	#125×C₆₇	C₂³	1	4	yes	—
117	#124×C₆₇	—	2	4	no	—
118	#124×C₆₇	C₂³	1	4	yes	—
119	#123×C₆₇	—	2	2	no	—
120	#123×C₆₇	—	2	4	no	—
121	#123×C₆₇	—	2	4	no	—
122	#123×C₆₇	—	2	4	no	—
123	C₂²⋊Q₈×C₆₇	—	2	2	no	—
124	C₂²⋊Q₈×C₆₇	—	2	2	no	—
125	C₂²⋊Q₈×C₆₇	—	2	4	no	—
126	C₂²⋊Q₈×C₆₇	—	2	2	no	—
127	C₂²⋊Q₈×C₆₇	—	2	2	no	—
128	C₂²⋊Q₈×C₆₇	—	2	4	no	—
129	C₂²⋊Q₈×C₆₇	—	2	4	no	—
130	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
131	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
132	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
133	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
134	#121×C₆₇	C₂³	1	2	yes	—
135	#121×C₆₇	C₂³	1	2	yes	—
136	#120×C₆₇	—	2	2	no	—
137	C₄×Q₈×C₆₇	—	2	2	no	—
138	C₄×Q₈×C₆₇	—	2	2	no	—
139	C₄×Q₈×C₆₇	—	2	4	no	—
140	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
141	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
142	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
143	C₄×D₄×C₆₇	—	2	2	no	—
144	C₄×D₄×C₆₇	—	2	2	no	—
145	C₄×D₄×C₆₇	—	2	4	no	—
146	C₄×D₄×C₆₇	—	2	4	no	—
147	C₄×D₄×C₆₇	C₂³	1	4	yes	—
148	#117×C₆₇	—	2	4	no	—
149	#117×C₆₇	—	2	2	no	—
150	#117×C₆₇	—	2	4	no	—
151	#117×C₆₇	—	2	4	no	—
152	#117×C₆₇	D₄	1	1	yes	characteristic
153	#117×C₆₇	D₄	1	2	yes	—
154	#117×C₆₇	D₄	1	1	yes	characteristic
155	#117×C₆₇	C₂³	1	2	yes	—
156	#117×C₆₇	C₂³	1	2	yes	—
157	#117×C₆₇	C₂³	1	2	yes	—
158	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
159	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
160	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
161	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
162	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
163	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
164	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
165	#107×C₆₇	C₂³	1	4	yes	—
166	#107×C₆₇	C₂³	1	2	yes	—
167	#106×C₆₇	C₂³	1	4	yes	—
168	#106×C₆₇	C₂³	1	1	yes	characteristic
169	#106×C₆₇	C₂³	1	1	yes	characteristic
170	#106×C₆₇	C₂³	1	2	yes	—
171	#106×C₆₇	C₂³	1	2	yes	—
172	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
173	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
174	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
175	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
176	#103×C₆₇	C₂³	1	4	yes	—
177	#103×C₆₇	C₂³	1	4	yes	—
178	#103×C₆₇	C₂³	1	2	yes	—
179	#103×C₆₇	C₂³	1	2	yes	—
180	#103×C₆₇	C₂³	1	4	yes	—
181	#103×C₆₇	C₂³	1	4	yes	—
182	#103×C₆₇	C₂³	1	4	yes	—
183	#102×C₆₇	C₂³	1	4	yes	—
184	#102×C₆₇	C₂³	1	2	yes	—
185	#102×C₆₇	C₂³	1	2	yes	—
186	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
187	#41×C₆₇	—	2	4	no	—
188	#41×C₆₇	—	2	4	no	—
189	C₂×Q₈×C₆₇	—	2	2	no	—
190	C₂×Q₈×C₆₇	—	2	4	no	—
191	C₂×Q₈×C₆₇	—	2	4	no	—
192	C₂×Q₈×C₆₇	—	2	4	no	—
193	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
194	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
195	C₂×D₄×C₆₇	—	2	2	no	—
196	C₂×D₄×C₆₇	—	2	2	no	—
197	C₂×D₄×C₆₇	—	2	2	no	—
198	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
199	C₂²×C₄×C₆₇	—	2	2	no	—
200	C₂²×C₄×C₆₇	—	2	2	no	—
201	C₂²×C₄×C₆₇	—	2	2	no	—
202	C₂²×C₄×C₆₇	—	2	2	no	—
203	C₂²×C₄×C₆₇	—	2	2	no	—
