Subgroups — C₃₁⋊C₃×#3874⋊C₂ #42124690

Order 27900.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁⋊C₃×(C₅⋊D₅)⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁⋊C₃×#3874	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3874⋊C₂	C₃	1	1	yes	maximal, characteristic, complemented
4	D₅²×C₃₁⋊C₃	—	3	1	no	maximal
5	C₅²⋊C₃×C₃₁⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	(C₅⋊D₅)⋊(C₃₁⋊C₃)	—	2	1	no	—
7	C₃₁×(C₅⋊D₅)⋊C₃	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃₁×#3874	C₂×C₃	1	2	yes	complemented
9	C₅⋊D₅×C₃₁⋊C₃	D₃	1	1	yes	characteristic, complemented
10	C₅×D₅×C₃₁⋊C₃	—	3	2	no	—
11	D₅²×C₃₁	—	3	1	no	Hall
12	C₅²⋊(C₃₁⋊C₃)	—	2	1	no	—
13	C₃₁×C₅²⋊C₃	C₂²×C₃	1	1	yes	characteristic, complemented
14	C₅²×C₃₁⋊C₃	C₂×D₃	1	1	yes	characteristic, complemented
15	C₂×D₅×C₃₁⋊C₃	—	15	2	no	—
16	C₃₁×C₅⋊D₅	C₃×D₃	1	1	yes	characteristic, complemented
17	C₅×D₅×C₃₁	—	3	2	no	—
18	C₂×D₃×C₃₁⋊C₃	—	25	1	no	maximal, Hall
19	C₂×C₅×C₃₁⋊C₃	—	15	2	no	—
20	D₅×C₃₁⋊C₃	—	3	2	no	—
21	D₅×C₃₁⋊C₃	—	15	2	no	—
22	C₃×#3874⋊C₂	—	31	1	no	maximal, Hall
23	C₅²×C₃₁	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
24	C₂×D₅×C₃₁	—	15	2	no	—
25	C₂×C₃×C₃₁⋊C₃	—	25	1	no	—
26	D₃×C₃₁⋊C₃	—	25	2	no	—
27	C₅×C₃₁⋊C₃	—	3	2	no	—
28	C₃×(C₅⋊D₅)⋊C₃	—	31	1	no	—
29	C₃×#3874	—	31	2	no	—
30	C₂×D₃×C₃₁	—	25	1	no	—
31	C₂²×C₃₁⋊C₃	—	75	1	no	—
32	C₂×C₅×C₃₁	—	15	2	no	—
33	D₅×C₃₁	—	3	2	no	—
34	D₅×C₃₁	—	15	2	no	—
35	C₃×D₅²	—	93	1	no	—
36	#3874⋊C₂	C₃₁⋊C₃	1	1	yes	characteristic, direct product factor
37	C₃×C₃₁⋊C₃	—	25	1	no	Hall
38	C₃×C₅²⋊C₃	—	31	1	no	Hall
39	C₂×C₃×C₃₁	—	25	1	no	—
40	D₃×C₃₁	—	25	2	no	—
41	C₂×C₃₁⋊C₃	—	15	2	no	—
42	C₂×C₃₁⋊C₃	—	25	1	no	—
43	C₂×C₃₁⋊C₃	—	50	1	no	—
44	C₅×C₃₁	—	3	2	no	—
45	C₃×C₅⋊D₅	—	31	1	no	—
46	C₃×C₅×D₅	—	93	2	no	—
47	(C₅⋊D₅)⋊C₃	—	62	1	no	—
48	(C₅⋊D₅)⋊C₃	C₂×C₃₁⋊C₃	1	1	yes	characteristic, complemented
49	#3874	C₂×C₃₁⋊C₃	1	2	yes	complemented
50	C₂²×C₃₁	—	75	1	no	Hall
51	D₅²	—	3	1	no	Hall
52	C₃×C₃₁	—	25	1	no	—
53	C₃₁⋊C₃	—	50	1	no	—
54	C₃₁⋊C₃	#3874⋊C₂	1	1	yes	characteristic, direct product factor
55	C₃×C₅²	—	31	1	no	—
56	C₅²⋊C₃	—	62	1	no	—
57	C₅²⋊C₃	C₂²×C₃₁⋊C₃	1	1	yes	characteristic, complemented
58	C₂×C₃₁	—	15	2	no	—
59	C₂×C₃₁	—	25	1	no	—
60	C₂×C₃×D₅	—	465	2	no	—
61	C₅⋊D₅	D₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented
62	C₅×D₅	—	3	2	no	—
63	C₂×C₃×D₃	—	775	1	no	Hall
64	C₃₁	C₃×#3874⋊C₂	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₃×C₅	—	465	2	no	—
66	C₃×D₅	—	93	2	no	—
67	C₃×D₅	—	465	2	no	—
68	C₅²	C₂×D₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
69	C₂×D₅	—	15	2	no	—
70	C₂×C₃²	—	775	1	no	—
71	C₃×D₃	—	775	2	no	—
72	C₃×C₅	—	93	2	no	—
73	C₂²×C₃	—	2325	1	no	—
74	C₂×D₃	—	25	1	no	—
75	C₂×C₅	—	15	2	no	—
76	D₅	—	3	2	no	—
77	D₅	—	15	2	no	—
78	C₃²	—	775	1	no	Sylow
79	C₂×C₃	—	775	1	no	—
80	C₂×C₃	—	1550	1	no	—
81	C₂×C₃	—	25	1	no	—
82	C₂×C₃	—	465	2	no	—
83	D₃	—	25	2	no	—
84	C₅	—	3	2	no	—
85	C₂²	—	75	1	no	Sylow
86	C₃	—	1550	1	no	—
87	C₃	—	25	1	no	—
88	C₃	—	31	1	no	—
89	C₂	—	15	2	no	—
90	C₂	—	25	1	no	—

