Group #34193261 — subgroups

Subgroups — #34193261

Order 20224.

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

normal subgroups only

This group has 320 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	#23764340	C₂	1	1	yes	maximal, characteristic
2	#23763575	C₂	1	1	yes	maximal, characteristic
3	#23762171	C₂	1	2	yes	maximal
4	#23761830	C₂	1	2	yes	maximal
5	C₇₉×#1275	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
7	C₂×#14579097	—	2	2	no	—
8	#19098587	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
10	#19098317	—	2	2	no	—
11	#19098314	—	2	4	no	—
12	#19098295	—	2	4	no	—
13	#431×C₇₉	C₂²	1	2	yes	—
14	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₅₈	Q₈	1	1	yes	characteristic
16	C₂×#11929608	—	2	2	no	—
17	C₂²×C₇₉⋊C₈	—	2	1	no	—
18	C₄×Dic₁₅₈	—	2	2	no	—
19	C₄×Dic₁₅₈	—	2	2	no	—
20	C₂×C₄⋊Dic₇₉	Q₈	1	1	yes	characteristic
21	C₂×Dic₇₉⋊C₄	—	2	1	no	—
22	C₂×C₄×Dic₇₉	—	2	1	no	—
23	#14579097	—	4	4	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
26	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
27	C₂²×C₈×C₇₉	D₄	1	2	yes	—
28	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₇₉	—	2	2	no	—
30	C₂²⋊C₈×C₇₉	—	2	2	no	—
31	#97×C₇₉	—	2	2	no	—
32	C₄×C₈×C₇₉	D₄	1	4	yes	—
33	C₂×Dic₁₅₈	—	2	2	no	—
34	C₂×Dic₁₅₈	—	4	2	no	—
35	C₂²×Dic₇₉	—	2	1	no	—
36	C₂²×Dic₇₉	—	2	1	no	—
37	#11929608	—	4	4	no	—
38	C₂×C₇₉⋊C₈	—	4	1	no	—
39	C₂×C₇₉⋊C₈	—	4	2	no	—
40	C₂×C₇₉⋊C₈	—	4	2	no	—
41	C₄⋊Dic₇₉	—	2	2	no	—
42	Dic₇₉⋊C₄	—	4	2	no	—
43	C₄×Dic₇₉	—	4	1	no	—
44	C₄×Dic₇₉	—	4	1	no	—
45	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
48	C₂×C₈×C₇₉	—	2	2	no	—
49	C₂×C₈×C₇₉	—	2	4	no	—
50	C₂×C₈×C₇₉	—	2	2	no	—
51	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
52	C₄²×C₇₉	C₂×D₄	1	2	yes	—
53	C₄²×C₇₉	#41	1	2	yes	—
54	C₂³×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₇₉	—	4	1	no	—
56	C₂×C₄×C₇₉	—	4	1	no	—
57	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
58	C₂×C₄×C₇₉	#123	1	2	yes	—
59	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
60	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
62	C₂×C₄×C₇₉	#137	1	2	yes	—
63	C₂×Dic₇₉	—	2	2	no	—
64	C₂×Dic₇₉	—	4	1	no	—
65	C₂×Dic₇₉	—	4	1	no	—
66	C₂×Dic₇₉	—	4	2	no	—
67	C₂×Dic₇₉	—	4	1	no	—
68	Dic₁₅₈	—	2	4	no	—
69	Dic₁₅₈	—	4	2	no	—
70	C₈×C₇₉	—	2	4	no	—
71	C₈×C₇₉	—	4	4	no	—
72	C₇₉⋊C₈	—	8	2	no	—
73	C₇₉⋊C₈	—	8	1	no	—
74	C₂²×C₇₉	—	2	1	no	—
75	C₂²×C₇₉	—	2	1	no	—
76	C₂²×C₇₉	#399	1	1	yes	characteristic
77	C₂²×C₇₉	#475	1	2	yes	—
78	C₄×C₇₉	—	2	2	no	—
79	C₄×C₇₉	—	2	2	no	—
80	C₄×C₇₉	#483	1	2	yes	—
81	C₄×C₇₉	#492	1	1	yes	characteristic
82	C₄×C₇₉	#497	1	1	yes	characteristic
83	Dic₇₉	—	4	2	no	—
84	Dic₇₉	—	8	1	no	—
85	#10663	—	79	1	no	maximal, Sylow
