Subgroups — C₃⋊#92301 #13091391

Order 1458.

2603 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃²⋊H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃⋊(C₃⋊(C₃⋊(C₃⋊D₃)))	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃⋊(C₃⋊((C₃⋊D₃)⋊C₃))	—	3	4	no	maximal
4	#92301	—	3	9	no	maximal
5	C₃⁵	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃²×H(3,𝔽₃)	—	2	4	no	—
7	C₃²×H(3,𝔽₃)	D₃	1	4	yes	complemented
8	C₃²⋊H(3,𝔽₃)	—	2	9	no	—
9	C₃²⋊H(3,𝔽₃)	D₃	1	9	yes	complemented
10	C₃⋊(C₃⋊(C₃⋊D₃))	—	3	4	no	—
11	C₃⋊(C₃⋊(C₃⋊D₃))	—	3	9	no	—
12	C₃⋊(C₃⋊(C₃⋊D₃))	—	9	36	no	—
13	C₃×C₃⋊(C₃⋊D₃)	—	9	1	no	—
14	C₃⋊((C₃⋊D₃)⋊C₃)	—	9	36	no	—
15	C₃⋊((C₃⋊D₃)⋊C₃)	—	9	12	no	—
16	C₃⁴	—	2	4	no	—
17	C₃⁴	—	3	36	no	—
18	C₃⁴	C₃×D₃	1	4	yes	—
19	C₃⁴	C₃×D₃	1	9	yes	complemented
20	C₃⁴	C₃⋊D₃	1	1	yes	characteristic, complemented
21	C₃×H(3,𝔽₃)	—	2	36	no	—
22	C₃×H(3,𝔽₃)	—	2	12	no	—
23	C₃×H(3,𝔽₃)	—	3	36	no	—
24	C₃×H(3,𝔽₃)	—	6	36	no	—
25	C₃×H(3,𝔽₃)	C₃⋊D₃	1	12	yes	complemented
26	C₃⋊(C₃⋊D₃)	—	9	1	no	—
27	C₃⋊(C₃⋊D₃)	—	9	36	no	—
28	C₃⋊(C₃⋊D₃)	—	9	12	no	—
29	C₃⋊(C₃⋊D₃)	—	27	36	no	—
30	C₃⋊(C₃⋊D₃)	—	27	108	no	—
31	C₃⋊(C₃⋊D₃)	—	27	243	no	—
32	C₃×C₃⋊D₃	—	27	12	no	—
33	C₃×C₃⋊D₃	—	27	1	no	—
34	(C₃⋊D₃)⋊C₃	—	27	36	no	—
35	C₃³	—	2	1	no	—
36	C₃³	—	2	12	no	—
37	C₃³	—	3	243	no	—
38	C₃³	—	3	36	no	—
39	C₃³	—	3	108	no	—
40	C₃³	—	3	12	no	—
41	C₃³	—	6	12	no	—
42	C₃³	—	6	36	no	—
43	C₃³	(C₃⋊D₃)⋊C₃	1	36	yes	complemented
44	C₃³	C₃×C₃⋊D₃	1	12	yes	—
45	C₃³	C₃×C₃⋊D₃	1	1	yes	characteristic
46	C₃³	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
47	H(3,𝔽₃)	—	3	36	no	—
48	H(3,𝔽₃)	—	6	36	no	—
49	H(3,𝔽₃)	—	6	108	no	—
50	C₃⋊D₃	—	27	12	no	—
51	C₃⋊D₃	—	27	1	no	—
52	C₃⋊D₃	—	27	36	no	—
53	C₃⋊D₃	—	81	108	no	—
54	C₃⋊D₃	—	81	36	no	—
55	C₃⋊D₃	—	81	243	no	—
56	C₃×D₃	—	81	4	no	—
57	C₃×D₃	—	81	9	no	—
58	C₃²	—	3	243	no	—
59	C₃²	—	3	108	no	—
60	C₃²	—	3	36	no	—
61	C₃²	—	3	4	no	—
62	C₃²	—	6	12	no	—
63	C₃²	—	6	4	no	—
64	C₃²	—	6	36	no	—
65	C₃²	—	9	9	no	—
66	C₃²	—	18	36	no	—
67	C₃²	C₃⋊((C₃⋊D₃)⋊C₃)	1	12	yes	—
68	C₃²	C₃⋊((C₃⋊D₃)⋊C₃)	1	36	yes	complemented
69	C₃²	C₃×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
70	C₂×C₃	—	243	1	no	—
71	D₃	—	81	9	no	—
72	D₃	—	81	4	no	—
73	D₃	—	243	36	no	—
74	C₃	—	3	36	no	—
75	C₃	—	9	1	no	—
76	C₃	—	18	1	no	—
77	C₃	—	18	12	no	—
78	C₃	#92301	1	9	yes	complemented
79	C₃	C₃⋊(C₃⋊((C₃⋊D₃)⋊C₃))	1	4	yes	—
80	C₂	—	243	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃⁵	C₂×C₃
Frattini	C₃²	C₃×C₃⋊(C₃⋊D₃)
Fitting	C₃×C₃²⋊H(3,𝔽₃)	C₂
socle	C₃³	C₃×C₃⋊D₃
LCS	G ▷ C₃⁵
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃×C₃²⋊H(3,𝔽₃)	C₂	semidirect
2. C₃⋊(C₃⋊(C₃⋊(C₃⋊D₃)))	C₃	semidirect
5. C₃⁵	C₂×C₃	semidirect
7. C₃²×H(3,𝔽₃)	D₃	semidirect
9. C₃²⋊H(3,𝔽₃)	D₃	semidirect
19. C₃⁴	C₃×D₃	semidirect
20. C₃⁴	C₃⋊D₃	semidirect
25. C₃×H(3,𝔽₃)	C₃⋊D₃	semidirect
43. C₃³	(C₃⋊D₃)⋊C₃	semidirect
46. C₃³	C₃⋊(C₃⋊D₃)	semidirect
68. C₃²	C₃⋊((C₃⋊D₃)⋊C₃)	semidirect
78. C₃	#92301	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃²⋊H(3,𝔽₃)	2
C₃⋊(C₃⋊(C₃⋊(C₃⋊D₃)))	3
C₃⋊(C₃⋊((C₃⋊D₃)⋊C₃))	3
#92301	3

Subgroups — C3⋊#92301 #13091391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊#92301 #13091391