Subgroups — #37582623

Order 24448.

392 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15690287	C₂	1	1	yes	maximal, characteristic, complemented
2	#27133001	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132984	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132967	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132960	C₂	1	1	yes	maximal, characteristic, complemented
6	#27132882	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132875	C₂	1	1	yes	maximal, characteristic, complemented
8	#27132823	C₂	1	1	yes	maximal, characteristic, complemented
9	#27132806	C₂	1	1	yes	maximal, characteristic
10	#27132773	C₂	1	1	yes	maximal, characteristic, complemented
11	#27132768	C₂	1	1	yes	maximal, characteristic, complemented
12	#27132763	C₂	1	1	yes	maximal, characteristic, complemented
13	#27132761	C₂	1	1	yes	maximal, characteristic, complemented
14	#27132720	C₂	1	1	yes	maximal, characteristic, complemented
15	#529×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15690287	C₂²	1	1	yes	characteristic
17	C₂×#15690277	C₂²	1	1	yes	characteristic
18	C₂×#15690275	C₂²	1	1	yes	characteristic
19	C₂×#15690274	C₂²	1	1	yes	characteristic
20	C₂×#15690272	C₂²	1	1	yes	characteristic
21	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
22	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
23	C₄×D₇₆₄	C₂²	1	1	yes	characteristic
24	C₄×D₇₆₄	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊D₁₉₁	—	2	1	no	—
26	C₄×C₂²⋊D₁₉₁	—	2	1	no	—
27	#20259281	C₂²	1	1	yes	characteristic
28	#20259279	—	2	1	no	—
29	C₄²×D₁₉₁	—	2	1	no	—
30	#20259255	C₂²	1	1	yes	characteristic
31	#20259255	C₂²	1	1	yes	characteristic
32	#20259253	C₂²	1	1	yes	characteristic
33	#20259253	C₂²	1	1	yes	characteristic
34	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
35	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
36	#20259246	C₂²	1	1	yes	characteristic
37	#20259246	C₂²	1	1	yes	characteristic
38	#20259245	C₂²	1	1	yes	characteristic
39	#20259245	C₂²	1	1	yes	characteristic
40	#20259243	C₂²	1	1	yes	characteristic
41	#20259243	C₂²	1	1	yes	characteristic
42	#20259242	C₂²	1	1	yes	characteristic
43	#20259242	C₂²	1	1	yes	characteristic
44	C₂²⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
45	C₂²⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
46	#20259233	C₂²	1	1	yes	characteristic
47	#20259233	C₂²	1	1	yes	characteristic
48	#126×C₁₉₁	—	2	1	no	—
49	#125×C₁₉₁	C₂²	1	1	yes	characteristic
50	#124×C₁₉₁	C₂²	1	1	yes	characteristic
51	#123×C₁₉₁	—	2	1	no	—
52	C₂²⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
53	#121×C₁₉₁	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₁₉₁	—	2	1	no	—
56	C₄×D₄×C₁₉₁	C₂²	1	1	yes	characteristic
57	#117×C₁₉₁	—	2	1	no	—
58	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
59	#15690287	—	2	2	no	—
60	#15690287	—	2	2	no	—
61	C₂×Dic₃₈₂	C₂³	1	1	yes	characteristic
62	C₂×D₇₆₄	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₉₁	—	2	1	no	—
64	C₂×C₄×D₁₉₁	—	2	1	no	—
65	C₂×C₄×D₁₉₁	—	2	1	no	—
66	C₂×C₄×D₁₉₁	—	2	1	no	—
67	#15690277	—	2	2	no	—
68	#15690275	—	2	2	no	—
69	#15690274	—	2	2	no	—
70	#15690272	—	2	2	no	—
71	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
77	Dic₁₉₁⋊C₄	—	2	1	no	—
78	Dic₁₉₁⋊C₄	—	2	1	no	—
79	C₂²⋊Dic₁₉₁	—	2	1	no	—
80	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
81	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
82	C₄×Dic₁₉₁	—	2	1	no	—
83	C₄×Dic₁₉₁	—	2	1	no	—
84	C₄×Dic₁₉₁	—	2	1	no	—
85	C₂×Q₈×C₁₉₁	C₂³	1	1	yes	characteristic
86	C₂×D₄×C₁₉₁	—	2	1	no	—
87	C₂×D₄×C₁₉₁	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₁₉₁	—	2	1	no	—
