Subgroups — #14550274

Order 2187.

1961 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₉	C₃	1	1	yes	maximal, characteristic
2	C₃×C₃²⋊C₂₇	C₃	1	12	yes	maximal, complemented
3	C₃⁵	C₉	1	1	yes	characteristic
4	C₃³×C₉	—	3	12	no	—
5	C₃³×C₉	—	3	24	no	—
6	C₃³×C₉	C₉	1	8	yes	—
7	C₃³×C₉	C₃²	1	4	yes	—
8	C₃²×C₂₇	C₃²	1	9	yes	complemented
9	C₃²⋊C₂₇	—	3	36	no	—
10	C₃⁴	—	3	12	no	—
11	C₃⁴	—	3	24	no	—
12	C₃⁴	C₂₇	1	9	yes	complemented
13	C₃⁴	C₃×C₉	1	4	yes	—
14	C₃²×C₉	—	3	27	no	—
15	C₃²×C₉	—	3	216	no	—
16	C₃²×C₉	—	3	24	no	—
17	C₃²×C₉	—	3	72	no	—
18	C₃²×C₉	—	3	12	no	—
19	C₃²×C₉	—	3	24	no	—
20	C₃²×C₉	C₃×C₉	1	8	yes	—
21	C₃²×C₉	H(3,𝔽₃)	1	12	yes	—
22	C₃²×C₉	3₋¹⁺²	1	24	yes	—
23	C₃²×C₉	C₃³	1	1	yes	characteristic
24	C₃×C₂₇	—	3	36	no	—
25	C₃³	—	3	27	no	—
26	C₃³	—	3	216	no	—
27	C₃³	—	3	108	no	—
28	C₃³	—	3	12	no	—
29	C₃³	—	3	24	no	—
30	C₃³	C₃²⋊C₉	1	12	yes	—
31	C₃³	C₃×C₂₇	1	12	yes	—
32	C₃³	C₂₇⋊C₃	1	24	yes	—
33	C₃³	C₃²×C₉	1	1	yes	characteristic
34	C₃×C₉	—	3	216	no	—
35	C₃×C₉	—	3	24	no	—
36	C₃×C₉	—	3	12	no	—
37	C₃×C₉	—	3	24	no	—
38	C₃×C₉	—	3	72	no	—
39	C₃×C₉	C₃²⋊C₉	1	24	yes	—
40	C₃×C₉	C₃×H(3,𝔽₃)	1	4	yes	—
41	C₃×C₉	C₃×3₋¹⁺²	1	8	yes	—
42	C₂₇	—	9	9	no	—
43	C₃²	—	3	243	no	—
44	C₃²	—	3	12	no	—
45	C₃²	—	3	36	no	—
46	C₃²	—	3	24	no	—
47	C₃²	—	3	72	no	—
48	C₃²	C₃²⋊C₂₇	1	36	yes	complemented
49	C₃²	C₃×C₃²⋊C₉	1	4	yes	—
50	C₃²	C₃²×C₂₇	1	1	yes	characteristic
51	C₃²	C₃×C₂₇⋊C₃	1	8	yes	—
52	C₉	—	3	24	no	—
53	C₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
54	C₉	#6891	1	8	yes	—
55	C₃	—	3	36	no	—
56	C₃	#10624547	1	1	yes	characteristic
57	C₃	C₃×C₃²⋊C₂₇	1	4	yes	—
58	C₃	#10624683	1	8	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₃²×C₉	C₃³
commutator	C₃²	C₃²×C₂₇
socle	C₃³	C₃²×C₉
prime radical	C₃⁵	C₉
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₃×C₃²⋊C₂₇	C₃	semidirect
8. C₃²×C₂₇	C₃²	semidirect
12. C₃⁴	C₂₇	semidirect
48. C₃²	C₃²⋊C₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₃⁴×C₉	3
C₃×C₃²⋊C₂₇	3