Group #10700179 — subgroups

Subgroups — #10700179

Order 768.

320 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 101 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	#70770	C₂	1	2	yes	maximal
2	#70749	C₂	1	2	yes	maximal
3	#69842	C₂	1	1	yes	maximal, characteristic
4	#69838	C₂	1	1	yes	maximal, characteristic
5	C₃×#1275	C₂	1	1	yes	maximal, characteristic
6	#10663	—	3	1	no	maximal, Sylow
7	C₂×C₄×Dic₆	C₂²	1	1	yes	characteristic
8	C₃×#431	C₂²	1	2	yes	—
9	C₂×C₃×#97	C₂²	1	1	yes	characteristic
10	C₂×#819	—	2	2	no	—
11	#4924	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
13	#4550	—	2	2	no	—
14	#4457	—	2	4	no	—
15	#4456	—	2	4	no	—
16	#1985	—	3	1	no	—
17	#1849	—	3	1	no	—
18	#1570	—	3	2	no	—
19	#1487	—	3	2	no	—
20	#1275	D₃	1	1	yes	characteristic
21	C₂²×Dic₆	Q₈	1	1	yes	characteristic
22	C₂²×C₃×C₈	D₄	1	2	yes	—
23	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₃	Q₈	1	1	yes	characteristic
25	C₂×Dic₃⋊C₄	—	2	1	no	—
26	C₂×C₄×Dic₃	—	2	1	no	—
27	C₂×#208	—	2	2	no	—
28	C₂²×C₃⋊C₈	—	2	1	no	—
29	C₄×Dic₆	—	2	2	no	—
30	C₄×Dic₆	—	2	2	no	—
31	C₃×C₄⋊C₈	—	2	2	no	—
32	C₃×C₂²⋊C₈	—	2	2	no	—
33	C₃×#97	—	2	2	no	—
34	C₃×C₄×C₈	D₄	1	4	yes	—
35	#819	—	4	4	no	—
36	C₄⋊(C₃⋊C₈)	—	2	2	no	—
37	C₄⋊(C₃⋊C₈)	—	2	2	no	—
38	C₄⋊(C₃⋊C₈)	—	2	2	no	—
39	C₂×C₄×Q₈	—	3	1	no	—
40	#431	C₂×D₃	1	2	yes	—
41	C₂×#108	—	6	2	no	—
42	#423	—	3	2	no	—
43	C₂×C₄⋊C₈	—	3	1	no	—
44	C₂×#97	C₂×D₃	1	1	yes	characteristic
45	#336	—	6	2	no	—
46	#334	—	6	4	no	—
47	Q₈⋊C₈	—	6	4	no	—
48	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₃×C₄	#41	1	1	yes	characteristic
51	C₂²×Dic₃	—	2	1	no	—
52	C₂²×Dic₃	—	2	1	no	—
53	C₂×Dic₆	—	2	2	no	—
54	C₂×Dic₆	—	4	2	no	—
55	C₂×C₃×C₈	—	2	2	no	—
56	C₂×C₃×C₈	—	2	2	no	—
57	C₂×C₃×C₈	—	2	4	no	—
58	C₂×C₃×C₈	SD₁₆	1	4	yes	—
59	C₃×C₄²	C₂×D₄	1	2	yes	—
60	C₃×C₄²	#41	1	2	yes	—
61	C₄⋊Dic₃	—	2	2	no	—
62	Dic₃⋊C₄	—	4	2	no	—
63	C₄×Dic₃	—	4	1	no	—
64	C₄×Dic₃	—	4	1	no	—
65	#208	—	4	4	no	—
66	C₂×C₃⋊C₈	—	4	1	no	—
67	C₂×C₃⋊C₈	—	4	2	no	—
68	C₂×C₃⋊C₈	—	4	2	no	—
69	C₂²×Q₈	—	3	1	no	—
70	C₂×M₁₆	—	6	2	no	—
71	C₂²×C₈	—	6	1	no	—
72	C₂²×C₈	D₁₂	1	2	yes	—
73	C₄×Q₈	—	6	2	no	—
74	C₄×Q₈	—	6	2	no	—
75	C₂×C₄⋊C₄	—	3	1	no	—
76	C₂×C₄⋊C₄	—	6	1	no	—
77	C₂×C₄²	—	6	1	no	—
78	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
79	#108	—	12	4	no	—
80	C₄⋊C₈	—	2	2	no	—
81	C₄⋊C₈	—	6	2	no	—
82	C₄⋊C₈	—	6	2	no	—
83	C₄⋊C₈	—	6	2	no	—
84	C₂²⋊C₈	—	2	2	no	—
85	#97	—	2	2	no	—
86	C₄×C₈	D₁₂	1	4	yes	—
87	C₂³×C₃	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₃×C₄	—	4	1	no	—
