Subgroups — #37582484

Order 24448.

386 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133236	C₂	1	1	yes	maximal, characteristic
2	#27133143	C₂	1	1	yes	maximal, characteristic
3	#27133006	C₂	1	1	yes	maximal, characteristic, complemented
4	#27133004	C₂	1	1	yes	maximal, characteristic, complemented
5	#27133003	C₂	1	1	yes	maximal, characteristic, complemented
6	#27132999	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132997	C₂	1	1	yes	maximal, characteristic, complemented
8	#27132995	C₂	1	1	yes	maximal, characteristic, complemented
9	#27132991	C₂	1	1	yes	maximal, characteristic, complemented
10	#27132702	C₂	1	1	yes	maximal, characteristic, complemented
11	#27132700	C₂	1	1	yes	maximal, characteristic, complemented
12	#27132670	C₂	1	1	yes	maximal, characteristic, complemented
13	#103×D₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
14	#27132628	C₂	1	1	yes	maximal, characteristic
15	#485×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15690287	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₉₁	C₂²	1	1	yes	characteristic
18	C₂×#15690270	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
20	#20259321	—	2	1	no	—
21	#20259308	—	2	1	no	—
22	#20259307	—	2	1	no	—
23	#20259304	—	2	1	no	—
24	#20259302	C₂²	1	1	yes	characteristic
25	#20259301	C₂²	1	1	yes	characteristic
26	#20259298	C₂²	1	1	yes	characteristic
27	C₄×Dic₃₈₂	—	2	1	no	—
28	C₄×D₇₆₄	—	2	1	no	—
29	(C₂²⋊D₁₉₁)⋊C₄	—	2	1	no	—
30	(C₂²⋊D₁₉₁)⋊C₄	—	2	1	no	—
31	#20259280	C₂²	1	1	yes	characteristic
32	#20259280	C₂²	1	1	yes	characteristic
33	#20259280	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₁₉₁	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₉₁	C₂²	1	1	yes	characteristic
37	#20259256	C₂²	1	1	yes	characteristic
38	#20259256	C₂²	1	1	yes	characteristic
39	#20259255	C₂²	1	1	yes	characteristic
40	#20259253	C₂²	1	1	yes	characteristic
41	#20259253	C₂²	1	1	yes	characteristic
42	#20259253	C₂²	1	1	yes	characteristic
43	#20259247	C₂²	1	1	yes	characteristic
44	#20259247	C₂²	1	1	yes	characteristic
45	#20259246	C₂²	1	1	yes	characteristic
46	#20259246	C₂²	1	1	yes	characteristic
47	#20259245	C₂²	1	1	yes	characteristic
48	#20259245	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₁₉₁)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₁₉₁)⋊C₈	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
53	#117×C₁₉₁	C₂²	1	1	yes	characteristic
54	#107×C₁₉₁	C₂²	1	1	yes	characteristic
55	#106×C₁₉₁	C₂²	1	1	yes	characteristic
56	#103×C₁₉₁	C₂²	1	1	yes	characteristic
57	#103×C₁₉₁	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
59	#15690287	—	2	2	no	—
60	#15690287	—	2	2	no	—
61	C₂×Dic₃₈₂	—	2	1	no	—
62	C₂×Dic₃₈₂	D₄	1	1	yes	characteristic
