Group #23764647 — subgroups

Subgroups — #23764647

Order 10112.

298 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 91 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	C₂×#14579161	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579086	C₂	1	2	yes	maximal
3	C₂×#14579086	C₂	1	2	yes	maximal
4	#19098461	C₂	1	1	yes	maximal, characteristic
5	#336×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₅₈	C₂²	1	2	yes	—
7	#14579161	—	2	4	no	—
8	C₂×C₄⋊Dic₇₉	—	2	1	no	—
9	C₂×C₄⋊Dic₇₉	—	2	1	no	—
10	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₇₉⋊C₄	—	2	1	no	—
13	#14579086	—	2	4	no	—
14	#14579086	—	2	4	no	—
15	C₇₉⋊#95	—	2	1	no	—
16	C₂²×C₈×C₇₉	C₂²	1	2	yes	—
17	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₅₈	—	2	2	no	—
19	C₂×Dic₁₅₈	—	2	2	no	—
20	C₂×Dic₁₅₈	—	2	2	no	—
21	C₂×Dic₁₅₈	—	2	2	no	—
22	C₂×Dic₁₅₈	—	2	2	no	—
23	C₂×Dic₁₅₈	D₄	1	4	yes	—
24	C₂²×Dic₇₉	—	2	2	no	—
25	C₂²×Dic₇₉	—	2	1	no	—
26	C₂²×Dic₇₉	—	2	1	no	—
27	C₄⋊Dic₇₉	—	2	4	no	—
28	C₄⋊Dic₇₉	—	2	4	no	—
29	C₄⋊Dic₇₉	—	2	2	no	—
30	C₄⋊Dic₇₉	—	2	1	no	—
31	C₄⋊Dic₇₉	D₄	1	2	yes	—
32	Dic₇₉⋊C₄	—	4	2	no	—
33	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₇₉	—	2	2	no	—
37	C₂×C₈×C₇₉	—	2	2	no	—
38	C₂×C₈×C₇₉	—	2	2	no	—
39	C₄²×C₇₉	—	2	2	no	—
40	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₇₉	—	2	1	no	—
42	C₂×C₄×C₇₉	—	2	1	no	—
43	C₂×C₄×C₇₉	—	2	1	no	—
44	C₂×C₄×C₇₉	—	2	1	no	—
45	C₂×C₄×C₇₉	SD₁₆	1	2	yes	—
46	C₂×C₄×C₇₉	Q₁₆	1	2	yes	—
47	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₇₉	#41	1	2	yes	—
50	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
51	C₂×Dic₇₉	—	2	4	no	—
52	C₂×Dic₇₉	—	2	2	no	—
53	C₂×Dic₇₉	—	4	2	no	—
54	C₂×Dic₇₉	—	4	2	no	—
55	C₂×Dic₇₉	—	4	1	no	—
56	C₂×Dic₇₉	—	4	1	no	—
57	C₂×Dic₇₉	—	4	1	no	—
58	C₂×Dic₇₉	—	4	1	no	—
59	C₂×Dic₇₉	—	4	1	no	—
60	Dic₁₅₈	—	2	4	no	—
61	Dic₁₅₈	—	2	4	no	—
62	Dic₁₅₈	—	4	2	no	—
63	Dic₁₅₈	—	4	2	no	—
64	C₈×C₇₉	—	4	2	no	—
65	C₂²×C₇₉	#120	1	1	yes	characteristic
66	C₂²×C₇₉	#121	1	1	yes	characteristic
67	C₂²×C₇₉	#124	1	1	yes	characteristic
68	C₂²×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
69	C₂²×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
70	C₂²×C₇₉	#137	1	1	yes	characteristic
71	C₂²×C₇₉	#137	1	1	yes	characteristic
72	C₄×C₇₉	—	2	2	no	—
73	C₄×C₇₉	—	2	2	no	—
74	C₄×C₇₉	#121	1	2	yes	—
75	C₄×C₇₉	#123	1	2	yes	—
76	Dic₇₉	—	4	4	no	—
77	Dic₇₉	—	4	2	no	—
78	Dic₇₉	—	8	1	no	—
79	C₂×C₇₉	#394	1	1	yes	characteristic
80	C₂×C₇₉	#448	1	1	yes	characteristic
81	C₂×C₇₉	#451	1	1	yes	characteristic
82	C₂×C₇₉	#461	1	1	yes	characteristic
