Subgroups — C₆₇×#61224 #30740475

Order 17152.

2020 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3512	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3320	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3315	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3312	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3307	C₂	1	4	yes	maximal, complemented
7	C₂×#491×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#3097	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3080	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#3063	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3039	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#3013	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2943	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#2932	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#2898	C₂	1	2	yes	maximal, complemented
16	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	2	yes	—
17	C₂×#137×C₆₇	C₂²	1	2	yes	—
18	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#131×C₆₇	C₂²	1	2	yes	—
20	#553×C₆₇	—	2	4	no	—
21	#553×C₆₇	C₂²	1	4	yes	—
22	#552×C₆₇	C₂²	1	4	yes	—
23	D₄×Q₈×C₆₇	C₂²	1	4	yes	—
24	#547×C₆₇	—	2	4	no	—
25	#547×C₆₇	C₂²	1	4	yes	—
26	#541×C₆₇	—	2	1	no	—
27	#540×C₆₇	—	2	1	no	—
28	#537×C₆₇	C₂²	1	2	yes	—
29	#536×C₆₇	—	2	2	no	—
30	#536×C₆₇	C₂²	1	2	yes	—
31	#532×C₆₇	—	2	1	no	—
32	#532×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
34	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
35	#529×C₆₇	—	2	2	no	—
36	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
37	#520×C₆₇	C₂²	1	2	yes	—
38	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
40	#491×C₆₇	C₂²	1	4	yes	—
41	#491×C₆₇	C₂²	1	4	yes	—
42	#485×C₆₇	C₂²	1	4	yes	—
43	#485×C₆₇	C₂²	1	4	yes	—
44	#484×C₆₇	C₂²	1	4	yes	—
45	#483×C₆₇	C₂²	1	4	yes	—
46	#479×C₆₇	C₂²	1	4	yes	—
47	#478×C₆₇	C₂²	1	4	yes	—
48	#470×C₆₇	C₂²	1	4	yes	—
49	#468×C₆₇	C₂²	1	4	yes	—
50	#467×C₆₇	C₂²	1	4	yes	—
51	#465×C₆₇	C₂²	1	4	yes	—
52	#462×C₆₇	C₂²	1	4	yes	—
53	#461×C₆₇	C₂²	1	4	yes	—
54	#452×C₆₇	C₂²	1	4	yes	—
55	#451×C₆₇	C₂²	1	4	yes	—
56	#449×C₆₇	C₂²	1	4	yes	—
57	#448×C₆₇	C₂²	1	4	yes	—
58	#442×C₆₇	C₂²	1	2	yes	—
59	#441×C₆₇	C₂²	1	2	yes	—
60	#440×C₆₇	C₂²	1	2	yes	—
61	#440×C₆₇	C₂²	1	2	yes	—
62	#435×C₆₇	C₂²	1	4	yes	—
63	#435×C₆₇	C₂²	1	4	yes	—
64	#428×C₆₇	C₂²	1	2	yes	—
65	#427×C₆₇	C₂²	1	1	yes	characteristic
66	#423×C₆₇	C₂²	1	1	yes	characteristic
67	#419×C₆₇	C₂²	1	2	yes	—
68	#419×C₆₇	C₂²	1	4	yes	—
69	#417×C₆₇	C₂²	1	2	yes	—
70	#417×C₆₇	C₂²	1	4	yes	—
71	#416×C₆₇	C₂²	1	4	yes	—
72	#416×C₆₇	C₂²	1	1	yes	characteristic
73	#416×C₆₇	C₂²	1	1	yes	characteristic
74	C₂×#103×C₆₇	C₂²	1	2	yes	—
75	C₂×#103×C₆₇	C₂²	1	4	yes	—
76	C₂×#103×C₆₇	C₂²	1	2	yes	—
77	#408×C₆₇	C₂²	1	2	yes	—
78	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
79	2₋¹⁺⁴×C₆₇	—	2	8	no	—
80	2₋¹⁺⁴×C₆₇	—	4	4	no	—
81	C₂×#41×C₆₇	—	2	2	no	—
82	C₂×#41×C₆₇	—	2	4	no	—
83	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
84	C₂×#41×C₆₇	C₂³	1	4	yes	—
85	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×#41×C₆₇	C₂³	1	2	yes	—
