Group #13144757 — subgroups

Subgroups — #13144757

Order 1728.

671 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 671 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂×Q₁₆×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#11721861	C₂	1	2	yes	maximal
3	C₂×#88550	C₂	1	2	yes	maximal
4	C₂×#88530	C₂	1	1	yes	maximal, characteristic
5	C₈×#1084	C₂	1	1	yes	maximal, characteristic
6	C₃×#5183	—	3	4	no	maximal
7	C₂×Q₈×H(3,𝔽₃)	C₂²	1	2	yes	—
8	C₂×#6229	C₂²	1	2	yes	—
9	Q₁₆×H(3,𝔽₃)	—	2	2	no	—
10	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
11	#88550	—	2	4	no	—
12	#88540	—	2	4	no	—
13	C₄×#1084	C₂²	1	1	yes	characteristic
14	C₂×#6180	C₂²	1	1	yes	characteristic
15	#88530	—	2	2	no	—
16	C₂×C₃²×Q₁₆	D₃	1	4	yes	—
17	C₃×#944	—	3	8	no	—
18	C₂×C₃×#216	—	3	8	no	—
19	C₂×C₃×Dic₁₂	—	3	4	no	—
20	C₃×C₈×Dic₃	—	3	4	no	—
21	Q₈×H(3,𝔽₃)	—	2	2	no	—
22	Q₈×H(3,𝔽₃)	—	2	2	no	—
23	C₂×C₄×H(3,𝔽₃)	—	2	2	no	—
24	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
25	C₂×#1084	—	2	2	no	—
26	C₂×#1084	D₄	1	2	yes	—
27	#6229	—	2	2	no	—
28	#6229	—	2	2	no	—
29	C₈×H(3,𝔽₃)	D₄	1	2	yes	—
30	#6180	D₄	1	2	yes	—
31	C₃×#494	—	9	1	no	—
32	#5183	—	3	4	no	—
33	C₂×C₃²×Q₈	C₂×D₃	1	8	yes	—
34	C₂×C₃×Dic₆	—	3	8	no	—
35	C₃²×Q₁₆	—	2	8	no	—
36	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
37	C₃×#216	—	6	16	no	—
38	C₃×Dic₃⋊C₄	—	6	16	no	—
39	C₃×C₄×Dic₃	—	3	4	no	—
40	C₂×C₃×C₃⋊C₈	—	3	4	no	—
41	C₃×Dic₁₂	—	6	8	no	—
42	C₂²×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
43	C₄×H(3,𝔽₃)	—	4	2	no	—
44	C₄×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
45	C₄×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
46	#1084	—	4	2	no	—
47	#1084	Q₁₆	1	4	yes	complemented
48	C₂×C₃×Q₁₆	—	3	4	no	—
49	C₂×C₃×Q₁₆	—	9	2	no	—
50	C₂×C₃×Q₁₆	—	9	1	no	—
51	C₂×C₃×Q₁₆	C₃⋊D₃	1	1	yes	characteristic
52	C₃×C₄⋊Q₈	—	9	2	no	—
53	#944	—	3	8	no	—
54	C₂×#216	—	3	8	no	—
55	C₂×Dic₁₂	—	3	4	no	—
56	C₃×C₄×C₈	—	9	1	no	—
57	C₈×Dic₃	—	3	4	no	—
58	C₃²×Q₈	—	2	8	no	—
59	C₃²×Q₈	—	2	8	no	—
60	C₂×C₃²×C₄	—	2	8	no	—
61	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
62	C₂×C₃×Dic₃	—	3	8	no	—
63	C₂×C₃×Dic₃	—	6	8	no	—
64	C₃×Dic₆	—	6	8	no	—
65	C₃×Dic₆	—	6	8	no	—
66	C₃²×C₈	#67	1	8	yes	—
67	C₃×C₃⋊C₈	—	3	8	no	—
68	#494	—	9	1	no	Sylow
69	C₂×H(3,𝔽₃)	#127	1	1	yes	characteristic
70	C₂×H(3,𝔽₃)	C₂×Q₁₆	1	2	yes	—
71	C₂×C₃×Q₈	—	3	8	no	—
72	C₂×C₃×Q₈	—	9	2	no	—
73	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	2	yes	—
74	C₂×Dic₆	—	3	8	no	—
75	C₃×Q₁₆	—	2	2	no	—
76	C₃×Q₁₆	—	6	8	no	—
