Subgroups — C₂×C₃×(C₅⋊D₅)⋊(C₁₃⋊C₃) #26043697

Order 11700.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₁₃⋊(C₅²⋊C₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×(C₅⋊D₅)⋊(C₁₃⋊C₃)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₁₃×C₅⋊D₅	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₅⋊D₅)⋊(C₁₃⋊C₃)	C₃	1	3	yes	maximal, direct product factor
5	C₃×C₁₃⋊(C₅²⋊C₃)	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×C₃×C₅²×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×C₁₃×C₅⋊D₅	C₂×C₃	1	2	yes	complemented
8	C₂×C₁₃⋊(C₅²⋊C₃)	C₂×C₃	1	3	yes	complemented
9	(C₅⋊D₅)⋊(C₁₃⋊C₃)	C₂×C₃	1	6	yes	direct product factor
10	C₂×C₁₃×C₅⋊D₅	C₃²	1	1	yes	characteristic, complemented, Hall
11	C₃×C₅²×C₁₃	C₂²×C₃	1	1	yes	characteristic, complemented
12	C₁₃⋊(C₅²⋊C₃)	C₂²×C₃	1	3	yes	complemented
13	C₂×C₃×(C₅⋊D₅)⋊C₃	—	13	1	no	maximal, Hall
14	C₂×C₃×D₅×C₁₃	—	15	2	no	—
15	C₂×C₅²×C₁₃	C₂×C₃²	1	1	yes	characteristic, complemented
16	C₁₃×C₅⋊D₅	C₂×C₃²	1	2	yes	complemented
17	C₂²×C₃×C₁₃⋊C₃	—	25	1	no	maximal, Hall
18	C₂×C₃×C₅²⋊C₃	—	13	1	no	—
19	C₃×(C₅⋊D₅)⋊C₃	—	13	2	no	—
20	C₂×C₃×C₅×C₁₃	—	3	2	no	—
21	C₃×D₅×C₁₃	—	15	4	no	—
22	C₅²×C₁₃	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
23	C₂×C₃×C₅⋊D₅	C₁₃⋊C₃	1	1	yes	characteristic, complemented
24	C₂×(C₅⋊D₅)⋊C₃	—	13	3	no	—
25	C₂×D₅×C₁₃	—	15	2	no	—
26	C₂×C₃×C₁₃⋊C₃	—	25	2	no	—
27	C₂×C₃×C₁₃⋊C₃	—	25	1	no	—
28	C₃×C₅²⋊C₃	—	13	1	no	Hall
29	C₃×C₅×C₁₃	—	3	2	no	—
30	C₂²×C₃×C₁₃	—	25	1	no	—
31	C₂²×C₁₃⋊C₃	—	25	3	no	—
32	C₂×C₃×C₅²	C₂×C₁₃⋊C₃	1	1	yes	characteristic, complemented
33	C₃×C₅⋊D₅	C₂×C₁₃⋊C₃	1	2	yes	complemented
34	C₂×C₅²⋊C₃	—	13	3	no	—
35	(C₅⋊D₅)⋊C₃	—	13	6	no	—
36	C₂×C₅×C₁₃	—	3	2	no	—
37	D₅×C₁₃	—	15	4	no	—
38	C₃×C₁₃⋊C₃	—	25	1	no	Hall
39	C₂×C₅⋊D₅	C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
40	C₂×C₃×C₁₃	—	25	2	no	—
41	C₂×C₃×C₁₃	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
42	C₂×C₁₃⋊C₃	—	25	6	no	—
43	C₂×C₁₃⋊C₃	—	25	3	no	—
44	C₃×C₅²	C₂²×C₁₃⋊C₃	1	1	yes	characteristic, complemented
45	C₅²⋊C₃	—	13	3	no	—
46	C₅×C₁₃	—	3	2	no	—
47	C₂×C₃×D₅	—	15	2	no	—
48	C₂²×C₁₃	—	25	1	no	Hall
49	C₂×C₅²	C₂×C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented
50	C₅⋊D₅	C₂×C₃×C₁₃⋊C₃	1	2	yes	complemented
51	C₃×C₁₃	C₂×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
52	C₁₃⋊C₃	—	25	3	no	—
53	C₂²×C₃²	—	325	1	no	Hall
54	C₂×C₃×C₅	—	3	2	no	—
55	C₃×D₅	—	15	4	no	—
