Group #34182058 — subgroups

Subgroups — #34182058

Order 20224.

438 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 227 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	#23763560	C₂	1	1	yes	maximal, characteristic
2	#23763560	C₂	1	2	yes	maximal
3	#23763557	C₂	1	1	yes	maximal, characteristic
4	#23763528	C₂	1	2	yes	maximal
5	C₂×#322×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579095	—	2	4	no	—
7	C₂×#14579095	—	2	2	no	—
8	C₂×#14579093	—	2	2	no	—
9	#19098433	C₂²	1	1	yes	characteristic
10	#19098433	C₂²	1	2	yes	—
11	#19098433	C₂²	1	1	yes	characteristic
12	#19098326	—	2	4	no	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	#322×C₇₉	C₄	1	4	yes	complemented
16	C₂×C₄⋊Dic₇₉	—	2	2	no	—
17	C₂×C₄⋊Dic₇₉	—	2	2	no	—
18	C₂×C₄⋊Dic₇₉	—	2	1	no	—
19	C₂×C₄⋊Dic₇₉	—	2	1	no	—
20	C₂×C₄⋊Dic₇₉	—	2	2	no	—
21	C₂×C₄⋊Dic₇₉	—	2	2	no	—
22	C₂×C₄⋊Dic₇₉	—	2	2	no	—
23	#14579095	—	4	8	no	—
24	#14579095	—	4	4	no	—
25	#14579093	—	4	4	no	—
26	C₂²×C₈×C₇₉	D₄	1	4	yes	—
27	C₂²×C₈×C₇₉	D₄	1	2	yes	—
28	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₇₉	Q₈	1	4	yes	—
32	C₄×C₈×C₇₉	Q₈	1	2	yes	—
33	C₂²×Dic₇₉	—	4	2	no	—
34	C₂²×Dic₇₉	—	4	1	no	—
35	C₂²×Dic₇₉	—	4	1	no	—
36	C₄⋊Dic₇₉	—	4	2	no	—
37	C₄⋊Dic₇₉	—	4	2	no	—
38	C₄⋊Dic₇₉	—	4	4	no	—
39	C₄⋊Dic₇₉	—	4	4	no	—
40	C₄⋊Dic₇₉	—	4	4	no	—
41	C₄⋊Dic₇₉	—	4	4	no	—
42	C₄⋊Dic₇₉	—	4	4	no	—
43	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₇₉	—	2	4	no	—
46	C₂×C₈×C₇₉	—	2	4	no	—
47	C₂×C₈×C₇₉	C₄⋊C₄	1	8	yes	—
48	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
49	C₂×C₈×C₇₉	D₈	1	4	yes	—
50	C₂×C₈×C₇₉	Q₁₆	1	4	yes	—
51	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
54	C₂³×C₇₉	#127	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
59	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
60	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
61	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	4	yes	—
62	C₂×C₄×C₇₉	#137	1	4	yes	—
63	C₂×Dic₇₉	—	8	1	no	—
64	C₂×Dic₇₉	—	8	2	no	—
65	C₂×Dic₇₉	—	8	1	no	—
66	C₂×Dic₇₉	—	8	2	no	—
67	C₂×Dic₇₉	—	8	2	no	—
68	C₂×Dic₇₉	—	8	2	no	—
69	C₂×Dic₇₉	—	8	2	no	—
70	C₈×C₇₉	—	2	8	no	—
71	C₈×C₇₉	#107	1	8	yes	complemented
72	C₂²×C₇₉	#384	1	1	yes	characteristic
73	C₂²×C₇₉	#493	1	1	yes	characteristic
74	C₂²×C₇₉	#494	1	1	yes	characteristic
75	C₂²×C₇₉	#496	1	2	yes	—
76	C₂²×C₇₉	#497	1	2	yes	—
77	C₄×C₇₉	C₂×#107	1	2	yes	—
78	C₄×C₇₉	#428	1	4	yes	—
79	C₄×C₇₉	#500	1	2	yes	—
80	C₄×C₇₉	#501	1	4	yes	—
81	Dic₇₉	—	16	1	no	—
82	Dic₇₉	—	16	2	no	—
83	Dic₇₉	—	16	1	no	—
84	#8820	—	79	1	no	maximal, Sylow
85	C₂×C₇₉	#1721	1	1	yes	characteristic
86	C₂×C₇₉	#1722	1	1	yes	characteristic
87	C₂×C₇₉	#1723	1	2	yes	—
88	C₂×C₇₉	#1845	1	1	yes	characteristic
89	C₂×C₇₉	#1847	1	2	yes	—
90	#1845	—	79	1	no	—
91	#1845	—	79	2	no	—
92	#1844	—	79	1	no	—
