Subgroups — D₅₅⋊D₁₁₅ #38760799

Order 25300.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₂₃×D₅₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×D₁₂₆₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₁×C₅⋊D₁₁₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅²×C₁₁×C₂₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	D₅₅⋊D₂₃	—	5	1	no	maximal
6	D₁₁×D₁₁₅	—	5	1	no	maximal
7	D₁₂₆₅	D₅	1	1	yes	characteristic, complemented
8	C₂₃×D₅₅	D₅	1	1	yes	characteristic, complemented
9	C₁₁×D₁₁₅	—	5	1	no	—
10	C₁₁×D₁₁₅	—	5	1	no	—
11	C₁₁×D₁₁₅	—	10	2	no	—
12	C₅×D₂₅₃	—	5	1	no	—
13	C₅×D₁₁×C₂₃	—	5	1	no	—
14	D₅×D₁₁₅	—	11	1	no	maximal, Hall
15	C₅×C₁₁×C₂₃	—	2	2	no	—
16	C₅×C₁₁×C₂₃	C₂×D₅	1	1	yes	characteristic, complemented
17	C₅×C₁₁×C₂₃	C₂×D₅	1	1	yes	characteristic, complemented
18	C₅⋊D₁₁₅	D₁₁	1	1	yes	characteristic, complemented
19	C₅×D₁₁₅	—	11	1	no	—
20	C₅×D₅×C₂₃	—	11	1	no	—
21	(C₅⋊D₅)⋊D₁₁	—	23	1	no	maximal, Hall
22	D₁₁×D₂₃	—	25	1	no	Hall
23	C₅²×C₂₃	C₂×D₁₁	1	1	yes	characteristic, complemented, Hall
24	C₁₁×C₅⋊D₅	—	23	1	no	—
25	C₅×D₅₅	—	23	1	no	—
26	C₅×D₅₅	D₂₃	1	1	yes	characteristic, complemented
27	D₂₅₃	—	5	1	no	—
28	C₁₁×D₂₃	—	25	1	no	—
29	D₁₁×C₂₃	—	5	1	no	—
30	C₂×D₁₁₅	—	55	1	no	—
31	D₅×D₂₃	—	55	1	no	—
32	C₅²×C₁₁	C₂×D₂₃	1	1	yes	characteristic, complemented, Hall
33	C₁₁×C₂₃	D₅²	1	1	yes	characteristic, complemented, Hall
34	C₂×C₅×C₂₃	—	55	1	no	—
35	D₁₁₅	—	5	1	no	—
36	D₁₁₅	—	5	1	no	—
37	D₁₁₅	—	10	2	no	—
38	D₁₁₅	—	11	1	no	—
39	C₅×D₂₃	—	55	1	no	—
40	D₅×C₂₃	—	11	1	no	—
41	D₅×D₁₁	—	115	1	no	—
42	D₅×D₁₁	—	115	1	no	—
43	C₅×C₂₃	—	2	2	no	—
44	C₅×C₂₃	D₅×D₁₁	1	1	yes	characteristic, complemented
45	C₅×C₂₃	D₅×D₁₁	1	1	yes	characteristic, complemented
46	D₅₅	—	23	1	no	—
47	D₅₅	D₁₁₅	1	1	yes	characteristic, complemented
48	C₅×D₁₁	—	5	1	no	—
49	C₅×D₁₁	—	115	1	no	—
50	D₅×C₁₁	—	115	1	no	—
51	D₅×C₁₁	—	115	1	no	—
52	D₅×C₁₁	—	230	2	no	—
53	D₅²	—	253	1	no	Hall
54	C₂×D₂₃	—	275	1	no	Hall
55	C₅×C₁₁	—	2	2	no	—
56	C₅×C₁₁	D₅×D₂₃	1	1	yes	characteristic, complemented
57	C₅×C₁₁	C₂×D₁₁₅	1	1	yes	characteristic, complemented
58	C₅⋊D₅	—	23	1	no	—
59	C₅×D₅	—	11	1	no	—
60	C₅×D₅	—	253	1	no	—
61	C₂×C₂₃	—	55	1	no	—
62	D₂₃	—	25	1	no	—
63	D₂₃	—	55	1	no	—
64	C₂×D₁₁	—	575	1	no	Hall
65	C₅²	D₁₁×D₂₃	1	1	yes	characteristic, complemented, Sylow
66	C₂₃	(C₅⋊D₅)⋊D₁₁	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₁₁	—	575	1	no	—
68	D₁₁	—	5	1	no	—
69	D₁₁	—	115	1	no	—
70	C₂×D₅	—	1265	1	no	—
71	C₂×D₅	—	1265	1	no	—
72	C₁₁	D₅×D₁₁₅	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₅	—	55	1	no	—
74	C₂×C₅	—	1265	1	no	—
75	D₅	—	11	1	no	—
76	D₅	—	115	1	no	—
77	D₅	—	115	1	no	—
78	D₅	—	230	2	no	—
79	D₅	—	253	1	no	—
80	C₅	—	2	2	no	—
81	C₅	D₁₁×D₁₁₅	1	1	yes	characteristic, complemented
82	C₅	D₅₅⋊D₂₃	1	1	yes	characteristic, complemented
83	C₂²	—	6325	1	no	Sylow
84	C₂	—	55	1	no	—
85	C₂	—	575	1	no	—
86	C₂	—	1265	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅²×C₁₁×C₂₃	C₂²
LCS	G ▷ C₅²×C₁₁×C₂₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₅×C₂₃×D₅₅	C₂	semidirect
2. C₅×D₁₂₆₅	C₂	semidirect
3. C₁₁×C₅⋊D₁₁₅	C₂	semidirect
4. C₅²×C₁₁×C₂₃	C₂²	semidirect
7. D₁₂₆₅	D₅	semidirect
8. C₂₃×D₅₅	D₅	semidirect
16. C₅×C₁₁×C₂₃	C₂×D₅	semidirect
17. C₅×C₁₁×C₂₃	C₂×D₅	semidirect
18. C₅⋊D₁₁₅	D₁₁	semidirect
23. C₅²×C₂₃	C₂×D₁₁	semidirect
26. C₅×D₅₅	D₂₃	semidirect
32. C₅²×C₁₁	C₂×D₂₃	semidirect
33. C₁₁×C₂₃	D₅²	semidirect
44. C₅×C₂₃	D₅×D₁₁	semidirect
45. C₅×C₂₃	D₅×D₁₁	semidirect
47. D₅₅	D₁₁₅	semidirect
56. C₅×C₁₁	D₅×D₂₃	semidirect
57. C₅×C₁₁	C₂×D₁₁₅	semidirect
65. C₅²	D₁₁×D₂₃	semidirect
66. C₂₃	(C₅⋊D₅)⋊D₁₁	semidirect
72. C₁₁	D₅×D₁₁₅	semidirect
81. C₅	D₁₁×D₁₁₅	semidirect
82. C₅	D₅₅⋊D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₂₃×D₅₅	2
C₅×D₁₂₆₅	2
C₁₁×C₅⋊D₁₁₅	2
D₅₅⋊D₂₃	5
D₁₁×D₁₁₅	5
D₅×D₁₁₅	11
(C₅⋊D₅)⋊D₁₁	23

Subgroups — D55⋊D115 #38760799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₅⋊D₁₁₅ #38760799