Group #34443750 — subgroups

Subgroups — C₂×#23765678 #34443750

Order 20224.

494 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 203 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×#19098964	C₂	1	2	yes	maximal
2	#23765678	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19098391	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098388	C₂	1	2	yes	maximal
5	C₂×C₁₆×Dic₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#367×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#14579152	—	2	2	no	—
8	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
9	#19098964	—	2	4	no	—
10	#19098964	C₂²	1	8	yes	—
11	C₂×#14579095	C₂²	1	2	yes	—
12	C₂×#14579092	—	2	2	no	—
13	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
14	#19098391	C₂²	1	4	yes	—
15	#19098388	C₂²	1	8	yes	—
16	C₁₆×Dic₇₉	C₂²	1	4	yes	—
17	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×#107×C₇₉	C₂²	1	2	yes	—
19	#367×C₇₉	C₂²	1	4	yes	—
20	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
21	#14579152	—	2	8	no	—
22	#14579152	—	4	4	no	—
23	C₂×C₄⋊Dic₇₉	—	2	2	no	—
24	C₂×C₄⋊Dic₇₉	—	4	2	no	—
25	C₂×Dic₇₉⋊C₄	—	4	2	no	—
26	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
27	C₂×C₇₉⋊C₁₆	Q₈	1	4	yes	—
28	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
29	#14579095	—	2	2	no	—
30	#14579095	C₂³	1	4	yes	—
31	#14579092	—	2	8	no	—
32	C₈×Dic₇₉	D₄	1	2	yes	—
33	C₈×Dic₇₉	C₂³	1	2	yes	—
34	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
36	C₂×C₁₆×C₇₉	Q₈	1	4	yes	—
37	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
38	#107×C₇₉	—	2	2	no	—
39	#107×C₇₉	C₂³	1	4	yes	—
40	C₂²×Dic₇₉	—	4	2	no	—
41	C₂²×Dic₇₉	D₈	1	2	yes	—
42	C₂×C₇₉⋊C₈	Q₁₆	1	4	yes	—
43	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
44	C₄⋊Dic₇₉	—	2	4	no	—
45	C₄⋊Dic₇₉	—	4	8	no	—
46	C₄⋊Dic₇₉	—	4	2	no	—
47	Dic₇₉⋊C₄	—	4	8	no	—
48	C₄×Dic₇₉	D₈	1	2	yes	—
49	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
50	C₇₉⋊C₁₆	C₂×Q₈	1	4	yes	—
51	C₂²×C₄×C₇₉	—	4	2	no	—
52	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₇₉	—	2	4	no	—
58	C₄⋊C₄×C₇₉	—	4	2	no	—
59	C₁₆×C₇₉	C₂×Q₈	1	4	yes	—
60	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₇₉	—	4	4	no	—
62	C₂×C₄×C₇₉	—	8	4	no	—
63	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
64	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
66	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
67	C₂×Dic₇₉	—	2	4	no	—
68	C₂×Dic₇₉	—	4	4	no	—
69	C₂×Dic₇₉	—	8	4	no	—
70	C₂×Dic₇₉	C₂×D₈	1	4	yes	—
71	C₈×C₇₉	C₄⋊Q₈	1	2	yes	—
72	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
73	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
74	C₇₉⋊C₈	C₂×Q₁₆	1	4	yes	—
75	C₂²×C₇₉	#500	1	2	yes	—
76	C₂²×C₇₉	#508	1	4	yes	—
77	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
78	C₄×C₇₉	—	8	4	no	—
79	C₄×C₇₉	#500	1	2	yes	—
80	C₄×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
