Group #10694684 — subgroups

Subgroups — #10694684

Order 768.

390 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 127 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×C₃×#500	C₂	1	1	yes	maximal, characteristic
2	#71775	C₂	1	2	yes	maximal
3	#71774	C₂	1	2	yes	maximal
4	#69791	C₂	1	1	yes	maximal, characteristic
5	#69726	C₂	1	1	yes	maximal, characteristic
6	#9778	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₃×#500	—	2	2	no	—
9	C₃×#500	C₄	1	4	yes	—
10	C₂×C₃×#107	—	2	2	no	—
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	#4925	C₂²	1	1	yes	characteristic
13	#4925	C₂²	1	1	yes	characteristic
14	#4924	C₂²	1	2	yes	—
15	#4552	—	2	2	no	—
16	#4547	—	2	4	no	—
17	#4546	—	2	4	no	—
18	#4454	—	2	2	no	—
19	C₂×#500	D₃	1	1	yes	characteristic
20	#1837	—	3	1	no	—
21	#1549	—	3	2	no	—
22	#1548	—	3	2	no	—
23	#1452	—	3	1	no	—
24	C₂²×C₃×Q₈	—	2	2	no	—
25	C₂²×C₃×C₈	D₄	1	2	yes	—
26	C₃×C₄⋊Q₈	—	2	2	no	—
27	C₃×C₄⋊Q₈	—	2	2	no	—
28	C₃×C₄⋊Q₈	—	2	2	no	—
29	C₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
30	C₂×C₃×C₄⋊C₄	—	2	2	no	—
31	C₂×C₃×C₄⋊C₄	—	2	2	no	—
32	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
33	C₂×#208	—	2	2	no	—
34	C₃×#107	—	2	8	no	—
35	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
36	C₃×C₄×C₈	D₄	1	2	yes	—
37	#830	—	2	1	no	—
38	#830	—	2	1	no	—
39	C₄⋊(C₃⋊C₈)	—	2	2	no	—
40	C₄⋊(C₃⋊C₈)	—	2	2	no	—
41	C₄⋊(C₃⋊C₈)	D₄	1	2	yes	—
42	C₄⋊(C₃⋊C₈)	D₄	1	2	yes	—
43	C₄×C₃⋊C₈	—	2	1	no	—
44	C₄×C₃⋊C₈	—	2	1	no	—
45	C₂×C₄⋊Q₈	C₂×D₃	1	2	yes	—
46	#500	—	2	2	no	—
47	#500	Dic₃	1	4	yes	—
48	#431	—	3	1	no	—
49	#431	—	3	1	no	—
50	C₂×#107	—	2	2	no	—
51	#423	—	3	2	no	—
52	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
53	#341	—	6	2	no	—
54	#333	—	6	4	no	—
55	#332	—	6	4	no	—
56	#322	—	6	2	no	—
57	C₂×C₃×Q₈	—	2	4	no	—
58	C₂×C₃×Q₈	—	4	2	no	—
59	C₂×C₃×Q₈	—	8	4	no	—
60	C₂²×C₃×C₄	—	4	2	no	—
61	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₃×C₄	#41	1	1	yes	characteristic
63	C₂²×C₃×C₄	#41	1	1	yes	characteristic
64	C₂×C₃×C₈	—	2	4	no	—
65	C₂×C₃×C₈	C₂²⋊C₄	1	4	yes	—
66	C₃×C₄⋊C₄	—	2	4	no	—
67	C₃×C₄⋊C₄	—	2	4	no	—
68	C₃×C₄⋊C₄	—	4	2	no	—
69	C₃×C₄⋊C₄	—	4	2	no	—
70	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
71	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
72	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
73	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
74	#208	—	4	4	no	—
75	C₂×C₃⋊C₈	—	2	2	no	—
76	C₂×C₃⋊C₈	—	2	2	no	—
77	C₂×C₃⋊C₈	—	4	2	no	—
78	C₂²×Q₈	—	2	2	no	—
79	C₂×M₁₆	—	6	2	no	—
80	C₂²×C₈	#67	1	2	yes	—
81	C₄⋊Q₈	—	2	2	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	—	2	2	no	—
84	C₄⋊Q₈	C₂×Dic₃	1	4	yes	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄⋊C₄	—	2	2	no	—
87	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
