Group #23764267 — subgroups

Subgroups — #23764267

Order 10112.

384 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 117 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×#14579095	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14579082	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098466	C₂	1	2	yes	maximal, complemented
5	C₂²×D₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
8	C₂×C₂²⋊Dic₇₉	—	2	4	no	—
9	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
10	#14579095	—	2	2	no	—
11	#14579082	—	2	4	no	—
12	C₈×Dic₇₉	—	2	2	no	—
13	C₂²×D₄×C₇₉	C₂²	1	2	yes	—
14	C₂×D₈×C₇₉	—	2	2	no	—
15	C₂×D₈×C₇₉	—	2	2	no	—
16	C₂×D₈×C₇₉	C₄	1	4	yes	complemented
17	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₂²×Dic₇₉	—	2	2	no	—
19	C₂²×Dic₇₉	D₄	1	2	yes	—
20	C₂×C₇₉⋊C₈	—	2	1	no	—
21	C₂×C₇₉⋊C₈	—	2	1	no	—
22	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
23	C₄⋊Dic₇₉	—	2	2	no	—
24	C₄⋊Dic₇₉	—	2	2	no	—
25	C₂²⋊Dic₇₉	—	4	8	no	—
26	C₄×Dic₇₉	—	2	1	no	—
27	C₄×Dic₇₉	—	2	1	no	—
28	C₂⁴×C₇₉	—	2	2	no	—
29	C₂×D₄×C₇₉	—	2	2	no	—
30	C₂×D₄×C₇₉	—	2	2	no	—
31	C₂×D₄×C₇₉	—	2	2	no	—
32	C₂×D₄×C₇₉	—	2	2	no	—
33	C₂×D₄×C₇₉	—	2	2	no	—
34	C₂×D₄×C₇₉	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
36	D₈×C₇₉	—	2	8	no	—
37	C₂×C₈×C₇₉	C₂×C₄	1	2	yes	—
38	C₂×C₈×C₇₉	D₄	1	2	yes	—
39	C₂×C₈×C₇₉	D₄	1	2	yes	—
40	C₂³×C₇₉	—	2	4	no	—
41	C₂³×C₇₉	—	4	2	no	—
42	C₂³×C₇₉	—	4	2	no	—
43	C₂³×C₇₉	—	4	4	no	—
44	C₂³×C₇₉	—	4	4	no	—
45	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
46	D₄×C₇₉	—	2	4	no	—
47	D₄×C₇₉	—	2	4	no	—
48	D₄×C₇₉	—	4	2	no	—
49	D₄×C₇₉	—	4	2	no	—
50	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
55	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
56	C₂×Dic₇₉	—	2	2	no	—
57	C₂×Dic₇₉	—	2	2	no	—
58	C₂×Dic₇₉	—	4	2	no	—
59	C₂×Dic₇₉	—	4	4	no	—
60	C₂×Dic₇₉	D₈	1	4	yes	complemented
61	C₈×C₇₉	C₂²⋊C₄	1	4	yes	complemented
62	C₇₉⋊C₈	—	2	2	no	—
63	C₂²×C₇₉	—	4	4	no	—
64	C₂²×C₇₉	—	4	8	no	—
65	C₂²×C₇₉	—	8	4	no	—
66	C₂²×C₇₉	—	8	2	no	—
67	C₂²×C₇₉	—	8	2	no	—
68	C₂²×C₇₉	C₄×D₄	1	1	yes	characteristic
69	C₂²×C₇₉	#121	1	1	yes	characteristic
70	C₂²×C₇₉	#127	1	1	yes	characteristic
71	C₂²×C₇₉	C₂×D₈	1	2	yes	—
72	C₂²×C₇₉	#135	1	2	yes	—
73	C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
75	C₄×C₇₉	#121	1	1	yes	characteristic
76	C₄×C₇₉	#124	1	1	yes	characteristic
77	Dic₇₉	—	2	4	no	—
78	Dic₇₉	—	8	2	no	—
79	C₂×C₇₉	—	8	4	no	—
80	C₂×C₇₉	#390	1	1	yes	characteristic
81	C₂×C₇₉	C₄×D₈	1	2	yes	—
82	C₂×C₇₉	#466	1	2	yes	—
