Subgroups — #30860769

Order 17152.

780 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568842	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11761612	C₂	1	1	yes	maximal, characteristic
3	#22560985	C₂	1	4	yes	maximal, complemented
4	#22560416	C₂	1	4	yes	maximal
5	C₄×#14523979	C₂	1	4	yes	maximal, complemented
6	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₂²	1	2	yes	—
8	#16863453	—	2	1	no	—
9	#16863453	—	2	1	no	—
10	#16863449	—	2	1	no	—
11	#16863449	—	2	1	no	—
12	C₂×#14523979	C₂²	1	8	yes	—
13	#16863420	C₂²	1	2	yes	—
14	#16863360	C₂²	1	2	yes	—
15	C₄×#11761612	—	2	1	no	—
16	C₄×#11761612	—	2	1	no	—
17	C₄×#11761612	C₂²	1	2	yes	—
18	C₄×#11761612	C₂²	1	2	yes	—
19	C₄×#11761612	C₂²	1	2	yes	—
20	C₂×#14523954	—	2	1	no	—
21	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
22	#16863161	C₂²	1	8	yes	—
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	#431×C₆₇	C₄	1	2	yes	—
25	#431×C₆₇	C₄	1	2	yes	—
26	#431×C₆₇	C₄	1	2	yes	—
27	#431×C₆₇	C₄	1	2	yes	—
28	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
31	C₂×#11761612	—	2	8	no	—
32	C₂×#11761612	—	2	2	no	—
33	C₂×#11761612	C₂³	1	4	yes	—
34	C₂×#11761612	C₂³	1	4	yes	—
35	C₂²×C₆₇⋊C₈	—	2	2	no	—
36	#14523979	—	2	16	no	—
37	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
38	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
39	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
40	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
41	#14523954	—	2	2	no	—
42	#14523954	—	2	2	no	—
43	#14523954	—	2	2	no	—
44	C₄×C₆₇⋊C₈	—	2	2	no	—
45	C₄×C₆₇⋊C₈	—	2	2	no	—
46	C₄×C₆₇⋊C₈	—	2	2	no	—
47	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
49	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
51	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
61	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
64	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
65	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
66	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
67	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
68	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
69	#11761612	—	2	8	no	—
70	#11761612	—	2	4	no	—
71	#11761612	—	2	8	no	—
72	#11761612	#41	1	8	yes	—
73	C₂×C₆₇⋊C₈	—	2	8	no	—
74	C₂×C₆₇⋊C₈	—	2	4	no	—
75	C₂×C₆₇⋊C₈	—	2	4	no	—
76	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	—	2	4	no	—
78	C₂²×C₄×C₆₇	—	2	4	no	—
79	C₂²×C₄×C₆₇	—	4	2	no	—
80	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
98	C₂×C₈×C₆₇	—	2	8	no	—
99	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
100	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂³×C₆₇	—	2	2	no	—
117	C₂³×C₆₇	—	2	2	no	—
118	C₂³×C₆₇	—	4	1	no	—
119	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	4	1	no	—
136	C₂×C₄×C₆₇	—	4	1	no	—
137	C₂×C₄×C₆₇	—	4	1	no	—
138	C₂×C₄×C₆₇	#108	1	4	yes	complemented
139	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
152	C₈×C₆₇	—	2	8	no	—
153	C₆₇⋊C₈	—	2	8	no	—
154	C₆₇⋊C₈	—	4	4	no	—
155	C₂²×C₆₇	—	2	1	no	—
156	C₂²×C₆₇	—	2	1	no	—
157	C₂²×C₆₇	—	2	1	no	—
158	C₂²×C₆₇	—	2	1	no	—
159	C₂²×C₆₇	—	2	1	no	—
160	C₂²×C₆₇	—	2	1	no	—
161	C₂²×C₆₇	—	2	1	no	—
162	C₂²×C₆₇	—	4	4	no	—
163	C₂²×C₆₇	C₂×#108	1	2	yes	—
164	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
165	C₂²×C₆₇	#533	1	2	yes	—
166	C₄×C₆₇	—	2	1	no	—
167	C₄×C₆₇	—	2	1	no	—
168	C₄×C₆₇	—	2	1	no	—
169	C₄×C₆₇	—	2	1	no	—
170	C₄×C₆₇	—	2	1	no	—
171	C₄×C₆₇	—	2	1	no	—
172	C₄×C₆₇	—	2	1	no	—
173	C₄×C₆₇	C₂×#108	1	2	yes	—
174	C₄×C₆₇	C₂×#108	1	2	yes	—
175	C₄×C₆₇	#430	1	2	yes	—
176	C₄×C₆₇	C₄×#41	1	2	yes	—
177	C₄×C₆₇	#525	1	2	yes	—
178	#21370	—	67	1	no	maximal, Sylow
179	C₂×C₆₇	—	2	2	no	—
180	C₂×C₆₇	—	2	2	no	—
181	C₂×C₆₇	—	4	1	no	—
182	C₂×C₆₇	#2296	1	1	yes	characteristic
183	C₂×C₆₇	C₂²×#108	1	1	yes	characteristic
184	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
185	#2920	—	67	1	no	—
186	C₂×#431	D₆₇	1	1	yes	characteristic
187	C₂×C₄×M₁₆	—	67	1	no	—
188	#1950	—	67	4	no	—
189	#1855	—	67	4	no	—
190	C₄×#108	—	67	4	no	—
191	C₆₇	#21370	1	1	yes	characteristic, complemented, Sylow
192	C₂²×M₁₆	—	67	2	no	—
193	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
194	#432	—	134	1	no	—
195	#432	—	134	1	no	—
196	#431	—	134	1	no	—
197	#431	—	134	1	no	—
198	#431	Dic₆₇	1	2	yes	—
199	#431	Dic₆₇	1	2	yes	—
200	#431	Dic₆₇	1	2	yes	—
201	#431	Dic₆₇	1	2	yes	—
202	C₂×#108	—	67	8	no	—
203	#423	—	67	2	no	—
204	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
205	#407	—	67	2	no	—
206	C₂×C₂²⋊C₈	C₂×D₆₇	1	2	yes	—
207	C₄×M₁₆	—	67	2	no	—
208	C₄×M₁₆	—	67	2	no	—
209	C₄×M₁₆	—	67	2	no	—
210	C₄×M₁₆	—	134	1	no	—
211	C₄×M₁₆	—	134	1	no	—
212	C₂×#97	—	134	1	no	—
213	C₂×C₄×C₈	—	134	1	no	—
214	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
215	#341	—	67	8	no	—
216	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
217	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
218	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
219	C₂×M₁₆	—	67	4	no	—
220	C₂×M₁₆	—	67	4	no	—
221	C₂×M₁₆	—	134	8	no	—
222	C₂×M₁₆	—	134	2	no	—
223	C₂²×C₈	—	134	2	no	—
224	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
225	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
226	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