63	C₂×Dic₃₈₂	C₂³	1	1	yes	characteristic
64	Q₈×D₁₉₁	—	2	2	no	—
65	Q₈×D₁₉₁	—	2	2	no	—
66	C₂×D₇₆₄	D₄	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₉₁	D₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₁₉₁	D₄	1	1	yes	characteristic
69	C₂²×Dic₁₉₁	—	2	1	no	—
70	C₂×C₄×D₁₉₁	—	2	1	no	—
71	C₂×C₄×D₁₉₁	—	2	1	no	—
72	C₂×C₄×D₁₉₁	—	2	1	no	—
73	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
75	#15690270	—	2	2	no	—
76	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
78	C₈×D₁₉₁	—	2	2	no	—
79	C₄⋊Dic₁₉₁	—	2	1	no	—
80	C₄⋊Dic₁₉₁	—	2	1	no	—
81	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
84	Dic₁₉₁⋊C₄	—	2	1	no	—
85	Dic₁₉₁⋊C₄	—	2	1	no	—
86	Dic₁₉₁⋊C₄	—	2	1	no	—
87	Dic₁₉₁⋊C₄	—	2	1	no	—
88	C₂²⋊Dic₁₉₁	—	2	1	no	—
89	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
90	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
91	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
92	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
93	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
94	C₄×Dic₁₉₁	—	2	1	no	—
95	C₄×Dic₁₉₁	—	2	1	no	—
96	C₄×Dic₁₉₁	—	2	1	no	—
97	C₂×Q₈×C₁₉₁	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₉₁	—	2	1	no	—
102	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₉₁	—	2	1	no	—
106	C₂²⋊C₄×C₁₉₁	—	2	1	no	—
107	C₄²×C₁₉₁	—	2	1	no	—
108	C₂²×D₁₉₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₁₉₁	C₂×D₄	1	1	yes	characteristic
110	Dic₃₈₂	—	2	2	no	—
111	Dic₃₈₂	—	2	1	no	—
112	Dic₃₈₂	—	2	1	no	—
113	Dic₃₈₂	—	4	2	no	—
114	D₇₆₄	—	2	2	no	—
115	C₂²⋊D₁₉₁	—	2	2	no	—
116	C₂²⋊D₁₉₁	—	2	2	no	—
117	C₂×Dic₁₉₁	—	2	1	no	—
118	C₂×Dic₁₉₁	—	2	1	no	—
119	C₂×Dic₁₉₁	—	2	1	no	—
120	C₂×Dic₁₉₁	—	2	1	no	—
121	C₂×Dic₁₉₁	—	8	1	no	—
122	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
124	C₄×D₁₉₁	—	2	1	no	—
125	C₄×D₁₉₁	—	2	1	no	—
126	C₄×D₁₉₁	—	4	2	no	—
127	C₄×D₁₉₁	—	8	1	no	—
128	C₄×D₁₉₁	—	8	1	no	—
129	C₄×D₁₉₁	#41	1	2	yes	—
130	C₄×D₁₉₁	#41	1	2	yes	—
131	C₁₉₁⋊C₈	—	2	1	no	—
132	C₁₉₁⋊C₈	—	2	1	no	—
133	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
134	Q₈×C₁₉₁	—	2	1	no	—
135	Q₈×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	—	2	1	no	—
140	C₂×C₄×C₁₉₁	—	2	1	no	—
141	C₂×C₄×C₁₉₁	—	2	1	no	—
142	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
144	C₈×C₁₉₁	—	2	1	no	—
145	C₈×C₁₉₁	—	2	1	no	—
146	C₂²×C₁₉₁	—	2	1	no	—
147	C₂²×C₁₉₁	—	2	1	no	—
148	C₂²×C₁₉₁	—	2	1	no	—
149	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₁₉₁	—	2	1	no	—
151	C₂×D₁₉₁	—	2	1	no	—
152	C₂×D₁₉₁	—	2	1	no	—
153	C₂×D₁₉₁	—	2	1	no	—
154	C₂×D₁₉₁	—	2	1	no	—
155	C₂×D₁₉₁	#123	1	2	yes	—
156	C₄×C₁₉₁	—	2	1	no	—
157	C₄×C₁₉₁	—	4	1	no	—
158	C₄×C₁₉₁	—	4	1	no	—
159	C₄×C₁₉₁	—	4	1	no	—
160	C₄×C₁₉₁	—	4	1	no	—
161	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
