Group #34206317 — subgroups

Subgroups — #34206317

Order 20224.

286 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 115 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	#23765184	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	#23764146	C₂	1	2	yes	maximal
4	#23761497	C₂	1	2	yes	maximal
5	C₂×#368×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	#19098601	C₂²	1	2	yes	—
8	#19098591	—	2	2	no	—
9	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
10	#19098391	C₄	1	2	yes	complemented
11	#19098389	C₄	1	2	yes	complemented
12	C₁₆×Dic₇₉	—	2	2	no	—
13	#19098324	—	2	2	no	—
14	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×#108×C₇₉	C₂²	1	2	yes	—
16	#368×C₇₉	—	2	2	no	—
17	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
18	#14579120	—	2	2	no	—
19	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
20	C₂×C₇₉⋊C₁₆	—	2	1	no	—
21	C₂×C₇₉⋊C₁₆	—	2	1	no	—
22	C₂×C₇₉⋊C₁₆	Q₈	1	2	yes	—
23	#14579095	C₂×C₄	1	4	yes	—
24	#14579093	—	2	2	no	—
25	Dic₇₉⋊C₈	—	4	4	no	—
26	C₈×Dic₇₉	—	2	1	no	—
27	C₈×Dic₇₉	—	2	1	no	—
28	C₂×M₁₆×C₇₉	—	2	2	no	—
29	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
33	C₂×C₁₆×C₇₉	Q₈	1	2	yes	—
34	#108×C₇₉	—	2	2	no	—
35	#108×C₇₉	—	2	2	no	—
36	C₂²×Dic₇₉	—	2	1	no	—
37	C₂×C₇₉⋊C₈	—	2	1	no	—
38	C₂×C₇₉⋊C₈	—	2	1	no	—
39	C₂×C₇₉⋊C₈	D₈	1	2	yes	—
40	C₄⋊Dic₇₉	—	2	4	no	—
41	C₂²⋊Dic₇₉	—	4	2	no	—
42	C₄×Dic₇₉	—	2	1	no	—
43	C₄×Dic₇₉	—	4	2	no	—
44	C₄×Dic₇₉	Q₁₆	1	2	yes	—
45	C₇₉⋊C₁₆	—	2	2	no	—
46	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
47	M₁₆×C₇₉	—	4	2	no	—
48	M₁₆×C₇₉	—	4	2	no	—
49	C₂×C₈×C₇₉	—	4	2	no	—
50	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
55	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
56	C₁₆×C₇₉	C₄⋊C₄	1	2	yes	—
57	C₁₆×C₇₉	C₄⋊C₄	1	2	yes	—
58	C₂³×C₇₉	#135	1	1	yes	characteristic
59	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
65	C₂×Dic₇₉	—	2	4	no	—
66	C₂×Dic₇₉	—	2	2	no	—
67	C₂×Dic₇₉	—	4	4	no	—
68	C₈×C₇₉	—	8	2	no	—
69	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₈×C₇₉	C₄×Q₈	1	1	yes	characteristic
71	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
72	C₈×C₇₉	#125	1	1	yes	characteristic
73	C₇₉⋊C₈	—	2	2	no	—
74	C₂²×C₇₉	—	2	2	no	—
75	C₂²×C₇₉	C₄×D₈	1	1	yes	characteristic
76	C₂²×C₇₉	#467	1	1	yes	characteristic
77	C₂²×C₇₉	#499	1	1	yes	characteristic
78	C₄×C₇₉	#389	1	1	yes	characteristic
79	C₄×C₇₉	C₄×Q₁₆	1	1	yes	characteristic
80	C₄×C₇₉	#466	1	1	yes	characteristic
81	C₄×C₇₉	#500	1	1	yes	characteristic
82	Dic₇₉	—	2	4	no	—
83	Dic₇₉	—	8	4	no	—
84	#13169	—	79	1	no	maximal, Sylow
85	C₂×C₇₉	—	2	1	no	—
86	C₂×C₇₉	—	2	1	no	—
87	C₂×C₇₉	#1942	1	1	yes	characteristic
88	C₂×C₇₉	#2178	1	2	yes	—
89	C₂×#368	D₇₉	1	1	yes	characteristic
90	#2158	—	79	1	no	—
91	C₂×C₄×C₁₆	—	79	1	no	—
92	#1945	—	79	2	no	—
93	#1382	—	79	2	no	—
94	C₇₉	#13169	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₁₆	—	79	1	no	—
96	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
97	C₂×#108	C₂×D₇₉	1	2	yes	—
98	#427	—	79	2	no	—
99	#424	—	158	2	no	—
100	C₂×C₄×C₈	—	79	1	no	—
101	#368	—	2	2	no	—
102	#367	—	79	2	no	—
103	#366	—	79	2	no	—
104	C₄×C₁₆	—	158	2	no	—
105	#339	—	158	2	no	—
106	C₂×M₁₆	—	2	2	no	—
107	C₂²×C₈	—	79	1	no	—
108	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
109	#117	—	158	2	no	—
110	C₂×C₄²	—	79	1	no	—
111	C₂×C₁₆	—	79	2	no	—
112	C₂×C₁₆	—	158	1	no	—
113	C₂×C₁₆	—	158	1	no	—
114	C₂×C₁₆	Dic₁₅₈	1	2	yes	—
115	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
116	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
117	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
118	#108	—	2	2	no	—
119	#108	—	2	2	no	—
120	#107	—	79	4	no	—
121	#106	—	158	2	no	—
122	C₄⋊C₈	—	316	4	no	—
123	C₄×C₈	—	158	1	no	—
124	C₄×C₈	—	158	1	no	—
125	C₂²×C₄	—	158	1	no	—
126	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
127	M₁₆	—	4	2	no	—
128	M₁₆	—	4	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	79	2	no	—
131	C₂×C₈	—	158	1	no	—
132	C₂×C₈	—	158	1	no	—
133	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
134	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
136	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
137	C₂×C₈	#11929623	1	1	yes	characteristic
138	C₂×C₈	#11929625	1	1	yes	characteristic
139	C₄⋊C₄	—	158	4	no	—
140	C₂²⋊C₄	—	316	2	no	—
141	C₄²	—	316	2	no	—
142	C₄²	—	79	2	no	—
143	C₄²	—	158	1	no	—
144	C₁₆	—	158	2	no	—
145	C₁₆	Dic₇₉⋊C₄	1	2	yes	—
146	C₁₆	Dic₇₉⋊C₄	1	2	yes	—
147	C₂³	#14579184	1	1	yes	characteristic
148	C₂×C₄	—	316	4	no	—
149	C₂×C₄	—	158	4	no	—
150	C₂×C₄	—	158	2	no	—
151	C₂×C₄	#14579123	1	1	yes	characteristic
152	C₂×C₄	#14579132	1	1	yes	characteristic
153	C₂×C₄	#14579160	1	1	yes	characteristic
154	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
155	C₂×C₄	Q₁₆×D₇₉	1	1	yes	characteristic
156	C₂×C₄	#14579180	1	1	yes	characteristic
157	C₈	—	8	2	no	—
158	C₈	—	158	2	no	—
159	C₈	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
160	C₈	#14579126	1	1	yes	characteristic
161	C₈	#14579137	1	1	yes	characteristic
162	C₈	#14579152	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	#19098645	1	1	yes	characteristic
165	C₂²	#19098801	1	1	yes	characteristic
166	C₂²	#19098964	1	1	yes	characteristic
167	C₄	—	632	4	no	—
168	C₄	—	158	4	no	—
169	C₄	#19098459	1	1	yes	characteristic
170	C₄	#19098657	1	1	yes	characteristic
171	C₄	#19098796	1	1	yes	characteristic
172	C₄	#19098967	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#23764131	1	1	yes	characteristic
176	C₂	#23765280	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765184

