Group #34175019 — subgroups

Subgroups — #34175019

Order 20224.

268 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 71 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	#23761484	C₂	1	2	yes	maximal, complemented
2	C₇₉×#1955	C₂	1	1	yes	maximal, characteristic
3	C₂×#14579101	—	2	2	no	—
4	C₂×C₄×D₄×C₇₉	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₈×C₇₉	C₄	1	1	yes	characteristic
6	C₂×C₂²⋊C₈×C₇₉	—	2	2	no	—
7	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
8	#14579101	—	2	4	no	—
9	#14579101	—	4	2	no	—
10	C₂×C₇₉⋊C₁₆	—	4	2	no	—
11	C₂²×D₄×C₇₉	C₈	1	1	yes	characteristic
12	C₂³×C₄×C₇₉	—	2	1	no	—
13	C₂²×C₈×C₇₉	—	2	1	no	—
14	C₂²×C₈×C₇₉	D₄	1	2	yes	—
15	C₄×D₄×C₇₉	—	4	1	no	—
16	C₄×D₄×C₇₉	—	4	1	no	—
17	C₂×C₄⋊C₄×C₇₉	C₈	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
19	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
20	C₄⋊C₈×C₇₉	—	4	1	no	—
21	C₂²⋊C₈×C₇₉	—	4	2	no	—
22	C₂²⋊C₈×C₇₉	—	4	2	no	—
23	#97×C₇₉	—	4	1	no	—
24	C₇₉⋊C₁₆	—	4	4	no	—
25	C₂⁴×C₇₉	—	2	1	no	—
26	C₂×D₄×C₇₉	—	4	1	no	—
27	C₂×D₄×C₇₉	—	4	2	no	—
28	C₂²×C₄×C₇₉	—	2	1	no	—
29	C₂²×C₄×C₇₉	—	4	1	no	—
30	C₂²×C₄×C₇₉	—	4	1	no	—
31	C₂²×C₄×C₇₉	—	4	2	no	—
32	C₂²×C₄×C₇₉	—	4	2	no	—
33	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₇₉	—	2	2	no	—
37	C₂×C₈×C₇₉	—	2	2	no	—
38	C₂×C₈×C₇₉	—	4	1	no	—
39	C₂×C₈×C₇₉	—	4	2	no	—
40	C₂×C₈×C₇₉	D₈	1	2	yes	—
41	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
42	C₄⋊C₄×C₇₉	—	4	1	no	—
43	C₂²⋊C₄×C₇₉	—	4	2	no	—
44	C₄²×C₇₉	—	2	1	no	—
45	C₄²×C₇₉	—	2	1	no	—
46	C₂³×C₇₉	—	2	2	no	—
47	C₂³×C₇₉	—	4	1	no	—
48	C₂³×C₇₉	—	4	2	no	—
49	C₂³×C₇₉	—	4	1	no	—
50	C₂³×C₇₉	—	4	2	no	—
51	C₂³×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
52	D₄×C₇₉	—	4	2	no	—
53	D₄×C₇₉	—	4	2	no	—
54	C₂×C₄×C₇₉	—	2	1	no	—
55	C₂×C₄×C₇₉	—	2	1	no	—
56	C₂×C₄×C₇₉	—	4	1	no	—
57	C₂×C₄×C₇₉	—	4	1	no	—
58	C₂×C₄×C₇₉	—	4	1	no	—
59	C₂×C₄×C₇₉	—	4	2	no	—
60	C₂×C₄×C₇₉	—	8	2	no	—
61	C₂×C₄×C₇₉	—	8	1	no	—
62	C₂×C₄×C₇₉	—	8	1	no	—
63	C₂×C₄×C₇₉	#99	1	1	yes	characteristic
64	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
65	C₂×C₄×C₇₉	#102	1	2	yes	—
66	C₂×C₄×C₇₉	#104	1	2	yes	—
67	C₈×C₇₉	—	2	2	no	—
68	C₈×C₇₉	—	2	2	no	—
69	C₈×C₇₉	—	8	1	no	—
70	C₂²×C₇₉	—	2	1	no	—
71	C₂²×C₇₉	—	2	1	no	—
72	C₂²×C₇₉	—	4	4	no	—
73	C₂²×C₇₉	—	4	2	no	—
74	C₂²×C₇₉	—	8	2	no	—
75	C₂²×C₇₉	—	8	1	no	—
76	C₂²×C₇₉	—	8	1	no	—
77	C₂²×C₇₉	#323	1	1	yes	characteristic
78	C₂²×C₇₉	D₄⋊C₈	1	2	yes	—
79	C₄×C₇₉	—	4	1	no	—
80	C₄×C₇₉	—	4	1	no	—
81	C₄×C₇₉	—	8	1	no	—
82	C₄×C₇₉	#327	1	2	yes	—
83	C₄×C₇₉	#329	1	1	yes	characteristic
