Subgroups — D₅₅⋊D₁₁₁ #37567627

Order 24420.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×D₅₅₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×D₁₂₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃₇×D₅₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅₅⋊D₃₇	—	3	1	no	maximal, Hall
5	C₃×C₅×C₁₁×C₃₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₁₁×D₁₁₁	—	5	1	no	maximal, Hall
7	C₃₇×D₅₅	D₃	1	1	yes	characteristic, complemented
8	C₁₁×D₁₈₅	—	3	1	no	—
9	C₅×D₄₀₇	—	3	1	no	—
10	D₁₂₂₁	D₅	1	1	yes	characteristic, complemented
11	C₁₁×D₁₁₁	—	5	1	no	—
12	C₃×D₁₁×C₃₇	—	5	1	no	—
13	D₅×D₁₁₁	—	11	1	no	maximal, Hall
14	C₅×C₁₁×C₃₇	C₂×D₃	1	1	yes	characteristic, complemented, Hall
15	D₁₁×D₃₇	—	15	1	no	Hall
16	C₃×C₁₁×C₃₇	C₂×D₅	1	1	yes	characteristic, complemented, Hall
17	D₅₅₅	D₁₁	1	1	yes	characteristic, complemented
18	C₅×D₁₁₁	—	11	1	no	—
19	C₃×D₅×C₃₇	—	11	1	no	—
20	D₄₀₇	—	3	1	no	—
21	C₁₁×D₃₇	—	15	1	no	—
22	D₁₁×C₃₇	—	5	1	no	—
23	D₅×D₃₇	—	33	1	no	Hall
24	D₁₅⋊D₁₁	—	37	1	no	maximal, Hall
25	C₃×C₅×C₃₇	C₂×D₁₁	1	1	yes	characteristic, complemented, Hall
26	C₂×D₁₁₁	—	55	1	no	Hall
27	C₁₁×C₃₇	D₃×D₅	1	1	yes	characteristic, complemented, Hall
28	D₁₈₅	—	3	1	no	—
29	C₅×D₃₇	—	33	1	no	—
30	D₅×C₃₇	—	11	1	no	—
31	C₁₁×D₁₅	—	37	1	no	—
32	C₅×D₃₃	—	37	1	no	—
33	C₃×D₅₅	D₃₇	1	1	yes	characteristic, complemented
34	C₂×C₃×C₃₇	—	55	1	no	—
35	D₁₁₁	—	5	1	no	—
36	D₁₁₁	—	11	1	no	—
37	D₅×D₁₁	—	111	1	no	Hall
38	C₅×C₃₇	D₃×D₁₁	1	1	yes	characteristic, complemented, Hall
39	C₃×C₅×C₁₁	C₂×D₃₇	1	1	yes	characteristic, complemented, Hall
40	C₂×D₃₇	—	165	1	no	Hall
41	D₃×D₁₁	—	185	1	no	Hall
42	C₃×C₃₇	D₅×D₁₁	1	1	yes	characteristic, complemented, Hall
43	D₅₅	D₁₁₁	1	1	yes	characteristic, complemented
44	C₅×D₁₁	—	111	1	no	—
45	D₅×C₁₁	—	111	1	no	—
46	C₂×C₃₇	—	55	1	no	—
47	D₃₇	—	15	1	no	—
48	D₃₇	—	33	1	no	—
49	D₃₃	—	37	1	no	—
50	C₃×D₁₁	—	5	1	no	—
51	D₃×C₁₁	—	185	1	no	—
52	D₃×D₅	—	407	1	no	Hall
53	C₅×C₁₁	C₂×D₁₁₁	1	1	yes	characteristic, complemented, Hall
54	C₂×D₁₁	—	555	1	no	Hall
55	C₃₇	D₁₅⋊D₁₁	1	1	yes	characteristic, complemented, Sylow
56	C₃×C₁₁	D₅×D₃₇	1	1	yes	characteristic, complemented, Hall
57	D₁₅	—	37	1	no	—
58	C₃×D₅	—	11	1	no	—
59	C₅×D₃	—	407	1	no	—
60	C₂×C₁₁	—	555	1	no	—
61	D₁₁	—	5	1	no	—
62	D₁₁	—	111	1	no	—
63	C₂×D₅	—	1221	1	no	Hall
64	C₃×C₅	D₁₁×D₃₇	1	1	yes	characteristic, complemented, Hall
65	C₂×D₃	—	2035	1	no	Hall
66	C₁₁	D₅×D₁₁₁	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₅	—	1221	1	no	—
68	D₅	—	11	1	no	—
69	D₅	—	111	1	no	—
70	C₂×C₃	—	55	1	no	—
71	D₃	—	407	1	no	—
72	D₃	—	185	1	no	—
73	C₅	D₁₁×D₁₁₁	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	6105	1	no	Sylow
75	C₃	D₅₅⋊D₃₇	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	555	1	no	—
77	C₂	—	55	1	no	—
78	C₂	—	1221	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₁₁×C₃₇	C₂²
LCS	G ▷ C₃×C₅×C₁₁×C₃₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₁₁×D₅₅₅	C₂	semidirect
2. C₅×D₁₂₂₁	C₂	semidirect
3. C₃×C₃₇×D₅₅	C₂	semidirect
5. C₃×C₅×C₁₁×C₃₇	C₂²	semidirect
7. C₃₇×D₅₅	D₃	semidirect
10. D₁₂₂₁	D₅	semidirect
14. C₅×C₁₁×C₃₇	C₂×D₃	semidirect
16. C₃×C₁₁×C₃₇	C₂×D₅	semidirect
17. D₅₅₅	D₁₁	semidirect
25. C₃×C₅×C₃₇	C₂×D₁₁	semidirect
27. C₁₁×C₃₇	D₃×D₅	semidirect
33. C₃×D₅₅	D₃₇	semidirect
38. C₅×C₃₇	D₃×D₁₁	semidirect
39. C₃×C₅×C₁₁	C₂×D₃₇	semidirect
42. C₃×C₃₇	D₅×D₁₁	semidirect
43. D₅₅	D₁₁₁	semidirect
53. C₅×C₁₁	C₂×D₁₁₁	semidirect
55. C₃₇	D₁₅⋊D₁₁	semidirect
56. C₃×C₁₁	D₅×D₃₇	semidirect
64. C₃×C₅	D₁₁×D₃₇	semidirect
66. C₁₁	D₅×D₁₁₁	semidirect
73. C₅	D₁₁×D₁₁₁	semidirect
75. C₃	D₅₅⋊D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₁×D₅₅₅	2
C₅×D₁₂₂₁	2
C₃×C₃₇×D₅₅	2
D₅₅⋊D₃₇	3
D₁₁×D₁₁₁	5
D₅×D₁₁₁	11
D₁₅⋊D₁₁	37

Subgroups — D55⋊D111 #37567627

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₅⋊D₁₁₁ #37567627