Subgroups — C₃₁×((C₅⋊D₅)⋊C₃)⋊D₃ #42124793

Order 27900.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×C₃⋊#3874	C₂	1	2	yes	maximal, complemented
2	C₃×C₃₁×(C₅⋊D₅)⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×D₁₅⋊D₅	—	3	1	no	maximal
4	C₃₁×#3874⋊C₂	—	3	3	no	maximal
5	C₃×C₃₁×C₅²⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₅×D₁₅×C₃₁	—	3	2	no	—
7	C₃×C₃₁×C₅⋊D₅	D₃	1	1	yes	characteristic, complemented
8	C₃₁×(C₅⋊D₅)⋊C₃	D₃	1	3	yes	complemented
9	C₃₁×#3874	—	3	6	no	—
10	D₅²×C₃₁	—	9	1	no	Hall
11	C₃₁×C₅²⋊C₃	C₂×D₃	1	3	yes	complemented
12	C₃×C₅²×C₃₁	C₂×D₃	1	1	yes	characteristic, complemented
13	D₃×D₅×C₃₁	—	15	2	no	—
14	C₃₁×C₅⋊D₅	C₃⋊D₃	1	1	yes	characteristic, complemented
15	C₅×D₅×C₃₁	—	9	2	no	—
16	C₂×C₃⋊D₃×C₃₁	—	25	1	no	maximal, Hall
17	D₁₅×C₃₁	—	3	2	no	—
18	C₅×D₃×C₃₁	—	15	2	no	—
19	C₃×D₅×C₃₁	—	15	2	no	—
20	((C₅⋊D₅)⋊C₃)⋊D₃	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅²×C₃₁	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
22	C₂×D₅×C₃₁	—	45	2	no	—
23	C₂×C₃²×C₃₁	—	25	1	no	—
24	C₃⋊D₃×C₃₁	—	25	2	no	—
25	C₃×C₅×C₃₁	—	3	2	no	—
26	C₃×(C₅⋊D₅)⋊C₃	C₂×C₃₁	1	1	yes	characteristic, complemented
27	C₃⋊#3874	C₂×C₃₁	1	2	yes	complemented
28	C₂×D₃×C₃₁	—	75	1	no	—
29	C₂×D₃×C₃₁	—	75	3	no	—
30	C₂×C₅×C₃₁	—	45	2	no	—
31	D₅×C₃₁	—	9	2	no	—
32	D₅×C₃₁	—	15	2	no	—
33	D₁₅⋊D₅	—	3	1	no	—
34	#3874⋊C₂	—	3	3	no	—
35	C₃²×C₃₁	—	25	1	no	Hall
36	C₃×C₅²⋊C₃	C₂²×C₃₁	1	1	yes	characteristic, complemented, Hall
37	C₂×C₃×C₃₁	—	25	3	no	—
38	C₂×C₃×C₃₁	—	25	1	no	—
39	D₃×C₃₁	—	15	2	no	—
40	D₃×C₃₁	—	75	6	no	—
41	C₅×C₃₁	—	3	2	no	—
42	C₅×D₁₅	—	3	2	no	—
43	C₃×C₅⋊D₅	D₃×C₃₁	1	1	yes	characteristic, complemented
44	(C₅⋊D₅)⋊C₃	D₃×C₃₁	1	3	yes	complemented
45	#3874	—	3	6	no	—
46	C₂²×C₃₁	—	225	1	no	Hall
47	D₅²	—	9	1	no	Hall
48	C₃×C₃₁	—	25	3	no	—
49	C₃×C₃₁	#3874⋊C₂	1	1	yes	characteristic, complemented
50	C₃×C₅²	C₂×D₃×C₃₁	1	1	yes	characteristic, complemented
51	C₅²⋊C₃	C₂×D₃×C₃₁	1	3	yes	complemented
52	C₂×C₃₁	—	25	1	no	—
53	C₂×C₃₁	—	45	2	no	—
54	D₃×D₅	—	15	2	no	—
55	C₅⋊D₅	C₃⋊D₃×C₃₁	1	1	yes	characteristic, complemented
56	C₅×D₅	—	9	2	no	—
57	C₂×C₃⋊D₃	—	25	1	no	Hall
58	C₃₁	((C₅⋊D₅)⋊C₃)⋊D₃	1	1	yes	characteristic, direct product factor, Sylow
59	D₁₅	—	3	2	no	—
60	C₃×D₅	—	15	2	no	—
61	C₅×D₃	—	15	2	no	—
62	C₅²	C₂×C₃⋊D₃×C₃₁	1	1	yes	characteristic, complemented, Sylow
63	C₂×D₅	—	45	2	no	—
64	C₂×C₃²	—	25	1	no	—
65	C₃⋊D₃	—	25	2	no	—
66	C₃×C₅	—	3	2	no	—
67	C₂×D₃	—	75	1	no	—
68	C₂×D₃	—	75	3	no	—
69	C₂×C₅	—	45	2	no	—
70	D₅	—	9	2	no	—
71	D₅	—	15	2	no	—
72	C₃²	—	25	1	no	Sylow
73	C₂×C₃	—	25	3	no	—
74	C₂×C₃	—	25	1	no	—
75	D₃	—	15	2	no	—
76	D₃	—	75	6	no	—
77	C₅	—	3	2	no	—
78	C₂²	—	225	1	no	Sylow
79	C₃	—	25	3	no	—
80	C₃	C₃₁×#3874⋊C₂	1	1	yes	characteristic, complemented
81	C₂	—	25	1	no	—
82	C₂	—	45	2	no	—

