Subgroups — #27132921

Order 12224.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#20259301	C₂	1	2	yes	maximal
3	#20259254	C₂	1	1	yes	maximal, characteristic, complemented
4	#20259247	C₂	1	2	yes	maximal, complemented
5	C₂×M₁₆×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₈₂	C₂²	1	2	yes	—
7	Dic₇₆₄	—	2	2	no	—
8	C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
9	Dic₁₉₁⋊C₄	—	2	2	no	—
10	C₂²⋊Dic₁₉₁	—	2	2	no	—
11	C₂²×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
12	M₁₆×C₁₉₁	—	2	2	no	—
13	C₂×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
15	Dic₃₈₂	—	2	2	no	—
16	Dic₃₈₂	—	2	2	no	—
17	C₂×Dic₁₉₁	—	2	2	no	—
18	C₂×Dic₁₉₁	—	2	2	no	—
19	C₂³×C₁₉₁	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₁	—	2	1	no	—
21	C₂×C₄×C₁₉₁	—	2	1	no	—
22	C₂×C₄×C₁₉₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
24	C₈×C₁₉₁	—	2	1	no	—
25	C₈×C₁₉₁	D₄	1	2	yes	—
26	C₂²×C₁₉₁	—	2	1	no	—
27	C₂²×C₁₉₁	—	2	2	no	—
28	C₂²×C₁₉₁	C₂×D₄	1	1	yes	characteristic
29	C₄×C₁₉₁	—	2	1	no	—
30	C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
31	C₄×C₁₉₁	#41	1	1	yes	characteristic
32	Dic₁₉₁	—	4	2	no	—
33	Dic₁₉₁	—	4	2	no	—
34	C₂×C₁₉₁	—	4	1	no	—
35	C₂×C₁₉₁	#121	1	1	yes	characteristic
36	C₂×C₁₉₁	#137	1	2	yes	—
37	C₁₉₁	#470	1	1	yes	characteristic, complemented, Sylow
38	#470	—	191	1	no	maximal, Sylow
39	C₂×Q₁₆	—	191	1	no	—
40	C₂×M₁₆	D₁₉₁	1	1	yes	characteristic
41	C₂²⋊Q₈	—	191	2	no	—
42	#106	—	191	1	no	—
43	#103	—	191	2	no	—
44	C₂×Q₈	—	191	2	no	—
45	C₂²×C₄	C₂×D₁₉₁	1	1	yes	characteristic
46	Q₁₆	—	382	2	no	—
47	M₁₆	—	2	2	no	—
48	C₂×C₈	C₂×D₁₉₁	1	1	yes	characteristic
49	C₂×C₈	C₂×D₁₉₁	1	1	yes	characteristic
50	C₄⋊C₄	—	382	2	no	—
51	C₄⋊C₄	—	191	2	no	—
52	C₂²⋊C₄	—	382	2	no	—
53	C₂³	D₇₆₄	1	1	yes	characteristic
54	Q₈	—	382	2	no	—
55	Q₈	—	382	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	382	2	no	—
59	C₂×C₄	—	382	2	no	—
60	C₂×C₄	D₇₆₄	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
62	C₈	—	2	1	no	—
63	C₈	C₂²⋊D₁₉₁	1	2	yes	—
64	C₂²	—	2	2	no	—
65	C₂²	—	2	1	no	—
66	C₂²	C₂×D₇₆₄	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	764	2	no	—
69	C₄	—	764	2	no	—
70	C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
71	C₄	#15690287	1	1	yes	characteristic
72	C₂	—	4	1	no	—
73	C₂	#20259296	1	1	yes	characteristic
74	C₂	#20259355	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₇₆₄
commutator, cosocle	C₂×C₄×C₁₉₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₁
Fitting	C₂×M₁₆×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×D₄
prime radical	C₂³×C₁₉₁	D₄
hypercenter	C₂×M₁₆	D₁₉₁
hypocenter	C₁₉₁	#470
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₇₆₄	C₂	semidirect
3. #20259254	C₂	semidirect
4. #20259247	C₂	semidirect
37. C₁₉₁	#470	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇₆₄	2
#20259301	2
#20259254	2
#20259247	2
C₂×M₁₆×C₁₉₁	2
#470	191