Subgroups — D₅₁⋊D₇₇ #30628161

Order 15708.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×D₂₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×C₁₁×D₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₁₃₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	D₁₇×D₇₇	—	3	1	no	maximal, Hall
5	C₃×C₇×C₁₁×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₁₃₀₉	D₃	1	1	yes	characteristic, complemented
7	C₁₇×D₇₇	—	3	1	no	—
8	C₇×C₁₁×D₁₇	—	3	1	no	—
9	D₃₃⋊D₁₇	—	7	1	no	maximal, Hall
10	D₂₁⋊D₁₇	—	11	1	no	maximal, Hall
11	C₇×C₁₁×C₁₇	C₂×D₃	1	1	yes	characteristic, complemented, Hall
12	C₁₇×D₃₃	—	7	1	no	—
13	C₁₁×D₅₁	D₇	1	1	yes	characteristic, complemented
14	C₃×D₁₈₇	—	7	1	no	—
15	D₃×D₇₇	—	17	1	no	maximal, Hall
16	D₁₁×D₁₇	—	21	1	no	Hall
17	C₁₇×D₂₁	—	11	1	no	—
18	C₇×D₅₁	D₁₁	1	1	yes	characteristic, complemented
19	C₃×D₁₁₉	—	11	1	no	—
20	C₃×C₁₁×C₁₇	C₂×D₇	1	1	yes	characteristic, complemented, Hall
21	D₇×D₁₇	—	33	1	no	Hall
22	D₂₃₁	D₁₇	1	1	yes	characteristic, complemented
23	D₃×C₇×C₁₁	—	17	1	no	—
24	C₃×D₇₇	—	17	1	no	—
25	D₁₈₇	—	7	1	no	—
26	C₁₁×D₁₇	—	3	1	no	—
27	C₁₇×D₁₁	—	21	1	no	—
28	C₃×C₇×C₁₇	C₂×D₁₁	1	1	yes	characteristic, complemented, Hall
29	C₂×D₇₇	—	51	1	no	Hall
30	D₁₁₉	—	11	1	no	—
31	C₇×D₁₇	—	3	1	no	—
32	D₇×C₁₇	—	33	1	no	—
33	C₃×C₇×C₁₁	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
34	D₃×D₁₇	—	77	1	no	Hall
35	C₁₁×C₁₇	D₃×D₇	1	1	yes	characteristic, complemented, Hall
36	C₂×C₇×C₁₁	—	51	1	no	—
37	D₇₇	—	3	1	no	—
38	D₇₇	—	17	1	no	—
39	D₃×D₁₁	—	119	1	no	Hall
40	C₇×C₁₇	D₃×D₁₁	1	1	yes	characteristic, complemented, Hall
41	D₅₁	D₇₇	1	1	yes	characteristic, complemented
42	C₃×D₁₇	—	77	1	no	—
43	D₃×C₁₇	—	77	1	no	—
44	D₃×D₇	—	187	1	no	Hall
45	C₇×C₁₁	D₃×D₁₇	1	1	yes	characteristic, complemented, Hall
46	C₂×D₁₇	—	231	1	no	Hall
47	D₃₃	—	7	1	no	—
48	C₃×D₁₁	—	119	1	no	—
49	D₃×C₁₁	—	17	1	no	—
50	C₃×C₁₇	C₂×D₇₇	1	1	yes	characteristic, complemented, Hall
51	C₂×D₁₁	—	357	1	no	Hall
52	D₂₁	—	11	1	no	—
53	C₃×D₇	—	187	1	no	—
54	D₃×C₇	—	17	1	no	—
55	C₂×C₁₇	—	231	1	no	—
56	D₁₇	—	3	1	no	—
57	D₁₇	—	77	1	no	—
58	C₃×C₁₁	D₇×D₁₇	1	1	yes	characteristic, complemented, Hall
59	C₂×D₇	—	561	1	no	Hall
60	C₂×C₁₁	—	51	1	no	—
61	D₁₁	—	21	1	no	—
62	D₁₁	—	119	1	no	—
63	C₃×C₇	D₁₁×D₁₇	1	1	yes	characteristic, complemented, Hall
64	C₁₇	D₃×D₇₇	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₇	—	51	1	no	—
66	D₇	—	33	1	no	—
67	D₇	—	187	1	no	—
68	C₂×D₃	—	1309	1	no	Hall
69	C₁₁	D₂₁⋊D₁₇	1	1	yes	characteristic, complemented, Sylow
70	C₇	D₃₃⋊D₁₇	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₃	—	1309	1	no	—
72	D₃	—	17	1	no	—
73	D₃	—	77	1	no	—
74	C₂²	—	3927	1	no	Sylow
75	C₃	D₁₇×D₇₇	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	1309	1	no	—
77	C₂	—	51	1	no	—
78	C₂	—	231	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₁₁×C₁₇	C₂²
LCS	G ▷ C₃×C₇×C₁₁×C₁₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₁₇×D₂₃₁	C₂	semidirect
2. C₇×C₁₁×D₅₁	C₂	semidirect
3. C₃×D₁₃₀₉	C₂	semidirect
5. C₃×C₇×C₁₁×C₁₇	C₂²	semidirect
6. D₁₃₀₉	D₃	semidirect
11. C₇×C₁₁×C₁₇	C₂×D₃	semidirect
13. C₁₁×D₅₁	D₇	semidirect
18. C₇×D₅₁	D₁₁	semidirect
20. C₃×C₁₁×C₁₇	C₂×D₇	semidirect
22. D₂₃₁	D₁₇	semidirect
28. C₃×C₇×C₁₇	C₂×D₁₁	semidirect
33. C₃×C₇×C₁₁	C₂×D₁₇	semidirect
35. C₁₁×C₁₇	D₃×D₇	semidirect
40. C₇×C₁₇	D₃×D₁₁	semidirect
41. D₅₁	D₇₇	semidirect
45. C₇×C₁₁	D₃×D₁₇	semidirect
50. C₃×C₁₇	C₂×D₇₇	semidirect
58. C₃×C₁₁	D₇×D₁₇	semidirect
63. C₃×C₇	D₁₁×D₁₇	semidirect
64. C₁₇	D₃×D₇₇	semidirect
69. C₁₁	D₂₁⋊D₁₇	semidirect
70. C₇	D₃₃⋊D₁₇	semidirect
75. C₃	D₁₇×D₇₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₇×D₂₃₁	2
C₇×C₁₁×D₅₁	2
C₃×D₁₃₀₉	2
D₁₇×D₇₇	3
D₃₃⋊D₁₇	7
D₂₁⋊D₁₇	11
D₃×D₇₇	17

Subgroups — D51⋊D77 #30628161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₁⋊D₇₇ #30628161