204	C₂²×C₄×C₆₇	—	2	4	no	—
205	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
206	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
207	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
208	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
209	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
210	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
211	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
212	Q₁₆×C₆₇	—	4	2	no	—
213	Q₁₆×C₆₇	—	4	2	no	—
214	Q₁₆×C₆₇	—	4	1	no	—
215	Q₁₆×C₆₇	—	4	4	no	—
216	SD₁₆×C₆₇	—	4	2	no	—
217	SD₁₆×C₆₇	—	4	4	no	—
218	D₈×C₆₇	—	4	1	no	—
219	M₁₆×C₆₇	—	2	4	no	—
220	M₁₆×C₆₇	—	4	1	no	—
221	M₁₆×C₆₇	—	4	1	no	—
222	M₁₆×C₆₇	—	4	1	no	—
223	M₁₆×C₆₇	—	4	1	no	—
224	M₁₆×C₆₇	—	4	2	no	—
225	C₂×C₈×C₆₇	—	2	2	no	—
226	C₂×C₈×C₆₇	—	2	2	no	—
227	C₂×C₈×C₆₇	—	4	2	no	—
228	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
229	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
230	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
231	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
232	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
233	C₄⋊C₄×C₆₇	—	2	2	no	—
234	C₄⋊C₄×C₆₇	—	2	4	no	—
235	C₄⋊C₄×C₆₇	—	2	4	no	—
236	C₄⋊C₄×C₆₇	—	2	4	no	—
237	C₄⋊C₄×C₆₇	—	2	4	no	—
238	C₄⋊C₄×C₆₇	—	2	4	no	—
239	C₄⋊C₄×C₆₇	—	2	4	no	—
240	C₄⋊C₄×C₆₇	—	2	4	no	—
241	C₄⋊C₄×C₆₇	—	2	4	no	—
242	C₄⋊C₄×C₆₇	—	2	2	no	—
243	C₄⋊C₄×C₆₇	—	2	4	no	—
244	C₄⋊C₄×C₆₇	—	2	2	no	—
245	C₄⋊C₄×C₆₇	—	2	4	no	—
246	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
247	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
248	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
249	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
250	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
251	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
252	C₂²⋊C₄×C₆₇	—	2	4	no	—
253	C₂²⋊C₄×C₆₇	—	2	4	no	—
254	C₂²⋊C₄×C₆₇	—	2	2	no	—
255	C₂²⋊C₄×C₆₇	—	2	2	no	—
256	C₂²⋊C₄×C₆₇	—	2	2	no	—
257	C₂²⋊C₄×C₆₇	—	2	2	no	—
258	C₂²⋊C₄×C₆₇	—	2	4	no	—
259	C₂²⋊C₄×C₆₇	—	2	4	no	—
260	C₂²⋊C₄×C₆₇	—	2	2	no	—
261	C₂²⋊C₄×C₆₇	—	2	4	no	—
262	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
263	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
264	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
265	C₄²×C₆₇	—	2	4	no	—
266	C₄²×C₆₇	—	2	4	no	—
267	C₄²×C₆₇	—	2	4	no	—
268	C₄²×C₆₇	—	2	2	no	—
269	C₄²×C₆₇	C₂×D₄	1	2	yes	—
270	C₄²×C₆₇	C₂×D₄	1	2	yes	—
271	C₄²×C₆₇	C₂⁴	1	4	yes	—
272	C₂³×C₆₇	—	2	2	no	—
273	C₂³×C₆₇	—	2	2	no	—
274	C₂³×C₆₇	—	4	2	no	—
275	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
276	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
277	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
278	Q₈×C₆₇	—	2	4	no	—
279	Q₈×C₆₇	—	2	2	no	—
280	Q₈×C₆₇	—	2	4	no	—
281	Q₈×C₆₇	—	2	2	no	—
282	Q₈×C₆₇	—	4	4	no	—
283	Q₈×C₆₇	—	8	4	no	—
284	D₄×C₆₇	—	2	2	no	—
285	D₄×C₆₇	—	2	2	no	—
286	D₄×C₆₇	—	4	4	no	—
287	D₄×C₆₇	—	4	4	no	—
288	D₄×C₆₇	—	8	2	no	—
289	C₂×C₄×C₆₇	—	2	2	no	—
290	C₂×C₄×C₆₇	—	2	2	no	—
291	C₂×C₄×C₆₇	—	2	2	no	—
292	C₂×C₄×C₆₇	—	2	1	no	—
293	C₂×C₄×C₆₇	—	2	1	no	—
294	C₂×C₄×C₆₇	—	2	1	no	—
295	C₂×C₄×C₆₇	—	2	1	no	—
