Group #21435242 — subgroups

Subgroups — #21435242

Order 7552.

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

normal subgroups only

This group has 360 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	#16837959	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273727	C₂	1	1	yes	maximal, characteristic
3	#16837800	C₂	1	2	yes	maximal, complemented
4	#16837753	C₂	1	2	yes	maximal, complemented
5	#16837733	C₂	1	4	yes	maximal
6	#16837688	C₂	1	4	yes	maximal, complemented
7	C₅₉×#431	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₁₈	—	2	1	no	—
9	#14273727	—	2	1	no	—
10	#14273727	—	2	1	no	—
11	#14273727	C₂²	1	1	yes	characteristic
12	#14273727	C₂²	1	2	yes	—
13	#14273727	C₂²	1	2	yes	—
14	#14273727	C₂²	1	1	yes	characteristic
15	#14273719	—	2	1	no	—
16	#14273706	—	2	1	no	—
17	#14273706	C₂²	1	2	yes	—
18	C₄×Dic₁₁₈	—	2	1	no	—
19	C₄×Dic₁₁₈	C₂²	1	2	yes	—
20	#14273686	—	2	1	no	—
21	C₂×C₄⋊Dic₅₉	—	2	1	no	—
22	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
23	#14273659	C₂²	1	8	yes	—
24	#14273652	C₂²	1	8	yes	—
25	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
27	C₂²⋊C₈×C₅₉	C₂²	1	2	yes	—
28	C₄×C₈×C₅₉	C₂²	1	2	yes	—
29	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
32	C₂²×Dic₅₉	—	2	2	no	—
33	C₂×Dic₁₁₈	—	2	1	no	—
34	C₂×Dic₁₁₈	—	2	2	no	—
35	C₂×Dic₁₁₈	—	2	2	no	—
36	C₂×Dic₁₁₈	—	4	4	no	—
37	C₂×Dic₁₁₈	C₂³	1	2	yes	—
38	C₂×C₈×C₅₉	C₂³	1	4	yes	—
39	C₄²×C₅₉	D₄	1	1	yes	characteristic
40	C₄²×C₅₉	D₄	1	1	yes	characteristic
41	C₄²×C₅₉	C₂³	1	1	yes	characteristic
42	C₄²×C₅₉	C₂³	1	1	yes	characteristic
43	C₂²⋊Dic₅₉	—	2	2	no	—
44	C₄⋊Dic₅₉	—	2	2	no	—
45	C₄⋊Dic₅₉	—	2	2	no	—
46	C₄⋊Dic₅₉	—	2	2	no	—
47	C₄⋊Dic₅₉	—	2	1	no	—
48	C₄⋊Dic₅₉	—	2	1	no	—
49	C₄⋊Dic₅₉	—	2	1	no	—
50	C₄⋊Dic₅₉	C₂³	1	4	yes	—
51	C₄⋊Dic₅₉	C₂³	1	2	yes	—
52	Dic₅₉⋊C₄	—	2	2	no	—
53	Dic₅₉⋊C₄	—	2	2	no	—
54	C₄×Dic₅₉	—	2	2	no	—
55	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	—	2	2	no	—
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	SD₁₆	1	4	yes	—
59	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₅₉	—	2	2	no	—
66	C₂×Dic₅₉	—	2	2	no	—
67	C₂×Dic₅₉	—	2	4	no	—
68	C₂×Dic₅₉	—	4	4	no	—
69	Dic₁₁₈	—	2	4	no	—
70	Dic₁₁₈	—	4	4	no	—
71	Dic₁₁₈	—	4	4	no	—
72	Dic₁₁₈	—	4	2	no	—
73	C₈×C₅₉	—	2	4	no	—
74	C₂²×C₅₉	—	2	1	no	—
75	C₂²×C₅₉	—	2	1	no	—
76	C₂²×C₅₉	C₂×SD₁₆	1	2	yes	—
77	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₅₉	—	2	1	no	—
79	C₄×C₅₉	—	2	1	no	—
80	C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
81	C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
82	C₄×C₅₉	#137	1	2	yes	—
83	C₄×C₅₉	2₋¹⁺⁴	1	2	yes	—
84	Dic₅₉	—	4	2	no	—
85	Dic₅₉	—	4	4	no	—
86	Dic₅₉	—	4	2	no	—
87	#3229	—	59	1	no	maximal, Sylow
88	C₂×C₅₉	—	2	2	no	—
89	C₂×C₅₉	#536	1	1	yes	characteristic
90	C₂×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