89	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₁₉₁	—	2	1	no	—
93	C₄⋊C₄×C₁₉₁	—	2	1	no	—
94	C₄⋊C₄×C₁₉₁	—	2	1	no	—
95	C₄⋊C₄×C₁₉₁	—	2	1	no	—
96	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
98	C₂²⋊C₄×C₁₉₁	—	2	1	no	—
99	C₂²⋊C₄×C₁₉₁	—	2	1	no	—
100	C₂²⋊C₄×C₁₉₁	—	2	1	no	—
101	C₂²⋊C₄×C₁₉₁	—	2	1	no	—
102	C₂²⋊C₄×C₁₉₁	—	2	1	no	—
103	C₄²×C₁₉₁	—	2	1	no	—
104	C₄²×C₁₉₁	D₄	1	1	yes	characteristic
105	C₄²×C₁₉₁	D₄	1	1	yes	characteristic
106	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
107	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
108	C₂²×D₁₉₁	—	2	1	no	—
109	Dic₃₈₂	—	2	1	no	—
110	Dic₃₈₂	#41	1	2	yes	—
111	D₇₆₄	—	2	1	no	—
112	D₇₆₄	#41	1	2	yes	—
113	C₂²⋊D₁₉₁	—	2	4	no	—
114	C₂×Dic₁₉₁	—	2	1	no	—
115	C₂×Dic₁₉₁	—	2	1	no	—
116	C₂×Dic₁₉₁	—	2	1	no	—
117	C₄×D₁₉₁	—	2	4	no	—
118	C₄×D₁₉₁	—	4	2	no	—
119	C₄×D₁₉₁	—	4	2	no	—
120	C₁₉₁⋊C₈	—	2	1	no	—
121	C₁₉₁⋊C₈	—	2	1	no	—
122	C₁₉₁⋊C₈	—	2	1	no	—
123	C₁₉₁⋊C₈	—	2	1	no	—
124	C₂³×C₁₉₁	—	2	1	no	—
125	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
126	Q₈×C₁₉₁	—	2	1	no	—
127	Q₈×C₁₉₁	—	2	1	no	—
128	D₄×C₁₉₁	—	2	1	no	—
129	D₄×C₁₉₁	—	2	1	no	—
130	D₄×C₁₉₁	—	4	2	no	—
131	C₂×C₄×C₁₉₁	—	2	1	no	—
132	C₂×C₄×C₁₉₁	—	2	1	no	—
133	C₂×C₄×C₁₉₁	—	2	1	no	—
134	C₂×C₄×C₁₉₁	—	2	1	no	—
135	C₂×C₄×C₁₉₁	—	2	1	no	—
136	C₂×C₄×C₁₉₁	—	4	1	no	—
137	C₂×C₄×C₁₉₁	—	4	1	no	—
138	C₂×C₄×C₁₉₁	—	4	2	no	—
139	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
145	C₂²×C₁₉₁	—	2	1	no	—
146	C₂²×C₁₉₁	—	2	1	no	—
147	C₂²×C₁₉₁	—	2	1	no	—
148	C₂²×C₁₉₁	—	4	1	no	—
149	C₂²×C₁₉₁	—	4	1	no	—
150	C₂²×C₁₉₁	—	4	1	no	—
151	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₁₉₁	—	2	2	no	—
153	C₂×D₁₉₁	—	4	1	no	—
154	C₂×D₁₉₁	—	4	1	no	—
155	C₄×C₁₉₁	—	2	1	no	—
156	C₄×C₁₉₁	—	2	1	no	—
157	C₄×C₁₉₁	—	2	1	no	—
158	C₄×C₁₉₁	—	2	1	no	—
159	C₄×C₁₉₁	—	2	1	no	—
160	C₄×C₁₉₁	—	4	1	no	—
161	C₄×C₁₉₁	—	4	1	no	—
162	C₄×C₁₉₁	—	4	1	no	—
163	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
164	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
165	Dic₁₉₁	—	2	2	no	—
166	Dic₁₉₁	—	4	1	no	—
167	Dic₁₉₁	—	4	1	no	—
168	C₂×C₁₉₁	—	4	1	no	—
169	C₂×C₁₉₁	—	8	1	no	—
170	C₂×C₁₉₁	#525	1	1	yes	characteristic
171	C₂×C₁₉₁	#575	1	1	yes	characteristic
172	C₂×C₁₉₁	#575	1	1	yes	characteristic
173	D₁₉₁	—	4	2	no	—
174	C₁₉₁	#3111	1	1	yes	characteristic, complemented, Sylow
175	#3111	—	191	1	no	maximal, Sylow
176	#529	D₁₉₁	1	1	yes	characteristic, complemented
177	C₄×#41	—	191	1	no	—
178	#485	—	191	1	no	—
179	#482	—	191	1	no	—
180	#479	—	191	1	no	—
181	#478	—	191	1	no	—
182	#464	—	191	1	no	—
183	#463	—	191	1	no	—
184	#452	—	191	1	no	—
185	#449	—	191	1	no	—
186	#442	—	191	1	no	—
187	#441	—	191	1	no	—
188	#440	—	191	1	no	—
189	#440	—	191	1	no	—
190	#432	—	191	1	no	—
191	C₂×#41	—	191	1	no	—
192	C₂×Q₁₆	—	191	1	no	—
193	C₂×SD₁₆	—	191	1	no	—
194	C₂×SD₁₆	—	191	1	no	—
195	C₂×D₈	—	191	1	no	—
196	#126	—	2	1	no	—
197	#125	C₂×D₁₉₁	1	1	yes	characteristic
198	#124	C₂×D₁₉₁	1	1	yes	characteristic
199	#123	—	2	1	no	—
200	C₂²⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
201	#121	C₂×D₁₉₁	1	1	yes	characteristic
202	C₄×Q₈	—	191	1	no	—
203	C₄×Q₈	—	191	1	no	—
204	C₄×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
205	C₄×D₄	—	2	1	no	—
206	C₄×D₄	—	382	1	no	—
207	C₄×D₄	—	382	1	no	—
208	C₄×D₄	—	191	1	no	—
209	C₄×D₄	—	191	1	no	—
210	C₄×D₄	C₂×D₁₉₁	1	1	yes	characteristic
211	#117	—	2	1	no	—
212	#117	—	382	1	no	—
213	#117	—	191	1	no	—
214	C₂×C₄²	—	382	1	no	—
215	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
216	#107	—	191	1	no	—
217	#107	—	191	1	no	—
218	#106	—	191	1	no	—
219	#106	—	191	1	no	—
220	C₄⋊C₈	—	191	1	no	—
221	C₄⋊C₈	—	191	1	no	—
222	#103	—	191	1	no	—
223	#103	—	191	1	no	—
