Subgroups — #37576216

Order 24448.

608 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20259321	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#20259301	C₂	1	2	yes	maximal
3	#27132596	C₂	1	2	yes	maximal, complemented
4	#27132583	C₂	1	2	yes	maximal, complemented
5	#27132562	C₂	1	1	yes	maximal, characteristic, complemented
6	#27132559	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132536	C₂	1	2	yes	maximal
8	#27132532	C₂	1	1	yes	maximal, characteristic, complemented
9	#27132523	C₂	1	1	yes	maximal, characteristic
10	C₄×C₂²⋊Dic₁₉₁	C₂	1	1	yes	maximal, characteristic
11	C₂²×C₄⋊C₄×C₁₉₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₃₈₂	C₂²	1	2	yes	—
13	#20259321	—	2	4	no	—
14	#20259301	—	2	8	no	—
15	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
18	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
19	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
20	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
21	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
27	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
28	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
29	C₁₉₁⋊#95	C₂²	1	2	yes	—
30	C₁₉₁⋊#95	C₂²	1	2	yes	—
31	C₁₉₁⋊#95	C₂²	1	2	yes	—
32	C₂³×C₄×C₁₉₁	C₂²	1	2	yes	—
33	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₉₁	—	2	4	no	—
35	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
37	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
38	C₂×Dic₃₈₂	—	2	4	no	—
39	C₂×Dic₃₈₂	—	2	4	no	—
40	C₂×Dic₃₈₂	—	4	2	no	—
41	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
43	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
44	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
46	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
47	C₄⋊Dic₁₉₁	—	2	4	no	—
48	C₄⋊Dic₁₉₁	—	2	2	no	—
49	C₄⋊Dic₁₉₁	—	2	2	no	—
50	Dic₁₉₁⋊C₄	—	2	4	no	—
51	Dic₁₉₁⋊C₄	—	2	4	no	—
52	Dic₁₉₁⋊C₄	—	2	4	no	—
53	Dic₁₉₁⋊C₄	—	4	2	no	—
54	C₂²⋊Dic₁₉₁	—	2	4	no	—
55	C₂²⋊Dic₁₉₁	D₄	1	4	yes	—
56	C₂²⋊Dic₁₉₁	Q₈	1	4	yes	complemented
57	C₄×Dic₁₉₁	—	2	2	no	—
58	C₄×Dic₁₉₁	—	2	2	no	—
59	C₂⁴×C₁₉₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₉₁	—	2	2	no	—
61	C₂²×C₄×C₁₉₁	—	2	2	no	—
62	C₂²×C₄×C₁₉₁	—	2	4	no	—
63	C₂²×C₄×C₁₉₁	—	2	4	no	—
64	C₂²×C₄×C₁₉₁	—	4	2	no	—
65	C₂²×C₄×C₁₉₁	Q₈	1	4	yes	complemented
66	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
68	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₉₁	—	2	4	no	—
71	C₄⋊C₄×C₁₉₁	—	2	2	no	—
72	C₄⋊C₄×C₁₉₁	D₄	1	4	yes	—
73	Dic₃₈₂	—	4	8	no	—
74	C₂×Dic₁₉₁	—	2	1	no	—
75	C₂×Dic₁₉₁	—	2	1	no	—
76	C₂×Dic₁₉₁	—	2	1	no	—
77	C₂×Dic₁₉₁	—	2	1	no	—
78	C₂×Dic₁₉₁	—	2	1	no	—
79	C₂×Dic₁₉₁	—	2	1	no	—
80	C₂×Dic₁₉₁	—	2	1	no	—
81	C₂×Dic₁₉₁	—	2	1	no	—
82	C₂×Dic₁₉₁	—	2	2	no	—
83	C₂×Dic₁₉₁	—	2	2	no	—
84	C₂×Dic₁₉₁	—	2	2	no	—
85	C₂×Dic₁₉₁	—	2	2	no	—
86	C₂×Dic₁₉₁	—	2	2	no	—
87	C₂×Dic₁₉₁	—	2	2	no	—
88	C₂×Dic₁₉₁	C₂×D₄	1	2	yes	—
89	C₂×Dic₁₉₁	C₂×D₄	1	2	yes	—
90	C₂×Dic₁₉₁	C₂×Q₈	1	2	yes	—
91	C₂×Dic₁₉₁	C₂×Q₈	1	2	yes	—
92	C₂³×C₁₉₁	—	2	1	no	—
93	C₂³×C₁₉₁	—	2	1	no	—
94	C₂³×C₁₉₁	—	2	1	no	—
95	C₂³×C₁₉₁	—	2	1	no	—
96	C₂³×C₁₉₁	C₂×D₄	1	2	yes	—
97	C₂³×C₁₉₁	C₂×Q₈	1	2	yes	—
98	C₂³×C₁₉₁	C₂×Q₈	1	2	yes	—
