Group #23765324 — subgroups

Subgroups — #23765324

Order 10112.

230 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 51 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×#14579180	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14579106	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14579102	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098800	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098366	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₇₉)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11929623	—	2	1	no	—
9	#14579180	—	2	2	no	—
10	#14579180	—	2	2	no	—
11	C₂×Dic₃₁₆	C₂²	1	1	yes	characteristic
12	C₂×#11929612	—	2	1	no	—
13	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₇₉	C₂²	1	1	yes	characteristic
15	#14579136	—	2	1	no	—
16	#14579106	—	2	2	no	—
17	#14579102	—	2	2	no	—
18	C₂×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
19	#14579081	—	2	1	no	—
20	C₂×D₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
22	#107×C₇₉	C₂²	1	1	yes	characteristic
23	#11929623	—	2	4	no	—
24	#11929623	—	4	2	no	—
25	C₂×Dic₁₅₈	—	2	1	no	—
26	C₂×D₃₁₆	—	2	1	no	—
27	C₂×D₃₁₆	—	4	1	no	—
28	C₂×C₂²⋊D₇₉	—	4	1	no	—
29	C₂²×Dic₇₉	—	4	1	no	—
30	C₂×C₄×D₇₉	D₄	1	1	yes	characteristic
31	Dic₃₁₆	—	2	1	no	—
32	Dic₃₁₆	D₄	1	2	yes	complemented
33	#11929612	—	2	4	no	—
34	D₆₃₂	—	2	1	no	—
35	D₆₃₂	—	2	1	no	—
36	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
37	C₈×D₇₉	—	2	1	no	—
38	C₈×D₇₉	—	2	1	no	—
39	(C₂×D₇₉)⋊C₄	—	4	1	no	—
40	C₇₉⋊C₁₆	—	2	1	no	—
41	C₂×D₄×C₇₉	—	2	1	no	—
42	D₈×C₇₉	—	2	1	no	—
43	D₈×C₇₉	D₄	1	2	yes	complemented
44	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₇₉	—	2	1	no	—
46	C₁₆×C₇₉	—	2	1	no	—
47	C₂³×C₇₉	—	4	1	no	—
48	C₂²×D₇₉	—	4	1	no	—
49	C₂²×D₇₉	D₈	1	1	yes	characteristic
50	D₄×C₇₉	—	2	2	no	—
51	D₄×C₇₉	—	4	1	no	—
52	C₂×C₄×C₇₉	—	4	1	no	—
53	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
54	C₂²⋊D₇₉	—	4	4	no	—
55	C₂×Dic₇₉	—	4	4	no	—
56	C₂×Dic₇₉	—	4	1	no	—
57	C₂×Dic₇₉	D₈	1	1	yes	characteristic
58	D₃₁₆	—	4	1	no	—
59	D₃₁₆	—	4	1	no	—
60	D₃₁₆	—	8	2	no	—
61	C₄×D₇₉	—	2	1	no	—
62	C₄×D₇₉	—	2	1	no	—
63	Dic₁₅₈	—	2	2	no	—
64	Dic₁₅₈	—	4	1	no	—
65	C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
67	C₇₉⋊C₈	—	2	1	no	—
68	C₂²×C₇₉	—	4	2	no	—
69	C₂²×C₇₉	—	8	1	no	—
70	C₂²×C₇₉	—	8	1	no	—
71	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
72	C₂×D₇₉	—	2	1	no	—
73	C₂×D₇₉	—	2	1	no	—
74	C₂×D₇₉	—	2	1	no	—
75	C₂×D₇₉	—	8	1	no	—
76	C₂×D₇₉	—	8	1	no	—
77	C₂×D₇₉	—	8	1	no	—
78	C₄×C₇₉	—	8	1	no	—
79	C₄×C₇₉	#120	1	1	yes	characteristic
80	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
