Subgroups — #37582696

Order 24448.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133198	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#27133000	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132983	C₂	1	1	yes	maximal, characteristic
5	#27132967	C₂	1	1	yes	maximal, characteristic, complemented
6	#27132886	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132885	C₂	1	1	yes	maximal, characteristic, complemented
8	#27132820	C₂	1	1	yes	maximal, characteristic, complemented
9	#27132803	C₂	1	1	yes	maximal, characteristic
10	#27132769	C₂	1	1	yes	maximal, characteristic, complemented
11	#27132767	C₂	1	1	yes	maximal, characteristic, complemented
12	#27132766	C₂	1	1	yes	maximal, characteristic, complemented
13	#27132762	C₂	1	1	yes	maximal, characteristic, complemented
14	#27132719	C₂	1	1	yes	maximal, characteristic, complemented
15	#479×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₉₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₇₆₄	C₂²	1	1	yes	characteristic
18	C₂×#15690277	C₂²	1	1	yes	characteristic
19	C₂×#15690276	C₂²	1	1	yes	characteristic
20	C₂×#15690274	C₂²	1	1	yes	characteristic
21	#20259317	—	2	1	no	—
22	#20259310	C₂²	1	1	yes	characteristic
23	#20259307	—	2	1	no	—
24	#20259302	C₂²	1	1	yes	characteristic
25	#20259297	C₂²	1	1	yes	characteristic
26	Q₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
27	C₄×Dic₃₈₂	—	2	1	no	—
28	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
29	#20259287	—	2	1	no	—
30	#20259287	C₂²	1	1	yes	characteristic
31	#20259287	C₂²	1	1	yes	characteristic
32	#20259284	—	2	1	no	—
33	#20259284	C₂²	1	1	yes	characteristic
34	#20259280	—	2	1	no	—
35	C₄⋊C₄×D₁₉₁	—	2	1	no	—
36	C₄⋊C₄×D₁₉₁	C₂²	1	1	yes	characteristic
37	C₄²×D₁₉₁	—	2	1	no	—
38	C₄²×D₁₉₁	C₂²	1	1	yes	characteristic
39	#20259255	C₂²	1	1	yes	characteristic
40	#20259253	C₂²	1	1	yes	characteristic
41	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
42	#20259247	C₂²	1	1	yes	characteristic
43	#20259245	C₂²	1	1	yes	characteristic
44	#20259245	C₂²	1	1	yes	characteristic
45	#20259245	C₂²	1	1	yes	characteristic
46	#20259244	C₂²	1	1	yes	characteristic
47	#20259242	C₂²	1	1	yes	characteristic
48	(C₂×D₁₉₁)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₁₉₁)⋊C₈	C₂²	1	1	yes	characteristic
50	#20259232	C₂²	1	1	yes	characteristic
51	#20259232	C₂²	1	1	yes	characteristic
52	#125×C₁₉₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
54	#107×C₁₉₁	C₂²	1	1	yes	characteristic
55	#106×C₁₉₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
57	#103×C₁₉₁	C₂²	1	1	yes	characteristic
58	#103×C₁₉₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₈₂	—	2	1	no	—
60	C₂×Dic₃₈₂	C₂³	1	1	yes	characteristic
