Subgroups — ΓL(1,𝔽₈)×(C₆₇⋊C₃)⋊C₄ #30683099

Order 16884.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×ΓL(1,𝔽₈)×C₆₇⋊C₃	C₂	1	1	yes	maximal, characteristic
2	#19150588	C₃	1	1	yes	maximal, characteristic, complemented
3	#19150587	C₃	1	1	yes	maximal, characteristic, complemented
4	ΓL(1,𝔽₈)×Dic₆₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₇×(C₆₇⋊C₃)⋊C₄	C₃	1	1	yes	maximal, characteristic, complemented
6	ΓL(1,𝔽₈)×C₆₇⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
7	C₂×#13064104	C₂×C₃	1	1	yes	characteristic
8	C₂×#13064105	C₂×C₃	1	1	yes	characteristic
9	C₂×ΓL(1,𝔽₈)×C₆₇	C₂×C₃	1	1	yes	characteristic
10	C₂×C₇×C₆₇⋊C₃	C₂×C₃	1	1	yes	characteristic
11	C₃×(C₆₇⋊C₃)⋊C₄	—	7	1	no	maximal, Hall
12	C₇×Dic₆₇	C₃²	1	1	yes	characteristic, complemented, Hall
13	#13064105	C₃×C₄	1	1	yes	characteristic, complemented
14	#13064104	C₃×C₄	1	1	yes	characteristic, complemented
15	C₇×C₆₇⋊C₃	C₃×C₄	1	1	yes	characteristic, complemented
16	ΓL(1,𝔽₈)×C₆₇	C₃×C₄	1	1	yes	characteristic, complemented
17	C₂×C₃×C₆₇⋊C₃	—	7	1	no	—
18	C₂×C₇×C₆₇	C₂×C₃²	1	1	yes	characteristic
19	C₃×Dic₆₇	—	7	1	no	—
20	(C₆₇⋊C₃)⋊C₄	—	7	1	no	—
21	(C₆₇⋊C₃)⋊C₄	—	7	1	no	—
22	(C₆₇⋊C₃)⋊C₄	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
23	C₃×C₆₇⋊C₃	—	7	1	no	Hall
24	C₇×C₆₇	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃×C₆₇	—	7	1	no	—
26	C₂×C₆₇⋊C₃	—	7	1	no	—
27	C₂×C₆₇⋊C₃	—	7	1	no	—
28	C₂×C₆₇⋊C₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
29	Dic₆₇	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
30	C₃×C₄×ΓL(1,𝔽₈)	—	67	1	no	maximal, Hall
31	C₃×C₆₇	—	7	1	no	—
32	C₆₇⋊C₃	—	7	1	no	—
33	C₆₇⋊C₃	—	7	1	no	—
34	C₆₇⋊C₃	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
35	C₂×C₆₇	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
36	C₂×C₃×ΓL(1,𝔽₈)	—	67	1	no	—
37	C₃×C₄×C₇	—	67	1	no	—
38	C₄×ΓL(1,𝔽₈)	—	67	1	no	—
39	C₄×ΓL(1,𝔽₈)	—	67	1	no	—
40	C₄×ΓL(1,𝔽₈)	—	201	1	no	—
41	C₆₇	C₃×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
42	C₃×ΓL(1,𝔽₈)	—	67	1	no	Hall
43	C₂×C₃×C₇	—	67	1	no	—
44	C₂×ΓL(1,𝔽₈)	—	67	1	no	—
45	C₂×ΓL(1,𝔽₈)	—	67	1	no	—
46	C₂×ΓL(1,𝔽₈)	D₆₇⋊C₃	1	1	yes	characteristic
47	C₃²×C₄	—	469	1	no	Hall
48	C₄×C₇	—	201	1	no	Hall
49	C₃×C₇	—	67	1	no	—
50	ΓL(1,𝔽₈)	—	67	1	no	—
51	ΓL(1,𝔽₈)	—	67	1	no	—
52	ΓL(1,𝔽₈)	(C₆₇⋊C₃)⋊C₄	1	1	yes	characteristic, direct product factor
53	C₂×C₃²	—	469	1	no	—
54	C₂×C₇	C₃×D₆₇⋊C₃	1	1	yes	characteristic
55	C₃×C₄	—	67	1	no	—
56	C₃×C₄	—	1407	1	no	—
57	C₃×C₄	—	469	1	no	—
58	C₃×C₄	—	469	1	no	—
59	C₃²	—	469	1	no	Sylow
60	C₇	C₃×(C₆₇⋊C₃)⋊C₄	1	1	yes	characteristic, complemented, Sylow
61	C₂×C₃	—	7	1	no	—
62	C₂×C₃	—	67	1	no	—
63	C₂×C₃	—	469	1	no	—
64	C₂×C₃	—	469	1	no	—
65	C₄	—	201	1	no	Sylow
66	C₃	—	7	1	no	—
67	C₃	—	67	1	no	—
68	C₃	—	469	1	no	—
69	C₃	—	469	1	no	—
70	C₂	ΓL(1,𝔽₈)×D₆₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	ΓL(1,𝔽₈)×D₆₇⋊C₃
commutator, hypocenter	C₇×C₆₇	C₃²×C₄
Fitting, socle, cosocle	C₂×C₇×C₆₇	C₂×C₃²
prime radical	C₂×ΓL(1,𝔽₈)×C₆₇⋊C₃	C₂
LCS	G ▷ C₇×C₆₇
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. #19150588	C₃	semidirect
3. #19150587	C₃	semidirect
4. ΓL(1,𝔽₈)×Dic₆₇	C₃	semidirect
5. C₇×(C₆₇⋊C₃)⋊C₄	C₃	semidirect
6. ΓL(1,𝔽₈)×C₆₇⋊C₃	C₄	semidirect
12. C₇×Dic₆₇	C₃²	semidirect
13. #13064105	C₃×C₄	semidirect
14. #13064104	C₃×C₄	semidirect
15. C₇×C₆₇⋊C₃	C₃×C₄	semidirect
16. ΓL(1,𝔽₈)×C₆₇	C₃×C₄	semidirect
22. (C₆₇⋊C₃)⋊C₄	ΓL(1,𝔽₈)	direct (N × H)
24. C₇×C₆₇	C₃²×C₄	semidirect
29. Dic₆₇	C₃×ΓL(1,𝔽₈)	semidirect
34. C₆₇⋊C₃	C₄×ΓL(1,𝔽₈)	semidirect
41. C₆₇	C₃×C₄×ΓL(1,𝔽₈)	semidirect
52. ΓL(1,𝔽₈)	(C₆₇⋊C₃)⋊C₄	direct (N × H)
60. C₇	C₃×(C₆₇⋊C₃)⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×ΓL(1,𝔽₈)×C₆₇⋊C₃	2
#19150588	3
#19150587	3
ΓL(1,𝔽₈)×Dic₆₇	3
C₇×(C₆₇⋊C₃)⋊C₄	3
C₃×(C₆₇⋊C₃)⋊C₄	7
C₃×C₄×ΓL(1,𝔽₈)	67

Subgroups — ΓL(1,𝔽8)×(C67⋊C3)⋊C4 #30683099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×(C₆₇⋊C₃)⋊C₄ #30683099