Group #23763416 — subgroups

Subgroups — #23763416

Order 10112.

292 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₄×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579086	C₂	1	2	yes	maximal
3	#19098429	C₂	1	1	yes	maximal, characteristic
4	#19098326	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
7	C₄×Dic₁₅₈	—	2	2	no	—
8	C₄×Dic₁₅₈	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₇₉⋊C₄	—	2	1	no	—
12	C₂×C₄×Dic₇₉	—	2	1	no	—
13	#14579086	—	2	4	no	—
14	C₇₉⋊#95	—	2	1	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	C₄⋊C₈×C₇₉	—	2	2	no	—
19	C₂×Dic₁₅₈	—	2	1	no	—
20	C₂×Dic₁₅₈	—	2	1	no	—
21	C₂×Dic₁₅₈	—	2	1	no	—
22	C₂×Dic₁₅₈	C₂×C₄	1	2	yes	—
23	C₂×Dic₁₅₈	D₄	1	2	yes	—
24	C₂×Dic₁₅₈	Q₈	1	2	yes	—
25	C₂²×Dic₇₉	—	2	1	no	—
26	C₂²×Dic₇₉	—	2	1	no	—
27	C₂²×Dic₇₉	—	2	2	no	—
28	C₄⋊Dic₇₉	—	2	1	no	—
29	C₄⋊Dic₇₉	—	2	2	no	—
30	C₄⋊Dic₇₉	—	2	2	no	—
31	C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
32	Dic₇₉⋊C₄	—	2	4	no	—
33	C₄×Dic₇₉	—	2	4	no	—
34	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₇₉	—	2	2	no	—
38	C₂×C₈×C₇₉	—	2	2	no	—
39	C₂×C₈×C₇₉	—	2	2	no	—
40	C₄²×C₇₉	C₂×C₄	1	2	yes	—
41	C₄²×C₇₉	D₄	1	2	yes	—
42	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₇₉	—	2	2	no	—
44	C₂×C₄×C₇₉	—	2	2	no	—
45	C₂×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
52	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
53	C₂×Dic₇₉	—	2	2	no	—
54	C₂×Dic₇₉	—	2	2	no	—
55	C₂×Dic₇₉	—	2	2	no	—
56	C₂×Dic₇₉	—	2	2	no	—
57	C₂×Dic₇₉	—	4	2	no	—
58	C₂×Dic₇₉	—	4	2	no	—
59	C₂×Dic₇₉	—	4	2	no	—
60	C₂×Dic₇₉	—	4	2	no	—
61	Dic₁₅₈	—	2	2	no	—
62	Dic₁₅₈	—	2	2	no	—
63	Dic₁₅₈	—	2	2	no	—
64	Dic₁₅₈	C₄⋊C₄	1	4	yes	—
65	C₈×C₇₉	—	4	2	no	—
66	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₇₉	#121	1	1	yes	characteristic
68	C₂²×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂²×C₇₉	#135	1	2	yes	—
70	C₂²×C₇₉	#137	1	2	yes	—
71	C₄×C₇₉	—	2	2	no	—
72	C₄×C₇₉	—	2	2	no	—
73	C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₄×C₇₉	#117	1	1	yes	characteristic
75	C₄×C₇₉	#121	1	1	yes	characteristic
76	C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
77	Dic₇₉	—	2	4	no	—
78	Dic₇₉	—	4	2	no	—
79	Dic₇₉	—	8	2	no	—
80	C₂×C₇₉	#380	1	1	yes	characteristic
81	C₂×C₇₉	#416	1	1	yes	characteristic
82	C₂×C₇₉	#419	1	1	yes	characteristic
83	C₂×C₇₉	#464	1	2	yes	—
84	C₂×C₇₉	#479	1	2	yes	—
85	#1805	—	79	1	no	maximal, Sylow
86	C₇₉	#1805	1	1	yes	characteristic, complemented, Sylow
87	C₂×C₄×Q₈	—	79	1	no	—
88	C₂×C₄⋊C₈	D₇₉	1	1	yes	characteristic
89	C₂×#103	—	79	2	no	—
90	#382	—	79	1	no	—
91	#340	—	79	2	no	—
92	C₂²×Q₈	—	79	1	no	—
93	C₂²×C₈	C₂×D₇₉	1	2	yes	—
94	C₄×Q₈	—	79	4	no	—
95	C₄×Q₈	—	158	2	no	—
96	C₂×C₄⋊C₄	—	79	1	no	—
97	C₂×C₄⋊C₄	—	79	2	no	—
98	C₂×C₄⋊C₄	—	158	1	no	—
99	C₂×C₄²	—	158	1	no	—
100	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
101	C₄⋊C₈	—	2	2	no	—
102	#103	—	158	4	no	—
103	#95	—	158	1	no	—
104	#95	—	158	1	no	—
105	C₂×Q₈	—	79	2	no	—
106	C₂×Q₈	—	79	2	no	—
107	C₂×Q₈	—	79	2	no	—
108	C₂×Q₈	—	158	1	no	—
109	C₂×Q₈	—	158	1	no	—
110	C₂×Q₈	—	158	1	no	—
111	C₂²×C₄	—	158	2	no	—
112	C₂²×C₄	—	158	1	no	—
113	C₂²×C₄	—	158	1	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	—	79	2	no	—
121	C₄⋊C₄	—	158	2	no	—
122	C₄⋊C₄	—	158	2	no	—
123	C₄⋊C₄	—	158	4	no	—
124	C₄⋊C₄	—	158	1	no	—
125	C₄²	—	158	4	no	—
126	C₄²	C₄×D₇₉	1	2	yes	—
127	C₄²	C₂²⋊D₇₉	1	2	yes	—
128	C₂³	C₂×D₃₁₆	1	1	yes	characteristic
129	Q₈	—	79	4	no	—
130	Q₈	—	158	2	no	—
131	Q₈	—	158	2	no	—
132	Q₈	—	158	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	316	2	no	—
136	C₂×C₄	—	316	2	no	—
137	C₂×C₄	—	316	2	no	—
138	C₂×C₄	—	316	2	no	—
139	C₂×C₄	—	158	2	no	—
140	C₂×C₄	—	158	2	no	—
141	C₂×C₄	—	158	2	no	—
142	C₂×C₄	—	158	2	no	—
143	C₂×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
144	C₂×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
145	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
146	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
148	C₂×C₄	Q₈×D₇₉	1	1	yes	characteristic
149	C₂×C₄	#11929623	1	1	yes	characteristic
150	C₂×C₄	#11929625	1	1	yes	characteristic
151	C₈	—	4	2	no	—
152	C₂²	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
153	C₂²	#14579136	1	1	yes	characteristic
154	C₂²	#14579141	1	1	yes	characteristic
155	C₂²	#14579183	1	2	yes	—
156	C₂²	#14579194	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	316	2	no	—
160	C₄	—	632	2	no	—
161	C₄	—	158	4	no	—
162	C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
163	C₄	#14579119	1	1	yes	characteristic
164	C₄	#14579132	1	1	yes	characteristic
165	C₄	#14579137	1	1	yes	characteristic
166	C₂	#19098420	1	1	yes	characteristic
167	C₂	#19098560	1	1	yes	characteristic
168	C₂	#19098573	1	1	yes	characteristic
169	C₂	#19098786	1	2	yes	—
170	C₂	#19098871	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₅₈

