Subgroups — C₃₁⋊C₃×C₂²⋊(C₅²⋊C₃) #42124833

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	1	yes	maximal, characteristic, complemented
2	C₃₁×C₂²⋊(C₅²⋊C₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊(C₅²⋊(C₃₁⋊C₃))	C₃	1	2	yes	maximal, complemented
4	C₅²⋊C₃×C₃₁⋊C₃	—	4	1	no	maximal, Hall
5	C₂×C₅²×C₃₁⋊C₃	—	3	1	no	—
6	C₂²×C₅²×C₃₁	C₃²	1	1	yes	characteristic, complemented, Hall
7	C₅²⋊(C₃₁⋊C₃)	—	4	2	no	—
8	C₃₁×C₅²⋊C₃	—	4	1	no	—
9	C₅²×C₃₁⋊C₃	A₄	1	1	yes	characteristic, complemented
10	C₂²×C₅×C₃₁⋊C₃	—	3	2	no	—
11	C₂×C₅²×C₃₁	—	3	1	no	—
12	A₄×C₃₁⋊C₃	—	25	1	no	maximal, Hall
13	C₂×C₅×C₃₁⋊C₃	—	3	6	no	—
14	C₃×C₂²⋊(C₅²⋊C₃)	—	31	1	no	maximal, Hall
15	C₅²×C₃₁	C₃×A₄	1	1	yes	characteristic, complemented, Hall
16	C₂²×C₅×C₃₁	—	3	2	no	—
17	C₅×C₃₁⋊C₃	—	3	2	no	—
18	C₂²⋊(C₃₁⋊C₃)	—	25	2	no	—
19	A₄×C₃₁	—	25	1	no	—
20	C₂²×C₃₁⋊C₃	C₅²⋊C₃	1	1	yes	characteristic, complemented
21	C₂×C₅×C₃₁	—	3	6	no	—
22	C₂²×C₃×C₅²	—	31	1	no	—
23	C₂²⋊(C₅²⋊C₃)	—	31	2	no	—
24	C₂²⋊(C₅²⋊C₃)	C₃₁⋊C₃	1	1	yes	characteristic, direct product factor
25	C₃×C₃₁⋊C₃	—	100	1	no	Hall
26	C₃×C₅²⋊C₃	—	124	1	no	Hall
27	C₂×C₃₁⋊C₃	—	3	1	no	—
28	C₅×C₃₁	—	3	2	no	—
29	C₂×C₃×C₅²	—	93	1	no	—
30	C₂²×C₃₁	C₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Hall
31	C₂²×C₅²	C₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Hall
32	C₃×C₃₁	—	100	1	no	—
33	C₃₁⋊C₃	—	100	2	no	—
34	C₃₁⋊C₃	C₂²⋊(C₅²⋊C₃)	1	1	yes	characteristic, direct product factor
35	C₃×C₅²	—	31	1	no	—
36	C₅²⋊C₃	—	4	1	no	—
37	C₅²⋊C₃	—	124	2	no	—
38	C₂×C₃₁	—	3	1	no	—
39	C₂²×C₃×C₅	—	93	2	no	—
40	C₂×C₅²	—	3	1	no	—
41	C₃×A₄	—	775	1	no	Hall
42	C₃₁	C₃×C₂²⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃×C₅	—	93	6	no	—
44	C₅²	A₄×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₅	—	3	2	no	—
46	C₃×C₅	—	93	2	no	—
47	C₂²×C₃	—	31	1	no	—
48	A₄	—	775	2	no	—
49	A₄	—	25	1	no	—
50	C₂×C₅	—	3	6	no	—
51	C₃²	—	3100	1	no	Sylow
52	C₂×C₃	—	93	1	no	—
53	C₅	—	3	2	no	—
54	C₂²	C₅²⋊C₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
55	C₃	—	3100	2	no	—
56	C₃	—	31	1	no	—
57	C₃	—	100	1	no	—
58	C₂	—	3	1	no	—

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

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅²×C₃₁⋊C₃	C₃	semidirect
2. C₃₁×C₂²⋊(C₅²⋊C₃)	C₃	semidirect
3. C₂²⋊(C₅²⋊(C₃₁⋊C₃))	C₃	semidirect
6. C₂²×C₅²×C₃₁	C₃²	semidirect
9. C₅²×C₃₁⋊C₃	A₄	semidirect
15. C₅²×C₃₁	C₃×A₄	semidirect
20. C₂²×C₃₁⋊C₃	C₅²⋊C₃	semidirect
24. C₂²⋊(C₅²⋊C₃)	C₃₁⋊C₃	direct (N × H)
30. C₂²×C₃₁	C₃×C₅²⋊C₃	semidirect
31. C₂²×C₅²	C₃×C₃₁⋊C₃	semidirect
34. C₃₁⋊C₃	C₂²⋊(C₅²⋊C₃)	direct (N × H)
42. C₃₁	C₃×C₂²⋊(C₅²⋊C₃)	semidirect
44. C₅²	A₄×C₃₁⋊C₃	semidirect
54. C₂²	C₅²⋊C₃×C₃₁⋊C₃	semidirect

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
C₃₁×C₂²⋊(C₅²⋊C₃)	3
C₂²⋊(C₅²⋊(C₃₁⋊C₃))	3
C₅²⋊C₃×C₃₁⋊C₃	4
A₄×C₃₁⋊C₃	25
C₃×C₂²⋊(C₅²⋊C₃)	31

Subgroups — C31⋊C3×C22⋊(C52⋊C3) #42124833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁⋊C₃×C₂²⋊(C₅²⋊C₃) #42124833