Subgroups — C₂×C₃₁×C₃⋊(C₇⋊D₇) #32912303

Order 18228.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×C₃⋊(C₇⋊D₇)	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃×C₇²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×C₇⋊D₇	—	3	1	no	maximal, Hall
4	C₃×C₇²×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₃₁×C₇⋊D₇	—	3	2	no	—
6	C₂×C₇²×C₃₁	D₃	1	1	yes	characteristic, complemented
7	C₂×C₃₁×D₂₁	—	7	8	no	maximal
8	C₇²×C₃₁	C₂×D₃	1	1	yes	characteristic, complemented, Hall
9	C₂×C₃×C₇×C₃₁	D₇	1	8	yes	complemented
10	C₃₁×D₂₁	—	7	16	no	—
11	C₂×D₇×C₃₁	—	21	8	no	—
12	C₃×C₇×C₃₁	C₂×D₇	1	8	yes	complemented
13	C₂×C₃⋊(C₇⋊D₇)	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₇×C₃₁	D₂₁	1	8	yes	complemented
15	D₇×C₃₁	—	21	16	no	—
16	C₂×D₃×C₃₁	—	49	1	no	Hall
17	C₂×C₃×C₇²	C₂×C₃₁	1	1	yes	characteristic, complemented
18	C₃⋊(C₇⋊D₇)	C₂×C₃₁	1	2	yes	direct product factor
19	C₇×C₃₁	C₂×D₂₁	1	8	yes	complemented
20	C₂×C₇⋊D₇	—	3	1	no	Hall
21	C₂×C₃×C₃₁	C₇⋊D₇	1	1	yes	characteristic, complemented
22	D₃×C₃₁	—	49	2	no	—
23	C₃×C₇²	C₂²×C₃₁	1	1	yes	characteristic, complemented, Hall
24	C₂²×C₃₁	—	147	1	no	Hall
25	C₂×C₇²	D₃×C₃₁	1	1	yes	characteristic, complemented
26	C₇⋊D₇	—	3	2	no	—
27	C₃×C₃₁	C₂×C₇⋊D₇	1	1	yes	characteristic, complemented, Hall
28	C₂×D₂₁	—	7	8	no	—
29	C₂×C₃₁	—	147	2	no	—
30	C₂×C₃₁	C₃⋊(C₇⋊D₇)	1	1	yes	characteristic, direct product factor
31	C₇²	C₂×D₃×C₃₁	1	1	yes	characteristic, complemented, Sylow
32	C₂×C₃×C₇	D₇×C₃₁	1	8	yes	complemented
33	D₂₁	—	7	16	no	—
34	C₃₁	C₂×C₃⋊(C₇⋊D₇)	1	1	yes	characteristic, direct product factor, Sylow
35	C₂×D₇	—	21	8	no	—
36	C₃×C₇	C₂×D₇×C₃₁	1	8	yes	complemented
37	C₂×C₇	C₃₁×D₂₁	1	8	yes	complemented
38	D₇	—	21	16	no	—
39	C₂×D₃	—	49	1	no	Hall
40	C₇	C₂×C₃₁×D₂₁	1	8	yes	complemented
41	C₂×C₃	C₃₁×C₇⋊D₇	1	1	yes	characteristic, complemented
42	D₃	—	49	2	no	—
43	C₂²	—	147	1	no	Sylow
44	C₃	C₂×C₃₁×C₇⋊D₇	1	1	yes	characteristic, complemented, Sylow
45	C₂	—	147	2	no	—
46	C₂	C₃₁×C₃⋊(C₇⋊D₇)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃₁	C₃⋊(C₇⋊D₇)
commutator, cosocle, hypocenter	C₃×C₇²	C₂²×C₃₁
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇²×C₃₁	C₂
LCS	G ▷ C₃×C₇²
UCS	C₁ ◁ C₂×C₃₁

Complements

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

N	H ≅ G/N	type
1. C₃₁×C₃⋊(C₇⋊D₇)	C₂	direct (N × H)
2. C₂×C₃×C₇²×C₃₁	C₂	semidirect
4. C₃×C₇²×C₃₁	C₂²	semidirect
6. C₂×C₇²×C₃₁	D₃	semidirect
8. C₇²×C₃₁	C₂×D₃	semidirect
9. C₂×C₃×C₇×C₃₁	D₇	semidirect
12. C₃×C₇×C₃₁	C₂×D₇	semidirect
13. C₂×C₃⋊(C₇⋊D₇)	C₃₁	direct (N × H)
14. C₂×C₇×C₃₁	D₂₁	semidirect
17. C₂×C₃×C₇²	C₂×C₃₁	semidirect
18. C₃⋊(C₇⋊D₇)	C₂×C₃₁	direct (N × H)
19. C₇×C₃₁	C₂×D₂₁	semidirect
21. C₂×C₃×C₃₁	C₇⋊D₇	semidirect
23. C₃×C₇²	C₂²×C₃₁	semidirect
25. C₂×C₇²	D₃×C₃₁	semidirect
27. C₃×C₃₁	C₂×C₇⋊D₇	semidirect
30. C₂×C₃₁	C₃⋊(C₇⋊D₇)	direct (N × H)
31. C₇²	C₂×D₃×C₃₁	semidirect
32. C₂×C₃×C₇	D₇×C₃₁	semidirect
34. C₃₁	C₂×C₃⋊(C₇⋊D₇)	direct (N × H)
36. C₃×C₇	C₂×D₇×C₃₁	semidirect
37. C₂×C₇	C₃₁×D₂₁	semidirect
40. C₇	C₂×C₃₁×D₂₁	semidirect
41. C₂×C₃	C₃₁×C₇⋊D₇	semidirect
44. C₃	C₂×C₃₁×C₇⋊D₇	semidirect
46. C₂	C₃₁×C₃⋊(C₇⋊D₇)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×C₃⋊(C₇⋊D₇)	2
C₂×C₃×C₇²×C₃₁	2
C₂×C₃₁×C₇⋊D₇	3
C₂×C₃₁×D₂₁	7
C₂×C₃⋊(C₇⋊D₇)	31

Subgroups — C2×C31×C3⋊(C7⋊D7) #32912303

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×C₃⋊(C₇⋊D₇) #32912303