Subgroups — #27212939

Order 13504.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269892	C₂	1	2	yes	maximal, complemented
2	#127×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊Dic₂₁₁	—	2	2	no	—
4	C₂×D₄×C₂₁₁	—	2	2	no	—
5	C₂×D₄×C₂₁₁	C₄	1	1	yes	characteristic, complemented
6	C₂×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
7	C₄²×C₂₁₁	C₄	1	1	yes	characteristic, complemented
8	C₂×Dic₂₁₁	—	4	2	no	—
9	C₂³×C₂₁₁	—	2	1	no	—
10	C₂³×C₂₁₁	D₄	1	2	yes	—
11	D₄×C₂₁₁	—	4	1	no	—
12	D₄×C₂₁₁	—	4	1	no	—
13	D₄×C₂₁₁	—	4	4	no	—
14	C₂×C₄×C₂₁₁	—	2	1	no	—
15	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₂₁₁	—	2	2	no	—
17	C₂²×C₂₁₁	—	4	2	no	—
18	C₂²×C₂₁₁	—	4	2	no	—
19	C₂²×C₂₁₁	—	4	1	no	—
20	C₂²×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₂₁₁	—	2	2	no	—
22	C₄×C₂₁₁	—	2	1	no	—
23	Dic₂₁₁	—	8	2	no	—
24	C₂×C₂₁₁	—	2	1	no	—
25	C₂×C₂₁₁	—	4	2	no	—
26	C₂×C₂₁₁	—	8	1	no	—
27	C₂×C₂₁₁	#99	1	1	yes	characteristic
28	C₂₁₁	#353	1	1	yes	characteristic, complemented, Sylow
29	#353	—	211	1	no	maximal, Sylow
30	#127	D₂₁₁	1	1	yes	characteristic
31	#99	—	211	2	no	—
32	C₂×D₄	—	2	2	no	—
33	C₂×D₄	Dic₂₁₁	1	1	yes	characteristic, complemented
34	C₂×D₄	C₂×D₂₁₁	1	1	yes	characteristic
35	C₂²⋊C₄	—	422	2	no	—
36	C₄²	Dic₂₁₁	1	1	yes	characteristic, complemented
37	C₂³	—	2	1	no	—
38	C₂³	C₂²⋊D₂₁₁	1	2	yes	—
39	D₄	—	4	1	no	—
40	D₄	—	4	1	no	—
41	D₄	—	4	4	no	—
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	—	844	2	no	—
44	C₂×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
45	C₂²	—	2	2	no	—
46	C₂²	—	4	2	no	—
47	C₂²	—	4	2	no	—
48	C₂²	—	4	1	no	—
49	C₂²	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
50	C₄	—	2	2	no	—
51	C₄	—	2	1	no	—
52	C₄	—	1688	2	no	—
53	C₂	—	2	1	no	—
54	C₂	—	4	2	no	—
55	C₂	—	8	1	no	—
56	C₂	#20269892	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20269892
commutator	C₂×C₄×C₂₁₁	C₂×C₄
Frattini	C₂×D₄	C₂×D₂₁₁
Fitting, prime radical	#127×C₂₁₁	C₂
socle	C₂×C₂₁₁	#99
cosocle	C₂×D₄×C₂₁₁	C₂²
hypercenter	#127	D₂₁₁
hypocenter	C₂₁₁	#353
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #127

Complements

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

N	H ≅ G/N	type
1. #20269892	C₂	semidirect
5. C₂×D₄×C₂₁₁	C₄	semidirect
7. C₄²×C₂₁₁	C₄	semidirect
28. C₂₁₁	#353	semidirect
33. C₂×D₄	Dic₂₁₁	semidirect
36. C₄²	Dic₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20269892	2
#127×C₂₁₁	2
#353	211