Subgroups — D₇×ΓL(1,𝔽₈)×D₃₁ #32912182

Order 18228.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₂₁₇	C₂	1	1	yes	maximal, characteristic, complemented
2	D₇×ΓL(1,𝔽₈)×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×ΓL(1,𝔽₈)×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×D₇×D₃₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₇×ΓL(1,𝔽₈)×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇×D₂₁₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×D₇×C₃₁	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇²×D₃₁	C₂×C₃	1	1	yes	characteristic, complemented
9	C₃×D₇×D₃₁	—	7	1	no	maximal
10	C₂×ΓL(1,𝔽₈)×D₃₁	—	7	1	no	maximal
11	C₇²×C₃₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	C₃×D₂₁₇	—	7	1	no	—
13	C₃×D₇×C₃₁	—	7	1	no	—
14	C₃×C₇×D₃₁	—	7	1	no	—
15	C₂×ΓL(1,𝔽₈)×C₃₁	—	7	1	no	—
16	ΓL(1,𝔽₈)×D₃₁	—	7	1	no	—
17	ΓL(1,𝔽₈)×D₃₁	D₇	1	1	yes	characteristic, direct product factor
18	C₂×C₇×D₃₁	—	7	1	no	—
19	D₇×D₃₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
20	C₃×C₇×C₃₁	—	7	1	no	—
21	ΓL(1,𝔽₈)×C₃₁	C₂×D₇	1	1	yes	characteristic, complemented
22	C₂×D₇×ΓL(1,𝔽₈)	—	31	1	no	maximal, Hall
23	C₂×C₇×C₃₁	—	7	1	no	—
24	D₂₁₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
25	C₇×D₃₁	—	6	1	no	—
26	C₇×D₃₁	—	7	1	no	—
27	C₇×D₃₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
28	C₇×D₃₁	C₃×D₇	1	1	yes	characteristic, complemented
29	D₇×C₃₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
30	C₂×C₃×D₃₁	—	49	1	no	Hall
31	C₂×C₇×ΓL(1,𝔽₈)	—	31	1	no	—
32	D₇×ΓL(1,𝔽₈)	—	31	1	no	—
33	D₇×ΓL(1,𝔽₈)	D₃₁	1	1	yes	characteristic, direct product factor
34	C₇×C₃₁	—	6	1	no	—
35	C₇×C₃₁	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
36	C₇×C₃₁	C₂×C₃×D₇	1	1	yes	characteristic, complemented
37	C₂×C₇×D₇	—	31	1	no	Hall
38	C₂×C₃×C₃₁	—	49	1	no	—
39	C₃×D₃₁	—	7	1	no	—
40	C₃×D₃₁	—	49	1	no	—
41	C₇×ΓL(1,𝔽₈)	C₂×D₃₁	1	1	yes	characteristic, complemented, Hall
42	C₂×D₃₁	—	7	1	no	Hall
43	C₂×C₇²	—	31	1	no	—
44	C₇×D₇	—	31	1	no	—
45	C₇×D₇	C₃×D₃₁	1	1	yes	characteristic, complemented
46	C₃×C₃₁	—	7	1	no	Hall
47	C₂×C₃×D₇	—	217	1	no	—
48	C₂²×ΓL(1,𝔽₈)	—	217	1	no	—
49	C₂×C₃₁	—	7	1	no	—
50	D₃₁	—	7	1	no	—
51	D₃₁	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
