Subgroups — #13091049

Order 1458.

449 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624732	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃⋊(C₃⋊(C₃⋊D₉))	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃⋊#4179	—	3	3	no	maximal
4	#92301	—	3	1	no	maximal
5	C₃³×C₉	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃×#732	—	2	3	no	—
7	C₃×#732	D₃	1	3	yes	complemented
8	C₃²⋊H(3,𝔽₃)	—	2	1	no	—
9	C₃²⋊H(3,𝔽₃)	D₃	1	1	yes	characteristic
10	C₃⋊(C₃⋊(C₃⋊D₃))	—	3	1	no	—
11	C₃⋊(C₃⋊D₉)	—	3	3	no	—
12	C₃⋊(C₃⋊D₉)	—	9	9	no	—
13	C₃⋊(C₃⋊D₉)	—	9	3	no	—
14	C₃⋊((C₃⋊D₃)⋊C₃)	—	9	3	no	—
15	C₃⋊((C₃⋊D₃)⋊C₃)	—	9	1	no	—
16	#4179	—	9	9	no	—
17	C₃⁴	C₃×D₃	1	1	yes	characteristic
18	C₃×H(3,𝔽₃)	—	2	3	no	—
19	C₃×H(3,𝔽₃)	—	2	1	no	—
20	C₃×H(3,𝔽₃)	—	3	3	no	—
21	C₃×H(3,𝔽₃)	—	6	3	no	—
22	C₃×H(3,𝔽₃)	C₃⋊D₃	1	1	yes	characteristic
23	C₃²×C₉	—	3	3	no	—
24	C₃²×C₉	—	3	9	no	—
25	C₃²×C₉	C₃×D₃	1	3	yes	—
26	#732	—	3	9	no	—
27	#732	—	6	9	no	—
28	C₃⋊(C₃⋊D₃)	—	9	3	no	—
29	C₃⋊(C₃⋊D₃)	—	9	1	no	—
30	C₃⋊(C₃⋊D₃)	—	27	3	no	—
31	C₃⋊(C₃⋊D₃)	—	27	9	no	—
32	C₃×C₃⋊D₃	—	27	1	no	—
33	C₃⋊D₉	—	9	9	no	—
34	C₃⋊D₉	—	27	3	no	—
35	C₃⋊D₉	—	27	6	no	—
36	C₃⋊D₉	—	27	27	no	—
37	(C₃⋊D₃)⋊C₃	—	27	3	no	—
38	(C₃⋊D₃)⋊C₃	—	27	9	no	—
39	C₃³	—	3	9	no	—
40	C₃³	—	3	3	no	—
41	C₃³	—	3	1	no	—
42	C₃³	—	6	1	no	—
43	C₃³	—	6	3	no	—
44	C₃³	(C₃⋊D₃)⋊C₃	1	3	yes	—
45	C₃³	C₃×C₃⋊D₃	1	1	yes	characteristic
46	H(3,𝔽₃)	—	3	3	no	—
47	H(3,𝔽₃)	—	6	3	no	—
48	H(3,𝔽₃)	—	6	6	no	—
49	H(3,𝔽₃)	—	6	3	no	—
50	H(3,𝔽₃)	—	9	9	no	—
51	H(3,𝔽₃)	—	18	9	no	—
52	C₃×C₉	—	3	27	no	—
53	C₃×C₉	—	3	6	no	—
54	C₃×C₉	—	3	3	no	—
55	C₃×C₉	(C₃⋊D₃)⋊C₃	1	9	yes	complemented
56	C₃⋊D₃	—	27	1	no	—
57	C₃⋊D₃	—	27	9	no	—
58	C₃⋊D₃	—	27	3	no	—
59	C₃⋊D₃	—	81	3	no	—
60	C₃⋊D₃	—	81	6	no	—
61	C₃⋊D₃	—	81	3	no	—
62	C₃⋊D₃	—	81	27	no	—
63	C₃×D₃	—	81	1	no	—
64	C₃×D₃	—	81	3	no	—
65	D₉	—	81	9	no	—
66	C₃²	—	3	27	no	—
67	C₃²	—	3	3	no	—
68	C₃²	—	3	6	no	—
69	C₃²	—	3	3	no	—
70	C₃²	—	9	1	no	—
71	C₃²	—	9	3	no	—
72	C₃²	—	18	3	no	—
73	C₃²	—	18	1	no	—
74	C₃²	—	18	3	no	—
75	C₃²	—	18	9	no	—
76	C₃²	#4179	1	9	yes	complemented
77	C₃²	C₃⋊((C₃⋊D₃)⋊C₃)	1	3	yes	—
78	C₃²	C₃⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic
79	C₉	—	3	9	no	—
80	C₂×C₃	—	243	1	no	—
81	D₃	—	81	1	no	—
82	D₃	—	81	3	no	—
83	D₃	—	243	9	no	—
84	D₃	—	243	3	no	—
85	C₃	—	3	9	no	—
86	C₃	—	3	3	no	—
87	C₃	—	27	1	no	—
88	C₃	—	54	1	no	—
89	C₃	—	54	3	no	—
90	C₃	#92301	1	1	yes	characteristic
91	C₃	C₃⋊#4179	1	3	yes	—
92	C₂	—	243	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃³×C₉	C₂×C₃
Frattini	C₃³	C₃×C₃⋊D₃
Fitting	#10624732	C₂
socle	C₃²	C₃⋊((C₃⋊D₃)⋊C₃)
LCS	G ▷ C₃³×C₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. #10624732	C₂	semidirect
2. C₃⋊(C₃⋊(C₃⋊D₉))	C₃	semidirect
5. C₃³×C₉	C₂×C₃	semidirect
7. C₃×#732	D₃	semidirect
55. C₃×C₉	(C₃⋊D₃)⋊C₃	semidirect
76. C₃²	#4179	semidirect

Maximal subgroups

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

subgroup	index
#10624732	2
C₃⋊(C₃⋊(C₃⋊D₉))	3
C₃⋊#4179	3
#92301	3