Group #14550707 — subgroups

Subgroups — #14550707

Order 2187.

582 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624880	C₃	1	1	yes	maximal, characteristic
2	#10624698	C₃	1	4	yes	maximal, complemented
3	C₃×H(3,𝔽₃)⋊C₉	C₃	1	2	yes	maximal, complemented
4	#10624689	C₃	1	4	yes	maximal, complemented
5	#10624555	C₃	1	2	yes	maximal, complemented
6	C₃²×H(3,𝔽₃)	C₃²	1	2	yes	—
7	C₃²×H(3,𝔽₃)	C₃²	1	2	yes	—
8	C₃³×C₉	C₃²	1	1	yes	characteristic
9	C₃²⋊H(3,𝔽₃)	—	3	6	no	—
10	C₃²⋊H(3,𝔽₃)	—	3	6	no	—
11	C₃×C₃²⋊C₉	—	3	6	no	—
12	C₃×C₃²⋊C₉	C₃²	1	4	yes	—
13	C₃×C₃²⋊C₉	C₃²	1	4	yes	—
14	H(3,𝔽₃)⋊C₉	—	3	6	no	—
15	H(3,𝔽₃)⋊C₉	—	3	12	no	—
16	#6866	—	3	12	no	—
17	C₃⁴	—	3	2	no	—
18	C₃⁴	H(3,𝔽₃)	1	6	yes	—
19	C₃⁴	C₃³	1	1	yes	characteristic
20	C₃×H(3,𝔽₃)	—	3	2	no	—
21	C₃×H(3,𝔽₃)	—	3	8	no	—
22	C₃×H(3,𝔽₃)	—	3	4	no	—
23	C₃×H(3,𝔽₃)	—	3	4	no	—
24	C₃×H(3,𝔽₃)	—	3	2	no	—
25	C₃×H(3,𝔽₃)	—	3	4	no	—
26	C₃×H(3,𝔽₃)	—	3	4	no	—
27	C₃×H(3,𝔽₃)	—	3	2	no	—
28	C₃×H(3,𝔽₃)	—	3	4	no	—
29	C₃×H(3,𝔽₃)	—	3	4	no	—
30	C₃×H(3,𝔽₃)	—	3	8	no	—
31	C₃×H(3,𝔽₃)	—	3	9	no	—
32	C₃×H(3,𝔽₃)	—	3	18	no	—
33	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	6	yes	—
34	C₃²×C₉	—	3	2	no	—
35	C₃²×C₉	—	3	2	no	—
36	C₃²×C₉	—	3	4	no	—
37	C₃²×C₉	—	3	4	no	—
38	C₃²×C₉	—	9	3	no	—
39	C₃²⋊C₉	—	3	4	no	—
40	C₃²⋊C₉	—	3	8	no	—
41	C₃²⋊C₉	—	3	4	no	—
42	C₃²⋊C₉	—	3	8	no	—
43	C₃²⋊C₉	—	9	18	no	—
44	C₃³	—	3	6	no	—
45	C₃³	—	3	6	no	—
46	C₃³	—	3	6	no	—
47	C₃³	—	3	12	no	—
48	C₃³	—	9	1	no	—
49	C₃³	—	9	2	no	—
50	C₃³	—	9	6	no	—
51	C₃³	—	9	12	no	—
52	C₃³	—	9	2	no	—
53	C₃³	—	9	4	no	—
54	C₃³	—	27	2	no	—
55	C₃³	—	27	2	no	—
56	C₃³	—	27	2	no	—
57	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
58	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
59	C₃³	#737	1	1	yes	characteristic
60	C₃³	#737	1	4	yes	—
61	C₃³	#737	1	4	yes	—
62	H(3,𝔽₃)	—	3	6	no	—
63	H(3,𝔽₃)	—	3	12	no	—
64	H(3,𝔽₃)	—	9	4	no	—
65	H(3,𝔽₃)	—	9	8	no	—
66	H(3,𝔽₃)	—	9	4	no	—
67	H(3,𝔽₃)	—	9	8	no	—
68	H(3,𝔽₃)	—	9	2	no	—
69	H(3,𝔽₃)	—	9	4	no	—
70	H(3,𝔽₃)	—	27	9	no	—
71	H(3,𝔽₃)	—	27	18	no	—
72	C₃×C₉	—	3	6	no	—
73	C₃×C₉	—	9	4	no	—
74	C₃×C₉	—	9	1	no	—
75	C₃×C₉	—	9	6	no	—
76	C₃×C₉	—	9	12	no	—
77	C₃×C₉	—	9	4	no	—
78	C₃×C₉	—	9	4	no	—
79	C₃×C₉	—	27	4	no	—
80	C₃²	—	3	2	no	—
81	C₃²	—	3	2	no	—
82	C₃²	—	3	2	no	—
83	C₃²	—	3	4	no	—
84	C₃²	—	3	2	no	—
85	C₃²	—	9	1	no	—
86	C₃²	—	9	4	no	—
87	C₃²	—	9	4	no	—
88	C₃²	—	9	18	no	—
89	C₃²	—	9	6	no	—
90	C₃²	—	9	6	no	—
91	C₃²	—	9	6	no	—
92	C₃²	—	9	6	no	—
93	C₃²	—	9	12	no	—
94	C₃²	—	9	6	no	—
95	C₃²	—	9	6	no	—
96	C₃²	—	27	12	no	—
97	C₃²	—	27	6	no	—
98	C₃²	—	27	12	no	—
99	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
100	C₃²	#6893	1	2	yes	—
101	C₃²	#6893	1	4	yes	—
102	C₃²	#6893	1	2	yes	—
103	C₃²	#6893	1	4	yes	—
104	C₉	—	9	3	no	—
105	C₉	—	27	4	no	—
106	C₉	—	27	4	no	—
107	C₃	—	3	2	no	—
108	C₃	—	3	4	no	—
109	C₃	—	3	2	no	—
110	C₃	—	3	4	no	—
111	C₃	—	27	6	no	—
112	C₃	—	27	6	no	—
113	C₃	#10624564	1	1	yes	characteristic
114	C₃	UT(4,𝔽₃)	1	1	yes	characteristic
115	C₃	#10624748	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10624880

