Group #34203671 — subgroups

Subgroups — #34203671

Order 20224.

366 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 159 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×#19098390	C₂	1	2	yes	maximal
2	C₂×C₁₆×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	#107×Dic₇₉	C₂	1	2	yes	maximal
4	#23761492	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1380	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14579095	C₂²	1	2	yes	—
8	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×Dic₇₉	—	2	2	no	—
10	#19098390	C₄	1	8	yes	complemented
11	C₁₆×Dic₇₉	C₄	1	4	yes	complemented
12	#19098325	—	2	2	no	—
13	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×#107×C₇₉	C₂²	1	2	yes	—
15	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
16	C₂×C₄⋊Dic₇₉	—	2	2	no	—
17	C₂×C₄×Dic₇₉	—	4	2	no	—
18	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
19	C₂×C₇₉⋊C₁₆	C₂×C₄	1	4	yes	—
20	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
21	#14579095	—	2	2	no	—
22	#14579095	C₂×C₄	1	4	yes	—
23	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
24	C₈×Dic₇₉	Q₈	1	2	yes	—
25	C₇₉⋊#95	—	4	2	no	—
26	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
28	C₂×C₁₆×C₇₉	—	2	2	no	—
29	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
30	#107×C₇₉	—	2	2	no	—
31	#107×C₇₉	C₂×C₄	1	4	yes	—
32	C₂²×Dic₇₉	—	4	2	no	—
33	C₂²×Dic₇₉	D₈	1	1	yes	characteristic
34	C₂²×Dic₇₉	D₈	1	1	yes	characteristic
35	C₂×C₇₉⋊C₈	—	2	2	no	—
36	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
37	C₄⋊Dic₇₉	—	2	4	no	—
38	C₄⋊Dic₇₉	—	4	2	no	—
39	C₄×Dic₇₉	—	8	4	no	—
40	C₄×Dic₇₉	C₄⋊C₄	1	2	yes	—
41	C₄×Dic₇₉	Q₁₆	1	2	yes	—
42	C₇₉⋊C₁₆	C₄²	1	4	yes	complemented
43	C₂²×C₄×C₇₉	—	4	2	no	—
44	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₇₉	#41	1	2	yes	—
49	C₄⋊C₄×C₇₉	—	2	4	no	—
50	C₄⋊C₄×C₇₉	—	4	2	no	—
51	C₁₆×C₇₉	—	2	2	no	—
52	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₇₉	—	4	4	no	—
54	C₂×C₄×C₇₉	—	8	2	no	—
55	C₂×C₄×C₇₉	—	8	2	no	—
56	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
58	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×C₄×C₇₉	#135	1	2	yes	—
60	C₂×Dic₇₉	—	2	1	no	—
61	C₂×Dic₇₉	—	2	1	no	—
62	C₂×Dic₇₉	—	4	4	no	—
63	C₂×Dic₇₉	—	8	2	no	—
64	C₂×Dic₇₉	—	8	2	no	—
65	C₂×Dic₇₉	#107	1	2	yes	—
66	C₂×Dic₇₉	#107	1	2	yes	—
67	C₂×Dic₇₉	SD₃₂	1	4	yes	—
68	C₈×C₇₉	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₇₉	#117	1	1	yes	characteristic
70	C₈×C₇₉	C₄×Q₈	1	2	yes	—
71	C₇₉⋊C₈	—	2	2	no	—
72	C₂²×C₇₉	C₂×#107	1	1	yes	characteristic
73	C₂²×C₇₉	C₄×D₈	1	2	yes	—
74	C₂²×C₇₉	C₂×SD₃₂	1	2	yes	—
75	C₂²×C₇₉	#508	1	1	yes	characteristic
76	C₂²×C₇₉	#508	1	1	yes	characteristic
77	C₄×C₇₉	—	8	4	no	—
78	C₄×C₇₉	C₄×C₄⋊C₄	1	1	yes	characteristic
79	C₄×C₇₉	#427	1	1	yes	characteristic
80	C₄×C₇₉	C₄×Q₁₆	1	2	yes	—
81	Dic₇₉	—	2	2	no	—
82	Dic₇₉	—	8	4	no	—
83	Dic₇₉	#367	1	4	yes	complemented
