Group #23765094 — subgroups

Subgroups — #23765094

Order 10112.

230 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 65 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	#19099134	C₂	1	1	yes	maximal, characteristic
2	#19098547	C₂	1	1	yes	maximal, characteristic
3	#19098361	C₂	1	4	yes	maximal, complemented
4	#413×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₅₈	C₄	1	1	yes	characteristic
6	C₂×#11929608	—	2	1	no	—
7	#14579159	C₂²	1	2	yes	—
8	#14579148	C₄	1	2	yes	—
9	#14579142	—	2	1	no	—
10	#14579142	—	2	1	no	—
11	#14579140	—	2	1	no	—
12	#14579140	—	2	1	no	—
13	#14579120	C₄	1	1	yes	characteristic
14	#14579080	C₂²	1	2	yes	—
15	#14579078	—	2	1	no	—
16	C₂×#41×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₇₉	—	2	1	no	—
18	#101×C₇₉	C₂²	1	2	yes	—
19	#100×C₇₉	—	2	1	no	—
20	C₂×Dic₁₅₈	—	2	1	no	—
21	C₂×Dic₁₅₈	—	4	1	no	—
22	C₂×Dic₁₅₈	—	4	1	no	—
23	C₂×Dic₁₅₈	C₂×C₄	1	2	yes	—
24	C₂²×Dic₇₉	—	2	1	no	—
25	#11929608	—	2	2	no	—
26	#11929608	—	4	1	no	—
27	C₂×C₇₉⋊C₈	—	4	1	no	—
28	C₄⋊Dic₇₉	—	2	1	no	—
29	Dic₇₉⋊C₄	—	2	2	no	—
30	Dic₇₉⋊C₄	—	2	2	no	—
31	C₂²⋊Dic₇₉	—	2	1	no	—
32	C₂²⋊Dic₇₉	—	2	2	no	—
33	C₂²⋊Dic₇₉	—	2	2	no	—
34	C₄×Dic₇₉	C₂×C₄	1	2	yes	—
35	#41×C₇₉	—	4	1	no	—
36	#41×C₇₉	—	4	1	no	—
37	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
38	C₂×D₄×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₂×D₄×C₇₉	D₄	1	1	yes	characteristic
40	C₂×D₄×C₇₉	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
44	M₁₆×C₇₉	—	2	2	no	—
45	M₁₆×C₇₉	—	4	1	no	—
46	C₂×C₈×C₇₉	—	4	1	no	—
47	C₂³×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂³×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂³×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
50	Q₈×C₇₉	—	4	1	no	—
51	D₄×C₇₉	—	4	1	no	—
52	D₄×C₇₉	—	4	1	no	—
53	D₄×C₇₉	—	4	1	no	—
54	C₂×C₄×C₇₉	—	2	1	no	—
55	C₂×C₄×C₇₉	—	2	1	no	—
56	C₂×C₄×C₇₉	—	2	1	no	—
57	C₂×C₄×C₇₉	—	2	1	no	—
58	C₂×C₄×C₇₉	—	2	1	no	—
59	C₂×C₄×C₇₉	—	2	1	no	—
60	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
64	C₂×Dic₇₉	—	2	2	no	—
65	C₂×Dic₇₉	—	2	2	no	—
66	C₂×Dic₇₉	—	4	2	no	—
67	Dic₁₅₈	—	4	1	no	—
68	Dic₁₅₈	—	4	1	no	—
69	Dic₁₅₈	—	4	2	no	—
70	C₈×C₇₉	—	4	2	no	—
71	C₇₉⋊C₈	—	4	2	no	—
72	C₂²×C₇₉	—	2	1	no	—
73	C₂²×C₇₉	—	2	1	no	—
74	C₂²×C₇₉	—	2	1	no	—
75	C₂²×C₇₉	—	4	1	no	—
76	C₂²×C₇₉	—	4	1	no	—
77	C₂²×C₇₉	—	4	1	no	—
78	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₄×C₇₉	—	2	1	no	—
80	C₄×C₇₉	—	2	1	no	—
81	C₄×C₇₉	—	2	1	no	—
82	C₄×C₇₉	#99	1	1	yes	characteristic
83	C₄×C₇₉	#99	1	1	yes	characteristic
84	Dic₇₉	—	4	2	no	—
85	Dic₇₉	—	4	2	no	—
86	C₂×C₇₉	—	2	1	no	—
