Subgroups — #14552616

Order 2187.

2375 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×#6895	C₃	1	3	yes	maximal, complemented
3	#10624556	C₃	1	27	yes	maximal, complemented
4	#10624549	C₃	1	6	yes	maximal, complemented
5	#10624547	C₃	1	3	yes	maximal
6	C₃⁵	C₃²	1	1	yes	characteristic
7	C₃³×C₉	—	3	9	no	—
8	C₃³×C₉	—	3	9	no	—
9	C₃³×C₉	—	3	18	no	—
10	C₃³×C₉	C₃²	1	9	yes	—
11	C₃³×C₉	C₃²	1	1	yes	characteristic
12	C₃³×C₉	C₃²	1	2	yes	—
13	#6895	—	3	27	no	—
14	C₃×C₉⋊C₉	C₃²	1	27	yes	complemented
15	C₃×C₃²⋊C₉	C₃²	1	9	yes	—
16	C₃×C₃²⋊C₉	C₃²	1	27	yes	—
17	#6855	C₃²	1	54	yes	complemented
18	C₃⁴	—	3	3	no	—
19	C₃⁴	—	3	6	no	—
20	C₃⁴	—	3	27	no	—
21	C₃⁴	3₋¹⁺²	1	9	yes	—
22	C₃⁴	C₃³	1	1	yes	characteristic
23	C₃⁴	C₃³	1	3	yes	—
24	C₃²×C₉	—	3	243	no	—
25	C₃²×C₉	—	3	6	no	—
26	C₃²×C₉	—	3	27	no	—
27	C₃²×C₉	—	3	3	no	—
28	C₃²×C₉	—	3	6	no	—
29	C₃²×C₉	—	3	6	no	—
30	C₃²×C₉	—	3	6	no	—
31	C₃²×C₉	—	3	27	no	—
32	C₃²×C₉	—	3	54	no	—
33	C₃²×C₉	H(3,𝔽₃)	1	9	yes	—
34	C₃²×C₉	3₋¹⁺²	1	18	yes	—
35	C₃²×C₉	C₃³	1	27	yes	—
36	C₃²×C₉	C₃³	1	3	yes	—
37	C₃²×C₉	C₃³	1	6	yes	—
38	C₉⋊C₉	—	3	81	no	—
39	C₃²⋊C₉	—	3	27	no	—
40	C₃²⋊C₉	—	3	81	no	—
41	C₃³	—	3	81	no	—
42	C₃³	—	3	162	no	—
43	C₃³	—	3	3	no	—
44	C₃³	—	3	6	no	—
45	C₃³	—	3	9	no	—
46	C₃³	—	3	18	no	—
47	C₃³	—	3	27	no	—
48	C₃³	—	3	81	no	—
49	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
50	C₃³	C₃×3₋¹⁺²	1	3	yes	—
51	C₃³	C₃×3₋¹⁺²	1	6	yes	—
52	C₃³	C₃×3₋¹⁺²	1	9	yes	—
53	C₃³	#737	1	27	yes	—
54	C₃³	C₃⁴	1	1	yes	characteristic
55	C₃×C₉	—	3	243	no	—
56	C₃×C₉	—	3	6	no	—
57	C₃×C₉	—	3	3	no	—
58	C₃×C₉	—	3	6	no	—
59	C₃×C₉	—	3	6	no	—
60	C₃×C₉	—	3	6	no	—
61	C₃×C₉	—	3	27	no	—
62	C₃×C₉	—	3	54	no	—
63	C₃×C₉	—	3	27	no	—
64	C₃×C₉	—	3	81	no	—
65	C₃×C₉	C₃×H(3,𝔽₃)	1	9	yes	—
66	C₃×C₉	C₃×3₋¹⁺²	1	18	yes	—
67	C₃²	—	3	243	no	—
68	C₃²	—	3	9	no	—
69	C₃²	—	3	9	no	—
70	C₃²	—	3	27	no	—
71	C₃²	—	3	9	no	—
72	C₃²	—	3	18	no	—
73	C₃²	—	3	18	no	—
74	C₃²	—	3	54	no	—
75	C₃²	#6891	1	9	yes	—
76	C₃²	#6895	1	27	yes	complemented
77	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
78	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
79	C₃²	C₃²×3₋¹⁺²	1	2	yes	—
80	C₃²	C₃×#737	1	9	yes	—
81	C₉	—	3	9	no	—
82	C₉	—	3	18	no	—
83	C₉	—	9	27	no	—
84	C₃	—	3	9	no	—
85	C₃	—	3	27	no	—
86	C₃	C₃×#6891	1	1	yes	characteristic
87	C₃	#10624869	1	3	yes	—
88	C₃	C₃×#6895	1	3	yes	—
89	C₃	#10624872	1	6	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃³	C₃⁴
prime radical	C₃⁵	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
2. C₃×#6895	C₃	semidirect
3. #10624556	C₃	semidirect
4. #10624549	C₃	semidirect
14. C₃×C₉⋊C₉	C₃²	semidirect
17. #6855	C₃²	semidirect
76. C₃²	#6895	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×#6895	3
#10624556	3
#10624549	3
#10624547	3