Subgroups — C₇×ΓL(1,𝔽₈)×D₁₅ #17984386

Order 4410.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₇×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇²×D₁₅	C₃	1	1	yes	maximal, characteristic, complemented
3	C₇×D₅×ΓL(1,𝔽₈)	—	3	1	no	maximal
4	D₃×C₇×ΓL(1,𝔽₈)	—	5	1	no	maximal, Hall
5	C₃×C₅×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
6	C₅×C₇×ΓL(1,𝔽₈)	—	2	1	no	—
7	C₅×C₇×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic, complemented
8	C₃×C₇×D₁₅	—	7	1	no	maximal
9	ΓL(1,𝔽₈)×D₁₅	C₇	1	1	yes	maximal, characteristic, direct product factor
10	C₇²×D₅	—	3	1	no	Hall
11	C₃×C₇×ΓL(1,𝔽₈)	D₅	1	1	yes	characteristic, complemented, Hall
12	C₃²×C₅×C₇	—	7	1	no	—
13	C₃×C₅×ΓL(1,𝔽₈)	C₂×C₇	1	1	yes	characteristic, complemented
14	D₃×C₇²	—	5	1	no	—
15	C₂×C₇×ΓL(1,𝔽₈)	—	15	1	no	—
16	C₅×C₇²	C₃×D₃	1	1	yes	characteristic, complemented, Hall
17	C₇×D₁₅	—	3	2	no	—
18	C₇×D₁₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
19	C₇×D₁₅	C₃×C₇	1	1	yes	characteristic, complemented
20	C₃×C₇×D₅	—	21	1	no	—
21	D₅×ΓL(1,𝔽₈)	—	3	1	no	—
22	C₃×C₇²	C₃×D₅	1	1	yes	characteristic, complemented
23	C₇×ΓL(1,𝔽₈)	—	2	1	no	—
24	C₇×ΓL(1,𝔽₈)	D₁₅	1	1	yes	characteristic, direct product factor
25	C₃×D₃×C₇	—	35	1	no	—
26	D₃×ΓL(1,𝔽₈)	—	5	1	no	—
27	C₃×C₅×C₇	—	3	2	no	—
28	C₃×C₅×C₇	—	7	1	no	—
29	C₃×C₅×C₇	—	14	1	no	—
30	C₃×C₅×C₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
31	C₃×C₅×C₇	C₂×C₃×C₇	1	1	yes	characteristic, complemented
32	C₅×ΓL(1,𝔽₈)	—	2	1	no	—
33	C₅×ΓL(1,𝔽₈)	D₃×C₇	1	1	yes	characteristic, complemented
34	C₂×C₇²	—	15	1	no	Hall
35	C₃×D₁₅	—	7	1	no	Hall
36	C₇×D₅	—	3	1	no	—
37	C₇×D₅	—	3	1	no	—
38	C₇×D₅	—	9	2	no	—
39	C₃²×C₇	—	7	1	no	—
40	C₃×ΓL(1,𝔽₈)	C₇×D₅	1	1	yes	characteristic, complemented
41	C₇²	C₃×D₁₅	1	1	yes	characteristic, complemented, Sylow
42	C₃²×C₅	—	7	1	no	Hall
43	C₂×C₃×C₇	—	105	1	no	—
44	D₃×C₇	—	5	1	no	—
45	D₃×C₇	—	5	1	no	—
46	D₃×C₇	—	15	2	no	—
47	C₂×ΓL(1,𝔽₈)	—	15	1	no	—
48	C₅×C₇	—	3	2	no	—
49	C₅×C₇	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
50	C₅×C₇	C₃×D₃×C₇	1	1	yes	characteristic, complemented
51	D₁₅	C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
52	C₃×D₅	—	21	1	no	—
53	C₃×C₇	—	3	2	no	—
54	C₃×C₇	—	7	1	no	—
55	C₃×C₇	—	14	1	no	—
