Subgroups — #44478329

Order 30420.

960 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅⋊(C₁₃⋊D₁₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×C₃⋊(C₃⋊(C₁₃⋊D₁₃))	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃²×C₃⋊D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₁₅⋊(C₁₃⋊D₁₃)	—	3	4	no	maximal
5	C₃²×C₅×C₁₃²	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃⋊D₃×C₁₃⋊D₁₃	—	5	1	no	maximal, Hall
7	C₁₃²×D₁₅	—	3	4	no	—
8	C₅×C₃⋊(C₁₃⋊D₁₃)	—	3	4	no	—
9	C₃×C₅⋊(C₁₃⋊D₁₃)	D₃	1	4	yes	complemented
10	D₅×C₁₃⋊D₁₃	—	9	1	no	Hall
11	C₃⋊(C₃⋊(C₁₃⋊D₁₃))	D₅	1	1	yes	characteristic, complemented
12	C₁₃²×C₃⋊D₃	—	5	1	no	—
13	C₃²×C₁₃⋊D₁₃	—	5	1	no	—
14	C₃×C₅×C₁₃²	C₂×D₃	1	4	yes	complemented
15	D₆₅⋊(C₃⋊D₃)	—	13	14	no	maximal
16	D₃×C₁₃⋊D₁₃	—	15	4	no	—
17	C₅⋊(C₁₃⋊D₁₃)	C₃⋊D₃	1	1	yes	characteristic, complemented
18	D₅×C₁₃²	—	9	1	no	—
19	C₅×C₁₃⋊D₁₃	—	9	1	no	—
20	C₃²×C₁₃²	C₂×D₅	1	1	yes	characteristic, complemented, Hall
21	C₁₃×C₃⋊D₁₅	D₁₃	1	14	yes	complemented
22	C₅×C₃⋊D₃₉	—	13	14	no	—
23	C₃²×D₆₅	—	13	14	no	—
24	C₃⋊(C₁₃⋊D₁₃)	—	3	4	no	—
25	D₃×C₁₃²	—	15	4	no	—
26	C₃×C₁₃⋊D₁₃	—	5	4	no	—
27	C₅×C₁₃²	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
28	D₁₅⋊D₁₃	—	39	56	no	—
29	C₂×C₁₃⋊D₁₃	—	45	1	no	Hall
30	C₃²×C₅×C₁₃	C₂×D₁₃	1	14	yes	complemented
31	C₃×C₁₃²	D₃×D₅	1	4	yes	complemented
32	C₃⋊D₃×D₁₃	—	65	14	no	—
33	C₁₃×D₁₅	—	3	56	no	—
34	C₅×D₃₉	—	39	56	no	—
35	C₃×D₆₅	—	13	56	no	—
36	C₂×C₁₃²	—	45	1	no	—
37	C₁₃⋊D₁₃	—	5	1	no	—
38	C₁₃⋊D₁₃	—	9	1	no	—
39	D₅×D₁₃	—	117	14	no	—
40	C₃⋊D₃₉	—	13	14	no	—
41	C₁₃×C₃⋊D₃	—	5	14	no	—
42	C₃²×D₁₃	—	65	14	no	—
43	C₃×C₅×C₁₃	D₃×D₁₃	1	56	yes	complemented
44	D₅×C₃⋊D₃	—	169	1	no	Hall
45	C₁₃²	D₅×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
46	D₃×D₁₃	—	195	56	no	—
47	D₆₅	—	13	14	no	—
48	C₅×D₁₃	—	117	14	no	—
49	D₅×C₁₃	—	9	14	no	—
50	C₃²×C₁₃	D₅×D₁₃	1	14	yes	complemented
51	C₃⋊D₁₅	C₁₃⋊D₁₃	1	1	yes	characteristic, complemented
52	C₅×C₃⋊D₃	—	169	1	no	—
53	C₃²×D₅	—	169	1	no	—
54	D₃₉	—	39	56	no	—
55	C₃×D₁₃	—	65	56	no	—
56	D₃×C₁₃	—	15	56	no	—
