Subgroups — C₂²×C₃₁⋊#3862 #32912293

Order 18228.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁⋊#3862	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₇²×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₃₁⋊#3862	C₂²	1	1	yes	characteristic, direct product factor, Hall
4	C₂×C₇²×C₃₁	C₂×C₃	1	3	yes	complemented
5	C₂²×#10620253	—	7	1	no	maximal
6	C₂²×#10620252	—	7	1	no	maximal
7	C₇²×C₃₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
8	C₂×#10620253	—	7	3	no	—
9	C₂×#10620252	—	7	3	no	—
10	C₂²×C₇×C₃₁	—	3	2	no	—
11	C₂²×C₇×C₃₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
12	C₂²×C₇×C₃₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
13	#10620253	—	7	1	no	—
14	#10620252	—	7	1	no	—
15	C₂²×#3862	—	31	1	no	maximal, Hall
16	C₂×C₇×C₃₁	—	3	6	no	—
17	C₂×C₇×C₃₁	C₂×ΓL(1,𝔽₈)	1	3	yes	complemented
18	C₂×C₇×C₃₁	C₂×ΓL(1,𝔽₈)	1	3	yes	complemented
19	C₂²×C₃₁⋊C₃	—	49	1	no	Hall
20	C₂×#3862	—	31	3	no	—
21	C₇×C₃₁	—	3	2	no	—
22	C₇×C₃₁	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
23	C₇×C₃₁	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
24	C₂²×C₇²	C₃₁⋊C₃	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃₁⋊C₃	—	49	3	no	—
26	#3862	—	31	1	no	Hall
27	C₂²×C₃₁	#3862	1	1	yes	characteristic, complemented, Hall
28	C₂×C₇²	C₂×C₃₁⋊C₃	1	3	yes	complemented
29	C₃₁⋊C₃	—	49	1	no	Hall
30	C₂²×ΓL(1,𝔽₈)	—	217	1	no	—
31	C₂²×ΓL(1,𝔽₈)	—	217	1	no	—
32	C₂×C₃₁	C₂×#3862	1	3	yes	complemented
33	C₇²	C₂²×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
34	C₂×ΓL(1,𝔽₈)	—	217	3	no	—
35	C₂×ΓL(1,𝔽₈)	—	217	3	no	—
36	C₃₁	C₂²×#3862	1	1	yes	characteristic, complemented, Sylow
37	C₂²×C₇	—	3	2	no	—
38	C₂²×C₇	#10620252	1	1	yes	characteristic, complemented
39	C₂²×C₇	#10620253	1	1	yes	characteristic, complemented
40	ΓL(1,𝔽₈)	—	217	1	no	—
41	ΓL(1,𝔽₈)	—	217	1	no	—
42	C₂×C₇	—	3	6	no	—
43	C₂×C₇	C₂×#10620252	1	3	yes	complemented
44	C₂×C₇	C₂×#10620253	1	3	yes	complemented
45	C₂²×C₃	—	1519	1	no	Hall
46	C₇	—	3	2	no	—
47	C₇	C₂²×#10620252	1	1	yes	characteristic, complemented
48	C₇	C₂²×#10620253	1	1	yes	characteristic, complemented
49	C₂×C₃	—	1519	3	no	—
50	C₂²	C₃₁⋊#3862	1	1	yes	characteristic, direct product factor, Sylow
51	C₃	—	1519	1	no	Sylow
52	C₂	C₂×C₃₁⋊#3862	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²	C₃₁⋊#3862
commutator, cosocle, hypocenter	C₇²×C₃₁	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂²×C₇²×C₃₁	C₃
LCS	G ▷ C₇²×C₃₁
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁⋊#3862	C₂	direct (N × H)
2. C₂²×C₇²×C₃₁	C₃	semidirect
3. C₃₁⋊#3862	C₂²	direct (N × H)
4. C₂×C₇²×C₃₁	C₂×C₃	semidirect
7. C₇²×C₃₁	C₂²×C₃	semidirect
11. C₂²×C₇×C₃₁	ΓL(1,𝔽₈)	semidirect
12. C₂²×C₇×C₃₁	ΓL(1,𝔽₈)	semidirect
17. C₂×C₇×C₃₁	C₂×ΓL(1,𝔽₈)	semidirect
18. C₂×C₇×C₃₁	C₂×ΓL(1,𝔽₈)	semidirect
22. C₇×C₃₁	C₂²×ΓL(1,𝔽₈)	semidirect
23. C₇×C₃₁	C₂²×ΓL(1,𝔽₈)	semidirect
24. C₂²×C₇²	C₃₁⋊C₃	semidirect
27. C₂²×C₃₁	#3862	semidirect
28. C₂×C₇²	C₂×C₃₁⋊C₃	semidirect
32. C₂×C₃₁	C₂×#3862	semidirect
33. C₇²	C₂²×C₃₁⋊C₃	semidirect
36. C₃₁	C₂²×#3862	semidirect
38. C₂²×C₇	#10620252	semidirect
39. C₂²×C₇	#10620253	semidirect
43. C₂×C₇	C₂×#10620252	semidirect
44. C₂×C₇	C₂×#10620253	semidirect
47. C₇	C₂²×#10620252	semidirect
48. C₇	C₂²×#10620253	semidirect
50. C₂²	C₃₁⋊#3862	direct (N × H)
52. C₂	C₂×C₃₁⋊#3862	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁⋊#3862	2
C₂²×C₇²×C₃₁	3
C₂²×#10620253	7
C₂²×#10620252	7
C₂²×#3862	31

Subgroups — C22×C31⋊#3862 #32912293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁⋊#3862 #32912293