Group #13144761 — subgroups

Subgroups — #13144761

Order 1728.

715 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 113 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, complemented
2	#11721864	C₂	1	2	yes	maximal, complemented
3	C₂×#88549	C₂	1	2	yes	maximal, complemented
4	C₂×#88529	C₂	1	1	yes	maximal, characteristic
5	C₈×#1084	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#5191	—	3	4	no	maximal
7	C₂×Q₈×H(3,𝔽₃)	C₂²	1	2	yes	—
8	C₂×C₄⋊#272	C₂²	1	2	yes	—
9	Q₁₆×H(3,𝔽₃)	—	2	2	no	—
10	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
11	#88549	—	2	4	no	—
12	#88542	—	2	4	no	—
13	C₄×#1084	C₂²	1	1	yes	characteristic
14	C₂×#6180	C₂²	1	1	yes	characteristic
15	#88529	—	2	2	no	—
16	C₂×C₃²×Q₁₆	D₃	1	4	yes	complemented
17	C₃×#947	—	3	8	no	—
18	C₂×C₃×#215	—	3	8	no	—
19	C₂×C₃×D₂₄	—	3	4	no	—
20	C₃×C₈×Dic₃	—	3	4	no	—
21	C₂²×#272	—	2	2	no	—
22	Q₈×H(3,𝔽₃)	—	2	2	no	—
23	Q₈×H(3,𝔽₃)	—	2	2	no	—
24	C₂×C₄×H(3,𝔽₃)	—	2	2	no	—
25	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
26	C₂×#1084	D₄	1	2	yes	—
27	C₄⋊#272	—	2	2	no	—
28	C₄⋊#272	—	2	2	no	—
29	C₈×H(3,𝔽₃)	D₄	1	2	yes	—
30	#6180	D₄	1	2	yes	—
31	C₃×#495	—	9	1	no	—
32	#5191	—	3	4	no	—
33	C₂×C₃²×Q₈	C₂×D₃	1	8	yes	—
34	C₂×C₃×D₁₂	—	3	8	no	—
35	C₃²×Q₁₆	—	2	8	no	—
36	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
37	C₃×#215	—	6	16	no	—
38	C₃×(C₂×D₃)⋊C₄	—	6	16	no	—
39	C₃×C₄×Dic₃	—	3	4	no	—
40	C₂×C₃×C₃⋊C₈	—	3	4	no	—
41	C₃×D₂₄	—	6	8	no	—
42	C₂²×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
43	C₂×#272	—	4	2	no	—
44	C₂×#272	—	4	4	no	—
45	C₄×H(3,𝔽₃)	—	4	2	no	—
46	C₄×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
47	C₄×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
48	#1084	—	2	2	no	—
49	C₂×C₃×Q₁₆	—	3	4	no	—
50	C₂×C₃×Q₁₆	C₃⋊D₃	1	1	yes	characteristic
51	C₂×C₃×SD₁₆	—	9	2	no	—
52	C₂×C₃×D₈	—	9	1	no	—
53	C₃×#124	—	9	2	no	—
54	#947	—	3	8	no	—
55	C₂×#215	—	3	8	no	—
56	C₂×D₂₄	—	3	4	no	—
57	C₃×C₄×C₈	—	9	1	no	—
58	C₈×Dic₃	—	3	4	no	—
59	C₂²×C₃×D₃	—	6	8	no	—
60	C₃²×Q₈	—	2	8	no	—
61	C₃²×Q₈	—	2	8	no	—
62	C₂×C₃²×C₄	—	2	8	no	—
63	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
64	C₂×C₃×Dic₃	—	3	8	no	—
65	C₃×D₁₂	—	6	8	no	—
66	C₃×D₁₂	—	6	8	no	—
67	C₃²×C₈	#67	1	8	yes	—
68	C₃×C₃⋊C₈	—	3	8	no	—
69	#495	—	9	1	no	Sylow
70	C₂×H(3,𝔽₃)	#127	1	1	yes	characteristic
71	C₂×H(3,𝔽₃)	#135	1	2	yes	—
72	#272	—	8	2	no	—
73	C₂×C₃×Q₈	—	3	8	no	—
74	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	2	yes	—
75	C₂×C₃×D₄	—	9	2	no	—
76	C₂×D₁₂	—	3	8	no	—
77	C₃×Q₁₆	—	2	2	no	—
78	C₃×Q₁₆	—	6	8	no	—
79	C₃×SD₁₆	—	18	4	no	—
80	C₃×D₈	—	18	2	no	—
81	C₂×C₃×C₈	—	3	4	no	—