52	C₇²	C₂×C₃×D₃₁	1	1	yes	characteristic, complemented, Sylow
53	C₂×C₃×C₇	—	217	1	no	—
54	C₃×D₇	—	7	1	no	—
55	C₃×D₇	—	217	1	no	—
56	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
57	C₂×ΓL(1,𝔽₈)	—	31	1	no	—
58	C₂×ΓL(1,𝔽₈)	—	217	1	no	—
59	C₃₁	C₂×D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₇	—	217	1	no	—
61	C₂×D₇	—	31	1	no	—
62	C₃×C₇	—	7	1	no	—
63	ΓL(1,𝔽₈)	D₇×D₃₁	1	1	yes	characteristic, direct product factor
64	C₂×C₇	—	7	1	no	—
65	C₂×C₇	—	31	1	no	—
66	C₂×C₇	—	31	1	no	—
67	C₂×C₇	—	186	1	no	—
68	C₂×C₇	—	217	1	no	—
69	D₇	—	31	1	no	—
70	D₇	ΓL(1,𝔽₈)×D₃₁	1	1	yes	characteristic, direct product factor
71	C₂²×C₃	—	1519	1	no	Hall
72	C₇	—	6	1	no	—
73	C₇	C₂×ΓL(1,𝔽₈)×D₃₁	1	1	yes	characteristic, complemented
74	C₇	C₃×D₇×D₃₁	1	1	yes	characteristic, complemented
75	C₂×C₃	—	49	1	no	—
76	C₂×C₃	—	217	1	no	—
77	C₂×C₃	—	1519	1	no	—
78	C₂²	—	217	1	no	Sylow
79	C₃	—	7	1	no	Sylow
80	C₂	—	7	1	no	—
81	C₂	—	31	1	no	—
82	C₂	—	217	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₃₁	C₂²×C₃
LCS	G ▷ C₇²×C₃₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×D₂₁₇	C₂	semidirect
2. D₇×ΓL(1,𝔽₈)×C₃₁	C₂	semidirect
3. C₇×ΓL(1,𝔽₈)×D₃₁	C₂	semidirect
4. C₇×D₇×D₃₁	C₃	semidirect
5. C₇×ΓL(1,𝔽₈)×C₃₁	C₂²	semidirect
6. C₇×D₂₁₇	C₂×C₃	semidirect
7. C₇×D₇×C₃₁	C₂×C₃	semidirect
8. C₇²×D₃₁	C₂×C₃	semidirect
11. C₇²×C₃₁	C₂²×C₃	semidirect
17. ΓL(1,𝔽₈)×D₃₁	D₇	direct (N × H)
19. D₇×D₃₁	ΓL(1,𝔽₈)	direct (N × H)
21. ΓL(1,𝔽₈)×C₃₁	C₂×D₇	semidirect
24. D₂₁₇	C₂×ΓL(1,𝔽₈)	semidirect
27. C₇×D₃₁	C₂×ΓL(1,𝔽₈)	semidirect
28. C₇×D₃₁	C₃×D₇	semidirect
29. D₇×C₃₁	C₂×ΓL(1,𝔽₈)	semidirect
33. D₇×ΓL(1,𝔽₈)	D₃₁	direct (N × H)
35. C₇×C₃₁	C₂²×ΓL(1,𝔽₈)	semidirect
36. C₇×C₃₁	C₂×C₃×D₇	semidirect
41. C₇×ΓL(1,𝔽₈)	C₂×D₃₁	semidirect
45. C₇×D₇	C₃×D₃₁	semidirect
51. D₃₁	D₇×ΓL(1,𝔽₈)	direct (N × H)
52. C₇²	C₂×C₃×D₃₁	semidirect
59. C₃₁	C₂×D₇×ΓL(1,𝔽₈)	semidirect
63. ΓL(1,𝔽₈)	D₇×D₃₁	direct (N × H)
70. D₇	ΓL(1,𝔽₈)×D₃₁	direct (N × H)
73. C₇	C₂×ΓL(1,𝔽₈)×D₃₁	semidirect
74. C₇	C₃×D₇×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×D₂₁₇	2
D₇×ΓL(1,𝔽₈)×C₃₁	2
C₇×ΓL(1,𝔽₈)×D₃₁	2
C₇×D₇×D₃₁	3
C₃×D₇×D₃₁	7
C₂×ΓL(1,𝔽₈)×D₃₁	7
C₂×D₇×ΓL(1,𝔽₈)	31

Subgroups — D7×ΓL(1,𝔽8)×D31 #32912182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×ΓL(1,𝔽₈)×D₃₁ #32912182