77	C₃×Q₁₆	—	18	4	no	—
78	C₃×Q₁₆	—	18	2	no	—
79	C₂×C₃×C₈	—	3	4	no	—
80	C₂×C₃×C₈	—	9	1	no	—
81	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
82	C₃×C₄⋊C₄	—	18	4	no	—
83	C₃×C₄²	—	9	1	no	—
84	#216	—	6	16	no	—
85	Dic₃⋊C₄	—	6	16	no	—
86	C₄×Dic₃	—	3	4	no	—
87	C₂×C₃⋊C₈	—	3	4	no	—
88	Dic₁₂	—	6	8	no	—
89	C₂²×C₃²	D₃×D₄	1	4	yes	—
90	C₃²×C₄	—	4	8	no	—
91	C₃²×C₄	D₃×D₄	1	4	yes	—
92	C₃²×C₄	C₂×#67	1	4	yes	—
93	C₃×Dic₃	—	3	16	no	—
94	C₃×Dic₃	—	12	8	no	—
95	C₂×Q₁₆	—	9	2	no	—
96	C₂×Q₁₆	—	9	1	no	—
97	C₂×Q₁₆	#272	1	1	yes	characteristic
98	C₄⋊Q₈	—	9	2	no	—
99	C₄×C₈	—	9	1	no	—
100	H(3,𝔽₃)	#494	1	1	yes	characteristic, complemented, Sylow
101	C₃×Q₈	—	2	2	no	—
102	C₃×Q₈	—	2	2	no	—
103	C₃×Q₈	—	6	8	no	—
104	C₃×Q₈	—	6	8	no	—
105	C₃×Q₈	—	18	2	no	—
106	C₃×Q₈	—	18	2	no	—
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	—	3	4	no	—
109	C₂×C₃×C₄	—	6	8	no	—
110	C₂×C₃×C₄	—	9	2	no	—
111	C₂×C₃×C₄	—	18	2	no	—
112	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
113	C₂×Dic₃	—	3	8	no	—
114	C₂×Dic₃	—	6	8	no	—
115	Dic₆	—	6	8	no	—
116	Dic₆	—	6	8	no	—
117	C₃×C₈	—	3	8	no	—
118	C₃×C₈	—	9	2	no	—
119	C₃×C₈	#638	1	2	yes	—
120	C₃⋊C₈	—	3	8	no	—
121	C₂×C₃²	D₃×Q₁₆	1	8	yes	—
122	C₂×C₃²	#940	1	4	yes	—
123	C₂×Q₈	—	9	2	no	—
124	C₂×Q₈	C₂×#272	1	2	yes	—
125	Q₁₆	—	2	2	no	—
126	Q₁₆	—	18	4	no	—
127	Q₁₆	—	18	2	no	—
128	C₂×C₈	—	9	1	no	—
129	C₂×C₈	C₂×#272	1	1	yes	characteristic
130	C₄⋊C₄	—	18	4	no	—
131	C₄²	—	9	1	no	—
132	C₂²×C₃	—	3	4	no	—
133	C₂²×C₃	D₄×C₃⋊D₃	1	1	yes	characteristic
134	C₃×C₄	—	3	4	no	—
135	C₃×C₄	—	3	4	no	—
136	C₃×C₄	—	4	2	no	—
137	C₃×C₄	—	9	4	no	—
138	C₃×C₄	—	12	8	no	—
139	C₃×C₄	—	36	2	no	—
140	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
141	C₃×C₄	C₂×#638	1	1	yes	characteristic
142	Dic₃	—	3	16	no	—
143	Dic₃	—	12	8	no	—
144	C₃²	#5183	1	4	yes	complemented
145	Q₈	—	2	2	no	—
146	Q₈	—	2	2	no	—
147	Q₈	—	18	2	no	—
148	Q₈	—	18	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	9	2	no	—
151	C₂×C₄	—	18	2	no	—
152	C₂×C₄	C₂²×#272	1	1	yes	characteristic
153	C₈	—	9	2	no	—
154	C₈	#6235	1	2	yes	—
155	C₂×C₃	—	3	8	no	—
156	C₂×C₃	—	3	4	no	—
157	C₂×C₃	Q₁₆×C₃⋊D₃	1	2	yes	—
158	C₂×C₃	#64149	1	1	yes	characteristic
159	C₂²	D₄×#272	1	1	yes	characteristic
160	C₄	—	4	2	no	—
161	C₄	—	9	4	no	—
162	C₄	—	36	2	no	—
163	C₄	D₄×#272	1	1	yes	characteristic
164	C₄	C₂×#6235	1	1	yes	characteristic
165	C₃	—	3	4	no	—
166	C₃	#10592882	1	1	yes	characteristic
167	C₂	Q₁₆×#272	1	2	yes	—
168	C₂	#11721850	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₁₆×H(3,𝔽₃)

