Group #34208228 — subgroups

Subgroups — #34208228

Order 20224.

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

normal subgroups only

This group has 268 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	#23765188	C₂	1	2	yes	maximal
2	#23764148	C₂	1	1	yes	maximal, characteristic
3	#23763564	C₂	1	1	yes	maximal, characteristic
4	#23761509	C₂	1	2	yes	maximal
5	C₇₉×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579097	C₂²	1	1	yes	characteristic
7	#19098601	C₂²	1	1	yes	characteristic
8	C₂×#14579095	C₂²	1	2	yes	—
9	C₂×#14579093	—	2	1	no	—
10	#19098589	—	2	1	no	—
11	#19098429	—	2	1	no	—
12	#19098391	—	2	2	no	—
13	#19098389	—	2	2	no	—
14	#19098323	—	2	1	no	—
15	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#11929608	—	2	1	no	—
18	#14579120	—	2	1	no	—
19	C₂×C₄⋊Dic₇₉	—	2	2	no	—
20	#14579097	—	2	2	no	—
21	#14579095	—	2	2	no	—
22	#14579095	—	2	2	no	—
23	#14579095	D₄	1	2	yes	—
24	#14579093	—	2	1	no	—
25	#14579093	—	4	2	no	—
26	Dic₇₉⋊C₈	—	4	2	no	—
27	C₇₉⋊#95	—	4	2	no	—
28	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
31	M₃₂×C₇₉	—	2	2	no	—
32	M₃₂×C₇₉	Q₈	1	4	yes	—
33	C₂×C₁₆×C₇₉	—	2	2	no	—
34	C₄×C₈×C₇₉	—	2	2	no	—
35	C₂²×Dic₇₉	—	4	2	no	—
36	#11929608	—	4	1	no	—
37	#11929608	—	4	1	no	—
38	C₂×C₇₉⋊C₈	—	4	1	no	—
39	C₄⋊Dic₇₉	—	2	2	no	—
40	C₄⋊Dic₇₉	—	4	2	no	—
41	C₄⋊Dic₇₉	—	4	2	no	—
42	C₂²⋊Dic₇₉	—	4	1	no	—
43	C₄×Dic₇₉	—	4	1	no	—
44	C₂²×C₄×C₇₉	—	2	1	no	—
45	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₇₉	—	2	1	no	—
47	C₂×C₈×C₇₉	—	2	1	no	—
48	C₂×C₈×C₇₉	D₈	1	2	yes	—
49	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
52	C₂×C₈×C₇₉	#41	1	2	yes	—
53	C₄²×C₇₉	—	2	1	no	—
54	C₄²×C₇₉	Q₁₆	1	2	yes	—
55	C₁₆×C₇₉	—	2	4	no	—
56	C₂³×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
57	C₂×C₄×C₇₉	—	2	2	no	—
58	C₂×C₄×C₇₉	—	4	2	no	—
59	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
60	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
65	C₂×Dic₇₉	—	4	2	no	—
66	C₂×Dic₇₉	—	8	4	no	—
67	C₂×Dic₇₉	—	8	2	no	—
68	C₈×C₇₉	—	2	2	no	—
69	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
70	C₈×C₇₉	#125	1	2	yes	—
71	C₇₉⋊C₈	—	8	1	no	—
72	C₂²×C₇₉	—	2	2	no	—
73	C₂²×C₇₉	#459	1	1	yes	characteristic
74	C₂²×C₇₉	#484	1	1	yes	characteristic
75	C₂²×C₇₉	#501	1	1	yes	characteristic
76	C₄×C₇₉	—	4	2	no	—
77	C₄×C₇₉	#398	1	1	yes	characteristic
78	C₄×C₇₉	#462	1	1	yes	characteristic
79	C₄×C₇₉	#480	1	1	yes	characteristic
80	C₄×C₇₉	#500	1	1	yes	characteristic
81	Dic₇₉	—	8	2	no	—
82	Dic₇₉	—	16	2	no	—
83	#13509	—	79	1	no	maximal, Sylow
84	C₂×C₇₉	—	2	1	no	—
85	C₂×C₇₉	—	2	1	no	—
86	C₂×C₇₉	#2083	1	1	yes	characteristic
87	C₂×C₇₉	#2212	1	2	yes	—
88	#2159	—	79	2	no	—
89	#1945	—	79	1	no	—
