Subgroups — C₃²×C₃⋊#168 #11761908

Order 1080.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₃×C₃⋊#168	C₃	1	4	yes	maximal, direct product factor
3	C₃²×#168	—	3	1	no	maximal
4	C₂×C₃³×C₅	C₄	1	1	yes	characteristic
5	C₃²×C₃⋊C₈	—	5	1	no	maximal, Hall
6	C₃²×C₄×C₅	—	2	4	no	—
7	C₃²×C₄×C₅	D₃	1	1	yes	characteristic
8	C₃²×C₄×C₅	C₂×C₃	1	4	yes	—
9	C₃³×C₅	C₈	1	1	yes	characteristic, complemented, Hall
10	C₃⋊#168	C₃²	1	1	yes	characteristic, direct product factor
11	C₃×#168	—	3	4	no	—
12	C₃³×C₄	D₅	1	1	yes	characteristic
13	C₂×C₃²×C₅	—	2	4	no	—
14	C₂×C₃²×C₅	Dic₃	1	1	yes	characteristic
15	C₂×C₃²×C₅	C₃×C₄	1	4	yes	—
16	C₃²×C₈	—	15	1	no	—
17	C₃×C₃⋊C₈	—	5	4	no	—
18	C₃×C₄×C₅	—	2	4	no	—
19	C₃×C₄×C₅	C₃×D₃	1	4	yes	—
20	C₃×C₄×C₅	C₂×C₃²	1	1	yes	characteristic
21	C₂×C₃³	Dic₅	1	1	yes	characteristic
22	C₃²×C₅	—	2	4	no	—
23	C₃²×C₅	C₃⋊C₈	1	1	yes	characteristic, complemented
24	C₃²×C₅	C₃×C₈	1	4	yes	complemented
25	#168	—	3	1	no	Hall
26	C₃²×C₄	—	2	4	no	—
27	C₃²×C₄	C₃×D₅	1	4	yes	—
28	C₃²×C₄	D₁₅	1	1	yes	characteristic
29	C₂×C₃×C₅	—	2	4	no	—
30	C₂×C₃×C₅	C₃×Dic₃	1	4	yes	—
31	C₂×C₃×C₅	C₃²×C₄	1	1	yes	characteristic
32	C₃³	#168	1	1	yes	characteristic, complemented, Sylow
33	C₃×C₈	—	15	4	no	—
34	C₃⋊C₈	—	5	1	no	—
35	C₄×C₅	C₃²×D₃	1	1	yes	characteristic
36	C₂×C₃²	—	2	4	no	—
37	C₂×C₃²	C₃×Dic₅	1	4	yes	—
38	C₂×C₃²	Dic₁₅	1	1	yes	characteristic
39	C₃×C₅	—	2	4	no	—
40	C₃×C₅	C₃×C₃⋊C₈	1	4	yes	complemented
41	C₃×C₅	C₃²×C₈	1	1	yes	characteristic, complemented
42	C₃×C₄	—	2	4	no	—
43	C₃×C₄	C₃²×D₅	1	1	yes	characteristic
44	C₃×C₄	C₃×D₁₅	1	4	yes	—
45	C₂×C₅	C₃²×Dic₃	1	1	yes	characteristic
46	C₃²	—	2	4	no	—
47	C₃²	C₃×#168	1	4	yes	complemented
48	C₃²	C₃⋊#168	1	1	yes	characteristic, direct product factor
49	C₈	—	15	1	no	Sylow
50	C₂×C₃	—	2	4	no	—
51	C₂×C₃	C₃²×Dic₅	1	1	yes	characteristic
52	C₂×C₃	C₃×Dic₁₅	1	4	yes	—
53	C₅	C₃²×C₃⋊C₈	1	1	yes	characteristic, complemented, Sylow
54	C₄	C₃²×D₁₅	1	1	yes	characteristic
55	C₃	—	2	4	no	—
56	C₃	C₃²×#168	1	1	yes	characteristic, complemented
57	C₃	C₃×C₃⋊#168	1	4	yes	direct product factor
58	C₂	C₃²×Dic₁₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃²×C₄	D₁₅
commutator, hypocenter	C₃×C₅	C₃²×C₈
Frattini	C₄	C₃²×D₁₅
Fitting	C₃³×C₄×C₅	C₂
socle, prime radical	C₂×C₃³×C₅	C₄
cosocle	C₃×C₄×C₅	C₂×C₃²
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₃²×C₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₃⋊#168	C₃	direct (N × H)
9. C₃³×C₅	C₈	semidirect
10. C₃⋊#168	C₃²	direct (N × H)
23. C₃²×C₅	C₃⋊C₈	semidirect
24. C₃²×C₅	C₃×C₈	semidirect
32. C₃³	#168	semidirect
40. C₃×C₅	C₃×C₃⋊C₈	semidirect
41. C₃×C₅	C₃²×C₈	semidirect
47. C₃²	C₃×#168	semidirect
48. C₃²	C₃⋊#168	direct (N × H)
53. C₅	C₃²×C₃⋊C₈	semidirect
56. C₃	C₃²×#168	semidirect
57. C₃	C₃×C₃⋊#168	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₄×C₅	2
C₃×C₃⋊#168	3
C₃²×#168	3
C₃²×C₃⋊C₈	5

Subgroups — C32×C3⋊#168 #11761908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₃⋊#168 #11761908