83	C₂×C₇₉	#493	1	1	yes	characteristic
84	C₂×C₇₉	#495	1	1	yes	characteristic
85	#1970	—	79	1	no	maximal, Sylow
86	C₇₉	#1970	1	1	yes	characteristic, complemented, Sylow
87	C₂²×D₈	D₇₉	1	1	yes	characteristic
88	C₂×#107	—	79	1	no	—
89	C₂×#102	—	79	2	no	—
90	C₂×C₄×C₈	—	79	1	no	—
91	#390	—	79	2	no	—
92	C₂²×D₄	C₂×D₇₉	1	2	yes	—
93	C₂×D₈	—	2	2	no	—
94	C₂×D₈	—	2	2	no	—
95	C₂×D₈	Dic₇₉	1	4	yes	complemented
96	C₂²×C₈	—	79	1	no	—
97	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
98	C₂×C₄⋊C₄	—	79	2	no	—
99	C₂×C₂²⋊C₄	—	158	4	no	—
100	C₂×C₄²	—	79	1	no	—
101	#107	—	158	2	no	—
102	#102	—	158	4	no	—
103	C₄×C₈	—	158	2	no	—
104	C₂⁴	—	2	2	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	—	2	2	no	—
110	C₂×D₄	C₂×Dic₇₉	1	4	yes	—
111	C₂²×C₄	—	79	2	no	—
112	C₂²×C₄	—	158	2	no	—
113	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
114	D₈	—	2	8	no	—
115	C₂×C₈	—	79	2	no	—
116	C₂×C₈	—	158	1	no	—
117	C₂×C₈	—	158	1	no	—
118	C₂×C₈	C₂×Dic₇₉	1	2	yes	—
119	C₂×C₈	C₂²⋊D₇₉	1	2	yes	—
120	C₂×C₈	C₂²⋊D₇₉	1	2	yes	—
121	C₄⋊C₄	—	158	2	no	—
122	C₄⋊C₄	—	158	2	no	—
123	C₂²⋊C₄	—	316	8	no	—
124	C₄²	—	158	1	no	—
125	C₄²	—	158	1	no	—
126	C₂³	—	2	4	no	—
127	C₂³	—	4	2	no	—
128	C₂³	—	4	2	no	—
129	C₂³	—	4	4	no	—
130	C₂³	—	4	4	no	—
131	C₂³	D₄×D₇₉	1	1	yes	characteristic
132	D₄	—	2	4	no	—
133	D₄	—	2	4	no	—
134	D₄	—	4	2	no	—
135	D₄	—	4	2	no	—
136	C₂×C₄	—	316	2	no	—
137	C₂×C₄	—	316	4	no	—
138	C₂×C₄	—	79	4	no	—
139	C₂×C₄	—	158	2	no	—
140	C₂×C₄	—	158	2	no	—
141	C₂×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
142	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
145	C₂×C₄	#11929623	1	1	yes	characteristic
146	C₂×C₄	#11929623	1	1	yes	characteristic
147	C₈	—	158	2	no	—
148	C₈	C₂²⋊Dic₇₉	1	4	yes	complemented
149	C₂²	—	4	8	no	—
150	C₂²	—	4	4	no	—
151	C₂²	—	8	2	no	—
152	C₂²	—	8	2	no	—
153	C₂²	—	8	4	no	—
154	C₂²	D₄×Dic₇₉	1	1	yes	characteristic
155	C₂²	#14579132	1	1	yes	characteristic
156	C₂²	#14579157	1	1	yes	characteristic
157	C₂²	D₈×D₇₉	1	2	yes	—
158	C₂²	#14579180	1	2	yes	—
159	C₄	—	632	2	no	—
160	C₄	—	158	4	no	—
161	C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
162	C₄	D₄×Dic₇₉	1	1	yes	characteristic
163	C₄	#14579132	1	1	yes	characteristic
164	C₄	#14579148	1	1	yes	characteristic
165	C₂	—	8	4	no	—
166	C₂	#19098466	1	1	yes	characteristic
167	C₂	D₈×Dic₇₉	1	2	yes	—
168	C₂	#19098796	1	2	yes	—
169	C₂	#19098938	1	1	yes	characteristic
170	C₂	#19098946	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579095