89	C₂×C₃×C₄	—	4	1	no	—
90	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
91	C₂×C₃×C₄	#123	1	2	yes	—
92	C₂×C₃×C₄	#123	1	1	yes	characteristic
93	C₂×C₃×C₄	#127	1	1	yes	characteristic
94	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
95	C₂×C₃×C₄	#137	1	2	yes	—
96	C₂×Dic₃	—	2	2	no	—
97	C₂×Dic₃	—	4	1	no	—
98	C₂×Dic₃	—	4	1	no	—
99	C₂×Dic₃	—	4	2	no	—
100	C₂×Dic₃	—	4	1	no	—
101	Dic₆	—	2	4	no	—
102	Dic₆	—	4	2	no	—
103	C₃×C₈	—	2	4	no	—
104	C₃×C₈	—	4	4	no	—
105	C₃⋊C₈	—	8	2	no	—
106	C₃⋊C₈	—	8	1	no	—
107	C₂×Q₈	—	6	2	no	—
108	C₂×Q₈	—	12	2	no	—
109	C₂²×C₄	—	6	1	no	—
110	C₂²×C₄	—	6	1	no	—
111	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
112	C₂²×C₄	#237	1	1	yes	characteristic
113	C₂²×C₄	D₃×Q₈	1	1	yes	characteristic
114	M₁₆	—	12	4	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	—	12	1	no	—
119	C₂×C₈	—	12	2	no	—
120	C₂×C₈	—	12	2	no	—
121	C₂×C₈	#204	1	4	yes	—
122	C₄⋊C₄	—	6	2	no	—
123	C₄⋊C₄	—	12	2	no	—
124	C₄²	—	12	1	no	—
125	C₄²	—	12	1	no	—
126	C₄²	C₂×D₁₂	1	2	yes	—
127	C₄²	#237	1	2	yes	—
128	C₂²×C₃	—	2	1	no	—
129	C₂²×C₃	—	2	1	no	—
130	C₂²×C₃	#399	1	1	yes	characteristic
131	C₂²×C₃	#475	1	2	yes	—
132	C₃×C₄	—	2	2	no	—
133	C₃×C₄	—	2	2	no	—
134	C₃×C₄	#483	1	2	yes	—
135	C₃×C₄	#492	1	1	yes	characteristic
136	C₃×C₄	#497	1	1	yes	characteristic
137	Dic₃	—	4	2	no	—
138	Dic₃	—	8	1	no	—
139	C₂³	#897	1	1	yes	characteristic
140	Q₈	—	6	4	no	—
141	Q₈	—	12	2	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	6	2	no	—
145	C₂×C₄	—	12	1	no	—
146	C₂×C₄	—	12	1	no	—
147	C₂×C₄	—	12	2	no	—
148	C₂×C₄	—	12	1	no	—
149	C₂×C₄	#874	1	1	yes	characteristic
150	C₂×C₄	#886	1	2	yes	—
151	C₂×C₄	#886	1	1	yes	characteristic
152	C₂×C₄	#897	1	2	yes	—
153	C₂×C₄	C₂×#204	1	2	yes	—
154	C₂×C₄	#909	1	2	yes	—
155	C₈	—	2	4	no	—
156	C₈	—	4	4	no	—
157	C₈	—	24	2	no	—
158	C₈	—	24	1	no	—
159	C₂×C₃	—	2	2	no	—
160	C₂×C₃	#2091	1	1	yes	characteristic
161	C₂×C₃	#2096	1	1	yes	characteristic
162	C₂×C₃	#2098	1	1	yes	characteristic
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	#4677	1	1	yes	characteristic
166	C₂²	#4851	1	2	yes	—
167	C₄	—	2	2	no	—
168	C₄	—	2	2	no	—
169	C₄	—	12	2	no	—
170	C₄	—	24	1	no	—
171	C₄	#4693	1	1	yes	characteristic
172	C₄	#4715	1	1	yes	characteristic
173	C₄	#4721	1	2	yes	—
174	C₃	#10663	1	1	yes	characteristic, complemented, Sylow
175	C₂	—	2	2	no	—
176	C₂	#68692	1	1	yes	characteristic
177	C₂	#68715	1	1	yes	characteristic
178	C₂	#68721	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#70770

