Subgroups — #14550579

Order 2187.

302 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉⋊C₂₇	C₃	1	3	yes	maximal
2	C₃×C₉⋊C₂₇	C₃	1	1	yes	maximal, characteristic
3	C₃×C₉⋊C₂₇	C₃	1	1	yes	maximal, characteristic
4	C₃×C₉×C₂₇	C₃	1	1	yes	maximal, characteristic
5	C₉×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
6	#10624463	C₃	1	6	yes	maximal
7	C₃²×C₂₇	C₃²	1	1	yes	characteristic
8	C₃²×C₂₇	C₃²	1	3	yes	—
9	C₃²×C₂₇	C₃²	1	2	yes	—
10	C₃²×C₂₇	C₃²	1	3	yes	—
11	C₃×C₉⋊C₉	C₉	1	2	yes	—
12	C₃×C₉⋊C₉	C₉	1	1	yes	characteristic
13	C₃×C₉²	C₃²	1	3	yes	—
14	C₃×C₉²	C₃²	1	1	yes	characteristic
15	C₉⋊C₂₇	—	3	9	no	—
16	C₉⋊C₂₇	—	3	3	no	—
17	C₉⋊C₂₇	—	3	3	no	—
18	C₉×C₂₇	—	3	3	no	—
19	#6855	C₉	1	6	yes	—
20	C₃²×C₉	C₃×C₉	1	6	yes	—
21	C₃²×C₉	C₃×C₉	1	2	yes	—
22	C₃²×C₉	C₃×C₉	1	3	yes	—
23	C₃²×C₉	C₃×C₉	1	1	yes	characteristic
24	C₃²×C₉	C₃³	1	1	yes	characteristic
25	C₃×C₂₇	—	3	3	no	—
26	C₃×C₂₇	—	3	6	no	—
27	C₃×C₂₇	—	3	3	no	—
28	C₃×C₂₇	—	3	3	no	—
29	C₃×C₂₇	—	3	2	no	—
30	C₃×C₂₇	—	3	3	no	—
31	C₃×C₂₇	—	3	1	no	—
32	C₃×C₂₇	—	3	6	no	—
33	C₃×C₂₇	—	3	6	no	—
34	C₃×C₂₇	3₋¹⁺²	1	9	yes	—
35	C₉⋊C₉	—	3	6	no	—
36	C₉⋊C₉	C₂₇	1	9	yes	complemented
37	C₉²	—	3	3	no	—
38	C₉²	—	3	6	no	—
39	C₉²	3₋¹⁺²	1	3	yes	—
40	C₉²	3₋¹⁺²	1	3	yes	—
41	C₉²	3₋¹⁺²	1	3	yes	—
42	C₃³	C₃²×C₉	1	1	yes	characteristic
43	C₃×C₉	—	3	3	no	—
44	C₃×C₉	—	3	6	no	—
45	C₃×C₉	—	3	6	no	—
46	C₃×C₉	—	3	6	no	—
47	C₃×C₉	—	3	6	no	—
48	C₃×C₉	—	3	12	no	—
49	C₃×C₉	C₉⋊C₉	1	6	yes	—
50	C₃×C₉	C₉⋊C₉	1	3	yes	—
51	C₃×C₉	C₃×C₂₇	1	9	yes	—
52	C₃×C₉	C₃×C₂₇	1	3	yes	—
53	C₃×C₉	C₂₇⋊C₃	1	6	yes	—
54	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
55	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
56	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
57	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
58	C₃×C₉	#737	1	6	yes	—
59	C₂₇	—	3	9	no	—
60	C₂₇	—	9	3	no	—
61	C₂₇	—	9	2	no	—
62	C₂₇	—	9	1	no	—
63	C₃²	C₃×C₉⋊C₉	1	1	yes	characteristic
64	C₃²	C₉×3₋¹⁺²	1	3	yes	—
65	C₃²	C₃²×C₂₇	1	1	yes	characteristic
66	C₃²	C₃×C₂₇⋊C₃	1	2	yes	—
67	C₃²	#6903	1	6	yes	—
68	C₉	—	3	6	no	—
69	C₉	—	3	9	no	—
70	C₉	—	9	6	no	—
71	C₉	C₉⋊C₂₇	1	9	yes	complemented
72	C₉	#6894	1	3	yes	—
73	C₉	#6896	1	3	yes	—
74	C₉	#6899	1	2	yes	—
75	C₉	#6900	1	1	yes	characteristic
76	C₃	#10624561	1	1	yes	characteristic
77	C₃	C₃×C₉⋊C₂₇	1	1	yes	characteristic
78	C₃	#10624707	1	2	yes	—
79	C₃	C₂₇×3₋¹⁺²	1	3	yes	—
80	C₃	#10624729	1	6	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₃²×C₉	C₃³
commutator	C₃²	C₃²×C₂₇
socle, prime radical	C₃³	C₃²×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
36. C₉⋊C₉	C₂₇	semidirect
71. C₉	C₉⋊C₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉⋊C₂₇	3
C₃×C₉×C₂₇	3
C₉×C₉⋊C₉	3
#10624463	3