91	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
92	#533	—	59	1	no	—
93	C₂×C₄⋊Q₈	—	59	1	no	—
94	#498	—	59	2	no	—
95	#487	—	59	2	no	—
96	#483	—	59	4	no	—
97	#475	—	59	4	no	—
98	#431	D₅₉	1	1	yes	characteristic
99	C₅₉	#3229	1	1	yes	characteristic, complemented, Sylow
100	C₂²×Q₈	—	118	1	no	—
101	C₄⋊Q₈	—	59	2	no	—
102	C₄⋊Q₈	—	59	2	no	—
103	C₄⋊Q₈	—	59	1	no	—
104	C₄⋊Q₈	—	59	1	no	—
105	C₄⋊Q₈	—	118	1	no	—
106	C₄⋊Q₈	—	118	1	no	—
107	#125	—	118	1	no	—
108	C₂²⋊Q₈	—	59	2	no	—
109	C₂²⋊Q₈	—	118	1	no	—
110	C₄×Q₈	—	59	2	no	—
111	C₄×Q₈	—	118	1	no	—
112	#117	—	118	1	no	—
113	C₂×C₄⋊C₄	—	59	2	no	—
114	C₂×C₄⋊C₄	—	118	1	no	—
115	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
116	#106	—	59	8	no	—
117	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
118	#103	—	59	8	no	—
119	C₂²⋊C₈	C₂×D₅₉	1	2	yes	—
120	C₄×C₈	C₂×D₅₉	1	2	yes	—
121	C₂×Q₈	—	59	2	no	—
122	C₂×Q₈	—	118	1	no	—
123	C₂×Q₈	—	118	2	no	—
124	C₂×Q₈	—	118	2	no	—
125	C₂×Q₈	—	236	4	no	—
126	C₂²×C₄	—	118	2	no	—
127	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
128	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
129	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
130	C₂×C₈	C₂²×D₅₉	1	4	yes	—
131	C₄⋊C₄	—	59	4	no	—
132	C₄⋊C₄	—	59	2	no	—
133	C₄⋊C₄	—	118	2	no	—
134	C₄⋊C₄	—	118	2	no	—
135	C₄⋊C₄	—	118	2	no	—
136	C₄⋊C₄	—	118	1	no	—
137	C₄⋊C₄	—	118	1	no	—
138	C₄⋊C₄	—	118	2	no	—
139	C₄⋊C₄	—	118	2	no	—
140	C₄⋊C₄	—	118	1	no	—
141	C₂²⋊C₄	—	118	2	no	—
142	C₄²	—	118	2	no	—
143	C₄²	D₂₃₆	1	1	yes	characteristic
144	C₄²	D₂₃₆	1	1	yes	characteristic
145	C₄²	C₂²×D₅₉	1	1	yes	characteristic
146	C₄²	C₂²×D₅₉	1	1	yes	characteristic
147	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
148	Q₈	—	118	4	no	—
149	Q₈	—	236	4	no	—
150	Q₈	—	236	4	no	—
151	Q₈	—	236	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	118	2	no	—
155	C₂×C₄	—	118	2	no	—
156	C₂×C₄	—	118	4	no	—
157	C₂×C₄	—	236	4	no	—
158	C₂×C₄	#11745660	1	4	yes	—
159	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
160	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
161	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
162	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
163	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
164	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
165	C₈	—	2	4	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	C₂×#11745660	1	2	yes	—
169	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	236	4	no	—
173	C₄	—	236	2	no	—
174	C₄	—	236	2	no	—
175	C₄	C₂×#11745660	1	2	yes	—
176	C₄	C₂×#11745660	1	2	yes	—
177	C₄	#14273760	1	2	yes	—
178	C₄	#14273774	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	#16837978	1	1	yes	characteristic
181	C₂	C₂²×#11745660	1	1	yes	characteristic
182	C₂	C₂×#14273760	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837959

