Subgroups — #14550900

Order 2187.

323 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#732	C₃	1	1	yes	maximal, characteristic
2	#10624766	C₃	1	2	yes	maximal, complemented
3	#10624688	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624485	C₃	1	9	yes	maximal, complemented
5	C₃²×H(3,𝔽₃)	—	3	1	no	—
6	C₃³×C₉	C₃²	1	1	yes	characteristic
7	C₃×#732	—	3	1	no	—
8	C₃×#732	—	3	1	no	—
9	C₃×#732	—	3	3	no	—
10	C₃×#732	—	3	6	no	—
11	C₃×#732	C₃²	1	3	yes	—
12	#6895	C₃²	1	6	yes	complemented
13	#6887	—	3	1	no	—
14	#6868	—	3	1	no	—
15	#6868	—	3	2	no	—
16	#6867	—	3	2	no	—
17	#6867	C₃²	1	3	yes	complemented
18	#6866	—	3	1	no	—
19	#6866	—	3	2	no	—
20	C₃⁴	—	9	1	no	—
21	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
22	C₃×H(3,𝔽₃)	—	3	9	no	—
23	C₃×H(3,𝔽₃)	—	3	3	no	—
24	C₃×H(3,𝔽₃)	—	9	6	no	—
25	C₃×H(3,𝔽₃)	—	9	2	no	—
26	C₃²×C₉	—	3	1	no	—
27	C₃²×C₉	—	3	2	no	—
28	C₃²×C₉	—	3	1	no	—
29	C₃²×C₉	—	3	2	no	—
30	C₃²×C₉	—	9	2	no	—
31	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
32	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
33	C₃²×C₉	C₃³	1	1	yes	characteristic
34	#732	—	3	9	no	—
35	#732	—	3	9	no	—
36	#732	—	3	3	no	—
37	#732	—	3	6	no	—
38	C₉⋊C₉	—	9	6	no	—
39	C₉⋊C₉	—	9	2	no	—
40	C₃²⋊C₉	—	3	6	no	—
41	C₃²⋊C₉	—	3	3	no	—
42	C₉²	—	9	1	no	—
43	C₃³	—	3	1	no	—
44	C₃³	—	3	2	no	—
45	C₃³	—	9	3	no	—
46	C₃³	—	9	9	no	—
47	C₃³	—	9	3	no	—
48	C₃³	—	27	9	no	—
49	C₃³	#731	1	1	yes	characteristic
50	C₃³	#731	1	2	yes	—
51	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
52	H(3,𝔽₃)	—	3	9	no	—
53	H(3,𝔽₃)	—	9	18	no	—
54	H(3,𝔽₃)	—	9	6	no	—
55	C₃×C₉	—	3	1	no	—
56	C₃×C₉	—	3	2	no	—
57	C₃×C₉	—	3	1	no	—
58	C₃×C₉	—	3	1	no	—
59	C₃×C₉	—	3	1	no	—
60	C₃×C₉	—	3	1	no	—
61	C₃×C₉	—	3	2	no	—
62	C₃×C₉	—	3	2	no	—
63	C₃×C₉	—	9	1	no	—
64	C₃×C₉	—	9	2	no	—
65	C₃×C₉	—	9	2	no	—
66	C₃×C₉	—	9	4	no	—
67	C₃×C₉	—	9	3	no	—
68	C₃×C₉	—	9	6	no	—
69	C₃×C₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
70	C₃×C₉	AΣL(1,𝔽₂₇)	1	2	yes	—
71	C₃²	—	3	3	no	—
72	C₃²	—	3	1	no	—
73	C₃²	—	3	2	no	—
74	C₃²	—	9	1	no	—
75	C₃²	—	9	3	no	—
76	C₃²	—	9	6	no	—
77	C₃²	—	9	2	no	—
78	C₃²	—	9	9	no	—
79	C₃²	—	27	27	no	—
80	C₃²	—	27	9	no	—
81	C₃²	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
82	C₃²	C₃×#731	1	1	yes	characteristic
83	C₃²	#6909	1	2	yes	—
84	C₉	—	3	1	no	—
85	C₉	—	3	2	no	—
86	C₉	—	9	1	no	—
87	C₉	—	9	1	no	—
88	C₉	—	27	3	no	—
89	C₉	—	27	6	no	—
90	C₃	—	3	1	no	—
91	C₃	—	3	2	no	—
92	C₃	—	3	3	no	—
93	C₃	—	3	1	no	—
94	C₃	—	9	2	no	—
95	C₃	—	27	9	no	—
96	C₃	#10624591	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃	#10624591
commutator, Frattini, cosocle	C₃²×C₉	C₃³
prime radical	C₃²×#732	C₃
LCS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #10624766	C₃	semidirect
3. #10624688	C₃	semidirect
4. #10624485	C₃	semidirect
12. #6895	C₃²	semidirect
17. #6867	C₃²	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#732	3
#10624766	3
#10624688	3
#10624485	3