Subgroups — C₃₁×(C₃⋊D₃)⋊D₂₅ #42124446

Order 27900.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊D₃×C₂₅×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₃₁×D₇₅	C₂	1	2	yes	maximal, complemented
3	D₃×C₃₁×D₂₅	—	3	2	no	maximal
4	C₃²×C₂₅×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₃₁×(C₃⋊D₃)⋊D₅	—	5	1	no	maximal
6	C₃₁×D₇₅	D₃	1	2	yes	complemented
7	D₃×C₂₅×C₃₁	—	3	2	no	—
8	D₃×C₂₅×C₃₁	—	6	1	no	—
9	C₃×C₃₁×D₂₅	—	3	2	no	—
10	C₂×C₃₁×D₂₅	—	9	1	no	Hall
11	C₅×C₃⋊D₃×C₃₁	D₅	1	1	yes	characteristic
12	C₃×D₁₅×C₃₁	—	5	2	no	—
13	C₃×C₂₅×C₃₁	—	2	1	no	—
14	C₃×C₂₅×C₃₁	C₂×D₃	1	2	yes	complemented
15	D₃×D₅×C₃₁	—	15	2	no	—
16	C₂×C₂₅×C₃₁	—	9	1	no	—
17	C₃₁×D₂₅	—	3	2	no	—
18	C₃²×C₅×C₃₁	C₂×D₅	1	1	yes	characteristic
19	D₃²×C₃₁	—	25	1	no	Hall
20	D₁₅×C₃₁	—	5	2	no	—
21	C₅×D₃×C₃₁	—	3	2	no	—
22	C₅×D₃×C₃₁	—	6	1	no	—
23	C₃×D₅×C₃₁	—	15	2	no	—
24	(C₃⋊D₃)⋊D₂₅	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂₅×C₃₁	D₃²	1	1	yes	characteristic, complemented, Hall
26	C₂×D₅×C₃₁	—	45	1	no	—
27	C₃⋊D₃×C₃₁	D₂₅	1	1	yes	characteristic, complemented
28	C₃×D₃×C₃₁	—	25	2	no	—
29	C₃×C₅×C₃₁	—	2	1	no	—
30	C₃×C₅×C₃₁	D₃×D₅	1	2	yes	—
31	C₃⋊D₃×C₂₅	C₂×C₃₁	1	1	yes	characteristic, complemented
32	C₃×D₇₅	C₂×C₃₁	1	2	yes	complemented
33	C₂×D₃×C₃₁	—	75	2	no	—
34	C₂×C₅×C₃₁	—	9	1	no	—
35	D₅×C₃₁	—	15	2	no	—
36	D₃×D₂₅	—	3	2	no	—
37	C₃²×C₃₁	C₂×D₂₅	1	1	yes	characteristic, complemented, Hall
38	C₃²×C₂₅	C₂²×C₃₁	1	1	yes	characteristic, complemented, Hall
39	C₂×C₃×C₃₁	—	75	2	no	—
40	D₃×C₃₁	—	3	2	no	—
41	D₃×C₃₁	—	6	1	no	—
42	D₃×C₃₁	—	25	2	no	—
43	(C₃⋊D₃)⋊D₅	—	5	1	no	—
44	C₅×C₃₁	(C₃⋊D₃)⋊D₅	1	1	yes	characteristic
45	D₇₅	D₃×C₃₁	1	2	yes	complemented
46	C₃×D₂₅	—	3	2	no	—
47	D₃×C₂₅	—	3	2	no	—
48	D₃×C₂₅	—	6	1	no	—
49	C₂²×C₃₁	—	225	1	no	Hall
50	C₂×D₂₅	—	9	1	no	Hall
51	C₃×C₃₁	—	2	1	no	—
52	C₃×C₃₁	D₃×D₂₅	1	2	yes	complemented
53	C₅×C₃⋊D₃	D₅×C₃₁	1	1	yes	characteristic
54	C₃×D₁₅	—	5	2	no	—
55	C₃×C₂₅	—	2	1	no	—
56	C₃×C₂₅	C₂×D₃×C₃₁	1	2	yes	complemented
57	C₂×C₃₁	—	9	1	no	—
58	C₂×C₃₁	—	75	2	no	—
59	D₃×D₅	—	15	2	no	—
60	C₂×C₂₅	—	9	1	no	—
61	D₂₅	—	3	2	no	—
62	C₃²×C₅	C₂×D₅×C₃₁	1	1	yes	characteristic
63	D₃²	—	25	1	no	Hall
64	C₃₁	(C₃⋊D₃)⋊D₂₅	1	1	yes	characteristic, direct product factor, Sylow
65	D₁₅	—	5	2	no	—
66	C₃×D₅	—	15	2	no	—
67	C₅×D₃	—	3	2	no	—
68	C₅×D₃	—	6	1	no	—
69	C₂₅	D₃²×C₃₁	1	1	yes	characteristic, complemented, Sylow
70	C₂×D₅	—	45	1	no	—
71	C₃⋊D₃	C₃₁×D₂₅	1	1	yes	characteristic, complemented
72	C₃×D₃	—	25	2	no	—
73	C₃×C₅	—	2	1	no	—
74	C₃×C₅	D₃×D₅×C₃₁	1	2	yes	—
75	C₂×D₃	—	75	2	no	—
76	C₂×C₅	—	9	1	no	—
77	D₅	—	15	2	no	—
78	C₃²	C₂×C₃₁×D₂₅	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₃	—	75	2	no	—
80	D₃	—	3	2	no	—
81	D₃	—	6	1	no	—
82	D₃	—	25	2	no	—
83	C₅	C₃₁×(C₃⋊D₃)⋊D₅	1	1	yes	characteristic
84	C₂²	—	225	1	no	Sylow
85	C₃	—	2	1	no	—
86	C₃	D₃×C₃₁×D₂₅	1	2	yes	complemented
87	C₂	—	9	1	no	—
88	C₂	—	75	2	no	—

