Group #10714840 — subgroups

Subgroups — #10714840

Order 768.

324 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 103 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×#4683	C₂	1	1	yes	maximal, characteristic
2	C₃×#1945	C₂	1	1	yes	maximal, characteristic
3	C₂×#4494	C₂	1	2	yes	maximal
4	#67726	C₂	1	2	yes	maximal
5	#67658	C₂	1	1	yes	maximal, characteristic
6	#13458	—	3	1	no	maximal, Sylow
7	C₂×#870	—	2	1	no	—
8	C₂×C₃×#108	C₂²	1	1	yes	characteristic
9	C₃×#427	C₂²	1	1	yes	characteristic
10	C₃×#424	—	2	1	no	—
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×#818	—	2	1	no	—
13	C₂×#817	C₂²	1	2	yes	—
14	C₂×#812	C₂²	1	2	yes	—
15	#4683	—	2	4	no	—
16	C₃×#339	—	2	1	no	—
17	#4494	—	2	4	no	—
18	C₂×#499	—	3	1	no	—
19	C₂×#365	—	3	2	no	—
20	#1945	D₃	1	1	yes	characteristic
21	#1386	—	3	2	no	—
22	#1376	—	3	1	no	—
23	C₂×C₃×M₁₆	—	2	1	no	—
24	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
26	C₃×#117	—	2	1	no	—
27	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
28	C₂×C₄⋊Dic₃	—	2	2	no	—
29	C₂×Dic₃⋊C₄	—	4	2	no	—
30	#870	—	2	4	no	—
31	#870	—	4	2	no	—
32	C₃×#108	—	2	2	no	—
33	C₃×#107	Q₈	1	2	yes	—
34	C₃×#106	—	2	1	no	—
35	C₃×C₄⋊C₈	—	4	2	no	—
36	C₃×C₄×C₈	—	2	2	no	—
37	#818	—	2	4	no	—
38	#817	—	2	2	no	—
39	#817	D₄	1	4	yes	—
40	#812	—	2	2	no	—
41	#812	Q₈	1	4	yes	—
42	C₂×C₃⋊C₁₆	—	2	2	no	—
43	C₂×#125	—	6	1	no	—
44	C₂×M₃₂	—	3	2	no	—
45	#499	—	6	4	no	—
46	C₂×#108	C₂×D₃	1	1	yes	characteristic
47	#427	C₂×D₃	1	1	yes	characteristic
48	C₂×#107	—	6	1	no	—
49	C₂×#106	—	3	2	no	—
50	#424	—	2	1	no	—
51	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
52	#365	—	6	4	no	—
53	#339	—	2	1	no	—
54	C₂²×C₃×C₄	—	2	1	no	—
55	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₃	—	4	2	no	—
57	C₃×M₁₆	—	4	1	no	—
58	C₃×M₁₆	—	4	1	no	—
59	C₂×C₃×C₈	—	2	1	no	—
60	C₂×C₃×C₈	—	2	1	no	—
61	C₂×C₃×C₈	—	4	1	no	—
62	C₂×C₃×C₈	SD₁₆	1	2	yes	—
63	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
64	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
66	C₂×C₃×C₈	#41	1	1	yes	characteristic
67	C₃×C₄⋊C₄	—	2	2	no	—
68	C₃×C₂²⋊C₄	—	4	1	no	—
69	C₃×C₄²	—	2	1	no	—
70	C₃×C₄²	—	4	1	no	—
71	C₃×C₄²	SD₁₆	1	2	yes	—
72	C₄⋊Dic₃	—	2	4	no	—
73	C₄⋊Dic₃	—	4	2	no	—
74	Dic₃⋊C₄	—	4	8	no	—
75	C₃⋊C₁₆	—	2	4	no	—
76	C₂×M₁₆	—	2	1	no	—
77	C₂²×C₈	#67	1	1	yes	characteristic
78	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
79	#125	—	6	4	no	—
80	#125	—	12	2	no	—
81	#117	—	2	1	no	—
82	C₂×C₄⋊C₄	—	6	2	no	—
83	C₂×C₄⋊C₄	—	12	2	no	—
84	C₂×C₄²	#67	1	1	yes	characteristic
85	M₃₂	—	3	4	no	—
86	M₃₂	—	6	2	no	—
87	C₂×C₁₆	—	6	2	no	—
88	#108	—	2	2	no	—