296	C₂×C₄×C₆₇	—	2	2	no	—
297	C₂×C₄×C₆₇	—	2	2	no	—
298	C₂×C₄×C₆₇	—	2	4	no	—
299	C₂×C₄×C₆₇	—	2	2	no	—
300	C₂×C₄×C₆₇	—	2	4	no	—
301	C₂×C₄×C₆₇	—	4	4	no	—
302	C₂×C₄×C₆₇	—	4	4	no	—
303	C₂×C₄×C₆₇	—	4	4	no	—
304	C₂×C₄×C₆₇	—	8	4	no	—
305	C₂×C₄×C₆₇	—	8	2	no	—
306	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
307	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
308	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
309	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
310	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
311	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
312	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
313	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
314	C₈×C₆₇	—	2	1	no	—
315	C₈×C₆₇	—	2	1	no	—
316	C₈×C₆₇	—	2	1	no	—
317	C₈×C₆₇	—	2	1	no	—
318	C₈×C₆₇	—	2	4	no	—
319	C₂²×C₆₇	—	2	2	no	—
320	C₂²×C₆₇	—	2	2	no	—
321	C₂²×C₆₇	—	2	2	no	—
322	C₂²×C₆₇	—	2	1	no	—
323	C₂²×C₆₇	—	2	1	no	—
324	C₂²×C₆₇	—	4	2	no	—
325	C₂²×C₆₇	—	4	2	no	—
326	C₂²×C₆₇	—	4	2	no	—
327	C₂²×C₆₇	—	8	2	no	—
328	C₂²×C₆₇	#537	1	2	yes	—
329	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
330	C₄×C₆₇	—	2	4	no	—
331	C₄×C₆₇	—	2	4	no	—
332	C₄×C₆₇	—	2	1	no	—
333	C₄×C₆₇	—	2	1	no	—
334	C₄×C₆₇	—	4	4	no	—
335	C₄×C₆₇	—	4	2	no	—
336	C₄×C₆₇	—	4	2	no	—
337	C₄×C₆₇	—	4	2	no	—
338	C₄×C₆₇	—	4	4	no	—
339	C₄×C₆₇	#537	1	2	yes	—
340	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
341	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
342	#60939	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
343	C₂×C₆₇	—	2	2	no	—
344	C₂×C₆₇	—	4	1	no	—
345	C₂×C₆₇	—	4	1	no	—
346	C₂×C₆₇	—	8	2	no	—
347	C₂×C₆₇	C₂×#537	1	1	yes	characteristic
348	C₂×C₆₇	#3586	1	2	yes	—
349	#3495	C₂×C₆₇	1	2	yes	complemented
350	#3263	C₂×C₆₇	1	2	yes	complemented
351	#3261	C₂×C₆₇	1	2	yes	complemented
352	#3258	C₂×C₆₇	1	4	yes	complemented
353	#3256	C₂×C₆₇	1	2	yes	complemented
354	#3255	C₂×C₆₇	1	2	yes	complemented
355	#3252	C₂×C₆₇	1	4	yes	complemented
356	#3084	C₂×C₆₇	1	2	yes	complemented
357	#3082	C₂×C₆₇	1	2	yes	complemented
358	#3056	C₂×C₆₇	1	2	yes	complemented
359	#3054	C₂×C₆₇	1	1	yes	characteristic
360	C₂×#470	C₂×C₆₇	1	1	yes	characteristic, complemented
361	#3046	C₂×C₆₇	1	2	yes	complemented
362	#3044	C₂×C₆₇	1	2	yes	complemented
363	#2905	C₂×C₆₇	1	1	yes	characteristic, complemented
364	C₆₇	#60939	1	1	yes	characteristic, direct product factor, Sylow
365	C₂×#137	C₂²×C₆₇	1	2	yes	—
366	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
367	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
368	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
369	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
370	#556	—	2	2	no	—
371	#554	—	2	2	no	—
372	#552	—	2	4	no	—
373	#552	—	2	2	no	—
374	#552	—	2	2	no	—
375	#551	—	2	4	no	—
376	#544	C₂²×C₆₇	1	2	yes	—
377	#542	C₂²×C₆₇	1	4	yes	—
378	#541	C₂²×C₆₇	1	4	yes	—
379	#536	C₂²×C₆₇	1	2	yes	—
380	#533	C₂²×C₆₇	1	4	yes	—
381	#529	C₂²×C₆₇	1	4	yes	—
382	#525	C₂²×C₆₇	1	2	yes	—
383	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
384	#520	C₂²×C₆₇	1	2	yes	—
385	#519	C₂²×C₆₇	1	2	yes	—
