Subgroups — #14546284

Order 2187.

174 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6869	C₃	1	1	yes	maximal, characteristic
2	C₃×#6869	C₃	1	2	yes	maximal
3	#10624548	C₃	1	1	yes	maximal, characteristic
4	C₃³×C₉	C₃²	1	1	yes	characteristic
5	C₃×C₂₇⋊C₃	—	3	1	no	—
6	C₃×C₂₇⋊C₃	—	3	2	no	—
7	C₃×C₉⋊C₉	—	3	1	no	—
8	C₃×C₃²⋊C₉	—	3	1	no	—
9	#6869	—	3	3	no	—
10	#6869	—	3	6	no	—
11	#6855	—	3	1	no	—
12	#6855	C₉	1	3	yes	—
13	C₃⁴	3₋¹⁺²	1	1	yes	characteristic
14	C₃²×C₉	—	3	1	no	—
15	C₃²×C₉	—	3	2	no	—
16	C₃²×C₉	—	3	1	no	—
17	C₃²×C₉	—	3	2	no	—
18	C₃²×C₉	—	3	1	no	—
19	C₃²×C₉	—	3	2	no	—
20	C₃²×C₉	—	3	1	no	—
21	C₃²×C₉	—	3	2	no	—
22	C₃²×C₉	—	3	3	no	—
23	C₃²×C₉	C₃×C₉	1	1	yes	characteristic
24	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
25	C₃²×C₉	3₋¹⁺²	1	1	yes	characteristic
26	C₂₇⋊C₃	—	3	3	no	—
27	C₂₇⋊C₃	—	3	6	no	—
28	C₂₇⋊C₃	—	9	2	no	—
29	C₂₇⋊C₃	—	9	2	no	—
30	C₂₇⋊C₃	—	9	2	no	—
31	C₃×C₂₇	—	9	1	no	—
32	C₃×C₂₇	—	9	2	no	—
33	C₉⋊C₉	—	9	1	no	—
34	C₉⋊C₉	—	9	1	no	—
35	C₉⋊C₉	—	9	1	no	—
36	C₃²⋊C₉	—	9	3	no	—
37	C₃³	—	3	3	no	—
38	C₃³	—	3	1	no	—
39	C₃³	—	3	1	no	—
40	C₃³	—	3	2	no	—
41	C₃³	—	9	1	no	—
42	C₃³	—	9	1	no	—
43	C₃³	C₃²⋊C₉	1	1	yes	characteristic
44	C₃×C₉	—	3	1	no	—
45	C₃×C₉	—	3	1	no	—
46	C₃×C₉	—	3	1	no	—
47	C₃×C₉	—	3	1	no	—
48	C₃×C₉	—	3	2	no	—
49	C₃×C₉	—	3	2	no	—
50	C₃×C₉	—	9	1	no	—
51	C₃×C₉	—	9	2	no	—
52	C₃×C₉	—	9	2	no	—
53	C₃×C₉	—	9	2	no	—
54	C₃×C₉	—	9	1	no	—
55	C₃×C₉	—	9	1	no	—
56	C₃×C₉	—	9	3	no	—
57	C₃×C₉	—	9	3	no	—
58	C₃×C₉	—	9	3	no	—
59	C₃×C₉	—	9	3	no	—
60	C₃×C₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
61	C₃×C₉	AΣL(1,𝔽₂₇)	1	2	yes	—
62	C₃×C₉	#731	1	1	yes	characteristic
63	C₃×C₉	#731	1	2	yes	—
64	C₃×C₉	#731	1	1	yes	characteristic
65	C₃×C₉	#731	1	2	yes	—
66	C₃×C₉	#732	1	1	yes	characteristic
67	C₃×C₉	#733	1	2	yes	—
68	C₂₇	—	9	3	no	—
69	C₂₇	—	9	6	no	—
70	C₃²	—	3	1	no	—
71	C₃²	—	3	1	no	—
72	C₃²	—	3	1	no	—
73	C₃²	—	3	3	no	—
74	C₃²	—	9	3	no	—
75	C₃²	—	9	6	no	—
76	C₃²	—	9	1	no	—
77	C₃²	—	9	1	no	—
78	C₃²	—	9	1	no	—
79	C₃²	#6867	1	1	yes	characteristic
80	C₃²	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
81	C₃²	3₋¹⁺²⋊C₉	1	2	yes	—
82	C₉	—	9	1	no	—
83	C₉	—	9	1	no	—
84	C₉	—	27	3	no	—
85	C₉	#6857	1	1	yes	characteristic
86	C₉	#6858	1	2	yes	—
87	C₉	#6858	1	2	yes	—
88	C₉	#6859	1	2	yes	—
89	C₉	#6861	1	1	yes	characteristic
90	C₉	#6862	1	1	yes	characteristic
91	C₃	—	3	1	no	—
92	C₃	—	3	2	no	—
93	C₃	—	9	3	no	—
94	C₃	#10624450	1	1	yes	characteristic
95	C₃	#10624514	1	1	yes	characteristic
96	C₃	#10624516	1	1	yes	characteristic
97	C₃	#10624517	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	#6867
commutator	C₃²×C₉	C₃×C₉
Frattini, cosocle	C₃³×C₉	C₃²
prime radical	C₃⁴	3₋¹⁺²
LCS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃³×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

Maximal subgroups

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

subgroup	index
C₃×#6869	3
#10624548	3