Subgroups — #37575556

Order 24448.

688 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20259296	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₇₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#20259281	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132565	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132563	C₂	1	2	yes	maximal, complemented
6	#27132561	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132552	C₂	1	2	yes	maximal, complemented
8	#27132539	C₂	1	2	yes	maximal, complemented
9	C₄×C₄⋊Dic₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#27132510	C₂	1	2	yes	maximal, complemented
11	C₄×C₂²⋊C₄×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₇₆₄	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊D₁₉₁	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₉₁	C₂²	1	2	yes	—
15	#20259296	C₄	1	8	yes	complemented
16	C₄×D₇₆₄	—	2	4	no	—
17	#20259281	—	2	4	no	—
18	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
23	C₂×(C₂×D₁₉₁)⋊C₄	C₂²	1	2	yes	—
24	C₂×(C₂×D₁₉₁)⋊C₄	C₂²	1	2	yes	—
25	C₂×(C₂×D₁₉₁)⋊C₄	C₂²	1	2	yes	—
26	C₂×(C₂×D₁₉₁)⋊C₄	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
28	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
29	C₁₉₁⋊#95	C₂²	1	2	yes	—
30	C₁₉₁⋊#95	C₂²	1	2	yes	—
31	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
36	#95×C₁₉₁	C₂²	1	1	yes	characteristic
37	#95×C₁₉₁	C₂²	1	1	yes	characteristic
38	C₂³×D₁₉₁	C₂³	1	2	yes	—
39	C₂×D₇₆₄	—	2	2	no	—
40	C₂×D₇₆₄	—	2	2	no	—
41	C₂×D₇₆₄	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊D₁₉₁	—	4	2	no	—
43	C₂×C₂²⋊D₁₉₁	—	4	2	no	—
44	C₂×C₂²⋊D₁₉₁	C₂×C₄	1	8	yes	complemented
45	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
46	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
47	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
48	C₂×C₄×D₁₉₁	—	2	4	no	—
49	C₂×C₄×D₁₉₁	—	2	4	no	—
50	C₂×C₄×D₁₉₁	—	4	2	no	—
51	C₄⋊Dic₁₉₁	—	2	2	no	—
52	C₄⋊Dic₁₉₁	C₂×C₄	1	4	yes	complemented
53	C₄⋊Dic₁₉₁	D₄	1	4	yes	complemented
54	Dic₁₉₁⋊C₄	—	4	2	no	—
55	C₂²⋊Dic₁₉₁	—	2	4	no	—
56	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
57	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
58	(C₂×D₁₉₁)⋊C₄	—	4	2	no	—
59	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	8	yes	complemented
60	C₄×Dic₁₉₁	—	2	4	no	—
61	C₄×Dic₁₉₁	—	4	2	no	—
62	C₂⁴×C₁₉₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₉₁	—	2	2	no	—
64	C₂²×C₄×C₁₉₁	—	2	2	no	—
65	C₂²×C₄×C₁₉₁	C₂×C₄	1	4	yes	complemented
66	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₁₉₁	—	2	2	no	—
73	C₂²⋊C₄×C₁₉₁	—	2	2	no	—
74	C₄²×C₁₉₁	—	2	2	no	—
75	C₄²×C₁₉₁	—	2	2	no	—