89	#107	—	6	4	no	—
90	#107	Dic₆	1	2	yes	—
91	#106	—	2	1	no	—
92	#106	—	3	4	no	—
93	#106	—	6	2	no	—
94	C₄⋊C₈	—	4	2	no	—
95	C₄×C₈	—	2	2	no	—
96	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	—	4	2	no	—
99	C₂×C₃×C₄	—	4	2	no	—
100	C₂×C₃×C₄	#120	1	1	yes	characteristic
101	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
103	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₃×C₄	#137	1	1	yes	characteristic
106	C₂×Dic₃	—	4	4	no	—
107	C₂×Dic₃	—	8	4	no	—
108	C₃×C₈	—	2	2	no	—
109	C₃×C₈	—	8	1	no	—
110	C₃×C₈	C₂²⋊Q₈	1	2	yes	—
111	C₃×C₈	C₄⋊Q₈	1	2	yes	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	12	2	no	—
114	C₂²×C₄	C₂×#67	1	1	yes	characteristic
115	M₁₆	—	4	1	no	—
116	M₁₆	—	4	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	#214	1	2	yes	—
121	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
122	C₂×C₈	#235	1	1	yes	characteristic
123	C₂×C₈	D₃×D₄	1	2	yes	—
124	C₂×C₈	D₃×Q₈	1	2	yes	—
125	C₄⋊C₄	—	2	2	no	—
126	C₄⋊C₄	—	6	4	no	—
127	C₄⋊C₄	—	12	8	no	—
128	C₄⋊C₄	—	12	2	no	—
129	C₂²⋊C₄	—	4	1	no	—
130	C₄²	—	2	1	no	—
131	C₄²	—	4	1	no	—
132	C₄²	#215	1	2	yes	—
133	C₁₆	—	6	4	no	—
134	C₂²×C₃	—	2	2	no	—
135	C₂²×C₃	#450	1	1	yes	characteristic
136	C₂²×C₃	#476	1	1	yes	characteristic
137	C₂²×C₃	#501	1	1	yes	characteristic
138	C₃×C₄	—	4	2	no	—
139	C₃×C₄	—	8	2	no	—
140	C₃×C₄	#395	1	1	yes	characteristic
141	C₃×C₄	#448	1	1	yes	characteristic
142	C₃×C₄	#475	1	1	yes	characteristic
143	C₃×C₄	#498	1	1	yes	characteristic
144	Dic₃	—	8	4	no	—
145	C₂³	#949	1	1	yes	characteristic
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	12	4	no	—
150	C₂×C₄	—	24	4	no	—
151	C₂×C₄	#924	1	1	yes	characteristic
152	C₂×C₄	C₂×#214	1	1	yes	characteristic
153	C₂×C₄	#937	1	1	yes	characteristic
154	C₂×C₄	C₂×#215	1	1	yes	characteristic
155	C₂×C₄	#944	1	1	yes	characteristic
156	C₂×C₄	#951	1	1	yes	characteristic
157	C₈	—	2	2	no	—
158	C₈	—	8	1	no	—
159	C₈	#888	1	2	yes	—
160	C₈	#896	1	2	yes	—
161	C₂×C₃	—	2	1	no	—
162	C₂×C₃	—	2	1	no	—
163	C₂×C₃	#2058	1	1	yes	characteristic
164	C₂×C₃	#2231	1	2	yes	—
165	C₂²	—	2	2	no	—
166	C₂²	#5008	1	1	yes	characteristic
167	C₂²	#5046	1	1	yes	characteristic
168	C₂²	#5064	1	1	yes	characteristic
169	C₄	—	4	2	no	—
170	C₄	—	8	2	no	—
171	C₄	—	24	4	no	—
172	C₄	#4979	1	1	yes	characteristic
173	C₄	#5020	1	1	yes	characteristic
174	C₄	#5040	1	1	yes	characteristic
175	C₄	#5080	1	1	yes	characteristic
176	C₃	#13458	1	1	yes	characteristic, complemented, Sylow
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#69605	1	1	yes	characteristic
180	C₂	#70920	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4683

