Subgroups — #27132874

Order 12224.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₆₄	C₂	1	2	yes	maximal
2	#20259322	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₃₈₂	C₂	1	1	yes	maximal, characteristic
4	#20259247	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₈₂	C₂²	1	2	yes	—
7	C₂×Dic₃₈₂	C₂²	1	1	yes	characteristic
8	Dic₇₆₄	—	2	4	no	—
9	C₄⋊Dic₁₉₁	—	2	1	no	—
10	C₄⋊Dic₁₉₁	—	2	1	no	—
11	C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
12	Dic₁₉₁⋊C₄	—	2	1	no	—
13	C₄×Dic₁₉₁	—	2	1	no	—
14	C₂×C₈×C₁₉₁	C₂²	1	2	yes	—
15	C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
16	Dic₃₈₂	—	2	1	no	—
17	Dic₃₈₂	—	2	2	no	—
18	Dic₃₈₂	—	2	2	no	—
19	Dic₃₈₂	D₄	1	2	yes	—
20	C₂×Dic₁₉₁	—	2	1	no	—
21	C₂×Dic₁₉₁	—	2	1	no	—
22	C₂×Dic₁₉₁	—	2	2	no	—
23	C₂×C₄×C₁₉₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₉₁	D₄	1	1	yes	characteristic
25	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
26	C₈×C₁₉₁	—	2	2	no	—
27	C₂²×C₁₉₁	C₂×D₄	1	1	yes	characteristic
28	C₄×C₁₉₁	—	2	1	no	—
29	C₄×C₁₉₁	Q₁₆	1	2	yes	complemented
30	C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
31	C₄×C₁₉₁	#41	1	1	yes	characteristic
32	Dic₁₉₁	—	2	2	no	—
33	Dic₁₉₁	—	4	2	no	—
34	Dic₁₉₁	—	4	1	no	—
35	C₂×C₁₉₁	#121	1	1	yes	characteristic
36	C₂×C₁₉₁	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₁₉₁	#137	1	1	yes	characteristic
38	C₁₉₁	#462	1	1	yes	characteristic, complemented, Sylow
39	#462	—	191	1	no	maximal, Sylow
40	C₂×Q₁₆	—	191	2	no	—
41	C₄⋊Q₈	—	191	1	no	—
42	C₄×Q₈	—	191	1	no	—
43	C₄⋊C₈	D₁₉₁	1	1	yes	characteristic
44	#103	—	191	2	no	—
45	C₂×Q₈	—	191	2	no	—
46	C₂×Q₈	—	191	1	no	—
47	Q₁₆	—	382	4	no	—
48	C₂×C₈	C₂×D₁₉₁	1	2	yes	—
49	C₄⋊C₄	—	382	1	no	—
50	C₄⋊C₄	—	382	1	no	—
51	C₄⋊C₄	—	382	1	no	—
52	C₄⋊C₄	—	191	1	no	—
53	C₄²	—	382	1	no	—
54	C₄²	C₂×D₁₉₁	1	1	yes	characteristic
55	Q₈	—	382	1	no	—
56	Q₈	—	382	2	no	—
57	Q₈	—	382	2	no	—
58	Q₈	—	191	2	no	—
59	C₂×C₄	—	382	1	no	—
60	C₂×C₄	—	382	1	no	—
61	C₂×C₄	—	382	2	no	—
62	C₂×C₄	D₇₆₄	1	1	yes	characteristic
63	C₂×C₄	D₇₆₄	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
65	C₈	—	2	2	no	—
66	C₂²	C₂×D₇₆₄	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	382	2	no	—
69	C₄	—	764	1	no	—
70	C₄	—	764	2	no	—
71	C₄	Dic₇₆₄	1	2	yes	complemented
72	C₄	D₄×D₁₉₁	1	1	yes	characteristic
73	C₄	#15690288	1	1	yes	characteristic
74	C₂	#20259297	1	1	yes	characteristic
75	C₂	C₂×Dic₇₆₄	1	1	yes	characteristic
76	C₂	#20259355	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₇₆₄
commutator, cosocle	C₂×C₄×C₁₉₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₁
Fitting	C₄⋊C₈×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	C₂×D₄
hypercenter	C₄⋊C₈	D₁₉₁
hypocenter	C₁₉₁	#462
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
29. C₄×C₁₉₁	Q₁₆	semidirect
38. C₁₉₁	#462	semidirect
71. C₄	Dic₇₆₄	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
#20259322	2
C₄×Dic₃₈₂	2
#20259247	2
C₄⋊C₈×C₁₉₁	2
#462	191