Group #23762759 — subgroups

Subgroups — #23762759

Order 10112.

188 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 53 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	#19098939	C₂	1	1	yes	maximal, characteristic
2	#19098764	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098760	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098687	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098315	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098306	C₂	1	1	yes	maximal, characteristic, complemented
7	#331×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11929613	—	2	1	no	—
9	C₂×#11929612	—	2	1	no	—
10	C₂×#11929609	—	2	1	no	—
11	C₂×#11929609	—	2	1	no	—
12	#14579158	C₂²	1	1	yes	characteristic
13	C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
14	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₇₉⋊C₈	—	2	1	no	—
16	#14579083	—	2	1	no	—
17	#14579082	—	2	1	no	—
18	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×Dic₇₉	—	2	1	no	—
20	#127×C₇₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₅₈	D₄	1	1	yes	characteristic
24	C₂×D₃₁₆	—	2	1	no	—
25	C₂×D₃₁₆	—	2	1	no	—
26	C₂×D₃₁₆	—	2	1	no	—
27	#11929613	—	4	2	no	—
28	#11929612	—	4	2	no	—
29	#11929609	—	4	2	no	—
30	#11929609	—	4	2	no	—
31	C₂×C₇₉⋊C₈	—	2	1	no	—
32	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
33	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
34	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
35	Dic₇₉⋊C₄	—	2	1	no	—
36	C₄×Dic₇₉	—	2	1	no	—
37	C₂×D₄×C₇₉	—	2	1	no	—
38	C₂×D₄×C₇₉	—	2	1	no	—
39	C₂×D₄×C₇₉	—	2	1	no	—
40	C₂×C₈×C₇₉	—	2	1	no	—
41	C₂×C₈×C₇₉	—	2	1	no	—
42	C₄²×C₇₉	C₂³	1	1	yes	characteristic
43	C₂³×C₇₉	—	4	1	no	—
44	C₂²×D₇₉	—	4	1	no	—
45	D₄×C₇₉	—	4	1	no	—
46	D₄×C₇₉	—	4	1	no	—
47	D₄×C₇₉	—	4	1	no	—
48	D₄×C₇₉	—	4	1	no	—
49	D₄×C₇₉	—	8	1	no	—
50	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
53	C₂×Dic₇₉	—	2	1	no	—
54	C₂×Dic₇₉	—	2	1	no	—
55	D₃₁₆	—	4	1	no	—
56	D₃₁₆	—	4	1	no	—
57	D₃₁₆	—	4	1	no	—
58	D₃₁₆	—	4	1	no	—
59	D₃₁₆	—	8	1	no	—
60	Dic₁₅₈	—	2	1	no	—
61	Dic₁₅₈	D₈	1	2	yes	complemented
62	C₈×C₇₉	—	4	1	no	—
63	C₈×C₇₉	—	4	1	no	—
64	C₇₉⋊C₈	—	2	1	no	—
65	C₇₉⋊C₈	—	4	1	no	—
66	C₇₉⋊C₈	SD₁₆	1	2	yes	complemented
67	C₂²×C₇₉	—	8	1	no	—
68	C₂²×C₇₉	—	8	1	no	—
69	C₂²×C₇₉	—	8	1	no	—
70	C₂²×C₇₉	#121	1	1	yes	characteristic
71	C₂×D₇₉	—	8	1	no	—
72	C₂×D₇₉	—	8	1	no	—
73	C₂×D₇₉	—	8	1	no	—
74	C₄×C₇₉	—	2	1	no	—
75	C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
76	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
77	C₄×C₇₉	#135	1	1	yes	characteristic