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

Complements

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

N	H ≅ G/N	type
1. C₃₁⋊C₃×(C₅⋊D₅)⋊C₃	C₂	semidirect
2. C₃₁⋊C₃×#3874	C₂	semidirect
3. C₃₁×#3874⋊C₂	C₃	semidirect
5. C₅²⋊C₃×C₃₁⋊C₃	C₂²	semidirect
7. C₃₁×(C₅⋊D₅)⋊C₃	C₂×C₃	semidirect
8. C₃₁×#3874	C₂×C₃	semidirect
9. C₅⋊D₅×C₃₁⋊C₃	D₃	semidirect
13. C₃₁×C₅²⋊C₃	C₂²×C₃	semidirect
14. C₅²×C₃₁⋊C₃	C₂×D₃	semidirect
16. C₃₁×C₅⋊D₅	C₃×D₃	semidirect
23. C₅²×C₃₁	C₂×C₃×D₃	semidirect
36. #3874⋊C₂	C₃₁⋊C₃	direct (N × H)
48. (C₅⋊D₅)⋊C₃	C₂×C₃₁⋊C₃	semidirect
49. #3874	C₂×C₃₁⋊C₃	semidirect
54. C₃₁⋊C₃	#3874⋊C₂	direct (N × H)
57. C₅²⋊C₃	C₂²×C₃₁⋊C₃	semidirect
61. C₅⋊D₅	D₃×C₃₁⋊C₃	semidirect
64. C₃₁	C₃×#3874⋊C₂	semidirect
68. C₅²	C₂×D₃×C₃₁⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃₁⋊C₃×(C₅⋊D₅)⋊C₃	2
C₃₁⋊C₃×#3874	2
C₃₁×#3874⋊C₂	3
D₅²×C₃₁⋊C₃	3
C₂×D₃×C₃₁⋊C₃	25
C₃×#3874⋊C₂	31

Subgroups — C31⋊C3×#3874⋊C2 #42124690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁⋊C₃×#3874⋊C₂ #42124690