Subgroups — #13182944

Order 1760.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊Dic₅₅	C₂	1	1	yes	maximal, characteristic
3	C₅×C₂²⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic
4	#11724889	C₂	1	2	yes	maximal, complemented
5	Dic₅×Dic₁₁	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×Dic₅₅	C₂²	1	2	yes	—
8	C₂×C₁₁×Dic₅	C₄	1	4	yes	complemented
9	C₂×C₁₁×Dic₅	C₂²	1	1	yes	characteristic
10	C₂×C₁₁×Dic₅	C₂²	1	1	yes	characteristic
11	C₂×C₅×Dic₁₁	C₂²	1	2	yes	—
12	#66979	—	5	1	no	maximal, Hall
13	C₂²×C₅×C₁₁	—	2	2	no	—
14	C₂²×C₅×C₁₁	C₂×C₄	1	2	yes	—
15	C₂²×C₅×C₁₁	C₂³	1	1	yes	characteristic
16	Dic₅₅	—	2	2	no	—
17	C₅×Dic₁₁	—	2	2	no	—
18	C₁₁×Dic₅	C₂×C₄	1	2	yes	—
19	C₁₁×Dic₅	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₁	—	5	1	no	—
21	C₂²⋊Dic₁₁	—	5	1	no	—
22	C₂²⋊Dic₁₁	D₅	1	1	yes	characteristic
23	C₄⋊Dic₁₁	—	5	2	no	—
24	C₄×Dic₁₁	—	5	2	no	—
25	#4023	—	11	1	no	maximal, Hall
26	C₂×C₅×C₁₁	—	2	2	no	—
27	C₂×C₅×C₁₁	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₅×C₁₁	#41	1	2	yes	—
29	C₂³×C₁₁	C₂×D₅	1	1	yes	characteristic
30	C₂×C₄×C₁₁	—	5	4	no	—
31	C₂×C₄×C₁₁	—	5	1	no	—
32	C₂×C₄×C₁₁	—	5	1	no	—
33	C₂×Dic₁₁	—	5	2	no	—
34	C₂×Dic₁₁	C₂×D₅	1	2	yes	—
35	C₂²×Dic₅	D₁₁	1	1	yes	characteristic
36	C₅×C₂²⋊C₄	—	11	1	no	—
37	C₂²⋊Dic₅	—	11	1	no	—
38	#683	—	11	2	no	—
39	C₄×Dic₅	—	11	2	no	—
40	C₅×C₁₁	#117	1	1	yes	characteristic, complemented, Hall
41	C₂²×C₁₁	—	2	2	no	—
42	C₂²×C₁₁	C₄×D₅	1	2	yes	—
43	C₂²×C₁₁	C₂²×D₅	1	1	yes	characteristic
44	C₄×C₁₁	—	5	2	no	—
45	C₄×C₁₁	—	5	2	no	—
46	Dic₁₁	—	2	2	no	—
47	Dic₁₁	—	10	2	no	—
48	C₂³×C₅	C₂×D₁₁	1	1	yes	characteristic
49	C₂×C₄×C₅	—	11	2	no	—
50	C₂×Dic₅	—	11	2	no	—
51	C₂×Dic₅	Dic₁₁	1	4	yes	complemented
52	C₂×Dic₅	C₂×D₁₁	1	1	yes	characteristic
53	C₂×Dic₅	C₂×D₁₁	1	1	yes	characteristic
54	#117	—	55	1	no	Sylow
55	C₂×C₁₁	—	2	2	no	—
56	C₂×C₁₁	C₂×C₄×D₅	1	1	yes	characteristic
57	C₂×C₁₁	#711	1	2	yes	—
58	C₂²×C₅	—	2	2	no	—
59	C₂²×C₅	C₂×Dic₁₁	1	2	yes	—
60	C₂²×C₅	C₂²×D₁₁	1	1	yes	characteristic
61	C₄×C₅	—	22	2	no	—
62	Dic₅	—	22	2	no	—
63	Dic₅	C₂×Dic₁₁	1	2	yes	—
64	Dic₅	C₂×Dic₁₁	1	2	yes	—
65	C₂²×C₄	—	5	1	no	—
66	C₄⋊C₄	—	55	2	no	—
67	C₂²⋊C₄	—	11	1	no	—
68	C₂²⋊C₄	—	55	1	no	—
69	C₄²	—	55	2	no	—
70	C₁₁	#4023	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₅	—	2	2	no	—
72	C₂×C₅	#4339	1	2	yes	—
73	C₂×C₅	C₂²×Dic₁₁	1	1	yes	characteristic
74	C₂³	D₅×D₁₁	1	1	yes	characteristic
75	C₂×C₄	—	5	4	no	—
76	C₂×C₄	—	5	1	no	—
77	C₂×C₄	—	5	1	no	—
78	C₂×C₄	—	11	2	no	—
79	C₂×C₄	—	55	2	no	—
80	C₅	#66979	1	1	yes	characteristic, complemented, Sylow
81	C₂²	—	2	2	no	—
82	C₂²	D₅×Dic₁₁	1	2	yes	—
83	C₂²	C₂×D₅×D₁₁	1	1	yes	characteristic
84	C₄	—	5	2	no	—
85	C₄	—	5	2	no	—
86	C₄	—	22	2	no	—
87	C₄	—	110	2	no	—
88	C₂	—	2	2	no	—
89	C₂	#11724996	1	2	yes	—
90	C₂	C₂×D₅×Dic₁₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₅×D₁₁
commutator	C₂×C₅×C₁₁	C₂²×C₄
Fitting, prime radical	C₂³×C₅×C₁₁	C₂²
socle, cosocle	C₂²×C₅×C₁₁	C₂³
hypercenter	C₂³	D₅×D₁₁
hypocenter	C₅×C₁₁	#117
LCS	G ▷ C₂×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
4. #11724889	C₂	semidirect
5. Dic₅×Dic₁₁	C₂	semidirect
8. C₂×C₁₁×Dic₅	C₄	semidirect
40. C₅×C₁₁	#117	semidirect
51. C₂×Dic₅	Dic₁₁	semidirect
70. C₁₁	#4023	semidirect
80. C₅	#66979	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₁×Dic₅	2
C₂²⋊Dic₅₅	2
C₅×C₂²⋊Dic₁₁	2
#11724889	2
Dic₅×Dic₁₁	2
#66979	5
#4023	11