Subgroups — #14551532

Order 2187.

161 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×C₂₇	C₃	1	1	yes	maximal, characteristic
2	C₉⋊C₈₁	C₃	1	6	yes	maximal, complemented
3	C₃²⋊C₈₁	C₃	1	3	yes	maximal
4	C₉×C₈₁	C₃	1	3	yes	maximal, complemented
5	C₃²×C₂₇	C₉	1	2	yes	—
6	C₃²×C₂₇	C₃²	1	1	yes	characteristic
7	C₃×C₉²	C₉	1	1	yes	characteristic
8	C₃×C₈₁	C₃²	1	9	yes	—
9	C₉×C₂₇	C₉	1	4	yes	—
10	C₉×C₂₇	C₉	1	2	yes	—
11	C₉×C₂₇	C₃²	1	2	yes	—
12	C₉×C₂₇	C₃²	1	1	yes	characteristic
13	C₃²×C₉	C₂₇	1	2	yes	—
14	C₃²×C₉	C₂₇	1	1	yes	characteristic
15	C₃²×C₉	C₃×C₉	1	1	yes	characteristic
16	C₃×C₂₇	—	3	6	no	—
17	C₃×C₂₇	—	3	3	no	—
18	C₃×C₂₇	C₃×C₉	1	2	yes	—
19	C₃×C₂₇	C₃×C₉	1	4	yes	—
20	C₃×C₂₇	C₃×C₉	1	2	yes	—
21	C₃×C₂₇	C₃³	1	1	yes	characteristic
22	C₉²	C₂₇	1	2	yes	—
23	C₉²	C₂₇	1	4	yes	—
24	C₉²	C₃×C₉	1	2	yes	—
25	C₉²	C₃×C₉	1	1	yes	characteristic
26	C₈₁	—	3	9	no	—
27	C₃³	C₃×C₂₇	1	1	yes	characteristic
28	C₃×C₉	—	3	6	no	—
29	C₃×C₉	—	3	1	no	—
30	C₃×C₉	—	3	2	no	—
31	C₃×C₉	C₈₁	1	9	yes	complemented
32	C₃×C₉	C₃×C₂₇	1	2	yes	—
33	C₃×C₉	C₃×C₂₇	1	4	yes	—
34	C₃×C₉	C₃×C₂₇	1	2	yes	—
35	C₃×C₉	C₃×C₂₇	1	1	yes	characteristic
36	C₃×C₉	C₃×C₂₇	1	2	yes	—
37	C₃×C₉	C₃²×C₉	1	1	yes	characteristic
38	C₂₇	—	3	2	no	—
39	C₂₇	—	3	4	no	—
40	C₂₇	C₂₇⋊C₃	1	6	yes	—
41	C₂₇	#737	1	3	yes	—
42	C₃²	—	3	2	no	—
43	C₃²	—	3	1	no	—
44	C₃²	C₃×C₈₁	1	3	yes	—
45	C₃²	C₃²×C₂₇	1	1	yes	characteristic
46	C₉	—	3	4	no	—
47	C₉	—	3	2	no	—
48	C₉	C₃×C₈₁	1	6	yes	—
49	C₉	C₃×C₈₁	1	3	yes	—
50	C₉	C₈₁⋊C₃	1	6	yes	—
51	C₉	#6887	1	1	yes	characteristic
52	C₉	C₃×C₂₇⋊C₃	1	2	yes	—
53	C₃	—	3	3	no	—
54	C₃	#10624710	1	1	yes	characteristic
55	C₃	C₃²×C₈₁	1	1	yes	characteristic
56	C₃	C₃×C₈₁⋊C₃	1	2	yes	—

	subgroup	G/H
center	C₉×C₂₇	C₃²
commutator	C₃	C₃²×C₈₁
Frattini, cosocle	C₃×C₂₇	C₃³
socle	C₃²	C₃²×C₂₇
prime radical	C₃³	C₃×C₂₇
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉×C₂₇ ◁ G
LECS	G ▷ C₃×C₂₇ ▷ C₉ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₉⋊C₈₁	C₃	semidirect
4. C₉×C₈₁	C₃	semidirect
31. C₃×C₉	C₈₁	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×C₂₇	3
C₉⋊C₈₁	3
C₃²⋊C₈₁	3
C₉×C₈₁	3