Subgroups — D₁₅⋊D₂₁₇ #27189261

Order 13020.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×D₁₅×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×D₆₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₁₀₈₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅×D₂₁₇	—	3	1	no	maximal, Hall
5	C₃×C₅×C₇×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₃×D₂₁₇	—	5	1	no	maximal, Hall
7	D₁₀₈₅	D₃	1	1	yes	characteristic, complemented
8	C₇×D₅×C₃₁	—	3	1	no	—
9	C₅×D₂₁₇	—	3	1	no	—
10	D₁₅⋊D₃₁	—	7	1	no	maximal, Hall
11	D₆₅₁	D₅	1	1	yes	characteristic, complemented
12	D₃×C₇×C₃₁	—	5	1	no	—
13	C₃×D₂₁₇	—	5	1	no	—
14	C₅×C₇×C₃₁	C₂×D₃	1	1	yes	characteristic, complemented, Hall
15	D₁₅×C₃₁	D₇	1	1	yes	characteristic, complemented
16	C₅×D₉₃	—	7	1	no	—
17	C₃×D₁₅₅	—	7	1	no	—
18	C₂×D₂₁₇	—	15	1	no	Hall
19	C₃×C₇×C₃₁	C₂×D₅	1	1	yes	characteristic, complemented, Hall
20	D₅×D₃₁	—	21	1	no	Hall
21	C₃×C₅×C₃₁	C₂×D₇	1	1	yes	characteristic, complemented, Hall
22	C₂×C₇×C₃₁	—	15	1	no	—
23	D₂₁₇	—	3	1	no	—
24	D₂₁₇	—	5	1	no	—
25	D₁₅⋊D₇	—	31	1	no	maximal, Hall
26	D₃×D₃₁	—	35	1	no	Hall
27	D₁₅₅	—	7	1	no	—
28	C₅×D₃₁	—	21	1	no	—
29	D₅×C₃₁	—	3	1	no	—
30	C₇×C₃₁	D₃×D₅	1	1	yes	characteristic, complemented, Hall
31	C₇×D₁₅	D₃₁	1	1	yes	characteristic, complemented
32	C₅×D₂₁	—	31	1	no	—
33	C₃×D₃₅	—	31	1	no	—
34	D₉₃	—	7	1	no	—
35	C₃×D₃₁	—	35	1	no	—
36	D₃×C₃₁	—	5	1	no	—
37	C₅×C₃₁	D₃×D₇	1	1	yes	characteristic, complemented, Hall
38	D₅×D₇	—	93	1	no	Hall
39	C₂×D₃₁	—	105	1	no	Hall
40	C₃×C₅×C₇	C₂×D₃₁	1	1	yes	characteristic, complemented, Hall
41	C₃×C₃₁	D₅×D₇	1	1	yes	characteristic, complemented, Hall
42	D₃×D₇	—	155	1	no	Hall
43	D₃₅	—	31	1	no	—
44	C₅×D₇	—	93	1	no	—
45	C₇×D₅	—	3	1	no	—
46	C₂×C₃₁	—	15	1	no	—
47	D₃₁	—	21	1	no	—
48	D₃₁	—	35	1	no	—
49	D₃×D₅	—	217	1	no	Hall
50	D₂₁	—	31	1	no	—
51	C₃×D₇	—	155	1	no	—
52	D₃×C₇	—	5	1	no	—
53	C₅×C₇	D₃×D₃₁	1	1	yes	characteristic, complemented, Hall
54	C₃₁	D₁₅⋊D₇	1	1	yes	characteristic, complemented, Sylow
55	D₁₅	D₂₁₇	1	1	yes	characteristic, complemented
56	C₃×D₅	—	217	1	no	—
57	C₅×D₃	—	217	1	no	—
58	C₂×D₇	—	465	1	no	Hall
59	C₃×C₇	D₅×D₃₁	1	1	yes	characteristic, complemented, Hall
60	C₂×D₅	—	651	1	no	Hall
61	C₃×C₅	C₂×D₂₁₇	1	1	yes	characteristic, complemented, Hall
62	C₂×C₇	—	15	1	no	—
63	D₇	—	93	1	no	—
64	D₇	—	155	1	no	—
65	C₂×D₃	—	1085	1	no	Hall
66	C₂×C₅	—	651	1	no	—
67	D₅	—	3	1	no	—
68	D₅	—	217	1	no	—
69	C₇	D₁₅⋊D₃₁	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	1085	1	no	—
71	D₃	—	5	1	no	—
72	D₃	—	217	1	no	—
73	C₅	D₃×D₂₁₇	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	3255	1	no	Sylow
75	C₃	D₅×D₂₁₇	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	15	1	no	—
77	C₂	—	1085	1	no	—
78	C₂	—	651	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₇×C₃₁	C₂²
LCS	G ▷ C₃×C₅×C₇×C₃₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₇×D₁₅×C₃₁	C₂	semidirect
2. C₅×D₆₅₁	C₂	semidirect
3. C₃×D₁₀₈₅	C₂	semidirect
5. C₃×C₅×C₇×C₃₁	C₂²	semidirect
7. D₁₀₈₅	D₃	semidirect
11. D₆₅₁	D₅	semidirect
14. C₅×C₇×C₃₁	C₂×D₃	semidirect
15. D₁₅×C₃₁	D₇	semidirect
19. C₃×C₇×C₃₁	C₂×D₅	semidirect
21. C₃×C₅×C₃₁	C₂×D₇	semidirect
30. C₇×C₃₁	D₃×D₅	semidirect
31. C₇×D₁₅	D₃₁	semidirect
37. C₅×C₃₁	D₃×D₇	semidirect
40. C₃×C₅×C₇	C₂×D₃₁	semidirect
41. C₃×C₃₁	D₅×D₇	semidirect
53. C₅×C₇	D₃×D₃₁	semidirect
54. C₃₁	D₁₅⋊D₇	semidirect
55. D₁₅	D₂₁₇	semidirect
59. C₃×C₇	D₅×D₃₁	semidirect
61. C₃×C₅	C₂×D₂₁₇	semidirect
69. C₇	D₁₅⋊D₃₁	semidirect
73. C₅	D₃×D₂₁₇	semidirect
75. C₃	D₅×D₂₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₇×D₁₅×C₃₁	2
C₅×D₆₅₁	2
C₃×D₁₀₈₅	2
D₅×D₂₁₇	3
D₃×D₂₁₇	5
D₁₅⋊D₃₁	7
D₁₅⋊D₇	31

Subgroups — D15⋊D217 #27189261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₅⋊D₂₁₇ #27189261