Group #21433971 — subgroups

Subgroups — C₂×#16837730 #21433971

Order 7552.

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	C₂²×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic
2	#16837730	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14273658	C₂	1	2	yes	maximal, complemented
4	C₂×#14273651	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂²⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₅₉	—	2	1	no	—
8	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₅₉	—	2	1	no	—
10	C₂×C₄⋊Dic₅₉	—	2	2	no	—
11	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₅₉	C₂²	1	4	yes	—
13	#14273658	C₂²	1	8	yes	complemented
14	#14273651	C₂²	1	8	yes	complemented
15	C₂²⋊(C₅₉⋊C₈)	C₂²	1	4	yes	—
16	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
18	C₂²⋊Q₈×C₅₉	—	2	2	no	—
19	C₂²⋊Q₈×C₅₉	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
21	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
23	C₂⁴×C₅₉	D₄	1	1	yes	characteristic
24	C₂×Q₈×C₅₉	—	2	1	no	—
25	C₂×Q₈×C₅₉	—	2	4	no	—
26	C₂×Q₈×C₅₉	C₂³	1	2	yes	—
27	C₂²×C₄×C₅₉	—	2	4	no	—
28	C₂²×C₄×C₅₉	—	2	1	no	—
29	C₂²×C₄×C₅₉	—	2	1	no	—
30	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₉	D₄	1	2	yes	—
32	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
33	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
34	C₂²×Dic₅₉	—	2	4	no	—
35	C₂²×Dic₅₉	—	2	2	no	—
36	C₂²×Dic₅₉	—	4	4	no	—
37	C₄⋊C₄×C₅₉	—	2	4	no	—
38	C₄⋊C₄×C₅₉	—	2	1	no	—
39	C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
40	C₂²⋊C₄×C₅₉	—	2	4	no	—
41	C₄⋊Dic₅₉	—	2	4	no	—
42	C₄⋊Dic₅₉	—	2	2	no	—
43	C₄⋊Dic₅₉	C₂³	1	4	yes	—
44	C₂×C₅₉⋊C₈	—	2	4	no	—
45	C₂×C₅₉⋊C₈	C₂³	1	4	yes	—
46	C₂³×C₅₉	—	2	2	no	—
47	C₂³×C₅₉	—	2	2	no	—
48	C₂³×C₅₉	SD₁₆	1	4	yes	—
49	C₂³×C₅₉	C₂×D₄	1	2	yes	—
50	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₅₉	—	2	4	no	—
52	Q₈×C₅₉	—	4	2	no	—
53	C₂×C₄×C₅₉	—	2	2	no	—
54	C₂×C₄×C₅₉	—	2	4	no	—
55	C₂×C₄×C₅₉	—	2	4	no	—
56	C₂×C₄×C₅₉	—	2	2	no	—
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	—	4	2	no	—
59	C₂×C₄×C₅₉	—	4	2	no	—
60	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#41	1	4	yes	—
63	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
64	C₂×Dic₅₉	—	2	4	no	—
65	C₂×Dic₅₉	—	4	8	no	—
66	C₂×Dic₅₉	—	4	4	no	—
67	C₅₉⋊C₈	—	2	4	no	—
68	C₂²×C₅₉	—	2	2	no	—
69	C₂²×C₅₉	—	2	2	no	—
70	C₂²×C₅₉	—	2	8	no	—
71	C₂²×C₅₉	#123	1	2	yes	—
72	C₂²×C₅₉	C₂×SD₁₆	1	4	yes	—
73	C₂²×C₅₉	C₂×SD₁₆	1	2	yes	—
74	C₂²×C₅₉	#137	1	2	yes	—
75	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
76	C₄×C₅₉	—	2	2	no	—
77	C₄×C₅₉	—	4	2	no	—
78	C₄×C₅₉	—	4	2	no	—
79	C₄×C₅₉	#123	1	2	yes	—
80	C₄×C₅₉	C₂×#41	1	2	yes	—
81	Dic₅₉	—	4	4	no	—
82	C₂×#483	—	59	1	no	maximal, Sylow