88	#107	—	2	8	no	—
89	C₄⋊C₈	—	3	2	no	—
90	C₄⋊C₈	—	3	2	no	—
91	C₄⋊C₈	—	6	2	no	—
92	C₄⋊C₈	—	6	2	no	—
93	C₂²⋊C₈	—	6	1	no	—
94	C₂²⋊C₈	—	6	1	no	—
95	C₄×C₈	—	6	1	no	—
96	C₄×C₈	—	6	1	no	—
97	C₄×C₈	C₂×Dic₃	1	2	yes	—
98	C₄×C₈	#67	1	2	yes	—
99	C₂³×C₃	#124	1	1	yes	characteristic
100	C₃×Q₈	—	8	4	no	—
101	C₃×Q₈	—	8	4	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	4	4	no	—
105	C₂×C₃×C₄	—	8	4	no	—
106	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
108	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
109	C₂×C₃×C₄	#121	1	1	yes	characteristic
110	C₂×C₃×C₄	#121	1	1	yes	characteristic
111	C₂×C₃×C₄	#127	1	1	yes	characteristic
112	C₂×C₃×C₄	#137	1	4	yes	—
113	C₃×C₈	—	2	4	no	—
114	C₃⋊C₈	—	4	2	no	—
115	C₃⋊C₈	—	4	2	no	—
116	C₃⋊C₈	—	8	2	no	—
117	C₂×Q₈	—	2	4	no	—
118	C₂×Q₈	—	4	2	no	—
119	C₂×Q₈	—	8	4	no	—
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	#237	1	1	yes	characteristic
122	C₂²×C₄	#237	1	1	yes	characteristic
123	C₂²×C₄	C₂×#67	1	1	yes	characteristic
124	M₁₆	—	12	4	no	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	6	2	no	—
127	C₂×C₈	—	6	2	no	—
128	C₂×C₈	—	12	2	no	—
129	C₂×C₈	#217	1	4	yes	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄²	D₃×D₄	1	1	yes	characteristic
135	C₄²	D₃×D₄	1	1	yes	characteristic
136	C₄²	C₂²×Dic₃	1	1	yes	characteristic
137	C₄²	C₂×#67	1	1	yes	characteristic
138	C₂²×C₃	—	2	1	no	—
139	C₂²×C₃	—	2	1	no	—
140	C₂²×C₃	#390	1	1	yes	characteristic
141	C₂²×C₃	#491	1	2	yes	—
142	C₃×C₄	—	2	1	no	—
143	C₃×C₄	—	2	1	no	—
144	C₃×C₄	—	8	4	no	—
145	C₃×C₄	#419	1	2	yes	—
146	C₃×C₄	#472	1	2	yes	—
147	C₃×C₄	#473	1	2	yes	—
148	C₃×C₄	#497	1	2	yes	—
149	C₂³	#936	1	1	yes	characteristic
150	Q₈	—	8	4	no	—
151	Q₈	—	8	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	4	no	—
156	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
157	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
158	C₂×C₄	#938	1	1	yes	characteristic
159	C₂×C₄	#938	1	1	yes	characteristic
160	C₂×C₄	#940	1	1	yes	characteristic
161	C₂×C₄	#942	1	4	yes	—
162	C₂×C₄	C₂×#217	1	1	yes	characteristic
163	C₈	—	2	4	no	—
164	C₈	—	12	2	no	—
165	C₈	—	12	2	no	—
166	C₈	—	24	2	no	—
167	C₂×C₃	—	2	2	no	—
168	C₂×C₃	#1960	1	1	yes	characteristic
169	C₂×C₃	#1978	1	1	yes	characteristic
170	C₂×C₃	#1979	1	1	yes	characteristic
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	#5082	1	2	yes	—
174	C₂²	#5221	1	1	yes	characteristic
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	8	4	no	—
178	C₄	#5086	1	2	yes	—
179	C₄	#5160	1	2	yes	—
180	C₄	#5192	1	2	yes	—
181	C₄	#5225	1	2	yes	—
182	C₃	#9778	1	1	yes	characteristic, complemented, Sylow
183	C₂	—	2	2	no	—
184	C₂	#71786	1	1	yes	characteristic
185	C₂	#72041	1	1	yes	characteristic
186	C₂	#72069	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#500

