Subgroups — #13183206

Order 1760.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11725031	C₂	1	1	yes	maximal, characteristic
2	C₅×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₁⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#11724923	C₂	1	2	yes	maximal, complemented
5	C₈×D₅₅	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅×C₁₁	C₂²	1	1	yes	characteristic
7	C₂²⋊D₅₅	C₄	1	2	yes	—
8	D₂₂₀	C₄	1	1	yes	characteristic
9	C₄×D₅₅	C₂²	1	2	yes	—
10	Dic₁₁₀	C₄	1	1	yes	characteristic
11	C₅×C₈×C₁₁	C₂²	1	2	yes	—
12	C₁₁⋊#168	C₂²	1	1	yes	characteristic
13	C₁₁⋊#168	C₂²	1	1	yes	characteristic
14	#66960	—	5	1	no	maximal, Hall
15	C₂²×C₅×C₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂×D₅₅	C₂×C₄	1	2	yes	—
17	C₄×C₅×C₁₁	C₂×C₄	1	1	yes	characteristic
18	C₄×C₅×C₁₁	C₂³	1	1	yes	characteristic
19	Dic₅₅	C₂×C₄	1	2	yes	—
20	#4339	—	5	1	no	—
21	C₁₁×M₁₆	D₅	1	1	yes	characteristic, complemented
22	C₂×C₁₁⋊C₈	—	5	1	no	—
23	#4313	—	5	2	no	—
24	C₈×D₁₁	—	5	2	no	—
25	#4053	—	11	1	no	maximal, Hall
26	C₂×C₅×C₁₁	—	2	1	no	—
27	C₂×C₅×C₁₁	C₂²×C₄	1	1	yes	characteristic
28	D₅₅	—	2	2	no	—
29	C₂×C₄×C₁₁	C₂×D₅	1	1	yes	characteristic
30	C₂²⋊D₁₁	—	5	2	no	—
31	D₄₄	—	5	1	no	—
32	C₄×D₁₁	—	5	2	no	—
33	Dic₂₂	—	5	1	no	—
34	C₈×C₁₁	C₂×D₅	1	2	yes	—
35	C₁₁⋊C₈	—	5	1	no	—
36	C₁₁⋊C₈	—	5	1	no	—
37	#709	—	11	1	no	—
38	C₅×M₁₆	D₁₁	1	1	yes	characteristic, complemented
39	C₂×#168	—	11	1	no	—
40	#676	—	11	2	no	—
41	C₈×D₅	—	11	2	no	—
42	C₅×C₁₁	#131	1	1	yes	characteristic, complemented, Hall
43	C₂²×C₁₁	C₄×D₅	1	1	yes	characteristic
44	C₂×D₁₁	—	5	2	no	—
45	C₄×C₁₁	C₄×D₅	1	1	yes	characteristic
46	C₄×C₁₁	C₂²×D₅	1	1	yes	characteristic
47	Dic₁₁	—	5	2	no	—
48	C₂×C₄×C₅	C₂×D₁₁	1	1	yes	characteristic
49	C₂²⋊D₅	—	11	2	no	—
50	D₂₀	—	11	1	no	—
51	C₄×D₅	—	11	2	no	—
52	Dic₁₀	—	11	1	no	—
53	C₅×C₈	C₂×D₁₁	1	2	yes	—
54	#168	—	11	1	no	—
55	#168	—	11	1	no	—
56	#131	—	55	1	no	Sylow
57	C₂×C₁₁	—	2	1	no	—
58	C₂×C₁₁	C₂×C₄×D₅	1	1	yes	characteristic
59	D₁₁	—	10	2	no	—
60	C₂²×C₅	C₄×D₁₁	1	1	yes	characteristic
61	C₂×D₅	—	11	2	no	—
62	C₄×C₅	C₄×D₁₁	1	1	yes	characteristic
63	C₄×C₅	C₂²×D₁₁	1	1	yes	characteristic
64	Dic₅	—	11	2	no	—
65	#41	—	55	1	no	—
66	M₁₆	—	55	2	no	—
67	M₁₆	D₅₅	1	1	yes	characteristic, complemented
68	C₂×C₈	—	55	1	no	—
69	C₂×C₈	—	55	2	no	—
70	C₁₁	#4053	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₅	—	2	1	no	—
72	C₂×C₅	C₂×C₄×D₁₁	1	1	yes	characteristic
73	D₅	—	22	2	no	—
74	Q₈	—	55	1	no	—
75	D₄	—	55	2	no	—
76	D₄	—	55	1	no	—
77	C₂×C₄	—	55	2	no	—
78	C₂×C₄	C₂×D₅₅	1	1	yes	characteristic
79	C₈	—	55	1	no	—
80	C₈	—	55	1	no	—
81	C₈	C₂×D₅₅	1	2	yes	—
82	C₅	#66960	1	1	yes	characteristic, complemented, Sylow
83	C₂²	—	55	2	no	—
84	C₂²	C₄×D₅₅	1	1	yes	characteristic
85	C₄	—	55	2	no	—
86	C₄	C₄×D₅₅	1	1	yes	characteristic
87	C₄	C₂²×D₅₅	1	1	yes	characteristic
88	C₂	—	2	1	no	—
89	C₂	—	110	2	no	—
90	C₂	C₂×C₄×D₅₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×D₅₅
commutator, socle	C₂×C₅×C₁₁	C₂²×C₄
Fitting	C₅×C₁₁×M₁₆	C₂
prime radical	#11725031	C₂
cosocle	C₄×C₅×C₁₁	C₂³
hypercenter	M₁₆	D₅₅
hypocenter	C₅×C₁₁	#131
LCS	G ▷ C₂×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. C₅×C₁₁×M₁₆	C₂	semidirect
3. C₂×C₁₁⋊#168	C₂	semidirect
4. #11724923	C₂	semidirect
5. C₈×D₅₅	C₂	semidirect
21. C₁₁×M₁₆	D₅	semidirect
38. C₅×M₁₆	D₁₁	semidirect
42. C₅×C₁₁	#131	semidirect
67. M₁₆	D₅₅	semidirect
70. C₁₁	#4053	semidirect
82. C₅	#66960	semidirect

Maximal subgroups

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

subgroup	index
#11725031	2
C₅×C₁₁×M₁₆	2
C₂×C₁₁⋊#168	2
#11724923	2
C₈×D₅₅	2
#66960	5
#4053	11