Subgroups — (C₇⋊C₈)⋊H(3,𝔽₃) #13094150

Order 1512.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇⋊H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₇⋊C₈	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×ΓL(1,𝔽₈)⋊C₈	C₃	1	3	yes	maximal, complemented
4	C₂×C₇⋊H(3,𝔽₃)	C₄	1	1	yes	characteristic
5	C₃²×C₄×C₇	C₂×C₃	1	1	yes	characteristic
6	C₃×C₄×ΓL(1,𝔽₈)	C₂×C₃	1	3	yes	—
7	C₈×H(3,𝔽₃)	—	7	1	no	maximal, Hall
8	C₇⋊H(3,𝔽₃)	C₈	1	1	yes	characteristic, complemented, Hall
9	C₃×C₇⋊C₈	—	3	1	no	—
10	C₃×C₇⋊C₈	C₃²	1	1	yes	characteristic
11	ΓL(1,𝔽₈)⋊C₈	—	3	3	no	—
12	C₂×C₃²×C₇	C₃×C₄	1	1	yes	characteristic
13	C₂×C₃×ΓL(1,𝔽₈)	C₃×C₄	1	3	yes	—
14	C₄×H(3,𝔽₃)	—	7	1	no	—
15	C₃×C₄×C₇	—	3	1	no	—
16	C₃×C₄×C₇	C₂×C₃²	1	1	yes	characteristic
17	C₄×ΓL(1,𝔽₈)	—	3	3	no	—
18	C₃²×C₈	—	7	3	no	—
19	C₃²×C₈	—	7	1	no	—
20	C₃²×C₇	C₃×C₈	1	1	yes	characteristic, complemented
21	C₃×ΓL(1,𝔽₈)	C₃×C₈	1	3	yes	complemented
22	C₇⋊C₈	H(3,𝔽₃)	1	1	yes	characteristic, complemented, Hall
23	C₂×H(3,𝔽₃)	—	7	1	no	—
24	C₂×C₃×C₇	—	3	1	no	—
25	C₂×C₃×C₇	C₃²×C₄	1	1	yes	characteristic
26	C₂×ΓL(1,𝔽₈)	—	3	3	no	—
27	C₃²×C₄	—	7	3	no	—
28	C₃²×C₄	AGL(1,𝔽₇)	1	1	yes	characteristic
29	C₄×C₇	C₂×H(3,𝔽₃)	1	1	yes	characteristic
30	H(3,𝔽₃)	—	7	1	no	Sylow
31	C₃×C₈	—	7	1	no	—
32	C₃×C₈	—	21	1	no	—
33	C₃×C₈	—	21	3	no	—
34	C₃×C₇	—	3	1	no	—
35	C₃×C₇	C₃²×C₈	1	1	yes	characteristic
36	ΓL(1,𝔽₈)	—	3	3	no	—
37	C₂×C₃²	—	7	3	no	—
38	C₂×C₃²	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
39	C₂×C₇	C₄×H(3,𝔽₃)	1	1	yes	characteristic
40	C₃×C₄	—	3	1	no	—
41	C₃×C₄	—	21	3	no	—
42	C₃×C₄	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic
43	C₃²	—	7	3	no	—
44	C₃²	ΓL(1,𝔽₈)⋊C₈	1	1	yes	characteristic, complemented
45	C₈	—	7	1	no	Sylow
46	C₇	C₈×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃	—	3	1	no	—
48	C₂×C₃	—	21	3	no	—
49	C₂×C₃	C₃×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
50	C₄	D₇⋊H(3,𝔽₃)	1	1	yes	characteristic
51	C₃	—	3	1	no	—
52	C₃	—	21	3	no	—
53	C₃	C₃×ΓL(1,𝔽₈)⋊C₈	1	1	yes	characteristic
54	C₂	Dic₇⋊H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃×C₄	C₃×AGL(1,𝔽₇)
commutator	C₃×C₇	C₃²×C₈
Fitting	C₃²×C₄×C₇	C₂×C₃
socle	C₂×C₃×C₇	C₃²×C₄
prime radical	C₂×C₇⋊H(3,𝔽₃)	C₄
cosocle	C₃×C₄×C₇	C₂×C₃²
hypercenter	C₃²×C₄	AGL(1,𝔽₇)
hypocenter	C₇	C₈×H(3,𝔽₃)
LCS	G ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃×C₄ ◁ C₃²×C₄

Complements

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

N	H ≅ G/N	type
2. C₃²×C₇⋊C₈	C₃	semidirect
3. C₃×ΓL(1,𝔽₈)⋊C₈	C₃	semidirect
8. C₇⋊H(3,𝔽₃)	C₈	semidirect
20. C₃²×C₇	C₃×C₈	semidirect
21. C₃×ΓL(1,𝔽₈)	C₃×C₈	semidirect
22. C₇⋊C₈	H(3,𝔽₃)	semidirect
44. C₃²	ΓL(1,𝔽₈)⋊C₈	semidirect
46. C₇	C₈×H(3,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇⋊H(3,𝔽₃)	2
C₃²×C₇⋊C₈	3
C₃×ΓL(1,𝔽₈)⋊C₈	3
C₈×H(3,𝔽₃)	7

Subgroups — (C7⋊C8)⋊H(3,𝔽3) #13094150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C₇⋊C₈)⋊H(3,𝔽₃) #13094150