87	C₂²×Q₈×C₆₇	—	2	2	no	—
88	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
89	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
90	#137×C₆₇	—	4	4	no	—
91	#135×C₆₇	—	2	4	no	—
92	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
93	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
94	C₂×SD₁₆×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×SD₁₆×C₆₇	C₂³	1	1	yes	characteristic
96	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
97	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
98	#131×C₆₇	—	4	4	no	—
99	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
100	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
101	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
102	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
103	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
104	C₄⋊Q₈×C₆₇	—	2	2	no	—
105	C₄⋊Q₈×C₆₇	—	2	4	no	—
106	C₄⋊Q₈×C₆₇	—	2	1	no	—
107	C₄⋊Q₈×C₆₇	—	2	1	no	—
108	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
109	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
110	#127×C₆₇	D₄	1	1	yes	characteristic
111	#126×C₆₇	—	2	4	no	—
112	#126×C₆₇	—	2	4	no	—
113	#125×C₆₇	—	2	4	no	—
114	#125×C₆₇	—	2	1	no	—
115	#125×C₆₇	—	2	1	no	—
116	#125×C₆₇	C₂³	1	4	yes	—
117	#124×C₆₇	—	2	2	no	—
118	#124×C₆₇	D₄	1	2	yes	—
119	#123×C₆₇	—	2	4	no	—
120	#123×C₆₇	—	2	4	no	—
121	#123×C₆₇	—	2	4	no	—
122	#123×C₆₇	—	2	4	no	—
123	#123×C₆₇	—	2	4	no	—
124	C₂²⋊Q₈×C₆₇	—	2	2	no	—
125	C₂²⋊Q₈×C₆₇	—	2	4	no	—
126	C₂²⋊Q₈×C₆₇	—	2	4	no	—
127	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
128	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
129	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
130	#121×C₆₇	—	2	2	no	—
131	#121×C₆₇	—	2	4	no	—
132	#121×C₆₇	—	2	4	no	—
133	#121×C₆₇	—	2	4	no	—
134	#121×C₆₇	D₄	1	4	yes	—
135	#121×C₆₇	C₂³	1	4	yes	—
136	C₄×Q₈×C₆₇	—	2	2	no	—
137	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
138	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
139	C₄×D₄×C₆₇	—	2	2	no	—
140	C₄×D₄×C₆₇	—	2	4	no	—
141	C₄×D₄×C₆₇	—	2	4	no	—
142	C₄×D₄×C₆₇	—	2	2	no	—
143	C₄×D₄×C₆₇	—	2	2	no	—
144	C₄×D₄×C₆₇	—	2	4	no	—
145	C₄×D₄×C₆₇	C₂³	1	2	yes	—
146	C₄×D₄×C₆₇	C₂³	1	2	yes	—
147	C₄×D₄×C₆₇	C₂³	1	4	yes	—
148	#117×C₆₇	—	2	4	no	—
149	#117×C₆₇	—	2	1	no	—
150	#117×C₆₇	—	2	1	no	—
151	#117×C₆₇	C₂³	1	4	yes	—
152	#117×C₆₇	C₂³	1	1	yes	characteristic
153	#117×C₆₇	C₂³	1	1	yes	characteristic
154	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
155	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
156	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
157	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
158	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
159	C₂×C₄²×C₆₇	—	2	1	no	—
160	C₂×C₄²×C₆₇	—	2	1	no	—
161	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
162	#107×C₆₇	C₂³	1	4	yes	—
163	#107×C₆₇	C₂³	1	2	yes	—
164	#106×C₆₇	C₂³	1	4	yes	—
165	#106×C₆₇	C₂³	1	2	yes	—
166	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
167	#103×C₆₇	C₂³	1	4	yes	—
168	#103×C₆₇	C₂³	1	4	yes	—
169	#103×C₆₇	C₂³	1	4	yes	—