C₃

yes

maximal, characteristic

#10624698

C₃

yes

maximal, complemented

C₃×H(3,𝔽₃)⋊C₉

C₃

yes

maximal, complemented

#10624689

C₃

yes

maximal, complemented

#10624555

C₃

yes

maximal, complemented

C₃²×H(3,𝔽₃)

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

—

C₃³×C₉

C₃²

yes

characteristic

C₃²⋊H(3,𝔽₃)

—

C₃²⋊H(3,𝔽₃)

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

H(3,𝔽₃)⋊C₉

—

H(3,𝔽₃)⋊C₉

—

#6866

—

C₃⁴

—

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

C₃³

yes

characteristic

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

H(3,𝔽₃)

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

#737

yes

characteristic

C₃³

#737

yes

—

C₃³

#737

yes

—

—

—

—

—

—

—

—

—

—

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

100

C₃²

#6893

yes

—

101

C₃²

#6893

yes

—

102

C₃²

#6893

yes

—

103

C₃²

#6893

yes

—

104

C₉

—

105

C₉

—

106

C₉

—

107

C₃

—

108

C₃

—

109

C₃

—

110

C₃

—

111

C₃

—

112

C₃

—

113

C₃

#10624564

yes

characteristic

114

C₃

UT(4,𝔽₃)

yes

characteristic

115

C₃

#10624748

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624880	C₃
LCS	G ▷ C₃⁴ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²⋊H(3,𝔽₃)

commutator, Frattini, cosocle

C₃⁴

C₃³

C₃

G ▷ C₃⁴ ▷ C₃ ▷ C₁

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

G ▷ C₃⁴ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
2. #10624698	C₃	semidirect
3. C₃×H(3,𝔽₃)⋊C₉	C₃	semidirect
4. #10624689	C₃	semidirect
5. #10624555	C₃	semidirect

H ≅ G/N

type

2. #10624698

C₃

semidirect

3. C₃×H(3,𝔽₃)⋊C₉

C₃

semidirect

4. #10624689

C₃

semidirect

5. #10624555

C₃

semidirect

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

subgroup	index
#10624880	3
#10624698	3
C₃×H(3,𝔽₃)⋊C₉	3
#10624689	3
#10624555	3

subgroup

index

C₃×H(3,𝔽₃)⋊C₉

Subgroups — #14550707

Conjugacy classes of subgroups

Complements

Maximal subgroups