Subgroups — H(3,𝔽₃)×C₇⋊C₈ #13094168

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	4	yes	maximal, complemented
3	C₂×C₇×H(3,𝔽₃)	C₄	1	1	yes	characteristic
4	C₃²×C₄×C₇	C₂×C₃	1	4	yes	—
5	C₈×H(3,𝔽₃)	—	7	1	no	maximal, Hall
6	C₇×H(3,𝔽₃)	C₈	1	1	yes	characteristic, complemented, Hall
7	C₃×C₇⋊C₈	—	3	4	no	—
8	C₃×C₇⋊C₈	C₃²	1	1	yes	characteristic
9	C₂×C₃²×C₇	C₃×C₄	1	4	yes	—
10	C₄×H(3,𝔽₃)	D₇	1	1	yes	characteristic
11	C₃×C₄×C₇	—	3	4	no	—
12	C₃×C₄×C₇	C₂×C₃²	1	1	yes	characteristic
13	C₃²×C₈	—	7	4	no	—
14	C₃²×C₇	C₃×C₈	1	4	yes	complemented
15	C₇⋊C₈	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Hall
16	C₂×H(3,𝔽₃)	Dic₇	1	1	yes	characteristic
17	C₂×C₃×C₇	—	3	4	no	—
18	C₂×C₃×C₇	C₃²×C₄	1	1	yes	characteristic
19	C₃²×C₄	C₃×D₇	1	4	yes	—
20	C₄×C₇	C₂×H(3,𝔽₃)	1	1	yes	characteristic
21	H(3,𝔽₃)	C₇⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
22	C₃×C₈	—	7	1	no	—
23	C₃×C₈	—	21	4	no	—
24	C₃×C₇	—	3	4	no	—
25	C₃×C₇	C₃²×C₈	1	1	yes	characteristic
26	C₂×C₃²	C₃×Dic₇	1	4	yes	—
27	C₂×C₇	C₄×H(3,𝔽₃)	1	1	yes	characteristic
28	C₃×C₄	—	3	4	no	—
29	C₃×C₄	C₃²×D₇	1	1	yes	characteristic
30	C₃²	C₃×C₇⋊C₈	1	4	yes	complemented
31	C₈	—	7	1	no	Sylow
32	C₇	C₈×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₃	—	3	4	no	—
34	C₂×C₃	C₃²×Dic₇	1	1	yes	characteristic
35	C₄	D₇×H(3,𝔽₃)	1	1	yes	characteristic
36	C₃	—	3	4	no	—
37	C₃	C₃²×C₇⋊C₈	1	1	yes	characteristic
38	C₂	H(3,𝔽₃)×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃×C₄	C₃²×D₇
commutator	C₃×C₇	C₃²×C₈
Fitting	C₄×C₇×H(3,𝔽₃)	C₂
socle	C₂×C₃×C₇	C₃²×C₄
prime radical	C₂×C₇×H(3,𝔽₃)	C₄
cosocle	C₃×C₄×C₇	C₂×C₃²
hypercenter	C₄×H(3,𝔽₃)	D₇
hypocenter	C₇	C₈×H(3,𝔽₃)
LCS	G ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃×C₄ ◁ C₄×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
2. C₃²×C₇⋊C₈	C₃	semidirect
6. C₇×H(3,𝔽₃)	C₈	semidirect
14. C₃²×C₇	C₃×C₈	semidirect
15. C₇⋊C₈	H(3,𝔽₃)	direct (N × H)
21. H(3,𝔽₃)	C₇⋊C₈	direct (N × H)
30. C₃²	C₃×C₇⋊C₈	semidirect
32. 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
C₈×H(3,𝔽₃)	7

Subgroups — H(3,𝔽3)×C7⋊C8 #13094168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×C₇⋊C₈ #13094168