C₂

yes

maximal

#70749

C₂

yes

maximal

#69842

C₂

yes

maximal, characteristic

#69838

C₂

yes

maximal, characteristic

C₃×#1275

C₂

yes

maximal, characteristic

#10663

—

maximal, Sylow

C₂×C₄×Dic₆

C₂²

yes

characteristic

C₃×#431

C₂²

yes

—

C₂×C₃×#97

C₂²

yes

characteristic

C₂×#819

—

#4924

C₂²

yes

—

C₂×C₄⋊(C₃⋊C₈)

C₂²

yes

—

—

—

—

—

—

—

D₃

yes

characteristic

C₂²×Dic₆

Q₈

yes

characteristic

C₂²×C₃×C₈

D₄

yes

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×C₄⋊Dic₃

Q₈

yes

characteristic

C₂×Dic₃⋊C₄

—

C₂×C₄×Dic₃

—

C₂×#208

—

C₂²×C₃⋊C₈

—

C₄×Dic₆

—

C₄×Dic₆

—

C₃×C₄⋊C₈

—

C₃×C₂²⋊C₈

—

C₃×#97

—

C₃×C₄×C₈

D₄

yes

—

#819

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₂×C₄×Q₈

—

#431

C₂×D₃

yes

—

C₂×#108

—

#423

—

C₂×C₄⋊C₈

—

C₂×#97

C₂×D₃

yes

characteristic

#336

—

#334

—

Q₈⋊C₈

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×C₃×C₄

C₂×Q₈

yes

characteristic

C₂²×C₃×C₄

#41

yes

characteristic

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

SD₁₆

yes

—

C₃×C₄²

C₂×D₄

yes

—

C₃×C₄²

#41

yes

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₁₂

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₃

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

D₁₂

yes

—

C₂³×C₃

C₂²⋊Q₈

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊Q₈

yes

—

C₂×C₃×C₄

#123

yes

—

C₂×C₃×C₄

yes

C₂×C₃×C₄

yes

C₂×C₃×C₄

C₂×SD₁₆

yes

—

C₂×C₃×C₄

#137

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

100

C₂×Dic₃

—

101

Dic₆

—

102

Dic₆

—

103

C₃×C₈

—

104

C₃×C₈

—

105

C₃⋊C₈

—

106

C₃⋊C₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂×D₁₂

yes

characteristic

112

C₂²×C₄

#237

yes

characteristic

113

C₂²×C₄

D₃×Q₈

yes

characteristic

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

#204

yes

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄²

—

125

C₄²

—

126

C₄²

C₂×D₁₂

yes

—

127

C₄²

#237

yes

—

128

C₂²×C₃

—

129

C₂²×C₃

—

130

C₂²×C₃

#399

yes

characteristic

131

C₂²×C₃

#475

yes

—

132

C₃×C₄

—

133

C₃×C₄

—

134

C₃×C₄

#483

yes

—

135

C₃×C₄

#492

yes

characteristic

136

C₃×C₄

#497

yes

characteristic

137

Dic₃

—

138

Dic₃

—

139

C₂³

#897

yes

characteristic

140

Q₈

—

141

Q₈

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

#874

yes

characteristic

150

C₂×C₄

#886

yes

—

151

C₂×C₄

#886

yes

characteristic

152

C₂×C₄

#897

yes

—

153

C₂×C₄

C₂×#204

yes

—

154

C₂×C₄

#909

yes

—

155

C₈

—

156

C₈

—

157

C₈

—

158

C₈

—

159

C₂×C₃

—

160

C₂×C₃

#2091

yes

characteristic

161

C₂×C₃

#2096

yes

characteristic

162

C₂×C₃

#2098

yes

characteristic

163

C₂²

—

164

C₂²

—

165

C₂²

#4677

yes

characteristic

166

C₂²

#4851

yes

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

#4693

yes

characteristic

172

C₄

#4715

yes

characteristic

173

C₄

#4721

yes

—

174

C₃

#10663

yes

characteristic, complemented, Sylow

175

C₂

—

176

C₂

#68692

yes

characteristic

177

C₂

#68715

yes

characteristic

178

C₂

#68721

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#874
commutator, cosocle	C₂×C₃×C₄²	C₂³
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1275	C₂
socle	C₂²×C₃	#399
prime radical	C₂³×C₃	C₂²⋊Q₈
hypercenter	#1275	D₃
hypocenter	C₃	#10663
LCS	G ▷ C₂×C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

subgroup

G/H

center

C₂×C₄

#874

commutator, cosocle

C₂×C₃×C₄²

C₂³

Frattini

C₂×C₄²

C₂²×D₃

C₂

C₂²×C₃

C₂³×C₃

C₂²⋊Q₈

D₃

C₃

G ▷ C₂×C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

N	H ≅ G/N	type
174. C₃	#10663	semidirect

H ≅ G/N

type

174. C₃

#10663

semidirect

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

subgroup	index
#70770	2
#70749	2
#69842	2
#69838	2
C₃×#1275	2
#10663	3

subgroup

index

Subgroups — #10700179

Conjugacy classes of subgroups

Complements

Maximal subgroups