Group #14517107 — subgroups

Subgroups — #14517107

Order 1944.

276 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 46 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	C₉×#1084	C₂	1	1	yes	maximal, characteristic
2	#11757601	C₂	1	2	yes	maximal
3	#10619900	—	3	1	no	maximal
4	#10619868	—	3	2	no	maximal
5	C₂×C₉×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₃⋊(C₃⋊Dic₉)	D₃	1	2	yes	—
7	C₃×#1084	D₃	1	1	yes	characteristic
8	C₃⋊((C₃⋊Dic₃)⋊C₃)	—	3	2	no	—
9	C₃×C₃⋊Dic₉	—	3	2	no	—
10	C₃×C₉×Dic₃	—	3	2	no	—
11	C₃×C₉×Dic₃	—	6	1	no	—
12	#66316	—	6	2	no	—
13	C₉×H(3,𝔽₃)	Q₈	1	1	yes	characteristic, complemented, Sylow
14	#6293	—	9	2	no	—
15	#6226	—	9	1	no	—
16	#6196	—	9	1	no	—
17	#6189	—	9	6	no	—
18	C₂×C₃×H(3,𝔽₃)	C₂×D₃	1	1	yes	characteristic
19	C₂×C₃²×C₉	—	2	1	no	—
20	C₂×C₃²×C₉	C₂×D₃	1	2	yes	—
21	C₂×C₃²⋊C₉	—	2	2	no	—
22	C₂×C₃²⋊C₉	—	4	1	no	—
23	C₃⋊(C₃⋊Dic₃)	—	3	2	no	—
24	C₃×C₃⋊Dic₃	—	9	2	no	—
25	C₃²×Dic₃	—	3	2	no	—
26	C₃²×Dic₃	—	6	1	no	—
27	C₃×C₄×C₉	—	9	1	no	—
28	#1084	D₉	1	1	yes	characteristic
29	C₃⋊Dic₉	—	3	2	no	—
30	C₃⋊Dic₉	—	6	2	no	—
31	C₃⋊Dic₉	—	9	6	no	—
32	(C₃⋊Dic₃)⋊C₃	—	9	2	no	—
33	(C₃⋊Dic₃)⋊C₃	—	18	2	no	—
34	C₉×Dic₃	—	3	6	no	—
35	C₉×Dic₃	—	6	3	no	—
36	C₃×Dic₉	—	9	6	no	—
37	C₃×H(3,𝔽₃)	Dic₆	1	1	yes	characteristic
38	C₃²×C₉	—	2	1	no	—
39	C₃²×C₉	Dic₆	1	2	yes	complemented
40	C₃²⋊C₉	—	2	2	no	—
41	C₃²⋊C₉	—	4	1	no	—
42	#634	—	27	1	no	—
43	#627	—	27	2	no	—
44	#627	—	27	2	no	—
45	#627	—	27	4	no	—
46	Dic₁₈	—	27	3	no	—
47	C₂×C₃³	—	2	1	no	—
48	C₂×C₃³	D₃²	1	2	yes	—
49	C₂×H(3,𝔽₃)	—	2	2	no	—
50	C₂×H(3,𝔽₃)	—	4	1	no	—
51	C₂×H(3,𝔽₃)	C₂×D₉	1	1	yes	characteristic
52	C₂×C₃×C₉	—	2	2	no	—
53	C₂×C₃×C₉	—	3	6	no	—
54	C₂×C₃×C₉	—	4	1	no	—
55	C₂×C₃×C₉	—	6	3	no	—
56	C₂×C₃×C₉	D₃²	1	1	yes	characteristic
57	C₃²×C₄	—	9	1	no	—
58	C₃⋊Dic₃	—	9	2	no	—
59	C₃⋊Dic₃	—	9	2	no	—
60	C₃⋊Dic₃	—	18	2	no	—
61	C₃⋊Dic₃	—	27	2	no	—
62	C₃⋊Dic₃	—	27	4	no	—
63	C₃×Dic₃	—	3	2	no	—