90	#1846	—	79	1	no	—
91	#1387	—	79	2	no	—
92	#1376	D₇₉	1	1	yes	characteristic
93	C₇₉	#13509	1	1	yes	characteristic, complemented, Sylow
94	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
95	C₂×#108	—	79	1	no	—
96	#427	—	79	1	no	—
97	C₂×#107	—	79	2	no	—
98	C₂×#106	—	158	1	no	—
99	#424	—	158	1	no	—
100	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
101	#382	—	158	1	no	—
102	#367	—	158	2	no	—
103	#366	—	158	2	no	—
104	#339	—	158	1	no	—
105	C₂×M₁₆	—	158	1	no	—
106	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
107	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
108	#117	—	158	1	no	—
109	C₂×C₄⋊C₄	—	158	2	no	—
110	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
111	M₃₂	—	2	2	no	—
112	M₃₂	Dic₁₅₈	1	4	yes	—
113	C₂×C₁₆	—	2	2	no	—
114	#108	—	158	2	no	—
115	#107	—	79	2	no	—
116	#107	—	158	2	no	—
117	#107	—	158	2	no	—
118	#106	—	316	2	no	—
119	#106	—	158	1	no	—
120	C₄⋊C₈	—	316	2	no	—
121	C₄×C₈	—	2	2	no	—
122	#95	—	316	2	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	316	2	no	—
125	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
126	M₁₆	—	316	1	no	—
127	M₁₆	—	316	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	316	1	no	—
131	C₂×C₈	D₆₃₂	1	2	yes	—
132	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
133	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
134	C₂×C₈	#11929623	1	2	yes	—
135	C₂×C₈	#11929625	1	1	yes	characteristic
136	C₄⋊C₄	—	316	2	no	—
137	C₄⋊C₄	—	316	2	no	—
138	C₄⋊C₄	—	158	2	no	—
139	C₂²⋊C₄	—	316	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	316	1	no	—
142	C₄²	Dic₃₁₆	1	2	yes	—
143	C₁₆	—	2	4	no	—
144	C₂³	#14579189	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	316	2	no	—
148	C₂×C₄	—	632	4	no	—
149	C₂×C₄	—	632	2	no	—
150	C₂×C₄	#14579136	1	1	yes	characteristic
151	C₂×C₄	#14579145	1	1	yes	characteristic
152	C₂×C₄	#14579161	1	1	yes	characteristic
153	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
154	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
155	C₂×C₄	#14579194	1	1	yes	characteristic
156	C₈	—	2	2	no	—
157	C₈	—	632	1	no	—
158	C₈	#14579138	1	2	yes	—
159	C₈	#14579152	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	#19098759	1	1	yes	characteristic
162	C₂²	#19098899	1	1	yes	characteristic
163	C₂²	#19098972	1	1	yes	characteristic
164	C₄	—	4	2	no	—
165	C₄	—	632	2	no	—
166	C₄	—	1264	2	no	—
167	C₄	#19098495	1	1	yes	characteristic
168	C₄	#19098774	1	1	yes	characteristic
169	C₄	#19098877	1	1	yes	characteristic
170	C₄	#19098968	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#23764873	1	1	yes	characteristic
174	C₂	#23765470	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765188

