Group #34208164 — subgroups

Subgroups — #34208164

Order 20224.

276 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 87 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	#23765195	C₂	1	1	yes	maximal, characteristic, complemented
2	#23763454	C₂	1	2	yes	maximal
3	#23761502	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761496	C₂	1	2	yes	maximal, complemented
5	C₇₉×#2113	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579097	C₂²	1	2	yes	—
7	#19098601	C₂²	1	2	yes	—
8	#19098580	—	2	2	no	—
9	#19098393	—	2	2	no	—
10	#19098335	—	2	2	no	—
11	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	C₂²⋊C₁₆×C₇₉	—	2	2	no	—
15	C₂×#11929608	—	2	2	no	—
16	#14579120	—	2	2	no	—
17	#14579097	—	2	2	no	—
18	#14579097	—	2	2	no	—
19	#14579095	—	2	2	no	—
20	#14579093	—	2	2	no	—
21	#14579088	—	4	4	no	—
22	C₂×#41×C₇₉	D₄	1	1	yes	characteristic
23	#131×C₇₉	—	2	2	no	—
24	#131×C₇₉	—	2	2	no	—
25	C₂×M₁₆×C₇₉	—	2	1	no	—
26	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₉	—	2	1	no	—
28	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	M₃₂×C₇₉	—	2	2	no	—
30	C₂×C₁₆×C₇₉	—	2	1	no	—
31	C₂×C₁₆×C₇₉	—	2	2	no	—
32	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
33	#11929608	—	4	2	no	—
34	#11929608	—	4	2	no	—
35	C₂×C₇₉⋊C₈	—	4	2	no	—
36	C₄⋊Dic₇₉	—	4	2	no	—
37	C₂²⋊Dic₇₉	—	4	2	no	—
38	C₄×Dic₇₉	—	4	2	no	—
39	#41×C₇₉	—	2	2	no	—
40	#41×C₇₉	—	2	2	no	—
41	C₂×Q₈×C₇₉	Q₁₆	1	1	yes	characteristic
42	C₂×D₄×C₇₉	—	2	1	no	—
43	C₂×D₄×C₇₉	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₇₉	—	2	1	no	—
45	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
46	M₁₆×C₇₉	—	2	2	no	—
47	M₁₆×C₇₉	—	4	2	no	—
48	C₂×C₈×C₇₉	—	2	1	no	—
49	C₂×C₈×C₇₉	—	4	1	no	—
50	C₂×C₈×C₇₉	—	4	1	no	—
51	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
53	C₂×C₈×C₇₉	#41	1	2	yes	—
54	C₂×C₈×C₇₉	#41	1	2	yes	—
55	C₁₆×C₇₉	—	2	2	no	—
56	C₁₆×C₇₉	—	4	1	no	—
57	C₂³×C₇₉	—	2	1	no	—
58	C₂³×C₇₉	#135	1	1	yes	characteristic
59	Q₈×C₇₉	—	2	2	no	—
60	D₄×C₇₉	—	2	2	no	—
61	D₄×C₇₉	—	4	2	no	—
62	C₂×C₄×C₇₉	—	2	1	no	—
63	C₂×C₄×C₇₉	—	4	1	no	—
64	C₂×C₄×C₇₉	—	4	1	no	—
65	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
66	C₂×C₄×C₇₉	#123	1	2	yes	—
67	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×Dic₇₉	—	8	2	no	—
70	C₈×C₇₉	—	4	1	no	—
71	C₈×C₇₉	#123	1	1	yes	characteristic
72	C₈×C₇₉	#124	1	1	yes	characteristic
73	C₈×C₇₉	#126	1	2	yes	—
74	C₇₉⋊C₈	—	8	2	no	—
75	C₂²×C₇₉	—	4	1	no	—
76	C₂²×C₇₉	—	4	2	no	—
77	C₂²×C₇₉	—	4	1	no	—
78	C₂²×C₇₉	#467	1	1	yes	characteristic
79	C₂²×C₇₉	#482	1	2	yes	—
80	C₄×C₇₉	—	4	1	no	—
81	C₄×C₇₉	#397	1	1	yes	characteristic
82	C₄×C₇₉	#469	1	1	yes	characteristic
83	C₄×C₇₉	#484	1	2	yes	—
84	Dic₇₉	—	16	2	no	—