64	C₃×Dic₃	—	3	4	no	—
65	C₃×Dic₃	—	3	2	no	—
66	C₃×Dic₃	—	6	1	no	—
67	C₃×Dic₃	—	6	1	no	—
68	C₃×Dic₃	—	6	2	no	—
69	C₃×Dic₃	—	27	2	no	—
70	C₃×Dic₃	—	27	4	no	—
71	C₃×Dic₃	—	27	2	no	—
72	C₄×C₉	—	9	3	no	—
73	Dic₉	—	9	6	no	—
74	Dic₉	—	54	2	no	—
75	C₃³	—	2	1	no	—
76	C₃³	#627	1	2	yes	—
77	H(3,𝔽₃)	—	2	2	no	—
78	H(3,𝔽₃)	—	4	1	no	—
79	H(3,𝔽₃)	Dic₁₈	1	1	yes	characteristic, complemented
80	C₃×C₉	—	2	2	no	—
81	C₃×C₉	—	3	6	no	—
82	C₃×C₉	—	4	1	no	—
83	C₃×C₉	—	6	3	no	—
84	C₃×C₉	#627	1	1	yes	characteristic
85	Dic₆	—	81	2	no	—
86	Dic₆	—	81	1	no	—
87	Dic₆	—	81	1	no	—
88	Dic₆	—	81	2	no	—
89	C₂×C₃²	—	2	2	no	—
90	C₂×C₃²	—	2	1	no	—
91	C₂×C₃²	—	3	2	no	—
92	C₂×C₃²	—	3	4	no	—
93	C₂×C₃²	—	4	1	no	—
94	C₂×C₃²	—	6	1	no	—
95	C₂×C₃²	—	6	2	no	—
96	C₂×C₃²	D₃×D₉	1	2	yes	—
97	C₂×C₃²	#1113	1	1	yes	characteristic
98	C₂×C₉	—	6	2	no	—
99	C₂×C₉	—	12	1	no	—
100	C₂×C₉	((C₃⋊D₃)⋊C₃)⋊C₂	1	3	yes	—
101	C₃×C₄	—	9	1	no	—
102	C₃×C₄	—	9	1	no	—
103	C₃×C₄	—	9	2	no	—
104	C₃×C₄	—	81	2	no	—
105	Dic₃	—	3	2	no	—
106	Dic₃	—	6	1	no	—
107	Dic₃	—	27	2	no	—
108	Dic₃	—	27	2	no	—
109	Dic₃	—	27	4	no	—
110	Dic₃	—	81	2	no	—
111	Dic₃	—	162	2	no	—
112	C₃²	—	2	2	no	—
113	C₃²	—	2	1	no	—
114	C₃²	—	3	4	no	—
115	C₃²	—	3	2	no	—
116	C₃²	—	4	1	no	—
117	C₃²	—	6	2	no	—
118	C₃²	—	6	1	no	—
119	C₃²	#6189	1	2	yes	complemented
120	C₃²	#6296	1	1	yes	characteristic
121	C₉	—	6	2	no	—
122	C₉	—	12	1	no	—
123	C₉	#6196	1	3	yes	complemented
124	Q₈	—	243	1	no	Sylow
125	C₂×C₃	—	3	2	no	—
126	C₂×C₃	—	6	1	no	—
127	C₂×C₃	—	6	2	no	—
128	C₂×C₃	—	12	1	no	—
129	C₂×C₃	#66431	1	1	yes	characteristic
130	C₂×C₃	#66432	1	1	yes	characteristic
131	C₂×C₃	#4207⋊C₂	1	2	yes	—
132	C₄	—	9	1	no	—
133	C₄	—	81	2	no	—
134	C₃	—	3	2	no	—
135	C₃	—	6	2	no	—
136	C₃	—	6	1	no	—
137	C₃	—	12	1	no	—
138	C₃	#10619895	1	1	yes	characteristic
139	C₃	#10619900	1	1	yes	characteristic
140	C₃	#10619905	1	2	yes	—
141	C₂	#11757748	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#1084