C₂

yes

maximal, characteristic

C₃×#1945

C₂

yes

maximal, characteristic

C₂×#4494

C₂

yes

maximal

#67726

C₂

yes

maximal

#67658

C₂

yes

maximal, characteristic

#13458

—

maximal, Sylow

C₂×#870

—

C₂×C₃×#108

C₂²

yes

characteristic

C₃×#427

C₂²

yes

characteristic

C₃×#424

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₂×#818

—

C₂×#817

C₂²

yes

—

C₂×#812

C₂²

yes

—

—

—

—

—

—

D₃

yes

characteristic

#1386

—

#1376

—

C₂×C₃×M₁₆

—

C₂²×C₃×C₈

D₄

yes

characteristic

C₂²×C₃×C₈

C₂³

yes

characteristic

C₃×#117

—

C₂×C₃×C₄²

D₄

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×Dic₃⋊C₄

—

—

—

—

Q₈

yes

—

C₃×#106

—

C₃×C₄⋊C₈

—

C₃×C₄×C₈

—

#818

—

#817

—

#817

D₄

yes

—

#812

—

#812

Q₈

yes

—

C₂×C₃⋊C₁₆

—

C₂×#125

—

C₂×M₃₂

—

#499

—

C₂×#108

C₂×D₃

yes

characteristic

#427

C₂×D₃

yes

—

—

—

C₂×C₄×C₈

C₂×D₃

yes

characteristic

#365

—

#339

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×Dic₃

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

SD₁₆

yes

—

C₂×C₃×C₈

C₂×D₄

yes

—

C₂×C₃×C₈

C₂×Q₈

yes

characteristic

C₂×C₃×C₈

C₂×Q₈

yes

—

C₂×C₃×C₈

#41

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

SD₁₆

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₃⋊C₁₆

—

C₂×M₁₆

—

C₂²×C₈

#67

yes

characteristic

C₂²×C₈

C₂²×D₃

yes

—

—

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

#67

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#108

—

#107

—

#107

Dic₆

yes

—

#106

—

#106

—

#106

—

C₄⋊C₈

—

C₄×C₈

—

C₂³×C₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

100

C₂×C₃×C₄

#120

yes

characteristic

101

C₂×C₃×C₄

C₂²⋊Q₈

yes

characteristic

102

C₂×C₃×C₄

C₄⋊Q₈

yes

characteristic

103

C₂×C₃×C₄

C₂×SD₁₆

yes

characteristic

104

C₂×C₃×C₄

C₂×SD₁₆

yes

characteristic

105

C₂×C₃×C₄

#137

yes

characteristic

106

C₂×Dic₃

—

107

C₂×Dic₃

—

108

C₃×C₈

—

109

C₃×C₈

—

110

C₃×C₈

C₂²⋊Q₈

yes

—

111

C₃×C₈

C₄⋊Q₈

yes

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×#67

yes

characteristic

115

M₁₆

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

#214

yes

—

121

C₂×C₈

C₂×Dic₆

yes

characteristic

122

C₂×C₈

#235

yes

characteristic

123

C₂×C₈

D₃×D₄

yes

—

124

C₂×C₈

D₃×Q₈

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₂²⋊C₄

—

130

C₄²

—

131

C₄²

—

132

C₄²

#215

yes

—

133

C₁₆

—

134

C₂²×C₃

—

135

C₂²×C₃

#450

yes

characteristic

136

C₂²×C₃

#476

yes

characteristic

137

C₂²×C₃

#501

yes

characteristic

138

C₃×C₄

—

139

C₃×C₄

—

140

C₃×C₄

#395

yes

characteristic

141

C₃×C₄

#448

yes

characteristic

142

C₃×C₄

#475

yes

characteristic

143

C₃×C₄

#498

yes

characteristic

144

Dic₃

—

145

C₂³

#949

yes

characteristic

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

#924

yes

characteristic

152

C₂×C₄

C₂×#214

yes

characteristic

153

C₂×C₄

#937

yes

characteristic

154

C₂×C₄

C₂×#215

yes

characteristic

155

C₂×C₄

#944

yes

characteristic

156

C₂×C₄

#951

yes

characteristic

157

C₈

—

158

C₈

—

159

C₈

#888

yes

—

160

C₈

#896

yes

—

161

C₂×C₃

—

162

C₂×C₃

—

163

C₂×C₃

#2058

yes

characteristic

164

C₂×C₃

#2231

yes

—

165

C₂²

—

166

C₂²

#5008

yes

characteristic

167

C₂²

#5046

yes

characteristic

168

C₂²

#5064

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

#4979

yes

characteristic

173

C₄

#5020

yes

characteristic

174

C₄

#5040

yes

characteristic

175

C₄

#5080

yes

characteristic

176

C₃

#13458

yes

characteristic, complemented, Sylow

177

C₂

—

178

C₂

—

179

C₂

#69605

yes

characteristic

180

C₂

#70920

yes

—

	subgroup	G/H
center	C₂²	#5008
commutator, cosocle	C₂²×C₃×C₈	C₂³
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#1945	C₂
socle	C₂²×C₃	#476
prime radical	C₂³×C₃	AGL(1,ℤ/8)
hypercenter	#1945	D₃
hypocenter	C₃	#13458
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1945

subgroup

G/H

center

C₂²

#5008

commutator, cosocle

C₂²×C₃×C₈

C₂³

Frattini

C₂²×C₈

C₂²×D₃

C₂

C₂²×C₃

C₂³×C₃

D₃

C₃

G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1945

N	H ≅ G/N	type
176. C₃	#13458	semidirect

H ≅ G/N

type

176. C₃

#13458

semidirect

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

subgroup	index
C₂×#4683	2
C₃×#1945	2
C₂×#4494	2
#67726	2
#67658	2
#13458	3

subgroup

index

Subgroups — #10714840

Conjugacy classes of subgroups

Complements

Maximal subgroups