Subgroups — #11725007

Order 880.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₅₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₅×D₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₁×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×C₂²⋊D₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#89169	C₂	1	1	yes	maximal, characteristic, complemented
6	#89168	C₂	1	1	yes	maximal, characteristic, complemented
7	#89166	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×C₁₁	C₂²	1	1	yes	characteristic, complemented
9	C₂×D₅₅	—	2	2	no	—
10	C₂×D₅₅	C₂²	1	1	yes	characteristic
11	C₂×D₅₅	C₂²	1	1	yes	characteristic, complemented
12	C₂×D₅×C₁₁	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₅×D₁₁	C₂²	1	1	yes	characteristic, complemented
14	D₅×D₁₁	—	2	2	no	—
15	C₅×Dic₁₁	C₂²	1	1	yes	characteristic, complemented
16	C₁₁×Dic₅	C₂²	1	1	yes	characteristic, complemented
17	D₄×D₁₁	—	5	1	no	maximal, Hall
18	C₂×C₅×C₁₁	—	2	1	no	—
19	C₂×C₅×C₁₁	C₂³	1	1	yes	characteristic
20	D₅₅	—	2	1	no	—
21	D₅₅	D₄	1	2	yes	complemented
22	C₅×D₁₁	—	2	1	no	—
23	D₅×C₁₁	—	2	1	no	—
24	C₂²×D₁₁	—	5	1	no	—
25	C₂²×D₁₁	—	5	1	no	—
26	D₄×C₁₁	—	5	1	no	—
27	C₂²⋊D₁₁	—	5	1	no	—
28	C₂²⋊D₁₁	D₅	1	1	yes	characteristic, complemented
29	D₄₄	—	5	1	no	—
30	C₄×D₁₁	—	5	1	no	—
31	D₄×D₅	—	11	1	no	maximal, Hall
32	C₅×C₁₁	C₂×D₄	1	1	yes	characteristic, complemented, Hall
33	C₂²×C₁₁	—	5	1	no	—
34	C₂²×C₁₁	C₂×D₅	1	1	yes	characteristic, complemented
35	C₂×D₁₁	—	5	1	no	—
36	C₂×D₁₁	—	5	1	no	—
37	C₂×D₁₁	—	10	2	no	—
38	C₂×D₁₁	—	10	2	no	—
39	C₂×D₁₁	C₂×D₅	1	1	yes	characteristic, complemented
40	C₄×C₁₁	—	5	1	no	—
41	Dic₁₁	C₂×D₅	1	1	yes	characteristic, complemented
42	C₂²×D₅	—	11	1	no	—
43	C₂²×D₅	—	11	1	no	—
44	C₅×D₄	—	11	1	no	—
45	C₂²⋊D₅	—	11	1	no	—
46	C₂²⋊D₅	D₁₁	1	1	yes	characteristic, complemented
47	D₂₀	—	11	1	no	—
48	C₄×D₅	—	11	1	no	—
49	C₂×C₁₁	—	2	1	no	—
50	C₂×C₁₁	—	10	1	no	—
51	C₂×C₁₁	C₂²×D₅	1	1	yes	characteristic
52	D₁₁	—	2	1	no	—
53	D₁₁	—	5	2	no	—
54	D₁₁	—	10	1	no	—
55	C₂²×C₅	—	11	1	no	—
56	C₂²×C₅	C₂×D₁₁	1	1	yes	characteristic, complemented
57	C₂×D₅	—	11	1	no	—
58	C₂×D₅	—	11	1	no	—
59	C₂×D₅	—	22	2	no	—
60	C₂×D₅	—	22	2	no	—
61	C₂×D₅	C₂×D₁₁	1	1	yes	characteristic, complemented
62	C₄×C₅	—	11	1	no	—
63	Dic₅	C₂×D₁₁	1	1	yes	characteristic, complemented
64	C₂×D₄	—	55	1	no	Sylow
65	C₁₁	D₄×D₅	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₅	—	2	1	no	—
67	C₂×C₅	—	22	1	no	—
68	C₂×C₅	C₂²×D₁₁	1	1	yes	characteristic
69	D₅	—	2	1	no	—
70	D₅	—	11	2	no	—
71	D₅	—	22	1	no	—
72	C₂³	—	55	1	no	—
73	C₂³	—	55	1	no	—
74	D₄	—	5	1	no	—
75	D₄	—	11	1	no	—
76	D₄	—	55	1	no	—
77	D₄	—	55	1	no	—
78	C₂×C₄	—	55	1	no	—
79	C₅	D₄×D₁₁	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	5	1	no	—
81	C₂²	—	11	1	no	—
82	C₂²	—	55	1	no	—
83	C₂²	—	55	1	no	—
84	C₂²	—	110	2	no	—
85	C₂²	—	110	2	no	—
86	C₂²	D₅×D₁₁	1	1	yes	characteristic, complemented
87	C₄	—	5	1	no	—
88	C₄	—	11	1	no	—
89	C₂	—	2	1	no	—
90	C₂	—	10	1	no	—
91	C₂	—	22	1	no	—
92	C₂	—	55	2	no	—
93	C₂	—	110	1	no	—
94	C₂	C₂×D₅×D₁₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₅×D₁₁
commutator, socle, cosocle	C₂×C₅×C₁₁	C₂³
Fitting	C₂²×C₅×C₁₁	C₂²
hypercenter	C₂²	D₅×D₁₁
hypocenter	C₅×C₁₁	C₂×D₄
LCS	G ▷ C₂×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂²×D₅₅	C₂	semidirect
2. C₂×D₅×D₁₁	C₂	semidirect
3. C₁₁×C₂²⋊D₅	C₂	semidirect
4. C₅×C₂²⋊D₁₁	C₂	semidirect
5. #89169	C₂	semidirect
6. #89168	C₂	semidirect
7. #89166	C₂	semidirect
8. C₂²×C₅×C₁₁	C₂²	semidirect
11. C₂×D₅₅	C₂²	semidirect
12. C₂×D₅×C₁₁	C₂²	semidirect
13. C₂×C₅×D₁₁	C₂²	semidirect
15. C₅×Dic₁₁	C₂²	semidirect
16. C₁₁×Dic₅	C₂²	semidirect
21. D₅₅	D₄	semidirect
28. C₂²⋊D₁₁	D₅	semidirect
32. C₅×C₁₁	C₂×D₄	semidirect
34. C₂²×C₁₁	C₂×D₅	semidirect
39. C₂×D₁₁	C₂×D₅	semidirect
41. Dic₁₁	C₂×D₅	semidirect
46. C₂²⋊D₅	D₁₁	semidirect
56. C₂²×C₅	C₂×D₁₁	semidirect
61. C₂×D₅	C₂×D₁₁	semidirect
63. Dic₅	C₂×D₁₁	semidirect
65. C₁₁	D₄×D₅	semidirect
79. C₅	D₄×D₁₁	semidirect
86. C₂²	D₅×D₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₅₅	2
C₂×D₅×D₁₁	2
C₁₁×C₂²⋊D₅	2
C₅×C₂²⋊D₁₁	2
#89169	2
#89168	2
#89166	2
D₄×D₁₁	5
D₄×D₅	11