C₂

yes

maximal

#23764148

C₂

yes

maximal, characteristic

#23763564

C₂

yes

maximal, characteristic

#23761509

C₂

yes

maximal

C₇₉×#1376

C₂

yes

maximal, characteristic

C₂×#14579097

C₂²

yes

characteristic

#19098601

C₂²

yes

characteristic

C₂×#14579095

C₂²

yes

—

—

—

—

—

—

—

C₂×M₃₂×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

#14579120

—

C₂×C₄⋊Dic₇₉

—

—

—

—

D₄

yes

—

#14579093

—

#14579093

—

Dic₇₉⋊C₈

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

M₃₂×C₇₉

—

M₃₂×C₇₉

Q₈

yes

—

C₂×C₁₆×C₇₉

—

C₄×C₈×C₇₉

—

C₂²×Dic₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

—

C₂×C₈×C₇₉

#41

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

Q₁₆

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₉

#137

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂²⋊Q₈

yes

—

C₈×C₇₉

#125

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

#13509

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#2083

yes

characteristic

C₂×C₇₉

#2212

yes

—

—

—

—

—

D₇₉

yes

characteristic

C₇₉

#13509

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₇₉

yes

—

—

—

—

—

158

—

#424

—

158

—

100

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

101

#382

—

158

—

102

#367

—

158

—

103

#366

—

158

—

104

#339

—

158

—

105

C₂×M₁₆

—

158

—

106

C₂²×C₈

D₃₁₆

yes

characteristic

107

C₂²×C₈

C₂²×D₇₉

yes

characteristic

108

#117

—

158

—

109

C₂×C₄⋊C₄

—

158

—

110

C₂×C₄²

D₃₁₆

yes

characteristic

111

M₃₂

—

112

M₃₂

Dic₁₅₈

yes

—

113

C₂×C₁₆

—

114

#108

—

158

—

115

#107

—

116

#107

—

158

—

117

#107

—

158

—

118

#106

—

316

—

119

#106

—

158

—

120

C₄⋊C₈

—

316

—

121

C₄×C₈

—

122

#95

—

316

—

123

C₂²×C₄

—

124

C₂²×C₄

—

316

—

125

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

126

M₁₆

—

316

—

127

M₁₆

—

316

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

316

—

131

C₂×C₈

D₆₃₂

yes

—

132

C₂×C₈

D₄×D₇₉

yes

characteristic

133

C₂×C₈

C₂×Dic₁₅₈

yes

—

134

C₂×C₈

#11929623

yes

—

135

C₂×C₈

#11929625

yes

characteristic

136

C₄⋊C₄

—

316

—

137

C₄⋊C₄

—

316

—

138

C₄⋊C₄

—

158

—

139

C₂²⋊C₄

—

316

—

140

C₄²

—

141

C₄²

—

316

—

142

C₄²

Dic₃₁₆

yes

—

143

C₁₆

—

144

C₂³

#14579189

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

316

—

148

C₂×C₄

—

632

—

149

C₂×C₄

—

632

—

150

C₂×C₄

#14579136

yes

characteristic

151

C₂×C₄

#14579145

yes

characteristic

152

C₂×C₄

#14579161

yes

characteristic

153

C₂×C₄

C₂×D₆₃₂

yes

characteristic

154

C₂×C₄

C₂×Dic₃₁₆

yes

characteristic

155

C₂×C₄

#14579194

yes

characteristic

156

C₈

—

157

C₈

—

632

—

158

C₈

#14579138

yes

—

159

C₈

#14579152

yes

—

160

C₂²

—

161

C₂²

#19098759

yes

characteristic

162

C₂²

#19098899

yes

characteristic

163

C₂²

#19098972

yes

characteristic

164

C₄

—

165

C₄

—

632

—

166

C₄

—

1264

—

167

C₄

#19098495

yes

characteristic

168

C₄

#19098774

yes

characteristic

169

C₄

#19098877

yes

characteristic

170

C₄

#19098968

yes

characteristic

171

C₂

—

172

C₂

—

173

C₂

#23764873

yes

characteristic

174

C₂

#23765470

yes

—

	subgroup	G/H
center	C₂²	#19098759
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1376	C₂
socle	C₂²×C₇₉	#459
prime radical	C₂³×C₇₉	AGL(1,ℤ/8)
hypercenter	#1376	D₇₉
hypocenter	C₇₉	#13509
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂²

#19098759

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1376

C₂

C₂²×C₇₉

C₂³×C₇₉

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

N	H ≅ G/N	type
93. C₇₉	#13509	semidirect

H ≅ G/N

type

93. C₇₉

#13509

semidirect

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

subgroup	index
#23765188	2
#23764148	2
#23763564	2
#23761509	2
C₇₉×#1376	2
#13509	79

subgroup

index

C₇₉×#1376

Subgroups — #34208228

Conjugacy classes of subgroups

Complements

Maximal subgroups