Subgroups — #1157⋊C₃ #66655

Order 336.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	(C₂²⋊D₇)⋊C₃	C₂	1	2	yes	maximal, complemented
3	C₄⋊AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×AGL(1,𝔽₇)	C₂	1	2	yes	maximal, complemented
5	ΓL(1,𝔽₈)⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#1157	C₃	1	1	yes	maximal, characteristic, complemented, Hall
7	C₂²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
8	C₂×AGL(1,𝔽₇)	C₂²	1	2	yes	—
9	C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
10	C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
11	ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
12	C₂×C₄×C₇	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂²⋊D₇	C₂×C₃	1	2	yes	complemented
14	D₂₈	C₂×C₃	1	1	yes	characteristic, complemented
15	C₄×D₇	C₂×C₃	1	2	yes	complemented
16	Dic₁₄	C₂×C₃	1	1	yes	characteristic, complemented
17	C₃×#41	—	7	1	no	maximal, Hall
18	C₂×ΓL(1,𝔽₈)	—	2	1	no	—
19	C₂×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
20	AGL(1,𝔽₇)	—	2	2	no	—
21	C₂²×C₇	C₂²×C₃	1	1	yes	characteristic
22	C₂×D₇	C₂²×C₃	1	2	yes	—
23	C₄×C₇	C₂²×C₃	1	1	yes	characteristic
24	C₄×C₇	C₂²×C₃	1	1	yes	characteristic
25	Dic₇	C₂²×C₃	1	2	yes	—
26	C₃×Q₈	—	7	1	no	—
27	C₃×D₄	—	7	1	no	—
28	C₃×D₄	—	7	2	no	—
29	C₂×C₃×C₄	—	7	2	no	—
30	C₂×C₃×C₄	—	7	1	no	—
31	ΓL(1,𝔽₈)	#41	1	1	yes	characteristic, complemented, Hall
32	#41	—	7	1	no	Sylow
33	C₂×C₇	—	2	1	no	—
34	C₂×C₇	C₂³×C₃	1	1	yes	characteristic
35	D₇	—	2	2	no	—
36	C₂²×C₃	—	7	2	no	—
37	C₂²×C₃	—	7	1	no	—
38	C₃×C₄	—	7	2	no	—
39	C₃×C₄	—	7	1	no	—
40	C₃×C₄	—	7	1	no	—
41	Q₈	—	7	1	no	—
42	D₄	—	7	2	no	—
43	D₄	—	7	1	no	—
44	C₂×C₄	—	7	2	no	—
45	C₂×C₄	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
46	C₇	C₃×#41	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃	—	7	1	no	—
48	C₂×C₃	—	14	1	no	—
49	C₂×C₃	—	14	2	no	—
50	C₂²	—	7	2	no	—
51	C₂²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
52	C₄	—	7	2	no	—
53	C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
54	C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
55	C₃	—	7	1	no	Sylow
56	C₂	—	2	1	no	—
57	C₂	—	14	2	no	—
58	C₂	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂×C₇	C₂³×C₃
Frattini	C₂	C₂²×AGL(1,𝔽₇)
Fitting	C₂×C₄×C₇	C₂×C₃
hypercenter	C₂×C₄	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#41
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×ΓL(1,𝔽₈)	C₂	semidirect
2. (C₂²⋊D₇)⋊C₃	C₂	semidirect
3. C₄⋊AGL(1,𝔽₇)	C₂	semidirect
4. C₄×AGL(1,𝔽₇)	C₂	semidirect
5. ΓL(1,𝔽₈)⋊Q₈	C₂	semidirect
6. #1157	C₃	semidirect
12. C₂×C₄×C₇	C₂×C₃	semidirect
13. C₂²⋊D₇	C₂×C₃	semidirect
14. D₂₈	C₂×C₃	semidirect
15. C₄×D₇	C₂×C₃	semidirect
16. Dic₁₄	C₂×C₃	semidirect
31. ΓL(1,𝔽₈)	#41	semidirect
45. C₂×C₄	AGL(1,𝔽₇)	semidirect
46. C₇	C₃×#41	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×ΓL(1,𝔽₈)	2
(C₂²⋊D₇)⋊C₃	2
C₄⋊AGL(1,𝔽₇)	2
C₄×AGL(1,𝔽₇)	2
ΓL(1,𝔽₈)⋊Q₈	2
#1157	3
C₃×#41	7

Subgroups — #1157⋊C3 #66655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1157⋊C₃ #66655