Subgroups — #14546262

Order 2187.

79 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₈₁⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₂₇×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊C₈₁	C₃	1	1	yes	maximal, characteristic
4	C₃²⋊C₈₁	C₃	1	1	yes	maximal, characteristic
5	C₃²×C₂₇	C₃²	1	1	yes	characteristic
6	C₉×3₋¹⁺²	C₉	1	1	yes	characteristic
7	C₈₁⋊C₃	—	3	3	no	—
8	C₃×C₈₁	—	3	1	no	—
9	C₃×C₈₁	—	3	1	no	—
10	C₃×C₈₁	—	3	1	no	—
11	C₉⋊C₂₇	—	3	2	no	—
12	C₃²⋊C₂₇	C₉	1	1	yes	characteristic
13	C₃²⋊C₂₇	C₉	1	1	yes	characteristic
14	C₉×C₂₇	—	3	1	no	—
15	C₃×3₋¹⁺²	C₂₇	1	1	yes	characteristic
16	C₃²×C₉	C₃×C₉	1	1	yes	characteristic
17	C₃×C₂₇	—	3	1	no	—
18	C₃×C₂₇	—	3	1	no	—
19	C₃×C₂₇	—	3	1	no	—
20	C₃×C₂₇	—	3	1	no	—
21	C₃×C₂₇	—	3	1	no	—
22	C₃×C₂₇	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃×C₂₇	3₋¹⁺²	1	2	yes	—
24	C₉⋊C₉	—	3	2	no	—
25	C₃²⋊C₉	C₂₇	1	1	yes	characteristic
26	C₃²⋊C₉	C₂₇	1	1	yes	characteristic
27	C₉²	—	3	1	no	—
28	C₈₁	—	3	3	no	—
29	C₈₁	—	9	1	no	—
30	C₈₁	—	9	1	no	—
31	C₃³	C₃×C₂₇	1	1	yes	characteristic
32	3₋¹⁺²	—	3	3	no	—
33	C₃×C₉	—	3	1	no	—
34	C₃×C₉	—	3	1	no	—
35	C₃×C₉	—	3	1	no	—
36	C₃×C₉	—	3	1	no	—
37	C₃×C₉	—	3	1	no	—
38	C₃×C₉	—	3	1	no	—
39	C₃×C₉	C₃²⋊C₉	1	1	yes	characteristic
40	C₃×C₉	C₂₇⋊C₃	1	2	yes	—
41	C₂₇	—	3	2	no	—
42	C₂₇	—	9	1	no	—
43	C₂₇	—	9	1	no	—
44	C₂₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
45	C₂₇	#731	1	1	yes	characteristic
46	C₂₇	#733	1	1	yes	characteristic
47	C₃²	—	3	1	no	—
48	C₃²	—	3	1	no	—
49	C₃²	—	3	1	no	—
50	C₃²	—	3	1	no	—
51	C₃²	C₃²⋊C₂₇	1	1	yes	characteristic
52	C₉	—	3	2	no	—
53	C₉	—	3	3	no	—
54	C₉	—	9	1	no	—
55	C₉	—	9	1	no	—
56	C₉	3₋¹⁺²⋊C₉	1	1	yes	characteristic
57	C₉	#6872	1	1	yes	characteristic
58	C₉	#6873	1	1	yes	characteristic
59	C₃	—	9	1	no	—
60	C₃	3₋¹⁺²⋊C₂₇	1	1	yes	characteristic
61	C₃	#10624502	1	1	yes	characteristic
62	C₃	#10624503	1	1	yes	characteristic
63	C₃	#10624504	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₂₇	H(3,𝔽₃)
commutator, prime radical	C₃³	C₃×C₂₇
Frattini, cosocle	C₃²×C₂₇	C₃²
socle	C₃²	C₃²⋊C₂₇
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃×C₂₇ ◁ C₃²×C₂₇ ◁ G
LECS	G ▷ C₃²×C₂₇ ▷ C₃×C₉ ▷ C₃ ▷ C₁

Maximal subgroups

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

subgroup	index
C₃×C₈₁⋊C₃	3
C₂₇×3₋¹⁺²	3
C₃²⋊C₈₁	3