Subgroups — (C₂²⋊(C₇⋊C₈))⋊C₃ #10620458

Order 672.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×ΓL(1,𝔽₈)⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂²⋊(C₇⋊C₈)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂³×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
5	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
6	C₂×C₄×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
7	C₂×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
8	ΓL(1,𝔽₈)⋊C₈	—	2	2	no	—
9	C₂²×C₄×C₇	C₂×C₃	1	1	yes	characteristic
10	C₂×C₇⋊C₈	C₂×C₃	1	2	yes	complemented
11	C₃×C₂²⋊C₈	—	7	1	no	maximal, Hall
12	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
13	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
14	C₂²×ΓL(1,𝔽₈)	C₈	1	2	yes	complemented
15	C₂²×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
16	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
17	C₄×ΓL(1,𝔽₈)	D₄	1	2	yes	—
18	C₂³×C₇	C₃×C₄	1	1	yes	characteristic
19	C₂×C₄×C₇	—	2	2	no	—
20	C₂×C₄×C₇	C₃×C₄	1	1	yes	characteristic
21	C₂×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
22	C₇⋊C₈	—	2	2	no	—
23	C₂²×C₃×C₄	—	7	1	no	—
24	C₂×C₃×C₈	—	7	2	no	—
25	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
26	C₂×ΓL(1,𝔽₈)	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×ΓL(1,𝔽₈)	C₂×C₈	1	1	yes	characteristic
28	C₂×ΓL(1,𝔽₈)	M₁₆	1	1	yes	characteristic
29	C₂²⋊C₈	—	7	1	no	Sylow
30	C₂²×C₇	—	2	1	no	—
31	C₂²×C₇	—	2	1	no	—
32	C₂²×C₇	C₃×C₈	1	2	yes	complemented
33	C₂²×C₇	C₂×C₃×C₄	1	1	yes	characteristic
34	C₄×C₇	—	2	1	no	—
35	C₄×C₇	C₃×D₄	1	2	yes	—
36	C₂³×C₃	—	7	1	no	—
37	C₂×C₃×C₄	—	7	1	no	—
38	C₂×C₃×C₄	—	7	1	no	—
39	C₂×C₃×C₄	—	14	2	no	—
40	C₃×C₈	—	14	2	no	—
41	ΓL(1,𝔽₈)	C₂²⋊C₈	1	1	yes	characteristic, complemented, Hall
42	C₂²×C₄	AGL(1,𝔽₇)	1	1	yes	characteristic
43	C₂×C₈	—	7	2	no	—
44	C₂×C₇	—	2	2	no	—
45	C₂×C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₇	C₂×C₃×C₈	1	1	yes	characteristic
47	C₂×C₇	C₃×M₁₆	1	1	yes	characteristic
48	C₂²×C₃	—	7	2	no	—
49	C₂²×C₃	—	7	1	no	—
50	C₂²×C₃	—	14	1	no	—
51	C₂²×C₃	—	14	1	no	—
52	C₃×C₄	—	7	2	no	—
53	C₃×C₄	—	14	1	no	—
54	C₂³	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
57	C₂×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
58	C₈	—	14	2	no	—
59	C₇	C₃×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₃	—	7	1	no	—
61	C₂×C₃	—	7	1	no	—
62	C₂×C₃	—	7	1	no	—
63	C₂×C₃	—	14	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	ΓL(1,𝔽₈)⋊C₈	1	2	yes	complemented
67	C₂²	C₂×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	(C₂²⋊D₇)⋊C₃	1	2	yes	—
70	C₃	—	7	1	no	Sylow
71	C₂	—	2	2	no	—
72	C₂	C₂×ΓL(1,𝔽₈)⋊C₈	1	1	yes	characteristic
73	C₂	#1136⋊C₃	1	1	yes	characteristic
74	C₂	(C₂²⋊Dic₇)⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×AGL(1,𝔽₇)
commutator	C₂×C₇	C₂×C₃×C₈
Fitting	C₂²×C₄×C₇	C₂×C₃
socle	C₂²×C₇	C₂×C₃×C₄
prime radical	C₂³×ΓL(1,𝔽₈)	C₄
cosocle	C₂×C₄×C₇	C₂²×C₃
hypercenter	C₂²×C₄	AGL(1,𝔽₇)
hypocenter	C₇	C₃×C₂²⋊C₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×ΓL(1,𝔽₈)⋊C₈	C₂	semidirect
3. C₂²⋊(C₇⋊C₈)	C₃	semidirect
10. C₂×C₇⋊C₈	C₂×C₃	semidirect
14. C₂²×ΓL(1,𝔽₈)	C₈	semidirect
32. C₂²×C₇	C₃×C₈	semidirect
41. ΓL(1,𝔽₈)	C₂²⋊C₈	semidirect
59. C₇	C₃×C₂²⋊C₈	semidirect
66. C₂²	ΓL(1,𝔽₈)⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×ΓL(1,𝔽₈)	2
C₂×ΓL(1,𝔽₈)⋊C₈	2
C₂²⋊(C₇⋊C₈)	3
C₃×C₂²⋊C₈	7

Subgroups — (C22⋊(C7⋊C8))⋊C3 #10620458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C₂²⋊(C₇⋊C₈))⋊C₃ #10620458