81	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
82	Dic₇₉	—	2	4	no	—
83	Dic₇₉	—	8	4	no	—
84	C₂×#2253	—	79	1	no	maximal, Sylow
85	C₂×C₇₉	#2253	1	4	yes	complemented
86	C₂×C₇₉	C₂×#500	1	1	yes	characteristic
87	C₂×C₇₉	C₂×#508	1	2	yes	—
88	C₂×#499	—	79	2	no	—
89	#2253	—	79	8	no	—
90	C₂×#367	—	79	1	no	—
91	C₂×#367	D₇₉	1	1	yes	characteristic
92	C₂×#366	—	79	2	no	—
93	C₂×C₄×C₁₆	—	79	1	no	—
94	C₇₉	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
95	C₂×#125	—	158	2	no	—
96	C₂²×C₁₆	—	79	1	no	—
97	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
98	#499	—	79	8	no	—
99	#499	—	158	4	no	—
100	C₂×#107	—	79	2	no	—
101	C₂×#107	C₂×D₇₉	1	2	yes	—
102	C₂×#106	—	158	2	no	—
103	C₂×C₄×C₈	—	79	1	no	—
104	#367	—	79	4	no	—
105	#367	C₂×D₇₉	1	4	yes	—
106	#366	—	79	8	no	—
107	C₄×C₁₆	—	79	4	no	—
108	C₂²×C₈	—	79	1	no	—
109	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
110	#125	—	316	4	no	—
111	#125	—	158	8	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	316	2	no	—
114	C₂×C₄⋊C₄	—	316	2	no	—
115	C₂×C₄⋊C₄	—	158	2	no	—
116	C₂×C₄²	—	79	1	no	—
117	C₂×C₁₆	—	79	4	no	—
118	C₂×C₁₆	—	79	2	no	—
119	C₂×C₁₆	Dic₁₅₈	1	4	yes	—
120	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
121	#107	—	2	2	no	—
122	#107	—	79	4	no	—
123	#107	—	158	2	no	—
124	#107	C₂²×D₇₉	1	4	yes	—
125	#106	—	158	8	no	—
126	C₄×C₈	—	79	2	no	—
127	C₄×C₈	—	79	2	no	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	—	316	2	no	—
130	C₂²×C₄	—	79	2	no	—
131	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
132	C₂×C₈	—	79	4	no	—
133	C₂×C₈	—	79	2	no	—
134	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₇₉	1	2	yes	—
136	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
137	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	316	8	no	—
141	C₄⋊C₄	—	316	8	no	—
142	C₄⋊C₄	—	316	2	no	—
143	C₄⋊C₄	—	158	4	no	—
144	C₄²	—	79	2	no	—
145	C₄²	—	79	2	no	—
146	C₁₆	—	79	4	no	—
147	C₁₆	C₂×Dic₁₅₈	1	4	yes	—
148	C₂³	D₈×D₇₉	1	1	yes	characteristic
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	8	4	no	—
151	C₂×C₄	—	316	4	no	—
152	C₂×C₄	—	79	4	no	—
153	C₂×C₄	—	632	4	no	—
154	C₂×C₄	—	158	4	no	—
155	C₂×C₄	#14579160	1	2	yes	—
156	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
157	C₂×C₄	Q₁₆×D₇₉	1	2	yes	—
158	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
159	C₈	—	79	4	no	—
160	C₈	#14579160	1	2	yes	—
161	C₈	C₂×Q₈×D₇₉	1	1	yes	characteristic
162	C₈	C₂²×Dic₁₅₈	1	1	yes	characteristic
163	C₂²	#19098967	1	2	yes	—
164	C₂²	#19098994	1	4	yes	—
165	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
166	C₄	—	8	4	no	—
167	C₄	—	632	4	no	—
168	C₄	—	158	4	no	—
169	C₄	#19098967	1	2	yes	—
170	C₄	C₂×#14579160	1	1	yes	characteristic
171	C₄	C₂×Q₁₆×D₇₉	1	1	yes	characteristic
172	C₂	#23765678	1	4	yes	direct product factor
173	C₂	C₂×#19098967	1	1	yes	characteristic
174	C₂	C₂×#19098994	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098964