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₇₉

C₂

yes

maximal, characteristic

#23764146

C₂

yes

maximal

#23761497

C₂

yes

maximal

C₂×#368×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₁₆

C₂²

yes

characteristic

#19098601

C₂²

yes

—

#19098591

—

C₂×C₈×Dic₇₉

C₂²

yes

characteristic

#19098391

C₄

yes

complemented

#19098389

C₄

yes

complemented

C₁₆×Dic₇₉

—

#19098324

—

C₂²×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×#108×C₇₉

C₂²

yes

—

#368×C₇₉

—

C₂²×C₇₉⋊C₈

D₄

yes

characteristic

#14579120

—

C₂×C₄×Dic₇₉

D₄

yes

characteristic

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

Q₈

yes

—

#14579095

C₂×C₄

yes

—

#14579093

—

Dic₇₉⋊C₈

—

C₈×Dic₇₉

—

C₈×Dic₇₉

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

D₄

yes

—

C₂×C₁₆×C₇₉

Q₈

yes

—

#108×C₇₉

—

#108×C₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

D₈

yes

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

Q₁₆

yes

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

C₂×C₈×C₇₉

yes

C₂×C₈×C₇₉

yes

C₁₆×C₇₉

C₄⋊C₄

yes

—

C₁₆×C₇₉

C₄⋊C₄

yes

—

C₂³×C₇₉

#135

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

characteristic

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₉

#135

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₉

C₄×Q₈

yes

characteristic

C₈×C₇₉

C₂²⋊Q₈

yes

characteristic

C₈×C₇₉

#125

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

C₄×D₈

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₄×Q₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

