Group #34181290 — subgroups

Subgroups — #34181290

Order 20224.

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

normal subgroups only

This group has 336 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	#23772330	C₂	1	1	yes	maximal, characteristic
2	#23762155	C₂	1	2	yes	maximal
3	#23762137	C₂	1	2	yes	maximal
4	C₇₉⋊#1309	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1309	C₂	1	1	yes	maximal, characteristic
6	#19099120	C₂²	1	1	yes	characteristic
7	#19098690	—	2	2	no	—
8	#19098687	—	2	2	no	—
9	#19098591	C₄	1	2	yes	—
10	#19098587	C₂²	1	2	yes	—
11	#19098521	C₄	1	2	yes	—
12	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
13	#19098313	—	2	2	no	—
14	#19098313	—	2	2	no	—
15	#19098310	—	2	2	no	—
16	#19098308	—	2	2	no	—
17	#423×C₇₉	C₂²	1	2	yes	—
18	#423×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×#11929608	—	2	2	no	—
20	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
22	#14579161	D₄	1	2	yes	—
23	#14579153	Q₈	1	2	yes	—
24	#14579140	—	2	2	no	—
25	C₄×Dic₁₅₈	—	2	2	no	—
26	#14579120	C₂×C₄	1	2	yes	—
27	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
28	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
29	Dic₇₉⋊C₈	—	2	4	no	—
30	#14579071	—	2	2	no	—
31	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
32	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
33	C₈×Dic₇₉	—	2	2	no	—
34	C₂×M₁₆×C₇₉	—	2	2	no	—
35	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₇₉	—	2	2	no	—
38	C₄⋊C₈×C₇₉	—	2	2	no	—
39	C₄⋊C₈×C₇₉	—	2	2	no	—
40	C₂×Dic₁₅₈	—	2	2	no	—
41	#11929608	—	8	2	no	—
42	C₂×C₇₉⋊C₈	—	2	2	no	—
43	C₂×C₇₉⋊C₈	—	4	1	no	—
44	C₂×C₇₉⋊C₈	—	4	2	no	—
45	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
47	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
48	C₄⋊Dic₇₉	—	2	2	no	—
49	C₄⋊Dic₇₉	C₄⋊C₄	1	4	yes	—
50	Dic₇₉⋊C₄	—	2	4	no	—
51	C₂²⋊Dic₇₉	—	2	2	no	—
52	C₄×Dic₇₉	—	2	2	no	—
53	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₉	#41	1	2	yes	—
55	M₁₆×C₇₉	—	4	2	no	—
56	M₁₆×C₇₉	—	8	2	no	—
57	C₂×C₈×C₇₉	—	2	2	no	—
58	C₂×C₈×C₇₉	—	4	2	no	—
59	C₄²×C₇₉	C₂×D₄	1	2	yes	—
60	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
62	C₂³×C₇₉	#117	1	1	yes	characteristic
63	C₂×C₄×C₇₉	—	2	1	no	—
64	C₂×C₄×C₇₉	—	2	1	no	—
65	C₂×C₄×C₇₉	—	2	2	no	—
66	C₂×C₄×C₇₉	—	2	2	no	—
67	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₉	#121	1	2	yes	—
70	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
71	C₂×Dic₇₉	—	2	4	no	—
72	Dic₁₅₈	—	4	2	no	—
73	Dic₁₅₈	—	4	2	no	—
74	C₈×C₇₉	—	4	2	no	—
75	C₈×C₇₉	—	8	2	no	—
76	C₇₉⋊C₈	—	2	1	no	—
77	C₇₉⋊C₈	—	2	2	no	—
78	C₇₉⋊C₈	—	8	2	no	—
79	C₇₉⋊C₈	#101	1	2	yes	—
80	C₂²×C₇₉	—	2	1	no	—
81	C₂²×C₇₉	—	2	2	no	—
82	C₂²×C₇₉	#380	1	1	yes	characteristic
83	C₄×C₇₉	—	2	1	no	—
