Group #23773277 — subgroups

Subgroups — C₂×#19098893 #23773277

Order 10112.

460 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14579140	C₂	1	1	yes	maximal, characteristic
2	#19098893	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14579092	C₂	1	2	yes	maximal, complemented
4	C₂×#14579084	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
8	C₂²×C₇₉⋊C₈	C₂²	1	2	yes	—
9	#14579140	—	2	2	no	—
10	#14579140	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₇₉⋊C₄	—	2	1	no	—
13	C₂×C₂²⋊Dic₇₉	—	2	1	no	—
14	#14579092	C₂²	1	8	yes	complemented
15	#14579084	C₂²	1	8	yes	complemented
16	C₂²⋊(C₇₉⋊C₈)	C₂²	1	4	yes	—
17	C₂³×C₄×C₇₉	—	2	1	no	—
18	C₂³×C₄×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
20	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
21	C₂×C₄⋊C₄×C₇₉	C₂²	1	4	yes	—
22	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
23	C₂×Dic₁₅₈	—	2	1	no	—
24	C₂×Dic₁₅₈	—	2	4	no	—
25	C₂×Dic₁₅₈	C₂³	1	2	yes	—
26	C₂²×Dic₇₉	—	2	1	no	—
27	C₂²×Dic₇₉	—	2	1	no	—
28	C₂×C₇₉⋊C₈	—	2	4	no	—
29	C₂×C₇₉⋊C₈	C₂³	1	4	yes	—
30	C₄⋊Dic₇₉	—	2	1	no	—
31	C₄⋊Dic₇₉	C₂³	1	2	yes	—
32	Dic₇₉⋊C₄	—	2	4	no	—
33	C₂²⋊Dic₇₉	—	2	4	no	—
34	C₂⁴×C₇₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	—	2	4	no	—
36	C₂²×C₄×C₇₉	—	2	4	no	—
37	C₂²×C₄×C₇₉	—	2	2	no	—
38	C₂²×C₄×C₇₉	—	4	4	no	—
39	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₉	D₄	1	2	yes	—
41	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
42	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₇₉	—	2	4	no	—
44	C₄⋊C₄×C₇₉	—	2	2	no	—
45	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
46	C₂³×C₇₉	—	2	2	no	—
47	C₂³×C₇₉	—	2	2	no	—
48	C₂³×C₇₉	SD₁₆	1	4	yes	—
49	C₂³×C₇₉	C₂×D₄	1	2	yes	—
50	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	—	2	2	no	—
52	C₂×C₄×C₇₉	—	2	4	no	—
53	C₂×C₄×C₇₉	—	2	4	no	—
54	C₂×C₄×C₇₉	—	2	4	no	—
55	C₂×C₄×C₇₉	—	4	8	no	—
56	C₂×C₄×C₇₉	—	4	4	no	—
57	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇₉	#41	1	4	yes	—
60	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
61	C₂×Dic₇₉	—	2	2	no	—
62	C₂×Dic₇₉	—	2	2	no	—
63	C₂×Dic₇₉	—	4	2	no	—
64	C₂×Dic₇₉	—	4	2	no	—
65	Dic₁₅₈	—	2	4	no	—
66	Dic₁₅₈	—	4	2	no	—
67	C₇₉⋊C₈	—	2	4	no	—
68	C₂²×C₇₉	—	2	2	no	—
69	C₂²×C₇₉	—	2	2	no	—
70	C₂²×C₇₉	—	2	8	no	—
71	C₂²×C₇₉	#123	1	2	yes	—
72	C₂²×C₇₉	C₂×SD₁₆	1	4	yes	—
73	C₂²×C₇₉	C₂×SD₁₆	1	2	yes	—
74	C₂²×C₇₉	#137	1	2	yes	—
75	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
76	C₄×C₇₉	—	2	2	no	—
77	C₄×C₇₉	—	4	4	no	—
78	C₄×C₇₉	#123	1	2	yes	—
79	C₄×C₇₉	C₂×#41	1	2	yes	—
80	Dic₇₉	—	4	2	no	—
81	Dic₇₉	—	4	2	no	—
82	C₂×C₇₉	—	2	4	no	—
83	C₂×C₇₉	#483	1	4	yes	complemented
84	C₂×C₇₉	C₂×#123	1	1	yes	characteristic
