Subgroups — #14552605

Order 2187.

1073 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6897	C₃	1	3	yes	maximal, complemented
2	C₃²×C₉²	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624557	C₃	1	24	yes	maximal, complemented
4	#10624556	C₃	1	8	yes	maximal, complemented
5	#10624555	C₃	1	4	yes	maximal
6	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
7	C₃³×C₉	C₃²	1	4	yes	—
8	#6897	—	3	27	no	—
9	C₃×C₉⋊C₉	C₃²	1	24	yes	—
10	C₃×C₃²⋊C₉	C₃²	1	8	yes	—
11	C₃×C₃²⋊C₉	C₃²	1	24	yes	—
12	C₃×C₃²⋊C₉	C₃²	1	12	yes	—
13	C₃×C₉²	—	3	9	no	—
14	C₃×C₉²	C₃²	1	8	yes	—
15	C₃×C₉²	C₃²	1	1	yes	characteristic
16	#6855	C₃²	1	48	yes	—
17	C₃⁴	C₃³	1	3	yes	—
18	C₃⁴	C₃³	1	1	yes	characteristic
19	C₃×H(3,𝔽₃)	—	3	12	no	—
20	C₃²×C₉	—	3	12	no	—
21	C₃²×C₉	—	3	12	no	—
22	C₃²×C₉	—	3	24	no	—
23	C₃²×C₉	C₃³	1	24	yes	—
24	C₃²×C₉	C₃³	1	4	yes	—
25	C₃²×C₉	C₃³	1	8	yes	—
26	C₉⋊C₉	—	3	72	no	—
27	C₃²⋊C₉	—	3	24	no	—
28	C₃²⋊C₉	—	3	36	no	—
29	C₃²⋊C₉	—	3	72	no	—
30	C₉²	—	3	24	no	—
31	C₉²	H(3,𝔽₃)	1	9	yes	complemented
32	C₃³	—	3	1	no	—
33	C₃³	—	3	8	no	—
34	C₃³	—	3	3	no	—
35	C₃³	—	3	24	no	—
36	C₃³	—	3	12	no	—
37	C₃³	#737	1	12	yes	—
38	C₃³	C₃⁴	1	1	yes	characteristic
39	H(3,𝔽₃)	—	9	9	no	—
40	C₃×C₉	—	3	108	no	—
41	C₃×C₉	—	3	8	no	—
42	C₃×C₉	—	3	4	no	—
43	C₃×C₉	—	3	8	no	—
44	C₃×C₉	—	3	8	no	—
45	C₃×C₉	—	3	8	no	—
46	C₃×C₉	—	3	24	no	—
47	C₃×C₉	—	3	24	no	—
48	C₃×C₉	—	3	48	no	—
49	C₃×C₉	C₃×H(3,𝔽₃)	1	12	yes	—
50	C₃×C₉	#737	1	24	yes	—
51	C₃²	—	3	12	no	—
52	C₃²	—	3	24	no	—
53	C₃²	—	9	12	no	—
54	C₃²	—	9	3	no	—
55	C₃²	—	9	24	no	—
56	C₃²	#6897	1	9	yes	complemented
57	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
58	C₃²	C₃×#737	1	4	yes	—
59	C₃²	C₃×#737	1	8	yes	—
60	C₉	—	3	12	no	—
61	C₉	—	3	24	no	—
62	C₉	—	9	24	no	—
63	C₃	—	3	9	no	—
64	C₃	—	27	3	no	—
65	C₃	#10624869	1	4	yes	—
66	C₃	C₃×#6897	1	1	yes	characteristic
67	C₃	#10624877	1	8	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃³	C₃⁴
prime radical	C₃²×H(3,𝔽₃)	C₃²
LCS	G ▷ C₃³ ▷ C₁
UCS	C₁ ◁ C₃³ ◁ G
LECS	G ▷ C₃³ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₃×#6897	C₃	semidirect
2. C₃²×C₉²	C₃	semidirect
3. #10624557	C₃	semidirect
4. #10624556	C₃	semidirect
31. C₉²	H(3,𝔽₃)	semidirect
56. C₃²	#6897	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6897	3
C₃²×C₉²	3
#10624557	3
#10624556	3
#10624555	3