170	#103×C₆₇	C₂³	1	4	yes	—
171	#103×C₆₇	C₂³	1	4	yes	—
172	#103×C₆₇	C₂³	1	4	yes	—
173	#103×C₆₇	C₂³	1	2	yes	—
174	#103×C₆₇	C₂³	1	4	yes	—
175	#103×C₆₇	C₂³	1	2	yes	—
176	#102×C₆₇	C₂³	1	4	yes	—
177	#102×C₆₇	C₂³	1	2	yes	—
178	#102×C₆₇	C₂³	1	2	yes	—
179	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
180	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
181	#97×C₆₇	C₂³	1	2	yes	—
182	#97×C₆₇	C₂³	1	2	yes	—
183	#41×C₆₇	—	2	4	no	—
184	#41×C₆₇	—	2	4	no	—
185	#41×C₆₇	—	2	2	no	—
186	#41×C₆₇	—	2	8	no	—
187	#41×C₆₇	—	2	2	no	—
188	#41×C₆₇	—	2	2	no	—
189	#41×C₆₇	—	4	4	no	—
190	#41×C₆₇	—	4	8	no	—
191	#41×C₆₇	—	4	16	no	—
192	#41×C₆₇	#41	1	4	yes	—
193	C₂×Q₈×C₆₇	—	2	4	no	—
194	C₂×Q₈×C₆₇	—	2	2	no	—
195	C₂×Q₈×C₆₇	—	2	8	no	—
196	C₂×Q₈×C₆₇	—	2	4	no	—
197	C₂×Q₈×C₆₇	—	2	4	no	—
198	C₂×Q₈×C₆₇	—	4	4	no	—
199	C₂×Q₈×C₆₇	—	4	8	no	—
200	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
201	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
202	C₂×Q₈×C₆₇	C₂⁴	1	1	yes	characteristic
203	C₂×Q₈×C₆₇	C₂⁴	1	1	yes	characteristic
204	C₂×D₄×C₆₇	—	2	1	no	—
205	C₂×D₄×C₆₇	—	2	1	no	—
206	C₂×D₄×C₆₇	—	2	4	no	—
207	C₂×D₄×C₆₇	—	2	4	no	—
208	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
209	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
210	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
211	C₂×D₄×C₆₇	C₂⁴	1	1	yes	characteristic
212	C₂×D₄×C₆₇	C₂⁴	1	1	yes	characteristic
213	C₂²×C₄×C₆₇	—	2	1	no	—
214	C₂²×C₄×C₆₇	—	2	2	no	—
215	C₂²×C₄×C₆₇	—	2	4	no	—
216	C₂²×C₄×C₆₇	—	2	2	no	—
217	C₂²×C₄×C₆₇	—	2	4	no	—
218	C₂²×C₄×C₆₇	—	2	2	no	—
219	C₂²×C₄×C₆₇	—	2	1	no	—
220	C₂²×C₄×C₆₇	—	2	4	no	—
221	C₂²×C₄×C₆₇	—	2	2	no	—
222	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
223	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
224	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
225	Q₁₆×C₆₇	—	2	2	no	—
226	Q₁₆×C₆₇	—	4	4	no	—
227	SD₁₆×C₆₇	—	2	2	no	—
228	SD₁₆×C₆₇	—	2	2	no	—
229	SD₁₆×C₆₇	—	4	4	no	—
230	D₈×C₆₇	—	2	2	no	—
231	M₁₆×C₆₇	—	2	4	no	—
232	M₁₆×C₆₇	—	4	4	no	—
233	C₂×C₈×C₆₇	—	2	2	no	—
234	C₂×C₈×C₆₇	—	2	2	no	—
235	C₂×C₈×C₆₇	—	4	4	no	—
236	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
237	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
238	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
239	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
240	C₄⋊C₄×C₆₇	—	2	4	no	—
241	C₄⋊C₄×C₆₇	—	2	4	no	—
242	C₄⋊C₄×C₆₇	—	2	4	no	—
243	C₄⋊C₄×C₆₇	—	2	4	no	—
244	C₄⋊C₄×C₆₇	—	2	4	no	—
245	C₄⋊C₄×C₆₇	—	2	4	no	—
246	C₄⋊C₄×C₆₇	—	2	2	no	—
247	C₄⋊C₄×C₆₇	—	2	4	no	—
248	C₄⋊C₄×C₆₇	—	2	2	no	—
249	C₄⋊C₄×C₆₇	—	2	2	no	—
250	C₄⋊C₄×C₆₇	—	2	4	no	—
251	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
252	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
253	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
254	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
255	C₂²⋊C₄×C₆₇	—	2	4	no	—
256	C₂²⋊C₄×C₆₇	—	2	1	no	—
257	C₂²⋊C₄×C₆₇	—	2	4	no	—
258	C₂²⋊C₄×C₆₇	—	2	1	no	—
259	C₂²⋊C₄×C₆₇	—	2	4	no	—
260	C₂²⋊C₄×C₆₇	—	2	4	no	—
261	C₂²⋊C₄×C₆₇	—	2	2	no	—
262	C₂²⋊C₄×C₆₇	—	2	2	no	—