	subgroup	G/H
center, hypercenter	C₃₁	(C₃⋊D₃)⋊D₂₅
commutator, cosocle, hypocenter	C₃²×C₂₅	C₂²×C₃₁
Frattini	C₅	C₃₁×(C₃⋊D₃)⋊D₅
Fitting	C₃²×C₂₅×C₃₁	C₂²
socle	C₃²×C₅×C₃₁	C₂×D₅
LCS	G ▷ C₃²×C₂₅
UCS	C₁ ◁ C₃₁

Complements

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

N	H ≅ G/N	type
1. C₃⋊D₃×C₂₅×C₃₁	C₂	semidirect
2. C₃×C₃₁×D₇₅	C₂	semidirect
4. C₃²×C₂₅×C₃₁	C₂²	semidirect
6. C₃₁×D₇₅	D₃	semidirect
14. C₃×C₂₅×C₃₁	C₂×D₃	semidirect
24. (C₃⋊D₃)⋊D₂₅	C₃₁	direct (N × H)
25. C₂₅×C₃₁	D₃²	semidirect
27. C₃⋊D₃×C₃₁	D₂₅	semidirect
31. C₃⋊D₃×C₂₅	C₂×C₃₁	semidirect
32. C₃×D₇₅	C₂×C₃₁	semidirect
37. C₃²×C₃₁	C₂×D₂₅	semidirect
38. C₃²×C₂₅	C₂²×C₃₁	semidirect
45. D₇₅	D₃×C₃₁	semidirect
52. C₃×C₃₁	D₃×D₂₅	semidirect
56. C₃×C₂₅	C₂×D₃×C₃₁	semidirect
64. C₃₁	(C₃⋊D₃)⋊D₂₅	direct (N × H)
69. C₂₅	D₃²×C₃₁	semidirect
71. C₃⋊D₃	C₃₁×D₂₅	semidirect
78. C₃²	C₂×C₃₁×D₂₅	semidirect
86. C₃	D₃×C₃₁×D₂₅	semidirect

Maximal subgroups

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

subgroup	index
C₃⋊D₃×C₂₅×C₃₁	2
C₃×C₃₁×D₇₅	2
D₃×C₃₁×D₂₅	3
C₃₁×(C₃⋊D₃)⋊D₅	5
(C₃⋊D₃)⋊D₂₅	31

Subgroups — C31×(C3⋊D3)⋊D25 #42124446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×(C₃⋊D₃)⋊D₂₅ #42124446