Subgroups — #14550366

Order 2187.

653 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃²⋊C₂₇	C₃	1	9	yes	maximal, complemented
4	C₃×C₃²⋊C₂₇	C₃	1	2	yes	maximal, complemented
5	C₃²×H(3,𝔽₃)	C₉	1	1	yes	characteristic
6	C₃³×C₉	C₃²	1	3	yes	—
7	C₃³×C₉	C₃²	1	1	yes	characteristic
8	C₃²×C₂₇	—	3	3	no	—
9	C₃²×C₂₇	—	3	9	no	—
10	C₃²×C₂₇	C₃²	1	6	yes	—
11	C₃²×C₂₇	C₃²	1	3	yes	—
12	C₉×H(3,𝔽₃)	—	3	3	no	—
13	C₉×H(3,𝔽₃)	—	3	6	no	—
14	C₃×C₃²⋊C₉	C₉	1	2	yes	—
15	C₃×C₃²⋊C₉	C₉	1	6	yes	—
16	C₃²⋊C₂₇	—	3	6	no	—
17	C₃²⋊C₂₇	—	3	27	no	—
18	C₃⁴	C₃×C₉	1	3	yes	—
19	C₃⁴	C₃×C₉	1	1	yes	characteristic
20	C₃×H(3,𝔽₃)	—	3	3	no	—
21	C₃×H(3,𝔽₃)	—	3	6	no	—
22	C₃×H(3,𝔽₃)	C₂₇	1	9	yes	complemented
23	C₃²×C₉	—	3	3	no	—
24	C₃²×C₉	—	3	6	no	—
25	C₃²×C₉	—	3	9	no	—
26	C₃²×C₉	—	3	3	no	—
27	C₃²×C₉	—	3	6	no	—
28	C₃²×C₉	—	3	18	no	—
29	C₃²×C₉	C₃×C₉	1	6	yes	—
30	C₃²×C₉	C₃×C₉	1	2	yes	—
31	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
32	C₃²×C₉	3₋¹⁺²	1	6	yes	—
33	C₃²×C₉	C₃³	1	1	yes	characteristic
34	C₃×C₂₇	—	3	27	no	—
35	C₃×C₂₇	—	3	6	no	—
36	C₃×C₂₇	—	3	3	no	—
37	C₃×C₂₇	—	3	18	no	—
38	C₃×C₂₇	—	3	6	no	—
39	C₃×C₂₇	H(3,𝔽₃)	1	9	yes	complemented
40	C₃²⋊C₉	—	3	6	no	—
41	C₃²⋊C₉	—	3	18	no	—
42	C₃³	—	3	3	no	—
43	C₃³	—	3	6	no	—
44	C₃³	—	3	3	no	—
45	C₃³	—	3	9	no	—
46	C₃³	—	3	6	no	—
47	C₃³	—	3	18	no	—
48	C₃³	C₃²⋊C₉	1	3	yes	—
49	C₃³	C₃×C₂₇	1	3	yes	—
50	C₃³	C₃×C₂₇	1	9	yes	complemented
51	C₃³	C₂₇⋊C₃	1	6	yes	—
52	C₃³	C₃²×C₉	1	1	yes	characteristic
53	H(3,𝔽₃)	—	3	27	no	—
54	C₃×C₉	—	3	6	no	—
55	C₃×C₉	—	3	3	no	—
56	C₃×C₉	—	3	18	no	—
57	C₃×C₉	—	3	6	no	—
58	C₃×C₉	—	3	6	no	—
59	C₃×C₉	—	3	18	no	—
60	C₃×C₉	—	9	3	no	—
61	C₃×C₉	—	9	18	no	—
62	C₃×C₉	—	9	6	no	—
63	C₃×C₉	C₃²⋊C₉	1	6	yes	—
64	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
65	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
66	C₃×C₉	C₃×3₋¹⁺²	1	2	yes	—
67	C₃×C₉	#737	1	6	yes	—
68	C₂₇	—	3	9	no	—
69	C₂₇	—	9	6	no	—
70	C₃²	—	3	3	no	—
71	C₃²	—	3	6	no	—
72	C₃²	—	3	9	no	—
73	C₃²	—	3	18	no	—
74	C₃²	—	3	9	no	—
75	C₃²	—	3	27	no	—
76	C₃²	—	9	3	no	—
77	C₃²	—	9	6	no	—
78	C₃²	—	9	18	no	—
79	C₃²	C₃²⋊C₂₇	1	9	yes	complemented
80	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
81	C₃²	C₉×H(3,𝔽₃)	1	3	yes	—
82	C₃²	C₃²×C₂₇	1	1	yes	characteristic
83	C₃²	C₃×C₂₇⋊C₃	1	2	yes	—
84	C₃²	#6903	1	6	yes	—
85	C₉	—	3	6	no	—
86	C₉	—	9	6	no	—
87	C₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
88	C₉	C₉⋊H(3,𝔽₃)	1	6	yes	—
89	C₉	#6893	1	2	yes	—
90	C₃	—	3	9	no	—
91	C₃	—	9	9	no	—
92	C₃	#10624555	1	1	yes	characteristic
93	C₃	C₃×C₃²⋊C₂₇	1	1	yes	characteristic
94	C₃	#10624684	1	2	yes	—
95	C₃	C₂₇×H(3,𝔽₃)	1	3	yes	—
96	C₃	C₂₇⋊H(3,𝔽₃)	1	6	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₃²×C₉	C₃³
commutator	C₃²	C₃²×C₂₇
socle	C₃³	C₃²×C₉
prime radical	C₃²×H(3,𝔽₃)	C₉
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₃³×C₂₇	C₃	semidirect
3. C₃×C₃²⋊C₂₇	C₃	semidirect
4. C₃×C₃²⋊C₂₇	C₃	semidirect
22. C₃×H(3,𝔽₃)	C₂₇	semidirect
39. C₃×C₂₇	H(3,𝔽₃)	semidirect
50. C₃³	C₃×C₂₇	semidirect
79. C₃²	C₃²⋊C₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₂₇	3
C₃×C₉×H(3,𝔽₃)	3
C₃×C₃²⋊C₂₇	3