84	C₄×C₇₉	—	2	2	no	—
85	C₄×C₇₉	C₂×#101	1	1	yes	characteristic
86	C₄×C₇₉	#413	1	1	yes	characteristic
87	C₄×C₇₉	#471	1	2	yes	—
88	C₄×C₇₉	#471	1	2	yes	—
89	Dic₇₉	—	4	4	no	—
90	#8637	—	79	1	no	maximal, Sylow
91	C₂×C₇₉	—	4	1	no	—
92	C₂×C₇₉	#1801	1	1	yes	characteristic
93	C₂×C₇₉	#1815	1	2	yes	—
94	#2960	—	79	1	no	—
95	#1566	—	79	2	no	—
96	#1561	—	79	2	no	—
97	#1309	—	79	1	no	—
98	#1309	D₇₉	1	1	yes	characteristic
99	C₇₉	#8637	1	1	yes	characteristic, complemented, Sylow
100	#533	—	79	1	no	—
101	#446	—	158	2	no	—
102	C₈×Q₈	—	158	2	no	—
103	#424	—	79	2	no	—
104	#423	—	79	2	no	—
105	#423	C₂×D₇₉	1	2	yes	—
106	#423	C₂×D₇₉	1	1	yes	characteristic
107	#405	—	79	2	no	—
108	C₂×C₄×C₈	—	79	1	no	—
109	#334	—	158	2	no	—
110	#334	—	158	2	no	—
111	#332	—	158	2	no	—
112	#332	—	158	2	no	—
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	158	2	no	—
115	C₂×M₁₆	C₄×D₇₉	1	2	yes	—
116	C₂²×C₈	—	79	1	no	—
117	C₂²×C₈	—	79	1	no	—
118	C₄⋊Q₈	—	79	2	no	—
119	#125	—	79	2	no	—
120	C₂²⋊Q₈	—	158	2	no	—
121	C₄×Q₈	—	158	2	no	—
122	#117	—	79	2	no	—
123	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	158	4	no	—
128	C₄⋊C₈	—	158	2	no	—
129	C₄⋊C₈	—	158	2	no	—
130	#97	—	158	2	no	—
131	C₄×C₈	—	79	2	no	—
132	C₄×C₈	—	79	2	no	—
133	C₄×C₈	—	158	2	no	—
134	C₂×Q₈	—	158	2	no	—
135	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
136	C₂²×C₄	#11929624	1	2	yes	—
137	M₁₆	—	4	2	no	—
138	M₁₆	—	8	2	no	—
139	M₁₆	—	632	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	316	2	no	—
143	C₂×C₈	—	316	1	no	—
144	C₂×C₈	—	79	1	no	—
145	C₂×C₈	—	79	1	no	—
146	C₂×C₈	—	79	2	no	—
147	C₂×C₈	—	158	2	no	—
148	C₄⋊C₄	—	79	4	no	—
149	C₄⋊C₄	—	158	2	no	—
150	C₄⋊C₄	—	158	4	no	—
151	C₂²⋊C₄	—	158	2	no	—
152	C₄²	—	158	2	no	—
153	C₄²	D₄×D₇₉	1	2	yes	—
154	C₄²	D₄×D₇₉	1	1	yes	characteristic
155	C₄²	Q₈×D₇₉	1	1	yes	characteristic
156	C₂³	#14579118	1	1	yes	characteristic
157	Q₈	—	316	2	no	—
158	Q₈	—	316	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	158	4	no	—
164	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
165	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
166	C₂×C₄	#14579134	1	2	yes	—
167	C₂×C₄	#14579139	1	2	yes	—
168	C₈	—	4	2	no	—
169	C₈	—	8	2	no	—
170	C₈	—	79	2	no	—
171	C₈	—	632	2	no	—
172	C₈	—	158	1	no	—
173	C₈	—	158	2	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	2	1	no	—
176	C₂²	#19098422	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	2	1	no	—
179	C₄	—	316	4	no	—
180	C₄	#101×D₇₉	1	1	yes	characteristic
181	C₄	#19098544	1	1	yes	characteristic
182	C₄	#19098825	1	2	yes	—
183	C₄	#19098829	1	2	yes	—
184	C₂	—	4	1	no	—
185	C₂	#23763395	1	1	yes	characteristic
186	C₂	#23763458	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23772330