386	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
387	#479	C₂²×C₆₇	1	4	yes	complemented
388	#479	C₂²×C₆₇	1	4	yes	complemented
389	#479	C₂²×C₆₇	1	4	yes	complemented
390	#478	C₂²×C₆₇	1	4	yes	complemented
391	#477	C₂²×C₆₇	1	4	yes	complemented
392	#476	C₂²×C₆₇	1	4	yes	complemented
393	#475	C₂²×C₆₇	1	4	yes	complemented
394	#475	C₂²×C₆₇	1	4	yes	complemented
395	#470	C₂²×C₆₇	1	2	yes	—
396	#470	C₂²×C₆₇	1	4	yes	—
397	#470	C₂²×C₆₇	1	1	yes	characteristic
398	#470	C₂²×C₆₇	1	2	yes	—
399	#470	C₂²×C₆₇	1	2	yes	—
400	#470	C₂²×C₆₇	1	1	yes	characteristic
401	#470	C₂²×C₆₇	1	2	yes	—
402	#469	C₂²×C₆₇	1	1	yes	characteristic
403	#469	C₂²×C₆₇	1	1	yes	characteristic
404	#468	C₂²×C₆₇	1	2	yes	—
405	#468	C₂²×C₆₇	1	4	yes	—
406	#468	C₂²×C₆₇	1	2	yes	—
407	#467	C₂²×C₆₇	1	4	yes	—
408	#465	C₂²×C₆₇	1	4	yes	—
409	#464	C₂²×C₆₇	1	4	yes	complemented
410	#464	C₂²×C₆₇	1	4	yes	complemented
411	#464	C₂²×C₆₇	1	4	yes	complemented
412	#463	C₂²×C₆₇	1	4	yes	complemented
413	#462	C₂²×C₆₇	1	4	yes	complemented
414	#462	C₂²×C₆₇	1	4	yes	complemented
415	#462	C₂²×C₆₇	1	4	yes	complemented
416	#461	C₂²×C₆₇	1	4	yes	complemented
417	#428	C₂²×C₆₇	1	2	yes	—
418	#427	C₂²×C₆₇	1	1	yes	characteristic
419	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
420	#424	C₂²×C₆₇	1	2	yes	—
421	#424	C₂²×C₆₇	1	2	yes	—
422	#424	C₂²×C₆₇	1	2	yes	—
423	#424	C₂²×C₆₇	1	2	yes	—
424	#423	C₂²×C₆₇	1	2	yes	—
425	#423	C₂²×C₆₇	1	2	yes	—
426	#419	C₂²×C₆₇	1	2	yes	—
427	#417	C₂²×C₆₇	1	2	yes	—
428	#416	C₂²×C₆₇	1	2	yes	—
429	C₂×#103	C₂²×C₆₇	1	2	yes	—
430	C₂×#41	C₂³×C₆₇	1	2	yes	—
431	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
432	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
433	#137	—	4	4	no	—
434	#135	—	4	2	no	—
435	C₂×Q₁₆	—	4	2	no	—
436	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
437	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
438	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
439	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
440	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
441	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
442	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
443	C₂×M₁₆	—	4	2	no	—
444	C₂×M₁₆	—	4	2	no	—
445	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
446	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
447	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
448	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
449	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
450	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
451	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
452	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
453	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
454	C₄⋊Q₈	—	2	2	no	—
455	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
456	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
457	#126	—	2	4	no	—
458	#126	—	2	4	no	—
459	#126	—	2	4	no	—
460	#126	—	2	4	no	—
461	#125	—	2	4	no	—
462	#125	—	2	4	no	—
463	#125	—	2	2	no	—
464	#125	—	2	4	no	—
465	#125	C₂³×C₆₇	1	4	yes	—
466	#124	—	2	4	no	—
467	#124	C₂³×C₆₇	1	4	yes	—