84	C₄×C₇₉	#331	1	1	yes	characteristic
85	#7105	—	79	1	no	maximal, Sylow
86	C₂×C₇₉	—	2	2	no	—
87	C₂×C₇₉	—	8	2	no	—
88	C₂×C₇₉	#1270	1	1	yes	characteristic
89	C₂×C₇₉	#1320	1	1	yes	characteristic
90	C₂×C₇₉	#1321	1	1	yes	characteristic
91	#1955	D₇₉	1	1	yes	characteristic
92	#1377	—	79	2	no	—
93	C₇₉	#7105	1	1	yes	characteristic, complemented, Sylow
94	C₂×C₄×D₄	Dic₇₉	1	1	yes	characteristic
95	C₂×M₃₂	—	158	2	no	—
96	C₂×C₄⋊C₈	Dic₇₉	1	1	yes	characteristic
97	C₂×C₂²⋊C₈	—	2	2	no	—
98	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
99	C₂²×D₄	C₇₉⋊C₈	1	1	yes	characteristic
100	C₂³×C₄	—	2	1	no	—
101	C₂²×C₈	—	2	1	no	—
102	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
103	C₄×D₄	—	4	1	no	—
104	C₄×D₄	—	4	1	no	—
105	C₂×C₄⋊C₄	C₇₉⋊C₈	1	1	yes	characteristic
106	C₂×C₂²⋊C₄	—	2	1	no	—
107	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
108	M₃₂	—	316	2	no	—
109	M₃₂	—	158	4	no	—
110	C₂×C₁₆	—	316	2	no	—
111	C₄⋊C₈	—	4	1	no	—
112	C₂²⋊C₈	—	4	2	no	—
113	C₂²⋊C₈	—	4	2	no	—
114	#97	—	4	1	no	—
115	C₂⁴	—	2	1	no	—
116	C₂×D₄	—	4	1	no	—
117	C₂×D₄	—	4	2	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	4	1	no	—
120	C₂²×C₄	—	4	1	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
125	C₂²×C₄	#11929608	1	1	yes	characteristic
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	#11929609	1	2	yes	—
131	C₂×C₈	#11929612	1	2	yes	—
132	C₄⋊C₄	—	4	1	no	—
133	C₂²⋊C₄	—	4	2	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	2	1	no	—
136	C₁₆	—	316	4	no	—
137	C₂³	—	2	2	no	—
138	C₂³	—	4	1	no	—
139	C₂³	—	4	2	no	—
140	C₂³	—	4	1	no	—
141	C₂³	—	4	2	no	—
142	C₂³	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
143	D₄	—	4	2	no	—
144	D₄	—	4	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	8	1	no	—
154	C₂×C₄	#14579076	1	1	yes	characteristic
155	C₂×C₄	#14579078	1	1	yes	characteristic
156	C₂×C₄	#14579082	1	2	yes	—
157	C₂×C₄	#14579088	1	2	yes	—
158	C₈	—	2	2	no	—
159	C₈	—	2	2	no	—
160	C₈	—	8	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	4	4	no	—
164	C₂²	—	4	2	no	—
165	C₂²	—	8	2	no	—
166	C₂²	—	8	1	no	—
167	C₂²	—	8	1	no	—
168	C₂²	#19098287	1	1	yes	characteristic
169	C₂²	D₄⋊(C₇₉⋊C₈)	1	2	yes	—
170	C₄	—	4	1	no	—
171	C₄	—	4	1	no	—
172	C₄	—	8	1	no	—
173	C₄	#19098297	1	2	yes	—
174	C₄	#19098303	1	1	yes	characteristic
175	C₄	#19098307	1	1	yes	characteristic
176	C₂	—	2	2	no	—
177	C₂	—	8	2	no	—
178	C₂	#23761221	1	1	yes	characteristic
179	C₂	#23761336	1	1	yes	characteristic
180	C₂	#23761338	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23761484