162	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
163	Dic₁₉₁	—	2	1	no	—
164	Dic₁₉₁	—	4	1	no	—
165	Dic₁₉₁	—	4	1	no	—
166	Dic₁₉₁	—	4	1	no	—
167	Dic₁₉₁	—	4	1	no	—
168	Dic₁₉₁	#123	1	2	yes	—
169	C₂×C₁₉₁	—	4	1	no	—
170	C₂×C₁₉₁	C₂×#123	1	1	yes	characteristic
171	C₂×C₁₉₁	#577	1	1	yes	characteristic
172	C₂×C₁₉₁	#578	1	1	yes	characteristic
173	D₁₉₁	—	2	2	no	—
174	D₁₉₁	—	4	1	no	—
175	C₁₉₁	#3097	1	1	yes	characteristic, complemented, Sylow
176	#3097	—	191	1	no	maximal, Sylow
177	#536	—	191	1	no	—
178	#520	—	191	1	no	—
179	#485	—	191	1	no	—
180	#485	—	191	1	no	—
181	#485	—	191	1	no	—
182	#485	D₁₉₁	1	1	yes	characteristic, complemented
183	#484	—	191	1	no	—
184	#484	—	191	1	no	—
185	#483	—	191	1	no	—
186	#483	—	191	1	no	—
187	#428	—	191	1	no	—
188	#427	—	191	1	no	—
189	#419	—	191	1	no	—
190	C₂×#103	—	191	1	no	—
191	#408	—	191	1	no	—
192	C₂×#41	—	191	1	no	—
193	C₂²×Q₈	—	191	1	no	—
194	C₂×M₁₆	—	191	1	no	—
195	C₂²×C₈	—	191	1	no	—
196	C₄⋊Q₈	—	382	1	no	—
197	#124	—	382	1	no	—
198	#123	—	382	1	no	—
199	#123	—	382	1	no	—
200	C₂²⋊Q₈	—	191	1	no	—
201	C₂²⋊Q₈	—	191	1	no	—
202	C₂²⋊Q₈	—	191	1	no	—
203	C₂²⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
204	C₄×Q₈	—	382	1	no	—
205	C₄×D₄	—	382	1	no	—
206	C₄×D₄	—	382	1	no	—
207	C₄×D₄	—	382	1	no	—
208	#117	—	191	1	no	—
209	#117	—	191	1	no	—
210	#117	—	191	1	no	—
211	#117	C₂×D₁₉₁	1	1	yes	characteristic
212	C₂×C₄⋊C₄	—	191	1	no	—
213	C₂×C₄⋊C₄	—	191	1	no	—
214	C₂×C₄⋊C₄	—	191	1	no	—
215	#107	—	191	1	no	—
216	#107	—	191	1	no	—
217	#107	—	191	1	no	—
218	#107	C₂×D₁₉₁	1	1	yes	characteristic
219	#106	—	191	1	no	—
220	#106	—	191	1	no	—
221	#106	—	191	1	no	—
222	#106	C₂×D₁₉₁	1	1	yes	characteristic
223	#103	—	191	1	no	—
224	#103	—	191	1	no	—
225	#103	—	191	1	no	—
226	#103	—	191	1	no	—
227	#103	—	191	1	no	—
228	#103	—	191	1	no	—
229	#103	C₂×D₁₉₁	1	1	yes	characteristic
230	#103	C₂×D₁₉₁	1	1	yes	characteristic
231	C₂²⋊C₈	—	191	1	no	—
232	C₂²⋊C₈	—	191	1	no	—
233	C₂²⋊C₈	—	191	1	no	—
234	C₂²⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
235	#41	—	382	2	no	—
236	#41	—	382	2	no	—
237	C₂×Q₈	—	382	1	no	—
238	C₂×Q₈	—	382	2	no	—
239	C₂×Q₈	—	382	2	no	—
240	C₂×Q₈	—	191	1	no	—
241	C₂×Q₈	—	191	1	no	—
242	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
243	C₂×D₄	—	191	1	no	—
244	C₂×D₄	—	191	1	no	—
245	C₂×D₄	—	191	1	no	—
246	C₂²×C₄	—	382	1	no	—
247	C₂²×C₄	—	382	1	no	—
248	C₂²×C₄	—	382	1	no	—
249	C₂²×C₄	—	382	1	no	—
250	C₂²×C₄	—	191	1	no	—
251	C₂²×C₄	—	191	1	no	—
252	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
253	M₁₆	—	382	2	no	—
254	C₂×C₈	—	382	2	no	—
255	C₂×C₈	—	191	1	no	—
256	C₂×C₈	—	191	1	no	—
257	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	382	1	no	—