86	C₂×C₇₉	—	2	2	no	—
87	C₂×C₇₉	#2091	1	1	yes	characteristic
88	C₂×C₇₉	#2096	1	1	yes	characteristic
89	C₂×C₇₉	#2098	1	1	yes	characteristic
90	#1985	—	79	1	no	—
91	#1849	—	79	1	no	—
92	#1570	—	79	2	no	—
93	#1487	—	79	2	no	—
94	#1275	D₇₉	1	1	yes	characteristic
95	C₇₉	#10663	1	1	yes	characteristic, complemented, Sylow
96	C₂×C₄×Q₈	—	79	1	no	—
97	#431	C₂×D₇₉	1	2	yes	—
98	C₂×#108	—	158	2	no	—
99	#423	—	79	2	no	—
100	C₂×C₄⋊C₈	—	79	1	no	—
101	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
102	#336	—	158	2	no	—
103	#334	—	158	4	no	—
104	Q₈⋊C₈	—	158	4	no	—
105	C₂²×Q₈	—	79	1	no	—
106	C₂×M₁₆	—	158	2	no	—
107	C₂²×C₈	—	158	1	no	—
108	C₂²×C₈	D₃₁₆	1	2	yes	—
109	C₄×Q₈	—	158	2	no	—
110	C₄×Q₈	—	158	2	no	—
111	C₂×C₄⋊C₄	—	79	1	no	—
112	C₂×C₄⋊C₄	—	158	1	no	—
113	C₂×C₄²	—	158	1	no	—
114	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
115	#108	—	316	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	158	2	no	—
118	C₄⋊C₈	—	158	2	no	—
119	C₄⋊C₈	—	158	2	no	—
120	C₂²⋊C₈	—	2	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	D₃₁₆	1	4	yes	—
123	C₂×Q₈	—	316	2	no	—
124	C₂×Q₈	—	158	2	no	—
125	C₂²×C₄	—	158	1	no	—
126	C₂²×C₄	—	158	1	no	—
127	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
128	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
129	C₂²×C₄	#11929623	1	1	yes	characteristic
130	M₁₆	—	316	4	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	316	1	no	—
135	C₂×C₈	—	316	2	no	—
136	C₂×C₈	—	316	2	no	—
137	C₂×C₈	#11929613	1	4	yes	—
138	C₄⋊C₄	—	316	2	no	—
139	C₄⋊C₄	—	158	2	no	—
140	C₄²	—	316	1	no	—
141	C₄²	—	316	1	no	—
142	C₄²	C₂×D₃₁₆	1	2	yes	—
143	C₄²	#11929623	1	2	yes	—
144	C₂³	#14579141	1	1	yes	characteristic
145	Q₈	—	316	2	no	—
146	Q₈	—	158	4	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	316	1	no	—
150	C₂×C₄	—	316	1	no	—
151	C₂×C₄	—	316	2	no	—
152	C₂×C₄	—	316	1	no	—
153	C₂×C₄	—	158	2	no	—
154	C₂×C₄	#14579141	1	2	yes	—
155	C₂×C₄	#14579145	1	2	yes	—
156	C₂×C₄	#14579145	1	1	yes	characteristic
157	C₂×C₄	#14579158	1	1	yes	characteristic
158	C₂×C₄	C₂×#11929613	1	2	yes	—
159	C₂×C₄	#14579194	1	2	yes	—
160	C₈	—	2	4	no	—
161	C₈	—	4	4	no	—
162	C₈	—	632	2	no	—
163	C₈	—	632	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#19098500	1	1	yes	characteristic
167	C₂²	#19098849	1	2	yes	—
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	316	2	no	—
171	C₄	—	632	1	no	—
172	C₄	#19098894	1	2	yes	—
173	C₄	#19098936	1	1	yes	characteristic
174	C₄	#19098957	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	#23764917	1	1	yes	characteristic
177	C₂	#23764942	1	1	yes	characteristic
178	C₂	#23764954	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23764340