61	C₂×Dic₃₈₂	C₂³	1	1	yes	characteristic
62	Q₈×D₁₉₁	—	2	2	no	—
63	Q₈×D₁₉₁	—	2	2	no	—
64	C₂×D₇₆₄	—	2	1	no	—
65	C₂×D₇₆₄	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₁₉₁	—	2	1	no	—
67	C₂×C₄×D₁₉₁	—	2	1	no	—
68	C₂×C₄×D₁₉₁	—	2	1	no	—
69	C₂×C₄×D₁₉₁	D₄	1	1	yes	characteristic
70	C₂×C₄×D₁₉₁	D₄	1	1	yes	characteristic
71	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
72	Dic₇₆₄	—	2	2	no	—
73	#15690277	—	2	2	no	—
74	#15690276	—	2	2	no	—
75	#15690274	—	2	2	no	—
76	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₉₁	—	2	1	no	—
79	C₄⋊Dic₁₉₁	—	2	1	no	—
80	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
81	Dic₁₉₁⋊C₄	—	2	1	no	—
82	Dic₁₉₁⋊C₄	—	2	1	no	—
83	Dic₁₉₁⋊C₄	—	2	1	no	—
84	Dic₁₉₁⋊C₄	—	2	1	no	—
85	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
86	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
87	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
88	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
89	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
90	C₄×Dic₁₉₁	—	2	1	no	—
91	C₄×Dic₁₉₁	—	2	1	no	—
92	C₄×Dic₁₉₁	—	2	1	no	—
93	C₄×Dic₁₉₁	—	2	1	no	—
94	C₄×Dic₁₉₁	D₄	1	1	yes	characteristic
95	C₄×Dic₁₉₁	D₄	1	1	yes	characteristic
96	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₉₁	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₉₁	—	2	1	no	—
101	C₄⋊C₄×C₁₉₁	—	2	1	no	—
102	C₄⋊C₄×C₁₉₁	—	2	1	no	—
103	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
106	C₄²×C₁₉₁	—	2	1	no	—
107	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
108	C₂²×D₁₉₁	—	2	1	no	—
109	C₂²×D₁₉₁	C₂×D₄	1	1	yes	characteristic
110	Dic₃₈₂	—	2	4	no	—
111	Dic₃₈₂	—	2	1	no	—
112	Dic₃₈₂	—	2	1	no	—
113	Dic₃₈₂	—	2	1	no	—
114	Dic₃₈₂	#41	1	2	yes	—
115	D₇₆₄	—	2	1	no	—
116	D₇₆₄	—	2	1	no	—
117	D₇₆₄	—	4	2	no	—
118	C₂×Dic₁₉₁	—	2	1	no	—
119	C₂×Dic₁₉₁	—	2	1	no	—
120	C₂×Dic₁₉₁	—	2	1	no	—
121	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
124	C₄×D₁₉₁	—	2	4	no	—
125	C₄×D₁₉₁	—	2	2	no	—
126	C₄×D₁₉₁	—	2	1	no	—
127	C₄×D₁₉₁	—	2	1	no	—
128	C₄×D₁₉₁	—	4	1	no	—
129	C₄×D₁₉₁	—	4	2	no	—
130	C₄×D₁₉₁	—	4	2	no	—
131	C₁₉₁⋊C₈	—	2	1	no	—
132	C₁₉₁⋊C₈	—	2	1	no	—
133	Q₈×C₁₉₁	—	2	1	no	—
134	Q₈×C₁₉₁	#41	1	2	yes	—
135	C₂×C₄×C₁₉₁	—	2	1	no	—
136	C₂×C₄×C₁₉₁	—	2	1	no	—
137	C₂×C₄×C₁₉₁	—	2	1	no	—
138	C₂×C₄×C₁₉₁	—	2	1	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	C₂⁴	1	1	yes	characteristic
142	C₈×C₁₉₁	—	2	1	no	—
143	C₈×C₁₉₁	—	2	1	no	—
144	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₁₉₁	—	2	1	no	—
146	C₂×D₁₉₁	—	2	1	no	—
147	C₂×D₁₉₁	—	4	1	no	—
148	C₂×D₁₉₁	—	4	1	no	—
149	C₂×D₁₉₁	—	4	1	no	—
150	C₂×D₁₉₁	#137	1	2	yes	—
151	C₄×C₁₉₁	—	2	2	no	—
152	C₄×C₁₉₁	—	2	1	no	—
153	C₄×C₁₉₁	—	2	1	no	—
154	C₄×C₁₉₁	—	4	1	no	—
155	C₄×C₁₉₁	—	4	1	no	—