78	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
79	Dic₇₉	—	2	2	no	—
80	Dic₇₉	—	4	1	no	—
81	C₂×C₇₉	—	16	1	no	—
82	C₂×C₇₉	#460	1	1	yes	characteristic
83	C₂×C₇₉	#466	1	1	yes	characteristic
84	C₂×C₇₉	#472	1	1	yes	characteristic
85	D₇₉	—	16	1	no	—
86	#1668	—	79	1	no	maximal, Sylow
87	C₇₉	#1668	1	1	yes	characteristic, complemented, Sylow
88	#493	—	79	1	no	—
89	#460	—	79	1	no	—
90	#460	—	79	1	no	—
91	C₈×Q₈	—	79	1	no	—
92	#335	—	79	1	no	—
93	#331	—	79	1	no	—
94	#331	D₇₉	1	1	yes	characteristic, complemented
95	C₂×SD₁₆	—	158	1	no	—
96	C₂×SD₁₆	—	158	1	no	—
97	C₂×D₈	—	158	1	no	—
98	C₂×D₈	—	158	1	no	—
99	#127	—	79	1	no	—
100	#127	C₂×D₇₉	1	1	yes	characteristic
101	C₄×Q₈	—	79	1	no	—
102	C₄⋊C₈	—	79	1	no	—
103	C₄⋊C₈	—	158	1	no	—
104	C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
105	C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
106	#102	—	158	1	no	—
107	#102	—	158	1	no	—
108	C₄×C₈	—	79	1	no	—
109	C₄×C₈	—	158	1	no	—
110	C₂×Q₈	—	79	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	—	158	1	no	—
115	C₂×D₄	—	158	1	no	—
116	C₂×D₄	—	158	1	no	—
117	SD₁₆	—	316	2	no	—
118	SD₁₆	—	316	2	no	—
119	D₈	—	316	2	no	—
120	D₈	—	316	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	79	1	no	—
124	C₂×C₈	—	79	1	no	—
125	C₂×C₈	—	158	1	no	—
126	C₄⋊C₄	—	79	1	no	—
127	C₄⋊C₄	—	158	1	no	—
128	C₄²	—	158	1	no	—
129	C₄²	C₂²×D₇₉	1	1	yes	characteristic
130	C₂³	—	4	1	no	—
131	C₂³	—	316	1	no	—
132	Q₈	—	79	2	no	—
133	Q₈	—	158	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	4	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	4	1	no	—
138	D₄	—	8	1	no	—
139	D₄	—	316	1	no	—
140	D₄	—	316	1	no	—
141	D₄	—	316	1	no	—
142	D₄	—	316	1	no	—
143	D₄	—	632	1	no	—
144	C₂×C₄	—	158	1	no	—
145	C₂×C₄	—	158	1	no	—
146	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
147	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
148	C₂×C₄	#11929624	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	316	1	no	—
152	C₈	—	79	2	no	—
153	C₈	—	158	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	8	1	no	—
156	C₂²	—	8	1	no	—
157	C₂²	—	632	1	no	—
158	C₂²	—	632	1	no	—
159	C₂²	—	632	1	no	—
160	C₂²	#14579134	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	316	1	no	—
163	C₄	—	158	2	no	—
164	C₄	D₈×D₇₉	1	1	yes	characteristic
165	C₄	SD₁₆×D₇₉	1	1	yes	characteristic
166	C₄	#14579182	1	1	yes	characteristic
167	C₄	#14579185	1	1	yes	characteristic
168	C₂	—	16	1	no	—
169	C₂	—	1264	1	no	—
170	C₂	#19098762	1	1	yes	characteristic
171	C₂	#19098798	1	1	yes	characteristic
172	C₂	#19098832	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098939