99	C₂³×C₁₉₁	C₂⁴	1	1	yes	characteristic
100	C₂×C₄×C₁₉₁	—	2	1	no	—
101	C₂×C₄×C₁₉₁	—	2	2	no	—
102	C₂×C₄×C₁₉₁	—	2	1	no	—
103	C₂×C₄×C₁₉₁	—	2	2	no	—
104	C₂×C₄×C₁₉₁	—	2	8	no	—
105	C₂×C₄×C₁₉₁	—	2	2	no	—
106	C₂×C₄×C₁₉₁	—	2	2	no	—
107	C₂×C₄×C₁₉₁	—	2	2	no	—
108	C₂×C₄×C₁₉₁	—	4	8	no	—
109	C₂×C₄×C₁₉₁	C₂×D₄	1	4	yes	—
110	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₁₉₁	C₂×Q₈	1	2	yes	—
112	C₂×C₄×C₁₉₁	C₂×Q₈	1	2	yes	—
113	C₂×C₄×C₁₉₁	#41	1	2	yes	—
114	C₂²×C₁₉₁	—	2	2	no	—
115	C₂²×C₁₉₁	—	2	2	no	—
116	C₂²×C₁₉₁	—	2	2	no	—
117	C₂²×C₁₉₁	—	2	2	no	—
118	C₂²×C₁₉₁	—	2	2	no	—
119	C₂²×C₁₉₁	—	2	2	no	—
120	C₂²×C₁₉₁	C₄⋊Q₈	1	4	yes	complemented
121	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
122	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
123	C₂²×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
124	C₂²×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
125	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
126	C₂²×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂²×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
128	C₄×C₁₉₁	—	2	2	no	—
129	C₄×C₁₉₁	—	2	4	no	—
130	C₄×C₁₉₁	—	4	2	no	—
131	C₄×C₁₉₁	C₂²⋊Q₈	1	4	yes	complemented
132	Dic₁₉₁	—	2	2	no	—
133	Dic₁₉₁	—	2	2	no	—
134	Dic₁₉₁	—	2	2	no	—
135	Dic₁₉₁	—	2	2	no	—
136	Dic₁₉₁	—	4	2	no	—
137	Dic₁₉₁	—	4	2	no	—
138	C₂×C₁₉₁	—	2	4	no	—
139	C₂×C₁₉₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₁₉₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
141	C₂×C₁₉₁	#544	1	1	yes	characteristic
142	C₂×C₁₉₁	D₄²	1	1	yes	characteristic
143	C₂×C₁₉₁	#547	1	1	yes	characteristic
144	C₂×C₁₉₁	D₄×Q₈	1	1	yes	characteristic
145	C₂×C₁₉₁	#554	1	1	yes	characteristic
146	C₁₉₁	#2440	1	1	yes	characteristic, complemented, Sylow
147	#2440	—	191	1	no	maximal, Sylow
148	C₂×C₄⋊Q₈	—	191	1	no	—
149	C₂×C₂²⋊Q₈	—	191	2	no	—
150	C₂²×C₄⋊C₄	D₁₉₁	1	1	yes	characteristic
151	#398	—	191	2	no	—
152	#395	—	191	2	no	—
153	#389	—	191	1	no	—
154	#389	—	191	1	no	—
155	#385	—	191	2	no	—
156	#384	—	191	1	no	—
157	#380	—	191	1	no	—
158	C₄×C₂²⋊C₄	—	191	1	no	—
159	C₂²×Q₈	—	191	2	no	—
160	C₂³×C₄	C₂×D₁₉₁	1	2	yes	—
161	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
162	C₄⋊Q₈	—	382	4	no	—
163	C₂²⋊Q₈	—	382	8	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	191	2	no	—
166	C₂×C₄⋊C₄	—	191	2	no	—
167	C₂×C₄⋊C₄	—	191	2	no	—
168	C₂×C₄⋊C₄	—	191	1	no	—
169	C₂×C₄⋊C₄	—	191	2	no	—
170	C₂×C₄⋊C₄	—	191	1	no	—
171	C₂×C₄⋊C₄	—	191	2	no	—
172	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
173	C₂×C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
174	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
175	C₂×C₂²⋊C₄	—	191	2	no	—
176	C₂×C₂²⋊C₄	—	191	1	no	—
177	C₂×C₂²⋊C₄	—	191	1	no	—
178	C₂×C₄²	—	191	1	no	—
179	C₂×C₄²	—	191	1	no	—
180	#95	—	191	1	no	—
181	#95	—	191	1	no	—
182	#95	—	191	2	no	—
183	#95	—	191	2	no	—
184	#95	—	191	2	no	—
185	C₂⁴	C₂²×D₁₉₁	1	1	yes	characteristic
186	C₂×Q₈	—	382	4	no	—
187	C₂×Q₈	—	382	4	no	—
188	C₂×Q₈	—	764	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	191	2	no	—
195	C₂²×C₄	—	191	2	no	—
196	C₂²×C₄	—	191	1	no	—
197	C₂²×C₄	—	191	1	no	—
198	C₂²×C₄	—	191	1	no	—
199	C₂²×C₄	—	191	1	no	—
200	C₂²×C₄	Dic₃₈₂	1	4	yes	complemented