87	C₂×C₇₉	—	4	1	no	—
88	C₂×C₇₉	—	4	1	no	—
89	C₂×C₇₉	—	4	1	no	—
90	C₂×C₇₉	C₂×#99	1	1	yes	characteristic
91	#2135	—	79	1	no	maximal, Sylow
92	C₇₉	#2135	1	1	yes	characteristic, complemented, Sylow
93	#536	—	79	1	no	—
94	#413	—	79	1	no	—
95	#413	D₇₉	1	1	yes	characteristic
96	#356	—	79	4	no	—
97	C₂×#41	C₂×D₇₉	1	1	yes	characteristic
98	C₂²×Q₈	—	79	1	no	—
99	C₂×M₁₆	—	2	1	no	—
100	C₂×M₁₆	—	158	1	no	—
101	C₄⋊Q₈	—	79	2	no	—
102	#124	—	79	2	no	—
103	#123	—	158	1	no	—
104	#123	—	158	1	no	—
105	C₂²⋊Q₈	—	158	1	no	—
106	C₂²⋊Q₈	—	158	1	no	—
107	#117	—	79	1	no	—
108	#101	—	79	2	no	—
109	#101	C₂×D₇₉	1	2	yes	—
110	#100	—	2	1	no	—
111	#100	—	158	1	no	—
112	#41	—	4	1	no	—
113	#41	—	4	1	no	—
114	C₂×Q₈	—	316	1	no	—
115	C₂×Q₈	—	316	1	no	—
116	C₂×Q₈	—	79	2	no	—
117	C₂×Q₈	—	158	1	no	—
118	C₂×Q₈	C₂²×D₇₉	1	1	yes	characteristic
119	C₂×D₄	C₄×D₇₉	1	1	yes	characteristic
120	C₂×D₄	D₃₁₆	1	1	yes	characteristic
121	C₂×D₄	C₂²⋊D₇₉	1	1	yes	characteristic
122	C₂²×C₄	—	158	1	no	—
123	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
124	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
125	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
126	M₁₆	—	2	2	no	—
127	M₁₆	—	4	1	no	—
128	M₁₆	—	316	1	no	—
129	M₁₆	—	158	2	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	316	1	no	—
132	C₄⋊C₄	—	158	1	no	—
133	C₄⋊C₄	—	158	2	no	—
134	C₄⋊C₄	—	158	2	no	—
135	C₂²⋊C₄	—	158	1	no	—
136	C₂²⋊C₄	—	158	2	no	—
137	C₂²⋊C₄	—	158	2	no	—
138	C₄²	—	79	2	no	—
139	C₂³	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
140	C₂³	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
141	C₂³	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
142	Q₈	—	4	1	no	—
143	Q₈	—	316	2	no	—
144	Q₈	—	316	1	no	—
145	Q₈	—	316	1	no	—
146	D₄	—	4	1	no	—
147	D₄	—	4	1	no	—
148	D₄	—	4	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	316	2	no	—
156	C₂×C₄	—	158	2	no	—
157	C₂×C₄	—	158	2	no	—
158	C₂×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
160	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
161	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
162	C₈	—	4	2	no	—
163	C₈	—	316	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	4	1	no	—
168	C₂²	—	4	1	no	—
169	C₂²	—	4	1	no	—
170	C₂²	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	316	2	no	—
175	C₄	—	316	2	no	—
176	C₄	#14579075	1	1	yes	characteristic
177	C₄	#14579075	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	4	1	no	—
180	C₂	—	4	1	no	—
181	C₂	—	4	1	no	—
182	C₂	C₂×#14579075	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099134