56	C₂×C₁₃	—	25	2	no	—
57	C₂×C₁₃	C₃×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
58	C₅²	C₂²×C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
59	C₂×D₅	—	15	2	no	—
60	C₂×C₃²	—	325	2	no	—
61	C₂×C₃²	—	325	1	no	—
62	C₃×C₅	—	3	2	no	—
63	C₁₃	C₂×C₃×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented, Sylow
64	C₂²×C₃	—	25	1	no	—
65	C₂²×C₃	—	325	3	no	—
66	C₂×C₅	—	3	2	no	—
67	D₅	—	15	4	no	—
68	C₃²	—	325	1	no	Sylow
69	C₂×C₃	—	25	2	no	—
70	C₂×C₃	—	325	6	no	—
71	C₂×C₃	—	325	3	no	—
72	C₂×C₃	(C₅⋊D₅)⋊(C₁₃⋊C₃)	1	1	yes	characteristic, direct product factor
73	C₅	—	3	2	no	—
74	C₂²	—	25	1	no	Sylow
75	C₃	—	325	3	no	—
76	C₃	C₂×(C₅⋊D₅)⋊(C₁₃⋊C₃)	1	1	yes	characteristic, direct product factor
77	C₂	—	25	2	no	—
78	C₂	C₃×(C₅⋊D₅)⋊(C₁₃⋊C₃)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	(C₅⋊D₅)⋊(C₁₃⋊C₃)
commutator, cosocle, hypocenter	C₅²×C₁₃	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅²×C₁₃	C₂×C₃
LCS	G ▷ C₅²×C₁₃
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₁₃⋊(C₅²⋊C₃)	C₂	semidirect
2. C₃×(C₅⋊D₅)⋊(C₁₃⋊C₃)	C₂	direct (N × H)
3. C₂×C₃×C₁₃×C₅⋊D₅	C₃	semidirect
4. C₂×(C₅⋊D₅)⋊(C₁₃⋊C₃)	C₃	direct (N × H)
5. C₃×C₁₃⋊(C₅²⋊C₃)	C₂²	semidirect
6. C₂×C₃×C₅²×C₁₃	C₂×C₃	semidirect
7. C₃×C₁₃×C₅⋊D₅	C₂×C₃	semidirect
8. C₂×C₁₃⋊(C₅²⋊C₃)	C₂×C₃	semidirect
9. (C₅⋊D₅)⋊(C₁₃⋊C₃)	C₂×C₃	direct (N × H)
10. C₂×C₁₃×C₅⋊D₅	C₃²	semidirect
11. C₃×C₅²×C₁₃	C₂²×C₃	semidirect
12. C₁₃⋊(C₅²⋊C₃)	C₂²×C₃	semidirect
15. C₂×C₅²×C₁₃	C₂×C₃²	semidirect
16. C₁₃×C₅⋊D₅	C₂×C₃²	semidirect
22. C₅²×C₁₃	C₂²×C₃²	semidirect
23. C₂×C₃×C₅⋊D₅	C₁₃⋊C₃	semidirect
32. C₂×C₃×C₅²	C₂×C₁₃⋊C₃	semidirect
33. C₃×C₅⋊D₅	C₂×C₁₃⋊C₃	semidirect
39. C₂×C₅⋊D₅	C₃×C₁₃⋊C₃	semidirect
41. C₂×C₃×C₁₃	(C₅⋊D₅)⋊C₃	semidirect
44. C₃×C₅²	C₂²×C₁₃⋊C₃	semidirect
49. C₂×C₅²	C₂×C₃×C₁₃⋊C₃	semidirect
50. C₅⋊D₅	C₂×C₃×C₁₃⋊C₃	semidirect
51. C₃×C₁₃	C₂×(C₅⋊D₅)⋊C₃	semidirect
57. C₂×C₁₃	C₃×(C₅⋊D₅)⋊C₃	semidirect
58. C₅²	C₂²×C₃×C₁₃⋊C₃	semidirect
63. C₁₃	C₂×C₃×(C₅⋊D₅)⋊C₃	semidirect
72. C₂×C₃	(C₅⋊D₅)⋊(C₁₃⋊C₃)	direct (N × H)
76. C₃	C₂×(C₅⋊D₅)⋊(C₁₃⋊C₃)	direct (N × H)
78. C₂	C₃×(C₅⋊D₅)⋊(C₁₃⋊C₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₁₃⋊(C₅²⋊C₃)	2
C₃×(C₅⋊D₅)⋊(C₁₃⋊C₃)	2
C₂×C₃×C₁₃×C₅⋊D₅	3
C₂×(C₅⋊D₅)⋊(C₁₃⋊C₃)	3
C₂×C₃×(C₅⋊D₅)⋊C₃	13
C₂²×C₃×C₁₃⋊C₃	25

Subgroups — C2×C3×(C5⋊D5)⋊(C13⋊C3) #26043697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×(C₅⋊D₅)⋊(C₁₃⋊C₃) #26043697