Subgroups — #27133048

Order 12224.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690274	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₅₂₈	C₂	1	1	yes	maximal, characteristic
3	#20259311	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₁₆×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₆₄	C₂²	1	2	yes	—
7	#15690274	—	2	4	no	—
8	D₁₅₂₈	—	2	2	no	—
9	C₂×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
10	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
11	C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
12	C₂×Q₈×C₁₉₁	C₂²	1	2	yes	—
13	Q₁₆×C₁₉₁	—	2	2	no	—
14	C₂×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₂²×D₁₉₁	—	2	2	no	—
16	D₇₆₄	—	2	2	no	—
17	D₇₆₄	—	2	2	no	—
18	C₂×Dic₁₉₁	D₄	1	2	yes	—
19	C₁₉₁⋊C₈	D₄	1	2	yes	—
20	Q₈×C₁₉₁	—	2	2	no	—
21	Q₈×C₁₉₁	—	2	2	no	—
22	C₂×C₄×C₁₉₁	—	2	2	no	—
23	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
24	C₈×C₁₉₁	D₄	1	2	yes	—
25	C₂²×C₁₉₁	C₂×D₄	1	1	yes	characteristic
26	C₂×D₁₉₁	—	4	2	no	—
27	C₂×D₁₉₁	—	4	4	no	—
28	C₄×C₁₉₁	—	4	2	no	—
29	C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
30	C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
31	Dic₁₉₁	—	2	2	no	—
32	C₂×C₁₉₁	#127	1	1	yes	characteristic
33	C₂×C₁₉₁	#135	1	2	yes	—
34	D₁₉₁	—	8	2	no	—
35	C₁₉₁	#495	1	1	yes	characteristic, complemented, Sylow
36	#495	—	191	1	no	maximal, Sylow
37	C₂×Q₁₆	D₁₉₁	1	1	yes	characteristic, complemented
38	C₂×SD₁₆	—	191	2	no	—
39	C₂×D₈	—	191	1	no	—
40	#124	—	191	2	no	—
41	C₄×C₈	—	191	1	no	—
42	C₂×Q₈	C₂×D₁₉₁	1	2	yes	—
43	C₂×D₄	—	191	2	no	—
44	Q₁₆	—	2	2	no	—
45	SD₁₆	—	382	4	no	—
46	D₈	—	382	2	no	—
47	C₂×C₈	—	191	1	no	—
48	C₂×C₈	C₂×D₁₉₁	1	1	yes	characteristic
49	C₂²⋊C₄	—	382	4	no	—
50	C₄²	—	191	1	no	—
51	C₂³	—	382	2	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	2	2	no	—
54	D₄	—	382	2	no	—
55	D₄	—	382	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	191	2	no	—
58	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
59	C₈	—	191	2	no	—
60	C₈	C₂²⋊D₁₉₁	1	2	yes	—
61	C₂²	—	764	4	no	—
62	C₂²	—	764	2	no	—
63	C₂²	D₄×D₁₉₁	1	1	yes	characteristic
64	C₄	—	4	2	no	—
65	C₄	—	382	2	no	—
66	C₄	D₄×D₁₉₁	1	1	yes	characteristic
67	C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
68	C₂	—	1528	2	no	—
69	C₂	#20259318	1	1	yes	characteristic
70	C₂	#20259345	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₁
Fitting	C₂×Q₁₆×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×D₄
prime radical	C₂×D₁₅₂₈	C₂
hypercenter	C₂×Q₁₆	D₁₉₁
hypocenter	C₁₉₁	#495
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#15690274	C₂	semidirect
3. #20259311	C₂	semidirect
4. C₈×Dic₁₉₁	C₂	semidirect
5. C₂×Q₁₆×C₁₉₁	C₂	semidirect
35. C₁₉₁	#495	semidirect
37. C₂×Q₁₆	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690274	2
C₂×D₁₅₂₈	2
#20259311	2
C₈×Dic₁₉₁	2
C₂×Q₁₆×C₁₉₁	2
#495	191