Subgroups — #6381⋊C₃ #10620435

Order 672.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂	1	1	yes	maximal, characteristic
3	C₂×ΓL(1,𝔽₈)⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#6381	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂×C₄×ΓL(1,𝔽₈)	—	2	1	no	—
6	C₂×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
7	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
8	ΓL(1,𝔽₈)⋊C₈	C₄	1	2	yes	complemented
9	C₇×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
10	C₄⋊Dic₇	C₂×C₃	1	1	yes	characteristic
11	C₂×C₇⋊C₈	C₂×C₃	1	1	yes	characteristic
12	C₃×#107	—	7	1	no	maximal, Hall
13	C₂²×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
14	C₄×ΓL(1,𝔽₈)	—	4	1	no	—
15	C₄×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
16	C₄×ΓL(1,𝔽₈)	Q₈	1	1	yes	characteristic
17	ΓL(1,𝔽₈)⋊C₄	—	4	1	no	—
18	C₂×C₄×C₇	—	2	1	no	—
19	C₂×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
20	C₂×Dic₇	—	2	1	no	—
21	C₇⋊C₈	C₃×C₄	1	2	yes	complemented
22	C₂×C₃×C₈	—	7	1	no	—
23	C₃×C₄⋊C₄	—	7	1	no	—
24	C₃×C₄⋊C₄	—	7	1	no	—
25	C₂×ΓL(1,𝔽₈)	C₄⋊C₄	1	1	yes	characteristic
26	C₂×ΓL(1,𝔽₈)	D₈	1	1	yes	characteristic
27	C₂×ΓL(1,𝔽₈)	Q₁₆	1	1	yes	characteristic
28	#107	—	7	1	no	Sylow
29	C₂²×C₇	C₃×D₄	1	1	yes	characteristic
30	C₄×C₇	—	4	1	no	—
31	C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
32	C₄×C₇	C₃×Q₈	1	1	yes	characteristic
33	Dic₇	—	4	1	no	—
34	C₂×C₃×C₄	—	7	1	no	—
35	C₂×C₃×C₄	—	14	1	no	—
36	C₂×C₃×C₄	—	14	1	no	—
37	C₃×C₈	—	7	2	no	—
38	ΓL(1,𝔽₈)	#107	1	1	yes	characteristic, complemented, Hall
39	C₂×C₈	—	7	1	no	—
40	C₄⋊C₄	—	7	1	no	—
41	C₄⋊C₄	AGL(1,𝔽₇)	1	1	yes	characteristic
42	C₂×C₇	C₃×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₇	C₃×D₈	1	1	yes	characteristic
44	C₂×C₇	C₃×Q₁₆	1	1	yes	characteristic
45	C₂²×C₃	—	7	1	no	—
46	C₃×C₄	—	7	1	no	—
47	C₃×C₄	—	7	1	no	—
48	C₃×C₄	—	28	1	no	—
49	C₃×C₄	—	28	1	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	14	1	no	—
52	C₂×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
53	C₈	—	7	2	no	—
54	C₇	C₃×#107	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₃	—	7	1	no	—
56	C₂×C₃	—	7	1	no	—
57	C₂×C₃	—	7	1	no	—
58	C₂²	(C₂²⋊D₇)⋊C₃	1	1	yes	characteristic
59	C₄	—	4	1	no	—
60	C₄	—	28	1	no	—
61	C₄	ΓL(1,𝔽₈)⋊Q₈	1	1	yes	characteristic
62	C₄	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
63	C₃	—	7	1	no	Sylow
64	C₂	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	1	1	yes	characteristic
65	C₂	#1141⋊C₃	1	1	yes	characteristic
66	C₂	#1144⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂²⋊D₇)⋊C₃
commutator	C₄×C₇	C₂×C₃×C₄
Frattini	C₂×C₄	C₂×AGL(1,𝔽₇)
Fitting	C₇×C₄⋊C₄	C₂×C₃
socle	C₂²×C₇	C₃×D₄
prime radical	C₂²×ΓL(1,𝔽₈)	D₄
cosocle	C₂×C₄×C₇	C₂²×C₃
hypercenter	C₄⋊C₄	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#107
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #6381	C₃	semidirect
8. ΓL(1,𝔽₈)⋊C₈	C₄	semidirect
21. C₇⋊C₈	C₃×C₄	semidirect
38. ΓL(1,𝔽₈)	#107	semidirect
54. C₇	C₃×#107	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×ΓL(1,𝔽₈)	2
C₄⋊(ΓL(1,𝔽₈)⋊C₄)	2
C₂×ΓL(1,𝔽₈)⋊C₈	2
#6381	3
C₃×#107	7

Subgroups — #6381⋊C3 #10620435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #6381⋊C₃ #10620435