Subgroups — #13183234

Order 1760.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×D₄×C₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊Dic₅₅	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₅₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₅×C₁₁	C₂²	1	2	yes	—
6	C₅×D₄×C₁₁	—	2	2	no	—
7	C₂×C₄×C₅×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×Dic₅₅	C₂²	1	2	yes	—
9	C₂×Dic₅₅	C₂²	1	2	yes	—
10	Dic₁₁₀	—	2	2	no	—
11	#66989	—	5	1	no	maximal, Hall
12	C₂²×C₅×C₁₁	—	2	2	no	—
13	C₂²×C₅×C₁₁	—	2	4	no	—
14	C₂²×C₅×C₁₁	C₂³	1	1	yes	characteristic
15	C₄×C₅×C₁₁	D₄	1	2	yes	—
16	Dic₅₅	—	2	2	no	—
17	Dic₅₅	—	2	2	no	—
18	C₂×D₄×C₁₁	D₅	1	1	yes	characteristic
19	C₂×Dic₂₂	—	5	1	no	—
20	C₂²⋊Dic₁₁	—	5	4	no	—
21	C₄×Dic₁₁	—	5	1	no	—
22	#4082	—	11	1	no	maximal, Hall
23	C₂×C₅×C₁₁	—	4	2	no	—
24	C₂×C₅×C₁₁	C₂×D₄	1	1	yes	characteristic
25	C₂×C₅×C₁₁	#41	1	2	yes	—
26	C₂³×C₁₁	C₂×D₅	1	2	yes	—
27	D₄×C₁₁	—	2	2	no	—
28	C₂×C₄×C₁₁	C₂×D₅	1	1	yes	characteristic
29	C₂×Dic₁₁	—	5	2	no	—
30	C₂×Dic₁₁	—	5	2	no	—
31	Dic₂₂	—	10	2	no	—
32	C₂×C₅×D₄	D₁₁	1	1	yes	characteristic
33	C₂×Dic₁₀	—	11	1	no	—
34	C₂²⋊Dic₅	—	11	4	no	—
35	C₄×Dic₅	—	11	1	no	—
36	C₅×C₁₁	#124	1	1	yes	characteristic, complemented, Hall
37	C₂²×C₁₁	—	2	4	no	—
38	C₂²×C₁₁	—	2	2	no	—
39	C₂²×C₁₁	C₂²×D₅	1	1	yes	characteristic
40	C₄×C₁₁	C₂²⋊D₅	1	2	yes	—
41	Dic₁₁	—	10	2	no	—
42	Dic₁₁	—	10	2	no	—
43	C₂³×C₅	C₂×D₁₁	1	2	yes	—
44	C₅×D₄	—	2	2	no	—
45	C₂×C₄×C₅	C₂×D₁₁	1	1	yes	characteristic
46	C₂×Dic₅	—	11	2	no	—
47	C₂×Dic₅	—	11	2	no	—
48	Dic₁₀	—	22	2	no	—
49	#124	—	55	1	no	Sylow
50	C₂×C₁₁	—	4	2	no	—
51	C₂×C₁₁	#711	1	2	yes	—
52	C₂×C₁₁	C₂×C₂²⋊D₅	1	1	yes	characteristic
53	C₂²×C₅	—	2	4	no	—
54	C₂²×C₅	—	2	2	no	—
55	C₂²×C₅	C₂²×D₁₁	1	1	yes	characteristic
56	C₄×C₅	C₂²⋊D₁₁	1	2	yes	—
57	Dic₅	—	22	2	no	—
58	Dic₅	—	22	2	no	—
59	C₂×Q₈	—	55	1	no	—
60	C₂×D₄	D₅₅	1	1	yes	characteristic
61	C₂²⋊C₄	—	55	4	no	—
62	C₄²	—	55	1	no	—
63	C₁₁	#4082	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₅	—	4	2	no	—
65	C₂×C₅	#4341	1	2	yes	—
66	C₂×C₅	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
67	C₂³	C₂×D₅₅	1	2	yes	—
68	Q₈	—	110	2	no	—
69	D₄	—	2	2	no	—
70	C₂×C₄	—	55	2	no	—
71	C₂×C₄	—	55	2	no	—
72	C₂×C₄	C₂×D₅₅	1	1	yes	characteristic
73	C₅	#66989	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	2	4	no	—
75	C₂²	—	2	2	no	—
76	C₂²	C₂²×D₅₅	1	1	yes	characteristic
77	C₄	—	110	2	no	—
78	C₄	—	110	2	no	—
79	C₄	C₂²⋊D₅₅	1	2	yes	—
80	C₂	—	4	2	no	—
81	C₂	#11725033	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, prime radical	C₂×C₅×D₄×C₁₁	C₂
hypercenter	C₂×D₄	D₅₅
hypocenter	C₅×C₁₁	#124
LCS	G ▷ C₂²×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₁₀	C₂	semidirect
3. C₂²⋊Dic₅₅	C₂	semidirect
4. C₄×Dic₅₅	C₂	semidirect
36. C₅×C₁₁	#124	semidirect
63. C₁₁	#4082	semidirect
73. C₅	#66989	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×D₄×C₁₁	2
C₂×Dic₁₁₀	2
C₂²⋊Dic₅₅	2
C₄×Dic₅₅	2
#66989	5
#4082	11