C₂

yes

maximal

#23765678

C₂

yes

maximal, direct product factor

C₂×#19098391

C₂

yes

maximal, characteristic

C₂×#19098388

C₂

yes

maximal

C₂×C₁₆×Dic₇₉

C₂

yes

maximal, characteristic

C₂×#367×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579152

—

C₂²×C₇₉⋊C₁₆

C₂²

yes

characteristic

#19098964

—

#19098964

C₂²

yes

—

C₂×#14579095

C₂²

yes

—

C₂×#14579092

—

C₂×C₈×Dic₇₉

C₂²

yes

characteristic

#19098391

C₂²

yes

—

#19098388

C₂²

yes

—

C₁₆×Dic₇₉

C₂²

yes

—

C₂²×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×#107×C₇₉

C₂²

yes

—

#367×C₇₉

C₂²

yes

—

C₂²×C₇₉⋊C₈

D₄

yes

characteristic

#14579152

—

#14579152

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×Dic₇₉⋊C₄

—

C₂×C₄×Dic₇₉

D₄

yes

characteristic

C₂×C₇₉⋊C₁₆

Q₈

yes

—

C₂×C₇₉⋊C₁₆

C₂³

yes

—

#14579095

—

#14579095

C₂³

yes

—

#14579092

—

C₈×Dic₇₉

D₄

yes

—

C₈×Dic₇₉

C₂³

yes

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₉

—

C₂×C₁₆×C₇₉

Q₈

yes

—

C₂×C₁₆×C₇₉

C₂³

yes

—

#107×C₇₉

—

#107×C₇₉

C₂³

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

D₈

yes

—

C₂×C₇₉⋊C₈

Q₁₆

yes

—

C₂×C₇₉⋊C₈

C₂×D₄

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

D₈

yes

—

C₄×Dic₇₉

C₂×D₄

yes

—

C₇₉⋊C₁₆

C₂×Q₈

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

—

C₂×C₈×C₇₉

C₂×Q₈

yes

—

C₂×C₈×C₇₉

C₂⁴

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₁₆×C₇₉

C₂×Q₈

yes

—

C₂³×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄⋊Q₈

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₁₆

yes

—

C₂×C₄×C₇₉

C₂²×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₈

yes

—

C₈×C₇₉

C₄⋊Q₈

yes

—

C₈×C₇₉

C₂²×Q₈

yes

characteristic

C₈×C₇₉

C₂²×Q₈

yes

characteristic

C₇₉⋊C₈

C₂×Q₁₆

yes

—

C₂²×C₇₉

#500

yes

—

C₂²×C₇₉

#508

yes

—

C₂²×C₇₉

C₂²×D₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

#500

yes

—

C₄×C₇₉

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₇₉

C₂²×Q₁₆

yes

characteristic

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

—

—

—

D₇₉

yes

characteristic

C₂×#366

—

C₂×C₄×C₁₆

—

C₇₉

C₂×#2253

yes

characteristic, complemented, Sylow

C₂×#125

—

158

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₉

yes

characteristic

#499

—

#499

—

158

—

100

C₂×#107

—

101

C₂×#107

C₂×D₇₉

yes

—

102

C₂×#106

—

158

—

103

C₂×C₄×C₈

—

104

#367

—

105

#367

C₂×D₇₉

yes

—

106

#366

—

107

C₄×C₁₆

—

108

C₂²×C₈

—

109

C₂²×C₈

C₂²×D₇₉

yes

characteristic

110

#125

—

316

—

111

#125

—

158

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

316

—

114

C₂×C₄⋊C₄

—

316

—

115

C₂×C₄⋊C₄

—

158

—

116

C₂×C₄²

—

117

C₂×C₁₆

—

118

C₂×C₁₆

—

119

C₂×C₁₆

Dic₁₅₈

yes

—

120

C₂×C₁₆

C₂²×D₇₉

yes

—

121

#107

—

122

#107

—

123

#107

—

158

—

124

#107

C₂²×D₇₉

yes

—

125

#106

—

158

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₂²×C₄

—

129

C₂²×C₄

—

316

—

130

C₂²×C₄

—

131

C₂²×C₄

D₄×D₇₉

yes

characteristic

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

D₄×D₇₉

yes

characteristic

135

C₂×C₈

Q₈×D₇₉

yes

—

136

C₂×C₈

C₂×Dic₁₅₈

yes

—

137

C₂×C₈

C₂³×D₇₉

yes

characteristic

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

316

—

141

C₄⋊C₄

—

316

—

142

C₄⋊C₄

—

316

—

143

C₄⋊C₄

—

158

—

144

C₄²

—

145

C₄²

—

146

C₁₆

—

147

C₁₆

C₂×Dic₁₅₈

yes

—

148

C₂³

D₈×D₇₉

yes

characteristic

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

316

—

152

C₂×C₄

—

153

C₂×C₄

—

632

—

154

C₂×C₄

—

158

—

155

C₂×C₄

#14579160

yes

—

156

C₂×C₄

D₈×D₇₉

yes

characteristic

157

C₂×C₄

Q₁₆×D₇₉

yes

—

158

C₂×C₄

C₂×D₄×D₇₉

yes

characteristic

159

C₈

—

160

C₈

#14579160

yes

—

161

C₈

C₂×Q₈×D₇₉

yes

characteristic

162

C₈

C₂²×Dic₁₅₈

yes

characteristic

163

C₂²

#19098967

yes

—

164

C₂²

#19098994

yes

—

165

C₂²

C₂×D₈×D₇₉

yes

characteristic

166

C₄

—

167

C₄

—

632

—

168

C₄

—

158

—

169

C₄

#19098967

yes

—

170

C₄

C₂×#14579160

yes

characteristic

171

C₄

C₂×Q₁₆×D₇₉

yes

characteristic

172

C₂

#23765678

yes

direct product factor

173

C₂

C₂×#19098967

yes

characteristic

174

C₂

C₂×#19098994

yes

—

	subgroup	G/H
center	C₂³	D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#367×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
hypercenter	C₂×#367	D₇₉
hypocenter	C₇₉	C₂×#2253
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

subgroup

G/H

center

C₂³

D₈×D₇₉

commutator, cosocle

C₂×C₈×C₇₉

C₂⁴

Frattini

C₂×C₈

C₂³×D₇₉

Fitting

C₂×#367×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₈

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

N	H ≅ G/N	type
2. #23765678	C₂	direct (N × H)
85. C₂×C₇₉	#2253	semidirect
94. C₇₉	C₂×#2253	semidirect
172. C₂	#23765678	direct (N × H)

H ≅ G/N

type

2. #23765678

C₂

direct (N × H)

85. C₂×C₇₉

#2253

semidirect

94. C₇₉

C₂×#2253

semidirect

172. C₂

#23765678

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#19098964	2
#23765678	2
C₂×#19098391	2
C₂×#19098388	2
C₂×C₁₆×Dic₇₉	2
C₂×#367×C₇₉	2
C₂×#2253	79

subgroup

index

C₂×C₁₆×Dic₇₉

C₂×#367×C₇₉

C₂×#2253

Subgroups — C₂×#23765678 #34443750

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#23765678 #34443750

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765678 #34443750