84	C₄×#367	—	79	1	no	maximal, Sylow
85	C₂×C₇₉	C₄×#107	1	1	yes	characteristic
86	C₂×C₇₉	C₂×#367	1	1	yes	characteristic
87	C₂×C₇₉	#2158	1	1	yes	characteristic
88	C₂×C₇₉	C₄×SD₃₂	1	2	yes	—
89	C₂×C₇₉	C₄×SD₃₂	1	2	yes	—
90	C₂×#367	—	79	2	no	—
91	C₂×C₄×C₁₆	—	79	1	no	—
92	C₄×#107	—	79	2	no	—
93	#1380	—	79	1	no	—
94	#1380	D₇₉	1	1	yes	characteristic
95	C₇₉	C₄×#367	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₁₆	—	79	1	no	—
97	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
98	C₂×#107	—	79	2	no	—
99	C₂×#107	C₂×D₇₉	1	2	yes	—
100	C₂×C₄×C₈	—	79	1	no	—
101	C₄×C₄⋊C₄	—	158	2	no	—
102	#367	—	79	8	no	—
103	C₄×C₁₆	—	79	4	no	—
104	#340	—	158	2	no	—
105	C₂²×C₈	—	79	1	no	—
106	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄⋊C₄	—	158	2	no	—
109	C₂×C₄²	—	316	2	no	—
110	C₂×C₄²	—	79	1	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	79	2	no	—
113	C₂×C₁₆	—	79	4	no	—
114	C₂×C₁₆	C₄×D₇₉	1	2	yes	—
115	#107	—	2	2	no	—
116	#107	—	79	4	no	—
117	#107	—	158	2	no	—
118	#107	C₄×D₇₉	1	4	yes	—
119	C₄×C₈	—	79	2	no	—
120	C₄×C₈	—	79	2	no	—
121	#95	—	316	2	no	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	—	316	2	no	—
124	C₂²×C₄	—	79	1	no	—
125	C₂²×C₄	—	79	1	no	—
126	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
127	C₂×C₈	—	79	2	no	—
128	C₂×C₈	—	158	2	no	—
129	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
131	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
132	C₂×C₈	#11929623	1	2	yes	—
133	C₄⋊C₄	—	2	4	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	316	2	no	—
136	C₄⋊C₄	—	158	4	no	—
137	C₄²	—	79	2	no	—
138	C₄²	—	79	2	no	—
139	C₄²	—	632	4	no	—
140	C₁₆	—	2	2	no	—
141	C₁₆	—	79	4	no	—
142	C₂³	D₈×D₇₉	1	1	yes	characteristic
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	316	4	no	—
147	C₂×C₄	—	79	4	no	—
148	C₂×C₄	—	79	2	no	—
149	C₂×C₄	—	79	2	no	—
150	C₂×C₄	—	632	2	no	—
151	C₂×C₄	—	632	2	no	—
152	C₂×C₄	—	158	1	no	—
153	C₂×C₄	—	158	1	no	—
154	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
155	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
156	C₂×C₄	Q₁₆×D₇₉	1	1	yes	characteristic
157	C₂×C₄	#14579182	1	2	yes	—
158	C₈	—	158	2	no	—
159	C₈	C₄²×D₇₉	1	1	yes	characteristic
160	C₈	#14579117	1	1	yes	characteristic
161	C₈	#14579126	1	2	yes	—
162	C₂²	#107×D₇₉	1	1	yes	characteristic
163	C₂²	#19098643	1	2	yes	—
164	C₂²	SD₃₂×D₇₉	1	2	yes	—
165	C₂²	#19098992	1	1	yes	characteristic
166	C₂²	#19098996	1	1	yes	characteristic
167	C₄	—	8	4	no	—
168	C₄	—	79	4	no	—
169	C₄	—	632	4	no	—
170	C₄	—	158	2	no	—
171	C₄	#19098414	1	1	yes	characteristic
172	C₄	#19098598	1	1	yes	characteristic
173	C₄	#19098655	1	2	yes	—
174	C₂	#23763292	1	1	yes	characteristic
175	C₂	#367×D₇₉	1	1	yes	characteristic
176	C₂	#23765181	1	1	yes	characteristic
177	C₂	#23765249	1	2	yes	—
178	C₂	#23765255	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098390