85	C₂×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
86	C₂×C₇₉	C₂×#137	1	1	yes	characteristic
87	C₂×#483	—	79	1	no	maximal, Sylow
88	C₇₉	C₂×#483	1	1	yes	characteristic, complemented, Sylow
89	C₂×C₂²⋊Q₈	—	79	1	no	—
90	C₂²×C₄⋊C₄	D₇₉	1	1	yes	characteristic
91	#483	—	79	8	no	—
92	C₂×#106	—	79	2	no	—
93	C₂×#103	—	79	2	no	—
94	C₂×C₂²⋊C₈	—	79	1	no	—
95	C₂²×Q₈	—	79	1	no	—
96	C₂³×C₄	—	2	1	no	—
97	C₂³×C₄	C₂×D₇₉	1	1	yes	characteristic
98	C₂²×C₈	—	79	2	no	—
99	C₂²⋊Q₈	—	79	4	no	—
100	C₂²⋊Q₈	—	158	2	no	—
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	—	2	1	no	—
103	C₂×C₄⋊C₄	—	79	1	no	—
104	C₂×C₄⋊C₄	—	158	1	no	—
105	C₂×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
106	C₂×C₄⋊C₄	C₂×D₇₉	1	4	yes	—
107	C₂×C₂²⋊C₄	—	158	1	no	—
108	#106	—	79	8	no	—
109	#103	—	79	8	no	—
110	C₂²⋊C₈	—	79	4	no	—
111	C₂⁴	C₂²⋊D₇₉	1	1	yes	characteristic
112	C₂×Q₈	—	79	2	no	—
113	C₂×Q₈	—	158	4	no	—
114	C₂×Q₈	—	158	1	no	—
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	2	4	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	4	4	no	—
119	C₂²×C₄	—	158	1	no	—
120	C₂²×C₄	—	158	1	no	—
121	C₂²×C₄	C₂²⋊D₇₉	1	2	yes	—
122	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
124	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
125	C₂×C₈	—	79	4	no	—
126	C₂×C₈	—	158	4	no	—
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	79	2	no	—
130	C₄⋊C₄	—	158	4	no	—
131	C₄⋊C₄	—	158	1	no	—
132	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
133	C₂²⋊C₄	—	158	4	no	—
134	C₂³	—	2	2	no	—
135	C₂³	—	2	2	no	—
136	C₂³	#11929612	1	4	yes	—
137	C₂³	C₂×C₂²⋊D₇₉	1	2	yes	—
138	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
139	Q₈	—	316	2	no	—
140	Q₈	—	158	4	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	8	no	—
146	C₂×C₄	—	4	4	no	—
147	C₂×C₄	—	316	2	no	—
148	C₂×C₄	—	316	2	no	—
149	C₂×C₄	—	158	2	no	—
150	C₂×C₄	—	158	2	no	—
151	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
152	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
153	C₂×C₄	#11929624	1	4	yes	—
154	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
155	C₈	—	158	4	no	—
156	C₂²	—	2	8	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	2	no	—
159	C₂²	#14579144	1	2	yes	—
160	C₂²	C₂×#11929612	1	4	yes	—
161	C₂²	C₂×#11929612	1	2	yes	—
162	C₂²	#14579192	1	2	yes	—
163	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	4	4	no	—
166	C₄	—	316	2	no	—
167	C₄	—	316	2	no	—
168	C₄	#14579144	1	2	yes	—
169	C₄	C₂×#11929624	1	2	yes	—
170	C₂	—	2	4	no	—
171	C₂	#19098893	1	4	yes	direct product factor
172	C₂	C₂×#14579144	1	1	yes	characteristic
173	C₂	C₂²×#11929612	1	1	yes	characteristic
174	C₂	C₂×#14579192	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579140

