Group #23776373 — subgroups

Subgroups — #23776373

Order 10112.

460 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 125 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	#19099233	C₂	1	2	yes	maximal, complemented
2	#19098938	C₂	1	2	yes	maximal, complemented
3	#19098756	C₂	1	4	yes	maximal, complemented
4	#19098755	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₆₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×Dic₇₉	C₂	1	2	yes	maximal, complemented
8	#493×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11929623	—	2	2	no	—
10	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
11	C₂×#11929609	C₂²	1	4	yes	—
12	#14579157	C₂²	1	4	yes	—
13	#14579133	—	2	4	no	—
14	#14579132	—	2	2	no	—
15	C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
16	C₄×D₃₁₆	C₂²	1	2	yes	—
17	D₄×Dic₇₉	C₂²	1	4	yes	—
18	#14579095	C₂²	1	1	yes	characteristic
19	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₇₉⋊C₈	C₂²	1	2	yes	—
21	#14579083	C₂²	1	2	yes	—
22	#14579082	C₂²	1	4	yes	—
23	C₈×Dic₇₉	C₂²	1	2	yes	—
24	C₂×D₈×C₇₉	C₂²	1	2	yes	—
25	C₂×D₈×C₇₉	C₂²	1	2	yes	—
26	#127×C₇₉	C₂²	1	2	yes	—
27	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
28	#11929623	—	4	8	no	—
29	C₂×Dic₁₅₈	D₄	1	1	yes	characteristic
30	C₂×D₃₁₆	C₂³	1	2	yes	—
31	C₂×C₂²⋊D₇₉	—	2	4	no	—
32	C₂×C₂²⋊D₇₉	—	2	4	no	—
33	C₂²×Dic₇₉	—	2	4	no	—
34	C₂×C₄×D₇₉	—	2	2	no	—
35	D₆₃₂	—	2	2	no	—
36	#11929609	—	4	4	no	—
37	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
38	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
39	C₄⋊Dic₇₉	C₂³	1	2	yes	—
40	Dic₇₉⋊C₄	D₄	1	2	yes	—
41	C₂²⋊Dic₇₉	—	2	4	no	—
42	(C₂×D₇₉)⋊C₄	—	2	2	no	—
43	C₄×Dic₇₉	C₂³	1	2	yes	—
44	C₂×D₄×C₇₉	—	2	2	no	—
45	C₂×D₄×C₇₉	—	2	2	no	—
46	C₂×D₄×C₇₉	C₂³	1	4	yes	—
47	D₈×C₇₉	—	2	4	no	—
48	D₈×C₇₉	—	4	2	no	—
49	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
51	C₄²×C₇₉	C₂³	1	1	yes	characteristic
52	C₂³×C₇₉	—	2	4	no	—
53	C₂²×D₇₉	—	2	2	no	—
54	D₄×C₇₉	—	2	4	no	—
55	D₄×C₇₉	—	2	4	no	—
56	D₄×C₇₉	—	4	4	no	—
57	D₄×C₇₉	—	8	2	no	—
58	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
61	C₂²⋊D₇₉	—	4	8	no	—
62	C₂²⋊D₇₉	—	8	4	no	—
63	C₂×Dic₇₉	—	2	2	no	—
64	C₂×Dic₇₉	—	4	8	no	—
65	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
66	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
67	D₃₁₆	—	2	2	no	—
68	D₃₁₆	—	2	2	no	—
69	C₄×D₇₉	—	4	4	no	—
70	Dic₁₅₈	D₈	1	4	yes	complemented
71	C₈×C₇₉	—	2	1	no	—
72	C₈×C₇₉	#41	1	2	yes	complemented
73	C₇₉⋊C₈	—	2	2	no	—
74	C₂²×C₇₉	—	4	4	no	—
75	C₂²×C₇₉	—	4	4	no	—
76	C₂²×C₇₉	—	4	4	no	—
77	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
78	C₂×D₇₉	—	4	2	no	—
79	C₂×D₇₉	—	4	2	no	—
80	C₂×D₇₉	—	4	2	no	—
81	C₄×C₇₉	—	2	1	no	—
82	C₄×C₇₉	C₂×D₈	1	2	yes	—
83	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
84	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