76	C₂²×D₁₉₁	—	2	2	no	—
77	C₂²×D₁₉₁	—	2	2	no	—
78	C₂²×D₁₉₁	—	2	2	no	—
79	C₂²×D₁₉₁	—	2	2	no	—
80	C₂²×D₁₉₁	—	2	2	no	—
81	C₂²×D₁₉₁	—	2	2	no	—
82	C₂²×D₁₉₁	C₂²×C₄	1	4	yes	—
83	D₇₆₄	—	2	8	no	—
84	C₂²⋊D₁₉₁	—	4	8	no	—
85	C₂×Dic₁₉₁	—	2	2	no	—
86	C₂×Dic₁₉₁	—	2	2	no	—
87	C₂×Dic₁₉₁	—	2	2	no	—
88	C₂×Dic₁₉₁	—	2	2	no	—
89	C₂×Dic₁₉₁	—	2	2	no	—
90	C₂×Dic₁₉₁	—	2	2	no	—
91	C₂×Dic₁₉₁	—	2	2	no	—
92	C₂×Dic₁₉₁	C₂²×C₄	1	4	yes	—
93	C₂×Dic₁₉₁	C₂×D₄	1	4	yes	—
94	C₄×D₁₉₁	—	4	8	no	—
95	C₂³×C₁₉₁	—	2	1	no	—
96	C₂³×C₁₉₁	—	2	1	no	—
97	C₂³×C₁₉₁	—	2	1	no	—
98	C₂³×C₁₉₁	—	2	1	no	—
99	C₂³×C₁₉₁	—	2	1	no	—
100	C₂³×C₁₉₁	—	2	1	no	—
101	C₂³×C₁₉₁	C₂²×C₄	1	2	yes	—
102	C₂³×C₁₉₁	C₂⁴	1	1	yes	characteristic
103	C₂×C₄×C₁₉₁	—	2	1	no	—
104	C₂×C₄×C₁₉₁	—	2	1	no	—
105	C₂×C₄×C₁₉₁	—	2	1	no	—
106	C₂×C₄×C₁₉₁	—	2	1	no	—
107	C₂×C₄×C₁₉₁	—	2	1	no	—
108	C₂×C₄×C₁₉₁	—	2	1	no	—
109	C₂×C₄×C₁₉₁	—	2	8	no	—
110	C₂×C₄×C₁₉₁	—	2	1	no	—
111	C₂×C₄×C₁₉₁	—	2	1	no	—
112	C₂×C₄×C₁₉₁	—	2	1	no	—
113	C₂×C₄×C₁₉₁	—	2	1	no	—
114	C₂×C₄×C₁₉₁	—	2	1	no	—
115	C₂×C₄×C₁₉₁	—	2	1	no	—
116	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
117	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
118	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₁₉₁	#41	1	2	yes	—
120	C₂×C₄×C₁₉₁	#41	1	2	yes	—
121	C₂×C₄×C₁₉₁	#41	1	2	yes	—
122	C₂²×C₁₉₁	—	2	2	no	—
123	C₂²×C₁₉₁	—	2	2	no	—
124	C₂²×C₁₉₁	—	2	2	no	—
125	C₂²×C₁₉₁	—	4	1	no	—
126	C₂²×C₁₉₁	—	4	1	no	—
127	C₂²×C₁₉₁	—	4	1	no	—
128	C₂²×C₁₉₁	—	4	1	no	—
129	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
130	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
131	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
132	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
133	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
134	C₂²×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂²×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂×D₁₉₁	—	2	4	no	—
137	C₂×D₁₉₁	—	2	4	no	—
138	C₂×D₁₉₁	—	2	4	no	—
139	C₂×D₁₉₁	—	4	2	no	—
140	C₂×D₁₉₁	—	4	2	no	—
141	C₂×D₁₉₁	—	4	2	no	—
142	C₂×D₁₉₁	—	4	2	no	—
143	C₄×C₁₉₁	—	2	2	no	—
144	C₄×C₁₉₁	—	2	2	no	—
145	C₄×C₁₉₁	—	2	2	no	—
146	C₄×C₁₉₁	—	4	1	no	—
147	C₄×C₁₉₁	—	4	1	no	—
148	C₄×C₁₉₁	C₄×D₄	1	4	yes	—
149	Dic₁₉₁	—	2	4	no	—
150	Dic₁₉₁	—	2	4	no	—
151	Dic₁₉₁	—	4	2	no	—
152	C₂×C₁₉₁	—	4	2	no	—
153	C₂×C₁₉₁	C₂×C₄×D₄	1	1	yes	characteristic
154	C₂×C₁₉₁	C₄×#41	1	1	yes	characteristic
155	C₂×C₁₉₁	#518	1	1	yes	characteristic
156	C₂×C₁₉₁	#546	1	1	yes	characteristic
157	C₂×C₁₉₁	#550	1	1	yes	characteristic
158	C₂×C₁₉₁	#553	1	1	yes	characteristic
159	C₂×C₁₉₁	#555	1	1	yes	characteristic
160	D₁₉₁	—	4	4	no	—
