Subgroups — #14546311

Order 2187.

111 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#732	C₃	1	1	yes	maximal, characteristic
2	C₃×C₉⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624446	C₃	1	2	yes	maximal, complemented
4	C₃×#732	C₉	1	1	yes	characteristic
5	C₃²×C₂₇	—	3	1	no	—
6	C₉×H(3,𝔽₃)	—	3	1	no	—
7	C₃×C₉²	C₃²	1	1	yes	characteristic
8	C₉⋊C₂₇	—	3	3	no	—
9	H(3,𝔽₃)⋊C₉	—	3	1	no	—
10	H(3,𝔽₃)⋊C₉	—	3	1	no	—
11	#6868	C₉	1	2	yes	—
12	C₃²⋊C₂₇	—	3	2	no	—
13	C₃×H(3,𝔽₃)	—	3	1	no	—
14	C₃²×C₉	—	9	1	no	—
15	C₃²×C₉	C₃×C₉	1	1	yes	characteristic
16	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
17	C₃²×C₉	3₋¹⁺²	1	1	yes	characteristic
18	C₃²×C₉	3₋¹⁺²	1	1	yes	characteristic
19	#732	—	3	3	no	—
20	C₃×C₂₇	—	3	3	no	—
21	C₃×C₂₇	—	9	2	no	—
22	C₃×C₂₇	—	9	1	no	—
23	C₃×C₂₇	—	9	2	no	—
24	C₃²⋊C₉	—	3	2	no	—
25	C₃²⋊C₉	—	9	1	no	—
26	C₃²⋊C₉	—	9	1	no	—
27	C₉²	—	3	1	no	—
28	C₉²	—	3	1	no	—
29	C₉²	—	3	1	no	—
30	C₃³	—	9	1	no	—
31	C₃³	C₃²⋊C₉	1	1	yes	characteristic
32	H(3,𝔽₃)	—	3	3	no	—
33	H(3,𝔽₃)	—	9	1	no	—
34	H(3,𝔽₃)	—	9	1	no	—
35	C₃×C₉	—	3	1	no	—
36	C₃×C₉	—	3	2	no	—
37	C₃×C₉	—	3	1	no	—
38	C₃×C₉	—	3	3	no	—
39	C₃×C₉	—	3	1	no	—
40	C₃×C₉	—	3	1	no	—
41	C₃×C₉	—	3	1	no	—
42	C₃×C₉	—	3	1	no	—
43	C₃×C₉	—	3	1	no	—
44	C₃×C₉	—	9	1	no	—
45	C₃×C₉	—	9	2	no	—
46	C₃×C₉	—	27	2	no	—
47	C₃×C₉	—	27	1	no	—
48	C₃×C₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
49	C₃×C₉	#731	1	1	yes	characteristic
50	C₃×C₉	#732	1	1	yes	characteristic
51	C₂₇	—	9	3	no	—
52	C₂₇	—	27	2	no	—
53	C₃²	—	3	1	no	—
54	C₃²	—	3	1	no	—
55	C₃²	—	3	1	no	—
56	C₃²	—	9	3	no	—
57	C₃²	—	27	1	no	—
58	C₃²	—	27	1	no	—
59	C₃²	—	27	1	no	—
60	C₃²	#6869	1	1	yes	characteristic
61	C₃²	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
62	C₃²	#6872	1	1	yes	characteristic
63	C₃²	#6873	1	1	yes	characteristic
64	C₉	—	3	3	no	—
65	C₉	—	3	1	no	—
66	C₉	—	3	1	no	—
67	C₉	—	9	1	no	—
68	C₉	—	9	1	no	—
69	C₉	—	27	2	no	—
70	C₉	#6881	1	2	yes	—
71	C₉	#6883	1	1	yes	characteristic
72	C₃	—	3	1	no	—
73	C₃	—	3	1	no	—
74	C₃	—	3	1	no	—
75	C₃	—	27	3	no	—
76	C₃	#10624453	1	1	yes	characteristic
77	C₃	#10624526	1	1	yes	characteristic
78	C₃	#10624527	1	1	yes	characteristic
79	C₃	#10624530	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₉	#732
commutator	C₃²×C₉	C₃×C₉
Frattini, cosocle	C₃×C₉²	C₃²
socle	C₃²	H(3,𝔽₃)⋊C₉
prime radical	C₃×#732	C₉
LCS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃×C₉² ◁ G
LECS	G ▷ C₃×C₉² ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₃×C₉⋊C₂₇	C₃	semidirect
3. #10624446	C₃	semidirect

Maximal subgroups

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

subgroup	index
C₉×#732	3
C₃×C₉⋊C₂₇	3
#10624446	3