Group #23762536 — subgroups

Subgroups — #23762536

Order 10112.

214 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19099270	C₂	1	1	yes	maximal, characteristic
2	#19098943	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098859	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098853	C₂	1	1	yes	maximal, characteristic
5	#19098311	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098295	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₁₆	—	2	2	no	—
9	#14579161	C₂²	1	1	yes	characteristic
10	#14579161	C₂²	1	1	yes	characteristic
11	#14579152	—	2	1	no	—
12	#14579140	—	2	1	no	—
13	#14579138	—	2	2	no	—
14	C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
15	D₄×Dic₇₉	—	2	1	no	—
16	C₂×C₄⋊Dic₇₉	—	2	1	no	—
17	#14579095	—	2	1	no	—
18	#14579092	—	2	1	no	—
19	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
20	#14579086	—	2	1	no	—
21	#14579082	—	2	1	no	—
22	C₄×D₄×C₇₉	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₅₈	—	2	1	no	—
26	C₂×Dic₁₅₈	D₄	1	1	yes	characteristic
27	C₂×Dic₁₅₈	D₄	1	2	yes	—
28	C₂²×Dic₇₉	—	2	2	no	—
29	Dic₃₁₆	—	4	4	no	—
30	C₂×C₇₉⋊C₈	—	2	1	no	—
31	C₄⋊Dic₇₉	—	2	2	no	—
32	C₄⋊Dic₇₉	—	2	1	no	—
33	C₄⋊Dic₇₉	—	2	1	no	—
34	C₄⋊Dic₇₉	—	2	1	no	—
35	C₄⋊Dic₇₉	—	2	1	no	—
36	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
37	Dic₇₉⋊C₄	—	2	2	no	—
38	Dic₇₉⋊C₄	—	2	1	no	—
39	C₂²⋊Dic₇₉	—	2	2	no	—
40	C₄×Dic₇₉	—	2	1	no	—
41	C₂×D₄×C₇₉	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₉	—	2	1	no	—
43	C₂×C₈×C₇₉	—	2	1	no	—
44	C₂×C₈×C₇₉	—	2	1	no	—
45	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
46	C₂²⋊C₄×C₇₉	—	2	1	no	—
47	C₄²×C₇₉	C₂³	1	1	yes	characteristic
48	C₂³×C₇₉	—	2	1	no	—
49	D₄×C₇₉	—	2	1	no	—
50	D₄×C₇₉	Q₁₆	1	2	yes	complemented
51	C₂×C₄×C₇₉	—	2	1	no	—
52	C₂×C₄×C₇₉	—	4	1	no	—
53	C₂×C₄×C₇₉	—	4	1	no	—
54	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₇₉	—	2	1	no	—
58	C₂×Dic₇₉	—	2	1	no	—
59	C₂×Dic₇₉	—	2	2	no	—
60	C₂×Dic₇₉	—	4	1	no	—
61	C₂×Dic₇₉	—	8	2	no	—
62	Dic₁₅₈	—	2	1	no	—
63	Dic₁₅₈	—	2	1	no	—
64	Dic₁₅₈	—	4	2	no	—
65	Dic₁₅₈	—	4	2	no	—
66	C₈×C₇₉	—	2	2	no	—
67	C₈×C₇₉	—	4	1	no	—
68	C₇₉⋊C₈	—	4	1	no	—
69	C₂²×C₇₉	—	2	2	no	—
70	C₂²×C₇₉	—	4	1	no	—
71	C₂²×C₇₉	—	4	1	no	—
72	C₂²×C₇₉	#120	1	1	yes	characteristic
73	C₄×C₇₉	—	2	1	no	—
74	C₄×C₇₉	—	4	1	no	—
75	C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
76	C₄×C₇₉	#135	1	1	yes	characteristic
77	C₄×C₇₉	#137	1	1	yes	characteristic
78	C₄×C₇₉	#137	1	1	yes	characteristic
79	Dic₇₉	—	4	1	no	—
80	Dic₇₉	—	4	2	no	—
81	Dic₇₉	—	4	1	no	—
82	Dic₇₉	—	8	1	no	—
83	C₂×C₇₉	—	4	2	no	—
84	C₂×C₇₉	#451	1	1	yes	characteristic