C₂

yes

maximal

C₂×C₁₆×Dic₇₉

C₂

yes

maximal, characteristic

#107×Dic₇₉

C₂

yes

maximal

#23761492

C₂

yes

maximal, characteristic

C₇₉×#1380

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₁₆

C₂²

yes

characteristic

C₂×#14579095

C₂²

yes

—

C₂×C₈×Dic₇₉

C₂²

yes

characteristic

C₄⋊C₄×Dic₇₉

—

#19098390

C₄

yes

complemented

C₁₆×Dic₇₉

C₄

yes

complemented

#19098325

—

C₂²×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×#107×C₇₉

C₂²

yes

—

C₂²×C₇₉⋊C₈

D₄

yes

characteristic

C₂×C₄⋊Dic₇₉

—

C₂×C₄×Dic₇₉

—

C₂×C₄×Dic₇₉

D₄

yes

characteristic

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

—

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

—

#14579095

—

#14579095

C₂×C₄

yes

—

C₈×Dic₇₉

C₂×C₄

yes

—

C₈×Dic₇₉

Q₈

yes

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

#107×C₇₉

—

#107×C₇₉

C₂×C₄

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

D₈

yes

characteristic

C₂²×Dic₇₉

D₈

yes

characteristic

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₄⋊C₄

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₄⋊C₄

yes

—

C₄×Dic₇₉

Q₁₆

yes

—

C₇₉⋊C₁₆

C₄²

yes

complemented

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂²×C₄

yes

—

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₉

#135

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

#107

yes

—

C₂×Dic₇₉

#107

yes

—

C₂×Dic₇₉

SD₃₂

yes

—

C₈×C₇₉

C₂×C₄²

yes

characteristic

C₈×C₇₉

#117

yes

characteristic

C₈×C₇₉

C₄×Q₈

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₂×#107

yes

characteristic

C₂²×C₇₉

C₄×D₈

yes

—

C₂²×C₇₉

C₂×SD₃₂

yes

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

C₄×C₄⋊C₄

yes

characteristic

C₄×C₇₉

#427

yes

characteristic

C₄×C₇₉

C₄×Q₁₆

yes

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

C₄×SD₃₂

yes

—

C₂×C₇₉

C₄×SD₃₂

yes

—

C₂×#367

—

C₂×C₄×C₁₆

—

C₄×#107

—

#1380

—

#1380

D₇₉

yes

characteristic

C₇₉

C₄×#367

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₉

yes

characteristic

C₂×#107

—

C₂×#107

C₂×D₇₉

yes

—

100

C₂×C₄×C₈

—

101

C₄×C₄⋊C₄

—

158

—

102

#367

—

103

C₄×C₁₆

—

104

#340

—

158

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂²×D₇₉

yes

characteristic

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

158

—

109

C₂×C₄²

—

316

—

110

C₂×C₄²

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

114

C₂×C₁₆

C₄×D₇₉

yes

—

115

#107

—

116

#107

—

117

#107

—

158

—

118

#107

C₄×D₇₉

yes

—

119

C₄×C₈

—

120

C₄×C₈

—

121

#95

—

316

—

122

C₂²×C₄

—

123

C₂²×C₄

—

316

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

D₄×D₇₉

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

158

—

129

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

130

C₂×C₈

C₂×C₄×D₇₉

yes

—

131

C₂×C₈

Q₈×D₇₉

yes

characteristic

132

C₂×C₈

#11929623

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

316

—

136

C₄⋊C₄

—

158

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

632

—

140

C₁₆

—

141

C₁₆

—

142

C₂³

D₈×D₇₉

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

316

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

632

—

151

C₂×C₄

—

632

—

152

C₂×C₄

—

158

—

153

C₂×C₄

—

158

—

154

C₂×C₄

C₄⋊C₄×D₇₉

yes

characteristic

155

C₂×C₄

(C₂²⋊D₇₉)⋊C₄

yes

—

156

C₂×C₄

Q₁₆×D₇₉

yes

characteristic

157

C₂×C₄

#14579182

yes

—

158

C₈

—

158

—

159

C₈

C₄²×D₇₉

yes

characteristic

160

C₈

#14579117

yes

characteristic

161

C₈

#14579126

yes

—

162

C₂²

#107×D₇₉

yes

characteristic

163

C₂²

#19098643

yes

—

164

C₂²

SD₃₂×D₇₉

yes

—

165

C₂²

#19098992

yes

characteristic

166

C₂²

#19098996

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

632

—

170

C₄

—

158

—

171

C₄

#19098414

yes

characteristic

172

C₄

#19098598

yes

characteristic

173

C₄

#19098655

yes

—

174

C₂

#23763292

yes

characteristic

175

C₂

#367×D₇₉

yes

characteristic

176

C₂

#23765181

yes

characteristic

177

C₂

#23765249

yes

—

178

C₂

#23765255

yes

—

	subgroup	G/H
center	C₂³	D₈×D₇₉
commutator	C₈×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1380	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1380	D₇₉
hypocenter	C₇₉	C₄×#367
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

subgroup

G/H

center

C₂³

D₈×D₇₉

commutator

C₈×C₇₉

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1380

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₈

cosocle

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

N	H ≅ G/N	type
10. #19098390	C₄	semidirect
11. C₁₆×Dic₇₉	C₄	semidirect
42. C₇₉⋊C₁₆	C₄²	semidirect
83. Dic₇₉	#367	semidirect
95. C₇₉	C₄×#367	semidirect

H ≅ G/N

type

10. #19098390

C₄

semidirect

11. C₁₆×Dic₇₉

C₄

semidirect

42. C₇₉⋊C₁₆

C₄²

semidirect

83. Dic₇₉

#367

semidirect

95. C₇₉

C₄×#367

semidirect

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

subgroup	index
C₂×#19098390	2
C₂×C₁₆×Dic₇₉	2
#107×Dic₇₉	2
#23761492	2
C₇₉×#1380	2
C₄×#367	79

subgroup

index

C₂×#19098390

C₂×C₁₆×Dic₇₉

#107×Dic₇₉

#23761492

C₇₉×#1380

C₄×#367

Subgroups — #34203671

Conjugacy classes of subgroups

Complements

Maximal subgroups