156	C₄×C₁₉₁	—	4	1	no	—
157	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
158	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
159	Dic₁₉₁	—	2	2	no	—
160	Dic₁₉₁	—	2	1	no	—
161	Dic₁₉₁	—	2	1	no	—
162	Dic₁₉₁	—	2	1	no	—
163	Dic₁₉₁	—	4	1	no	—
164	Dic₁₉₁	—	4	1	no	—
165	C₂×C₁₉₁	#525	1	1	yes	characteristic
166	C₂×C₁₉₁	C₂×#137	1	1	yes	characteristic
167	C₂×C₁₉₁	#575	1	1	yes	characteristic
168	D₁₉₁	—	2	2	no	—
169	D₁₉₁	—	8	1	no	—
170	C₁₉₁	#3117	1	1	yes	characteristic, complemented, Sylow
171	#3117	—	191	1	no	maximal, Sylow
172	#529	—	191	1	no	—
173	C₂×C₄×Q₈	—	191	1	no	—
174	#484	—	191	1	no	—
175	#481	—	191	1	no	—
176	#479	—	191	1	no	—
177	#479	D₁₉₁	1	1	yes	characteristic, complemented
178	#464	—	191	1	no	—
179	#464	—	191	1	no	—
180	#451	—	191	1	no	—
181	#448	—	191	1	no	—
182	#441	—	191	1	no	—
183	#441	—	191	1	no	—
184	#440	—	191	1	no	—
185	#440	—	191	1	no	—
186	#432	—	191	1	no	—
187	C₂²×Q₈	—	191	1	no	—
188	C₂×Q₁₆	—	191	1	no	—
189	C₂×Q₁₆	—	191	1	no	—
190	C₂×SD₁₆	—	191	1	no	—
191	C₂×SD₁₆	—	191	1	no	—
192	#126	—	382	1	no	—
193	#125	C₂×D₁₉₁	1	1	yes	characteristic
194	#124	—	191	1	no	—
195	#123	—	382	1	no	—
196	C₂²⋊Q₈	—	191	1	no	—
197	#121	—	191	1	no	—
198	C₄×Q₈	—	382	1	no	—
199	C₄×Q₈	—	382	1	no	—
200	C₄×Q₈	—	191	1	no	—
201	C₄×Q₈	—	191	1	no	—
202	C₄×Q₈	—	191	1	no	—
203	C₄×Q₈	—	191	1	no	—
204	C₄×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
205	C₄×D₄	—	382	1	no	—
206	C₄×D₄	—	191	1	no	—
207	#117	—	382	1	no	—
208	C₂×C₄⋊C₄	—	382	1	no	—
209	C₂×C₄⋊C₄	—	191	1	no	—
210	C₂×C₄²	—	382	1	no	—
211	C₂×C₄²	—	191	1	no	—
212	#107	—	191	1	no	—
213	#107	C₂×D₁₉₁	1	1	yes	characteristic
214	#106	—	191	1	no	—
215	#106	C₂×D₁₉₁	1	1	yes	characteristic
216	C₄⋊C₈	—	191	1	no	—
217	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
218	#103	—	191	1	no	—
219	#103	—	191	1	no	—
220	#103	—	191	1	no	—
221	#103	—	191	1	no	—
222	#103	C₂×D₁₉₁	1	1	yes	characteristic
223	#103	C₂×D₁₉₁	1	1	yes	characteristic
224	#102	—	191	1	no	—
225	#102	—	191	1	no	—
226	C₂²⋊C₈	—	191	1	no	—
227	C₂²⋊C₈	—	191	1	no	—
228	#97	—	191	1	no	—
229	#97	—	191	1	no	—
230	C₂×Q₈	—	382	1	no	—
231	C₂×Q₈	—	382	2	no	—
232	C₂×Q₈	—	382	2	no	—
233	C₂×Q₈	—	191	1	no	—
234	C₂×Q₈	—	191	1	no	—
235	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
236	C₂×D₄	—	382	1	no	—
237	C₂×D₄	—	191	1	no	—
238	C₂²×C₄	—	382	1	no	—
239	C₂²×C₄	—	382	1	no	—
240	C₂²×C₄	—	382	1	no	—
241	C₂²×C₄	—	191	1	no	—
242	C₂²×C₄	—	191	1	no	—
243	C₂²×C₄	—	191	1	no	—
244	Q₁₆	—	382	2	no	—
245	Q₁₆	—	382	2	no	—
246	SD₁₆	—	382	2	no	—
247	SD₁₆	—	382	2	no	—
248	C₂×C₈	—	191	1	no	—
249	C₂×C₈	—	191	1	no	—
250	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	382	1	no	—
256	C₄⋊C₄	—	382	1	no	—
257	C₄⋊C₄	—	382	1	no	—