93	#1834	—	79	2	no	—
94	C₂×#322	D₇₉	1	1	yes	characteristic
95	C₇₉	#8820	1	1	yes	characteristic, complemented, Sylow
96	C₂×#107	—	158	4	no	—
97	C₂×#107	—	158	2	no	—
98	C₂×#106	—	158	2	no	—
99	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
100	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
101	#384	—	79	1	no	—
102	#384	—	79	2	no	—
103	#384	—	79	1	no	—
104	#340	—	158	4	no	—
105	#322	Dic₇₉	1	4	yes	complemented
106	C₂²×C₈	D₃₁₆	1	4	yes	—
107	C₂²×C₈	D₃₁₆	1	2	yes	—
108	C₂×C₄⋊C₄	—	158	2	no	—
109	C₂×C₄⋊C₄	—	158	2	no	—
110	C₂×C₄⋊C₄	—	158	1	no	—
111	C₂×C₄⋊C₄	—	158	1	no	—
112	C₂×C₄⋊C₄	—	158	2	no	—
113	C₂×C₄⋊C₄	—	158	2	no	—
114	C₂×C₄⋊C₄	—	158	2	no	—
115	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
116	#107	—	316	8	no	—
117	#107	—	316	4	no	—
118	#106	—	316	4	no	—
119	C₄×C₈	Dic₁₅₈	1	4	yes	—
120	C₄×C₈	Dic₁₅₈	1	2	yes	—
121	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
122	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
123	C₂²×C₄	—	316	2	no	—
124	C₂²×C₄	—	316	1	no	—
125	C₂²×C₄	—	316	1	no	—
126	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
127	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	C₄⋊Dic₇₉	1	8	yes	—
131	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
132	C₂×C₈	D₆₃₂	1	4	yes	—
133	C₂×C₈	Dic₃₁₆	1	4	yes	—
134	C₄⋊C₄	—	316	2	no	—
135	C₄⋊C₄	—	316	2	no	—
136	C₄⋊C₄	—	316	4	no	—
137	C₄⋊C₄	—	316	4	no	—
138	C₄⋊C₄	—	316	4	no	—
139	C₄⋊C₄	—	316	4	no	—
140	C₄⋊C₄	—	316	4	no	—
141	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
142	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
143	C₄²	C₂×Dic₁₅₈	1	2	yes	—
144	C₂³	#14579158	1	1	yes	characteristic
145	C₂×C₄	—	632	1	no	—
146	C₂×C₄	—	632	2	no	—
147	C₂×C₄	—	632	1	no	—
148	C₂×C₄	—	632	2	no	—
149	C₂×C₄	—	632	2	no	—
150	C₂×C₄	—	632	2	no	—
151	C₂×C₄	—	632	2	no	—
152	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
154	C₂×C₄	#14579161	1	1	yes	characteristic
155	C₂×C₄	#14579161	1	2	yes	—
156	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
157	C₂×C₄	C₂×Dic₃₁₆	1	2	yes	—
158	C₂×C₄	#14579189	1	4	yes	—
159	C₂×C₄	#14579194	1	4	yes	—
160	C₈	—	2	8	no	—
161	C₈	#14579095	1	8	yes	complemented
162	C₂²	#19098433	1	1	yes	characteristic
163	C₂²	#19098940	1	1	yes	characteristic
164	C₂²	#19098943	1	1	yes	characteristic
165	C₂²	#19098952	1	2	yes	—
166	C₂²	#19098957	1	2	yes	—
167	C₄	—	1264	1	no	—
168	C₄	—	1264	2	no	—
169	C₄	—	1264	1	no	—
170	C₄	C₂×#14579095	1	2	yes	—
171	C₄	#19098605	1	4	yes	—
172	C₄	#19098968	1	2	yes	—
173	C₄	#19098972	1	4	yes	—
174	C₂	#23763066	1	1	yes	characteristic
175	C₂	#23763071	1	1	yes	characteristic
176	C₂	#23763077	1	2	yes	—
177	C₂	#23763560	1	1	yes	characteristic
178	C₂	#23763568	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763560

