Subgroups — #13183190

Order 1760.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₁₁×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Dic₅₅	C₂	1	2	yes	maximal
3	C₄⋊Dic₅₅	C₂	1	1	yes	maximal, characteristic
4	#11724930	C₂	1	2	yes	maximal
5	C₄×Dic₅₅	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₅×C₁₁	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×Dic₅₅	C₂²	1	1	yes	characteristic
9	C₂×Dic₅₅	C₂²	1	1	yes	characteristic
10	C₂×Dic₅₅	C₂²	1	2	yes	—
11	#66943	—	5	1	no	maximal, Hall
12	C₂²×C₅×C₁₁	C₂³	1	1	yes	characteristic
13	C₄×C₅×C₁₁	—	2	2	no	—
14	C₄×C₅×C₁₁	Q₈	1	2	yes	—
15	Dic₅₅	—	2	1	no	—
16	Dic₅₅	—	2	1	no	—
17	Dic₅₅	—	2	2	no	—
18	C₁₁×C₄⋊C₄	D₅	1	1	yes	characteristic
19	C₄⋊Dic₁₁	—	5	2	no	—
20	C₄⋊Dic₁₁	—	5	1	no	—
21	Dic₁₁⋊C₄	—	5	2	no	—
22	C₄×Dic₁₁	—	5	1	no	—
23	#4036	—	11	1	no	maximal, Hall
24	C₂×C₅×C₁₁	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₅×C₁₁	#41	1	1	yes	characteristic
26	C₂×C₅×C₁₁	#41	1	1	yes	characteristic
27	C₂×C₄×C₁₁	C₂×D₅	1	2	yes	—
28	C₂×C₄×C₁₁	C₂×D₅	1	1	yes	characteristic
29	C₂×Dic₁₁	—	5	1	no	—
30	C₂×Dic₁₁	—	5	1	no	—
31	C₂×Dic₁₁	—	5	2	no	—
32	C₅×C₄⋊C₄	D₁₁	1	1	yes	characteristic
33	C₄⋊Dic₅	—	11	2	no	—
34	C₄⋊Dic₅	—	11	1	no	—
35	#683	—	11	2	no	—
36	C₄×Dic₅	—	11	1	no	—
37	C₅×C₁₁	#125	1	1	yes	characteristic, complemented, Hall
38	C₂²×C₁₁	C₂²×D₅	1	1	yes	characteristic
39	C₄×C₁₁	—	2	2	no	—
40	C₄×C₁₁	Dic₁₀	1	2	yes	—
41	Dic₁₁	—	10	1	no	—
42	Dic₁₁	—	10	1	no	—
43	Dic₁₁	—	10	2	no	—
44	C₂×C₄×C₅	C₂×D₁₁	1	2	yes	—
45	C₂×C₄×C₅	C₂×D₁₁	1	1	yes	characteristic
46	C₂×Dic₅	—	11	1	no	—
47	C₂×Dic₅	—	11	1	no	—
48	C₂×Dic₅	—	11	2	no	—
49	#125	—	55	1	no	Sylow
50	C₂×C₁₁	C₂×Dic₁₀	1	1	yes	characteristic
51	C₂×C₁₁	#711	1	1	yes	characteristic
52	C₂×C₁₁	#713	1	1	yes	characteristic
53	C₂²×C₅	C₂²×D₁₁	1	1	yes	characteristic
54	C₄×C₅	—	2	2	no	—
55	C₄×C₅	Dic₂₂	1	2	yes	—
56	Dic₅	—	22	1	no	—
57	Dic₅	—	22	1	no	—
58	Dic₅	—	22	2	no	—
59	C₄⋊C₄	—	55	2	no	—
60	C₄⋊C₄	—	55	2	no	—
61	C₄⋊C₄	—	55	1	no	—
62	C₄⋊C₄	D₅₅	1	1	yes	characteristic
63	C₄²	—	55	1	no	—
64	C₁₁	#4036	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₅	C₂×Dic₂₂	1	1	yes	characteristic
66	C₂×C₅	#4341	1	1	yes	characteristic
67	C₂×C₅	#4343	1	1	yes	characteristic
68	C₂×C₄	—	55	1	no	—
69	C₂×C₄	—	55	1	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	#66943	1	1	yes	characteristic, complemented, Sylow
74	C₂²	C₂²×D₅₅	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	110	1	no	—
77	C₄	—	110	1	no	—
78	C₄	—	110	2	no	—
79	C₄	Dic₁₁₀	1	2	yes	—
80	C₂	C₂×Dic₁₁₀	1	1	yes	characteristic
81	C₂	#11725033	1	1	yes	characteristic
82	C₂	#11725035	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅₅
commutator, socle, prime radical, cosocle	C₂²×C₅×C₁₁	C₂³
Fitting	C₅×C₁₁×C₄⋊C₄	C₂
hypercenter	C₄⋊C₄	D₅₅
hypocenter	C₅×C₁₁	#125
LCS	G ▷ C₂²×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
37. C₅×C₁₁	#125	semidirect
64. C₁₁	#4036	semidirect
73. C₅	#66943	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₁₁×C₄⋊C₄	2
C₄⋊Dic₅₅	2
#11724930	2
C₄×Dic₅₅	2
#66943	5
#4036	11