C₂

yes

maximal, characteristic

#11721861

C₂

yes

maximal

C₂×#88550

C₂

yes

maximal

C₂×#88530

C₂

yes

maximal, characteristic

C₈×#1084

C₂

yes

maximal, characteristic

C₃×#5183

—

maximal

C₂×Q₈×H(3,𝔽₃)

C₂²

yes

—

C₂×#6229

C₂²

yes

—

Q₁₆×H(3,𝔽₃)

—

C₂×C₈×H(3,𝔽₃)

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×#6180

C₂²

yes

characteristic

#88530

—

C₂×C₃²×Q₁₆

D₃

yes

—

C₃×#944

—

C₂×C₃×#216

—

C₂×C₃×Dic₁₂

—

C₃×C₈×Dic₃

—

Q₈×H(3,𝔽₃)

—

Q₈×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

C₂³

yes

characteristic

C₂×#1084

—

C₂×#1084

D₄

yes

—

#6229

—

#6229

—

C₈×H(3,𝔽₃)

D₄

yes

—

#6180

D₄

yes

—

C₃×#494

—

#5183

—

C₂×C₃²×Q₈

C₂×D₃

yes

—

C₂×C₃×Dic₆

—

C₃²×Q₁₆

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃×#216

—

C₃×Dic₃⋊C₄

—

C₃×C₄×Dic₃

—

C₂×C₃×C₃⋊C₈

—

C₃×Dic₁₂

—

C₂²×H(3,𝔽₃)

C₂×D₄

yes

characteristic

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₂×D₄

yes

characteristic

C₄×H(3,𝔽₃)

C₂×D₄

yes

characteristic

#1084

—

#1084

Q₁₆

yes

complemented

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

C₃⋊D₃

yes

characteristic

C₃×C₄⋊Q₈

—

#944

—

C₂×#216

—

C₂×Dic₁₂

—

C₃×C₄×C₈

—

C₈×Dic₃

—

C₃²×Q₈

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃×Dic₆

—

C₃²×C₈

#67

yes

—

C₃×C₃⋊C₈

—

—

C₂×H(3,𝔽₃)

yes

C₂×H(3,𝔽₃)

C₂×Q₁₆

yes

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₂×C₃⋊D₃

yes

—

C₂×Dic₆

—

C₃×Q₁₆

—

C₃×Q₁₆

—

C₃×Q₁₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄²

—

#216

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

Dic₁₂

—

C₂²×C₃²

D₃×D₄

yes

—

C₃²×C₄

—

C₃²×C₄

D₃×D₄

yes

—

C₃²×C₄

C₂×#67

yes

—

C₃×Dic₃

—

C₃×Dic₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

#272

yes

characteristic

C₄⋊Q₈

—

C₄×C₈

—

100

H(3,𝔽₃)