85	C₂×C₇₉	#452	1	1	yes	characteristic
86	C₂×C₇₉	#453	1	1	yes	characteristic
87	#1632	—	79	1	no	maximal, Sylow
88	C₇₉	#1632	1	1	yes	characteristic, complemented, Sylow
89	#554	—	79	1	no	—
90	#494	—	79	1	no	—
91	#477	—	79	1	no	—
92	#476	—	79	1	no	—
93	#333	—	79	1	no	—
94	Q₈⋊C₈	—	79	1	no	—
95	D₄⋊C₈	D₇₉	1	1	yes	characteristic
96	C₂×Q₁₆	—	158	2	no	—
97	C₄⋊Q₈	—	79	1	no	—
98	C₄⋊Q₈	—	79	1	no	—
99	#125	—	158	1	no	—
100	C₂²⋊Q₈	—	158	2	no	—
101	C₂²⋊Q₈	—	158	1	no	—
102	C₄×Q₈	—	79	1	no	—
103	C₄×D₄	—	158	1	no	—
104	C₄×D₄	C₂×D₇₉	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	158	1	no	—
106	#107	—	158	1	no	—
107	#106	—	158	1	no	—
108	C₄⋊C₈	—	79	1	no	—
109	C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
110	#103	—	158	1	no	—
111	#102	—	158	1	no	—
112	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
113	C₂×Q₈	—	79	2	no	—
114	C₂×Q₈	—	79	1	no	—
115	C₂×Q₈	—	158	1	no	—
116	C₂×D₄	D₃₁₆	1	1	yes	characteristic
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	158	2	no	—
119	Q₁₆	—	316	4	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	158	1	no	—
123	C₄⋊C₄	—	79	1	no	—
124	C₄⋊C₄	—	158	2	no	—
125	C₄⋊C₄	—	158	2	no	—
126	C₄⋊C₄	—	158	1	no	—
127	C₄⋊C₄	—	158	1	no	—
128	C₄⋊C₄	—	158	1	no	—
129	C₄⋊C₄	—	158	1	no	—
130	C₄⋊C₄	—	158	1	no	—
131	C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
132	C₂²⋊C₄	—	2	1	no	—
133	C₂²⋊C₄	—	158	2	no	—
134	C₄²	—	158	1	no	—
135	C₄²	C₂²×D₇₉	1	1	yes	characteristic
136	C₂³	—	2	1	no	—
137	Q₈	—	316	2	no	—
138	Q₈	—	316	2	no	—
139	Q₈	—	158	1	no	—
140	Q₈	—	158	1	no	—
141	D₄	—	2	1	no	—
142	D₄	Dic₃₁₆	1	2	yes	complemented
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	316	1	no	—
147	C₂×C₄	—	632	2	no	—
148	C₂×C₄	—	158	1	no	—
149	C₂×C₄	—	158	1	no	—
150	C₂×C₄	—	158	2	no	—
151	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
152	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
153	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	4	1	no	—
156	C₈	—	316	1	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	4	1	no	—
159	C₂²	—	4	1	no	—
160	C₂²	#14579131	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	4	1	no	—
163	C₄	—	316	1	no	—
164	C₄	—	316	1	no	—
165	C₄	—	316	2	no	—
166	C₄	—	632	1	no	—
167	C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
168	C₄	#14579180	1	1	yes	characteristic
169	C₄	#14579190	1	1	yes	characteristic
170	C₄	#14579194	1	1	yes	characteristic
171	C₂	—	4	2	no	—
172	C₂	#19098721	1	1	yes	characteristic
173	C₂	#19098726	1	1	yes	characteristic
174	C₂	#19098732	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099270

