Subgroups — C₃×C₂²⋊(C₅²⋊(C₃₁⋊C₃)) #42124974

Order 27900.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊(C₅²⋊(C₃₁⋊C₃))	C₃	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₅²×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅²⋊(C₃₁⋊C₃)	—	4	1	no	maximal, Hall
4	C₂×C₃×C₅²×C₃₁	—	3	1	no	—
5	C₂²×C₅²×C₃₁	C₃²	1	1	yes	characteristic, complemented, Hall
6	C₅²⋊(C₃₁⋊C₃)	—	4	3	no	—
7	C₃×C₅²×C₃₁	A₄	1	1	yes	characteristic, complemented
8	C₂²×C₃×C₅×C₃₁	—	3	2	no	—
9	C₂×C₅²×C₃₁	—	3	1	no	—
10	C₃×C₂²⋊(C₃₁⋊C₃)	—	25	1	no	maximal, Hall
11	C₂×C₃×C₅×C₃₁	—	3	6	no	—
12	C₃×C₂²⋊(C₅²⋊C₃)	—	31	1	no	maximal, Hall
13	C₅²×C₃₁	C₃×A₄	1	1	yes	characteristic, complemented, Hall
14	C₂²×C₅×C₃₁	—	3	2	no	—
15	C₃×C₅×C₃₁	—	3	2	no	—
16	C₂²×C₃×C₃₁	C₅²⋊C₃	1	1	yes	characteristic, complemented
17	C₂²⋊(C₃₁⋊C₃)	—	25	3	no	—
18	C₂×C₅×C₃₁	—	3	6	no	—
19	C₂²×C₃×C₅²	C₃₁⋊C₃	1	1	yes	characteristic, complemented
20	C₂²⋊(C₅²⋊C₃)	—	31	3	no	—
21	C₃×C₃₁⋊C₃	—	100	1	no	Hall
22	C₃×C₅²⋊C₃	—	124	1	no	Hall
23	C₂×C₃×C₃₁	—	3	1	no	—
24	C₅×C₃₁	—	3	2	no	—
25	C₂×C₃×C₅²	—	3	1	no	—
26	C₂²×C₃₁	C₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Hall
27	C₂²×C₅²	C₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Hall
28	C₃×C₃₁	C₂²⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented
29	C₃₁⋊C₃	—	100	3	no	—
30	C₃×C₅²	C₂²⋊(C₃₁⋊C₃)	1	1	yes	characteristic, complemented
31	C₅²⋊C₃	—	124	3	no	—
32	C₂×C₃₁	—	3	1	no	—
33	C₂²×C₃×C₅	—	3	2	no	—
34	C₂×C₅²	—	3	1	no	—
35	C₃×A₄	—	775	1	no	Hall
36	C₃₁	C₃×C₂²⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₃×C₅	—	3	6	no	—
38	C₅²	C₃×C₂²⋊(C₃₁⋊C₃)	1	1	yes	characteristic, complemented, Sylow
39	C₂²×C₅	—	3	2	no	—
40	C₃×C₅	—	3	2	no	—
41	C₂²×C₃	C₅²⋊(C₃₁⋊C₃)	1	1	yes	characteristic, complemented
42	A₄	—	775	3	no	—
43	C₂×C₅	—	3	6	no	—
44	C₃²	—	3100	1	no	Sylow
45	C₂×C₃	—	3	1	no	—
46	C₅	—	3	2	no	—
47	C₂²	C₃×C₅²⋊(C₃₁⋊C₃)	1	1	yes	characteristic, complemented, Sylow
48	C₃	—	3100	3	no	—
49	C₃	C₂²⋊(C₅²⋊(C₃₁⋊C₃))	1	1	yes	characteristic, direct product factor
50	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃	C₂²⋊(C₅²⋊(C₃₁⋊C₃))
commutator, cosocle, hypocenter	C₂²×C₅²×C₃₁	C₃²
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅²×C₃₁	C₃
LCS	G ▷ C₂²×C₅²×C₃₁
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₂²⋊(C₅²⋊(C₃₁⋊C₃))	C₃	direct (N × H)
2. C₂²×C₃×C₅²×C₃₁	C₃	semidirect
5. C₂²×C₅²×C₃₁	C₃²	semidirect
7. C₃×C₅²×C₃₁	A₄	semidirect
13. C₅²×C₃₁	C₃×A₄	semidirect
16. C₂²×C₃×C₃₁	C₅²⋊C₃	semidirect
19. C₂²×C₃×C₅²	C₃₁⋊C₃	semidirect
26. C₂²×C₃₁	C₃×C₅²⋊C₃	semidirect
27. C₂²×C₅²	C₃×C₃₁⋊C₃	semidirect
28. C₃×C₃₁	C₂²⋊(C₅²⋊C₃)	semidirect
30. C₃×C₅²	C₂²⋊(C₃₁⋊C₃)	semidirect
36. C₃₁	C₃×C₂²⋊(C₅²⋊C₃)	semidirect
38. C₅²	C₃×C₂²⋊(C₃₁⋊C₃)	semidirect
41. C₂²×C₃	C₅²⋊(C₃₁⋊C₃)	semidirect
47. C₂²	C₃×C₅²⋊(C₃₁⋊C₃)	semidirect
49. C₃	C₂²⋊(C₅²⋊(C₃₁⋊C₃))	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²⋊(C₅²⋊(C₃₁⋊C₃))	3
C₂²×C₃×C₅²×C₃₁	3
C₃×C₅²⋊(C₃₁⋊C₃)	4
C₃×C₂²⋊(C₃₁⋊C₃)	25
C₃×C₂²⋊(C₅²⋊C₃)	31

Subgroups — C3×C22⋊(C52⋊(C31⋊C3)) #42124974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₂²⋊(C₅²⋊(C₃₁⋊C₃)) #42124974