Subgroups — D₃₁×C₃⋊(C₃⋊(C₅⋊D₅)) #42124964

Order 27900.

480 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅²×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×C₃⋊(C₃⋊(C₅⋊D₅))	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊(C₃⋊(C₅⋊D₁₅₅))	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃₁×C₃⋊(C₅⋊D₅)	—	3	4	no	maximal
5	C₃²×C₅²×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₃₁×C₃⋊D₁₅	—	5	6	no	maximal
7	C₃⋊(C₅⋊D₁₅₅)	—	3	4	no	—
8	C₃₁×C₃⋊(C₅⋊D₅)	—	3	4	no	—
9	C₃×C₅²×D₃₁	D₃	1	4	yes	complemented
10	C₅⋊D₅×D₃₁	—	9	1	no	Hall
11	C₃⋊D₄₆₅	—	5	6	no	—
12	C₃₁×C₃⋊D₁₅	—	5	6	no	—
13	C₃²×C₅×D₃₁	D₅	1	6	yes	complemented
14	C₃×C₅²×C₃₁	C₂×D₃	1	4	yes	complemented
15	D₁₅×D₃₁	—	15	24	no	—
16	C₅⋊D₁₅₅	—	9	1	no	—
17	C₃₁×C₅⋊D₅	—	9	1	no	—
18	C₅²×D₃₁	C₃⋊D₃	1	1	yes	characteristic, complemented
19	C₃²×C₅×C₃₁	C₂×D₅	1	6	yes	complemented
20	C₃⋊D₃×D₃₁	—	25	1	no	Hall
21	D₄₆₅	—	15	24	no	—
22	D₁₅×C₃₁	—	15	24	no	—
23	C₃×C₅×D₃₁	D₁₅	1	24	yes	complemented
24	C₂×C₃⋊(C₃⋊(C₅⋊D₅))	—	31	1	no	maximal, Hall
25	C₅²×C₃₁	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
26	D₅×D₃₁	—	45	6	no	—
27	C₃⋊D₉₃	—	25	1	no	—
28	C₃⋊D₃×C₃₁	—	25	1	no	—
29	C₃²×D₃₁	C₅⋊D₅	1	1	yes	characteristic, complemented
30	C₃×C₅×C₃₁	C₂×D₁₅	1	24	yes	complemented
31	C₂×C₃²×C₅²	—	31	1	no	—
32	C₃⋊(C₃⋊(C₅⋊D₅))	—	31	1	no	—
33	C₃⋊(C₃⋊(C₅⋊D₅))	D₃₁	1	1	yes	characteristic, direct product factor
34	D₃×D₃₁	—	75	4	no	—
35	D₁₅₅	—	45	6	no	—
36	C₅×D₃₁	C₃⋊D₁₅	1	6	yes	complemented
37	D₅×C₃₁	—	45	6	no	—
38	C₂×C₃⋊(C₅⋊D₅)	—	93	4	no	—
39	C₃²×C₃₁	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
40	C₃²×C₅²	C₂×D₃₁	1	1	yes	characteristic, complemented, Hall
41	D₉₃	—	75	4	no	—
42	C₃×D₃₁	C₃⋊(C₅⋊D₅)	1	4	yes	complemented
43	D₃×C₃₁	—	75	4	no	—
44	C₂×C₃⋊D₁₅	—	155	6	no	—
45	C₅×C₃₁	C₂×C₃⋊D₁₅	1	6	yes	complemented
46	C₂×C₃×C₅²	—	31	4	no	—
47	C₃⋊(C₅⋊D₅)	—	3	4	no	—
48	C₃⋊(C₅⋊D₅)	—	93	4	no	—
49	C₂×D₃₁	—	225	1	no	Hall
50	C₂×C₅⋊D₅	—	279	1	no	Hall
51	C₃×C₃₁	C₂×C₃⋊(C₅⋊D₅)	1	4	yes	complemented
52	C₂×C₃²×C₅	—	31	6	no	—
53	C₃⋊D₁₅	—	5	6	no	—
54	C₃⋊D₁₅	—	155	6	no	—
55	C₃×C₅²	D₃×D₃₁	1	4	yes	complemented
56	C₂×C₃₁	—	225	1	no	—
57	D₃₁	—	225	1	no	—
58	D₃₁	C₃⋊(C₃⋊(C₅⋊D₅))	1	1	yes	characteristic, direct product factor
59	C₂×D₁₅	—	465	24	no	—
60	C₂×C₅²	—	31	1	no	—
61	C₅⋊D₅	—	9	1	no	—
62	C₅⋊D₅	—	279	1	no	—
63	C₃²×C₅	D₅×D₃₁	1	6	yes	complemented
64	C₂×C₃⋊D₃	—	775	1	no	Hall
65	C₃₁	C₂×C₃⋊(C₃⋊(C₅⋊D₅))	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃×C₅	—	31	24	no	—
67	D₁₅	—	15	24	no	—
68	D₁₅	—	465	24	no	—
69	C₅²	C₃⋊D₃×D₃₁	1	1	yes	characteristic, complemented, Sylow
70	C₂×D₅	—	1395	6	no	—
71	C₂×C₃²	—	31	1	no	—
72	C₃⋊D₃	—	775	1	no	—
73	C₃⋊D₃	—	25	1	no	—
74	C₃×C₅	D₁₅×D₃₁	1	24	yes	complemented
75	C₂×D₃	—	2325	4	no	—
76	C₂×C₅	—	31	6	no	—
77	D₅	—	45	6	no	—
78	D₅	—	1395	6	no	—
79	C₃²	C₅⋊D₅×D₃₁	1	1	yes	characteristic, complemented, Sylow
80	C₂×C₃	—	31	4	no	—
81	D₃	—	2325	4	no	—
82	D₃	—	75	4	no	—
83	C₅	D₃₁×C₃⋊D₁₅	1	6	yes	complemented
84	C₂²	—	6975	1	no	Sylow
85	C₃	D₃₁×C₃⋊(C₅⋊D₅)	1	4	yes	complemented
86	C₂	—	31	1	no	—
87	C₂	—	6975	1	no	—
88	C₂	—	225	1	no	—