C₂

yes

maximal, characteristic, complemented

C₂×#14579082

C₂

yes

maximal, complemented

C₂×C₈×Dic₇₉

C₂

yes

maximal, characteristic, complemented

#19098466

C₂

yes

maximal, complemented

C₂²×D₈×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₂²⋊Dic₇₉

—

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

#14579095

—

#14579082

—

C₈×Dic₇₉

—

C₂²×D₄×C₇₉

C₂²

yes

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

C₄

yes

complemented

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂²×Dic₇₉

—

C₂²×Dic₇₉

D₄

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂⁴×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

D₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₈×C₇₉

D₄

yes

—

C₂×C₈×C₇₉

D₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

D₈

yes

complemented

C₈×C₇₉

C₂²⋊C₄

yes

complemented

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₄×D₄

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

C₂×D₈

yes

—

C₂²×C₇₉

#135

yes

—

C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

C₄×D₄

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#390

yes

characteristic

C₂×C₇₉

C₄×D₈

yes

—

C₂×C₇₉

#466

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂²×D₈

D₇₉

yes

characteristic

C₂×#107

—

C₂×#102

—

C₂×C₄×C₈

—

#390

—

C₂²×D₄

C₂×D₇₉

yes

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

Dic₇₉

yes

complemented

C₂²×C₈

—

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

158

—

100

C₂×C₄²

—

101

#107

—

158

—

102

#102

—

158

—

103

C₄×C₈

—

158

—

104

C₂⁴

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

C₂×Dic₇₉

yes

—

111

C₂²×C₄

—

112

C₂²×C₄

—

158

—

113

C₂²×C₄

C₂²×D₇₉

yes

characteristic

114

D₈

—

115

C₂×C₈

—

116

C₂×C₈

—

158

—

117

C₂×C₈

—

158

—

118

C₂×C₈

C₂×Dic₇₉

yes

—

119

C₂×C₈

C₂²⋊D₇₉

yes

—

120

C₂×C₈

C₂²⋊D₇₉

yes

—

121

C₄⋊C₄

—

158

—

122

C₄⋊C₄

—

158

—

123

C₂²⋊C₄

—

316

—

124

C₄²

—

158

—

125

C₄²

—

158

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

D₄×D₇₉

yes

characteristic

132

D₄

—

133

D₄

—

134

D₄

—

135

D₄

—

136

C₂×C₄

—

316

—

137

C₂×C₄

—

316

—

138

C₂×C₄

—

139

C₂×C₄

—

158

—

140

C₂×C₄

—

158

—

141

C₂×C₄

C₂²×Dic₇₉

yes

characteristic

142

C₂×C₄

D₄×D₇₉

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

145

C₂×C₄

#11929623

yes

characteristic

146

C₂×C₄

#11929623

yes

characteristic

147

C₈

—

158

—

148

C₈

C₂²⋊Dic₇₉

yes

complemented

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

D₄×Dic₇₉

yes

characteristic

155

C₂²

#14579132

yes

characteristic

156

C₂²

#14579157

yes

characteristic

157

C₂²

D₈×D₇₉

yes

—

158

C₂²

#14579180

yes

—

159

C₄

—

632

—

160

C₄

—

158

—

161

C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

162

C₄

D₄×Dic₇₉

yes

characteristic

163

C₄

#14579132

yes

characteristic

164

C₄

#14579148

yes

characteristic

165

C₂

—

166

C₂

#19098466

yes

characteristic

167

C₂

D₈×Dic₇₉

yes

—

168

C₂

#19098796

yes

—

169

C₂

#19098938

yes

characteristic

170

C₂

#19098946

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting, prime radical	C₂²×D₈×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	C₂²×D₈	D₇₉
hypocenter	C₇₉	#1970
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×D₈

subgroup

G/H

center

C₂³

D₄×D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting, prime radical

C₂²×D₈×C₇₉

C₂

socle

C₂³×C₇₉

C₂×D₄

cosocle

C₂²×C₄×C₇₉

C₂³

hypercenter

C₂²×D₈

D₇₉

hypocenter

C₇₉

#1970

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×D₈

N	H ≅ G/N	type
1. C₂×#14579095	C₂	semidirect
2. C₂×#14579082	C₂	semidirect
3. C₂×C₈×Dic₇₉	C₂	semidirect
4. #19098466	C₂	semidirect
16. C₂×D₈×C₇₉	C₄	semidirect
60. C₂×Dic₇₉	D₈	semidirect
61. C₈×C₇₉	C₂²⋊C₄	semidirect
86. C₇₉	#1970	semidirect
95. C₂×D₈	Dic₇₉	semidirect
148. C₈	C₂²⋊Dic₇₉	semidirect

H ≅ G/N

type

1. C₂×#14579095

C₂

semidirect

2. C₂×#14579082

C₂

semidirect

3. C₂×C₈×Dic₇₉

C₂

semidirect

4. #19098466

C₂

semidirect

16. C₂×D₈×C₇₉

C₄

semidirect

60. C₂×Dic₇₉

D₈

semidirect

61. C₈×C₇₉

C₂²⋊C₄

semidirect

86. C₇₉

#1970

semidirect

95. C₂×D₈

Dic₇₉

semidirect

148. C₈

C₂²⋊Dic₇₉

semidirect

Subgroups — #23764267

Conjugacy classes of subgroups

Complements