161	C₁₉₁	#2363	1	1	yes	characteristic, complemented, Sylow
162	#2363	—	191	1	no	maximal, Sylow
163	C₂×#121	—	191	1	no	—
164	C₂×C₄×D₄	—	191	1	no	—
165	C₂×#117	—	191	1	no	—
166	#390	—	191	2	no	—
167	#390	—	191	1	no	—
168	#389	—	191	1	no	—
169	#388	—	191	2	no	—
170	#386	—	191	2	no	—
171	C₄×C₄⋊C₄	—	191	1	no	—
172	C₄×C₂²⋊C₄	—	191	2	no	—
173	C₄×C₂²⋊C₄	D₁₉₁	1	1	yes	characteristic, complemented
174	C₂²×D₄	—	191	2	no	—
175	C₂²×D₄	—	191	1	no	—
176	C₂³×C₄	—	191	2	no	—
177	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
178	#121	—	191	8	no	—
179	C₄×D₄	—	382	4	no	—
180	#117	—	382	4	no	—
181	C₂×C₄⋊C₄	—	191	1	no	—
182	C₂×C₄⋊C₄	—	191	1	no	—
183	C₂×C₄⋊C₄	—	191	1	no	—
184	C₂×C₄⋊C₄	—	191	2	no	—
185	C₂×C₂²⋊C₄	—	191	2	no	—
186	C₂×C₂²⋊C₄	—	191	2	no	—
187	C₂×C₂²⋊C₄	—	191	2	no	—
188	C₂×C₂²⋊C₄	—	191	2	no	—
189	C₂×C₂²⋊C₄	—	191	2	no	—
190	C₂×C₂²⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
191	C₂×C₂²⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
192	C₂×C₄²	—	191	2	no	—
193	C₂×C₄²	—	191	2	no	—
194	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
195	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
196	#95	—	191	2	no	—
197	#95	—	191	2	no	—
198	#95	C₂×D₁₉₁	1	1	yes	characteristic
199	#95	C₂×D₁₉₁	1	1	yes	characteristic
200	C₂⁴	—	191	2	no	—
201	C₂⁴	C₂²×D₁₉₁	1	1	yes	characteristic
202	C₂×D₄	—	382	2	no	—
203	C₂×D₄	—	382	2	no	—
204	C₂×D₄	—	191	8	no	—
205	C₂×D₄	—	191	4	no	—
206	C₂×D₄	—	764	2	no	—
207	C₂×D₄	—	764	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	382	4	no	—
211	C₂²×C₄	—	382	4	no	—
212	C₂²×C₄	—	191	2	no	—
213	C₂²×C₄	—	191	2	no	—
214	C₂²×C₄	—	191	2	no	—
215	C₂²×C₄	—	764	2	no	—
216	C₂²×C₄	C₄×D₁₉₁	1	4	yes	complemented
217	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
218	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
219	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
220	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
221	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
222	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
223	C₄⋊C₄	—	382	2	no	—
224	C₄⋊C₄	—	191	4	no	—
225	C₄⋊C₄	—	191	4	no	—
226	C₄⋊C₄	—	764	2	no	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	382	4	no	—
230	C₂²⋊C₄	—	382	4	no	—
231	C₂²⋊C₄	—	382	4	no	—
232	C₂²⋊C₄	—	191	8	no	—
233	C₂²⋊C₄	—	764	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	382	4	no	—
237	C₄²	—	764	2	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	382	2	no	—
245	C₂³	—	382	2	no	—
246	C₂³	—	382	2	no	—
247	C₂³	—	382	2	no	—
248	C₂³	—	382	2	no	—
249	C₂³	—	382	2	no	—
250	C₂³	—	191	4	no	—
251	C₂³	C₂×C₄×D₁₉₁	1	2	yes	—