C₂

yes

maximal, characteristic

#19098547

C₂

yes

maximal, characteristic

#19098361

C₂

yes

maximal, complemented

#413×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₄

yes

characteristic

C₂×#11929608

—

#14579159

C₂²

yes

—

#14579148

C₄

yes

—

—

—

—

—

C₄

yes

characteristic

#14579080

C₂²

yes

—

#14579078

—

C₂×#41×C₇₉

C₂²

yes

characteristic

C₂×M₁₆×C₇₉

—

#101×C₇₉

C₂²

yes

—

#100×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂×C₄

yes

—

C₂²×Dic₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

C₂×C₄

yes

—

#41×C₇₉

—

#41×C₇₉

—

C₂×Q₈×C₇₉

C₂³

yes

characteristic

C₂×D₄×C₇₉

C₂×C₄

yes

characteristic

C₂×D₄×C₇₉

D₄

yes

characteristic

C₂×D₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂³×C₇₉

C₂²⋊C₄

yes

characteristic

C₂³×C₇₉

C₂²⋊C₄

yes

characteristic

C₂³×C₇₉

C₂²⋊C₄

yes

characteristic

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₂×#99

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#536

—

#413

—

#413

D₇₉

yes

characteristic

#356

—

C₂×#41

C₂×D₇₉

yes

characteristic

C₂²×Q₈

—

C₂×M₁₆

—

100

C₂×M₁₆

—

158

—

101

C₄⋊Q₈

—

102

#124

—

103

#123

—

158

—

104

#123

—

158

—

105

C₂²⋊Q₈

—

158

—

106

C₂²⋊Q₈

—

158

—

107

#117

—

108

#101

—

109

#101

C₂×D₇₉

yes

—

110

#100

—

111

#100

—

158

—

112

#41

—

113

#41

—

114

C₂×Q₈

—

316

—

115

C₂×Q₈

—

316

—

116

C₂×Q₈

—

117

C₂×Q₈

—

158

—

118

C₂×Q₈

C₂²×D₇₉

yes

characteristic

119

C₂×D₄

C₄×D₇₉

yes

characteristic

120

C₂×D₄

D₃₁₆

yes

characteristic

121

C₂×D₄

C₂²⋊D₇₉

yes

characteristic

122

C₂²×C₄

—

158

—

123

C₂²×C₄

C₄×D₇₉

yes

characteristic

124

C₂²×C₄

D₃₁₆

yes

characteristic

125

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

126

M₁₆

—

127

M₁₆

—

128

M₁₆

—

316

—

129

M₁₆

—

158

—

130

C₂×C₈

—

131

C₂×C₈

—

316

—

132

C₄⋊C₄

—

158

—

133

C₄⋊C₄

—

158

—

134

C₄⋊C₄

—

158

—

135

C₂²⋊C₄

—

158

—

136

C₂²⋊C₄

—

158

—

137

C₂²⋊C₄

—

158

—

138

C₄²

—

139

C₂³

(C₂×D₇₉)⋊C₄

yes

characteristic

140

C₂³

(C₂×D₇₉)⋊C₄

yes

characteristic

141

C₂³

(C₂×D₇₉)⋊C₄

yes

characteristic

142

Q₈

—

143

Q₈

—

316

—

144

Q₈

—

316

—

145

Q₈

—

316

—

146

D₄

—

147

D₄

—

148

D₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

316

—

156

C₂×C₄

—

158

—

157

C₂×C₄

—

158

—

158

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

159

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

160

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

161

C₂×C₄

C₂×D₃₁₆

yes

characteristic

162

C₈

—

163

C₈

—

316

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

316

—

175

C₄

—

316

—

176

C₄

#14579075

yes

characteristic

177

C₄

#14579075

yes

characteristic

178

C₂

—

179

C₂

—

180

C₂

—

181

C₂

—

182

C₂

C₂×#14579075

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#14579075
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂×Q₈	C₂²×D₇₉
Fitting	#413×C₇₉	C₂
socle	C₂×C₇₉	C₂×#99
prime radical	C₂×#41×C₇₉	C₂²
cosocle	C₂×Q₈×C₇₉	C₂³
hypercenter	#413	D₇₉
hypocenter	C₇₉	#2135
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #413

subgroup

G/H

center

C₂

C₂×#14579075

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂×Q₈

C₂²×D₇₉

Fitting

#413×C₇₉

C₂

socle

C₂×C₇₉

C₂×#99

prime radical

C₂×#41×C₇₉

C₂²

cosocle

C₂×Q₈×C₇₉

C₂³

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #413

N	H ≅ G/N	type
3. #19098361	C₂	semidirect
92. C₇₉	#2135	semidirect

H ≅ G/N

type

3. #19098361

C₂

semidirect

92. C₇₉

#2135

semidirect

Subgroups — #23765094

Conjugacy classes of subgroups

Complements