Subgroups — #37581039

Order 24448.

772 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15690271	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#20259281	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132672	C₂	1	2	yes	maximal, complemented
4	#27132668	C₂	1	2	yes	maximal, complemented
5	#27132664	C₂	1	4	yes	maximal, complemented
6	#27132659	C₂	1	4	yes	maximal, complemented
7	C₂²×#41×C₁₉₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#15690271	—	2	4	no	—
9	C₂×#15690271	C₂²	1	2	yes	—
10	C₂×#15690271	C₂²	1	2	yes	complemented
11	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	complemented
12	#20259281	—	2	1	no	—
13	#20259281	—	2	1	no	—
14	#20259281	C₂²	1	2	yes	—
15	#20259281	C₂²	1	2	yes	complemented
16	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	complemented
17	C₂×C₂²⋊Dic₁₉₁	—	2	1	no	—
18	C₂×C₄×Dic₁₉₁	—	2	1	no	—
19	#20259246	C₂²	1	8	yes	complemented
20	#20259243	C₂²	1	8	yes	complemented
21	C₂×#41×C₁₉₁	—	2	4	no	—
22	C₂×#41×C₁₉₁	—	2	1	no	—
23	C₂×#41×C₁₉₁	—	2	1	no	—
24	C₂×#41×C₁₉₁	C₄	1	8	yes	complemented
25	C₂×#41×C₁₉₁	C₂²	1	2	yes	—
26	C₂×#41×C₁₉₁	C₂²	1	2	yes	—
27	C₂²×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
28	C₂²×D₄×C₁₉₁	—	2	1	no	—
29	C₂²×D₄×C₁₉₁	C₂²	1	1	yes	characteristic
30	C₂³×C₄×C₁₉₁	—	2	1	no	—
31	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
32	C₂²×Dic₁₉₁	—	2	2	no	—
33	#15690271	—	2	4	no	—
34	#15690271	—	2	4	no	—
35	C₂×C₁₉₁⋊C₈	—	2	4	no	—
36	C₂×C₁₉₁⋊C₈	C₂³	1	4	yes	—
37	C₄⋊Dic₁₉₁	—	2	2	no	—
38	C₄⋊Dic₁₉₁	C₂³	1	4	yes	—
39	C₂²⋊Dic₁₉₁	—	2	2	no	—
40	C₂²⋊Dic₁₉₁	—	2	2	no	—
41	C₄×Dic₁₉₁	—	2	2	no	—
42	C₄×Dic₁₉₁	—	2	2	no	—
43	C₂⁴×C₁₉₁	—	2	1	no	—
44	C₂⁴×C₁₉₁	D₄	1	1	yes	characteristic
45	#41×C₁₉₁	—	2	8	no	—
46	#41×C₁₉₁	—	2	8	no	—
47	#41×C₁₉₁	—	2	4	no	—
48	#41×C₁₉₁	—	2	4	no	—
49	#41×C₁₉₁	C₂×C₄	1	8	yes	—
50	#41×C₁₉₁	C₂×C₄	1	8	yes	complemented
51	C₂×Q₈×C₁₉₁	—	2	2	no	—
52	C₂×Q₈×C₁₉₁	—	2	1	no	—
53	C₂×Q₈×C₁₉₁	C₂×C₄	1	4	yes	complemented
54	C₂×Q₈×C₁₉₁	C₂³	1	2	yes	—
55	C₂×D₄×C₁₉₁	—	2	8	no	—
56	C₂×D₄×C₁₉₁	—	2	2	no	—
57	C₂×D₄×C₁₉₁	—	2	2	no	—
58	C₂×D₄×C₁₉₁	—	2	1	no	—
59	C₂×D₄×C₁₉₁	—	2	2	no	—
60	C₂×D₄×C₁₉₁	C₂×C₄	1	4	yes	complemented
61	C₂×D₄×C₁₉₁	C₂³	1	2	yes	—
62	C₂²×C₄×C₁₉₁	—	2	1	no	—
63	C₂²×C₄×C₁₉₁	—	2	8	no	—
64	C₂²×C₄×C₁₉₁	—	2	2	no	—
65	C₂²×C₄×C₁₉₁	—	2	2	no	—
66	C₂²×C₄×C₁₉₁	C₂×C₄	1	4	yes	—
67	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉₁	D₄	1	4	yes	—
71	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
74	C₂×Dic₁₉₁	—	2	4	no	—
75	C₂×Dic₁₉₁	—	4	4	no	—
76	C₁₉₁⋊C₈	—	2	4	no	—
77	C₂³×C₁₉₁	—	2	4	no	—
78	C₂³×C₁₉₁	—	2	2	no	—
79	C₂³×C₁₉₁	—	4	2	no	—
80	C₂³×C₁₉₁	—	4	4	no	—
81	C₂³×C₁₉₁	C₂²⋊C₄	1	4	yes	—
82	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
83	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
84	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
85	Q₈×C₁₉₁	—	2	4	no	—
86	Q₈×C₁₉₁	—	2	2	no	—
87	Q₈×C₁₉₁	C₂²×C₄	1	4	yes	—
88	D₄×C₁₉₁	—	2	8	no	—
89	D₄×C₁₉₁	—	2	8	no	—
90	D₄×C₁₉₁	—	2	4	no	—
91	D₄×C₁₉₁	—	2	2	no	—
92	D₄×C₁₉₁	C₂²×C₄	1	4	yes	—
