Subgroups — Dic₃×C₂₃×C₃₁⋊C₃ #38787293

Order 25668.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₂₃×C₃₁⋊C₃	C₂	1	1	yes	maximal, characteristic
2	Dic₃×C₂₃×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂₃×C₃₁⋊C₃	—	3	1	no	maximal
4	C₃×C₂₃×C₃₁⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
5	C₂×C₂₃×C₃₁⋊C₃	—	2	1	no	—
6	C₂×C₂₃×C₃₁⋊C₃	D₃	1	1	yes	characteristic
7	C₂×C₃×C₂₃×C₃₁	C₂×C₃	1	1	yes	characteristic
8	C₄×C₂₃×C₃₁	—	3	1	no	Hall
9	C₃×C₂₃×C₃₁	C₃×C₄	1	1	yes	characteristic, complemented
10	C₂₃×C₃₁⋊C₃	—	2	1	no	—
11	C₂₃×C₃₁⋊C₃	Dic₃	1	1	yes	characteristic, direct product factor
12	C₂×C₂₃×C₃₁	C₃×D₃	1	1	yes	characteristic
13	Dic₃×C₃₁⋊C₃	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₃×Dic₃×C₂₃	—	31	1	no	maximal, Hall
15	C₂₃×C₃₁	C₃×Dic₃	1	1	yes	characteristic, complemented, Hall
16	C₂×C₃×C₃₁⋊C₃	C₂×C₂₃	1	1	yes	characteristic
17	C₂×C₃²×C₂₃	—	31	1	no	—
18	Dic₃×C₃₁	C₃×C₂₃	1	1	yes	characteristic, complemented
19	C₄×C₃₁⋊C₃	—	3	1	no	—
20	C₃×C₃₁⋊C₃	C₄×C₂₃	1	1	yes	characteristic, complemented, Hall
21	C₃×C₄×C₂₃	—	93	1	no	—
22	Dic₃×C₂₃	C₃₁⋊C₃	1	1	yes	characteristic, direct product factor
23	C₃²×C₂₃	—	31	1	no	Hall
24	C₂×C₃×C₃₁	C₂×C₃×C₂₃	1	1	yes	characteristic
25	C₂×C₃₁⋊C₃	—	2	1	no	—
26	C₂×C₃₁⋊C₃	D₃×C₂₃	1	1	yes	characteristic
27	C₂×C₃×C₂₃	—	31	1	no	—
28	C₂×C₃×C₂₃	—	62	1	no	—
29	C₂×C₃×C₂₃	C₂×C₃₁⋊C₃	1	1	yes	characteristic
30	C₄×C₃₁	—	3	1	no	Hall
31	C₃×C₃₁	C₃×C₄×C₂₃	1	1	yes	characteristic, complemented
32	C₃₁⋊C₃	—	2	1	no	—
33	C₃₁⋊C₃	Dic₃×C₂₃	1	1	yes	characteristic, direct product factor
34	C₄×C₂₃	—	3	1	no	Hall
35	C₃×C₂₃	—	31	1	no	—
36	C₃×C₂₃	—	62	1	no	—
37	C₃×C₂₃	C₄×C₃₁⋊C₃	1	1	yes	characteristic, complemented
38	C₂×C₃₁	C₃×D₃×C₂₃	1	1	yes	characteristic
39	C₂×C₂₃	D₃×C₃₁⋊C₃	1	1	yes	characteristic
40	C₃×Dic₃	—	31	1	no	Hall
41	C₃₁	C₃×Dic₃×C₂₃	1	1	yes	characteristic, complemented, Sylow
42	C₂₃	Dic₃×C₃₁⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₃²	—	31	1	no	—
44	C₃×C₄	—	93	1	no	—
45	Dic₃	C₂₃×C₃₁⋊C₃	1	1	yes	characteristic, direct product factor
46	C₃²	—	31	1	no	Sylow
47	C₂×C₃	—	31	1	no	—
48	C₂×C₃	—	62	1	no	—
49	C₂×C₃	C₂×C₂₃×C₃₁⋊C₃	1	1	yes	characteristic
50	C₄	—	3	1	no	Sylow
51	C₃	—	31	1	no	—
52	C₃	—	62	1	no	—
53	C₃	C₄×C₂₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented
54	C₂	D₃×C₂₃×C₃₁⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₂₃	D₃×C₃₁⋊C₃
commutator, hypocenter	C₃×C₃₁	C₃×C₄×C₂₃
Frattini	C₂	D₃×C₂₃×C₃₁⋊C₃
Fitting, socle	C₂×C₃×C₂₃×C₃₁	C₂×C₃
prime radical	C₂×C₃×C₂₃×C₃₁⋊C₃	C₂
cosocle	C₂×C₃×C₃₁	C₂×C₃×C₂₃
LCS	G ▷ C₃×C₃₁
UCS	C₁ ◁ C₂×C₂₃

Complements

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

N	H ≅ G/N	type
2. Dic₃×C₂₃×C₃₁	C₃	semidirect
4. C₃×C₂₃×C₃₁⋊C₃	C₄	semidirect
9. C₃×C₂₃×C₃₁	C₃×C₄	semidirect
11. C₂₃×C₃₁⋊C₃	Dic₃	direct (N × H)
13. Dic₃×C₃₁⋊C₃	C₂₃	direct (N × H)
15. C₂₃×C₃₁	C₃×Dic₃	semidirect
18. Dic₃×C₃₁	C₃×C₂₃	semidirect
20. C₃×C₃₁⋊C₃	C₄×C₂₃	semidirect
22. Dic₃×C₂₃	C₃₁⋊C₃	direct (N × H)
31. C₃×C₃₁	C₃×C₄×C₂₃	semidirect
33. C₃₁⋊C₃	Dic₃×C₂₃	direct (N × H)
37. C₃×C₂₃	C₄×C₃₁⋊C₃	semidirect
41. C₃₁	C₃×Dic₃×C₂₃	semidirect
42. C₂₃	Dic₃×C₃₁⋊C₃	direct (N × H)
45. Dic₃	C₂₃×C₃₁⋊C₃	direct (N × H)
53. C₃	C₄×C₂₃×C₃₁⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₂₃×C₃₁⋊C₃	2
Dic₃×C₂₃×C₃₁	3
C₄×C₂₃×C₃₁⋊C₃	3
Dic₃×C₃₁⋊C₃	23
C₃×Dic₃×C₂₃	31

Subgroups — Dic3×C23×C31⋊C3 #38787293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₃×C₂₃×C₃₁⋊C₃ #38787293