57	C₅×C₁₃	C₃⋊D₃×D₁₃	1	14	yes	complemented
58	D₃×D₅	—	507	4	no	—
59	C₂×D₁₃	—	585	14	no	—
60	C₃²×C₅	C₂×C₁₃⋊D₁₃	1	1	yes	characteristic, complemented, Hall
61	C₃×C₁₃	D₁₅⋊D₁₃	1	56	yes	complemented
62	C₂×C₃⋊D₃	—	845	1	no	Hall
63	D₁₅	—	3	4	no	—
64	C₃×D₅	—	169	4	no	—
65	C₅×D₃	—	507	4	no	—
66	C₂×C₁₃	—	45	14	no	—
67	D₁₃	—	65	14	no	—
68	D₁₃	—	117	14	no	—
69	C₂×D₅	—	1521	1	no	Hall
70	C₂×C₃²	—	845	1	no	—
71	C₃⋊D₃	—	5	1	no	—
72	C₃⋊D₃	—	169	1	no	—
73	C₃×C₅	D₃×C₁₃⋊D₁₃	1	4	yes	complemented
74	C₁₃	D₆₅⋊(C₃⋊D₃)	1	14	yes	complemented
75	C₂×D₃	—	2535	4	no	—
76	C₂×C₅	—	1521	1	no	—
77	D₅	—	9	1	no	—
78	D₅	—	169	1	no	—
79	C₃²	D₅×C₁₃⋊D₁₃	1	1	yes	characteristic, complemented, Sylow
80	C₂×C₃	—	845	4	no	—
81	D₃	—	15	4	no	—
82	D₃	—	507	4	no	—
83	C₅	C₃⋊D₃×C₁₃⋊D₁₃	1	1	yes	characteristic, complemented, Sylow
84	C₂²	—	7605	1	no	Sylow
85	C₃	D₁₅⋊(C₁₃⋊D₁₃)	1	4	yes	complemented
86	C₂	—	45	1	no	—
87	C₂	—	845	1	no	—
88	C₂	—	1521	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₅×C₁₃²	C₂²
LCS	G ▷ C₃²×C₅×C₁₃²
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅⋊(C₁₃⋊D₁₃)	C₂	semidirect
2. C₅×C₃⋊(C₃⋊(C₁₃⋊D₁₃))	C₂	semidirect
3. C₁₃²×C₃⋊D₁₅	C₂	semidirect
5. C₃²×C₅×C₁₃²	C₂²	semidirect
9. C₃×C₅⋊(C₁₃⋊D₁₃)	D₃	semidirect
11. C₃⋊(C₃⋊(C₁₃⋊D₁₃))	D₅	semidirect
14. C₃×C₅×C₁₃²	C₂×D₃	semidirect
17. C₅⋊(C₁₃⋊D₁₃)	C₃⋊D₃	semidirect
20. C₃²×C₁₃²	C₂×D₅	semidirect
21. C₁₃×C₃⋊D₁₅	D₁₃	semidirect
27. C₅×C₁₃²	C₂×C₃⋊D₃	semidirect
30. C₃²×C₅×C₁₃	C₂×D₁₃	semidirect
31. C₃×C₁₃²	D₃×D₅	semidirect
43. C₃×C₅×C₁₃	D₃×D₁₃	semidirect
45. C₁₃²	D₅×C₃⋊D₃	semidirect
50. C₃²×C₁₃	D₅×D₁₃	semidirect
51. C₃⋊D₁₅	C₁₃⋊D₁₃	semidirect
57. C₅×C₁₃	C₃⋊D₃×D₁₃	semidirect
60. C₃²×C₅	C₂×C₁₃⋊D₁₃	semidirect
61. C₃×C₁₃	D₁₅⋊D₁₃	semidirect
73. C₃×C₅	D₃×C₁₃⋊D₁₃	semidirect
74. C₁₃	D₆₅⋊(C₃⋊D₃)	semidirect
79. C₃²	D₅×C₁₃⋊D₁₃	semidirect
83. C₅	C₃⋊D₃×C₁₃⋊D₁₃	semidirect
85. C₃	D₁₅⋊(C₁₃⋊D₁₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₅⋊(C₁₃⋊D₁₃)	2
C₅×C₃⋊(C₃⋊(C₁₃⋊D₁₃))	2
C₁₃²×C₃⋊D₁₅	2
D₁₅⋊(C₁₃⋊D₁₃)	3
C₃⋊D₃×C₁₃⋊D₁₃	5
D₆₅⋊(C₃⋊D₃)	13