82	C₂×C₃×C₈	—	9	1	no	—
83	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
84	C₃×C₂²⋊C₄	—	18	4	no	—
85	C₃×C₄²	—	9	1	no	—
86	#215	—	6	16	no	—
87	(C₂×D₃)⋊C₄	—	6	16	no	—
88	C₄×Dic₃	—	3	4	no	—
89	C₂×C₃⋊C₈	—	3	4	no	—
90	D₂₄	—	6	8	no	—
91	C₂²×C₃²	D₃×D₄	1	4	yes	—
92	C₂×C₃×D₃	—	12	16	no	—
93	C₂×C₃×D₃	—	12	8	no	—
94	C₃²×C₄	—	4	8	no	—
95	C₃²×C₄	D₃×D₄	1	4	yes	—
96	C₃²×C₄	C₂×#67	1	4	yes	—
97	C₃×Dic₃	—	6	8	no	—
98	C₂×Q₁₆	#272	1	1	yes	characteristic, complemented
99	C₂×SD₁₆	—	9	2	no	—
100	C₂×D₈	—	9	1	no	—
101	#124	—	9	2	no	—
102	C₄×C₈	—	9	1	no	—
103	H(3,𝔽₃)	#495	1	1	yes	characteristic, complemented, Sylow
104	C₂³×C₃	—	18	2	no	—
105	C₂²×D₃	—	6	8	no	—
106	C₃×Q₈	—	2	2	no	—
107	C₃×Q₈	—	2	2	no	—
108	C₃×Q₈	—	6	8	no	—
109	C₃×Q₈	—	6	8	no	—
110	C₃×D₄	—	18	2	no	—
111	C₃×D₄	—	18	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	3	4	no	—
114	C₂×C₃×C₄	—	6	8	no	—
115	C₂×C₃×C₄	—	9	2	no	—
116	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
117	C₂×Dic₃	—	3	8	no	—
118	D₁₂	—	6	8	no	—
119	D₁₂	—	6	8	no	—
120	C₃×C₈	—	3	8	no	—
121	C₃×C₈	—	9	2	no	—
122	C₃×C₈	#638	1	2	yes	—
123	C₃⋊C₈	—	3	8	no	—
124	C₂×C₃²	#919	1	8	yes	—
125	C₂×C₃²	#940	1	4	yes	—
126	C₃×D₃	—	24	8	no	—
127	C₂×Q₈	C₂×#272	1	2	yes	—
128	C₂×D₄	—	9	2	no	—
129	Q₁₆	—	2	2	no	—
130	SD₁₆	—	18	4	no	—
131	D₈	—	18	2	no	—
132	C₂×C₈	—	9	1	no	—
133	C₂×C₈	C₂×#272	1	1	yes	characteristic
134	C₂²⋊C₄	—	18	4	no	—
135	C₄²	—	9	1	no	—
136	C₂²×C₃	—	3	4	no	—
137	C₂²×C₃	—	36	4	no	—
138	C₂²×C₃	—	36	2	no	—
139	C₂²×C₃	D₄×C₃⋊D₃	1	1	yes	characteristic
140	C₂×D₃	—	12	8	no	—
141	C₂×D₃	—	12	16	no	—
142	C₃×C₄	—	3	4	no	—
143	C₃×C₄	—	3	4	no	—
144	C₃×C₄	—	4	2	no	—
145	C₃×C₄	—	12	8	no	—
146	C₃×C₄	—	18	2	no	—
147	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
148	C₃×C₄	C₂×#638	1	1	yes	characteristic
149	Dic₃	—	6	8	no	—
150	C₃²	#5191	1	4	yes	complemented
151	C₂³	—	18	2	no	—
152	Q₈	—	2	2	no	—
153	Q₈	—	2	2	no	—
154	D₄	—	18	2	no	—
155	D₄	—	18	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	9	2	no	—
158	C₂×C₄	C₂²×#272	1	1	yes	characteristic
159	C₈	—	9	2	no	—
160	C₈	#6235	1	2	yes	—
161	C₂×C₃	—	3	8	no	—
162	C₂×C₃	—	3	4	no	—
163	C₂×C₃	—	72	2	no	—
164	C₂×C₃	#64128	1	2	yes	—
165	C₂×C₃	#64149	1	1	yes	characteristic
166	D₃	—	24	8	no	—
167	C₂²	—	36	4	no	—
168	C₂²	—	36	2	no	—
169	C₂²	D₄×#272	1	1	yes	characteristic
170	C₄	—	4	2	no	—
171	C₄	—	18	2	no	—
172	C₄	D₄×#272	1	1	yes	characteristic
173	C₄	C₂×#6235	1	1	yes	characteristic
174	C₃	—	3	4	no	—
175	C₃	#10592890	1	1	yes	characteristic
176	C₂	—	72	2	no	—
177	C₂	#11721808	1	2	yes	—
178	C₂	#11721850	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₁₆×H(3,𝔽₃)

