Subgroups — #14550552

Order 2187.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624709	C₃	1	2	yes	maximal
2	#10624709	C₃	1	1	yes	maximal, characteristic
3	#10624681	C₃	1	1	yes	maximal, characteristic
4	#10624473	C₃	1	2	yes	maximal
5	#10624473	C₃	1	1	yes	maximal, characteristic
6	#10624471	C₃	1	2	yes	maximal
7	#10624470	C₃	1	1	yes	maximal, characteristic
8	#10624469	C₃	1	2	yes	maximal
9	#10624469	C₃	1	1	yes	maximal, characteristic
10	C₃×C₂₇⋊C₃	C₃²	1	1	yes	characteristic
11	C₃×C₂₇⋊C₃	C₃²	1	2	yes	—
12	C₃×C₂₇⋊C₃	C₃²	1	2	yes	—
13	C₃×C₂₇⋊C₃	C₃²	1	1	yes	characteristic
14	C₃×C₂₇⋊C₃	C₃²	1	2	yes	—
15	C₃²×C₂₇	C₃²	1	1	yes	characteristic
16	#6887	C₃²	1	2	yes	—
17	#6887	C₃²	1	1	yes	characteristic
18	#6887	C₃²	1	1	yes	characteristic
19	#6875	—	3	6	no	—
20	#6875	C₉	1	9	yes	complemented
21	#6864	—	3	2	no	—
22	C₉×C₂₇	—	3	1	no	—
23	C₃²×C₉	C₃³	1	1	yes	characteristic
24	C₂₇⋊C₃	—	3	2	no	—
25	C₂₇⋊C₃	—	3	2	no	—
26	C₂₇⋊C₃	—	3	2	no	—
27	C₂₇⋊C₃	—	3	2	no	—
28	C₂₇⋊C₃	—	3	2	no	—
29	C₂₇⋊C₃	—	3	2	no	—
30	C₂₇⋊C₃	—	3	1	no	—
31	C₂₇⋊C₃	—	3	2	no	—
32	C₂₇⋊C₃	—	3	2	no	—
33	C₂₇⋊C₃	—	3	2	no	—
34	C₂₇⋊C₃	C₃×C₉	1	3	yes	—
35	C₂₇⋊C₃	C₃×C₉	1	6	yes	—
36	C₂₇⋊C₃	3₋¹⁺²	1	6	yes	—
37	C₃×C₂₇	—	3	2	no	—
38	C₃×C₂₇	—	3	1	no	—
39	C₃×C₂₇	—	3	2	no	—
40	C₃×C₂₇	—	3	1	no	—
41	C₃×C₂₇	—	3	2	no	—
42	C₃×C₂₇	—	3	1	no	—
43	C₃×C₂₇	—	3	2	no	—
44	C₃×C₂₇	3₋¹⁺²	1	1	yes	characteristic
45	C₃×C₂₇	3₋¹⁺²	1	2	yes	—
46	C₉⋊C₉	—	3	1	no	—
47	C₉⋊C₉	—	3	2	no	—
48	C₉⋊C₉	—	3	2	no	—
49	C₉⋊C₉	—	3	2	no	—
50	C₉⋊C₉	C₃×C₉	1	3	yes	—
51	C₃²⋊C₉	—	3	1	no	—
52	C₃²⋊C₉	—	3	1	no	—
53	C₃²⋊C₉	—	3	2	no	—
54	C₉²	—	3	1	no	—
55	C₉²	—	3	1	no	—
56	C₉²	—	3	2	no	—
57	C₃³	#737	1	1	yes	characteristic
58	C₃×C₉	—	3	3	no	—
59	C₃×C₉	—	9	1	no	—
60	C₃×C₉	—	9	2	no	—
61	C₃×C₉	C₃²×C₉	1	1	yes	characteristic
62	C₃×C₉	C₃×3₋¹⁺²	1	2	yes	—
63	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
64	C₃×C₉	#737	1	2	yes	—
65	C₃×C₉	#737	1	1	yes	characteristic
66	C₃×C₉	#737	1	2	yes	—
67	C₃×C₉	#737	1	2	yes	—
68	C₃×C₉	#737	1	1	yes	characteristic
69	C₂₇	—	3	3	no	—
70	C₂₇	—	3	6	no	—
71	C₂₇	—	3	6	no	—
72	C₂₇	—	3	1	no	—
73	C₂₇	—	3	2	no	—
74	C₂₇	—	9	1	no	—
75	C₂₇	—	9	2	no	—
76	C₃²	—	9	1	no	—
77	C₃²	C₉×3₋¹⁺²	1	1	yes	characteristic
78	C₃²	#6898	1	1	yes	characteristic
79	C₃²	#6899	1	2	yes	—
80	C₉	—	9	3	no	—
81	C₉	—	27	1	no	—
82	C₉	—	27	2	no	—
83	C₉	#6887	1	1	yes	characteristic
84	C₉	C₉×3₋¹⁺²	1	2	yes	—
85	C₉	#6894	1	2	yes	—
86	C₉	#6895	1	1	yes	characteristic
87	C₉	#6896	1	1	yes	characteristic
88	C₉	#6897	1	2	yes	—
89	C₃	—	3	1	no	—
90	C₃	—	3	1	no	—
91	C₃	—	3	2	no	—
92	C₃	#10624559	1	1	yes	characteristic

	subgroup	G/H
center	C₉	#6895
commutator	C₃×C₉	C₃²×C₉
Frattini, cosocle	C₃²×C₉	C₃³
socle	C₃	#10624559
prime radical	C₃³	#737
LCS	G ▷ C₃×C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉ ◁ 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
20. #6875	C₉	semidirect

Maximal subgroups

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

subgroup	index
#10624709	3
#10624681	3
#10624473	3
#10624471	3
#10624470	3
#10624469	3