Subgroups — #27132757

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
2	#20259287	C₂	1	2	yes	maximal
3	#20259245	C₂	1	2	yes	maximal
4	C₈×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic
5	#107×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₈₂	C₂²	1	2	yes	—
7	Dic₇₆₄	C₄	1	4	yes	complemented
8	C₂×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
9	Dic₁₉₁⋊C₄	—	2	2	no	—
10	C₄×Dic₁₉₁	—	2	2	no	—
11	C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
14	Dic₃₈₂	—	2	2	no	—
15	Dic₃₈₂	C₂×C₄	1	4	yes	—
16	C₂×Dic₁₉₁	—	2	2	no	—
17	C₂×Dic₁₉₁	D₄	1	1	yes	characteristic
18	C₂×Dic₁₉₁	D₄	1	1	yes	characteristic
19	C₁₉₁⋊C₈	—	2	1	no	—
20	C₂×C₄×C₁₉₁	—	2	2	no	—
21	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
22	C₈×C₁₉₁	C₂×C₄	1	2	yes	—
23	C₂²×C₁₉₁	C₂×D₄	1	1	yes	characteristic
24	C₄×C₁₉₁	—	4	2	no	—
25	C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
26	C₄×C₁₉₁	#41	1	1	yes	characteristic
27	Dic₁₉₁	—	2	4	no	—
28	Dic₁₉₁	—	2	1	no	—
29	Dic₁₉₁	Q₁₆	1	2	yes	complemented
30	C₂×C₁₉₁	C₄×D₄	1	1	yes	characteristic
31	C₂×C₁₉₁	C₂×Q₁₆	1	1	yes	characteristic
32	C₂×C₁₉₁	#135	1	1	yes	characteristic
33	C₁₉₁	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₄×Q₁₆	—	191	1	no	maximal, Sylow
35	C₂×Q₁₆	—	191	1	no	—
36	C₄×Q₈	—	191	2	no	—
37	#107	D₁₉₁	1	1	yes	characteristic
38	#103	—	191	2	no	—
39	C₄×C₈	—	191	1	no	—
40	C₂×Q₈	—	191	2	no	—
41	Q₁₆	—	191	4	no	—
42	C₂×C₈	—	191	1	no	—
43	C₂×C₈	C₂×D₁₉₁	1	1	yes	characteristic
44	C₄⋊C₄	—	382	2	no	—
45	C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
46	C₄²	—	382	2	no	—
47	C₄²	—	191	1	no	—
48	Q₈	—	382	2	no	—
49	Q₈	—	191	4	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	382	2	no	—
52	C₂×C₄	—	191	1	no	—
53	C₂×C₄	—	191	1	no	—
54	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
55	C₈	—	382	1	no	—
56	C₈	C₄×D₁₉₁	1	2	yes	—
57	C₂²	D₄×D₁₉₁	1	1	yes	characteristic
58	C₄	—	4	2	no	—
59	C₄	—	382	1	no	—
60	C₄	—	382	4	no	—
61	C₄	—	191	2	no	—
62	C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
63	C₄	#15690288	1	1	yes	characteristic
64	C₂	#20259284	1	1	yes	characteristic
65	C₂	Q₁₆×D₁₉₁	1	1	yes	characteristic
66	C₂	#20259341	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₁
commutator	C₄×C₁₉₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₉₁
Fitting	#107×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	C₂×D₄
cosocle	C₂×C₄×C₁₉₁	C₂³
hypercenter	#107	D₁₉₁
hypocenter	C₁₉₁	C₄×Q₁₆
LCS	G ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
7. Dic₇₆₄	C₄	semidirect
29. Dic₁₉₁	Q₁₆	semidirect
33. C₁₉₁	C₄×Q₁₆	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
#20259287	2
#20259245	2
C₈×Dic₁₉₁	2
#107×C₁₉₁	2
C₄×Q₁₆	191