Group #23773218 — subgroups

Subgroups — #23773218

Order 10112.

384 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₄×#11929624	C₂	1	1	yes	maximal, characteristic
2	#19098867	C₂	1	4	yes	maximal, complemented
3	#19098860	C₂	1	4	yes	maximal, complemented
4	#19098603	C₂	1	2	yes	maximal, complemented
5	#19098587	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098562	C₂	1	2	yes	maximal
7	C₂×#125×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11929624	C₂²	1	1	yes	characteristic
9	C₂×#11929608	C₂²	1	2	yes	—
10	C₄×Dic₁₅₈	C₂²	1	2	yes	—
11	C₄×D₃₁₆	C₂²	1	2	yes	—
12	C₄×C₂²⋊D₇₉	—	2	2	no	—
13	#14579120	C₂²	1	1	yes	characteristic
14	#14579118	—	2	1	no	—
15	C₄²×D₇₉	—	2	1	no	—
16	#14579094	C₂²	1	4	yes	—
17	#14579092	C₂²	1	4	yes	—
18	C₄⋊(C₇₉⋊C₈)	C₂²	1	4	yes	—
19	#14579084	C₂²	1	4	yes	—
20	#14579081	C₂²	1	4	yes	—
21	#125×C₇₉	—	2	1	no	—
22	#125×C₇₉	—	2	1	no	—
23	#125×C₇₉	C₂²	1	4	yes	—
24	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
26	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
27	#11929624	—	2	2	no	—
28	#11929624	Q₈	1	4	yes	—
29	C₂×Dic₁₅₈	C₂³	1	1	yes	characteristic
30	C₂×D₃₁₆	C₂³	1	1	yes	characteristic
31	C₂×C₂²⋊D₇₉	—	2	1	no	—
32	C₂×C₄×D₇₉	—	2	1	no	—
33	C₂×C₄×D₇₉	—	2	2	no	—
34	#11929608	—	2	4	no	—
35	C₂×C₇₉⋊C₈	C₂³	1	4	yes	—
36	C₄⋊Dic₇₉	C₂³	1	2	yes	—
37	Dic₇₉⋊C₄	—	2	2	no	—
38	C₂²⋊Dic₇₉	—	2	1	no	—
39	(C₂×D₇₉)⋊C₄	—	2	2	no	—
40	C₄×Dic₇₉	—	2	1	no	—
41	C₄×Dic₇₉	—	2	2	no	—
42	C₂²×C₄×C₇₉	—	2	2	no	—
43	C₂²×C₄×C₇₉	D₄	1	2	yes	—
44	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₇₉	—	2	4	no	—
46	C₄⋊C₄×C₇₉	—	2	4	no	—
47	C₄⋊C₄×C₇₉	—	2	2	no	—
48	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
49	C₄²×C₇₉	D₄	1	1	yes	characteristic
50	C₄²×C₇₉	D₄	1	1	yes	characteristic
51	C₄²×C₇₉	C₂³	1	2	yes	—
52	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×D₇₉	—	2	1	no	—
54	C₂×C₄×C₇₉	—	2	4	no	—
55	C₂×C₄×C₇₉	—	4	2	no	—
56	C₂×C₄×C₇₉	—	4	4	no	—
57	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₇₉	#41	1	2	yes	—
62	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
63	C₂²⋊D₇₉	—	2	4	no	—
64	C₂×Dic₇₉	—	2	1	no	—
65	C₂×Dic₇₉	—	2	2	no	—
66	D₃₁₆	—	2	1	no	—
67	D₃₁₆	C₂×Q₈	1	2	yes	—
68	C₄×D₇₉	—	2	4	no	—
69	C₄×D₇₉	—	4	4	no	—
70	Dic₁₅₈	—	2	1	no	—
71	Dic₁₅₈	C₂×Q₈	1	2	yes	—
72	C₇₉⋊C₈	—	2	4	no	—
73	C₂²×C₇₉	—	2	2	no	—
74	C₂²×C₇₉	C₂²⋊Q₈	1	2	yes	—
75	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
76	C₂×D₇₉	—	2	2	no	—
77	C₂×D₇₉	—	4	2	no	—
78	C₄×C₇₉	—	2	2	no	—
79	C₄×C₇₉	—	2	2	no	—
80	C₄×C₇₉	—	4	4	no	—
81	C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
82	C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
84	Dic₇₉	—	2	2	no	—
85	Dic₇₉	—	4	2	no	—
86	C₂×C₇₉	—	2	2	no	—
