Subgroups — #13091029

Order 1458.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624728	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃⋊(D₂₇⋊C₃)	—	3	1	no	maximal
3	#92299	—	3	1	no	maximal
4	C₉⋊D₂₇	C₃	1	1	yes	maximal, characteristic, complemented
5	#92172	—	3	2	no	maximal
6	C₃×C₂₇⋊C₃	D₃	1	1	yes	characteristic
7	C₉×3₋¹⁺²	D₃	1	1	yes	characteristic
8	C₉⋊C₂₇	—	2	2	no	—
9	C₉⋊C₂₇	—	2	1	no	—
10	C₃²⋊C₂₇	D₃	1	2	yes	—
11	#6864	—	2	1	no	—
12	C₉×C₂₇	C₂×C₃	1	1	yes	characteristic, complemented
13	C₃⋊AGL(1,ℤ/9)	—	9	1	no	—
14	C₃×C₃⋊D₉	—	9	1	no	—
15	C₃⋊D₂₇	—	3	1	no	—
16	C₃⋊D₂₇	—	3	2	no	—
17	C₉⋊D₉	—	3	1	no	—
18	D₂₇⋊C₃	—	9	3	no	—
19	#4169	—	9	2	no	—
20	C₃×3₋¹⁺²	D₉	1	1	yes	characteristic
21	C₃²×C₉	C₃⋊D₃	1	1	yes	characteristic
22	C₂₇⋊C₃	—	3	3	no	—
23	C₃×C₂₇	—	2	2	no	—
24	C₃×C₂₇	—	2	1	no	—
25	C₃×C₂₇	C₃×D₃	1	2	yes	—
26	C₃×C₂₇	C₃×D₃	1	1	yes	characteristic
27	C₉⋊C₉	—	2	1	no	—
28	C₉⋊C₉	—	2	2	no	—
29	C₃²⋊C₉	D₉	1	2	yes	—
30	C₉²	—	2	1	no	—
31	C₉²	C₃×D₃	1	1	yes	characteristic
32	C₃×C₃⋊D₃	—	27	1	no	—
33	C₃⋊D₉	—	9	1	no	—
34	C₃⋊D₉	—	9	1	no	—
35	C₃⋊D₉	—	9	2	no	—
36	AGL(1,ℤ/9)	—	27	3	no	—
37	C₃×D₉	—	27	1	no	—
38	C₃×D₉	—	27	2	no	—
39	D₂₇	—	9	3	no	—
40	D₂₇	—	27	2	no	—
41	C₃³	C₃⋊D₉	1	1	yes	characteristic
42	3₋¹⁺²	—	3	3	no	—
43	C₃×C₉	—	2	2	no	—
44	C₃×C₉	—	2	1	no	—
45	C₃×C₉	—	2	1	no	—
46	C₃×C₉	—	3	1	no	—
47	C₃×C₉	—	3	2	no	—
48	C₃×C₉	C₃×D₉	1	2	yes	—
49	C₃×C₉	C₃×D₉	1	1	yes	characteristic
50	C₃×C₉	C₃×C₃⋊D₃	1	1	yes	characteristic
51	C₂₇	—	3	2	no	—
52	C₂₇	—	6	1	no	—
53	C₂₇	—	6	2	no	—
54	C₂₇	AGL(1,ℤ/9)	1	3	yes	complemented
55	C₃⋊D₃	—	27	1	no	—
56	C₃×D₃	—	81	1	no	—
57	C₃×D₃	—	81	1	no	—
58	C₃×D₃	—	81	2	no	—
59	D₉	—	27	3	no	—
60	D₉	—	27	1	no	—
61	D₉	—	27	2	no	—
62	D₉	—	81	2	no	—
63	C₃²	—	3	1	no	—
64	C₃²	—	3	1	no	—
65	C₃²	—	3	2	no	—
66	C₃²	C₃×C₃⋊D₉	1	1	yes	characteristic
67	C₉	—	3	2	no	—
68	C₉	—	6	1	no	—
69	C₉	—	6	1	no	—
70	C₉	—	6	2	no	—
71	C₉	D₂₇⋊C₃	1	3	yes	complemented
72	C₉	C₃⋊AGL(1,ℤ/9)	1	1	yes	characteristic
73	C₉	#4207	1	2	yes	—
74	C₂×C₃	—	243	1	no	—
75	D₃	—	81	1	no	—
76	D₃	—	81	1	no	—
77	D₃	—	81	2	no	—
78	C₃	—	9	1	no	—
79	C₃	#92299	1	1	yes	characteristic
80	C₃	C₃⋊(D₂₇⋊C₃)	1	1	yes	characteristic
81	C₃	#92318	1	2	yes	—
82	C₂	—	243	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₉×C₂₇	C₂×C₃
Frattini	C₃×C₉	C₃×C₃⋊D₃
Fitting	#10624728	C₂
socle	C₃²	C₃×C₃⋊D₉
LCS	G ▷ C₉×C₂₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. #10624728	C₂	semidirect
4. C₉⋊D₂₇	C₃	semidirect
12. C₉×C₂₇	C₂×C₃	semidirect
54. C₂₇	AGL(1,ℤ/9)	semidirect
71. C₉	D₂₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
#10624728	2
C₃⋊(D₂₇⋊C₃)	3
#92299	3
C₉⋊D₂₇	3
#92172	3