81	Dic₇₉	—	2	1	no	—
82	Dic₇₉	—	4	2	no	—
83	C₂×C₇₉	—	8	2	no	—
84	C₂×C₇₉	#447	1	1	yes	characteristic
85	C₂×C₇₉	#508	1	1	yes	characteristic
86	C₂×C₇₉	#509	1	1	yes	characteristic
87	D₇₉	—	4	1	no	—
88	D₇₉	—	16	1	no	—
89	#2186	—	79	1	no	maximal, Sylow
90	C₇₉	#2186	1	1	yes	characteristic, complemented, Sylow
91	C₂×#135	—	79	1	no	—
92	C₂×SD₃₂	—	79	1	no	—
93	C₂×D₁₆	—	79	1	no	—
94	#466	—	79	1	no	—
95	#358	—	79	1	no	—
96	#357	D₇₉	1	1	yes	characteristic, complemented
97	C₂²⋊C₁₆	—	79	1	no	—
98	C₂×#41	—	158	1	no	—
99	#135	—	158	2	no	—
100	#135	—	158	2	no	—
101	C₂×Q₁₆	—	79	1	no	—
102	C₂×SD₁₆	—	158	1	no	—
103	C₂×D₈	—	79	1	no	—
104	C₂×D₈	C₂×D₇₉	1	1	yes	characteristic
105	C₂²×C₈	—	79	1	no	—
106	#121	—	158	1	no	—
107	SD₃₂	—	158	2	no	—
108	D₁₆	—	158	2	no	—
109	C₂×C₁₆	—	79	1	no	—
110	C₂×C₁₆	C₂×D₇₉	1	1	yes	characteristic
111	#107	C₂×D₇₉	1	1	yes	characteristic
112	#102	—	158	1	no	—
113	#41	—	316	2	no	—
114	#41	—	158	4	no	—
115	C₂×Q₈	—	158	1	no	—
116	C₂×D₄	—	2	1	no	—
117	C₂×D₄	—	316	1	no	—
118	C₂×D₄	—	316	1	no	—
119	C₂×D₄	—	158	1	no	—
120	C₂²×C₄	—	316	1	no	—
121	C₂²×C₄	—	79	1	no	—
122	Q₁₆	—	79	2	no	—
123	Q₁₆	—	158	1	no	—
124	SD₁₆	—	158	4	no	—
125	D₈	—	2	1	no	—
126	D₈	—	158	1	no	—
127	D₈	—	158	1	no	—
128	D₈	D₃₁₆	1	2	yes	complemented
129	C₂×C₈	—	79	1	no	—
130	C₂×C₈	—	158	1	no	—
131	C₂×C₈	—	158	1	no	—
132	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
133	C₄⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	316	1	no	—
135	C₁₆	—	2	1	no	—
136	C₁₆	—	158	1	no	—
137	C₂³	—	4	1	no	—
138	C₂³	—	316	1	no	—
139	C₂³	—	79	1	no	—
140	Q₈	—	316	1	no	—
141	Q₈	—	158	2	no	—
142	D₄	—	2	2	no	—
143	D₄	—	4	1	no	—
144	D₄	—	316	4	no	—
145	D₄	—	316	1	no	—
146	D₄	—	316	1	no	—
147	D₄	—	632	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	316	1	no	—
150	C₂×C₄	—	316	4	no	—
151	C₂×C₄	—	79	1	no	—
152	C₂×C₄	—	158	1	no	—
153	C₂×C₄	—	158	1	no	—
154	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
155	C₈	—	158	1	no	—
156	C₈	D₄×D₇₉	1	1	yes	characteristic
157	C₈	C₂×D₃₁₆	1	1	yes	characteristic
158	C₂²	—	4	2	no	—
159	C₂²	—	8	1	no	—
160	C₂²	—	8	1	no	—
161	C₂²	—	632	1	no	—
162	C₂²	—	632	1	no	—
163	C₂²	—	632	1	no	—
164	C₂²	—	158	1	no	—
165	C₂²	—	158	1	no	—
166	C₂²	—	158	1	no	—
167	C₂²	D₈×D₇₉	1	1	yes	characteristic
168	C₄	—	8	1	no	—
169	C₄	—	316	2	no	—
170	C₄	—	158	1	no	—
171	C₄	#14579131	1	1	yes	characteristic
172	C₄	#14579188	1	1	yes	characteristic
173	C₂	—	8	2	no	—
174	C₂	—	316	1	no	—
175	C₂	—	1264	1	no	—
176	C₂	#19098698	1	1	yes	characteristic
177	C₂	#19098994	1	1	yes	characteristic
178	C₂	#19098997	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579180

