Subgroups — C₇×D₄×H(3,𝔽₃) #13094521

Order 1512.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×H(3,𝔽₃)	C₂	1	2	yes	maximal, complemented
2	C₄×C₇×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₇×D₄	C₃	1	4	yes	maximal, complemented
4	C₂×C₇×H(3,𝔽₃)	—	2	2	no	—
5	C₂×C₇×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₂²×C₃²×C₇	C₂×C₃	1	8	yes	complemented
7	C₃²×C₄×C₇	C₂×C₃	1	4	yes	complemented
8	D₄×H(3,𝔽₃)	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₇×H(3,𝔽₃)	D₄	1	1	yes	characteristic, direct product factor, Hall
10	C₃×C₇×D₄	—	3	4	no	—
11	C₃×C₇×D₄	C₃²	1	1	yes	characteristic
12	C₂×C₃²×C₇	—	2	8	no	—
13	C₂×C₃²×C₇	C₂²×C₃	1	4	yes	—
14	C₂²×H(3,𝔽₃)	C₂×C₇	1	2	yes	complemented
15	C₄×H(3,𝔽₃)	C₂×C₇	1	1	yes	characteristic, complemented
16	C₂²×C₃×C₇	—	3	8	no	—
17	C₂²×C₃×C₇	C₂×C₃²	1	2	yes	—
18	C₃×C₄×C₇	—	3	4	no	—
19	C₃×C₄×C₇	C₂×C₃²	1	1	yes	characteristic
20	C₃²×D₄	C₃×C₇	1	4	yes	complemented
21	C₃²×C₇	C₃×D₄	1	4	yes	complemented
22	C₇×D₄	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Hall
23	C₂×H(3,𝔽₃)	—	2	2	no	—
24	C₂×H(3,𝔽₃)	C₂²×C₇	1	1	yes	characteristic
25	C₂×C₃×C₇	—	2	2	no	—
26	C₂×C₃×C₇	—	3	4	no	—
27	C₂×C₃×C₇	—	6	8	no	—
28	C₂×C₃×C₇	C₂²×C₃²	1	1	yes	characteristic
29	C₂²×C₃²	C₂×C₃×C₇	1	8	yes	complemented
30	C₃²×C₄	C₂×C₃×C₇	1	4	yes	complemented
31	C₂²×C₇	C₂×H(3,𝔽₃)	1	2	yes	complemented
32	C₄×C₇	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
33	H(3,𝔽₃)	C₇×D₄	1	1	yes	characteristic, direct product factor, Sylow
34	C₃×D₄	—	3	4	no	—
35	C₃×D₄	C₃²×C₇	1	1	yes	characteristic
36	C₃×C₇	—	3	4	no	—
37	C₃×C₇	C₃²×D₄	1	1	yes	characteristic
38	C₂×C₃²	—	2	8	no	—
39	C₂×C₃²	C₂²×C₃×C₇	1	4	yes	—
40	C₂×C₇	—	2	2	no	—
41	C₂×C₇	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
42	C₂²×C₃	—	3	8	no	—
43	C₂²×C₃	C₂×C₃²×C₇	1	2	yes	—
44	C₃×C₄	—	3	4	no	—
45	C₃×C₄	C₂×C₃²×C₇	1	1	yes	characteristic
46	C₃²	C₃×C₇×D₄	1	4	yes	complemented
47	D₄	C₇×H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
48	C₇	D₄×H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
49	C₂×C₃	—	2	2	no	—
50	C₂×C₃	—	3	4	no	—
51	C₂×C₃	—	6	8	no	—
52	C₂×C₃	C₂²×C₃²×C₇	1	1	yes	characteristic
53	C₂²	C₂×C₇×H(3,𝔽₃)	1	2	yes	complemented
54	C₄	C₂×C₇×H(3,𝔽₃)	1	1	yes	characteristic, complemented
55	C₃	—	3	4	no	—
56	C₃	C₃²×C₇×D₄	1	1	yes	characteristic
57	C₂	—	2	2	no	—
58	C₂	C₂²×C₇×H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₇	C₂²×C₃²
commutator, Frattini, cosocle	C₂×C₃	C₂²×C₃²×C₇
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×H(3,𝔽₃)	C₂	semidirect
2. C₄×C₇×H(3,𝔽₃)	C₂	semidirect
3. C₃²×C₇×D₄	C₃	semidirect
6. C₂²×C₃²×C₇	C₂×C₃	semidirect
7. C₃²×C₄×C₇	C₂×C₃	semidirect
8. D₄×H(3,𝔽₃)	C₇	direct (N × H)
9. C₇×H(3,𝔽₃)	D₄	direct (N × H)
14. C₂²×H(3,𝔽₃)	C₂×C₇	semidirect
15. C₄×H(3,𝔽₃)	C₂×C₇	semidirect
20. C₃²×D₄	C₃×C₇	semidirect
21. C₃²×C₇	C₃×D₄	semidirect
22. C₇×D₄	H(3,𝔽₃)	direct (N × H)
29. C₂²×C₃²	C₂×C₃×C₇	semidirect
30. C₃²×C₄	C₂×C₃×C₇	semidirect
31. C₂²×C₇	C₂×H(3,𝔽₃)	semidirect
32. C₄×C₇	C₂×H(3,𝔽₃)	semidirect
33. H(3,𝔽₃)	C₇×D₄	direct (N × H)
46. C₃²	C₃×C₇×D₄	semidirect
47. D₄	C₇×H(3,𝔽₃)	direct (N × H)
48. C₇	D₄×H(3,𝔽₃)	direct (N × H)
53. C₂²	C₂×C₇×H(3,𝔽₃)	semidirect
54. C₄	C₂×C₇×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
C₂²×C₇×H(3,𝔽₃)	2
C₄×C₇×H(3,𝔽₃)	2
C₃²×C₇×D₄	3
D₄×H(3,𝔽₃)	7

Subgroups — C7×D4×H(3,𝔽3) #13094521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×D₄×H(3,𝔽₃) #13094521