85	Dic₇₉	—	2	2	no	—
86	Dic₇₉	—	4	2	no	—
87	Dic₇₉	C₂×D₈	1	4	yes	—
88	C₂×C₇₉	—	8	4	no	—
89	C₂×C₇₉	#550	1	1	yes	characteristic
90	C₂×C₇₉	C₂²×D₈	1	1	yes	characteristic
91	C₂×C₇₉	#576	1	1	yes	characteristic
92	D₇₉	—	8	2	no	—
93	#3391	—	79	1	no	maximal, Sylow
94	C₇₉	#3391	1	1	yes	characteristic, complemented, Sylow
95	#550	—	79	2	no	—
96	#493	—	79	2	no	—
97	#493	D₇₉	1	1	yes	characteristic, complemented
98	#459	—	79	4	no	—
99	#459	—	79	2	no	—
100	C₈×Q₈	—	79	1	no	—
101	C₄×D₈	—	79	2	no	—
102	C₄×D₈	—	79	1	no	—
103	C₂×#41	—	158	2	no	—
104	C₂×D₈	—	79	4	no	—
105	C₂×D₈	—	79	1	no	—
106	C₂×D₈	C₂×D₇₉	1	2	yes	—
107	C₂×D₈	C₂×D₇₉	1	2	yes	—
108	#127	—	79	4	no	—
109	#127	C₂×D₇₉	1	2	yes	—
110	#121	—	158	2	no	—
111	#121	—	158	4	no	—
112	C₄×Q₈	—	79	1	no	—
113	C₄×D₄	—	79	4	no	—
114	C₄×D₄	—	79	2	no	—
115	#107	—	79	1	no	—
116	C₄⋊C₈	—	79	2	no	—
117	C₄⋊C₈	—	79	1	no	—
118	#102	—	79	4	no	—
119	#102	—	79	2	no	—
120	C₄×C₈	—	79	2	no	—
121	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
122	#41	—	316	8	no	—
123	C₂×Q₈	—	79	1	no	—
124	C₂×D₄	—	2	2	no	—
125	C₂×D₄	—	2	2	no	—
126	C₂×D₄	—	79	2	no	—
127	C₂×D₄	—	158	4	no	—
128	C₂×D₄	—	158	4	no	—
129	C₂×D₄	C₂²×D₇₉	1	4	yes	—
130	C₂²×C₄	—	158	4	no	—
131	C₂²×C₄	—	158	2	no	—
132	D₈	—	2	4	no	—
133	D₈	—	4	2	no	—
134	D₈	—	316	4	no	—
135	D₈	—	158	2	no	—
136	C₂×C₈	—	79	2	no	—
137	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
138	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
139	C₄⋊C₄	—	79	2	no	—
140	C₄⋊C₄	—	79	1	no	—
141	C₄⋊C₄	—	79	2	no	—
142	C₂²⋊C₄	—	158	4	no	—
143	C₂²⋊C₄	—	158	2	no	—
144	C₄²	—	79	2	no	—
145	C₄²	C₂²×D₇₉	1	1	yes	characteristic
146	C₂³	—	2	4	no	—
147	C₂³	—	158	2	no	—
148	Q₈	—	79	4	no	—
149	D₄	—	2	4	no	—
150	D₄	—	2	4	no	—
151	D₄	—	4	4	no	—
152	D₄	—	8	2	no	—
153	D₄	—	316	8	no	—
154	D₄	—	632	4	no	—
155	D₄	—	158	2	no	—
156	D₄	—	158	2	no	—
157	C₂×C₄	—	316	8	no	—
158	C₂×C₄	—	316	4	no	—
159	C₂×C₄	—	79	2	no	—
160	C₂×C₄	—	79	2	no	—
161	C₂×C₄	—	158	2	no	—
162	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
163	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
164	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	158	2	no	—
167	C₈	#11929623	1	2	yes	complemented
168	C₂²	—	4	4	no	—
169	C₂²	—	4	4	no	—
170	C₂²	—	4	4	no	—
171	C₂²	—	316	2	no	—
172	C₂²	—	316	2	no	—
173	C₂²	—	316	2	no	—
174	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
175	C₄	—	2	1	no	—
176	C₄	—	316	2	no	—
177	C₄	—	79	4	no	—
178	C₄	—	158	2	no	—
179	C₄	D₈×D₇₉	1	2	yes	—
180	C₄	C₂×#11929623	1	1	yes	characteristic
181	C₄	#14579206	1	1	yes	characteristic
182	C₂	—	8	4	no	—
183	C₂	—	632	2	no	—
184	C₂	#19099233	1	1	yes	characteristic
185	C₂	C₂×D₈×D₇₉	1	1	yes	characteristic
186	C₂	#19099364	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099233