263	C₂²⋊C₄×C₆₇	—	2	4	no	—
264	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
265	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
266	C₄²×C₆₇	—	2	2	no	—
267	C₄²×C₆₇	—	2	1	no	—
268	C₄²×C₆₇	—	2	1	no	—
269	C₄²×C₆₇	—	2	2	no	—
270	C₄²×C₆₇	—	2	4	no	—
271	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
272	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
273	C₄²×C₆₇	C₂⁴	1	2	yes	—
274	C₂³×C₆₇	—	2	1	no	—
275	C₂³×C₆₇	—	2	1	no	—
276	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
277	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
278	Q₈×C₆₇	—	2	4	no	—
279	Q₈×C₆₇	—	2	1	no	—
280	Q₈×C₆₇	—	2	1	no	—
281	Q₈×C₆₇	—	2	4	no	—
282	Q₈×C₆₇	—	2	4	no	—
283	Q₈×C₆₇	—	2	1	no	—
284	Q₈×C₆₇	—	4	8	no	—
285	Q₈×C₆₇	—	4	4	no	—
286	Q₈×C₆₇	C₂×#41	1	2	yes	—
287	D₄×C₆₇	—	2	4	no	—
288	D₄×C₆₇	—	2	8	no	—
289	D₄×C₆₇	—	2	4	no	—
290	D₄×C₆₇	—	2	1	no	—
291	D₄×C₆₇	—	2	1	no	—
292	D₄×C₆₇	—	2	1	no	—
293	D₄×C₆₇	—	4	4	no	—
294	D₄×C₆₇	—	4	8	no	—
295	D₄×C₆₇	—	4	2	no	—
296	D₄×C₆₇	—	8	4	no	—
297	D₄×C₆₇	C₂×#41	1	2	yes	—
298	C₂×C₄×C₆₇	—	2	1	no	—
299	C₂×C₄×C₆₇	—	2	4	no	—
300	C₂×C₄×C₆₇	—	2	4	no	—
301	C₂×C₄×C₆₇	—	2	4	no	—
302	C₂×C₄×C₆₇	—	2	2	no	—
303	C₂×C₄×C₆₇	—	2	2	no	—
304	C₂×C₄×C₆₇	—	2	1	no	—
305	C₂×C₄×C₆₇	—	2	4	no	—
306	C₂×C₄×C₆₇	—	2	4	no	—
307	C₂×C₄×C₆₇	—	4	2	no	—
308	C₂×C₄×C₆₇	—	4	8	no	—
309	C₂×C₄×C₆₇	—	4	8	no	—
310	C₂×C₄×C₆₇	—	4	4	no	—
311	C₂×C₄×C₆₇	—	4	4	no	—
312	C₂×C₄×C₆₇	—	4	2	no	—
313	C₂×C₄×C₆₇	—	8	2	no	—
314	C₂×C₄×C₆₇	—	8	4	no	—
315	C₂×C₄×C₆₇	—	8	2	no	—
316	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
317	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
318	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
319	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
320	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
321	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
322	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
323	C₈×C₆₇	—	2	1	no	—
324	C₈×C₆₇	—	2	1	no	—
325	C₈×C₆₇	—	2	2	no	—
326	C₈×C₆₇	—	2	4	no	—
327	C₂²×C₆₇	—	2	2	no	—
328	C₂²×C₆₇	—	2	4	no	—
329	C₂²×C₆₇	—	2	4	no	—
330	C₂²×C₆₇	—	2	2	no	—
331	C₂²×C₆₇	—	2	4	no	—
332	C₂²×C₆₇	—	4	2	no	—
333	C₂²×C₆₇	—	4	2	no	—
334	C₂²×C₆₇	—	4	1	no	—
335	C₂²×C₆₇	#546	1	2	yes	—
336	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
337	C₄×C₆₇	—	2	2	no	—
338	C₄×C₆₇	—	2	2	no	—
339	C₄×C₆₇	—	2	2	no	—
340	C₄×C₆₇	—	2	4	no	—
341	C₄×C₆₇	—	4	2	no	—
342	C₄×C₆₇	—	4	2	no	—
343	C₄×C₆₇	—	4	4	no	—
344	C₄×C₆₇	—	4	4	no	—
345	C₄×C₆₇	—	4	1	no	—
346	C₄×C₆₇	#546	1	2	yes	—
347	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
348	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
349	#61224	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
350	C₂×C₆₇	—	2	2	no	—
351	C₂×C₆₇	—	4	4	no	—
352	C₂×C₆₇	—	4	2	no	—
353	C₂×C₆₇	—	8	1	no	—
354	C₂×C₆₇	C₂×#546	1	1	yes	characteristic
355	C₂×C₆₇	#3586	1	2	yes	—
356	#3512	C₂×C₆₇	1	1	yes	characteristic, complemented
357	#3499	C₂×C₆₇	1	1	yes	characteristic, complemented
358	#3320	C₂×C₆₇	1	4	yes	complemented
359	#3315	C₂×C₆₇	1	4	yes	complemented
360	#3312	C₂×C₆₇	1	4	yes	complemented