	subgroup	G/H
center, hypercenter	C₃₁	((C₅⋊D₅)⋊C₃)⋊D₃
commutator, cosocle, hypocenter	C₃×C₅²⋊C₃	C₂²×C₃₁
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₃₁	C₂×D₃
LCS	G ▷ C₃×C₅²⋊C₃
UCS	C₁ ◁ C₃₁

Complements

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

N	H ≅ G/N	type
1. C₃₁×C₃⋊#3874	C₂	semidirect
2. C₃×C₃₁×(C₅⋊D₅)⋊C₃	C₂	semidirect
5. C₃×C₃₁×C₅²⋊C₃	C₂²	semidirect
7. C₃×C₃₁×C₅⋊D₅	D₃	semidirect
8. C₃₁×(C₅⋊D₅)⋊C₃	D₃	semidirect
11. C₃₁×C₅²⋊C₃	C₂×D₃	semidirect
12. C₃×C₅²×C₃₁	C₂×D₃	semidirect
14. C₃₁×C₅⋊D₅	C₃⋊D₃	semidirect
20. ((C₅⋊D₅)⋊C₃)⋊D₃	C₃₁	direct (N × H)
21. C₅²×C₃₁	C₂×C₃⋊D₃	semidirect
26. C₃×(C₅⋊D₅)⋊C₃	C₂×C₃₁	semidirect
27. C₃⋊#3874	C₂×C₃₁	semidirect
36. C₃×C₅²⋊C₃	C₂²×C₃₁	semidirect
43. C₃×C₅⋊D₅	D₃×C₃₁	semidirect
44. (C₅⋊D₅)⋊C₃	D₃×C₃₁	semidirect
49. C₃×C₃₁	#3874⋊C₂	semidirect
50. C₃×C₅²	C₂×D₃×C₃₁	semidirect
51. C₅²⋊C₃	C₂×D₃×C₃₁	semidirect
55. C₅⋊D₅	C₃⋊D₃×C₃₁	semidirect
58. C₃₁	((C₅⋊D₅)⋊C₃)⋊D₃	direct (N × H)
62. C₅²	C₂×C₃⋊D₃×C₃₁	semidirect
80. C₃	C₃₁×#3874⋊C₂	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×C₃⋊#3874	2
C₃×C₃₁×(C₅⋊D₅)⋊C₃	2
C₃₁×D₁₅⋊D₅	3
C₃₁×#3874⋊C₂	3
C₂×C₃⋊D₃×C₃₁	25
((C₅⋊D₅)⋊C₃)⋊D₃	31

Subgroups — C31×((C5⋊D5)⋊C3)⋊D3 #42124793

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×((C₅⋊D₅)⋊C₃)⋊D₃ #42124793