Subgroups — #67180⋊C₃ #11761863

Order 1080.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	(C₃⋊Dic₁₅)⋊C₃	C₂	1	2	yes	maximal
3	#67180	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×Dic₃₀	—	3	1	no	maximal
5	C₂×C₅×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	#634⋊C₃	—	5	1	no	maximal, Hall
7	C₃²×C₄×C₅	—	2	1	no	—
8	C₃²×C₄×C₅	D₃	1	1	yes	characteristic
9	C₃²×C₄×C₅	C₂×C₃	1	1	yes	characteristic
10	C₃⋊Dic₁₅	C₂×C₃	1	2	yes	—
11	C₃×Dic₁₅	—	3	2	no	—
12	C₅×H(3,𝔽₃)	Q₈	1	1	yes	characteristic, complemented, Hall
13	Dic₃₀	—	3	1	no	—
14	Dic₃₀	—	9	1	no	—
15	C₃×Dic₁₀	—	9	1	no	—
16	C₄×H(3,𝔽₃)	D₅	1	1	yes	characteristic
17	(C₃⋊Dic₃)⋊C₃	—	5	2	no	—
18	C₂×C₃²×C₅	—	2	1	no	—
19	C₂×C₃²×C₅	C₂×D₃	1	1	yes	characteristic
20	C₂×C₃²×C₅	C₂²×C₃	1	1	yes	characteristic
21	#634	—	5	1	no	—
22	C₃×Dic₆	—	15	1	no	—
23	C₃×C₄×C₅	—	3	1	no	—
24	C₃×C₄×C₅	—	3	1	no	—
25	C₃×C₄×C₅	—	6	1	no	—
26	C₃×C₄×C₅	C₃×D₃	1	1	yes	characteristic
27	Dic₁₅	—	3	2	no	—
28	Dic₁₅	—	9	2	no	—
29	C₃×Dic₅	—	9	2	no	—
30	C₂×H(3,𝔽₃)	C₂×D₅	1	1	yes	characteristic
31	C₃²×C₅	—	2	1	no	—
32	C₃²×C₅	Dic₆	1	1	yes	characteristic, complemented
33	C₃²×C₅	C₃×Q₈	1	1	yes	characteristic, complemented
34	Dic₁₀	—	9	1	no	Hall
35	C₃²×C₄	—	2	1	no	—
36	C₃²×C₄	C₃×D₅	1	1	yes	characteristic
37	C₃²×C₄	D₁₅	1	1	yes	characteristic
38	C₃⋊Dic₃	—	5	2	no	—
39	C₃×Dic₃	—	15	2	no	—
40	C₂×C₃×C₅	—	3	1	no	—
41	C₂×C₃×C₅	—	3	1	no	—
42	C₂×C₃×C₅	—	6	1	no	—
43	C₂×C₃×C₅	C₂×C₃×D₃	1	1	yes	characteristic
44	H(3,𝔽₃)	Dic₁₀	1	1	yes	characteristic, complemented, Sylow
45	C₃×Q₈	—	45	1	no	—
46	Dic₆	—	15	1	no	—
47	Dic₆	—	45	1	no	—
48	C₄×C₅	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
49	Dic₅	—	9	2	no	—
50	C₂×C₃²	—	2	1	no	—
51	C₂×C₃²	C₂×C₃×D₅	1	1	yes	characteristic
52	C₂×C₃²	C₂×D₁₅	1	1	yes	characteristic
53	C₃×C₅	—	3	1	no	—
54	C₃×C₅	—	3	1	no	—
55	C₃×C₅	—	6	1	no	—
56	C₃×C₅	C₃×Dic₆	1	1	yes	characteristic
57	C₃×C₄	—	3	1	no	—
58	C₃×C₄	—	3	1	no	—
59	C₃×C₄	—	6	1	no	—
60	C₃×C₄	—	45	2	no	—
61	C₃×C₄	C₃×D₁₅	1	1	yes	characteristic
62	Dic₃	—	15	2	no	—
63	Dic₃	—	45	2	no	—
64	C₂×C₅	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
65	C₃²	—	2	1	no	—
66	C₃²	C₃×Dic₁₀	1	1	yes	characteristic, complemented
67	C₃²	Dic₃₀	1	1	yes	characteristic, complemented
68	Q₈	—	45	1	no	Sylow
69	C₂×C₃	—	3	1	no	—
70	C₂×C₃	—	3	1	no	—
71	C₂×C₃	—	6	1	no	—
72	C₂×C₃	C₂×C₃×D₁₅	1	1	yes	characteristic
73	C₅	#634⋊C₃	1	1	yes	characteristic, complemented, Sylow
74	C₄	—	45	2	no	—
75	C₄	C₅⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic
76	C₃	—	3	1	no	—
77	C₃	—	3	1	no	—
78	C₃	—	6	1	no	—
79	C₃	C₃×Dic₃₀	1	1	yes	characteristic
80	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
prime radical	C₂×C₅×H(3,𝔽₃)	C₂²
hypercenter	C₄	C₅⋊((C₃⋊D₃)⋊C₃)
hypocenter	C₃²×C₅	C₃×Q₈
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
3. #67180	C₃	semidirect
12. C₅×H(3,𝔽₃)	Q₈	semidirect
32. C₃²×C₅	Dic₆	semidirect
33. C₃²×C₅	C₃×Q₈	semidirect
44. H(3,𝔽₃)	Dic₁₀	semidirect
66. C₃²	C₃×Dic₁₀	semidirect
67. C₃²	Dic₃₀	semidirect
73. C₅	#634⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×H(3,𝔽₃)	2
(C₃⋊Dic₁₅)⋊C₃	2
#67180	3
C₃×Dic₃₀	3
#634⋊C₃	5

Subgroups — #67180⋊C3 #11761863

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #67180⋊C₃ #11761863