Subgroups — #27212956

Order 13504.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269917	C₂	1	1	yes	maximal, characteristic, complemented
2	#20269909	C₂	1	1	yes	maximal, characteristic, complemented
3	#135×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Dic₂₁₁	—	2	1	no	—
5	C₂₁₁⋊C₁₆	—	2	1	no	—
6	#41×C₂₁₁	—	2	1	no	—
7	Q₁₆×C₂₁₁	C₄	1	1	yes	characteristic, complemented
8	SD₁₆×C₂₁₁	—	2	1	no	—
9	D₈×C₂₁₁	C₄	1	1	yes	characteristic, complemented
10	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₁₁	—	4	1	no	—
12	Q₈×C₂₁₁	—	2	1	no	—
13	D₄×C₂₁₁	—	2	1	no	—
14	D₄×C₂₁₁	—	4	1	no	—
15	C₂×C₄×C₂₁₁	—	4	1	no	—
16	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
17	C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
18	C₈×C₂₁₁	D₄	1	1	yes	characteristic
19	C₂²×C₂₁₁	—	4	1	no	—
20	C₂²×C₂₁₁	D₈	1	1	yes	characteristic
21	C₄×C₂₁₁	—	4	1	no	—
22	C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₂₁₁	SD₁₆	1	1	yes	characteristic
24	Dic₂₁₁	—	8	1	no	—
25	C₂×C₂₁₁	—	2	1	no	—
26	C₂×C₂₁₁	—	8	1	no	—
27	C₂×C₂₁₁	#102	1	1	yes	characteristic
28	C₂₁₁	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	211	1	no	maximal, Sylow
30	#135	D₂₁₁	1	1	yes	characteristic
31	M₃₂	—	211	1	no	—
32	#106	—	211	1	no	—
33	#41	—	2	1	no	—
34	Q₁₆	Dic₂₁₁	1	1	yes	characteristic, complemented
35	SD₁₆	—	2	1	no	—
36	D₈	Dic₂₁₁	1	1	yes	characteristic, complemented
37	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
38	C₄⋊C₄	—	422	1	no	—
39	C₁₆	—	422	1	no	—
40	Q₈	—	2	1	no	—
41	D₄	—	2	1	no	—
42	D₄	—	4	1	no	—
43	C₂×C₄	—	4	1	no	—
44	C₂×C₄	—	844	1	no	—
45	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
46	C₈	C₂×Dic₂₁₁	1	1	yes	characteristic
47	C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
48	C₂²	—	4	1	no	—
49	C₂²	#16809947	1	1	yes	characteristic
50	C₄	—	4	1	no	—
51	C₄	—	1688	1	no	—
52	C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
53	C₄	#16809950	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	8	1	no	—
56	C₂	#20269898	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20269898
commutator	C₈×C₂₁₁	C₂×C₄
Frattini	C₂×C₈	C₂×D₂₁₁
Fitting, prime radical	#135×C₂₁₁	C₂
socle	C₂×C₂₁₁	#102
cosocle	C₂×C₈×C₂₁₁	C₂²
hypercenter	#135	D₂₁₁
hypocenter	C₂₁₁	#360
LCS	G ▷ C₈×C₂₁₁ ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #20269917	C₂	semidirect
2. #20269909	C₂	semidirect
7. Q₁₆×C₂₁₁	C₄	semidirect
9. D₈×C₂₁₁	C₄	semidirect
28. C₂₁₁	#360	semidirect
34. Q₁₆	Dic₂₁₁	semidirect
36. D₈	Dic₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20269917	2
#20269909	2
#135×C₂₁₁	2
#360	211