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

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×D₃₁	C₂	semidirect
2. C₃₁×C₃⋊(C₃⋊(C₅⋊D₅))	C₂	semidirect
3. C₃⋊(C₃⋊(C₅⋊D₁₅₅))	C₂	semidirect
5. C₃²×C₅²×C₃₁	C₂²	semidirect
9. C₃×C₅²×D₃₁	D₃	semidirect
13. C₃²×C₅×D₃₁	D₅	semidirect
14. C₃×C₅²×C₃₁	C₂×D₃	semidirect
18. C₅²×D₃₁	C₃⋊D₃	semidirect
19. C₃²×C₅×C₃₁	C₂×D₅	semidirect
23. C₃×C₅×D₃₁	D₁₅	semidirect
25. C₅²×C₃₁	C₂×C₃⋊D₃	semidirect
29. C₃²×D₃₁	C₅⋊D₅	semidirect
30. C₃×C₅×C₃₁	C₂×D₁₅	semidirect
33. C₃⋊(C₃⋊(C₅⋊D₅))	D₃₁	direct (N × H)
36. C₅×D₃₁	C₃⋊D₁₅	semidirect
39. C₃²×C₃₁	C₂×C₅⋊D₅	semidirect
40. C₃²×C₅²	C₂×D₃₁	semidirect
42. C₃×D₃₁	C₃⋊(C₅⋊D₅)	semidirect
45. C₅×C₃₁	C₂×C₃⋊D₁₅	semidirect
51. C₃×C₃₁	C₂×C₃⋊(C₅⋊D₅)	semidirect
55. C₃×C₅²	D₃×D₃₁	semidirect
58. D₃₁	C₃⋊(C₃⋊(C₅⋊D₅))	direct (N × H)
63. C₃²×C₅	D₅×D₃₁	semidirect
65. C₃₁	C₂×C₃⋊(C₃⋊(C₅⋊D₅))	semidirect
69. C₅²	C₃⋊D₃×D₃₁	semidirect
74. C₃×C₅	D₁₅×D₃₁	semidirect
79. C₃²	C₅⋊D₅×D₃₁	semidirect
83. C₅	D₃₁×C₃⋊D₁₅	semidirect
85. C₃	D₃₁×C₃⋊(C₅⋊D₅)	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₅²×D₃₁	2
C₃₁×C₃⋊(C₃⋊(C₅⋊D₅))	2
C₃⋊(C₃⋊(C₅⋊D₁₅₅))	2
D₃₁×C₃⋊(C₅⋊D₅)	3
D₃₁×C₃⋊D₁₅	5
C₂×C₃⋊(C₃⋊(C₅⋊D₅))	31

Subgroups — D31×C3⋊(C3⋊(C5⋊D5)) #42124964

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃₁×C₃⋊(C₃⋊(C₅⋊D₅)) #42124964