C₂

yes

maximal, characteristic

#19098943

C₂

yes

maximal, characteristic, complemented

#19098859

C₂

yes

maximal, characteristic, complemented

#19098853

C₂

yes

maximal, characteristic

#19098311

C₂

yes

maximal, characteristic, complemented

#19098295

C₂

yes

maximal, characteristic, complemented

D₄⋊C₈×C₇₉

C₂

yes

maximal, characteristic

C₂×Dic₃₁₆

—

#14579161

C₂²

yes

characteristic

#14579161

C₂²

yes

—

—

—

C₄×Dic₁₅₈

C₂²

yes

characteristic

D₄×Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

#14579095

—

#14579092

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

#14579086

—

#14579082

—

C₄×D₄×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

D₄

yes

characteristic

C₂×Dic₁₅₈

D₄

yes

—

C₂²×Dic₇₉

—

Dic₃₁₆

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

characteristic

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂×D₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

Q₁₆

yes

complemented

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#120

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×Q₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

—

—

—

—

—

Q₈⋊C₈

—

D₄⋊C₈

D₇₉

yes

characteristic

C₂×Q₁₆

—

158

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

158

—

100

C₂²⋊Q₈

—

158

—

101

C₂²⋊Q₈

—

158

—

102

C₄×Q₈

—

103

C₄×D₄

—

158

—

104

C₄×D₄

C₂×D₇₉

yes

characteristic

105

C₂×C₄⋊C₄

—

158

—

106

#107

—

158

—

107

#106

—

158

—

108

C₄⋊C₈

—

109

C₄⋊C₈

C₂×D₇₉

yes

characteristic

110

#103

—

158

—

111

#102

—

158

—

112

C₄×C₈

C₂×D₇₉

yes

characteristic

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

158

—

116

C₂×D₄

D₃₁₆

yes

characteristic

117

C₂²×C₄

—

118

C₂²×C₄

—

158

—

119

Q₁₆

—

316

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

158

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

158

—

125

C₄⋊C₄

—

158

—

126

C₄⋊C₄

—

158

—

127

C₄⋊C₄

—

158

—

128

C₄⋊C₄

—

158

—

129

C₄⋊C₄

—

158

—

130

C₄⋊C₄

—

158

—

131

C₄⋊C₄

D₃₁₆

yes

characteristic

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

158

—

134

C₄²

—

158

—

135

C₄²

C₂²×D₇₉

yes

characteristic

136

C₂³

—

137

Q₈

—

316

—

138

Q₈

—

316

—

139

Q₈

—

158

—

140

Q₈

—

158

—

141

D₄

—

142

D₄

Dic₃₁₆

yes

complemented

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

316

—

147

C₂×C₄

—

632

—

148

C₂×C₄

—

158

—

149

C₂×C₄

—

158

—

150

C₂×C₄

—

158

—

151

C₂×C₄

D₄×D₇₉

yes

characteristic

152

C₂×C₄

D₄×D₇₉

yes

characteristic

153

C₂×C₄

C₂×D₃₁₆

yes

characteristic

154

C₈

—

155

C₈

—

156

C₈

—

316

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

#14579131

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

316

—

164

C₄

—

316

—

165

C₄

—

316

—

166

C₄

—

632

—

167

C₄

C₂×Dic₃₁₆

yes

characteristic

168

C₄

#14579180

yes

characteristic

169

C₄

#14579190

yes

characteristic

170

C₄

#14579194

yes

characteristic

171

C₂

—

172

C₂

#19098721

yes

characteristic

173

C₂

#19098726

yes

characteristic

174

C₂

#19098732

yes

characteristic

	subgroup	G/H
center	C₂²	#14579131
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	D₄⋊C₈×C₇₉	C₂
socle	C₂²×C₇₉	#120
prime radical	C₂×D₄×C₇₉	D₄
hypercenter	D₄⋊C₈	D₇₉
hypocenter	C₇₉	#1632
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ D₄⋊C₈

subgroup

G/H

center

C₂²

#14579131

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

D₄⋊C₈×C₇₉

C₂

socle

C₂²×C₇₉

#120

prime radical

C₂×D₄×C₇₉

D₄

hypercenter

D₄⋊C₈

D₇₉

hypocenter

C₇₉

#1632

LCS

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ D₄⋊C₈

N	H ≅ G/N	type
2. #19098943	C₂	semidirect
3. #19098859	C₂	semidirect
5. #19098311	C₂	semidirect
6. #19098295	C₂	semidirect
50. D₄×C₇₉	Q₁₆	semidirect
88. C₇₉	#1632	semidirect
142. D₄	Dic₃₁₆	semidirect

H ≅ G/N

type

2. #19098943

C₂

semidirect

3. #19098859

C₂

semidirect

5. #19098311

C₂

semidirect

6. #19098295

C₂

semidirect

50. D₄×C₇₉

Q₁₆

semidirect

88. C₇₉

#1632

semidirect

142. D₄

Dic₃₁₆

semidirect

Subgroups — #23762536

Conjugacy classes of subgroups

Complements