83	C₂×C₇₉	#462	1	1	yes	characteristic
84	C₂×C₇₉	#487	1	1	yes	characteristic
85	C₂×C₇₉	#491	1	1	yes	characteristic
86	#2043	—	79	1	no	maximal, Sylow
87	C₇₉	#2043	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₄⋊Q₈	—	79	1	no	—
89	C₂×#103	—	79	2	no	—
90	C₂×#103	—	79	2	no	—
91	#389	—	79	1	no	—
92	#336	D₇₉	1	1	yes	characteristic
93	C₂²×Q₈	—	79	2	no	—
94	C₂²×C₈	C₂×D₇₉	1	2	yes	—
95	C₄⋊Q₈	—	158	4	no	—
96	C₂×C₄⋊C₄	—	79	1	no	—
97	C₂×C₄⋊C₄	—	79	1	no	—
98	C₂×C₄⋊C₄	—	158	1	no	—
99	C₂×C₄⋊C₄	—	158	1	no	—
100	C₂×C₄⋊C₄	—	158	1	no	—
101	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
102	#103	—	158	4	no	—
103	#103	—	158	4	no	—
104	#95	—	158	1	no	—
105	C₂×Q₈	—	79	4	no	—
106	C₂×Q₈	—	158	2	no	—
107	C₂×Q₈	—	158	2	no	—
108	C₂×Q₈	—	158	2	no	—
109	C₂×Q₈	—	158	2	no	—
110	C₂×Q₈	—	158	2	no	—
111	C₂²×C₄	—	158	1	no	—
112	C₂²×C₄	—	158	1	no	—
113	C₂²×C₄	—	158	2	no	—
114	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
115	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₄⋊C₄	—	316	2	no	—
121	C₄⋊C₄	—	79	2	no	—
122	C₄⋊C₄	—	158	4	no	—
123	C₄⋊C₄	—	158	4	no	—
124	C₄⋊C₄	—	158	2	no	—
125	C₄⋊C₄	—	158	1	no	—
126	C₄²	—	2	2	no	—
127	C₂³	C₂×D₃₁₆	1	1	yes	characteristic
128	Q₈	—	316	2	no	—
129	Q₈	—	316	2	no	—
130	Q₈	—	158	4	no	—
131	Q₈	—	158	4	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	316	2	no	—
137	C₂×C₄	—	316	2	no	—
138	C₂×C₄	—	316	1	no	—
139	C₂×C₄	—	316	1	no	—
140	C₂×C₄	—	316	1	no	—
141	C₂×C₄	—	316	1	no	—
142	C₂×C₄	—	316	1	no	—
143	C₂×C₄	—	158	4	no	—
144	C₂×C₄	—	158	2	no	—
145	C₂×C₄	#11929613	1	2	yes	—
146	C₂×C₄	Dic₃₁₆	1	2	yes	—
147	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
148	C₂×C₄	D₄×D₇₉	1	2	yes	—
149	C₂×C₄	#11929624	1	2	yes	—
150	C₂×C₄	#11929625	1	1	yes	characteristic
151	C₈	—	4	2	no	—
152	C₂²	#14579131	1	1	yes	characteristic
153	C₂²	#14579136	1	1	yes	characteristic
154	C₂²	#14579149	1	1	yes	characteristic
155	C₂²	C₂×#11929613	1	1	yes	characteristic
156	C₂²	C₂×Dic₃₁₆	1	1	yes	characteristic
157	C₂²	#14579194	1	1	yes	characteristic
158	C₂²	#14579194	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	316	4	no	—
162	C₄	—	316	2	no	—
163	C₄	—	632	1	no	—
164	C₄	#14579134	1	2	yes	—
165	C₄	#14579146	1	2	yes	—
166	C₂	#19098481	1	1	yes	characteristic
167	C₂	#19098704	1	1	yes	characteristic
168	C₂	#19098721	1	1	yes	characteristic
169	C₂	#19098769	1	1	yes	characteristic
170	C₂	#19098774	1	1	yes	characteristic
171	C₂	#19098914	1	1	yes	characteristic
172	C₂	#19098932	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579161

