Subgroups — #14548096

Order 2187.

351 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6887	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624767	C₃	1	2	yes	maximal, complemented
3	#10624731	C₃	1	1	yes	maximal, characteristic
4	#10624643	C₃	1	3	yes	maximal, complemented
5	#10624641	C₃	1	3	yes	maximal, complemented
6	#10624639	C₃	1	3	yes	maximal
7	C₃³×C₉	C₃²	1	1	yes	characteristic
8	C₃×#731	—	3	1	no	—
9	C₃×#731	—	3	2	no	—
10	#6891	C₃²	1	6	yes	—
11	#6891	C₃²	1	1	yes	characteristic
12	#6891	C₃²	1	1	yes	characteristic
13	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
14	#6887	—	3	9	no	—
15	C₃×C₉⋊C₉	—	3	2	no	—
16	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
17	C₃×C₉²	—	3	1	no	—
18	#6867	—	3	2	no	—
19	#6867	—	3	4	no	—
20	#6862	—	3	9	no	—
21	#6861	—	3	9	no	—
22	#6857	—	3	9	no	—
23	C₃⁴	C₃³	1	1	yes	characteristic
24	C₃×3₋¹⁺²	—	3	1	no	—
25	C₃×3₋¹⁺²	—	3	1	no	—
26	C₃×3₋¹⁺²	—	3	6	no	—
27	C₃×H(3,𝔽₃)	—	3	1	no	—
28	C₃×H(3,𝔽₃)	—	3	3	no	—
29	C₃²×C₉	—	3	3	no	—
30	C₃²×C₉	—	3	6	no	—
31	C₃²×C₉	—	3	3	no	—
32	C₃²×C₉	—	3	3	no	—
33	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
34	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
35	#731	—	9	3	no	—
36	#731	—	9	6	no	—
37	C₉⋊C₉	—	3	18	no	—
38	C₃²⋊C₉	—	3	6	no	—
39	C₃²⋊C₉	—	3	6	no	—
40	C₃²⋊C₉	—	3	3	no	—
41	C₃²⋊C₉	—	3	6	no	—
42	C₃²⋊C₉	—	3	3	no	—
43	C₃²⋊C₉	—	3	6	no	—
44	C₃²⋊C₉	H(3,𝔽₃)	1	9	yes	—
45	C₉²	—	3	9	no	—
46	C₃³	—	3	2	no	—
47	C₃³	—	3	1	no	—
48	C₃³	—	9	3	no	—
49	C₃³	—	9	1	no	—
50	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
51	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
52	3₋¹⁺²	—	9	2	no	—
53	3₋¹⁺²	—	9	1	no	—
54	3₋¹⁺²	—	9	2	no	—
55	3₋¹⁺²	—	9	1	no	—
56	3₋¹⁺²	—	27	6	no	—
57	H(3,𝔽₃)	—	9	2	no	—
58	H(3,𝔽₃)	—	9	1	no	—
59	H(3,𝔽₃)	—	27	3	no	—
60	C₃×C₉	—	3	27	no	—
61	C₃×C₉	—	3	1	no	—
62	C₃×C₉	—	3	2	no	—
63	C₃×C₉	—	3	2	no	—
64	C₃×C₉	—	3	2	no	—
65	C₃×C₉	—	3	2	no	—
66	C₃×C₉	—	3	9	no	—
67	C₃×C₉	—	9	9	no	—
68	C₃×C₉	—	9	1	no	—
69	C₃×C₉	—	9	1	no	—
70	C₃×C₉	—	9	6	no	—
71	C₃×C₉	#731	1	3	yes	—
72	C₃×C₉	#731	1	6	yes	—
73	C₃²	—	3	3	no	—
74	C₃²	—	3	3	no	—
75	C₃²	—	3	3	no	—
76	C₃²	—	3	6	no	—
77	C₃²	—	9	6	no	—
78	C₃²	—	9	3	no	—
79	C₃²	—	27	1	no	—
80	C₃²	—	27	3	no	—
81	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
82	C₃²	C₃×#731	1	2	yes	—
83	C₃²	C₃×#731	1	1	yes	characteristic
84	C₉	—	3	3	no	—
85	C₉	—	3	6	no	—
86	C₉	—	9	9	no	—
87	C₉	—	27	1	no	—
88	C₉	—	27	1	no	—
89	C₉	—	27	6	no	—
90	C₃	—	3	2	no	—
91	C₃	—	3	1	no	—
92	C₃	—	9	3	no	—
93	C₃	—	81	1	no	—
94	C₃	#10624731	1	1	yes	characteristic
95	C₃	#10624756	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃×#6887	C₃	semidirect
2. #10624767	C₃	semidirect
4. #10624643	C₃	semidirect
5. #10624641	C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6887	3
#10624767	3
#10624731	3
#10624643	3
#10624641	3
#10624639	3