C₂

yes

maximal, characteristic

#11757601

C₂

yes

—

—

C₂×C₉×H(3,𝔽₃)

C₂²

yes

characteristic

C₃⋊(C₃⋊Dic₉)

D₃

yes

—

C₃×#1084

D₃

yes

characteristic

C₃⋊((C₃⋊Dic₃)⋊C₃)

—

C₃×C₃⋊Dic₉

—

C₃×C₉×Dic₃

—

C₃×C₉×Dic₃

—

#66316

—

C₉×H(3,𝔽₃)

Q₈

yes

characteristic, complemented, Sylow

—

—

—

—

C₂×C₃×H(3,𝔽₃)

C₂×D₃

yes

characteristic

C₂×C₃²×C₉

—

C₂×C₃²×C₉

C₂×D₃

yes

—

C₂×C₃²⋊C₉

—

C₂×C₃²⋊C₉

—

C₃⋊(C₃⋊Dic₃)

—

C₃×C₃⋊Dic₃

—

C₃²×Dic₃

—

C₃²×Dic₃

—

C₃×C₄×C₉

—

#1084

D₉

yes

characteristic

C₃⋊Dic₉

—

C₃⋊Dic₉

—

C₃⋊Dic₉

—

(C₃⋊Dic₃)⋊C₃

—

(C₃⋊Dic₃)⋊C₃

—

C₉×Dic₃

—

C₉×Dic₃

—

C₃×Dic₉

—

C₃×H(3,𝔽₃)

Dic₆

yes

characteristic

C₃²×C₉

—

C₃²×C₉

Dic₆

yes

complemented

C₃²⋊C₉

—

C₃²⋊C₉

—

—

—

—

—

Dic₁₈

—

C₂×C₃³

—

C₂×C₃³

D₃²

yes

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

C₂×D₉

yes

characteristic

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

D₃²

yes

characteristic

C₃²×C₄

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₄×C₉

—

Dic₉

—

Dic₉

—

C₃³

—

C₃³

#627

yes

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

Dic₁₈

yes

characteristic, complemented

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

#627

yes

characteristic

Dic₆

—

Dic₆

—

Dic₆

—

Dic₆

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

D₃×D₉

yes

—

C₂×C₃²

#1113

yes

characteristic

C₂×C₉

—

C₂×C₉

—

100

C₂×C₉

((C₃⋊D₃)⋊C₃)⋊C₂

yes

—

101

C₃×C₄

—

102

C₃×C₄

—

103

C₃×C₄

—

104

C₃×C₄

—

105

Dic₃

—

106

Dic₃

—

107

Dic₃

—

108

Dic₃

—

109

Dic₃

—

110

Dic₃

—

111

Dic₃

—

162

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

#6189

yes

complemented

120

C₃²

#6296

yes

characteristic

121

C₉

—

122

C₉

—

123

C₉

#6196

yes

complemented

124

Q₈

—

243

Sylow

125

C₂×C₃

—

126

C₂×C₃

—

127

C₂×C₃

—

128

C₂×C₃

—

129

C₂×C₃

#66431

yes

characteristic

130

C₂×C₃

#66432

yes

characteristic

131

C₂×C₃

#4207⋊C₂

yes

—

132

C₄

—

133

C₄

—

134

C₃

—

135

C₃

—

136

C₃

—

137

C₃

—

138

C₃

#10619895

yes

characteristic

139

C₃

#10619900

yes

characteristic

140

C₃

#10619905

yes

—

141

C₂

#11757748

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂	#11757748
commutator, Fitting, cosocle	C₂×C₉×H(3,𝔽₃)	C₂²
Frattini, socle	C₂×C₃²	#1113
prime radical	C₂×C₃×H(3,𝔽₃)	C₂×D₃
hypocenter	C₉×H(3,𝔽₃)	Q₈
LCS	G ▷ C₂×C₉×H(3,𝔽₃) ▷ C₉×H(3,𝔽₃)
UCS	C₁ ◁ C₂

subgroup

G/H

center, hypercenter

C₂

#11757748

commutator, Fitting, cosocle

C₂×C₉×H(3,𝔽₃)

C₂²

Frattini, socle

C₂×C₃²

#1113

prime radical

C₂×C₃×H(3,𝔽₃)

C₂×D₃

hypocenter

C₉×H(3,𝔽₃)

Q₈

LCS

G ▷ C₂×C₉×H(3,𝔽₃) ▷ C₉×H(3,𝔽₃)

UCS

C₁ ◁ C₂

N	H ≅ G/N	type
13. C₉×H(3,𝔽₃)	Q₈	semidirect
39. C₃²×C₉	Dic₆	semidirect
79. H(3,𝔽₃)	Dic₁₈	semidirect
119. C₃²	#6189	semidirect
123. C₉	#6196	semidirect

H ≅ G/N

type

13. C₉×H(3,𝔽₃)

Q₈

semidirect

39. C₃²×C₉

Dic₆

semidirect

79. H(3,𝔽₃)

Dic₁₈

semidirect

119. C₃²

#6189

semidirect

123. C₉

#6196

semidirect

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

subgroup	index
C₉×#1084	2
#11757601	2
#10619900	3
#10619868	3

subgroup

index

Subgroups — #14517107

Conjugacy classes of subgroups

Complements

Maximal subgroups