Subgroups — #13183449

Order 1760.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#11725035	C₂	1	4	yes	maximal, complemented
3	Q₈×D₅₅	C₂	1	4	yes	maximal, complemented
4	#11725031	C₂	1	6	yes	maximal, complemented
5	C₅×Q₈×C₁₁	C₂²	1	4	yes	—
6	C₂×C₄×C₅×C₁₁	C₂²	1	3	yes	—
7	C₂²⋊D₅₅	C₂²	1	4	yes	—
8	D₂₂₀	C₂²	1	6	yes	—
9	C₄×D₅₅	C₂²	1	12	yes	—
10	Dic₁₁₀	C₂²	1	6	yes	—
11	#67040	—	5	1	no	maximal, Hall
12	C₂²×C₅×C₁₁	C₂³	1	1	yes	characteristic
13	C₂×D₅₅	C₂³	1	4	yes	—
14	C₄×C₅×C₁₁	C₂³	1	6	yes	—
15	Dic₅₅	C₂³	1	4	yes	—
16	C₂×Q₈×C₁₁	D₅	1	1	yes	characteristic, complemented
17	#4343	—	5	4	no	—
18	Q₈×D₁₁	—	5	4	no	—
19	#4339	—	5	6	no	—
20	#4147	—	11	1	no	maximal, Hall
21	C₂×C₅×C₁₁	—	2	1	no	—
22	C₂×C₅×C₁₁	C₂⁴	1	1	yes	characteristic
23	D₅₅	—	2	4	no	—
24	Q₈×C₁₁	C₂×D₅	1	4	yes	—
25	C₂×C₄×C₁₁	C₂×D₅	1	3	yes	—
26	C₂²⋊D₁₁	—	5	4	no	—
27	D₄₄	—	5	6	no	—
28	C₄×D₁₁	—	5	12	no	—
29	Dic₂₂	—	5	6	no	—
30	C₂×C₅×Q₈	D₁₁	1	1	yes	characteristic, complemented
31	#713	—	11	4	no	—
32	Q₈×D₅	—	11	4	no	—
33	#709	—	11	6	no	—
34	C₅×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Hall
35	C₂²×C₁₁	C₂²×D₅	1	1	yes	characteristic
36	C₂×D₁₁	—	5	4	no	—
37	C₄×C₁₁	C₂²×D₅	1	6	yes	—
38	Dic₁₁	—	5	4	no	—
39	C₅×Q₈	C₂×D₁₁	1	4	yes	—
40	C₂×C₄×C₅	C₂×D₁₁	1	3	yes	—
41	C₂²⋊D₅	—	11	4	no	—
42	D₂₀	—	11	6	no	—
43	C₄×D₅	—	11	12	no	—
44	Dic₁₀	—	11	6	no	—
45	2₋¹⁺⁴	—	55	1	no	Sylow
46	C₂×C₁₁	—	2	1	no	—
47	C₂×C₁₁	C₂³×D₅	1	1	yes	characteristic
48	D₁₁	—	10	4	no	—
49	C₂²×C₅	C₂²×D₁₁	1	1	yes	characteristic
50	C₂×D₅	—	11	4	no	—
51	C₄×C₅	C₂²×D₁₁	1	6	yes	—
52	Dic₅	—	11	4	no	—
53	#41	—	55	4	no	—
54	#41	—	55	6	no	—
55	C₂×Q₈	—	55	4	no	—
56	C₂×Q₈	D₅₅	1	1	yes	characteristic, complemented
57	C₁₁	#4147	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₅	—	2	1	no	—
59	C₂×C₅	C₂³×D₁₁	1	1	yes	characteristic
60	D₅	—	22	4	no	—
61	Q₈	—	55	6	no	—
62	Q₈	C₂×D₅₅	1	4	yes	—
63	D₄	—	55	4	no	—
64	D₄	—	55	6	no	—
65	C₂×C₄	—	55	12	no	—
66	C₂×C₄	C₂×D₅₅	1	3	yes	—
67	C₅	#67040	1	1	yes	characteristic, complemented, Sylow
68	C₂²	—	55	4	no	—
69	C₂²	C₂²×D₅₅	1	1	yes	characteristic
70	C₄	—	55	4	no	—
71	C₄	C₂²×D₅₅	1	6	yes	—
72	C₂	—	2	1	no	—
73	C₂	—	110	4	no	—
74	C₂	C₂³×D₅₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₅₅
commutator, socle, cosocle	C₂×C₅×C₁₁	C₂⁴
Fitting	C₂×C₅×Q₈×C₁₁	C₂
hypercenter	C₂×Q₈	D₅₅
hypocenter	C₅×C₁₁	2₋¹⁺⁴
LCS	G ▷ C₂×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×Q₈×C₁₁	C₂	semidirect
2. #11725035	C₂	semidirect
3. Q₈×D₅₅	C₂	semidirect
4. #11725031	C₂	semidirect
16. C₂×Q₈×C₁₁	D₅	semidirect
30. C₂×C₅×Q₈	D₁₁	semidirect
34. C₅×C₁₁	2₋¹⁺⁴	semidirect
56. C₂×Q₈	D₅₅	semidirect
57. C₁₁	#4147	semidirect
67. C₅	#67040	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×Q₈×C₁₁	2
#11725035	2
Q₈×D₅₅	2
#11725031	2
#67040	5
#4147	11