361	#3307	C₂×C₆₇	1	4	yes	complemented
362	C₂×#491	C₂×C₆₇	1	1	yes	characteristic, complemented
363	#3097	C₂×C₆₇	1	2	yes	complemented
364	#3080	C₂×C₆₇	1	1	yes	characteristic, complemented
365	#3063	C₂×C₆₇	1	2	yes	complemented
366	#3039	C₂×C₆₇	1	1	yes	characteristic, complemented
367	#3013	C₂×C₆₇	1	2	yes	complemented
368	#2943	C₂×C₆₇	1	1	yes	characteristic, complemented
369	#2932	C₂×C₆₇	1	1	yes	characteristic, complemented
370	#2898	C₂×C₆₇	1	2	yes	complemented
371	C₆₇	#61224	1	1	yes	characteristic, direct product factor, Sylow
372	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	2	yes	—
373	C₂×#137	C₂²×C₆₇	1	2	yes	—
374	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
375	C₂×#131	C₂²×C₆₇	1	2	yes	—
376	#553	—	2	4	no	—
377	#553	C₂²×C₆₇	1	4	yes	—
378	#552	C₂²×C₆₇	1	4	yes	—
379	D₄×Q₈	C₂²×C₆₇	1	4	yes	—
380	#547	—	2	4	no	—
381	#547	C₂²×C₆₇	1	4	yes	—
382	#541	—	2	1	no	—
383	#540	—	2	1	no	—
384	#537	C₂²×C₆₇	1	2	yes	—
385	#536	—	2	2	no	—
386	#536	C₂²×C₆₇	1	2	yes	—
387	#532	—	2	1	no	—
388	#532	C₂²×C₆₇	1	1	yes	characteristic
389	C₂×C₄⋊Q₈	—	2	1	no	—
390	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
391	#529	—	2	2	no	—
392	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
393	#520	C₂²×C₆₇	1	2	yes	—
394	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
395	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
396	#491	C₂²×C₆₇	1	4	yes	—
397	#491	C₂²×C₆₇	1	4	yes	—
398	#485	C₂²×C₆₇	1	4	yes	—
399	#485	C₂²×C₆₇	1	4	yes	—
400	#484	C₂²×C₆₇	1	4	yes	—
401	#483	C₂²×C₆₇	1	4	yes	—
402	#479	C₂²×C₆₇	1	4	yes	—
403	#478	C₂²×C₆₇	1	4	yes	—
404	#470	C₂²×C₆₇	1	4	yes	—
405	#468	C₂²×C₆₇	1	4	yes	—
406	#467	C₂²×C₆₇	1	4	yes	—
407	#465	C₂²×C₆₇	1	4	yes	—
408	#462	C₂²×C₆₇	1	4	yes	—
409	#461	C₂²×C₆₇	1	4	yes	—
410	#452	C₂²×C₆₇	1	4	yes	—
411	#451	C₂²×C₆₇	1	4	yes	—
412	#449	C₂²×C₆₇	1	4	yes	—
413	#448	C₂²×C₆₇	1	4	yes	—
414	#442	C₂²×C₆₇	1	2	yes	—
415	#441	C₂²×C₆₇	1	2	yes	—
416	#440	C₂²×C₆₇	1	2	yes	—
417	#440	C₂²×C₆₇	1	2	yes	—
418	#435	C₂²×C₆₇	1	4	yes	—
419	#435	C₂²×C₆₇	1	4	yes	—
420	#428	C₂²×C₆₇	1	2	yes	—
421	#427	C₂²×C₆₇	1	1	yes	characteristic
422	#423	C₂²×C₆₇	1	1	yes	characteristic
423	#419	C₂²×C₆₇	1	2	yes	—
424	#419	C₂²×C₆₇	1	4	yes	—
425	#417	C₂²×C₆₇	1	2	yes	—
426	#417	C₂²×C₆₇	1	4	yes	—
427	#416	C₂²×C₆₇	1	1	yes	characteristic
428	#416	C₂²×C₆₇	1	4	yes	—
429	#416	C₂²×C₆₇	1	1	yes	characteristic
430	C₂×#103	C₂²×C₆₇	1	2	yes	—
431	C₂×#103	C₂²×C₆₇	1	2	yes	—
432	C₂×#103	C₂²×C₆₇	1	4	yes	—
433	#408	C₂²×C₆₇	1	2	yes	—
434	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
435	2₋¹⁺⁴	—	2	8	no	—
436	2₋¹⁺⁴	—	4	4	no	—
437	C₂×#41	—	2	2	no	—
438	C₂×#41	—	2	4	no	—
439	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
440	C₂×#41	C₂³×C₆₇	1	4	yes	—
441	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
442	C₂×#41	C₂³×C₆₇	1	2	yes	—
443	C₂²×Q₈	—	2	2	no	—
444	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
445	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
446	#137	—	4	4	no	—
447	#135	—	2	4	no	—
448	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