C₂

yes

maximal, characteristic

#19098764

C₂

yes

maximal, characteristic, complemented

#19098760

C₂

yes

maximal, characteristic, complemented

#19098687

C₂

yes

maximal, characteristic, complemented

#19098315

C₂

yes

maximal, characteristic, complemented

#19098306

C₂

yes

maximal, characteristic, complemented

#331×C₇₉

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₁₅₈

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

Dic₇₉⋊C₈

—

#14579083

—

#14579082

—

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₈×Dic₇₉

—

#127×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

D₄

yes

characteristic

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

—

—

—

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₉

D₄

yes

characteristic

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

C₂²×D₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

Dic₁₅₈

—

Dic₁₅₈

D₈

yes

complemented

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

SD₁₆

yes

complemented

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#121

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×D₈

yes

characteristic

C₄×C₇₉

C₂×SD₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

#1668

—

maximal, Sylow

C₇₉

#1668

yes

characteristic, complemented, Sylow

#493

—

#460

—

#460

—

C₈×Q₈

—

#335

—

#331

—

#331

D₇₉

yes

characteristic, complemented

C₂×SD₁₆

—

158

—

C₂×SD₁₆

—

158

—

C₂×D₈

—

158

—

C₂×D₈

—

158

—

#127

—

100

#127

C₂×D₇₉

yes

characteristic

101

C₄×Q₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

158

—

104

C₄⋊C₈

C₂×D₇₉

yes

characteristic

105

C₄⋊C₈

C₂×D₇₉

yes

characteristic

106

#102

—

158

—

107

#102

—

158

—

108

C₄×C₈

—

109

C₄×C₈

—

158

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

158

—

115

C₂×D₄

—

158

—

116

C₂×D₄

—

158

—

117

SD₁₆

—

316

—

118

SD₁₆

—

316

—

119

D₈

—

316

—

120

D₈

—

316

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

158

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

158

—

128

C₄²

—

158

—

129

C₄²

C₂²×D₇₉

yes

characteristic

130

C₂³

—

131

C₂³

—

316

—

132

Q₈

—

133

Q₈

—

158

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

316

—

140

D₄

—

316

—

141

D₄

—

316

—

142

D₄

—

316

—

143

D₄

—

632

—

144

C₂×C₄

—

158

—

145

C₂×C₄

—

158

—

146

C₂×C₄

D₄×D₇₉

yes

characteristic

147

C₂×C₄

D₄×D₇₉

yes

characteristic

148

C₂×C₄

#11929624

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

316

—

152

C₈

—

153

C₈

—

158

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

632

—

158

C₂²

—

632

—

159

C₂²

—

632

—

160

C₂²

#14579134

yes

characteristic

161

C₄

—

162

C₄

—

316

—

163

C₄

—

158

—

164

C₄

D₈×D₇₉

yes

characteristic

165

C₄

SD₁₆×D₇₉

yes

characteristic

166

C₄

#14579182

yes

characteristic

167

C₄

#14579185

yes

characteristic

168

C₂

—

169

C₂

—

1264

—

170

C₂

#19098762

yes

characteristic

171

C₂

#19098798

yes

characteristic

172

C₂

#19098832

yes

characteristic

	subgroup	G/H
center	C₂²	#14579134
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	#331×C₇₉	C₂
socle	C₂²×C₇₉	#121
prime radical	#19098939	C₂
hypercenter	#331	D₇₉
hypocenter	C₇₉	#1668
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#14579134

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

#331×C₇₉

C₂

C₂²×C₇₉

C₂

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #19098764	C₂	semidirect
3. #19098760	C₂	semidirect
4. #19098687	C₂	semidirect
5. #19098315	C₂	semidirect
6. #19098306	C₂	semidirect
7. #331×C₇₉	C₂	semidirect
61. Dic₁₅₈	D₈	semidirect
66. C₇₉⋊C₈	SD₁₆	semidirect
87. C₇₉	#1668	semidirect
94. #331	D₇₉	semidirect

H ≅ G/N

type

2. #19098764

C₂

semidirect

3. #19098760

C₂

semidirect

4. #19098687

C₂

semidirect

5. #19098315

C₂

semidirect

6. #19098306

C₂

semidirect

7. #331×C₇₉

C₂

semidirect

61. Dic₁₅₈

D₈

semidirect

66. C₇₉⋊C₈

SD₁₆

semidirect

87. C₇₉

#1668

semidirect

94. #331

D₇₉

semidirect

Subgroups — #23762759

Conjugacy classes of subgroups

Complements