83	C₂×C₅₉	—	2	4	no	—
84	C₂×C₅₉	#483	1	4	yes	complemented
85	C₂×C₅₉	C₂×#123	1	1	yes	characteristic
86	C₂×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
87	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
88	C₂×C₂²⋊Q₈	D₅₉	1	1	yes	characteristic
89	C₂²×C₄⋊C₄	—	59	1	no	—
90	#483	—	59	8	no	—
91	C₂×#106	—	59	2	no	—
92	C₂×#103	—	59	2	no	—
93	C₂×C₂²⋊C₈	—	59	1	no	—
94	C₅₉	C₂×#483	1	1	yes	characteristic, complemented, Sylow
95	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
96	C₂³×C₄	—	118	1	no	—
97	C₂³×C₄	C₂×D₅₉	1	1	yes	characteristic
98	C₂²×C₈	—	59	2	no	—
99	C₂²⋊Q₈	—	2	2	no	—
100	C₂²⋊Q₈	C₂×D₅₉	1	4	yes	—
101	C₂×C₄⋊C₄	—	2	1	no	—
102	C₂×C₄⋊C₄	—	59	2	no	—
103	C₂×C₄⋊C₄	—	59	4	no	—
104	C₂×C₄⋊C₄	—	118	2	no	—
105	C₂×C₄⋊C₄	—	118	1	no	—
106	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
107	C₂×C₂²⋊C₄	—	2	1	no	—
108	#106	—	59	8	no	—
109	#103	—	59	8	no	—
110	C₂²⋊C₈	—	59	4	no	—
111	C₂⁴	C₂²⋊D₅₉	1	1	yes	characteristic
112	C₂×Q₈	—	2	1	no	—
113	C₂×Q₈	—	2	4	no	—
114	C₂×Q₈	C₂²×D₅₉	1	2	yes	—
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	118	4	no	—
119	C₂²×C₄	—	118	2	no	—
120	C₂²×C₄	—	236	4	no	—
121	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
122	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₅₉	1	2	yes	—
124	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
125	C₂×C₈	—	59	4	no	—
126	C₂×C₈	—	118	4	no	—
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	59	4	no	—
130	C₄⋊C₄	—	118	4	no	—
131	C₄⋊C₄	—	118	2	no	—
132	C₄⋊C₄	C₂²×D₅₉	1	2	yes	—
133	C₂²⋊C₄	—	2	4	no	—
134	C₂³	—	2	2	no	—
135	C₂³	—	2	2	no	—
136	C₂³	#11745671	1	4	yes	—
137	C₂³	C₂×C₂²⋊D₅₉	1	2	yes	—
138	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
139	Q₈	—	2	4	no	—
140	Q₈	—	4	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	118	4	no	—
149	C₂×C₄	—	236	8	no	—
150	C₂×C₄	—	236	4	no	—
151	C₂×C₄	#11745687	1	4	yes	—
152	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
153	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
154	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
155	C₈	—	118	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	8	no	—
159	C₂²	#14273708	1	2	yes	—
160	C₂²	C₂×#11745671	1	4	yes	—
161	C₂²	C₂×#11745671	1	2	yes	—
162	C₂²	#14273757	1	2	yes	—
163	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	4	2	no	—
166	C₄	—	4	2	no	—
167	C₄	—	236	4	no	—
168	C₄	#14273708	1	2	yes	—
169	C₄	C₂×#11745687	1	2	yes	—
170	C₂	—	2	4	no	—
171	C₂	#16837730	1	4	yes	direct product factor
172	C₂	C₂×#14273708	1	1	yes	characteristic
173	C₂	C₂²×#11745671	1	1	yes	characteristic
174	C₂	C₂×#14273757	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₅₉