468	#123	—	2	2	no	—
469	#123	—	2	4	no	—
470	#123	—	2	4	no	—
471	#123	—	2	4	no	—
472	C₂²⋊Q₈	—	2	2	no	—
473	C₂²⋊Q₈	—	2	2	no	—
474	C₂²⋊Q₈	—	2	4	no	—
475	C₂²⋊Q₈	—	2	2	no	—
476	C₂²⋊Q₈	—	2	2	no	—
477	C₂²⋊Q₈	—	2	4	no	—
478	C₂²⋊Q₈	—	2	4	no	—
479	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
480	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
481	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
482	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
483	#121	C₂³×C₆₇	1	2	yes	—
484	#121	C₂³×C₆₇	1	2	yes	—
485	#120	—	2	2	no	—
486	C₄×Q₈	—	2	2	no	—
487	C₄×Q₈	—	2	2	no	—
488	C₄×Q₈	—	2	4	no	—
489	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
490	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
491	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
492	C₄×D₄	—	2	2	no	—
493	C₄×D₄	—	2	2	no	—
494	C₄×D₄	—	2	4	no	—
495	C₄×D₄	—	2	4	no	—
496	C₄×D₄	C₂³×C₆₇	1	4	yes	—
497	#117	—	2	4	no	—
498	#117	—	2	2	no	—
499	#117	—	2	4	no	—
500	#117	—	2	4	no	—
501	#117	D₄×C₆₇	1	1	yes	characteristic
502	#117	D₄×C₆₇	1	2	yes	—
503	#117	D₄×C₆₇	1	1	yes	characteristic
504	#117	C₂³×C₆₇	1	2	yes	—
505	#117	C₂³×C₆₇	1	2	yes	—
506	#117	C₂³×C₆₇	1	2	yes	—
507	C₂×C₄⋊C₄	—	2	2	no	—
508	C₂×C₄⋊C₄	—	2	2	no	—
509	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
510	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
511	C₂×C₂²⋊C₄	—	2	2	no	—
512	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
513	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
514	#107	C₂³×C₆₇	1	2	yes	—
515	#107	C₂³×C₆₇	1	4	yes	—
516	#106	C₂³×C₆₇	1	2	yes	—
517	#106	C₂³×C₆₇	1	2	yes	—
518	#106	C₂³×C₆₇	1	1	yes	characteristic
519	#106	C₂³×C₆₇	1	1	yes	characteristic
520	#106	C₂³×C₆₇	1	4	yes	—
521	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
522	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
523	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
524	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
525	#103	C₂³×C₆₇	1	4	yes	—
526	#103	C₂³×C₆₇	1	4	yes	—
527	#103	C₂³×C₆₇	1	2	yes	—
528	#103	C₂³×C₆₇	1	2	yes	—
529	#103	C₂³×C₆₇	1	4	yes	—
530	#103	C₂³×C₆₇	1	4	yes	—
531	#103	C₂³×C₆₇	1	4	yes	—
532	#102	C₂³×C₆₇	1	2	yes	—
533	#102	C₂³×C₆₇	1	2	yes	—
534	#102	C₂³×C₆₇	1	4	yes	—
535	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
536	#41	—	2	4	no	—
537	#41	—	2	4	no	—
538	C₂×Q₈	—	2	2	no	—
539	C₂×Q₈	—	2	4	no	—
540	C₂×Q₈	—	2	4	no	—
541	C₂×Q₈	—	2	4	no	—
542	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
543	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
544	C₂×D₄	—	2	2	no	—
545	C₂×D₄	—	2	2	no	—
546	C₂×D₄	—	2	2	no	—
547	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
548	C₂²×C₄	—	2	2	no	—
549	C₂²×C₄	—	2	2	no	—
550	C₂²×C₄	—	2	2	no	—
551	C₂²×C₄	—	2	2	no	—
552	C₂²×C₄	—	2	2	no	—
553	C₂²×C₄	—	2	4	no	—
554	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
555	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
556	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
557	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
558	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