85	#13499	—	79	1	no	maximal, Sylow
86	C₂×C₇₉	—	2	2	no	—
87	C₂×C₇₉	—	2	1	no	—
88	C₂×C₇₉	—	8	1	no	—
89	C₂×C₇₉	#2076	1	1	yes	characteristic
90	#2161	—	79	1	no	—
91	#2113	D₇₉	1	1	yes	characteristic
92	#1814	—	79	2	no	—
93	#1384	—	79	1	no	—
94	#1382	—	79	2	no	—
95	C₇₉	#13499	1	1	yes	characteristic, complemented, Sylow
96	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
97	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
98	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
99	C₂×#108	—	79	2	no	—
100	#427	—	79	2	no	—
101	#421	—	158	2	no	—
102	#368	—	158	2	no	—
103	C₂²⋊C₁₆	—	2	2	no	—
104	#344	—	158	2	no	—
105	C₂×#41	D₃₁₆	1	1	yes	characteristic
106	#131	—	2	2	no	—
107	#131	—	2	2	no	—
108	C₂×M₁₆	—	2	1	no	—
109	C₂×M₁₆	—	158	2	no	—
110	C₂×M₁₆	D₃₁₆	1	1	yes	characteristic
111	C₂²×C₈	—	2	1	no	—
112	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
113	#117	—	158	2	no	—
114	M₃₂	—	2	2	no	—
115	C₂×C₁₆	—	2	1	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
118	#108	—	158	2	no	—
119	#108	—	158	2	no	—
120	#107	—	158	2	no	—
121	#106	—	158	2	no	—
122	#104	—	316	4	no	—
123	#41	—	2	2	no	—
124	#41	—	2	2	no	—
125	C₂×Q₈	Dic₃₁₆	1	1	yes	characteristic
126	C₂×D₄	—	2	1	no	—
127	C₂×D₄	Dic₃₁₆	1	1	yes	characteristic
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
130	M₁₆	—	2	2	no	—
131	M₁₆	—	4	2	no	—
132	M₁₆	—	316	2	no	—
133	M₁₆	—	316	2	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	—	316	2	no	—
138	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
139	C₂×C₈	#11929623	1	2	yes	—
140	C₂×C₈	#11929623	1	2	yes	—
141	C₂×C₈	#11929624	1	1	yes	characteristic
142	C₄⋊C₄	—	316	2	no	—
143	C₂²⋊C₄	—	316	2	no	—
144	C₄²	—	316	2	no	—
145	C₁₆	—	2	2	no	—
146	C₁₆	—	4	1	no	—
147	C₂³	—	2	1	no	—
148	C₂³	#14579183	1	1	yes	characteristic
149	Q₈	—	2	2	no	—
150	D₄	—	2	2	no	—
151	D₄	—	4	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	632	2	no	—
156	C₂×C₄	#14579135	1	1	yes	characteristic
157	C₂×C₄	#14579145	1	2	yes	—
158	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
159	C₂×C₄	#14579189	1	2	yes	—
160	C₈	—	4	1	no	—
161	C₈	—	632	2	no	—
162	C₈	#14579142	1	1	yes	characteristic
163	C₈	#14579148	1	1	yes	characteristic
164	C₈	#14579154	1	2	yes	—
165	C₂²	—	4	1	no	—
166	C₂²	—	4	2	no	—
167	C₂²	—	4	1	no	—
168	C₂²	#19098804	1	1	yes	characteristic
169	C₂²	#19098888	1	2	yes	—
170	C₄	—	4	1	no	—
171	C₄	—	1264	2	no	—
172	C₄	#19098488	1	1	yes	characteristic
173	C₄	#19098816	1	1	yes	characteristic
174	C₄	#19098899	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	—	2	1	no	—
177	C₂	—	8	1	no	—
178	C₂	#23764828	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765195

