Subgroups — C₃×C₂²⋊((C₃₁⋊C₃)⋊C₂₅) #42124393

Order 27900.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₃₁⋊C₂₅	C₃	1	1	yes	maximal, characteristic, complemented
2	C₂²⋊((C₃₁⋊C₃)⋊C₂₅)	C₃	1	3	yes	maximal, direct product factor
3	C₃×(C₃₁⋊C₃)⋊C₂₅	—	4	1	no	maximal, Hall
4	C₃×C₅×C₂²⋊(C₃₁⋊C₃)	C₅	1	1	yes	maximal, characteristic
5	C₂×C₃×C₃₁⋊C₂₅	—	3	1	no	—
6	C₂²×C₃₁⋊C₂₅	C₃²	1	1	yes	characteristic, complemented, Hall
7	C₃×C₃₁⋊C₂₅	A₄	1	1	yes	characteristic, complemented
8	(C₃₁⋊C₃)⋊C₂₅	—	4	3	no	—
9	C₂²×C₃×C₅×C₃₁	C₃×C₅	1	1	yes	characteristic
10	C₅×C₂²⋊(C₃₁⋊C₃)	C₃×C₅	1	3	yes	—
11	C₂×C₃₁⋊C₂₅	—	3	1	no	—
12	C₃×C₅×C₃₁⋊C₃	—	4	1	no	—
13	C₃×C₂²⋊(C₃₁⋊C₃)	C₂₅	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃×C₅×C₃₁	—	3	1	no	—
15	C₃×A₄×C₂₅	—	31	1	no	maximal, Hall
16	C₃₁⋊C₂₅	C₃×A₄	1	1	yes	characteristic, complemented, Hall
17	C₂²×C₅×C₃₁	C₃²×C₅	1	1	yes	characteristic
18	C₃×C₅×C₃₁	C₅×A₄	1	1	yes	characteristic
19	C₅×C₃₁⋊C₃	—	4	3	no	—
20	C₂²×C₃×C₃₁	C₃×C₂₅	1	1	yes	characteristic, complemented
21	C₂²⋊(C₃₁⋊C₃)	C₃×C₂₅	1	3	yes	complemented
22	C₂×C₅×C₃₁	—	3	1	no	—
23	C₂²×C₃×C₂₅	—	31	1	no	—
24	A₄×C₂₅	—	31	3	no	—
25	C₃×C₃₁⋊C₃	—	4	1	no	Hall
26	C₃²×C₂₅	—	124	1	no	Hall
27	C₂×C₃×C₃₁	—	3	1	no	—
28	C₃×C₅×A₄	—	31	1	no	—
29	C₅×C₃₁	C₃×C₅×A₄	1	1	yes	characteristic
30	C₂×C₃×C₂₅	—	93	1	no	—
31	C₂²×C₃₁	C₃²×C₂₅	1	1	yes	characteristic, complemented, Hall
32	C₂²×C₂₅	—	31	1	no	Hall
33	C₃×C₃₁	A₄×C₂₅	1	1	yes	characteristic, complemented
34	C₃₁⋊C₃	—	4	3	no	—
35	C₃×C₂₅	—	31	1	no	—
36	C₃×C₂₅	—	124	3	no	—
37	C₂×C₃₁	—	3	1	no	—
38	C₂²×C₃×C₅	(C₃₁⋊C₃)⋊C₅	1	1	yes	characteristic
39	C₅×A₄	—	31	3	no	—
40	C₂×C₂₅	—	93	1	no	—
41	C₃²×C₅	—	124	1	no	—
42	C₃×A₄	—	31	1	no	Hall
43	C₃₁	C₃×A₄×C₂₅	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₃×C₅	—	3	1	no	—
45	C₂₅	—	31	1	no	Sylow
46	C₂²×C₅	C₃×(C₃₁⋊C₃)⋊C₅	1	1	yes	characteristic
47	C₃×C₅	—	124	3	no	—
48	C₃×C₅	C₂²⋊((C₃₁⋊C₃)⋊C₅)	1	1	yes	characteristic
49	C₂²×C₃	(C₃₁⋊C₃)⋊C₂₅	1	1	yes	characteristic, complemented
50	A₄	—	31	3	no	—
51	C₂×C₅	—	3	1	no	—
52	C₃²	—	124	1	no	Sylow
53	C₂×C₃	—	3	1	no	—
54	C₅	C₃×C₂²⋊((C₃₁⋊C₃)⋊C₅)	1	1	yes	characteristic
55	C₂²	C₃×(C₃₁⋊C₃)⋊C₂₅	1	1	yes	characteristic, complemented, Sylow
56	C₃	—	124	3	no	—
57	C₃	C₂²⋊((C₃₁⋊C₃)⋊C₂₅)	1	1	yes	characteristic, direct product factor
58	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅	C₂²⋊((C₃₁⋊C₃)⋊C₅)
commutator, hypocenter	C₂²×C₃₁	C₃²×C₂₅
Frattini	C₅	C₃×C₂²⋊((C₃₁⋊C₃)⋊C₅)
Fitting, socle	C₂²×C₃×C₅×C₃₁	C₃×C₅
prime radical	C₃×C₅×C₂²⋊(C₃₁⋊C₃)	C₅
cosocle	C₂²×C₅×C₃₁	C₃²×C₅
LCS	G ▷ C₂²×C₃₁
UCS	C₁ ◁ C₃×C₅

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₃₁⋊C₂₅	C₃	semidirect
2. C₂²⋊((C₃₁⋊C₃)⋊C₂₅)	C₃	direct (N × H)
6. C₂²×C₃₁⋊C₂₅	C₃²	semidirect
7. C₃×C₃₁⋊C₂₅	A₄	semidirect
13. C₃×C₂²⋊(C₃₁⋊C₃)	C₂₅	semidirect
16. C₃₁⋊C₂₅	C₃×A₄	semidirect
20. C₂²×C₃×C₃₁	C₃×C₂₅	semidirect
21. C₂²⋊(C₃₁⋊C₃)	C₃×C₂₅	semidirect
31. C₂²×C₃₁	C₃²×C₂₅	semidirect
33. C₃×C₃₁	A₄×C₂₅	semidirect
43. C₃₁	C₃×A₄×C₂₅	semidirect
49. C₂²×C₃	(C₃₁⋊C₃)⋊C₂₅	semidirect
55. C₂²	C₃×(C₃₁⋊C₃)⋊C₂₅	semidirect
57. C₃	C₂²⋊((C₃₁⋊C₃)⋊C₂₅)	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×C₃×C₃₁⋊C₂₅	3
C₂²⋊((C₃₁⋊C₃)⋊C₂₅)	3
C₃×(C₃₁⋊C₃)⋊C₂₅	4
C₃×C₅×C₂²⋊(C₃₁⋊C₃)	5
C₃×A₄×C₂₅	31

Subgroups — C3×C22⋊((C31⋊C3)⋊C25) #42124393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₂²⋊((C₃₁⋊C₃)⋊C₂₅) #42124393