C₂

yes

maximal, characteristic

#23763575

C₂

yes

maximal, characteristic

#23762171

C₂

yes

maximal

#23761830

C₂

yes

maximal

C₇₉×#1275

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₅₈

C₂²

yes

characteristic

C₂×#14579097

—

#19098587

C₂²

yes

—

C₂×C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

—

—

#431×C₇₉

C₂²

yes

—

C₂×#97×C₇₉

C₂²

yes

characteristic

C₂²×Dic₁₅₈

Q₈

yes

characteristic

C₂×#11929608

—

C₂²×C₇₉⋊C₈

—

C₄×Dic₁₅₈

—

C₄×Dic₁₅₈

—

C₂×C₄⋊Dic₇₉

Q₈

yes

characteristic

C₂×Dic₇₉⋊C₄

—

C₂×C₄×Dic₇₉

—

#14579097

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

#97×C₇₉

—

C₄×C₈×C₇₉

D₄

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#399

yes

characteristic

C₂²×C₇₉

#475

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#483

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

#10663

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

—

—

—

D₇₉

yes

characteristic

C₇₉

#10663

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

#431

C₂×D₇₉

yes

—

C₂×#108

—

158

—

#423

—

100

C₂×C₄⋊C₈

—

101

C₂×#97

C₂×D₇₉

yes

characteristic

102

#336

—

158

—

103

#334

—

158

—

104

Q₈⋊C₈

—

158

—

105

C₂²×Q₈

—

106

C₂×M₁₆

—

158

—

107

C₂²×C₈

—

158

—

108

C₂²×C₈

D₃₁₆

yes

—

109

C₄×Q₈

—

158

—

110

C₄×Q₈

—

158

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

158

—

113

C₂×C₄²

—

158

—

114

C₂×C₄²

C₂²×D₇₉

yes

characteristic

115

#108

—

316

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

158

—

118

C₄⋊C₈

—

158

—

119

C₄⋊C₈

—

158

—

120

C₂²⋊C₈

—

121

#97

—

122

C₄×C₈

D₃₁₆

yes

—

123

C₂×Q₈

—

316

—

124

C₂×Q₈

—

158

—

125

C₂²×C₄

—

158

—

126

C₂²×C₄

—

158

—

127

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

128

C₂²×C₄

Q₈×D₇₉

yes

characteristic

129

C₂²×C₄

#11929623

yes

characteristic

130

M₁₆

—

316

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

316

—

135

C₂×C₈

—

316

—

136

C₂×C₈

—

316

—

137

C₂×C₈

#11929613

yes

—

138

C₄⋊C₄

—

316

—

139

C₄⋊C₄

—

158

—

140

C₄²

—

316

—

141

C₄²

—

316

—

142

C₄²

C₂×D₃₁₆

yes

—

143

C₄²

#11929623

yes

—

144

C₂³

#14579141

yes

characteristic

145

Q₈

—

316

—

146

Q₈

—

158

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

316

—

150

C₂×C₄

—

316

—

151

C₂×C₄

—

316

—

152

C₂×C₄

—

316

—

153

C₂×C₄

—

158

—

154

C₂×C₄

#14579141

yes

—

155

C₂×C₄

#14579145

yes

—

156

C₂×C₄

#14579145

yes

characteristic

157

C₂×C₄

#14579158

yes

characteristic

158

C₂×C₄

C₂×#11929613

yes

—

159

C₂×C₄

#14579194

yes

—

160

C₈

—

161

C₈

—

162

C₈

—

632

—

163

C₈

—

632

—

164

C₂²

—

165

C₂²

—

166

C₂²

#19098500

yes

characteristic

167

C₂²

#19098849

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

316

—

171

C₄

—

632

—

172

C₄

#19098894

yes

—

173

C₄

#19098936

yes

characteristic

174

C₄

#19098957

yes

characteristic

175

C₂

—

176

C₂

#23764917

yes

characteristic

177

C₂

#23764942

yes

characteristic

178

C₂

#23764954

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579158
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1275	C₂
socle	C₂²×C₇₉	#399
prime radical	C₂³×C₇₉	C₂²⋊Q₈
hypercenter	#1275	D₇₉
hypocenter	C₇₉	#10663
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

subgroup

G/H

center

C₂×C₄

#14579158

commutator, cosocle

C₂×C₄²×C₇₉

C₂³

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1275

C₂

socle

C₂²×C₇₉

#399

prime radical

C₂³×C₇₉

C₂²⋊Q₈

D₇₉

C₇₉

G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

N	H ≅ G/N	type
95. C₇₉	#10663	semidirect

H ≅ G/N

type

95. C₇₉

#10663

semidirect

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

subgroup	index
#23764340	2
#23763575	2
#23762171	2
#23761830	2
C₇₉×#1275	2
#10663	79

subgroup

index

C₇₉×#1275

Subgroups — #34193261

Conjugacy classes of subgroups

Complements

Maximal subgroups