C₂

yes

maximal, characteristic

#23763560

C₂

yes

maximal

#23763557

C₂

yes

maximal, characteristic

#23763528

C₂

yes

maximal

C₂×#322×C₇₉

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

characteristic

#19098433

C₂²

yes

—

#19098433

C₂²

yes

characteristic

#19098326

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#322×C₇₉

C₄

yes

complemented

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

#14579095

—

#14579095

—

#14579093

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

Q₁₆

yes

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

—

C₂³×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×C₄×C₇₉

C₂×Q₁₆

yes

—

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

#497

yes

—

C₄×C₇₉

C₂×#107

yes

—

C₄×C₇₉

#428

yes

—

C₄×C₇₉

#500

yes

—

C₄×C₇₉

#501

yes

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#1845

yes

characteristic

C₂×C₇₉

#1847

yes

—

—

—

—

—

D₇₉

yes

characteristic

C₇₉

#8820

yes

characteristic, complemented, Sylow

C₂×#107

—

158

—

C₂×#107

—

158

—

C₂×#106

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

—

100

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

101

#384

—

102

#384

—

103

#384

—

104

#340

—

158

—

105

#322

Dic₇₉

yes

complemented

106

C₂²×C₈

D₃₁₆

yes

—

107

C₂²×C₈

D₃₁₆

yes

—

108

C₂×C₄⋊C₄

—

158

—

109

C₂×C₄⋊C₄

—

158

—

110

C₂×C₄⋊C₄

—

158

—

111

C₂×C₄⋊C₄

—

158

—

112

C₂×C₄⋊C₄

—

158

—

113

C₂×C₄⋊C₄

—

158

—

114

C₂×C₄⋊C₄

—

158

—

115

C₂×C₄²

C₂²×D₇₉

yes

characteristic

116

#107

—

316

—

117

#107

—

316

—

118

#106

—

316

—

119

C₄×C₈

Dic₁₅₈

yes

—

120

C₄×C₈

Dic₁₅₈

yes

—

121

C₄×C₈

C₂×Dic₇₉

yes

—

122

C₄×C₈

C₂×Dic₇₉

yes

—

123

C₂²×C₄

—

316

—

124

C₂²×C₄

—

316

—

125

C₂²×C₄

—

316

—

126

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

127

C₂²×C₄

C₂×D₃₁₆

yes

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

C₄⋊Dic₇₉

yes

—

131

C₂×C₈

C₄⋊Dic₇₉

yes

—

132

C₂×C₈

D₆₃₂

yes

—

133

C₂×C₈

Dic₃₁₆

yes

—

134

C₄⋊C₄

—

316

—

135

C₄⋊C₄

—

316

—

136

C₄⋊C₄

—

316

—

137

C₄⋊C₄

—

316

—

138

C₄⋊C₄

—

316

—

139

C₄⋊C₄

—

316

—

140

C₄⋊C₄

—

316

—

141

C₄²

C₂²×Dic₇₉

yes

characteristic

142

C₄²

C₂×Dic₁₅₈

yes

characteristic

143

C₄²

C₂×Dic₁₅₈

yes

—

144

C₂³

#14579158

yes

characteristic

145

C₂×C₄

—

632

—

146

C₂×C₄

—

632

—

147

C₂×C₄

—

632

—

148

C₂×C₄

—

632

—

149

C₂×C₄

—

632

—

150

C₂×C₄

—

632

—

151

C₂×C₄

—

632

—

152

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

153

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

—

154

C₂×C₄

#14579161

yes

characteristic

155

C₂×C₄

#14579161

yes

—

156

C₂×C₄

C₂×D₆₃₂

yes

—

157

C₂×C₄

C₂×Dic₃₁₆

yes

—

158

C₂×C₄

#14579189

yes

—

159

C₂×C₄

#14579194

yes

—

160

C₈

—

161

C₈

#14579095

yes

complemented

162

C₂²

#19098433

yes

characteristic

163

C₂²

#19098940

yes

characteristic

164

C₂²

#19098943

yes

characteristic

165

C₂²

#19098952

yes

—

166

C₂²

#19098957

yes

—

167

C₄

—

1264

—

168

C₄

—

1264

—

169

C₄

—

1264

—

170

C₄

C₂×#14579095

yes

—

171

C₄

#19098605

yes

—

172

C₄

#19098968

yes

—

173

C₄

#19098972

yes

—

174

C₂

#23763066

yes

characteristic

175

C₂

#23763071

yes

characteristic

176

C₂

#23763077

yes

—

177

C₂

#23763560

yes

characteristic

178

C₂

#23763568

yes

—

	subgroup	G/H
center	C₂³	#14579158
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#322×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#127
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#322	D₇₉
hypocenter	C₇₉	#8820
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂³

#14579158

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₂×#322×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

#127

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
15. #322×C₇₉	C₄	semidirect
71. C₈×C₇₉	#107	semidirect
95. C₇₉	#8820	semidirect
105. #322	Dic₇₉	semidirect
161. C₈	#14579095	semidirect

H ≅ G/N

type

15. #322×C₇₉

C₄

semidirect

71. C₈×C₇₉

#107

semidirect

95. C₇₉

#8820

semidirect

105. #322

Dic₇₉

semidirect

161. C₈

#14579095

semidirect

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

subgroup	index
#23763560	2
#23763557	2
#23763528	2
C₂×#322×C₇₉	2
#8820	79

subgroup

index

C₂×#322×C₇₉

Subgroups — #34182058

Conjugacy classes of subgroups

Complements

Maximal subgroups