C₂

yes

maximal, characteristic

#71775

C₂

yes

maximal

#71774

C₂

yes

maximal

#69791

C₂

yes

maximal, characteristic

#69726

C₂

yes

maximal, characteristic

#9778

—

maximal, Sylow

C₂×C₃×C₄⋊Q₈

C₂²

yes

—

C₃×#500

—

C₃×#500

C₄

yes

—

C₂×C₃×#107

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

#4925

C₂²

yes

characteristic

#4925

C₂²

yes

characteristic

#4924

C₂²

yes

—

—

—

—

—

D₃

yes

—

—

—

—

C₂²×C₃×Q₈

—

C₂²×C₃×C₈

D₄

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×#208

—

C₃×#107

—

C₃×C₄×C₈

C₂×C₄

yes

—

C₃×C₄×C₈

D₄

yes

—

#830

—

#830

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

D₄

yes

—

C₄⋊(C₃⋊C₈)

D₄

yes

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₂×C₄⋊Q₈

C₂×D₃

yes

—

#500

—

#500

Dic₃

yes

—

—

—

—

—

C₂×C₄×C₈

C₂×D₃

yes

—

—

—

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×D₄

yes

C₂²×C₃×C₄

yes

C₂²×C₃×C₄

yes

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

characteristic

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

#67

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×Dic₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₃

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Dic₃

yes

—

C₄×C₈

#67

yes

—

C₂³×C₃

#124

yes

characteristic

100

C₃×Q₈

—

101

C₃×Q₈

—

102

C₂×C₃×C₄

—

103

C₂×C₃×C₄

—

104

C₂×C₃×C₄

—

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

107

C₂×C₃×C₄

C₄×D₄

yes

characteristic

108

C₂×C₃×C₄

C₄×D₄

yes

characteristic

109

C₂×C₃×C₄

#121

yes

characteristic

110

C₂×C₃×C₄

#121

yes

characteristic

111

C₂×C₃×C₄

#127

yes

characteristic

112

C₂×C₃×C₄

#137

yes

—

113

C₃×C₈

—

114

C₃⋊C₈

—

115

C₃⋊C₈

—

116

C₃⋊C₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂²×C₄

—

121

C₂²×C₄

#237

yes

characteristic

122

C₂²×C₄

#237

yes

characteristic

123

C₂²×C₄

C₂×#67

yes

characteristic

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

#217

yes

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄²

D₃×D₄

yes

characteristic

135

C₄²

D₃×D₄

yes

characteristic

136

C₄²

C₂²×Dic₃

yes

characteristic

137

C₄²

C₂×#67

yes

characteristic

138

C₂²×C₃

—

139

C₂²×C₃

—

140

C₂²×C₃

#390

yes

characteristic

141

C₂²×C₃

#491

yes

—

142

C₃×C₄

—

143

C₃×C₄

—

144

C₃×C₄

—

145

C₃×C₄

#419

yes

—

146

C₃×C₄

#472

yes

—

147

C₃×C₄

#473

yes

—

148

C₃×C₄

#497

yes

—

149

C₂³

#936

yes

characteristic

150

Q₈

—

151

Q₈

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

D₄×Dic₃

yes

characteristic

157

C₂×C₄

D₄×Dic₃

yes

characteristic

158

C₂×C₄

#938

yes

characteristic

159

C₂×C₄

#938

yes

characteristic

160

C₂×C₄

#940

yes

characteristic

161

C₂×C₄

#942

yes

—

162

C₂×C₄

C₂×#217

yes

characteristic

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

—

167

C₂×C₃

—

168

C₂×C₃

#1960

yes

characteristic

169

C₂×C₃

#1978

yes

characteristic

170

C₂×C₃

#1979

yes

characteristic

171

C₂²

—

172

C₂²

—

173

C₂²

#5082

yes

—

174

C₂²

#5221

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

#5086

yes

—

179

C₄

#5160

yes

—

180

C₄

#5192

yes

—

181

C₄

#5225

yes

—

182

C₃

#9778

yes

characteristic, complemented, Sylow

183

C₂

—

184

C₂

#71786

yes

characteristic

185

C₂

#72041

yes

characteristic

186

C₂

#72069

yes

characteristic

	subgroup	G/H
center	C₂²	#5221
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#500	C₂
socle	C₂²×C₃	#390
prime radical	C₂³×C₃	#124
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#500	D₃
hypocenter	C₃	#9778
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂²

#5221

commutator

C₃×C₄²

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

Fitting

C₂×C₃×#500

C₂

C₂²×C₃

C₂³×C₃

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

N	H ≅ G/N	type
182. C₃	#9778	semidirect

H ≅ G/N

type

182. C₃

#9778

semidirect

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

subgroup	index
C₂×C₃×#500	2
#71775	2
#71774	2
#69791	2
#69726	2
#9778	3

subgroup

index

C₂×C₃×#500

Subgroups — #10694684

Conjugacy classes of subgroups

Complements

Maximal subgroups