C₂

yes

maximal, characteristic

#16837730

C₂

yes

maximal, direct product factor

C₂×#14273658

C₂

yes

maximal, complemented

C₂×#14273651

C₂

yes

maximal, complemented

C₂×C₂²⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic

C₂×C₂²⋊Q₈×C₅₉

C₂

yes

maximal, characteristic

C₂³×Dic₅₉

—

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

#14273658

C₂²

yes

complemented

#14273651

C₂²

yes

complemented

C₂²⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂²×Q₈×C₅₉

C₂²

yes

characteristic

C₂³×C₄×C₅₉

C₂²

yes

characteristic

C₂²⋊Q₈×C₅₉

—

C₂²⋊Q₈×C₅₉

C₂²

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₅₉

—

C₂⁴×C₅₉

D₄

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₂²⋊C₄×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

SD₁₆

yes

—

C₂³×C₅₉

C₂×D₄

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#123

yes

—

C₂²×C₅₉

C₂×SD₁₆

yes

—

C₂²×C₅₉

C₂×SD₁₆

yes

—

C₂²×C₅₉

#137

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#123

yes

—

C₄×C₅₉

C₂×#41

yes

—

Dic₅₉

—

C₂×#483

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

C₂²×SD₁₆

yes

characteristic

C₂×C₅₉

C₂×#137

yes

characteristic

C₂×C₂²⋊Q₈

D₅₉

yes

characteristic

C₂²×C₄⋊C₄

—

#483

—

C₂×#106

—

C₂×#103

—

C₂×C₂²⋊C₈

—

C₅₉

C₂×#483

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₉

yes

characteristic

C₂³×C₄

—

118

—

C₂³×C₄

C₂×D₅₉

yes

characteristic

C₂²×C₈

—

C₂²⋊Q₈

—

100

C₂²⋊Q₈

C₂×D₅₉

yes

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

118

—

105

C₂×C₄⋊C₄

—

118

—

106

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

107

C₂×C₂²⋊C₄

—

108

#106

—

109

#103

—

110

C₂²⋊C₈

—

111

C₂⁴

C₂²⋊D₅₉

yes

characteristic

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

C₂²×D₅₉

yes

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

118

—

119

C₂²×C₄

—

118

—

120

C₂²×C₄

—

236

—

121

C₂²×C₄

C₂²⋊D₅₉

yes

—

122

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

123

C₂²×C₄

C₂²×D₅₉

yes

—

124

C₂²×C₄

C₂²×D₅₉

yes

characteristic

125

C₂×C₈

—

126

C₂×C₈

—

118

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

118

—

131

C₄⋊C₄

—

118

—

132

C₄⋊C₄

C₂²×D₅₉

yes

—

133

C₂²⋊C₄

—

134

C₂³

—

135

C₂³

—

136

C₂³

#11745671

yes

—

137

C₂³

C₂×C₂²⋊D₅₉

yes

—

138

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

139

Q₈

—

140

Q₈

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

118

—

149

C₂×C₄

—

236

—

150

C₂×C₄

—

236

—

151

C₂×C₄

#11745687

yes

—

152

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

153

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

154

C₂×C₄

C₂³×D₅₉

yes

characteristic

155

C₈

—

118

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

#14273708

yes

—

160

C₂²

C₂×#11745671

yes

—

161

C₂²

C₂×#11745671

yes

—

162

C₂²

#14273757

yes

—

163

C₂²

C₂²×C₂²⋊D₅₉

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

236

—

168

C₄

#14273708

yes

—

169

C₄

C₂×#11745687

yes

—

170

C₂

—

171

C₂

#16837730

yes

direct product factor

172

C₂

C₂×#14273708

yes

characteristic

173

C₂

C₂²×#11745671

yes

characteristic

174

C₂

C₂×#14273757

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₂×C₂²⋊Q₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×C₂²⋊Q₈	D₅₉
hypocenter	C₅₉	C₂×#483
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₂×C₂²⋊Q₈×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂×C₂²⋊Q₈

D₅₉

hypocenter

C₅₉

C₂×#483

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊Q₈

N	H ≅ G/N	type
2. #16837730	C₂	direct (N × H)
3. C₂×#14273658	C₂	semidirect
4. C₂×#14273651	C₂	semidirect
13. #14273658	C₂²	semidirect
14. #14273651	C₂²	semidirect
84. C₂×C₅₉	#483	semidirect
94. C₅₉	C₂×#483	semidirect
171. C₂	#16837730	direct (N × H)

H ≅ G/N

type

2. #16837730

C₂

direct (N × H)

3. C₂×#14273658

C₂

semidirect

4. C₂×#14273651

C₂

semidirect

13. #14273658

C₂²

semidirect

14. #14273651

C₂²

semidirect

84. C₂×C₅₉

#483

semidirect

94. C₅₉

C₂×#483

semidirect

171. C₂

#16837730

direct (N × H)

Subgroups — C₂×#16837730 #21433971

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16837730 #21433971

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16837730 #21433971