C₂

yes

maximal, characteristic, complemented

#23763454

C₂

yes

maximal

#23761502

C₂

yes

maximal, characteristic, complemented

#23761496

C₂

yes

maximal, complemented

C₇₉×#2113

C₂

yes

maximal, characteristic

C₂×#14579097

C₂²

yes

—

#19098601

C₂²

yes

—

#19098580

—

#19098393

—

#19098335

—

C₂×#131×C₇₉

C₂²

yes

characteristic

C₂×M₃₂×C₇₉

C₂²

yes

characteristic

C₂²×C₁₆×C₇₉

C₂²

yes

characteristic

C₂²⋊C₁₆×C₇₉

—

—

—

—

—

—

—

—

C₂×#41×C₇₉

D₄

yes

characteristic

#131×C₇₉

—

#131×C₇₉

—

C₂×M₁₆×C₇₉

—

C₂×M₁₆×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

#41×C₇₉

—

#41×C₇₉

—

C₂×Q₈×C₇₉

Q₁₆

yes

characteristic

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

Q₁₆

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

C₂×C₈×C₇₉

yes

C₂×C₈×C₇₉

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

#135

yes

characteristic

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₈×C₇₉

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#467

yes

characteristic

C₂²×C₇₉

#482

yes

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

#13499

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

D₇₉

yes

—

—

—

C₇₉

#13499

yes

characteristic, complemented, Sylow

C₂×#131

C₂×D₇₉

yes

characteristic

C₂×M₃₂

C₂×D₇₉

yes

characteristic

C₂²×C₁₆

C₂×D₇₉

yes

characteristic

C₂×#108

—

100

#427

—

101

#421

—

158

—

102

#368

—

158

—

103

C₂²⋊C₁₆

—

104

#344

—

158

—

105

C₂×#41

D₃₁₆

yes

characteristic

106

#131

—

107

#131

—

108

C₂×M₁₆

—

109

C₂×M₁₆

—

158

—

110

C₂×M₁₆

D₃₁₆

yes

characteristic

111

C₂²×C₈

—

112

C₂²×C₈

C₂²×D₇₉

yes

characteristic

113

#117

—

158

—

114

M₃₂

—

115

C₂×C₁₆

—

116

C₂×C₁₆

—

117

C₂×C₁₆

C₂²⋊D₇₉

yes

—

118

#108

—

158

—

119

#108

—

158

—

120

#107

—

158

—

121

#106

—

158

—

122

#104

—

316

—

123

#41

—

124

#41

—

125

C₂×Q₈

Dic₃₁₆

yes

characteristic

126

C₂×D₄

—

127

C₂×D₄

Dic₃₁₆

yes

characteristic

128

C₂²×C₄

—

129

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

130

M₁₆

—

131

M₁₆

—

132

M₁₆

—

316

—

133

M₁₆

—

316

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

316

—

138

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

139

C₂×C₈

#11929623

yes

—

140

C₂×C₈

#11929623

yes

—

141

C₂×C₈

#11929624

yes

characteristic

142

C₄⋊C₄

—

316

—

143

C₂²⋊C₄

—

316

—

144

C₄²

—

316

—

145

C₁₆

—

146

C₁₆

—

147

C₂³

—

148

C₂³

#14579183

yes

characteristic

149

Q₈

—

150

D₄

—

151

D₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

632

—

156

C₂×C₄

#14579135

yes

characteristic

157

C₂×C₄

#14579145

yes

—

158

C₂×C₄

C₂×Dic₃₁₆

yes

characteristic

159

C₂×C₄

#14579189

yes

—

160

C₈

—

161

C₈

—

632

—

162

C₈

#14579142

yes

characteristic

163

C₈

#14579148

yes

characteristic

164

C₈

#14579154

yes

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

#19098804

yes

characteristic

169

C₂²

#19098888

yes

—

170

C₄

—

171

C₄

—

1264

—

172

C₄

#19098488

yes

characteristic

173

C₄

#19098816

yes

characteristic

174

C₄

#19098899

yes

—

175

C₂

—

176

C₂

—

177

C₂

—

178

C₂

#23764828

yes

characteristic

	subgroup	G/H
center	C₄	#19098816
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2113	C₂
socle	C₂×C₇₉	#2076
prime radical	C₂×#41×C₇₉	D₄
hypercenter	#2113	D₇₉
hypocenter	C₇₉	#13499
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

subgroup

G/H

center

C₄

#19098816

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#2113

C₂

socle

C₂×C₇₉

#2076

prime radical

C₂×#41×C₇₉

D₄

D₇₉

C₇₉

G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

N	H ≅ G/N	type
1. #23765195	C₂	semidirect
3. #23761502	C₂	semidirect
4. #23761496	C₂	semidirect
95. C₇₉	#13499	semidirect

H ≅ G/N

type

1. #23765195

C₂

semidirect

3. #23761502

C₂

semidirect

4. #23761496

C₂

semidirect

95. C₇₉

#13499

semidirect

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

subgroup	index
#23765195	2
#23763454	2
#23761502	2
#23761496	2
C₇₉×#2113	2
#13499	79

subgroup

index

C₇₉×#2113

Subgroups — #34208164

Conjugacy classes of subgroups

Complements

Maximal subgroups