C₂

yes

maximal, characteristic, complemented

C₂×#14579106

C₂

yes

maximal, characteristic, complemented

C₂×#14579102

C₂

yes

maximal, characteristic, complemented

#19098800

C₂

yes

maximal, characteristic, complemented

#19098366

C₂

yes

maximal, characteristic, complemented

(C₂×D₇₉)⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#357×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929623

—

#14579180

—

#14579180

—

C₂×Dic₃₁₆

C₂²

yes

characteristic

C₂×#11929612

—

C₂×D₆₃₂

C₂²

yes

characteristic

C₂×C₈×D₇₉

C₂²

yes

—

—

—

C₂×C₇₉⋊C₁₆

C₂²

yes

characteristic

#14579081

—

C₂×D₈×C₇₉

C₂²

yes

characteristic

C₂×C₁₆×C₇₉

C₂²

yes

characteristic

#107×C₇₉

C₂²

yes

characteristic

#11929623

—

#11929623

—

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×C₂²⋊D₇₉

—

C₂²×Dic₇₉

—

C₂×C₄×D₇₉

D₄

yes

characteristic

Dic₃₁₆

—

Dic₃₁₆

D₄

yes

complemented

#11929612

—

D₆₃₂

—

D₆₃₂

—

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₈×D₇₉

—

C₈×D₇₉

—

(C₂×D₇₉)⋊C₄

—

C₇₉⋊C₁₆

—

C₂×D₄×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

D₄

yes

complemented

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

D₈

yes

characteristic

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

D₈

yes

characteristic

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

C₂×D₄

yes

characteristic

C₈×C₇₉

C₂×D₄

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

D₇₉

—

#2186

—

maximal, Sylow

C₇₉

#2186

yes

characteristic, complemented, Sylow

C₂×#135

—

C₂×SD₃₂

—

C₂×D₁₆

—

#466

—

#358

—

#357

D₇₉

yes

characteristic, complemented

C₂²⋊C₁₆

—

C₂×#41

—

158

—

#135

—

158

—

100

#135

—

158

—

101

C₂×Q₁₆

—

102

C₂×SD₁₆

—

158

—

103

C₂×D₈

—

104

C₂×D₈

C₂×D₇₉

yes

characteristic

105

C₂²×C₈

—

106

#121

—

158

—

107

SD₃₂

—

158

—

108

D₁₆

—

158

—

109

C₂×C₁₆

—

110

C₂×C₁₆

C₂×D₇₉

yes

characteristic

111

#107

C₂×D₇₉

yes

characteristic

112

#102

—

158

—

113

#41

—

316

—

114

#41

—

158

—

115

C₂×Q₈

—

158

—

116

C₂×D₄

—

117

C₂×D₄

—

316

—

118

C₂×D₄

—

316

—

119

C₂×D₄

—

158

—

120

C₂²×C₄

—

316

—

121

C₂²×C₄

—

122

Q₁₆

—

123

Q₁₆

—

158

—

124

SD₁₆

—

158

—

125

D₈

—

126

D₈

—

158

—

127

D₈

—

158

—

128

D₈

D₃₁₆

yes

complemented

129

C₂×C₈

—

130

C₂×C₈

—

158

—

131

C₂×C₈

—

158

—

132

C₂×C₈

C₂²×D₇₉

yes

characteristic

133

C₄⋊C₄

—

134

C₂²⋊C₄

—

316

—

135

C₁₆

—

136

C₁₆

—

158

—

137

C₂³

—

138

C₂³

—

316

—

139

C₂³

—

140

Q₈

—

316

—

141

Q₈

—

158

—

142

D₄

—

143

D₄

—

144

D₄

—

316

—

145

D₄

—

316

—

146

D₄

—

316

—

147

D₄

—

632

—

148

C₂×C₄

—

149

C₂×C₄

—

316

—

150

C₂×C₄

—

316

—

151

C₂×C₄

—

152

C₂×C₄

—

158

—

153

C₂×C₄

—

158

—

154

C₂×C₄

D₄×D₇₉

yes

characteristic

155

C₈

—

158

—

156

C₈

D₄×D₇₉

yes

characteristic

157

C₈

C₂×D₃₁₆

yes

characteristic

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

632

—

162

C₂²

—

632

—

163

C₂²

—

632

—

164

C₂²

—

158

—

165

C₂²

—

158

—

166

C₂²

—

158

—

167

C₂²

D₈×D₇₉

yes

characteristic

168

C₄

—

169

C₄

—

316

—

170

C₄

—

158

—

171

C₄

#14579131

yes

characteristic

172

C₄

#14579188

yes

characteristic

173

C₂

—

174

C₂

—

316

—

175

C₂

—

1264

—

176

C₂

#19098698

yes

characteristic

177

C₂

#19098994

yes

characteristic

178

C₂

#19098997

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂³
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	#357×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₈
hypercenter	#357	D₇₉
hypocenter	C₇₉	#2186
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₇₉

commutator, cosocle

C₂×C₈×C₇₉

C₂³

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

#357×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₈

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. C₂×#14579180	C₂	semidirect
2. C₂×#14579106	C₂	semidirect
3. C₂×#14579102	C₂	semidirect
4. #19098800	C₂	semidirect
5. #19098366	C₂	semidirect
6. (C₂×D₇₉)⋊C₁₆	C₂	semidirect
7. #357×C₇₉	C₂	semidirect
32. Dic₃₁₆	D₄	semidirect
43. D₈×C₇₉	D₄	semidirect
90. C₇₉	#2186	semidirect
96. #357	D₇₉	semidirect
128. D₈	D₃₁₆	semidirect

H ≅ G/N

type

1. C₂×#14579180

C₂

semidirect

2. C₂×#14579106

C₂

semidirect

3. C₂×#14579102

C₂

semidirect

4. #19098800

C₂

semidirect

5. #19098366

C₂

semidirect

6. (C₂×D₇₉)⋊C₁₆

C₂

semidirect

7. #357×C₇₉

C₂

semidirect

32. Dic₃₁₆

D₄

semidirect

43. D₈×C₇₉

D₄

semidirect

90. C₇₉

#2186

semidirect

96. #357

D₇₉

semidirect

128. D₈

D₃₁₆

semidirect

Subgroups — #23765324

Conjugacy classes of subgroups

Complements