Subgroups — #11725031

Order 880.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²⋊D₅₅	C₂	1	2	yes	maximal, complemented
3	D₂₂₀	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×D₅₅	C₂	1	2	yes	maximal, complemented
5	Dic₁₁₀	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×D₅₅	C₂²	1	2	yes	—
8	C₄×C₅×C₁₁	C₂²	1	1	yes	characteristic
9	C₄×C₅×C₁₁	C₂²	1	1	yes	characteristic
10	Dic₅₅	C₂²	1	2	yes	—
11	#4339	—	5	1	no	maximal, Hall
12	C₂×C₅×C₁₁	—	2	1	no	—
13	C₂×C₅×C₁₁	C₂³	1	1	yes	characteristic
14	D₅₅	—	2	2	no	—
15	C₂×C₄×C₁₁	D₅	1	1	yes	characteristic, complemented
16	C₂²⋊D₁₁	—	5	2	no	—
17	D₄₄	—	5	1	no	—
18	C₄×D₁₁	—	5	2	no	—
19	Dic₂₂	—	5	1	no	—
20	#709	—	11	1	no	maximal, Hall
21	C₅×C₁₁	#41	1	1	yes	characteristic, complemented, Hall
22	C₂²×C₁₁	C₂×D₅	1	1	yes	characteristic
23	C₂×D₁₁	—	5	2	no	—
24	C₄×C₁₁	C₂×D₅	1	1	yes	characteristic
25	C₄×C₁₁	C₂×D₅	1	1	yes	characteristic
26	Dic₁₁	—	5	2	no	—
27	C₂×C₄×C₅	D₁₁	1	1	yes	characteristic, complemented
28	C₂²⋊D₅	—	11	2	no	—
29	D₂₀	—	11	1	no	—
30	C₄×D₅	—	11	2	no	—
31	Dic₁₀	—	11	1	no	—
32	C₂×C₁₁	—	2	1	no	—
33	C₂×C₁₁	C₂²×D₅	1	1	yes	characteristic
34	D₁₁	—	10	2	no	—
35	C₂²×C₅	C₂×D₁₁	1	1	yes	characteristic
36	C₂×D₅	—	11	2	no	—
37	C₄×C₅	C₂×D₁₁	1	1	yes	characteristic
38	C₄×C₅	C₂×D₁₁	1	1	yes	characteristic
39	Dic₅	—	11	2	no	—
40	#41	—	55	1	no	Sylow
41	C₁₁	#709	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₅	—	2	1	no	—
43	C₂×C₅	C₂²×D₁₁	1	1	yes	characteristic
44	D₅	—	22	2	no	—
45	Q₈	—	55	1	no	—
46	D₄	—	55	2	no	—
47	D₄	—	55	1	no	—
48	C₂×C₄	—	55	2	no	—
49	C₂×C₄	D₅₅	1	1	yes	characteristic, complemented
50	C₅	#4339	1	1	yes	characteristic, complemented, Sylow
51	C₂²	—	55	2	no	—
52	C₂²	C₂×D₅₅	1	1	yes	characteristic
53	C₄	—	55	2	no	—
54	C₄	C₂×D₅₅	1	1	yes	characteristic
55	C₄	C₂×D₅₅	1	1	yes	characteristic
56	C₂	—	2	1	no	—
57	C₂	—	110	2	no	—
58	C₂	C₂²×D₅₅	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₅₅
commutator, socle, cosocle	C₂×C₅×C₁₁	C₂³
Frattini	C₂	C₂²×D₅₅
Fitting	C₂×C₄×C₅×C₁₁	C₂
hypercenter	C₂×C₄	D₅₅
hypocenter	C₅×C₁₁	#41
LCS	G ▷ C₂×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×C₅×C₁₁	C₂	semidirect
2. C₂²⋊D₅₅	C₂	semidirect
3. D₂₂₀	C₂	semidirect
4. C₄×D₅₅	C₂	semidirect
5. Dic₁₁₀	C₂	semidirect
15. C₂×C₄×C₁₁	D₅	semidirect
21. C₅×C₁₁	#41	semidirect
27. C₂×C₄×C₅	D₁₁	semidirect
41. C₁₁	#709	semidirect
49. C₂×C₄	D₅₅	semidirect
50. C₅	#4339	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×C₁₁	2
C₂²⋊D₅₅	2
D₂₂₀	2
C₄×D₅₅	2
Dic₁₁₀	2
#4339	5
#709	11