Subgroups — C₃₁×#11744719 #42124961

Order 27900.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃₁×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×C₃₁×C₃⋊D₁₅	C₂	1	2	yes	maximal, complemented
3	C₃₁×D₁₅⋊D₅	—	3	4	no	maximal
4	C₃²×C₅²×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	D₅×C₃⋊D₃×C₃₁	—	5	2	no	maximal
6	C₅×D₁₅×C₃₁	—	3	8	no	—
7	C₃×C₃₁×C₅⋊D₅	D₃	1	4	yes	complemented
8	D₅²×C₃₁	—	9	1	no	Hall
9	C₃₁×C₃⋊D₁₅	D₅	1	2	yes	complemented
10	C₅×C₃⋊D₃×C₃₁	—	5	2	no	—
11	C₃²×D₅×C₃₁	—	5	2	no	—
12	C₃²×D₅×C₃₁	—	10	2	no	—
13	C₃×C₅²×C₃₁	C₂×D₃	1	4	yes	complemented
14	D₃×D₅×C₃₁	—	15	8	no	—
15	C₃₁×C₅⋊D₅	C₃⋊D₃	1	1	yes	characteristic, complemented
16	C₅×D₅×C₃₁	—	9	2	no	—
17	C₃²×C₅×C₃₁	—	2	2	no	—
18	C₃²×C₅×C₃₁	C₂×D₅	1	2	yes	complemented
19	C₂×C₃⋊D₃×C₃₁	—	25	1	no	Hall
20	D₁₅×C₃₁	—	3	8	no	—
21	C₅×D₃×C₃₁	—	15	8	no	—
22	C₃×D₅×C₃₁	—	5	8	no	—
23	C₃×D₅×C₃₁	—	10	8	no	—
24	#11744719	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅²×C₃₁	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
26	C₂×D₅×C₃₁	—	45	2	no	—
27	C₂×C₃²×C₃₁	—	25	1	no	—
28	C₃⋊D₃×C₃₁	—	5	2	no	—
29	C₃×C₅×C₃₁	—	2	8	no	—
30	C₃×C₅×C₃₁	D₃×D₅	1	8	yes	complemented
31	C₅×C₃⋊D₁₅	C₂×C₃₁	1	2	yes	complemented
32	C₃²×C₅⋊D₅	C₂×C₃₁	1	1	yes	characteristic, complemented
33	C₂×D₃×C₃₁	—	75	4	no	—
34	C₂×C₅×C₃₁	—	45	2	no	—
35	D₅×C₃₁	—	5	2	no	—
36	D₅×C₃₁	—	9	2	no	—
37	D₅×C₃₁	—	10	2	no	—
38	D₁₅⋊D₅	—	3	4	no	—
39	C₃²×C₃₁	D₅²	1	1	yes	characteristic, complemented, Hall
40	C₃²×C₅²	C₂²×C₃₁	1	1	yes	characteristic, complemented, Hall
41	C₂×C₃×C₃₁	—	25	4	no	—
42	D₃×C₃₁	—	15	8	no	—
43	D₅×C₃⋊D₃	—	5	2	no	—
44	C₅×C₃₁	—	2	2	no	—
45	C₅×C₃₁	D₅×C₃⋊D₃	1	2	yes	complemented
46	C₅×D₁₅	—	3	8	no	—
47	C₃×C₅⋊D₅	D₃×C₃₁	1	4	yes	complemented
48	C₂²×C₃₁	—	225	1	no	Hall
49	D₅²	—	9	1	no	Hall
50	C₃×C₃₁	D₁₅⋊D₅	1	4	yes	complemented
51	C₃⋊D₁₅	D₅×C₃₁	1	2	yes	complemented
52	C₅×C₃⋊D₃	—	5	2	no	—
53	C₃²×D₅	—	5	2	no	—
54	C₃²×D₅	—	10	2	no	—
55	C₃×C₅²	C₂×D₃×C₃₁	1	4	yes	complemented
56	C₂×C₃₁	—	25	1	no	—
57	C₂×C₃₁	—	45	2	no	—
58	D₃×D₅	—	15	8	no	—
59	C₅⋊D₅	C₃⋊D₃×C₃₁	1	1	yes	characteristic, complemented
60	C₅×D₅	—	9	2	no	—
61	C₃²×C₅	—	2	2	no	—
62	C₃²×C₅	C₂×D₅×C₃₁	1	2	yes	complemented
63	C₂×C₃⋊D₃	—	25	1	no	Hall
64	C₃₁	#11744719	1	1	yes	characteristic, direct product factor, Sylow
65	D₁₅	—	3	8	no	—
66	C₃×D₅	—	5	8	no	—
67	C₃×D₅	—	10	8	no	—
68	C₅×D₃	—	15	8	no	—
69	C₅²	C₂×C₃⋊D₃×C₃₁	1	1	yes	characteristic, complemented, Sylow
70	C₂×D₅	—	45	2	no	—
71	C₂×C₃²	—	25	1	no	—
72	C₃⋊D₃	—	5	2	no	—
73	C₃×C₅	—	2	8	no	—
74	C₃×C₅	D₃×D₅×C₃₁	1	8	yes	complemented
75	C₂×D₃	—	75	4	no	—
76	C₂×C₅	—	45	2	no	—
77	D₅	—	5	2	no	—
78	D₅	—	9	2	no	—
79	D₅	—	10	2	no	—
80	C₃²	D₅²×C₃₁	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₃	—	25	4	no	—
82	D₃	—	15	8	no	—
83	C₅	—	2	2	no	—
84	C₅	D₅×C₃⋊D₃×C₃₁	1	2	yes	complemented
85	C₂²	—	225	1	no	Sylow
86	C₃	C₃₁×D₁₅⋊D₅	1	4	yes	complemented
87	C₂	—	25	1	no	—
88	C₂	—	45	2	no	—