449	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
450	C₂×SD₁₆	C₂³×C₆₇	1	1	yes	characteristic
451	C₂×SD₁₆	C₂³×C₆₇	1	1	yes	characteristic
452	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
453	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
454	#131	—	4	4	no	—
455	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
456	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
457	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
458	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
459	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
460	C₄⋊Q₈	—	2	2	no	—
461	C₄⋊Q₈	—	2	4	no	—
462	C₄⋊Q₈	—	2	1	no	—
463	C₄⋊Q₈	—	2	1	no	—
464	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
465	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
466	#127	D₄×C₆₇	1	1	yes	characteristic
467	#126	—	2	4	no	—
468	#126	—	2	4	no	—
469	#125	—	2	4	no	—
470	#125	—	2	1	no	—
471	#125	—	2	1	no	—
472	#125	C₂³×C₆₇	1	4	yes	—
473	#124	—	2	2	no	—
474	#124	D₄×C₆₇	1	2	yes	—
475	#123	—	2	4	no	—
476	#123	—	2	4	no	—
477	#123	—	2	4	no	—
478	#123	—	2	4	no	—
479	#123	—	2	4	no	—
480	C₂²⋊Q₈	—	2	2	no	—
481	C₂²⋊Q₈	—	2	4	no	—
482	C₂²⋊Q₈	—	2	4	no	—
483	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
484	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
485	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
486	#121	—	2	2	no	—
487	#121	—	2	4	no	—
488	#121	—	2	4	no	—
489	#121	—	2	4	no	—
490	#121	D₄×C₆₇	1	4	yes	—
491	#121	C₂³×C₆₇	1	4	yes	—
492	C₄×Q₈	—	2	2	no	—
493	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
494	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
495	C₄×D₄	—	2	2	no	—
496	C₄×D₄	—	2	4	no	—
497	C₄×D₄	—	2	4	no	—
498	C₄×D₄	—	2	2	no	—
499	C₄×D₄	—	2	2	no	—
500	C₄×D₄	—	2	4	no	—
501	C₄×D₄	C₂³×C₆₇	1	2	yes	—
502	C₄×D₄	C₂³×C₆₇	1	2	yes	—
503	C₄×D₄	C₂³×C₆₇	1	4	yes	—
504	#117	—	2	4	no	—
505	#117	—	2	1	no	—
506	#117	—	2	1	no	—
507	#117	C₂³×C₆₇	1	1	yes	characteristic
508	#117	C₂³×C₆₇	1	1	yes	characteristic
509	#117	C₂³×C₆₇	1	4	yes	—
510	C₂×C₄⋊C₄	—	2	2	no	—
511	C₂×C₄⋊C₄	—	2	2	no	—
512	C₂×C₄⋊C₄	—	2	4	no	—
513	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
514	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
515	C₂×C₄²	—	2	1	no	—
516	C₂×C₄²	—	2	1	no	—
517	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
518	#107	C₂³×C₆₇	1	2	yes	—
519	#107	C₂³×C₆₇	1	4	yes	—
520	#106	C₂³×C₆₇	1	2	yes	—
521	#106	C₂³×C₆₇	1	4	yes	—
522	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
523	#103	C₂³×C₆₇	1	2	yes	—
524	#103	C₂³×C₆₇	1	4	yes	—
525	#103	C₂³×C₆₇	1	4	yes	—
526	#103	C₂³×C₆₇	1	4	yes	—
527	#103	C₂³×C₆₇	1	4	yes	—
528	#103	C₂³×C₆₇	1	4	yes	—
529	#103	C₂³×C₆₇	1	4	yes	—
530	#103	C₂³×C₆₇	1	4	yes	—
531	#103	C₂³×C₆₇	1	2	yes	—
532	#102	C₂³×C₆₇	1	2	yes	—
533	#102	C₂³×C₆₇	1	4	yes	—
534	#102	C₂³×C₆₇	1	2	yes	—
535	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
536	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
537	#97	C₂³×C₆₇	1	2	yes	—
538	#97	C₂³×C₆₇	1	2	yes	—
539	#41	—	2	4	no	—
540	#41	—	2	4	no	—
541	#41	—	2	2	no	—
542	#41	—	2	8	no	—
543	#41	—	2	2	no	—
544	#41	—	2	2	no	—
