Subgroups — #14550375

Order 2187.

476 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624869	C₃	1	1	yes	maximal, characteristic
3	C₃×#6873	C₃	1	9	yes	maximal, complemented
4	#10624683	C₃	1	2	yes	maximal, complemented
5	C₃×#737	C₃²	1	3	yes	—
6	C₃³×C₉	C₃²	1	1	yes	characteristic
7	C₃×C₂₇⋊C₃	—	3	3	no	—
8	C₃×C₂₇⋊C₃	—	3	9	no	—
9	C₃×C₂₇⋊C₃	C₃²	1	6	yes	—
10	C₃²×C₂₇	C₃²	1	3	yes	—
11	#6893	C₉	1	6	yes	—
12	C₉⋊H(3,𝔽₃)	—	3	6	no	—
13	#6891	C₉	1	2	yes	—
14	C₃²⋊H(3,𝔽₃)	C₉	1	1	yes	characteristic
15	C₉×H(3,𝔽₃)	—	3	3	no	—
16	#6873	—	3	27	no	—
17	C₃²⋊C₂₇	—	3	6	no	—
18	C₃⁴	C₃×C₉	1	1	yes	characteristic
19	#737	—	3	9	no	—
20	#737	—	3	18	no	—
21	C₃×3₋¹⁺²	—	3	6	no	—
22	C₃×3₋¹⁺²	C₃×C₉	1	6	yes	—
23	C₃×H(3,𝔽₃)	—	3	3	no	—
24	C₃×H(3,𝔽₃)	C₃×C₉	1	3	yes	—
25	C₃²×C₉	—	3	3	no	—
26	C₃²×C₉	—	3	6	no	—
27	C₃²×C₉	—	3	3	no	—
28	C₃²×C₉	C₃×C₉	1	2	yes	—
29	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
30	C₃²×C₉	3₋¹⁺²	1	6	yes	—
31	C₃²×C₉	C₃³	1	1	yes	characteristic
32	C₂₇⋊C₃	—	3	27	no	—
33	C₂₇⋊C₃	—	3	18	no	—
34	C₃×C₂₇	—	3	6	no	—
35	C₃×C₂₇	—	3	3	no	—
36	C₃×C₂₇	—	3	6	no	—
37	C₃×C₂₇	H(3,𝔽₃)	1	9	yes	complemented
38	C₃²⋊C₉	—	3	6	no	—
39	C₃²⋊C₉	—	3	18	no	—
40	C₃³	—	3	3	no	—
41	C₃³	—	3	3	no	—
42	C₃³	—	9	3	no	—
43	C₃³	C₃²⋊C₉	1	3	yes	—
44	C₃³	C₃²×C₉	1	1	yes	characteristic
45	3₋¹⁺²	—	3	18	no	—
46	3₋¹⁺²	—	9	18	no	—
47	H(3,𝔽₃)	—	3	9	no	—
48	H(3,𝔽₃)	—	9	9	no	—
49	C₃×C₉	—	3	6	no	—
50	C₃×C₉	—	3	3	no	—
51	C₃×C₉	—	3	18	no	—
52	C₃×C₉	—	3	6	no	—
53	C₃×C₉	—	3	6	no	—
54	C₃×C₉	—	9	3	no	—
55	C₃×C₉	—	9	6	no	—
56	C₃×C₉	C₃²⋊C₉	1	6	yes	—
57	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
58	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
59	C₃×C₉	C₃×3₋¹⁺²	1	2	yes	—
60	C₃×C₉	#737	1	6	yes	—
61	C₂₇	—	3	9	no	—
62	C₂₇	—	9	6	no	—
63	C₃²	—	3	3	no	—
64	C₃²	—	3	3	no	—
65	C₃²	—	3	3	no	—
66	C₃²	—	3	6	no	—
67	C₃²	—	9	9	no	—
68	C₃²	—	9	3	no	—
69	C₃²	—	9	9	no	—
70	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
71	C₃²	C₉×H(3,𝔽₃)	1	3	yes	—
72	C₉	—	3	6	no	—
73	C₉	—	9	6	no	—
74	C₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
75	C₉	C₉⋊H(3,𝔽₃)	1	6	yes	—
76	C₉	#6893	1	2	yes	—
77	C₃	—	3	3	no	—
78	C₃	—	9	3	no	—
79	C₃	—	27	3	no	—
80	C₃	#10624555	1	1	yes	characteristic
81	C₃	#10624704	1	3	yes	—

	subgroup	G/H
center	C₃×C₉	C₃×H(3,𝔽₃)
commutator	C₃³	C₃²×C₉
Frattini, cosocle	C₃²×C₉	C₃³
socle	C₃²	C₃×C₃²⋊C₉
prime radical	C₃²⋊H(3,𝔽₃)	C₉
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉ ◁ C₃³×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. C₃²×C₂₇⋊C₃	C₃	semidirect
3. C₃×#6873	C₃	semidirect
4. #10624683	C₃	semidirect
37. C₃×C₂₇	H(3,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₂₇⋊C₃	3
#10624869	3
C₃×#6873	3
#10624683	3