93	C₂×C₄×C₁₉₁	—	2	4	no	—
94	C₂×C₄×C₁₉₁	—	2	2	no	—
95	C₂×C₄×C₁₉₁	—	2	8	no	—
96	C₂×C₄×C₁₉₁	—	2	8	no	—
97	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	4	yes	—
98	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	4	yes	—
99	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	4	yes	—
100	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
101	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
110	C₂²×C₁₉₁	—	2	2	no	—
111	C₂²×C₁₉₁	—	2	2	no	—
112	C₂²×C₁₉₁	—	2	2	no	—
113	C₂²×C₁₉₁	—	2	4	no	—
114	C₂²×C₁₉₁	—	4	4	no	—
115	C₂²×C₁₉₁	—	4	8	no	—
116	C₂²×C₁₉₁	—	4	4	no	—
117	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	2	yes	—
118	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	2	yes	—
120	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₁₉₁	—	2	4	no	—
122	C₄×C₁₉₁	C₂×C₂²⋊C₄	1	2	yes	—
123	C₄×C₁₉₁	C₂×C₂²⋊C₄	1	2	yes	—
124	C₄×C₁₉₁	C₂×C₂²⋊C₄	1	2	yes	—
125	C₄×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
126	C₄×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
127	Dic₁₉₁	—	4	4	no	—
128	C₂×C₁₉₁	—	2	2	no	—
129	C₂×C₁₉₁	—	2	2	no	—
130	C₂×C₁₉₁	—	2	2	no	—
131	C₂×C₁₉₁	—	4	4	no	—
132	C₂×C₁₉₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₁₉₁	#575	1	2	yes	—
134	C₁₉₁	#2896	1	1	yes	characteristic, complemented, Sylow
135	#2896	—	191	1	no	maximal, Sylow
136	C₂²×#41	D₁₉₁	1	1	yes	characteristic
137	C₂²×M₁₆	—	191	1	no	—
138	C₂×#117	—	191	1	no	—
139	#419	—	191	2	no	—
140	#418	—	191	2	no	—
141	#417	—	191	4	no	—
142	#416	—	191	4	no	—
143	C₂×#41	—	2	4	no	—
144	C₂×#41	—	2	1	no	—
145	C₂×#41	—	2	1	no	—
146	C₂×#41	Dic₁₉₁	1	8	yes	complemented
147	C₂×#41	C₂×D₁₉₁	1	2	yes	—
148	C₂×#41	C₂×D₁₉₁	1	2	yes	—
149	C₂²×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
150	C₂²×D₄	—	2	1	no	—
151	C₂²×D₄	C₂×D₁₉₁	1	1	yes	characteristic
152	C₂³×C₄	—	2	1	no	—
153	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
154	C₂×M₁₆	—	382	4	no	—
155	C₂×M₁₆	—	191	2	no	—
156	C₂×M₁₆	—	191	2	no	—
157	C₂²×C₈	—	191	2	no	—
158	#117	—	382	1	no	—
159	#117	—	382	1	no	—
160	#117	—	191	2	no	—
161	#117	—	191	2	no	—
162	C₂×C₄⋊C₄	—	191	2	no	—
163	C₂×C₂²⋊C₄	—	382	1	no	—
164	C₂×C₄²	—	382	1	no	—
165	#103	—	191	8	no	—
166	#102	—	191	8	no	—
167	C₂⁴	—	2	1	no	—
168	C₂⁴	C₂²⋊D₁₉₁	1	1	yes	characteristic
169	#41	—	2	8	no	—
170	#41	—	2	8	no	—
171	#41	—	2	4	no	—
172	#41	—	2	4	no	—
173	#41	C₂×Dic₁₉₁	1	8	yes	—
174	#41	C₂×Dic₁₉₁	1	8	yes	complemented
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	C₂×Dic₁₉₁	1	4	yes	complemented
178	C₂×Q₈	C₂²×D₁₉₁	1	2	yes	—
179	C₂×D₄	—	2	8	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	C₂×Dic₁₉₁	1	4	yes	complemented
185	C₂×D₄	C₂²×D₁₉₁	1	2	yes	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	8	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	382	2	no	—
191	C₂²×C₄	C₂×Dic₁₉₁	1	4	yes	—
192	C₂²×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
193	C₂²×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
194	C₂²×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
195	C₂²×C₄	C₂²⋊D₁₉₁	1	4	yes	—