201	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
202	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
203	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	382	4	no	—
208	C₄⋊C₄	—	382	4	no	—
209	C₄⋊C₄	—	382	4	no	—
210	C₄⋊C₄	—	382	4	no	—
211	C₄⋊C₄	—	382	2	no	—
212	C₄⋊C₄	—	382	2	no	—
213	C₄⋊C₄	—	764	2	no	—
214	C₄⋊C₄	C₂²⋊D₁₉₁	1	4	yes	—
215	C₂²⋊C₄	—	382	4	no	—
216	C₂²⋊C₄	—	191	4	no	—
217	C₂²⋊C₄	—	191	4	no	—
218	C₄²	—	382	2	no	—
219	C₄²	—	382	2	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	1	no	—
224	C₂³	D₄×D₁₉₁	1	2	yes	—
225	C₂³	Q₈×D₁₉₁	1	2	yes	—
226	C₂³	C₂×Dic₃₈₂	1	2	yes	—
227	C₂³	C₂³×D₁₉₁	1	1	yes	characteristic
228	Q₈	—	764	8	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	—	382	1	no	—
239	C₂×C₄	—	382	1	no	—
240	C₂×C₄	—	382	1	no	—
241	C₂×C₄	—	382	1	no	—
242	C₂×C₄	—	382	1	no	—
243	C₂×C₄	—	382	1	no	—
244	C₂×C₄	—	382	1	no	—
245	C₂×C₄	—	382	1	no	—
246	C₂×C₄	—	382	2	no	—
247	C₂×C₄	—	382	2	no	—
248	C₂×C₄	—	382	2	no	—
249	C₂×C₄	—	382	2	no	—
250	C₂×C₄	—	382	2	no	—
251	C₂×C₄	—	382	2	no	—
252	C₂×C₄	—	191	2	no	—
253	C₂×C₄	—	191	2	no	—
254	C₂×C₄	—	191	2	no	—
255	C₂×C₄	—	191	2	no	—
256	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	4	yes	—
257	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
258	C₂×C₄	C₂×Dic₃₈₂	1	2	yes	—
259	C₂×C₄	C₂×Dic₃₈₂	1	2	yes	—
260	C₂×C₄	#15690287	1	2	yes	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	#20259321	1	4	yes	complemented
268	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
269	C₂²	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
270	C₂²	C₂×Q₈×D₁₉₁	1	1	yes	characteristic
271	C₂²	C₂²×Dic₃₈₂	1	1	yes	characteristic
272	C₂²	C₂×#15690287	1	1	yes	characteristic
273	C₂²	#20259367	1	1	yes	characteristic
274	C₂²	#20259370	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	2	no	—
278	C₄	—	382	2	no	—
279	C₄	—	382	2	no	—
280	C₄	—	382	2	no	—
281	C₄	—	382	2	no	—
282	C₄	—	764	2	no	—
283	C₄	—	764	2	no	—
284	C₄	#20259301	1	4	yes	complemented
285	C₂	—	2	4	no	—
286	C₂	C₂×#20259301	1	1	yes	characteristic
287	C₂	C₂×#20259321	1	1	yes	characteristic
288	C₂	#27133294	1	1	yes	characteristic
289	C₂	D₄×C₂²⋊D₁₉₁	1	1	yes	characteristic
290	C₂	#27133314	1	1	yes	characteristic
291	C₂	Q₈×C₂²⋊D₁₉₁	1	1	yes	characteristic
292	C₂	#27133368	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₁
commutator, socle, cosocle	C₂³×C₁₉₁	C₂⁴
Fitting	C₂²×C₄⋊C₄×C₁₉₁	C₂
prime radical	C₂⁴×C₁₉₁	C₂³
hypercenter	C₂²×C₄⋊C₄	D₁₉₁
hypocenter	C₁₉₁	#2440
LCS	G ▷ C₂³×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#20259321	C₂	semidirect
3. #27132596	C₂	semidirect
4. #27132583	C₂	semidirect
5. #27132562	C₂	semidirect
6. #27132559	C₂	semidirect
8. #27132532	C₂	semidirect
56. C₂²⋊Dic₁₉₁	Q₈	semidirect
65. C₂²×C₄×C₁₉₁	Q₈	semidirect
120. C₂²×C₁₉₁	C₄⋊Q₈	semidirect
131. C₄×C₁₉₁	C₂²⋊Q₈	semidirect
146. C₁₉₁	#2440	semidirect
200. C₂²×C₄	Dic₃₈₂	semidirect
267. C₂²	#20259321	semidirect
284. C₄	#20259301	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20259321	2
C₂×#20259301	2
#27132596	2
#27132583	2
#27132562	2
#27132559	2
#27132536	2
#27132532	2
#27132523	2
C₄×C₂²⋊Dic₁₉₁	2
C₂²×C₄⋊C₄×C₁₉₁	2
#2440	191