Subgroups — ΓL(1,𝔽₈)×C₃⋊#92150 #44479112

Order 30492.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×ΓL(1,𝔽₈)×C₁₁⋊D₁₁	C₂	1	1	yes	maximal, characteristic
2	C₇×C₃⋊#92150	C₃	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×#92150	—	3	1	no	maximal
4	C₃×C₁₁²×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×C₇×C₁₁⋊D₁₁	C₂×C₃	1	1	yes	characteristic
6	ΓL(1,𝔽₈)×C₁₁⋊D₁₁	—	2	1	no	—
7	ΓL(1,𝔽₈)×C₁₁⋊D₁₁	D₃	1	1	yes	characteristic
8	C₃×C₃⋊#92150	—	7	1	no	maximal, Hall
9	C₇×#92150	—	3	1	no	Hall
10	C₁₁²×ΓL(1,𝔽₈)	—	2	1	no	—
11	C₁₁²×ΓL(1,𝔽₈)	Dic₃	1	1	yes	characteristic, complemented
12	C₃×C₇×C₁₁²	C₃×C₄	1	1	yes	characteristic, complemented
13	C₃²×C₁₁⋊D₁₁	—	7	1	no	—
14	C₇×C₁₁⋊D₁₁	C₃×D₃	1	1	yes	characteristic
15	C₃⋊#92150	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
16	C₃×#92150	—	21	1	no	—
17	C₃×ΓL(1,𝔽₈)×D₁₁	—	22	6	no	—
18	C₃²×C₁₁²	—	7	1	no	Hall
19	C₇×C₁₁²	C₃×Dic₃	1	1	yes	characteristic, complemented, Hall
20	C₃×C₁₁⋊D₁₁	—	7	1	no	—
21	C₃×C₁₁⋊D₁₁	—	14	1	no	—
22	C₃×C₁₁⋊D₁₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
23	C₃×C₁₁×ΓL(1,𝔽₈)	—	2	6	no	—
24	#92150	—	3	1	no	Hall
25	C₃×C₇×D₁₁	—	22	6	no	—
26	ΓL(1,𝔽₈)×D₁₁	—	22	6	no	—
27	ΓL(1,𝔽₈)×D₁₁	—	22	12	no	—
28	C₃×C₁₁²	—	7	1	no	—
29	C₃×C₁₁²	—	14	1	no	—
30	C₃×C₁₁²	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
31	Dic₃×ΓL(1,𝔽₈)	—	121	1	no	maximal, Hall
32	C₁₁⋊D₁₁	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
33	C₃×C₇×C₁₁	—	2	6	no	—
34	C₁₁×ΓL(1,𝔽₈)	—	2	12	no	—
35	C₁₁×ΓL(1,𝔽₈)	—	2	6	no	—
36	C₃²×D₁₁	—	154	6	no	—
37	C₇×D₁₁	—	22	6	no	—
38	C₂×C₃×ΓL(1,𝔽₈)	—	121	1	no	—
39	C₁₁²	Dic₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
40	C₃²×C₁₁	—	14	6	no	—
41	C₇×Dic₃	—	121	1	no	—
42	C₄×ΓL(1,𝔽₈)	—	363	1	no	—
43	C₇×C₁₁	—	2	6	no	—
44	C₃×D₁₁	—	22	6	no	—
45	C₃×D₁₁	—	154	6	no	—
46	C₃×D₁₁	—	154	12	no	—
47	C₃×ΓL(1,𝔽₈)	#92150	1	1	yes	characteristic, complemented, Hall
48	C₂×C₃×C₇	—	121	1	no	—
49	C₂×ΓL(1,𝔽₈)	—	121	1	no	—
50	C₂×ΓL(1,𝔽₈)	—	242	1	no	—
51	C₃×Dic₃	—	847	1	no	Hall
52	C₃×C₁₁	—	2	6	no	—
53	C₃×C₁₁	—	14	6	no	—
54	C₃×C₁₁	—	14	12	no	—
55	C₄×C₇	—	363	1	no	Hall
56	D₁₁	—	22	6	no	—
57	C₃×C₇	C₃×#92150	1	1	yes	characteristic, complemented
58	ΓL(1,𝔽₈)	—	2	1	no	—
59	ΓL(1,𝔽₈)	C₃⋊#92150	1	1	yes	characteristic, direct product factor
60	C₂×C₃²	—	847	1	no	—
61	C₂×C₇	—	121	1	no	—
62	C₃×C₄	—	2541	1	no	—
63	Dic₃	—	121	1	no	—
64	C₁₁	—	2	6	no	—
65	C₃²	—	7	1	no	Sylow
66	C₇	C₃×C₃⋊#92150	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₃	—	847	1	no	—
68	C₂×C₃	—	121	1	no	—
69	C₂×C₃	—	1694	1	no	—
70	C₄	—	363	1	no	Sylow
71	C₃	—	7	1	no	—
72	C₃	—	14	1	no	—
73	C₃	ΓL(1,𝔽₈)×#92150	1	1	yes	characteristic, complemented
74	C₂	—	121	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₇×C₁₁²	C₃×C₄
prime radical	C₃×ΓL(1,𝔽₈)×C₁₁⋊D₁₁	C₂
cosocle	C₃×C₇×C₁₁⋊D₁₁	C₂×C₃
LCS	G ▷ C₃×C₇×C₁₁²
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₇×C₃⋊#92150	C₃	semidirect
4. C₃×C₁₁²×ΓL(1,𝔽₈)	C₄	semidirect
11. C₁₁²×ΓL(1,𝔽₈)	Dic₃	semidirect
12. C₃×C₇×C₁₁²	C₃×C₄	semidirect
15. C₃⋊#92150	ΓL(1,𝔽₈)	direct (N × H)
19. C₇×C₁₁²	C₃×Dic₃	semidirect
30. C₃×C₁₁²	C₄×ΓL(1,𝔽₈)	semidirect
39. C₁₁²	Dic₃×ΓL(1,𝔽₈)	semidirect
47. C₃×ΓL(1,𝔽₈)	#92150	semidirect
57. C₃×C₇	C₃×#92150	semidirect
59. ΓL(1,𝔽₈)	C₃⋊#92150	direct (N × H)
66. C₇	C₃×C₃⋊#92150	semidirect
73. C₃	ΓL(1,𝔽₈)×#92150	semidirect

Maximal subgroups

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

subgroup	index
C₃×ΓL(1,𝔽₈)×C₁₁⋊D₁₁	2
C₇×C₃⋊#92150	3
ΓL(1,𝔽₈)×#92150	3
C₃×C₃⋊#92150	7
Dic₃×ΓL(1,𝔽₈)	121

Subgroups — ΓL(1,𝔽8)×C3⋊#92150 #44479112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×C₃⋊#92150 #44479112