C₂

yes

maximal, characteristic

C₂×#14579086

C₂

yes

maximal

#19098429

C₂

yes

maximal, characteristic

#19098326

C₂

yes

maximal

C₂×C₄⋊C₈×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

characteristic

C₄×Dic₁₅₈

—

C₄×Dic₁₅₈

C₄

yes

complemented

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×Dic₇₉⋊C₄

—

C₂×C₄×Dic₇₉

—

#14579086

—

C₇₉⋊#95

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂×C₄

yes

—

C₂×Dic₁₅₈

D₄

yes

—

C₂×Dic₁₅₈

Q₈

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

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

C₂×C₄

yes

—

C₄²×C₇₉

D₄

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

—

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×Q₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

C₄⋊C₄

yes

—

C₈×C₇₉

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₉

#121

yes

characteristic

C₂²×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₉

#135

yes

—

C₂²×C₇₉

#137

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×C₄⋊C₄

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#479

yes

—

#1805

—

maximal, Sylow

C₇₉

#1805

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

D₇₉

yes

—

—

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₇₉

yes

—

C₄×Q₈

—

C₄×Q₈

—

158

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄²

—

158

—

100

C₂×C₄²

C₂×D₇₉

yes

characteristic

101

C₄⋊C₈

—

102

#103

—

158

—

103

#95

—

158

—

104

#95

—

158

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

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

—

121

C₄⋊C₄

—

158

—

122

C₄⋊C₄

—

158

—

123

C₄⋊C₄

—

158

—

124

C₄⋊C₄

—

158

—

125

C₄²

—

158

—

126

C₄²

C₄×D₇₉

yes

—

127

C₄²

C₂²⋊D₇₉

yes

—

128

C₂³

C₂×D₃₁₆

yes

characteristic

129

Q₈

—

130

Q₈

—

158

—

131

Q₈

—

158

—

132

Q₈

—

158

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

316

—

136

C₂×C₄

—

316

—

137

C₂×C₄

—

316

—

138

C₂×C₄

—

316

—

139

C₂×C₄

—

158

—

140

C₂×C₄

—

158

—

141

C₂×C₄

—

158

—

142

C₂×C₄

—

158

—

143

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

—

144

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

—

145

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

146

C₂×C₄

D₄×D₇₉

yes

characteristic

147

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

148

C₂×C₄

Q₈×D₇₉

yes

characteristic

149

C₂×C₄

#11929623

yes

characteristic

150

C₂×C₄

#11929625

yes

characteristic

151

C₈

—

152

C₂²

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

153

C₂²

#14579136

yes

characteristic

154

C₂²

#14579141

yes

characteristic

155

C₂²

#14579183

yes

—

156

C₂²

#14579194

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

316

—

160

C₄

—

632

—

161

C₄

—

158

—

162

C₄

C₄⋊C₄×D₇₉

yes

characteristic

163

C₄

#14579119

yes

characteristic

164

C₄

#14579132

yes

characteristic

165

C₄

#14579137

yes

characteristic

166

C₂

#19098420

yes

characteristic

167

C₂

#19098560

yes

characteristic

168

C₂

#19098573

yes

characteristic

169

C₂

#19098786

yes

—

170

C₂

#19098871

yes

—

	subgroup	G/H
center	C₂³	C₂×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	C₂×C₄⋊C₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₇₉
hypocenter	C₇₉	#1805
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₃₁₆

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₂×C₄⋊C₈×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

cosocle

C₂²×C₄×C₇₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₇₉

hypocenter

C₇₉

#1805

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
8. C₄×Dic₁₅₈	C₄	semidirect
86. C₇₉	#1805	semidirect

H ≅ G/N

type

8. C₄×Dic₁₅₈

C₄

semidirect

86. C₇₉

#1805

semidirect

Subgroups — #23763416

Conjugacy classes of subgroups

Complements