Subgroups — #13183175

Order 1760.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₅×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#11724932	C₂	1	3	yes	maximal
3	#11724930	C₂	1	3	yes	maximal, complemented
4	C₂²×D₅₅	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₅×C₁₁	C₂²	1	3	yes	—
6	C₂×Dic₅₅	C₂²	1	3	yes	—
7	#66929	—	5	1	no	maximal, Hall
8	C₂²×C₅×C₁₁	C₂³	1	1	yes	characteristic
9	C₂×D₅₅	—	2	3	no	—
10	C₄×C₅×C₁₁	—	2	3	no	—
11	Dic₅₅	—	2	3	no	—
12	C₄²×C₁₁	D₅	1	1	yes	characteristic, complemented
13	(C₂×D₁₁)⋊C₄	—	5	3	no	—
14	Dic₁₁⋊C₄	—	5	3	no	—
15	#4022	—	11	1	no	maximal, Hall
16	C₂×C₅×C₁₁	#41	1	3	yes	—
17	D₅₅	—	4	1	no	—
18	C₂²×D₁₁	—	5	1	no	—
19	C₂×C₄×C₁₁	C₂×D₅	1	3	yes	—
20	C₂×Dic₁₁	—	5	3	no	—
21	C₄²×C₅	D₁₁	1	1	yes	characteristic, complemented
22	#685	—	11	3	no	—
23	#683	—	11	3	no	—
24	C₅×C₁₁	#126	1	1	yes	characteristic, complemented, Hall
25	C₂²×C₁₁	C₂²×D₅	1	1	yes	characteristic
26	C₂×D₁₁	—	10	3	no	—
27	C₄×C₁₁	—	2	3	no	—
28	Dic₁₁	—	10	3	no	—
29	C₂²×D₅	—	11	1	no	—
30	C₂×C₄×C₅	C₂×D₁₁	1	3	yes	—
31	C₂×Dic₅	—	11	3	no	—
32	#126	—	55	1	no	Sylow
33	C₂×C₁₁	#709	1	3	yes	—
34	D₁₁	—	20	1	no	—
35	C₂²×C₅	C₂²×D₁₁	1	1	yes	characteristic
36	C₂×D₅	—	22	3	no	—
37	C₄×C₅	—	2	3	no	—
38	Dic₅	—	22	3	no	—
39	C₄⋊C₄	—	55	3	no	—
40	C₂²⋊C₄	—	55	3	no	—
41	C₄²	D₅₅	1	1	yes	characteristic, complemented
42	C₁₁	#4022	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₅	#4339	1	3	yes	—
44	D₅	—	44	1	no	—
45	C₂³	—	55	1	no	—
46	C₂×C₄	—	55	3	no	—
47	C₂×C₄	C₂×D₅₅	1	3	yes	—
48	C₅	#66929	1	1	yes	characteristic, complemented, Sylow
49	C₂²	—	110	3	no	—
50	C₂²	C₂²×D₅₅	1	1	yes	characteristic
51	C₄	—	2	3	no	—
52	C₄	—	110	3	no	—
53	C₂	—	220	1	no	—
54	C₂	#11725031	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅₅
commutator, socle, cosocle	C₂²×C₅×C₁₁	C₂³
Fitting	C₄²×C₅×C₁₁	C₂
prime radical	C₂²×D₅₅	C₂²
hypercenter	C₄²	D₅₅
hypocenter	C₅×C₁₁	#126
LCS	G ▷ C₂²×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. C₄²×C₅×C₁₁	C₂	semidirect
3. #11724930	C₂	semidirect
12. C₄²×C₁₁	D₅	semidirect
21. C₄²×C₅	D₁₁	semidirect
24. C₅×C₁₁	#126	semidirect
41. C₄²	D₅₅	semidirect
42. C₁₁	#4022	semidirect
48. C₅	#66929	semidirect

Maximal subgroups

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

subgroup	index
C₄²×C₅×C₁₁	2
#11724932	2
#11724930	2
#66929	5
#4022	11