Subgroups — C₄⋊(C₅⋊((C₃⋊D₃)⋊C₃)) #11761865

Order 1080.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅⋊((C₃⋊D₃)⋊C₃)	C₂	1	2	yes	maximal, complemented
2	C₄×C₅×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₃⋊D₁₅)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×D₆₀	—	3	1	no	maximal
5	C₂×C₅×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₅⋊((C₃⋊D₃)⋊C₃)	—	2	2	no	—
7	C₄⋊((C₃⋊D₃)⋊C₃)	—	5	1	no	maximal, Hall
8	C₂×C₃⋊D₁₅	C₂×C₃	1	2	yes	complemented
9	C₂×C₃×D₁₅	—	3	2	no	—
10	C₃²×C₄×C₅	—	2	1	no	—
11	C₃²×C₄×C₅	D₃	1	1	yes	characteristic, complemented
12	C₃²×C₄×C₅	C₂×C₃	1	1	yes	characteristic, complemented
13	C₅×H(3,𝔽₃)	D₄	1	1	yes	characteristic, complemented, Hall
14	D₆₀	—	3	1	no	—
15	D₆₀	—	9	1	no	—
16	C₃×D₂₀	—	9	1	no	—
17	C₂×(C₃⋊D₃)⋊C₃	—	5	2	no	—
18	C₄×H(3,𝔽₃)	D₅	1	1	yes	characteristic, complemented
19	C₂×C₃²×C₅	—	2	1	no	—
20	C₂×C₃²×C₅	C₂×D₃	1	1	yes	characteristic
21	C₂×C₃²×C₅	C₂²×C₃	1	1	yes	characteristic
22	C₃⋊D₁₅	—	2	2	no	—
23	C₃×D₁₅	—	6	2	no	—
24	C₄⋊(C₃⋊D₃)	—	5	1	no	—
25	C₃×D₁₂	—	15	1	no	—
26	C₂×D₁₅	—	3	2	no	—
27	C₂×D₁₅	—	9	2	no	—
28	C₂×C₃×D₅	—	9	2	no	—
29	C₃×C₄×C₅	—	3	1	no	—
30	C₃×C₄×C₅	—	3	1	no	—
31	C₃×C₄×C₅	—	6	1	no	—
32	C₃×C₄×C₅	C₃×D₃	1	1	yes	characteristic
33	C₂×H(3,𝔽₃)	C₂×D₅	1	1	yes	characteristic
34	(C₃⋊D₃)⋊C₃	—	10	2	no	—
35	C₃²×C₅	—	2	1	no	—
36	C₃²×C₅	D₁₂	1	1	yes	characteristic, complemented
37	C₃²×C₅	C₃×D₄	1	1	yes	characteristic, complemented
38	D₂₀	—	9	1	no	Hall
39	C₂×C₃⋊D₃	—	5	2	no	—
40	C₂×C₃×D₃	—	15	2	no	—
41	C₃²×C₄	—	2	1	no	—
42	C₃²×C₄	C₃×D₅	1	1	yes	characteristic, complemented
43	C₃²×C₄	D₁₅	1	1	yes	characteristic, complemented
44	C₂×C₃×C₅	—	3	1	no	—
45	C₂×C₃×C₅	—	3	1	no	—
46	C₂×C₃×C₅	—	6	1	no	—
47	C₂×C₃×C₅	C₂×C₃×D₃	1	1	yes	characteristic
48	D₁₅	—	6	2	no	—
49	D₁₅	—	18	2	no	—
50	C₃×D₅	—	18	2	no	—
51	H(3,𝔽₃)	D₂₀	1	1	yes	characteristic, complemented, Sylow
52	C₃×D₄	—	45	1	no	—
53	D₁₂	—	15	1	no	—
54	D₁₂	—	45	1	no	—
55	C₂×D₅	—	9	2	no	—
56	C₄×C₅	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, complemented
57	C₂×C₃²	—	2	1	no	—
58	C₂×C₃²	C₂×C₃×D₅	1	1	yes	characteristic
59	C₂×C₃²	C₂×D₁₅	1	1	yes	characteristic