559	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
560	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
561	Q₁₆	—	4	2	no	—
562	Q₁₆	—	4	2	no	—
563	Q₁₆	—	4	1	no	—
564	Q₁₆	—	4	4	no	—
565	SD₁₆	—	4	2	no	—
566	SD₁₆	—	4	4	no	—
567	D₈	—	4	1	no	—
568	M₁₆	—	2	4	no	—
569	M₁₆	—	4	1	no	—
570	M₁₆	—	4	1	no	—
571	M₁₆	—	4	1	no	—
572	M₁₆	—	4	1	no	—
573	M₁₆	—	4	2	no	—
574	C₂×C₈	—	2	2	no	—
575	C₂×C₈	—	2	2	no	—
576	C₂×C₈	—	4	2	no	—
577	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
578	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
579	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
580	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
581	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
582	C₄⋊C₄	—	2	2	no	—
583	C₄⋊C₄	—	2	4	no	—
584	C₄⋊C₄	—	2	4	no	—
585	C₄⋊C₄	—	2	4	no	—
586	C₄⋊C₄	—	2	4	no	—
587	C₄⋊C₄	—	2	4	no	—
588	C₄⋊C₄	—	2	4	no	—
589	C₄⋊C₄	—	2	4	no	—
590	C₄⋊C₄	—	2	4	no	—
591	C₄⋊C₄	—	2	2	no	—
592	C₄⋊C₄	—	2	4	no	—
593	C₄⋊C₄	—	2	2	no	—
594	C₄⋊C₄	—	2	4	no	—
595	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
596	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
597	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
598	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
599	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
600	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
601	C₂²⋊C₄	—	2	4	no	—
602	C₂²⋊C₄	—	2	4	no	—
603	C₂²⋊C₄	—	2	2	no	—
604	C₂²⋊C₄	—	2	2	no	—
605	C₂²⋊C₄	—	2	2	no	—
606	C₂²⋊C₄	—	2	2	no	—
607	C₂²⋊C₄	—	2	4	no	—
608	C₂²⋊C₄	—	2	4	no	—
609	C₂²⋊C₄	—	2	2	no	—
610	C₂²⋊C₄	—	2	4	no	—
611	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
612	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
613	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
614	C₄²	—	2	4	no	—
615	C₄²	—	2	4	no	—
616	C₄²	—	2	4	no	—
617	C₄²	—	2	2	no	—
618	C₄²	C₂×D₄×C₆₇	1	2	yes	—
619	C₄²	C₂×D₄×C₆₇	1	2	yes	—
620	C₄²	C₂⁴×C₆₇	1	4	yes	—
621	C₂³	—	2	2	no	—
622	C₂³	—	2	2	no	—
623	C₂³	—	4	2	no	—
624	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
625	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
626	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
627	Q₈	—	2	4	no	—
628	Q₈	—	2	2	no	—
629	Q₈	—	2	4	no	—
630	Q₈	—	2	2	no	—
631	Q₈	—	4	4	no	—
632	Q₈	—	8	4	no	—
633	D₄	—	2	2	no	—
634	D₄	—	2	2	no	—
635	D₄	—	4	4	no	—
636	D₄	—	4	4	no	—
637	D₄	—	8	2	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	2	1	no	—
642	C₂×C₄	—	2	1	no	—
643	C₂×C₄	—	2	1	no	—
644	C₂×C₄	—	2	1	no	—
645	C₂×C₄	—	2	2	no	—
646	C₂×C₄	—	2	2	no	—
647	C₂×C₄	—	2	4	no	—
648	C₂×C₄	—	2	2	no	—
649	C₂×C₄	—	2	4	no	—
650	C₂×C₄	—	4	4	no	—
651	C₂×C₄	—	4	4	no	—
652	C₂×C₄	—	4	4	no	—
653	C₂×C₄	—	8	4	no	—
654	C₂×C₄	—	8	2	no	—
655	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
656	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
657	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
658	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
659	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
660	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
661	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
