Subgroups — C₃⋊(C₃⋊(C₃⋊((C₂²⋊C₉)⋊C₂))) #14519230

Order 1944.

2220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₂²⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#10620179	—	3	1	no	maximal
3	C₃⋊(C₃⋊((C₂²⋊C₉)⋊C₂))	—	3	39	no	maximal
4	C₃⋊(C₃⋊(C₃⋊D₉))	—	4	1	no	maximal
5	C₂²×C₃⁴	D₃	1	1	yes	characteristic
6	C₂×C₃⋊(C₃⋊(C₃⋊D₃))	—	3	1	no	—
7	C₃⋊(C₃⋊(C₃⋊Dic₃))	—	3	1	no	—
8	C₃²×C₂²⋊C₉	D₃	1	39	yes	complemented
9	C₃³×C₉	—	4	1	no	Sylow
10	#6312	—	9	27	no	—
11	#6312	—	9	13	no	—
12	C₃⋊((C₂²⋊C₉)⋊C₂)	—	9	117	no	—
13	C₂×C₃⁴	—	3	1	no	—
14	C₃⋊(C₃⋊(C₃⋊D₃))	—	6	1	no	—
15	C₃⋊(C₃⋊D₉)	—	12	39	no	—
16	C₂²×C₃³	D₉	1	27	yes	complemented
17	C₂²×C₃³	C₃⋊D₃	1	13	yes	—
18	C₂×C₃⋊(C₃⋊D₃)	—	9	27	no	—
19	C₂×C₃⋊(C₃⋊D₃)	—	9	13	no	—
20	C₃⋊(C₃⋊Dic₃)	—	9	27	no	—
21	C₃⋊(C₃⋊Dic₃)	—	9	13	no	—
22	C₃×C₂²⋊C₉	C₃⋊D₃	1	117	yes	complemented
23	C₃⁴	S₄	1	1	yes	characteristic
24	C₃²×C₉	—	4	39	no	—
25	#638	—	27	117	no	—
26	#638	—	27	13	no	—
27	(C₂²⋊C₉)⋊C₂	—	27	27	no	—
28	C₂×C₃³	—	3	27	no	—
29	C₂×C₃³	—	3	13	no	—
30	C₃⋊(C₃⋊D₃)	—	18	27	no	—
31	C₃⋊(C₃⋊D₃)	—	18	13	no	—
32	C₃⋊D₉	—	36	117	no	—
33	C₂²×C₃²	C₃⋊D₉	1	117	yes	complemented
34	C₂²×C₃²	C₃⋊(C₃⋊D₃)	1	13	yes	—
35	C₂×C₃⋊D₃	—	27	117	no	—
36	C₂×C₃⋊D₃	—	27	13	no	—
37	C₃⋊Dic₃	—	27	117	no	—
38	C₃⋊Dic₃	—	27	13	no	—
39	C₂²⋊C₉	C₃⋊(C₃⋊D₃)	1	27	yes	complemented
40	C₃³	(C₂²⋊C₉)⋊C₂	1	27	yes	complemented
41	C₃³	C₃⋊S₄	1	13	yes	—
42	C₃×C₉	—	4	117	no	—
43	#67	—	81	39	no	—
44	#67	—	81	1	no	—
45	C₂×C₃²	—	3	117	no	—
46	C₂×C₃²	—	3	13	no	—
47	C₃⋊D₃	—	54	117	no	—
48	C₃⋊D₃	—	54	13	no	—
49	D₉	—	108	27	no	—
50	C₂²×C₃	C₃⋊(C₃⋊D₉)	1	39	yes	complemented
51	C₂²×C₃	C₃⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
52	C₂×D₃	—	81	1	no	—
53	C₂×D₃	—	81	39	no	—
54	Dic₃	—	81	1	no	—
55	Dic₃	—	81	39	no	—
56	C₃²	C₃⋊((C₂²⋊C₉)⋊C₂)	1	117	yes	complemented
57	C₃²	C₃⋊(C₃⋊S₄)	1	13	yes	—
58	C₉	—	4	27	no	—
59	D₄	—	243	1	no	Sylow
60	C₂×C₃	—	3	1	no	—
61	C₂×C₃	—	3	39	no	—
62	D₃	—	162	1	no	—
63	D₃	—	162	39	no	—
64	C₂²	—	243	1	no	—
65	C₂²	C₃⋊(C₃⋊(C₃⋊D₉))	1	1	yes	characteristic, complemented
66	C₄	—	243	1	no	—
67	C₃	C₃⋊(C₃⋊((C₂²⋊C₉)⋊C₂))	1	39	yes	complemented
68	C₃	C₃⋊(C₃⋊(C₃⋊S₄))	1	1	yes	characteristic
69	C₂	—	3	1	no	—
70	C₂	—	486	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃³×C₂²⋊C₉	C₂
Frattini	C₃	C₃⋊(C₃⋊(C₃⋊S₄))
Fitting, socle	C₂²×C₃⁴	D₃
LCS	G ▷ C₃³×C₂²⋊C₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃³×C₂²⋊C₉	C₂	semidirect
8. C₃²×C₂²⋊C₉	D₃	semidirect
16. C₂²×C₃³	D₉	semidirect
22. C₃×C₂²⋊C₉	C₃⋊D₃	semidirect
33. C₂²×C₃²	C₃⋊D₉	semidirect
39. C₂²⋊C₉	C₃⋊(C₃⋊D₃)	semidirect
40. C₃³	(C₂²⋊C₉)⋊C₂	semidirect
50. C₂²×C₃	C₃⋊(C₃⋊D₉)	semidirect
56. C₃²	C₃⋊((C₂²⋊C₉)⋊C₂)	semidirect
65. C₂²	C₃⋊(C₃⋊(C₃⋊D₉))	semidirect
67. C₃	C₃⋊(C₃⋊((C₂²⋊C₉)⋊C₂))	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₂²⋊C₉	2
#10620179	3
C₃⋊(C₃⋊((C₂²⋊C₉)⋊C₂))	3
C₃⋊(C₃⋊(C₃⋊D₉))	4

Subgroups — C3⋊(C3⋊(C3⋊((C22⋊C9)⋊C2))) #14519230

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊(C₃⋊(C₃⋊((C₂²⋊C₉)⋊C₂))) #14519230