227	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
228	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
229	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
230	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
231	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
232	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
233	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
234	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
236	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
237	#108	—	134	16	no	—
238	C₄⋊C₈	—	134	2	no	—
239	C₄⋊C₈	—	134	2	no	—
240	C₄⋊C₈	C₂×Dic₆₇	1	4	yes	—
241	C₄⋊C₈	C₂×Dic₆₇	1	4	yes	—
242	C₂²⋊C₈	—	134	2	no	—
243	C₂²⋊C₈	—	134	2	no	—
244	C₂²⋊C₈	C₂×Dic₆₇	1	4	yes	—
245	C₂²⋊C₈	C₂×Dic₆₇	1	4	yes	—
246	#97	—	134	2	no	—
247	#97	—	134	2	no	—
248	#97	—	134	2	no	—
249	C₄×C₈	—	134	2	no	—
250	C₄×C₈	—	134	2	no	—
251	C₄×C₈	—	134	2	no	—
252	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
253	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
254	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
259	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
260	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
261	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
262	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
263	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
264	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
265	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
266	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
267	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
270	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
271	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
272	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
273	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
274	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
275	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
276	M₁₆	—	67	8	no	—
277	M₁₆	—	134	8	no	—
278	M₁₆	—	134	8	no	—
279	M₁₆	—	134	4	no	—
280	C₂×C₈	—	2	8	no	—
281	C₂×C₈	—	134	4	no	—
282	C₂×C₈	—	134	8	no	—
283	C₂×C₈	—	134	4	no	—
284	C₂×C₈	C₂²×Dic₆₇	1	8	yes	—
285	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
286	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
287	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
288	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
289	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
290	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
291	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
292	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
293	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
294	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
295	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
296	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
297	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
298	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
299	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
300	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
301	C₂³	—	2	2	no	—
302	C₂³	—	2	2	no	—
303	C₂³	—	4	1	no	—
304	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
305	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
306	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	#14523979	1	4	yes	complemented
324	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
325	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
326	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
327	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
328	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
329	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
330	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
331	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
332	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
333	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
334	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
335	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
336	C₂×C₄	#41×D₆₇	1	4	yes	—
337	C₈	—	2	8	no	—
338	C₈	—	268	4	no	—
339	C₈	—	134	8	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	4	no	—
348	C₂²	C₂×#14523979	1	2	yes	—
349	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
350	C₂²	#16863949	1	2	yes	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	C₂×#14523979	1	2	yes	—
359	C₄	C₂×#14523979	1	2	yes	—
360	C₄	#16863445	1	2	yes	—
361	C₄	#41×Dic₆₇	1	2	yes	—
362	C₄	#16863908	1	2	yes	—
363	C₂	—	2	2	no	—
364	C₂	—	2	2	no	—
365	C₂	—	4	1	no	—
366	C₂	#22562693	1	1	yes	characteristic
367	C₂	C₂²×#14523979	1	1	yes	characteristic
368	C₂	C₂×#16863445	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#431×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#431	D₆₇
hypocenter	C₆₇	#21370
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #22560985	C₂	semidirect
5. C₄×#14523979	C₂	semidirect
138. C₂×C₄×C₆₇	#108	semidirect
191. C₆₇	#21370	semidirect
323. C₂×C₄	#14523979	semidirect

Maximal subgroups

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

subgroup	index
#22568842	2
C₂×C₄×#11761612	2
#22560985	2
#22560416	2
C₄×#14523979	2
C₂×#431×C₆₇	2
#21370	67