C₂

yes

maximal, characteristic

C₂×#14579086

C₂

yes

maximal

C₂×#14579086

C₂

yes

maximal

#19098461

C₂

yes

maximal, characteristic

#336×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

—

#14579161

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×Dic₇₉⋊C₄

—

#14579086

—

#14579086

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

D₄

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

—

Dic₇₉⋊C₄

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

SD₁₆

yes

—

C₂×C₄×C₇₉

Q₁₆

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

#41

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

C₂×SD₁₆

yes

characteristic

C₂²×C₇₉

C₂×Q₁₆

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#121

yes

—

C₄×C₇₉

#123

yes

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

—

—

—

D₇₉

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

C₂×D₇₉

yes

—

C₄⋊Q₈

—

158

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

—

158

—

100

C₂×C₄⋊C₄

—

158

—

101

C₂×C₄²

C₂×D₇₉

yes

characteristic

102

#103

—

158

—

103

#103

—

158

—

104

#95

—

158

—

105

C₂×Q₈

—

106

C₂×Q₈

—

158

—

107

C₂×Q₈

—

158

—

108

C₂×Q₈

—

158

—

109

C₂×Q₈

—

158

—

110

C₂×Q₈

—

158

—

111

C₂²×C₄

—

158

—

112

C₂²×C₄

—

158

—

113

C₂²×C₄

—

158

—

114

C₂²×C₄

D₃₁₆

yes

characteristic

115

C₂²×C₄

D₃₁₆

yes

characteristic

116

C₂²×C₄

C₂²×D₇₉

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

316

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

158

—

123

C₄⋊C₄

—

158

—

124

C₄⋊C₄

—

158

—

125

C₄⋊C₄

—

158

—

126

C₄²

—

127

C₂³

C₂×D₃₁₆

yes

characteristic

128

Q₈

—

316

—

129

Q₈

—

316

—

130

Q₈

—

158

—

131

Q₈

—

158

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

316

—

137

C₂×C₄

—

316

—

138

C₂×C₄

—

316

—

139

C₂×C₄

—

316

—

140

C₂×C₄

—

316

—

141

C₂×C₄

—

316

—

142

C₂×C₄

—

316

—

143

C₂×C₄

—

158

—

144

C₂×C₄

—

158

—

145

C₂×C₄

#11929613

yes

—

146

C₂×C₄

Dic₃₁₆

yes

—

147

C₂×C₄

D₄×D₇₉

yes

characteristic

148

C₂×C₄

D₄×D₇₉

yes

—

149

C₂×C₄

#11929624

yes

—

150

C₂×C₄

#11929625

yes

characteristic

151

C₈

—

152

C₂²

#14579131

yes

characteristic

153

C₂²

#14579136

yes

characteristic

154

C₂²

#14579149

yes

characteristic

155

C₂²

C₂×#11929613

yes

characteristic

156

C₂²

C₂×Dic₃₁₆

yes

characteristic

157

C₂²

#14579194

yes

characteristic

158

C₂²

#14579194

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

316

—

162

C₄

—

316

—

163

C₄

—

632

—

164

C₄

#14579134

yes

—

165

C₄

#14579146

yes

—

166

C₂

#19098481

yes

characteristic

167

C₂

#19098704

yes

characteristic

168

C₂

#19098721

yes

characteristic

169

C₂

#19098769

yes

characteristic

170

C₂

#19098774

yes

characteristic

171

C₂

#19098914

yes

characteristic

172

C₂

#19098932

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₁₆
commutator, cosocle	C₂²×C₄×C₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#336×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
hypercenter	#336	D₇₉
hypocenter	C₇₉	#2043
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₃₁₆

commutator, cosocle

C₂²×C₄×C₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#336×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
87. C₇₉	#2043	semidirect

H ≅ G/N

type

87. C₇₉

#2043

semidirect

Subgroups — #23764647

Conjugacy classes of subgroups

Complements