662	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
663	C₈	—	2	1	no	—
664	C₈	—	2	1	no	—
665	C₈	—	2	1	no	—
666	C₈	—	2	1	no	—
667	C₈	—	2	4	no	—
668	C₂²	—	2	2	no	—
669	C₂²	—	2	2	no	—
670	C₂²	—	2	2	no	—
671	C₂²	—	2	1	no	—
672	C₂²	—	2	1	no	—
673	C₂²	—	4	2	no	—
674	C₂²	—	4	2	no	—
675	C₂²	—	4	2	no	—
676	C₂²	—	8	2	no	—
677	C₂²	#537×C₆₇	1	2	yes	—
678	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
679	C₄	—	2	4	no	—
680	C₄	—	2	4	no	—
681	C₄	—	2	1	no	—
682	C₄	—	2	1	no	—
683	C₄	—	4	4	no	—
684	C₄	—	4	2	no	—
685	C₄	—	4	2	no	—
686	C₄	—	4	2	no	—
687	C₄	—	4	4	no	—
688	C₄	#537×C₆₇	1	2	yes	—
689	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
690	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
691	C₂	—	2	2	no	—
692	C₂	—	4	1	no	—
693	C₂	—	4	1	no	—
694	C₂	—	8	2	no	—
695	C₂	C₂×#537×C₆₇	1	1	yes	characteristic
696	C₂	C₆₇×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₆₇×#3054	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3495	C₂	semidirect
2. C₆₇×#3263	C₂	semidirect
3. C₆₇×#3261	C₂	semidirect
4. C₆₇×#3258	C₂	semidirect
5. C₆₇×#3256	C₂	semidirect
6. C₆₇×#3255	C₂	semidirect
7. C₆₇×#3252	C₂	semidirect
8. C₆₇×#3084	C₂	semidirect
9. C₆₇×#3082	C₂	semidirect
10. C₆₇×#3056	C₂	semidirect
12. C₂×#470×C₆₇	C₂	semidirect
13. C₆₇×#3046	C₂	semidirect
14. C₆₇×#3044	C₂	semidirect
15. C₆₇×#2905	C₂	semidirect
38. #479×C₆₇	C₂²	semidirect
39. #479×C₆₇	C₂²	semidirect
40. #479×C₆₇	C₂²	semidirect
41. #478×C₆₇	C₂²	semidirect
42. #477×C₆₇	C₂²	semidirect
43. #476×C₆₇	C₂²	semidirect
44. #475×C₆₇	C₂²	semidirect
45. #475×C₆₇	C₂²	semidirect
60. #464×C₆₇	C₂²	semidirect
61. #464×C₆₇	C₂²	semidirect
62. #464×C₆₇	C₂²	semidirect
63. #463×C₆₇	C₂²	semidirect
64. #462×C₆₇	C₂²	semidirect
65. #462×C₆₇	C₂²	semidirect
66. #462×C₆₇	C₂²	semidirect
67. #461×C₆₇	C₂²	semidirect
342. #60939	C₆₇	direct (N × H)
349. #3495	C₂×C₆₇	semidirect
350. #3263	C₂×C₆₇	semidirect
351. #3261	C₂×C₆₇	semidirect
352. #3258	C₂×C₆₇	semidirect
353. #3256	C₂×C₆₇	semidirect
354. #3255	C₂×C₆₇	semidirect
355. #3252	C₂×C₆₇	semidirect
356. #3084	C₂×C₆₇	semidirect
357. #3082	C₂×C₆₇	semidirect
358. #3056	C₂×C₆₇	semidirect
360. C₂×#470	C₂×C₆₇	semidirect
361. #3046	C₂×C₆₇	semidirect
362. #3044	C₂×C₆₇	semidirect
363. #2905	C₂×C₆₇	semidirect
364. C₆₇	#60939	direct (N × H)
387. #479	C₂²×C₆₇	semidirect
388. #479	C₂²×C₆₇	semidirect
389. #479	C₂²×C₆₇	semidirect
390. #478	C₂²×C₆₇	semidirect
391. #477	C₂²×C₆₇	semidirect
392. #476	C₂²×C₆₇	semidirect
393. #475	C₂²×C₆₇	semidirect
394. #475	C₂²×C₆₇	semidirect
409. #464	C₂²×C₆₇	semidirect
410. #464	C₂²×C₆₇	semidirect
411. #464	C₂²×C₆₇	semidirect
412. #463	C₂²×C₆₇	semidirect
413. #462	C₂²×C₆₇	semidirect
414. #462	C₂²×C₆₇	semidirect
415. #462	C₂²×C₆₇	semidirect
416. #461	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3495	2
C₆₇×#3263	2
C₆₇×#3261	2
C₆₇×#3258	2
C₆₇×#3256	2
C₆₇×#3255	2
C₆₇×#3252	2
C₆₇×#3084	2
C₆₇×#3082	2
C₆₇×#3056	2
C₆₇×#3054	2
C₂×#470×C₆₇	2
C₆₇×#3046	2
C₆₇×#3044	2
C₆₇×#2905	2
#60939	67

Subgroups — C67×#60939 #30740190

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60939 #30740190