87	C₂×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₇₉	#575	1	2	yes	—
89	D₇₉	—	4	2	no	—
90	#3080	—	79	1	no	maximal, Sylow
91	C₇₉	#3080	1	1	yes	characteristic, complemented, Sylow
92	C₂×#125	D₇₉	1	1	yes	characteristic, complemented
93	C₄×#41	—	79	1	no	—
94	#479	—	79	4	no	—
95	#478	—	79	4	no	—
96	#428	—	79	2	no	—
97	#423	—	79	1	no	—
98	#417	—	79	2	no	—
99	C₂×#41	—	79	1	no	—
100	C₂×M₁₆	—	79	2	no	—
101	#125	—	2	1	no	—
102	#125	—	2	1	no	—
103	#125	C₂×D₇₉	1	4	yes	—
104	C₄×Q₈	—	79	2	no	—
105	C₄×D₄	—	79	2	no	—
106	C₄×D₄	—	158	2	no	—
107	#117	—	79	1	no	—
108	#117	—	158	1	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
111	C₂×C₄²	—	158	1	no	—
112	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
113	#107	—	79	4	no	—
114	#106	—	79	4	no	—
115	C₄⋊C₈	—	79	4	no	—
116	#103	—	79	4	no	—
117	#102	—	79	4	no	—
118	#41	—	79	4	no	—
119	#41	—	158	2	no	—
120	C₂×Q₈	—	79	1	no	—
121	C₂×D₄	—	79	1	no	—
122	C₂×D₄	—	158	1	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	158	2	no	—
125	C₂²×C₄	—	158	1	no	—
126	C₂²×C₄	C₂²⋊D₇₉	1	2	yes	—
127	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
128	M₁₆	—	158	4	no	—
129	C₂×C₈	—	79	4	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	79	2	no	—
134	C₄⋊C₄	—	158	2	no	—
135	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
136	C₂²⋊C₄	—	158	1	no	—
137	C₂²⋊C₄	—	158	2	no	—
138	C₄²	—	158	1	no	—
139	C₄²	—	158	2	no	—
140	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
141	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
142	C₄²	C₂²×D₇₉	1	2	yes	—
143	C₂³	—	158	1	no	—
144	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
145	Q₈	—	79	2	no	—
146	Q₈	—	158	1	no	—
147	D₄	—	79	2	no	—
148	D₄	—	158	4	no	—
149	D₄	—	158	1	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	—	316	4	no	—
154	C₂×C₄	—	158	2	no	—
155	C₂×C₄	—	158	1	no	—
156	C₂×C₄	—	158	4	no	—
157	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
158	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
159	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
160	C₂×C₄	Q₈×D₇₉	1	2	yes	—
161	C₂×C₄	#11929625	1	2	yes	—
162	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
163	C₈	—	158	4	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	316	2	no	—
166	C₂²	—	158	2	no	—
167	C₂²	#14579137	1	2	yes	—
168	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	2	2	no	—
171	C₄	—	4	4	no	—
172	C₄	—	316	2	no	—
173	C₄	—	158	2	no	—
174	C₄	#14579137	1	2	yes	—
175	C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
176	C₄	C₂×#11929625	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	—	316	2	no	—
179	C₂	C₂×#14579137	1	1	yes	characteristic
180	C₂	#19099361	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11929624