C₂

yes

maximal, complemented

#19098938

C₂

yes

maximal, complemented

#19098756

C₂

yes

maximal, complemented

#19098755

C₂

yes

maximal, complemented

C₈×Dic₁₅₈

C₂

yes

maximal, characteristic, complemented

C₄×D₆₃₂

C₂

yes

maximal, characteristic, complemented

D₈×Dic₇₉

C₂

yes

maximal, complemented

#493×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929623

—

C₂×D₆₃₂

C₂²

yes

characteristic

C₂×#11929609

C₂²

yes

—

#14579157

C₂²

yes

—

#14579133

—

#14579132

—

C₄×Dic₁₅₈

C₂²

yes

characteristic

C₄×D₃₁₆

C₂²

yes

—

D₄×Dic₇₉

C₂²

yes

—

#14579095

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

Dic₇₉⋊C₈

C₂²

yes

—

#14579083

C₂²

yes

—

#14579082

C₂²

yes

—

C₈×Dic₇₉

C₂²

yes

—

C₂×D₈×C₇₉

C₂²

yes

—

C₂×D₈×C₇₉

C₂²

yes

—

#127×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

#11929623

—

C₂×Dic₁₅₈

D₄

yes

characteristic

C₂×D₃₁₆

C₂³

yes

—

C₂×C₂²⋊D₇₉

—

C₂×C₂²⋊D₇₉

—

C₂²×Dic₇₉

—

C₂×C₄×D₇₉

—

D₆₃₂

—

#11929609

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

D₄

yes

characteristic

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

D₄

yes

—

C₂²⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₄×Dic₇₉

C₂³

yes

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₂³

yes

—

D₈×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

C₂²×D₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂²⋊D₇₉

—

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₄

yes

—

C₂×Dic₇₉

C₂×D₄

yes

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

Dic₁₅₈

D₈

yes

complemented

C₈×C₇₉

—

C₈×C₇₉

#41

yes

complemented

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×D₈

yes

—

C₄×C₇₉

C₂×#41

yes

characteristic

C₄×C₇₉

H(5,𝔽₂)