545	#41	—	4	4	no	—
546	#41	—	4	8	no	—
547	#41	—	4	16	no	—
548	#41	#41×C₆₇	1	4	yes	—
549	C₂×Q₈	—	2	4	no	—
550	C₂×Q₈	—	2	2	no	—
551	C₂×Q₈	—	2	8	no	—
552	C₂×Q₈	—	2	4	no	—
553	C₂×Q₈	—	2	4	no	—
554	C₂×Q₈	—	4	4	no	—
555	C₂×Q₈	—	4	8	no	—
556	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
557	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
558	C₂×Q₈	C₂⁴×C₆₇	1	1	yes	characteristic
559	C₂×Q₈	C₂⁴×C₆₇	1	1	yes	characteristic
560	C₂×D₄	—	2	1	no	—
561	C₂×D₄	—	2	1	no	—
562	C₂×D₄	—	2	4	no	—
563	C₂×D₄	—	2	4	no	—
564	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
565	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
566	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
567	C₂×D₄	C₂⁴×C₆₇	1	1	yes	characteristic
568	C₂×D₄	C₂⁴×C₆₇	1	1	yes	characteristic
569	C₂²×C₄	—	2	1	no	—
570	C₂²×C₄	—	2	2	no	—
571	C₂²×C₄	—	2	4	no	—
572	C₂²×C₄	—	2	2	no	—
573	C₂²×C₄	—	2	4	no	—
574	C₂²×C₄	—	2	2	no	—
575	C₂²×C₄	—	2	1	no	—
576	C₂²×C₄	—	2	4	no	—
577	C₂²×C₄	—	2	2	no	—
578	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
579	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
580	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
581	Q₁₆	—	2	2	no	—
582	Q₁₆	—	4	4	no	—
583	SD₁₆	—	2	2	no	—
584	SD₁₆	—	2	2	no	—
585	SD₁₆	—	4	4	no	—
586	D₈	—	2	2	no	—
587	M₁₆	—	2	4	no	—
588	M₁₆	—	4	4	no	—
589	C₂×C₈	—	2	2	no	—
590	C₂×C₈	—	2	2	no	—
591	C₂×C₈	—	4	4	no	—
592	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
593	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
594	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
595	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
596	C₄⋊C₄	—	2	4	no	—
597	C₄⋊C₄	—	2	4	no	—
598	C₄⋊C₄	—	2	4	no	—
599	C₄⋊C₄	—	2	4	no	—
600	C₄⋊C₄	—	2	4	no	—
601	C₄⋊C₄	—	2	4	no	—
602	C₄⋊C₄	—	2	2	no	—
603	C₄⋊C₄	—	2	4	no	—
604	C₄⋊C₄	—	2	2	no	—
605	C₄⋊C₄	—	2	2	no	—
606	C₄⋊C₄	—	2	4	no	—
607	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
608	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
609	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
610	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
611	C₂²⋊C₄	—	2	4	no	—
612	C₂²⋊C₄	—	2	1	no	—
613	C₂²⋊C₄	—	2	4	no	—
614	C₂²⋊C₄	—	2	1	no	—
615	C₂²⋊C₄	—	2	4	no	—
616	C₂²⋊C₄	—	2	4	no	—
617	C₂²⋊C₄	—	2	2	no	—
618	C₂²⋊C₄	—	2	2	no	—
619	C₂²⋊C₄	—	2	4	no	—
620	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
621	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
622	C₄²	—	2	2	no	—
623	C₄²	—	2	1	no	—
624	C₄²	—	2	1	no	—
625	C₄²	—	2	2	no	—
626	C₄²	—	2	4	no	—
627	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
628	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
629	C₄²	C₂⁴×C₆₇	1	2	yes	—
630	C₂³	—	2	1	no	—
631	C₂³	—	2	1	no	—
632	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
633	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
634	Q₈	—	2	4	no	—
635	Q₈	—	2	1	no	—
636	Q₈	—	2	1	no	—
637	Q₈	—	2	4	no	—
638	Q₈	—	2	4	no	—
639	Q₈	—	2	1	no	—
640	Q₈	—	4	8	no	—
641	Q₈	—	4	4	no	—
642	Q₈	C₂×#41×C₆₇	1	2	yes	—
643	D₄	—	2	4	no	—
644	D₄	—	2	8	no	—
645	D₄	—	2	4	no	—
646	D₄	—	2	1	no	—