60	C₃⋊D₃	—	10	2	no	—
61	C₃×D₃	—	30	2	no	—
62	C₃×C₅	—	3	1	no	—
63	C₃×C₅	—	3	1	no	—
64	C₃×C₅	—	6	1	no	—
65	C₃×C₅	C₃×D₁₂	1	1	yes	characteristic
66	C₂²×C₃	—	45	2	no	—
67	C₂×D₃	—	15	2	no	—
68	C₂×D₃	—	45	2	no	—
69	C₃×C₄	—	3	1	no	—
70	C₃×C₄	—	3	1	no	—
71	C₃×C₄	—	6	1	no	—
72	C₃×C₄	C₃×D₁₅	1	1	yes	characteristic
73	C₂×C₅	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
74	D₅	—	18	2	no	—
75	C₃²	—	2	1	no	—
76	C₃²	C₃×D₂₀	1	1	yes	characteristic, complemented
77	C₃²	D₆₀	1	1	yes	characteristic, complemented
78	D₄	—	45	1	no	Sylow
79	C₂×C₃	—	3	1	no	—
80	C₂×C₃	—	3	1	no	—
81	C₂×C₃	—	6	1	no	—
82	C₂×C₃	—	90	2	no	—
83	C₂×C₃	C₂×C₃×D₁₅	1	1	yes	characteristic
84	D₃	—	30	2	no	—
85	D₃	—	90	2	no	—
86	C₅	C₄⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic, complemented, Sylow
87	C₂²	—	45	2	no	—
88	C₄	C₅⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic, complemented
89	C₃	—	3	1	no	—
90	C₃	—	3	1	no	—
91	C₃	—	6	1	no	—
92	C₃	C₃×D₆₀	1	1	yes	characteristic
93	C₂	—	90	2	no	—
94	C₂	C₂×C₅⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₅⋊((C₃⋊D₃)⋊C₃)
commutator, cosocle	C₂×C₃²×C₅	C₂²×C₃
Frattini	C₂×C₃	C₂×C₃×D₁₅
Fitting	C₄×C₅×H(3,𝔽₃)	C₂
socle	C₂×C₃×C₅	C₂×C₃×D₃
hypercenter	C₄	C₅⋊((C₃⋊D₃)⋊C₃)
hypocenter	C₃²×C₅	C₃×D₄
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₅⋊((C₃⋊D₃)⋊C₃)	C₂	semidirect
2. C₄×C₅×H(3,𝔽₃)	C₂	semidirect
3. C₄⋊(C₃⋊D₁₅)	C₃	semidirect
8. C₂×C₃⋊D₁₅	C₂×C₃	semidirect
11. C₃²×C₄×C₅	D₃	semidirect
12. C₃²×C₄×C₅	C₂×C₃	semidirect
13. C₅×H(3,𝔽₃)	D₄	semidirect
18. C₄×H(3,𝔽₃)	D₅	semidirect
36. C₃²×C₅	D₁₂	semidirect
37. C₃²×C₅	C₃×D₄	semidirect
42. C₃²×C₄	C₃×D₅	semidirect
43. C₃²×C₄	D₁₅	semidirect
51. H(3,𝔽₃)	D₂₀	semidirect
56. C₄×C₅	(C₃⋊D₃)⋊C₃	semidirect
76. C₃²	C₃×D₂₀	semidirect
77. C₃²	D₆₀	semidirect
86. C₅	C₄⋊((C₃⋊D₃)⋊C₃)	semidirect
88. C₄	C₅⋊((C₃⋊D₃)⋊C₃)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅⋊((C₃⋊D₃)⋊C₃)	2
C₄×C₅×H(3,𝔽₃)	2
C₄⋊(C₃⋊D₁₅)	3
C₃×D₆₀	3
C₄⋊((C₃⋊D₃)⋊C₃)	5

Subgroups — C4⋊(C5⋊((C3⋊D3)⋊C3)) #11761865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊(C₅⋊((C₃⋊D₃)⋊C₃)) #11761865