C₂

yes

maximal, characteristic

#19098867

C₂

yes

maximal, complemented

#19098860

C₂

yes

maximal, complemented

#19098603

C₂

yes

maximal, complemented

#19098587

C₂

yes

maximal, characteristic, complemented

#19098562

C₂

yes

maximal

C₂×#125×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929624

C₂²

yes

characteristic

C₂×#11929608

C₂²

yes

—

C₄×Dic₁₅₈

C₂²

yes

—

C₄×D₃₁₆

C₂²

yes

—

C₄×C₂²⋊D₇₉

—

#14579120

C₂²

yes

characteristic

#14579118

—

C₄²×D₇₉

—

#14579094

C₂²

yes

—

#14579092

C₂²

yes

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

#14579084

C₂²

yes

—

#14579081

C₂²

yes

—

#125×C₇₉

—

#125×C₇₉

—

#125×C₇₉

C₂²

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

#11929624

—

#11929624

Q₈

yes

—

C₂×Dic₁₅₈

C₂³

yes

characteristic

C₂×D₃₁₆

C₂³

yes

characteristic

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

C₂×C₄×D₇₉

—

#11929608

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

C₂³

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂²×D₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

C₂×Q₈

yes

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

C₂×Q₈

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²⋊Q₈

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂²⋊Q₈

yes

—

C₄×C₇₉

C₂²×Q₈

yes

characteristic

C₄×C₇₉

C₂×#41

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₇₉

#575

yes

—

D₇₉

—

#3080

—

maximal, Sylow

C₇₉

#3080

yes

characteristic, complemented, Sylow

C₂×#125

D₇₉

yes

characteristic, complemented

C₄×#41

—

—

—

—

—

—

C₂×#41

—

100

C₂×M₁₆

—

101

#125

—

102

#125

—

103

#125

C₂×D₇₉

yes

—

104

C₄×Q₈

—

105

C₄×D₄

—

106

C₄×D₄

—

158

—

107

#117

—

108

#117

—

158

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

111

C₂×C₄²

—

158

—

112

C₂×C₄²

C₂×D₇₉

yes

characteristic

113

#107

—

114

#106

—

115

C₄⋊C₈

—

116

#103

—

117

#102

—

118

#41

—

119

#41

—

158

—

120

C₂×Q₈

—

121

C₂×D₄

—

122

C₂×D₄

—

158

—

123

C₂²×C₄

—

124

C₂²×C₄

—

158

—

125

C₂²×C₄

—

158

—

126

C₂²×C₄

C₂²⋊D₇₉

yes

—

127

C₂²×C₄

C₂²×D₇₉

yes

characteristic

128

M₁₆

—

158

—

129

C₂×C₈

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

158

—

135

C₄⋊C₄

C₂²×D₇₉

yes

—

136

C₂²⋊C₄

—

158

—

137

C₂²⋊C₄

—

158

—

138

C₄²

—

158

—

139

C₄²

—

158

—

140

C₄²

C₂²⋊D₇₉

yes

characteristic

141

C₄²

C₂²⋊D₇₉

yes

characteristic

142

C₄²

C₂²×D₇₉

yes

—

143

C₂³

—

158

—

144

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

145

Q₈

—

146

Q₈

—

158

—

147

D₄

—

148

D₄

—

158

—

149

D₄

—

158

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

316

—

154

C₂×C₄

—

158

—

155

C₂×C₄

—

158

—

156

C₂×C₄

—

158

—

157

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

158

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

159

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

160

C₂×C₄

Q₈×D₇₉

yes

—

161

C₂×C₄

#11929625

yes

—

162

C₂×C₄

C₂³×D₇₉

yes

characteristic

163

C₈

—

158

—

164

C₂²

—

165

C₂²

—

316

—

166

C₂²

—

158

—

167

C₂²

#14579137

yes

—

168

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

316

—

173

C₄

—

158

—

174

C₄

#14579137

yes

—

175

C₄

C₂×Q₈×D₇₉

yes

characteristic

176

C₄

C₂×#11929625

yes

characteristic

177

C₂

—

178

C₂

—

316

—

179

C₂

C₂×#14579137

yes

characteristic

180

C₂

#19099361

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₂×#125×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	C₂×#11929624	C₂²
hypercenter	C₂×#125	D₇₉
hypocenter	C₇₉	#3080
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₂×#125×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

C₂²

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #19098867	C₂	semidirect
3. #19098860	C₂	semidirect
4. #19098603	C₂	semidirect
5. #19098587	C₂	semidirect
7. C₂×#125×C₇₉	C₂	semidirect
91. C₇₉	#3080	semidirect
92. C₂×#125	D₇₉	semidirect

H ≅ G/N

type

2. #19098867

C₂

semidirect

3. #19098860

C₂

semidirect

4. #19098603

C₂

semidirect

5. #19098587

C₂

semidirect

7. C₂×#125×C₇₉

C₂

semidirect

91. C₇₉

#3080

semidirect

92. C₂×#125

D₇₉

semidirect

Subgroups — #23773218

Conjugacy classes of subgroups

Complements