C₂

yes

maximal, characteristic, complemented

#11721864

C₂

yes

maximal, complemented

C₂×#88549

C₂

yes

maximal, complemented

C₂×#88529

C₂

yes

maximal, characteristic

C₈×#1084

C₂

yes

maximal, characteristic, complemented

C₃×#5191

—

maximal

C₂×Q₈×H(3,𝔽₃)

C₂²

yes

—

C₂×C₄⋊#272

C₂²

yes

—

Q₁₆×H(3,𝔽₃)

—

C₂×C₈×H(3,𝔽₃)

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×#6180

C₂²

yes

characteristic

#88529

—

C₂×C₃²×Q₁₆

D₃

yes

complemented

C₃×#947

—

C₂×C₃×#215

—

C₂×C₃×D₂₄

—

C₃×C₈×Dic₃

—

C₂²×#272

—

Q₈×H(3,𝔽₃)

—

Q₈×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

C₂³

yes

characteristic

C₂×#1084

D₄

yes

—

C₄⋊#272

—

C₄⋊#272

—

C₈×H(3,𝔽₃)

D₄

yes

—

#6180

D₄

yes

—

C₃×#495

—

#5191

—

C₂×C₃²×Q₈

C₂×D₃

yes

—

C₂×C₃×D₁₂

—

C₃²×Q₁₆

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃×#215

—

C₃×(C₂×D₃)⋊C₄

—

C₃×C₄×Dic₃

—

C₂×C₃×C₃⋊C₈

—

C₃×D₂₄

—

C₂²×H(3,𝔽₃)

C₂×D₄

yes

characteristic

C₂×#272

—

C₂×#272

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₂×D₄

yes

characteristic

C₄×H(3,𝔽₃)

C₂×D₄

yes

characteristic

#1084

—

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

C₃⋊D₃

yes

characteristic

C₂×C₃×SD₁₆

—

C₂×C₃×D₈

—

C₃×#124

—

#947

—

C₂×#215

—

C₂×D₂₄

—

C₃×C₄×C₈

—

C₈×Dic₃

—

C₂²×C₃×D₃

—

C₃²×Q₈

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃×Dic₃

—

C₃×D₁₂

—

C₃×D₁₂

—

C₃²×C₈

#67

yes

—

C₃×C₃⋊C₈

—

—

C₂×H(3,𝔽₃)

yes

C₂×H(3,𝔽₃)

yes

—

#272

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₂×C₃⋊D₃

yes

—

C₂×C₃×D₄

—

C₂×D₁₂

—

C₃×Q₁₆

—

C₃×Q₁₆

—

C₃×SD₁₆

—

C₃×D₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

characteristic

C₃×C₂²⋊C₄

—

C₃×C₄²

—

#215

—

(C₂×D₃)⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

D₂₄

—

C₂²×C₃²

D₃×D₄

yes

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃²×C₄

—

C₃²×C₄

D₃×D₄

yes

—

C₃²×C₄

C₂×#67

yes

—

C₃×Dic₃

—

C₂×Q₁₆

#272

yes

characteristic, complemented

C₂×SD₁₆

—

100

C₂×D₈

—

101

#124

—

102

C₄×C₈

—

103

H(3,𝔽₃)

