Subgroups — #14552173

Order 2187.

1010 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×3₋¹⁺²	C₃	1	4	yes	maximal
2	#10624577	C₃	1	2	yes	maximal, complemented
3	#10624573	C₃	1	16	yes	maximal, complemented
4	#10624571	C₃	1	1	yes	maximal, characteristic
5	#10624565	C₃	1	8	yes	maximal
6	C₉×C₉⋊C₉	C₃	1	8	yes	maximal, complemented
7	#10624544	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×3₋¹⁺²	C₃²	1	4	yes	—
9	C₃³×C₉	C₃²	1	1	yes	characteristic
10	C₉×3₋¹⁺²	—	3	36	no	—
11	C₃×C₉⋊C₉	C₃²	1	24	yes	—
12	C₃×C₉⋊C₉	C₃²	1	16	yes	—
13	C₃×C₉⋊C₉	C₃²	1	8	yes	—
14	C₃×C₃²⋊C₉	C₃²	1	8	yes	—
15	C₃×C₉²	C₃²	1	8	yes	—
16	C₃×C₉²	C₃²	1	4	yes	—
17	#6855	C₃²	1	8	yes	—
18	#6855	C₃²	1	1	yes	characteristic
19	#6855	C₃²	1	24	yes	—
20	#6855	C₃²	1	16	yes	—
21	#6855	C₃²	1	8	yes	—
22	C₃⁴	C₃³	1	1	yes	characteristic
23	C₃×3₋¹⁺²	—	3	12	no	—
24	C₃×3₋¹⁺²	—	3	12	no	—
25	C₃×3₋¹⁺²	—	3	24	no	—
26	C₃²×C₉	—	3	12	no	—
27	C₃²×C₉	C₃³	1	2	yes	—
28	C₃²×C₉	C₃³	1	16	yes	—
29	C₃²×C₉	C₃³	1	4	yes	—
30	C₃²×C₉	C₃³	1	8	yes	—
31	C₃²×C₉	C₃³	1	8	yes	—
32	C₃²×C₉	C₃³	1	1	yes	characteristic
33	C₉⋊C₉	—	3	48	no	—
34	C₉⋊C₉	—	3	24	no	—
35	C₉⋊C₉	—	3	72	no	—
36	C₃²⋊C₉	—	3	24	no	—
37	C₉²	—	3	24	no	—
38	C₉²	3₋¹⁺²	1	36	yes	complemented
39	C₃³	—	3	4	no	—
40	C₃³	—	3	8	no	—
41	C₃³	#737	1	3	yes	—
42	C₃³	C₃⁴	1	1	yes	characteristic
43	3₋¹⁺²	—	3	108	no	—
44	C₃×C₉	—	3	8	no	—
45	C₃×C₉	—	3	24	no	—
46	C₃×C₉	—	3	4	no	—
47	C₃×C₉	—	3	8	no	—
48	C₃×C₉	—	3	8	no	—
49	C₃×C₉	—	3	8	no	—
50	C₃×C₉	—	3	16	no	—
51	C₃×C₉	—	3	8	no	—
52	C₃×C₉	—	3	48	no	—
53	C₃×C₉	—	9	9	no	—
54	C₃×C₉	C₃×3₋¹⁺²	1	24	yes	—
55	C₃×C₉	C₃×3₋¹⁺²	1	12	yes	—
56	C₃×C₉	C₃×3₋¹⁺²	1	12	yes	—
57	C₃×C₉	#737	1	24	yes	—
58	C₃²	—	3	12	no	—
59	C₃²	—	9	1	no	—
60	C₃²	—	9	8	no	—
61	C₃²	C₃²×3₋¹⁺²	1	4	yes	—
62	C₃²	C₃×#737	1	1	yes	characteristic
63	C₃²	C₃×#737	1	8	yes	—
64	C₉	—	3	12	no	—
65	C₉	—	3	24	no	—
66	C₉	—	3	24	no	—
67	C₉	—	9	2	no	—
68	C₉	—	9	16	no	—
69	C₉	#6900	1	9	yes	—
70	C₃	—	9	3	no	—
71	C₃	#10624872	1	4	yes	—
72	C₃	C₃×#6900	1	1	yes	characteristic
73	C₃	#10624879	1	8	yes	—

	subgroup	G/H
center	C₃²×C₉	C₃³
commutator, Frattini, socle, cosocle	C₃³	C₃⁴
prime radical	C₃⁴	C₃³
LCS	G ▷ C₃³ ▷ C₁
UCS	C₁ ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃³ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #10624577	C₃	semidirect
3. #10624573	C₃	semidirect
6. C₉×C₉⋊C₉	C₃	semidirect
7. #10624544	C₃	semidirect
38. C₉²	3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×3₋¹⁺²	3
#10624577	3
#10624573	3
#10624571	3
#10624565	3
C₉×C₉⋊C₉	3
#10624544	3