56	C₃×C₇	D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
57	C₃×C₇	C₃×C₇×D₅	1	1	yes	characteristic, complemented
58	ΓL(1,𝔽₈)	—	2	1	no	—
59	ΓL(1,𝔽₈)	C₇×D₁₅	1	1	yes	characteristic, direct product factor
60	C₃×D₃	—	35	1	no	Hall
61	C₃×C₅	—	7	1	no	—
62	C₃×C₅	—	14	1	no	—
63	C₃×C₅	C₂×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
64	C₂×C₇	—	15	1	no	—
65	C₂×C₇	—	15	1	no	—
66	C₂×C₇	—	45	2	no	—
67	D₅	—	3	1	no	Hall
68	C₃²	—	7	1	no	Sylow
69	C₇	—	3	2	no	—
70	C₇	ΓL(1,𝔽₈)×D₁₅	1	1	yes	characteristic, direct product factor
71	C₇	C₃×C₇×D₁₅	1	1	yes	characteristic, complemented
72	C₂×C₃	—	105	1	no	—
73	D₃	—	5	1	no	—
74	C₅	D₃×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
75	C₃	—	7	1	no	—
76	C₃	—	14	1	no	—
77	C₃	C₇×D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
78	C₂	—	15	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇	ΓL(1,𝔽₈)×D₁₅
commutator, cosocle, hypocenter	C₃×C₅×C₇	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇²	C₂×C₃
LCS	G ▷ C₃×C₅×C₇
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₇×ΓL(1,𝔽₈)	C₂	semidirect
2. C₇²×D₁₅	C₃	semidirect
5. C₃×C₅×C₇²	C₂×C₃	semidirect
7. C₅×C₇×ΓL(1,𝔽₈)	D₃	semidirect
9. ΓL(1,𝔽₈)×D₁₅	C₇	direct (N × H)
11. C₃×C₇×ΓL(1,𝔽₈)	D₅	semidirect
13. C₃×C₅×ΓL(1,𝔽₈)	C₂×C₇	semidirect
16. C₅×C₇²	C₃×D₃	semidirect
18. C₇×D₁₅	ΓL(1,𝔽₈)	direct (N × H)
19. C₇×D₁₅	C₃×C₇	semidirect
22. C₃×C₇²	C₃×D₅	semidirect
24. C₇×ΓL(1,𝔽₈)	D₁₅	direct (N × H)
30. C₃×C₅×C₇	C₂×ΓL(1,𝔽₈)	semidirect
31. C₃×C₅×C₇	C₂×C₃×C₇	semidirect
33. C₅×ΓL(1,𝔽₈)	D₃×C₇	semidirect
40. C₃×ΓL(1,𝔽₈)	C₇×D₅	semidirect
41. C₇²	C₃×D₁₅	semidirect
49. C₅×C₇	D₃×ΓL(1,𝔽₈)	semidirect
50. C₅×C₇	C₃×D₃×C₇	semidirect
51. D₁₅	C₇×ΓL(1,𝔽₈)	direct (N × H)
56. C₃×C₇	D₅×ΓL(1,𝔽₈)	semidirect
57. C₃×C₇	C₃×C₇×D₅	semidirect
59. ΓL(1,𝔽₈)	C₇×D₁₅	direct (N × H)
63. C₃×C₅	C₂×C₇×ΓL(1,𝔽₈)	semidirect
70. C₇	ΓL(1,𝔽₈)×D₁₅	direct (N × H)
71. C₇	C₃×C₇×D₁₅	semidirect
74. C₅	D₃×C₇×ΓL(1,𝔽₈)	semidirect
77. C₃	C₇×D₅×ΓL(1,𝔽₈)	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₇×ΓL(1,𝔽₈)	2
C₇²×D₁₅	3
C₇×D₅×ΓL(1,𝔽₈)	3
D₃×C₇×ΓL(1,𝔽₈)	5
C₃×C₇×D₁₅	7
ΓL(1,𝔽₈)×D₁₅	7

Subgroups — C7×ΓL(1,𝔽8)×D15 #17984386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×ΓL(1,𝔽₈)×D₁₅ #17984386