C₂

yes

maximal, complemented

C₇₉×#1955

C₂

yes

maximal, characteristic

C₂×#14579101

—

C₂×C₄×D₄×C₇₉

C₄

yes

characteristic

C₂×C₄⋊C₈×C₇₉

C₄

yes

characteristic

C₂×C₂²⋊C₈×C₇₉

—

C₂×#97×C₇₉

C₂²

yes

characteristic

#14579101

—

#14579101

—

C₂×C₇₉⋊C₁₆

—

C₂²×D₄×C₇₉

C₈

yes

characteristic

C₂³×C₄×C₇₉

—

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

C₄×D₄×C₇₉

—

C₄×D₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₈

yes

characteristic

C₂×C₂²⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

#97×C₇₉

—

C₇₉⋊C₁₆

—

C₂⁴×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₉

M₁₆

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂²⋊C₈

yes

characteristic

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

—

C₂×C₄×C₇₉

#104

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#323

yes

characteristic

C₂²×C₇₉

D₄⋊C₈

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#327

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

yes

characteristic

#1377

—

C₇₉

#7105

yes

characteristic, complemented, Sylow

C₂×C₄×D₄

Dic₇₉

yes

characteristic

C₂×M₃₂

—

158

—

C₂×C₄⋊C₈

Dic₇₉

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×#97

C₂×D₇₉

yes

characteristic

C₂²×D₄

C₇₉⋊C₈

yes

characteristic

100

C₂³×C₄

—

101

C₂²×C₈

—

102

C₂²×C₈

C₂²⋊D₇₉

yes

—

103

C₄×D₄

—

104

C₄×D₄

—

105

C₂×C₄⋊C₄

C₇₉⋊C₈

yes

characteristic

106

C₂×C₂²⋊C₄

—

107

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

108

M₃₂

—

316

—

109

M₃₂

—

158

—

110

C₂×C₁₆

—

316

—

111

C₄⋊C₈

—

112

C₂²⋊C₈

—

113

C₂²⋊C₈

—

114

#97

—

115

C₂⁴

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

124

C₂²×C₄

C₂×C₇₉⋊C₈

yes

characteristic

125

C₂²×C₄

#11929608

yes

characteristic

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

#11929609

yes

—

131

C₂×C₈

#11929612

yes

—

132

C₄⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

—

135

C₄²

—

136

C₁₆

—

316

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

142

C₂³

C₂²⋊(C₇₉⋊C₈)

yes

characteristic

143

D₄

—

144

D₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

#14579076

yes

characteristic

155

C₂×C₄

#14579078

yes

characteristic

156

C₂×C₄

#14579082

yes

—

157

C₂×C₄

#14579088

yes

—

158

C₈

—

159

C₈

—

160

C₈

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

#19098287

yes

characteristic

169

C₂²

D₄⋊(C₇₉⋊C₈)

yes

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

#19098297

yes

—

174

C₄

#19098303

yes

characteristic

175

C₄

#19098307

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#23761221

yes

characteristic

179

C₂

#23761336

yes

characteristic

180

C₂

#23761338

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579076
commutator	C₂²×C₄×C₇₉	C₂×C₈
Frattini	C₂×#97	C₂×D₇₉
Fitting	C₇₉×#1955	C₂
socle	C₂²×C₇₉	#323
prime radical	C₂²×D₄×C₇₉	C₈
cosocle	C₂×#97×C₇₉	C₂²
hypercenter	#1955	D₇₉
hypocenter	C₇₉	#7105
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1955

subgroup

G/H

center

C₂×C₄

#14579076

commutator

C₂²×C₄×C₇₉

C₂×C₈

Frattini

C₂×#97

C₂×D₇₉

Fitting

C₇₉×#1955

C₂

socle

C₂²×C₇₉

#323

prime radical

C₂²×D₄×C₇₉

C₈

cosocle

C₂×#97×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1955

N	H ≅ G/N	type
1. #23761484	C₂	semidirect
93. C₇₉	#7105	semidirect

H ≅ G/N

type

1. #23761484

C₂

semidirect

93. C₇₉

#7105

semidirect

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

subgroup	index
#23761484	2
C₇₉×#1955	2
#7105	79

subgroup

index

#23761484

C₇₉×#1955

#7105

Subgroups — #34175019

Conjugacy classes of subgroups

Complements

Maximal subgroups