252	C₂³	C₂³×D₁₉₁	1	1	yes	characteristic
253	D₄	—	382	8	no	—
254	D₄	—	764	8	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	382	2	no	—
269	C₂×C₄	—	382	2	no	—
270	C₂×C₄	—	382	2	no	—
271	C₂×C₄	—	382	2	no	—
272	C₂×C₄	—	382	2	no	—
273	C₂×C₄	—	382	2	no	—
274	C₂×C₄	—	382	2	no	—
275	C₂×C₄	—	191	4	no	—
276	C₂×C₄	—	191	4	no	—
277	C₂×C₄	—	764	8	no	—
278	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
279	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
280	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
281	C₂×C₄	#15690286	1	2	yes	—
282	C₂×C₄	#15690287	1	2	yes	—
283	C₂×C₄	#15690287	1	2	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	382	4	no	—
292	C₂²	—	382	4	no	—
293	C₂²	—	382	4	no	—
294	C₂²	—	764	2	no	—
295	C₂²	—	764	2	no	—
296	C₂²	—	764	2	no	—
297	C₂²	—	764	2	no	—
298	C₂²	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
299	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
300	C₂²	C₂×#15690286	1	1	yes	characteristic
301	C₂²	C₂×#15690287	1	1	yes	characteristic
302	C₂²	C₂×#15690287	1	1	yes	characteristic
303	C₂²	#20259368	1	1	yes	characteristic
304	C₂²	#20259368	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	382	4	no	—
311	C₄	—	382	4	no	—
312	C₄	—	764	2	no	—
313	C₄	(C₂²⋊D₁₉₁)⋊C₄	1	4	yes	—
314	C₂	—	4	2	no	—
315	C₂	—	764	4	no	—
316	C₂	C₂×(C₂²⋊D₁₉₁)⋊C₄	1	1	yes	characteristic
317	C₂	C₄×#15690287	1	1	yes	characteristic
318	C₂	#27133137	1	1	yes	characteristic
319	C₂	#27133309	1	1	yes	characteristic
320	C₂	#27133334	1	1	yes	characteristic
321	C₂	#27133359	1	1	yes	characteristic
322	C₂	#27133378	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₁
commutator	C₂²×C₁₉₁	C₂³×C₄
Fitting	C₄×C₂²⋊C₄×C₁₉₁	C₂
socle, cosocle	C₂³×C₁₉₁	C₂⁴
prime radical	C₂×#20259296	C₂
hypercenter	C₄×C₂²⋊C₄	D₁₉₁
hypocenter	C₁₉₁	#2363
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₇₆₄	C₂	semidirect
3. C₂×#20259281	C₂	semidirect
4. #27132565	C₂	semidirect
5. #27132563	C₂	semidirect
6. #27132561	C₂	semidirect
7. #27132552	C₂	semidirect
8. #27132539	C₂	semidirect
9. C₄×C₄⋊Dic₁₉₁	C₂	semidirect
10. #27132510	C₂	semidirect
11. C₄×C₂²⋊C₄×C₁₉₁	C₂	semidirect
15. #20259296	C₄	semidirect
44. C₂×C₂²⋊D₁₉₁	C₂×C₄	semidirect
52. C₄⋊Dic₁₉₁	C₂×C₄	semidirect
53. C₄⋊Dic₁₉₁	D₄	semidirect
59. (C₂×D₁₉₁)⋊C₄	C₂×C₄	semidirect
65. C₂²×C₄×C₁₉₁	C₂×C₄	semidirect
161. C₁₉₁	#2363	semidirect
173. C₄×C₂²⋊C₄	D₁₉₁	semidirect
216. C₂²×C₄	C₄×D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20259296	2
C₂×C₄×D₇₆₄	2
C₂×#20259281	2
#27132565	2
#27132563	2
#27132561	2
#27132552	2
#27132539	2
C₄×C₄⋊Dic₁₉₁	2
#27132510	2
C₄×C₂²⋊C₄×C₁₉₁	2
#2363	191