#13169

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1942

yes

characteristic

C₂×C₇₉

#2178

yes

—

C₂×#368

D₇₉

yes

characteristic

#2158

—

C₂×C₄×C₁₆

—

#1945

—

#1382

—

C₇₉

#13169

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₉

yes

characteristic

C₂×#108

C₂×D₇₉

yes

—

#427

—

#424

—

158

—

100

C₂×C₄×C₈

—

101

#368

—

102

#367

—

103

#366

—

104

C₄×C₁₆

—

158

—

105

#339

—

158

—

106

C₂×M₁₆

—

107

C₂²×C₈

—

108

C₂²×C₈

C₂²×D₇₉

yes

characteristic

109

#117

—

158

—

110

C₂×C₄²

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

158

—

113

C₂×C₁₆

—

158

—

114

C₂×C₁₆

Dic₁₅₈

yes

—

115

C₂×C₁₆

C₄×D₇₉

yes

characteristic

116

C₂×C₁₆

C₄×D₇₉

yes

characteristic

117

C₂×C₁₆

C₂²⋊D₇₉

yes

—

118

#108

—

119

#108

—

120

#107

—

121

#106

—

158

—

122

C₄⋊C₈

—

316

—

123

C₄×C₈

—

158

—

124

C₄×C₈

—

158

—

125

C₂²×C₄

—

158

—

126

C₂²×C₄

D₄×D₇₉

yes

characteristic

127

M₁₆

—

128

M₁₆

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

158

—

132

C₂×C₈

—

158

—

133

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

134

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

135

C₂×C₈

Q₈×D₇₉

yes

characteristic

136

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

137

C₂×C₈

#11929623

yes

characteristic

138

C₂×C₈

#11929625

yes

characteristic

139

C₄⋊C₄

—

158

—

140

C₂²⋊C₄

—

316

—

141

C₄²

—

316

—

142

C₄²

—

143

C₄²

—

158

—

144

C₁₆

—

158

—

145

C₁₆

Dic₇₉⋊C₄

yes

—

146

C₁₆

Dic₇₉⋊C₄

yes

—

147

C₂³

#14579184

yes

characteristic

148

C₂×C₄

—

316

—

149

C₂×C₄

—

158

—

150

C₂×C₄

—

158

—

151

C₂×C₄

#14579123

yes

characteristic

152

C₂×C₄

#14579132

yes

characteristic

153

C₂×C₄

#14579160

yes

characteristic

154

C₂×C₄

D₈×D₇₉

yes

characteristic

155

C₂×C₄

Q₁₆×D₇₉

yes

characteristic

156

C₂×C₄

#14579180

yes

characteristic

157

C₈

—

158

C₈

—

158

—

159

C₈

C₂×Dic₇₉⋊C₄

yes

characteristic

160

C₈

#14579126

yes

characteristic

161

C₈

#14579137

yes

characteristic

162

C₈

#14579152

yes

characteristic

163

C₂²

—

164

C₂²

#19098645

yes

characteristic

165

C₂²

#19098801

yes

characteristic

166

C₂²

#19098964

yes

characteristic

167

C₄

—

632

—

168

C₄

—

158

—

169

C₄

#19098459

yes

characteristic

170

C₄

#19098657

yes

characteristic

171

C₄

#19098796

yes

characteristic

172

C₄

#19098967

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

#23764131

yes

characteristic

176

C₂

#23765280

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₇₉
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#368×C₇₉	C₂
socle	C₂²×C₇₉	C₄×D₈
prime radical	C₂³×C₇₉	#135
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#368	D₇₉
hypocenter	C₇₉	#13169
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

subgroup

G/H

center

C₂×C₄

D₈×D₇₉

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₂×#368×C₇₉

C₂

socle

C₂²×C₇₉

C₄×D₈

prime radical

C₂³×C₇₉

#135

cosocle

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

N	H ≅ G/N	type
10. #19098391	C₄	semidirect
11. #19098389	C₄	semidirect
94. C₇₉	#13169	semidirect

H ≅ G/N

type

10. #19098391

C₄

semidirect

11. #19098389

C₄

semidirect

94. C₇₉

#13169

semidirect

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

subgroup	index
#23765184	2
C₂×C₁₆×Dic₇₉	2
#23764146	2
#23761497	2
C₂×#368×C₇₉	2
#13169	79

subgroup

index

C₂×C₁₆×Dic₇₉

C₂×#368×C₇₉

Subgroups — #34206317

Conjugacy classes of subgroups

Complements

Maximal subgroups