261	C₄⋊C₄	—	382	1	no	—
262	C₄⋊C₄	—	382	1	no	—
263	C₄⋊C₄	—	382	1	no	—
264	C₄⋊C₄	—	382	1	no	—
265	C₄⋊C₄	—	382	1	no	—
266	C₄⋊C₄	—	191	1	no	—
267	C₄⋊C₄	—	191	1	no	—
268	C₄⋊C₄	—	191	1	no	—
269	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	382	1	no	—
275	C₂²⋊C₄	—	382	1	no	—
276	C₂²⋊C₄	—	382	1	no	—
277	C₂²⋊C₄	—	382	1	no	—
278	C₂²⋊C₄	—	382	1	no	—
279	C₂²⋊C₄	—	382	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	382	1	no	—
282	C₄²	—	382	1	no	—
283	C₄²	—	382	1	no	—
284	C₂³	—	191	1	no	—
285	C₂³	—	191	1	no	—
286	C₂³	D₄×D₁₉₁	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	382	2	no	—
290	Q₈	—	382	1	no	—
291	Q₈	—	382	1	no	—
292	Q₈	—	764	2	no	—
293	D₄	—	382	2	no	—
294	D₄	—	382	2	no	—
295	D₄	—	382	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	—	2	1	no	—
302	C₂×C₄	—	382	1	no	—
303	C₂×C₄	—	382	1	no	—
304	C₂×C₄	—	382	1	no	—
305	C₂×C₄	—	382	1	no	—
306	C₂×C₄	—	382	1	no	—
307	C₂×C₄	—	382	1	no	—
308	C₂×C₄	—	191	1	no	—
309	C₂×C₄	—	191	1	no	—
310	C₂×C₄	—	191	2	no	—
311	C₂×C₄	—	191	2	no	—
312	C₂×C₄	—	1528	1	no	—
313	C₂×C₄	—	1528	1	no	—
314	C₂×C₄	—	1528	1	no	—
315	C₂×C₄	—	764	2	no	—
316	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
317	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	382	1	no	—
321	C₈	—	382	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	382	1	no	—
326	C₂²	—	382	1	no	—
327	C₂²	—	382	1	no	—
328	C₂²	—	382	1	no	—
329	C₂²	—	382	1	no	—
330	C₂²	—	191	2	no	—
331	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	382	1	no	—
338	C₄	—	191	2	no	—
339	C₄	—	764	1	no	—
340	C₄	—	764	1	no	—
341	C₄	—	764	1	no	—
342	C₄	—	764	1	no	—
343	C₄	#41×D₁₉₁	1	1	yes	characteristic
344	C₄	#41×D₁₉₁	1	1	yes	characteristic
345	C₂	—	4	1	no	—
346	C₂	—	382	2	no	—
347	C₂	—	764	1	no	—
348	C₂	#123×D₁₉₁	1	1	yes	characteristic
349	C₂	#27133472	1	1	yes	characteristic
350	C₂	#27133477	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#485×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂×#15690287	C₂²
hypercenter	#485	D₁₉₁
hypocenter	C₁₉₁	#3097
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
3. #27133006	C₂	semidirect
4. #27133004	C₂	semidirect
5. #27133003	C₂	semidirect
6. #27132999	C₂	semidirect
7. #27132997	C₂	semidirect
8. #27132995	C₂	semidirect
9. #27132991	C₂	semidirect
10. #27132702	C₂	semidirect
11. #27132700	C₂	semidirect
12. #27132670	C₂	semidirect
13. #103×D₁₉₁	C₂	semidirect
15. #485×C₁₉₁	C₂	semidirect
175. C₁₉₁	#3097	semidirect
182. #485	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#27133236	2
#27133143	2
#27133006	2
#27133004	2
#27133003	2
#27132999	2
#27132997	2
#27132995	2
#27132991	2
#27132702	2
#27132700	2
#27132670	2
#103×D₁₉₁	2
#27132628	2
#485×C₁₉₁	2
#3097	191