C₂

yes

maximal, characteristic

#23762155

C₂

yes

maximal

#23762137

C₂

yes

maximal

C₇₉⋊#1309

C₂

yes

maximal, characteristic

C₇₉×#1309

C₂

yes

maximal, characteristic

#19099120

C₂²

yes

—

—

C₄

yes

—

#19098587

C₂²

yes

—

#19098521

C₄

yes

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

—

—

—

#423×C₇₉

C₂²

yes

—

#423×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

C₂²×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇₉⋊C₈

C₂×C₄

yes

characteristic

#14579161

D₄

yes

—

#14579153

Q₈

yes

—

#14579140

—

C₄×Dic₁₅₈

—

#14579120

C₂×C₄

yes

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

Dic₇₉⋊C₈

—

#14579071

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₈×Dic₇₉

—

C₂×M₁₆×C₇₉

—

C₂×M₁₆×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₂×Dic₁₅₈

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₇₉⋊C₈

C₂²⋊C₄

yes

—

C₂×C₇₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₄⋊C₄

yes

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₂³×C₇₉

#117

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#121

yes

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

#101

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#380

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₄×C₇₉

#471

yes

—

Dic₇₉

—

#8637

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1801

yes

characteristic

C₂×C₇₉

#1815

yes

—

—

—

—

—

D₇₉

yes

characteristic

C₇₉

#8637

yes

characteristic, complemented, Sylow

100

#533

—

101

#446

—

158

—

102

C₈×Q₈

—

158

—

103

#424

—

104

#423

—

105

#423

C₂×D₇₉

yes

—

106

#423

C₂×D₇₉

yes

characteristic

107

#405

—

108

C₂×C₄×C₈

—

109

#334

—

158

—

110

#334

—

158

—

111

#332

—

158

—

112

#332

—

158

—

113

C₂×M₁₆

—

114

C₂×M₁₆

—

158

—

115

C₂×M₁₆

C₄×D₇₉

yes

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₄⋊Q₈

—

119

#125

—

120

C₂²⋊Q₈

—

158

—

121

C₄×Q₈

—

158

—

122

#117

—

123

C₂×C₄²

C₂²×D₇₉

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

158

—

128

C₄⋊C₈

—

158

—

129

C₄⋊C₈

—

158

—

130

#97

—

158

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

158

—

134

C₂×Q₈

—

158

—

135

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

136

C₂²×C₄

#11929624

yes

—

137

M₁₆

—

138

M₁₆

—

139

M₁₆

—

632

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

316

—

143

C₂×C₈

—

316

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

158

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

158

—

150

C₄⋊C₄

—

158

—

151

C₂²⋊C₄

—

158

—

152

C₄²

—

158

—

153

C₄²

D₄×D₇₉

yes

—

154

C₄²

D₄×D₇₉

yes

characteristic

155

C₄²

Q₈×D₇₉

yes

characteristic

156

C₂³

#14579118

yes

characteristic

157

Q₈

—

316

—

158

Q₈

—

316

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

158

—

164

C₂×C₄

C₂²⋊C₄×D₇₉

yes

characteristic

165

C₂×C₄

C₄⋊C₄×D₇₉

yes

characteristic

166

C₂×C₄

#14579134

yes

—

167

C₂×C₄

#14579139

yes

—

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

632

—

172

C₈

—

158

—

173

C₈

—

158

—

174

C₂²

—

175

C₂²

—

176

C₂²

#19098422

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

316

—

180

C₄

#101×D₇₉

yes

characteristic

181

C₄

#19098544

yes

characteristic

182

C₄

#19098825

yes

—

183

C₄

#19098829

yes

—

184

C₂

—

185

C₂

#23763395

yes

characteristic

186

C₂

#23763458

yes

—

	subgroup	G/H
center	C₂²	#19098422
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1309	C₂
socle	C₂²×C₇₉	#380
prime radical	C₂³×C₇₉	#117
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1309	D₇₉
hypocenter	C₇₉	#8637
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#19098422

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1309

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
99. C₇₉	#8637	semidirect

H ≅ G/N

type

99. C₇₉

#8637

semidirect

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

subgroup	index
#23772330	2
#23762155	2
#23762137	2
C₇₉⋊#1309	2
C₇₉×#1309	2
#8637	79

subgroup

index

#23772330

#23762155

#23762137

C₇₉⋊#1309

C₇₉×#1309

#8637

Subgroups — #34181290

Conjugacy classes of subgroups

Complements

Maximal subgroups