#495

yes

characteristic, complemented, Sylow

104

C₂³×C₃

—

105

C₂²×D₃

—

106

C₃×Q₈

—

107

C₃×Q₈

—

108

C₃×Q₈

—

109

C₃×Q₈

—

110

C₃×D₄

—

111

C₃×D₄

—

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

—

114

C₂×C₃×C₄

—

115

C₂×C₃×C₄

—

116

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

117

C₂×Dic₃

—

118

D₁₂

—

119

D₁₂

—

120

C₃×C₈

—

121

C₃×C₈

—

122

C₃×C₈

#638

yes

—

123

C₃⋊C₈

—

124

C₂×C₃²

#919

yes

—

125

C₂×C₃²

#940

yes

—

126

C₃×D₃

—

127

C₂×Q₈

C₂×#272

yes

—

128

C₂×D₄

—

129

Q₁₆

—

130

SD₁₆

—

131

D₈

—

132

C₂×C₈

—

133

C₂×C₈

C₂×#272

yes

characteristic

134

C₂²⋊C₄

—

135

C₄²

—

136

C₂²×C₃

—

137

C₂²×C₃

—

138

C₂²×C₃

—

139

C₂²×C₃

D₄×C₃⋊D₃

yes

characteristic

140

C₂×D₃

—

141

C₂×D₃

—

142

C₃×C₄

—

143

C₃×C₄

—

144

C₃×C₄

—

145

C₃×C₄

—

146

C₃×C₄

—

147

C₃×C₄

D₄×C₃⋊D₃

yes

characteristic

148

C₃×C₄

C₂×#638

yes

characteristic

149

Dic₃

—

150

C₃²

#5191

yes

complemented

151

C₂³

—

152

Q₈

—

153

Q₈

—

154

D₄

—

155

D₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

C₂²×#272

yes

characteristic

159

C₈

—

160

C₈

#6235

yes

—

161

C₂×C₃

—

162

C₂×C₃

—

163

C₂×C₃

—

164

C₂×C₃

#64128

yes

—

165

C₂×C₃

#64149

yes

characteristic

166

D₃

—

167

C₂²

—

168

C₂²

—

169

C₂²

D₄×#272

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

D₄×#272

yes

characteristic

173

C₄

C₂×#6235

yes

characteristic

174

C₃

—

175

C₃

#10592890

yes

characteristic

176

C₂

—

177

C₂

#11721808

yes

—

178

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	C₂×#88529	C₂
hypercenter	C₂×C₃×Q₁₆	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#495
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

C₂×#88529

C₂

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
1. C₂×Q₁₆×H(3,𝔽₃)	C₂	semidirect
2. #11721864	C₂	semidirect
3. C₂×#88549	C₂	semidirect
5. C₈×#1084	C₂	semidirect
16. C₂×C₃²×Q₁₆	D₃	semidirect
98. C₂×Q₁₆	#272	semidirect
103. H(3,𝔽₃)	#495	semidirect
150. C₃²	#5191	semidirect

H ≅ G/N

type

1. C₂×Q₁₆×H(3,𝔽₃)

C₂

semidirect

2. #11721864

C₂

semidirect

3. C₂×#88549

C₂

semidirect

5. C₈×#1084

C₂

semidirect

16. C₂×C₃²×Q₁₆

D₃

semidirect

98. C₂×Q₁₆

#272

semidirect

103. H(3,𝔽₃)

#495

semidirect

150. C₃²

#5191

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
#11721864	2
C₂×#88549	2
C₂×#88529	2
C₈×#1084	2
C₃×#5191	3

subgroup

index

C₂×Q₁₆×H(3,𝔽₃)

Subgroups — #13144761

Conjugacy classes of subgroups

Complements

Maximal subgroups