	subgroup	G/H
center, hypercenter	C₃₁	#11744719
commutator, cosocle, hypocenter	C₃²×C₅²	C₂²×C₃₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₁	C₂²
LCS	G ▷ C₃²×C₅²
UCS	C₁ ◁ C₃₁

Complements

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

N	H ≅ G/N	type
1. C₃²×C₃₁×C₅⋊D₅	C₂	semidirect
2. C₅×C₃₁×C₃⋊D₁₅	C₂	semidirect
4. C₃²×C₅²×C₃₁	C₂²	semidirect
7. C₃×C₃₁×C₅⋊D₅	D₃	semidirect
9. C₃₁×C₃⋊D₁₅	D₅	semidirect
13. C₃×C₅²×C₃₁	C₂×D₃	semidirect
15. C₃₁×C₅⋊D₅	C₃⋊D₃	semidirect
18. C₃²×C₅×C₃₁	C₂×D₅	semidirect
24. #11744719	C₃₁	direct (N × H)
25. C₅²×C₃₁	C₂×C₃⋊D₃	semidirect
30. C₃×C₅×C₃₁	D₃×D₅	semidirect
31. C₅×C₃⋊D₁₅	C₂×C₃₁	semidirect
32. C₃²×C₅⋊D₅	C₂×C₃₁	semidirect
39. C₃²×C₃₁	D₅²	semidirect
40. C₃²×C₅²	C₂²×C₃₁	semidirect
45. C₅×C₃₁	D₅×C₃⋊D₃	semidirect
47. C₃×C₅⋊D₅	D₃×C₃₁	semidirect
50. C₃×C₃₁	D₁₅⋊D₅	semidirect
51. C₃⋊D₁₅	D₅×C₃₁	semidirect
55. C₃×C₅²	C₂×D₃×C₃₁	semidirect
59. C₅⋊D₅	C₃⋊D₃×C₃₁	semidirect
62. C₃²×C₅	C₂×D₅×C₃₁	semidirect
64. C₃₁	#11744719	direct (N × H)
69. C₅²	C₂×C₃⋊D₃×C₃₁	semidirect
74. C₃×C₅	D₃×D₅×C₃₁	semidirect
80. C₃²	D₅²×C₃₁	semidirect
84. C₅	D₅×C₃⋊D₃×C₃₁	semidirect
86. C₃	C₃₁×D₁₅⋊D₅	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₃₁×C₅⋊D₅	2
C₅×C₃₁×C₃⋊D₁₅	2
C₃₁×D₁₅⋊D₅	3
D₅×C₃⋊D₃×C₃₁	5
#11744719	31

Subgroups — C31×#11744719 #42124961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11744719 #42124961