Subgroups — #14552143

Order 2187.

758 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6900	C₃	1	1	yes	maximal, characteristic
2	#10624560	C₃	1	8	yes	maximal
3	#10624559	C₃	1	16	yes	maximal, complemented
4	#10624559	C₃	1	8	yes	maximal, complemented
5	#10624558	C₃	1	4	yes	maximal
6	#10624552	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624551	C₃	1	2	yes	maximal, complemented
8	C₃²×3₋¹⁺²	C₃²	1	4	yes	—
9	C₃³×C₉	C₃²	1	1	yes	characteristic
10	#6900	C₉	1	27	yes	complemented
11	C₃×C₉⋊C₉	C₃²	1	16	yes	—
12	C₃×C₉⋊C₉	C₃²	1	8	yes	—
13	C₃×C₃²⋊C₉	C₃²	1	8	yes	—
14	C₃×C₉²	C₃²	1	8	yes	—
15	C₃×C₉²	C₃²	1	1	yes	characteristic
16	#6855	C₃²	1	8	yes	—
17	#6855	C₃²	1	16	yes	—
18	#6855	C₃²	1	4	yes	—
19	#6855	C₃²	1	8	yes	—
20	#6855	C₃²	1	16	yes	—
21	#6855	C₃²	1	8	yes	—
22	#6855	C₃²	1	24	yes	—
23	C₃⁴	C₃³	1	1	yes	characteristic
24	C₃×3₋¹⁺²	—	9	8	no	—
25	C₃×3₋¹⁺²	—	9	4	no	—
26	C₃×3₋¹⁺²	C₃×C₉	1	36	yes	—
27	C₃²×C₉	—	9	4	no	—
28	C₃²×C₉	C₃³	1	2	yes	—
29	C₃²×C₉	C₃³	1	1	yes	characteristic
30	C₃²×C₉	C₃³	1	16	yes	—
31	C₃²×C₉	C₃³	1	8	yes	—
32	C₃²×C₉	C₃³	1	4	yes	—
33	C₃²×C₉	C₃³	1	8	yes	—
34	C₉⋊C₉	—	9	16	no	—
35	C₉⋊C₉	C₃×C₉	1	72	yes	complemented
36	C₃²⋊C₉	—	9	8	no	—
37	C₉²	—	9	8	no	—
38	C₉²	C₃×C₉	1	9	yes	complemented
39	C₃³	—	3	4	no	—
40	C₃³	—	3	8	no	—
41	C₃³	C₃²×C₉	1	3	yes	—
42	C₃³	C₃⁴	1	1	yes	characteristic
43	3₋¹⁺²	—	9	36	no	—
44	C₃×C₉	—	3	8	no	—
45	C₃×C₉	—	3	4	no	—
46	C₃×C₉	—	3	8	no	—
47	C₃×C₉	—	3	8	no	—
48	C₃×C₉	—	3	8	no	—
49	C₃×C₉	—	3	16	no	—
50	C₃×C₉	—	3	8	no	—
51	C₃×C₉	—	3	16	no	—
52	C₃×C₉	—	3	8	no	—
53	C₃×C₉	—	3	32	no	—
54	C₃×C₉	—	3	16	no	—
55	C₃×C₉	—	3	8	no	—
56	C₃×C₉	—	3	4	no	—
57	C₃×C₉	—	9	9	no	—
58	C₃×C₉	C₃²×C₉	1	12	yes	—
59	C₃×C₉	C₃²×C₉	1	24	yes	—
60	C₃²	—	3	12	no	—
61	C₃²	—	9	1	no	—
62	C₃²	—	9	8	no	—
63	C₃²	C₃³×C₉	1	1	yes	characteristic
64	C₃²	3₋¹⁺⁴	1	4	yes	—
65	C₃²	3₋¹⁺⁴	1	8	yes	—
66	C₉	—	3	24	no	—
67	C₉	—	3	12	no	—
68	C₉	—	9	2	no	—
69	C₉	—	9	1	no	—
70	C₉	—	9	16	no	—
71	C₉	—	9	8	no	—
72	C₃	—	9	3	no	—
73	C₃	#10624863	1	4	yes	—
74	C₃	#10624912	1	1	yes	characteristic
75	C₃	#10624914	1	8	yes	—

	subgroup	G/H
center, Frattini, socle, cosocle	C₃³	C₃⁴
commutator	C₃²	C₃³×C₉
prime radical	C₃⁴	C₃³
LCS	G ▷ C₃² ▷ 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
3. #10624559	C₃	semidirect
4. #10624559	C₃	semidirect
6. #10624552	C₃	semidirect
7. #10624551	C₃	semidirect
10. #6900	C₉	semidirect
35. C₉⋊C₉	C₃×C₉	semidirect
38. C₉²	C₃×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6900	3
#10624560	3
#10624559	3
#10624558	3
#10624552	3
#10624551	3