196	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
199	M₁₆	—	382	4	no	—
200	M₁₆	—	382	4	no	—
201	C₂×C₈	—	382	4	no	—
202	C₂×C₈	—	191	4	no	—
203	C₄⋊C₄	—	382	2	no	—
204	C₄⋊C₄	—	191	4	no	—
205	C₂²⋊C₄	—	382	2	no	—
206	C₂²⋊C₄	—	382	2	no	—
207	C₄²	—	382	2	no	—
208	C₄²	—	382	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	4	no	—
211	C₂³	—	4	2	no	—
212	C₂³	—	4	4	no	—
213	C₂³	C₂²⋊Dic₁₉₁	1	4	yes	—
214	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
215	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
216	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
217	Q₈	—	2	2	no	—
218	Q₈	—	2	4	no	—
219	Q₈	C₂²×Dic₁₉₁	1	4	yes	—
220	D₄	—	2	8	no	—
221	D₄	—	2	8	no	—
222	D₄	—	2	2	no	—
223	D₄	—	2	4	no	—
224	D₄	C₂²×Dic₁₉₁	1	4	yes	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	382	4	no	—
230	C₂×C₄	—	764	4	no	—
231	C₂×C₄	C₂²⋊Dic₁₉₁	1	4	yes	—
232	C₂×C₄	C₂²⋊Dic₁₉₁	1	4	yes	—
233	C₂×C₄	C₂²⋊Dic₁₉₁	1	4	yes	—
234	C₂×C₄	C₂²×Dic₁₉₁	1	2	yes	—
235	C₂×C₄	C₂²×Dic₁₉₁	1	2	yes	—
236	C₂×C₄	C₂²×Dic₁₉₁	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
238	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
239	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
241	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
244	C₈	—	382	4	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	4	4	no	—
250	C₂²	—	4	8	no	—
251	C₂²	—	4	4	no	—
252	C₂²	C₂×C₂²⋊Dic₁₉₁	1	2	yes	—
253	C₂²	C₂×C₂²⋊Dic₁₉₁	1	2	yes	—
254	C₂²	C₂×C₂²⋊Dic₁₉₁	1	2	yes	—
255	C₂²	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
256	C₄	—	2	4	no	—
257	C₄	—	764	4	no	—
258	C₄	C₂×C₂²⋊Dic₁₉₁	1	2	yes	—
259	C₄	C₂×C₂²⋊Dic₁₉₁	1	2	yes	—
260	C₄	C₂×C₂²⋊Dic₁₉₁	1	2	yes	—
261	C₄	C₂³×Dic₁₉₁	1	1	yes	characteristic
262	C₄	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	2	2	no	—
265	C₂	—	2	2	no	—
266	C₂	—	4	4	no	—
267	C₂	C₂²×C₂²⋊Dic₁₉₁	1	1	yes	characteristic
268	C₂	#27133461	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₉₁
commutator	C₄×C₁₉₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting, prime radical	C₂²×#41×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
cosocle	C₂×C₄×C₁₉₁	C₂⁴
hypercenter	C₂²×#41	D₁₉₁
hypocenter	C₁₉₁	#2896
LCS	G ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#15690271	C₂	semidirect
2. C₂×#20259281	C₂	semidirect
3. #27132672	C₂	semidirect
4. #27132668	C₂	semidirect
5. #27132664	C₂	semidirect
6. #27132659	C₂	semidirect
10. C₂×#15690271	C₂²	semidirect
11. C₂²×C₁₉₁⋊C₈	C₂²	semidirect
15. #20259281	C₂²	semidirect
16. C₂×C₄⋊Dic₁₉₁	C₂²	semidirect
19. #20259246	C₂²	semidirect
20. #20259243	C₂²	semidirect
24. C₂×#41×C₁₉₁	C₄	semidirect
50. #41×C₁₉₁	C₂×C₄	semidirect
53. C₂×Q₈×C₁₉₁	C₂×C₄	semidirect
60. C₂×D₄×C₁₉₁	C₂×C₄	semidirect
134. C₁₉₁	#2896	semidirect
146. C₂×#41	Dic₁₉₁	semidirect
174. #41	C₂×Dic₁₉₁	semidirect
177. C₂×Q₈	C₂×Dic₁₉₁	semidirect
184. C₂×D₄	C₂×Dic₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15690271	2
C₂×#20259281	2
#27132672	2
#27132668	2
#27132664	2
#27132659	2
C₂²×#41×C₁₉₁	2
#2896	191