#494

yes

characteristic, complemented, Sylow

101

C₃×Q₈

—

102

C₃×Q₈

—

103

C₃×Q₈

—

104

C₃×Q₈

—

105

C₃×Q₈

—

106

C₃×Q₈

—

107

C₂×C₃×C₄

—

108

C₂×C₃×C₄

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

—

112

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

113

C₂×Dic₃

—

114

C₂×Dic₃

—

115

Dic₆

—

116

Dic₆

—

117

C₃×C₈

—

118

C₃×C₈

—

119

C₃×C₈

#638

yes

—

120

C₃⋊C₈

—

121

C₂×C₃²

D₃×Q₁₆

yes

—

122

C₂×C₃²

#940

yes

—

123

C₂×Q₈

—

124

C₂×Q₈

C₂×#272

yes

—

125

Q₁₆

—

126

Q₁₆

—

127

Q₁₆

—

128

C₂×C₈

—

129

C₂×C₈

C₂×#272

yes

characteristic

130

C₄⋊C₄

—

131

C₄²

—

132

C₂²×C₃

—

133

C₂²×C₃

D₄×C₃⋊D₃

yes

characteristic

134

C₃×C₄

—

135

C₃×C₄

—

136

C₃×C₄

—

137

C₃×C₄

—

138

C₃×C₄

—

139

C₃×C₄

—

140

C₃×C₄

D₄×C₃⋊D₃

yes

characteristic

141

C₃×C₄

C₂×#638

yes

characteristic

142

Dic₃

—

143

Dic₃

—

144

C₃²

#5183

yes

complemented

145

Q₈

—

146

Q₈

—

147

Q₈

—

148

Q₈

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

C₂²×#272

yes

characteristic

153

C₈

—

154

C₈

#6235

yes

—

155

C₂×C₃

—

156

C₂×C₃

—

157

C₂×C₃

Q₁₆×C₃⋊D₃

yes

—

158

C₂×C₃

#64149

yes

characteristic

159

C₂²

D₄×#272

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

D₄×#272

yes

characteristic

164

C₄

C₂×#6235

yes

characteristic

165

C₃

—

166

C₃

#10592882

yes

characteristic

167

C₂

Q₁₆×#272

yes

—

168

C₂

#11721850

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₃	D₄×C₃⋊D₃
commutator, cosocle	C₂×C₄×H(3,𝔽₃)	C₂³
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₂×Q₁₆×H(3,𝔽₃)	C₂
prime radical	C₂²×H(3,𝔽₃)	C₂×D₄
hypercenter	C₂×C₃×Q₁₆	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#494
LCS	G ▷ C₂×C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×Q₁₆

subgroup

G/H

center, socle

C₂²×C₃

D₄×C₃⋊D₃

commutator, cosocle

C₂×C₄×H(3,𝔽₃)

C₂³

Frattini

C₂×C₃×C₄

C₂²×C₃⋊D₃

Fitting

C₂×Q₁₆×H(3,𝔽₃)

C₂

prime radical

C₂²×H(3,𝔽₃)

C₂×D₄

hypercenter

C₂×C₃×Q₁₆

C₃⋊D₃

G ▷ C₂×C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×Q₁₆

N	H ≅ G/N	type
47. #1084	Q₁₆	semidirect
100. H(3,𝔽₃)	#494	semidirect
144. C₃²	#5183	semidirect

H ≅ G/N

type

47. #1084

Q₁₆

semidirect

100. H(3,𝔽₃)

#494

semidirect

144. C₃²

#5183

semidirect

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

subgroup	index
C₂×Q₁₆×H(3,𝔽₃)	2
#11721861	2
C₂×#88550	2
C₂×#88530	2
C₈×#1084	2
C₃×#5183	3

subgroup

index

C₂×Q₁₆×H(3,𝔽₃)

Subgroups — #13144757

Conjugacy classes of subgroups

Complements

Maximal subgroups