Subgroups — ΓL(1,𝔽₈)×C₂₃×A₅ #44394686

Order 28980.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₂₃×A₅	C₃	1	1	yes	maximal, characteristic, complemented
2	A₄×ΓL(1,𝔽₈)×C₂₃	—	5	1	no	maximal, Hall
3	D₅×ΓL(1,𝔽₈)×C₂₃	—	6	1	no	maximal
4	C₃×C₂₃×A₅	—	7	1	no	maximal, Hall
5	D₃×ΓL(1,𝔽₈)×C₂₃	—	10	1	no	maximal
6	C₅×ΓL(1,𝔽₈)×C₂₃	—	6	1	no	—
7	C₂₃×C₂²⋊ΓL(1,𝔽₈)	—	5	2	no	—
8	C₇×A₄×C₂₃	—	5	1	no	—
9	C₂²×ΓL(1,𝔽₈)×C₂₃	—	5	1	no	—
10	C₇×D₅×C₂₃	—	6	1	no	—
11	C₃×ΓL(1,𝔽₈)×C₂₃	—	10	1	no	Hall
12	C₂₃×A₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
13	ΓL(1,𝔽₈)×A₅	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
14	D₃×C₇×C₂₃	—	10	1	no	—
15	C₂×ΓL(1,𝔽₈)×C₂₃	—	15	1	no	—
16	C₃×A₄×C₂₃	—	35	1	no	Hall
17	C₅×C₇×C₂₃	—	6	1	no	Hall
18	C₃×D₅×C₂₃	—	42	1	no	—
19	C₂²×C₇×C₂₃	—	5	1	no	Hall
20	C₃×C₇×C₂₃	—	10	1	no	—
21	ΓL(1,𝔽₈)×C₂₃	—	20	1	no	—
22	ΓL(1,𝔽₈)×C₂₃	A₅	1	1	yes	maximal normal, characteristic, direct product factor
23	C₇×A₅	C₃×C₂₃	1	1	yes	characteristic, complemented
24	C₃×D₃×C₂₃	—	70	1	no	—
25	C₃×C₅×C₂₃	—	42	1	no	—
26	C₂×C₇×C₂₃	—	15	1	no	—
27	C₂²×C₃×C₂₃	—	35	1	no	—
28	A₄×C₂₃	—	5	1	no	—
29	A₄×C₂₃	—	35	2	no	—
30	A₄×ΓL(1,𝔽₈)	—	5	1	no	Hall
31	D₅×C₂₃	—	6	1	no	—
32	D₅×ΓL(1,𝔽₈)	—	6	1	no	—
33	C₃²×C₂₃	—	70	1	no	Hall
34	C₃×A₅	—	7	1	no	Hall
35	C₇×C₂₃	C₃×A₅	1	1	yes	characteristic, complemented, Hall
36	C₂×C₃×C₂₃	—	105	1	no	—
37	D₃×C₂₃	—	10	1	no	—
38	D₃×ΓL(1,𝔽₈)	—	10	1	no	—
39	C₅×C₂₃	—	6	1	no	Hall
40	C₅×ΓL(1,𝔽₈)	—	6	1	no	—
41	C₂²×C₂₃	—	5	1	no	Hall
42	C₂²⋊ΓL(1,𝔽₈)	—	5	2	no	—
43	C₇×A₄	—	5	1	no	—
44	C₂²×ΓL(1,𝔽₈)	—	5	1	no	—
45	C₇×D₅	—	6	1	no	—
46	C₃×C₂₃	—	7	1	no	—
47	C₃×C₂₃	—	10	1	no	—
48	C₃×C₂₃	—	140	1	no	—
49	C₃×ΓL(1,𝔽₈)	—	10	1	no	Hall
50	A₅	ΓL(1,𝔽₈)×C₂₃	1	1	yes	characteristic, direct product factor
51	C₂×C₂₃	—	15	1	no	—
52	D₃×C₇	—	10	1	no	—
53	C₂×ΓL(1,𝔽₈)	—	15	1	no	—
54	C₃×A₄	—	35	1	no	Hall
55	C₅×C₇	—	6	1	no	Hall
56	C₃×D₅	—	42	1	no	—
57	C₂²×C₇	—	5	1	no	Hall
58	C₂₃	ΓL(1,𝔽₈)×A₅	1	1	yes	characteristic, direct product factor, Sylow
59	C₃×C₇	—	10	1	no	—
60	ΓL(1,𝔽₈)	—	20	1	no	—
61	ΓL(1,𝔽₈)	C₂₃×A₅	1	1	yes	characteristic, direct product factor
62	C₃×D₃	—	70	1	no	—
63	C₃×C₅	—	42	1	no	—
64	C₂×C₇	—	15	1	no	—
65	C₂²×C₃	—	35	1	no	—
66	A₄	—	5	1	no	—
67	A₄	—	35	2	no	—
68	D₅	—	6	1	no	—
69	C₃²	—	70	1	no	Sylow
70	C₇	C₃×C₂₃×A₅	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₃	—	105	1	no	—
72	D₃	—	10	1	no	—
73	C₅	—	6	1	no	Sylow
74	C₂²	—	5	1	no	Sylow
75	C₃	—	7	1	no	—
76	C₃	—	10	1	no	—
77	C₃	—	140	1	no	—
78	C₂	—	15	1	no	—

	subgroup	G/H
center, hypercenter	C₂₃	ΓL(1,𝔽₈)×A₅
commutator, hypocenter	C₇×A₅	C₃×C₂₃
Frattini	C₁	G
Fitting	C₇×C₂₃	C₃×A₅
socle	C₇×C₂₃×A₅	C₃
cosocle	C₇	C₃×C₂₃×A₅
LCS	G ▷ C₇×A₅
UCS	C₁ ◁ C₂₃

Complements

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

N	H ≅ G/N	type
1. C₇×C₂₃×A₅	C₃	semidirect
12. C₂₃×A₅	ΓL(1,𝔽₈)	direct (N × H)
13. ΓL(1,𝔽₈)×A₅	C₂₃	direct (N × H)
22. ΓL(1,𝔽₈)×C₂₃	A₅	direct (N × H)
23. C₇×A₅	C₃×C₂₃	semidirect
35. C₇×C₂₃	C₃×A₅	semidirect
50. A₅	ΓL(1,𝔽₈)×C₂₃	direct (N × H)
58. C₂₃	ΓL(1,𝔽₈)×A₅	direct (N × H)
61. ΓL(1,𝔽₈)	C₂₃×A₅	direct (N × H)
70. C₇	C₃×C₂₃×A₅	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₂₃×A₅	3
A₄×ΓL(1,𝔽₈)×C₂₃	5
D₅×ΓL(1,𝔽₈)×C₂₃	6
C₃×C₂₃×A₅	7
D₃×ΓL(1,𝔽₈)×C₂₃	10
ΓL(1,𝔽₈)×A₅	23

Subgroups — ΓL(1,𝔽8)×C23×A5 #44394686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×C₂₃×A₅ #44394686