224	#103	—	191	1	no	—
225	#103	—	191	1	no	—
226	#102	—	191	1	no	—
227	#102	—	191	1	no	—
228	#102	—	191	1	no	—
229	#102	—	191	1	no	—
230	C₂²⋊C₈	—	191	1	no	—
231	C₂²⋊C₈	—	191	1	no	—
232	#97	—	191	1	no	—
233	#97	—	191	1	no	—
234	#41	—	382	2	no	—
235	#41	—	382	2	no	—
236	C₂×Q₈	—	191	1	no	—
237	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	382	1	no	—
240	C₂×D₄	—	191	1	no	—
241	C₂×D₄	C₂²×D₁₉₁	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	382	1	no	—
244	C₂²×C₄	—	382	1	no	—
245	C₂²×C₄	—	382	1	no	—
246	C₂²×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
247	C₂²×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
248	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
249	Q₁₆	—	382	2	no	—
250	SD₁₆	—	382	2	no	—
251	SD₁₆	—	382	2	no	—
252	D₈	—	382	2	no	—
253	C₂×C₈	—	191	1	no	—
254	C₂×C₈	—	191	1	no	—
255	C₂×C₈	—	191	1	no	—
256	C₂×C₈	—	191	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	382	1	no	—
262	C₄⋊C₄	—	382	1	no	—
263	C₄⋊C₄	—	191	1	no	—
264	C₄⋊C₄	—	191	1	no	—
265	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
266	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	382	1	no	—
273	C₂²⋊C₄	—	382	1	no	—
274	C₂²⋊C₄	—	382	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	382	1	no	—
277	C₄²	—	382	1	no	—
278	C₄²	—	382	1	no	—
279	C₄²	C₂²⋊D₁₉₁	1	1	yes	characteristic
280	C₄²	C₂²⋊D₁₉₁	1	1	yes	characteristic
281	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
282	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	382	1	no	—
285	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	382	1	no	—
289	Q₈	—	191	2	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	4	2	no	—
293	D₄	—	382	4	no	—
294	D₄	—	382	1	no	—
295	D₄	—	191	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	382	4	no	—
305	C₂×C₄	—	382	1	no	—
306	C₂×C₄	—	382	1	no	—
307	C₂×C₄	—	382	1	no	—
308	C₂×C₄	—	764	2	no	—
309	C₂×C₄	—	764	2	no	—
310	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
311	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
312	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
313	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
314	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
316	C₈	—	382	1	no	—
317	C₈	—	382	1	no	—
318	C₈	—	382	1	no	—
319	C₈	—	382	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	382	2	no	—
327	C₂²	—	764	1	no	—
328	C₂²	—	764	1	no	—
329	C₂²	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	382	2	no	—
339	C₄	—	764	1	no	—
340	C₄	—	764	1	no	—
341	C₄	#41×D₁₉₁	1	1	yes	characteristic
342	C₄	#41×D₁₉₁	1	1	yes	characteristic
343	C₂	—	4	1	no	—
344	C₂	—	8	1	no	—
345	C₂	—	764	2	no	—
346	C₂	#27133170	1	1	yes	characteristic
347	C₂	#27133461	1	1	yes	characteristic
348	C₂	#27133461	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#529×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	#27132806	C₂
hypercenter	#529	D₁₉₁
hypocenter	C₁₉₁	#3111
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. C₄×#15690287	C₂	semidirect
2. #27133001	C₂	semidirect
3. #27132984	C₂	semidirect
4. #27132967	C₂	semidirect
5. #27132960	C₂	semidirect
6. #27132882	C₂	semidirect
7. #27132875	C₂	semidirect
8. #27132823	C₂	semidirect
10. #27132773	C₂	semidirect
11. #27132768	C₂	semidirect
12. #27132763	C₂	semidirect
13. #27132761	C₂	semidirect
14. #27132720	C₂	semidirect
15. #529×C₁₉₁	C₂	semidirect
174. C₁₉₁	#3111	semidirect
176. #529	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15690287	2
#27133001	2
#27132984	2
#27132967	2
#27132960	2
#27132882	2
#27132875	2
#27132823	2
#27132806	2
#27132773	2
#27132768	2
#27132763	2
#27132761	2
#27132720	2
#529×C₁₉₁	2
#3111	191