258	C₄⋊C₄	—	382	1	no	—
259	C₄⋊C₄	—	382	1	no	—
260	C₄⋊C₄	—	382	1	no	—
261	C₄⋊C₄	—	191	1	no	—
262	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
265	C₂²⋊C₄	—	382	1	no	—
266	C₂²⋊C₄	—	382	1	no	—
267	C₂²⋊C₄	—	382	1	no	—
268	C₂²⋊C₄	—	382	1	no	—
269	C₂²⋊C₄	—	382	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	382	1	no	—
272	C₄²	—	382	1	no	—
273	C₄²	—	382	1	no	—
274	C₄²	—	382	1	no	—
275	C₄²	—	191	1	no	—
276	C₄²	—	191	1	no	—
277	C₄²	—	191	1	no	—
278	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
279	C₂³	—	382	1	no	—
280	C₂³	—	191	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	382	4	no	—
283	Q₈	—	382	1	no	—
284	Q₈	—	382	1	no	—
285	Q₈	—	382	1	no	—
286	Q₈	—	191	2	no	—
287	Q₈	#15690288	1	2	yes	—
288	D₄	—	382	1	no	—
289	D₄	—	382	1	no	—
290	D₄	—	764	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	382	4	no	—
296	C₂×C₄	—	382	2	no	—
297	C₂×C₄	—	382	1	no	—
298	C₂×C₄	—	382	1	no	—
299	C₂×C₄	—	382	1	no	—
300	C₂×C₄	—	382	1	no	—
301	C₂×C₄	—	382	1	no	—
302	C₂×C₄	—	191	1	no	—
303	C₂×C₄	—	191	1	no	—
304	C₂×C₄	—	191	1	no	—
305	C₂×C₄	—	764	1	no	—
306	C₂×C₄	—	764	2	no	—
307	C₂×C₄	—	764	2	no	—
308	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
309	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	382	1	no	—
314	C₈	—	382	1	no	—
315	C₂²	—	382	1	no	—
316	C₂²	—	382	1	no	—
317	C₂²	—	191	2	no	—
318	C₂²	—	764	1	no	—
319	C₂²	—	764	1	no	—
320	C₂²	—	764	1	no	—
321	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	2	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	382	1	no	—
329	C₄	—	382	1	no	—
330	C₄	—	382	2	no	—
331	C₄	—	382	1	no	—
332	C₄	—	764	1	no	—
333	C₄	—	764	1	no	—
334	C₄	#41×D₁₉₁	1	1	yes	characteristic
335	C₄	C₂×#15690288	1	1	yes	characteristic
336	C₂	—	382	2	no	—
337	C₂	—	1528	1	no	—
338	C₂	#27133176	1	1	yes	characteristic
339	C₂	#137×D₁₉₁	1	1	yes	characteristic
340	C₂	#27133460	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#479×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	#20259297	C₂²
hypercenter	#479	D₁₉₁
hypocenter	C₁₉₁	#3117
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×D₁₉₁	C₂	semidirect
3. #27133000	C₂	semidirect
5. #27132967	C₂	semidirect
6. #27132886	C₂	semidirect
7. #27132885	C₂	semidirect
8. #27132820	C₂	semidirect
10. #27132769	C₂	semidirect
11. #27132767	C₂	semidirect
12. #27132766	C₂	semidirect
13. #27132762	C₂	semidirect
14. #27132719	C₂	semidirect
15. #479×C₁₉₁	C₂	semidirect
170. C₁₉₁	#3117	semidirect
177. #479	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#27133198	2
C₄×Q₈×D₁₉₁	2
#27133000	2
#27132983	2
#27132967	2
#27132886	2
#27132885	2
#27132820	2
#27132803	2
#27132769	2
#27132767	2
#27132766	2
#27132762	2
#27132719	2
#479×C₁₉₁	2
#3117	191