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

C₂×D₈

yes

—

C₂×C₇₉

—

C₂×C₇₉

#550

yes

characteristic

C₂×C₇₉

C₂²×D₈

yes

characteristic

C₂×C₇₉

#576

yes

characteristic

D₇₉

—

#3391

—

maximal, Sylow

C₇₉

#3391

yes

characteristic, complemented, Sylow

#550

—

#493

—

#493

D₇₉

yes

characteristic, complemented

#459

—

#459

—

100

C₈×Q₈

—

101

C₄×D₈

—

102

C₄×D₈

—

103

C₂×#41

—

158

—

104

C₂×D₈

—

105

C₂×D₈

—

106

C₂×D₈

C₂×D₇₉

yes

—

107

C₂×D₈

C₂×D₇₉

yes

—

108

#127

—

109

#127

C₂×D₇₉

yes

—

110

#121

—

158

—

111

#121

—

158

—

112

C₄×Q₈

—

113

C₄×D₄

—

114

C₄×D₄

—

115

#107

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

#102

—

119

#102

—

120

C₄×C₈

—

121

C₄×C₈

C₂×D₇₉

yes

characteristic

122

#41

—

316

—

123

C₂×Q₈

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

158

—

128

C₂×D₄

—

158

—

129

C₂×D₄

C₂²×D₇₉

yes

—

130

C₂²×C₄

—

158

—

131

C₂²×C₄

—

158

—

132

D₈

—

133

D₈

—

134

D₈

—

316

—

135

D₈

—

158

—

136

C₂×C₈

—

137

C₂×C₈

C₂²×D₇₉

yes

characteristic

138

C₂×C₈

C₂²×D₇₉

yes

characteristic

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₂²⋊C₄

—

158

—

143

C₂²⋊C₄

—

158

—

144

C₄²

—

145

C₄²

C₂²×D₇₉

yes

characteristic

146

C₂³

—

147

C₂³

—

158

—

148

Q₈

—

149

D₄

—

150

D₄

—

151

D₄

—

152

D₄

—

153

D₄

—

316

—

154

D₄

—

632

—

155

D₄

—

158

—

156

D₄

—

158

—

157

C₂×C₄

—

316

—

158

C₂×C₄

—

316

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

158

—

162

C₂×C₄

D₄×D₇₉

yes

characteristic

163

C₂×C₄

D₄×D₇₉

yes

characteristic

164

C₂×C₄

C₂³×D₇₉

yes

characteristic

165

C₈

—

166

C₈

—

158

—

167

C₈

#11929623

yes

complemented

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

316

—

172

C₂²

—

316

—

173

C₂²

—

316

—

174

C₂²

C₂×D₄×D₇₉

yes

characteristic

175

C₄

—

176

C₄

—

316

—

177

C₄

—

178

C₄

—

158

—

179

C₄

D₈×D₇₉

yes

—

180

C₄

C₂×#11929623

yes

characteristic

181

C₄

#14579206

yes

characteristic

182

C₂

—

183

C₂

—

632

—

184

C₂

#19099233

yes

characteristic

185

C₂

C₂×D₈×D₇₉

yes

characteristic

186

C₂

#19099364

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#493×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
hypercenter	#493	D₇₉
hypocenter	C₇₉	#3391
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

C₂×D₄×D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#493×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #19099233	C₂	semidirect
2. #19098938	C₂	semidirect
3. #19098756	C₂	semidirect
4. #19098755	C₂	semidirect
5. C₈×Dic₁₅₈	C₂	semidirect
6. C₄×D₆₃₂	C₂	semidirect
7. D₈×Dic₇₉	C₂	semidirect
8. #493×C₇₉	C₂	semidirect
70. Dic₁₅₈	D₈	semidirect
72. C₈×C₇₉	#41	semidirect
94. C₇₉	#3391	semidirect
97. #493	D₇₉	semidirect
167. C₈	#11929623	semidirect

H ≅ G/N

type

1. #19099233

C₂

semidirect

2. #19098938

C₂

semidirect

3. #19098756

C₂

semidirect

4. #19098755

C₂

semidirect

5. C₈×Dic₁₅₈

C₂

semidirect

6. C₄×D₆₃₂

C₂

semidirect

7. D₈×Dic₇₉

C₂

semidirect

8. #493×C₇₉

C₂

semidirect

70. Dic₁₅₈

D₈

semidirect

72. C₈×C₇₉

#41

semidirect

94. C₇₉

#3391

semidirect

97. #493

D₇₉

semidirect

167. C₈

#11929623

semidirect

Subgroups — #23776373

Conjugacy classes of subgroups

Complements