Subgroups — #13183246

Order 1760.

152 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	C₂×D₂₂₀	C₂	1	1	yes	maximal, characteristic
3	#11724932	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₅₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₅₅	C₂²	1	2	yes	—
6	C₅×Q₈×C₁₁	—	2	2	no	—
7	C₂×C₄×C₅×C₁₁	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₅₅	C₂²	1	2	yes	—
10	D₂₂₀	—	2	2	no	—
11	#67000	—	5	1	no	maximal, Hall
12	C₂²×C₅×C₁₁	C₂³	1	1	yes	characteristic
13	C₂×D₅₅	—	2	2	no	—
14	C₂×D₅₅	—	2	4	no	—
15	C₄×C₅×C₁₁	—	2	2	no	—
16	C₄×C₅×C₁₁	D₄	1	2	yes	—
17	Dic₅₅	—	2	2	no	—
18	C₂×Q₈×C₁₁	D₅	1	1	yes	characteristic, complemented
19	C₂×D₄₄	—	5	1	no	—
20	(C₂×D₁₁)⋊C₄	—	5	4	no	—
21	C₄×Dic₁₁	—	5	1	no	—
22	#4093	—	11	1	no	maximal, Hall
23	C₂×C₅×C₁₁	C₂×D₄	1	1	yes	characteristic
24	C₂×C₅×C₁₁	#41	1	2	yes	—
25	D₅₅	—	4	2	no	—
26	C₂²×D₁₁	—	5	2	no	—
27	Q₈×C₁₁	—	2	2	no	—
28	C₂×C₄×C₁₁	C₂×D₅	1	2	yes	—
29	C₂×C₄×C₁₁	C₂×D₅	1	1	yes	characteristic
30	C₂×Dic₁₁	—	5	2	no	—
31	D₄₄	—	10	2	no	—
32	C₂×C₅×Q₈	D₁₁	1	1	yes	characteristic, complemented
33	C₂×D₂₀	—	11	1	no	—
34	#685	—	11	4	no	—
35	C₄×Dic₅	—	11	1	no	—
36	C₅×C₁₁	#124	1	1	yes	characteristic, complemented, Hall
37	C₂²×C₁₁	C₂²×D₅	1	1	yes	characteristic
38	C₂×D₁₁	—	10	4	no	—
39	C₂×D₁₁	—	10	2	no	—
40	C₄×C₁₁	—	2	2	no	—
41	C₄×C₁₁	C₂²⋊D₅	1	2	yes	—
42	Dic₁₁	—	10	2	no	—
43	C₂²×D₅	—	11	2	no	—
44	C₅×Q₈	—	2	2	no	—
45	C₂×C₄×C₅	C₂×D₁₁	1	2	yes	—
46	C₂×C₄×C₅	C₂×D₁₁	1	1	yes	characteristic
47	C₂×Dic₅	—	11	2	no	—
48	D₂₀	—	22	2	no	—
49	#124	—	55	1	no	Sylow
50	C₂×C₁₁	#713	1	2	yes	—
51	C₂×C₁₁	C₂×C₂²⋊D₅	1	1	yes	characteristic
52	D₁₁	—	20	2	no	—
53	C₂²×C₅	C₂²×D₁₁	1	1	yes	characteristic
54	C₂×D₅	—	22	4	no	—
55	C₂×D₅	—	22	2	no	—
56	C₄×C₅	—	2	2	no	—
57	C₄×C₅	C₂²⋊D₁₁	1	2	yes	—
58	Dic₅	—	22	2	no	—
59	C₂×Q₈	D₅₅	1	1	yes	characteristic, complemented
60	C₂×D₄	—	55	1	no	—
61	C₂²⋊C₄	—	55	4	no	—
62	C₄²	—	55	1	no	—
63	C₁₁	#4093	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₅	#4343	1	2	yes	—
65	C₂×C₅	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
66	D₅	—	44	2	no	—
67	C₂³	—	55	2	no	—
68	Q₈	—	2	2	no	—
69	D₄	—	110	2	no	—
70	C₂×C₄	—	55	2	no	—
71	C₂×C₄	C₂×D₅₅	1	2	yes	—
72	C₂×C₄	C₂×D₅₅	1	1	yes	characteristic
73	C₅	#67000	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	110	4	no	—
75	C₂²	—	110	2	no	—
76	C₂²	C₂²×D₅₅	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	110	2	no	—
79	C₄	C₂²⋊D₅₅	1	2	yes	—
80	C₂	—	220	2	no	—
81	C₂	#11725035	1	2	yes	—
82	C₂	C₂×C₂²⋊D₅₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅₅
commutator, socle, cosocle	C₂²×C₅×C₁₁	C₂³
Fitting	C₂×C₅×Q₈×C₁₁	C₂
prime radical	C₂×D₂₂₀	C₂
hypercenter	C₂×Q₈	D₅₅
hypocenter	C₅×C₁₁	#124
LCS	G ▷ C₂²×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂² ◁ 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
3. #11724932	C₂	semidirect
4. C₄×Dic₅₅	C₂	semidirect
18. C₂×Q₈×C₁₁	D₅	semidirect
32. C₂×C₅×Q₈	D₁₁	semidirect
36. C₅×C₁₁	#124	semidirect
59. C₂×Q₈	D₅₅	semidirect
63. C₁₁	#4093	semidirect
73. C₅	#67000	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
C₂×D₂₂₀	2
#11724932	2
C₄×Dic₅₅	2
#67000	5
#4093	11