647	D₄	—	2	1	no	—
648	D₄	—	2	1	no	—
649	D₄	—	4	4	no	—
650	D₄	—	4	8	no	—
651	D₄	—	4	2	no	—
652	D₄	—	8	4	no	—
653	D₄	C₂×#41×C₆₇	1	2	yes	—
654	C₂×C₄	—	2	1	no	—
655	C₂×C₄	—	2	4	no	—
656	C₂×C₄	—	2	4	no	—
657	C₂×C₄	—	2	4	no	—
658	C₂×C₄	—	2	2	no	—
659	C₂×C₄	—	2	2	no	—
660	C₂×C₄	—	2	1	no	—
661	C₂×C₄	—	2	4	no	—
662	C₂×C₄	—	2	4	no	—
663	C₂×C₄	—	4	2	no	—
664	C₂×C₄	—	4	8	no	—
665	C₂×C₄	—	4	8	no	—
666	C₂×C₄	—	4	4	no	—
667	C₂×C₄	—	4	4	no	—
668	C₂×C₄	—	4	2	no	—
669	C₂×C₄	—	8	2	no	—
670	C₂×C₄	—	8	4	no	—
671	C₂×C₄	—	8	2	no	—
672	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
673	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
674	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
675	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
676	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
677	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
678	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
679	C₈	—	2	1	no	—
680	C₈	—	2	1	no	—
681	C₈	—	2	2	no	—
682	C₈	—	2	4	no	—
683	C₂²	—	2	2	no	—
684	C₂²	—	2	4	no	—
685	C₂²	—	2	4	no	—
686	C₂²	—	2	2	no	—
687	C₂²	—	2	4	no	—
688	C₂²	—	4	2	no	—
689	C₂²	—	4	2	no	—
690	C₂²	—	4	1	no	—
691	C₂²	#546×C₆₇	1	2	yes	—
692	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
693	C₄	—	2	2	no	—
694	C₄	—	2	2	no	—
695	C₄	—	2	2	no	—
696	C₄	—	2	4	no	—
697	C₄	—	4	2	no	—
698	C₄	—	4	2	no	—
699	C₄	—	4	4	no	—
700	C₄	—	4	4	no	—
701	C₄	—	4	1	no	—
702	C₄	#546×C₆₇	1	2	yes	—
703	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
704	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
705	C₂	—	2	2	no	—
706	C₂	—	4	4	no	—
707	C₂	—	4	2	no	—
708	C₂	—	8	1	no	—
709	C₂	C₂×#546×C₆₇	1	1	yes	characteristic
710	C₂	C₆₇×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
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×#3512	C₂	semidirect
2. C₆₇×#3499	C₂	semidirect
3. C₆₇×#3320	C₂	semidirect
4. C₆₇×#3315	C₂	semidirect
5. C₆₇×#3312	C₂	semidirect
6. C₆₇×#3307	C₂	semidirect
7. C₂×#491×C₆₇	C₂	semidirect
8. C₆₇×#3097	C₂	semidirect
9. C₆₇×#3080	C₂	semidirect
10. C₆₇×#3063	C₂	semidirect
11. C₆₇×#3039	C₂	semidirect
12. C₆₇×#3013	C₂	semidirect
13. C₆₇×#2943	C₂	semidirect
14. C₆₇×#2932	C₂	semidirect
15. C₆₇×#2898	C₂	semidirect
349. #61224	C₆₇	direct (N × H)
356. #3512	C₂×C₆₇	semidirect
357. #3499	C₂×C₆₇	semidirect
358. #3320	C₂×C₆₇	semidirect
359. #3315	C₂×C₆₇	semidirect
360. #3312	C₂×C₆₇	semidirect
361. #3307	C₂×C₆₇	semidirect
362. C₂×#491	C₂×C₆₇	semidirect
363. #3097	C₂×C₆₇	semidirect
364. #3080	C₂×C₆₇	semidirect
365. #3063	C₂×C₆₇	semidirect
366. #3039	C₂×C₆₇	semidirect
367. #3013	C₂×C₆₇	semidirect
368. #2943	C₂×C₆₇	semidirect
369. #2932	C₂×C₆₇	semidirect
370. #2898	C₂×C₆₇	semidirect
371. C₆₇	#61224	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×#3512	2
C₆₇×#3499	2
C₆₇×#3320	2
C₆₇×#3315	2
C₆₇×#3312	2
C₆₇×#3307	2
C₂×#491×C₆₇	2
C₆₇×#3097	2
C₆₇×#3080	2
C₆₇×#3063	2
C₆₇×#3039	2
C₆₇×#3013	2
C₆₇×#2943	2
C₆₇×#2932	2
C₆₇×#2898	2
#61224	67

Subgroups — C67×#61224 #30740475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61224 #30740475