C₂

yes

maximal, characteristic

#19098893

C₂

yes

maximal, direct product factor

C₂×#14579092

C₂

yes

maximal, complemented

C₂×#14579084

C₂

yes

maximal, complemented

C₂×C₂²⋊(C₇₉⋊C₈)

C₂

yes

maximal, characteristic

C₂²×C₄⋊C₄×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

—

#14579140

—

#14579140

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×Dic₇₉⋊C₄

—

C₂×C₂²⋊Dic₇₉

—

#14579092

C₂²

yes

complemented

#14579084

C₂²

yes

complemented

C₂²⋊(C₇₉⋊C₈)

C₂²

yes

—

C₂³×C₄×C₇₉

—

C₂³×C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂³

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₂⁴×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

SD₁₆

yes

—

C₂³×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#123

yes

—

C₂²×C₇₉

C₂×SD₁₆

yes

—

C₂²×C₇₉

C₂×SD₁₆

yes

—

C₂²×C₇₉

#137

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#123

yes

—

C₄×C₇₉

C₂×#41

yes

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

C₂²×SD₁₆

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×C₂²⋊Q₈

—

C₂²×C₄⋊C₄

D₇₉

yes

—

—

—

C₂×C₂²⋊C₈

—

C₂²×Q₈

—

C₂³×C₄

—

C₂³×C₄

C₂×D₇₉

yes

characteristic

C₂²×C₈

—

C₂²⋊Q₈

—

100

C₂²⋊Q₈

—

158

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

158

—

105

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

106

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

107

C₂×C₂²⋊C₄

—

158

—

108

#106

—

109

#103

—

110

C₂²⋊C₈

—

111

C₂⁴

C₂²⋊D₇₉

yes

characteristic

112

C₂×Q₈

—

113

C₂×Q₈

—

158

—

114

C₂×Q₈

—

158

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

158

—

120

C₂²×C₄

—

158

—

121

C₂²×C₄

C₂²⋊D₇₉

yes

—

122

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

123

C₂²×C₄

C₂²×D₇₉

yes

—

124

C₂²×C₄

C₂²×D₇₉

yes

characteristic

125

C₂×C₈

—

126

C₂×C₈

—

158

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

158

—

131

C₄⋊C₄

—

158

—

132

C₄⋊C₄

C₂²×D₇₉

yes

—

133

C₂²⋊C₄

—

158

—

134

C₂³

—

135

C₂³

—

136

C₂³

#11929612

yes

—

137

C₂³

C₂×C₂²⋊D₇₉

yes

—

138

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

139

Q₈

—

316

—

140

Q₈

—

158

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

316

—

148

C₂×C₄

—

316

—

149

C₂×C₄

—

158

—

150

C₂×C₄

—

158

—

151

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

152

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

153

C₂×C₄

#11929624

yes

—

154

C₂×C₄

C₂³×D₇₉

yes

characteristic

155

C₈

—

158

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

#14579144

yes

—

160

C₂²

C₂×#11929612

yes

—

161

C₂²

C₂×#11929612

yes

—

162

C₂²

#14579192

yes

—

163

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

316

—

167

C₄

—

316

—

168

C₄

#14579144

yes

—

169

C₄

C₂×#11929624

yes

—

170

C₂

—

171

C₂

#19098893

yes

direct product factor

172

C₂

C₂×#14579144

yes

characteristic

173

C₂

C₂²×#11929612

yes

characteristic

174

C₂

C₂×#14579192

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₂²×C₄⋊C₄×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₄
prime radical	C₂⁴×C₇₉	D₄
hypercenter	C₂²×C₄⋊C₄	D₇₉
hypocenter	C₇₉	C₂×#483
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₂²×C₄⋊C₄×C₇₉

C₂

socle

C₂³×C₇₉

C₂×D₄

prime radical

C₂⁴×C₇₉

D₄

hypercenter

C₂²×C₄⋊C₄

D₇₉

hypocenter

C₇₉

C₂×#483

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
2. #19098893	C₂	direct (N × H)
3. C₂×#14579092	C₂	semidirect
4. C₂×#14579084	C₂	semidirect
14. #14579092	C₂²	semidirect
15. #14579084	C₂²	semidirect
83. C₂×C₇₉	#483	semidirect
88. C₇₉	C₂×#483	semidirect
171. C₂	#19098893	direct (N × H)

H ≅ G/N

type

2. #19098893

C₂

direct (N × H)

3. C₂×#14579092

C₂

semidirect

4. C₂×#14579084

C₂

semidirect

14. #14579092

C₂²

semidirect

15. #14579084

C₂²

semidirect

83. C₂×C₇₉

#483

semidirect

88. C₇₉

C₂×#483

semidirect

171. C₂

#19098893

direct (N × H)

Subgroups — C₂×#19098893 #23773277

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19098893 #23773277

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19098893 #23773277