C₂

yes

maximal, characteristic

C₂×#14273727

C₂

yes

maximal, characteristic

#16837800

C₂

yes

maximal, complemented

#16837753

C₂

yes

maximal, complemented

#16837733

C₂

yes

maximal

#16837688

C₂

yes

maximal, complemented

C₅₉×#431

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

—

#14273727

—

#14273727

—

#14273727

C₂²

yes

characteristic

#14273727

C₂²

yes

—

#14273727

C₂²

yes

—

#14273727

C₂²

yes

—

—

C₂²

yes

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

C₂²

yes

—

#14273686

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

#14273659

C₂²

yes

—

#14273652

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₂²⋊C₈×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×SD₁₆

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×SD₁₆

yes

—

C₄×C₅₉

C₂×SD₁₆

yes

—

C₄×C₅₉

#137

yes

—

C₄×C₅₉

2₋¹⁺⁴

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#3229

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#536

yes

characteristic

C₂×C₅₉

C₂²×SD₁₆

yes

C₂×C₅₉

yes

—

C₂×C₄⋊Q₈

—

—

—

—

—

D₅₉

yes

characteristic

C₅₉

#3229

yes

characteristic, complemented, Sylow

100

C₂²×Q₈

—

118

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

105

C₄⋊Q₈

—

118

—

106

C₄⋊Q₈

—

118

—

107

#125

—

118

—

108

C₂²⋊Q₈

—

109

C₂²⋊Q₈

—

118

—

110

C₄×Q₈

—

111

C₄×Q₈

—

118

—

112

#117

—

118

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

118

—

115

C₂×C₄²

C₂×D₅₉

yes

characteristic

116

#106

—

117

C₄⋊C₈

C₂×D₅₉

yes

—

118

#103

—

119

C₂²⋊C₈

C₂×D₅₉

yes

—

120

C₄×C₈

C₂×D₅₉

yes

—

121

C₂×Q₈

—

122

C₂×Q₈

—

118

—

123

C₂×Q₈

—

118

—

124

C₂×Q₈

—

118

—

125

C₂×Q₈

—

236

—

126

C₂²×C₄

—

118

—

127

C₂²×C₄

D₂₃₆

yes

characteristic

128

C₂²×C₄

D₂₃₆

yes

characteristic

129

C₂²×C₄

C₂²×D₅₉

yes

characteristic

130

C₂×C₈

C₂²×D₅₉

yes

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

118

—

134

C₄⋊C₄

—

118

—

135

C₄⋊C₄

—

118

—

136

C₄⋊C₄

—

118

—

137

C₄⋊C₄

—

118

—

138

C₄⋊C₄

—

118

—

139

C₄⋊C₄

—

118

—

140

C₄⋊C₄

—

118

—

141

C₂²⋊C₄

—

118

—

142

C₄²

—

118

—

143

C₄²

D₂₃₆

yes

characteristic

144

C₄²

D₂₃₆

yes

characteristic

145

C₄²

C₂²×D₅₉

yes

characteristic

146

C₄²

C₂²×D₅₉

yes

characteristic

147

C₂³

C₂×D₂₃₆

yes

characteristic

148

Q₈

—

118

—

149

Q₈

—

236

—

150

Q₈

—

236

—

151

Q₈

—

236

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

118

—

155

C₂×C₄

—

118

—

156

C₂×C₄

—

118

—

157

C₂×C₄

—

236

—

158

C₂×C₄

#11745660

yes

—

159

C₂×C₄

C₂×D₂₃₆

yes

characteristic

160

C₂×C₄

C₂×D₂₃₆

yes

characteristic

161

C₂×C₄

C₂×D₂₃₆

yes

characteristic

162

C₂×C₄

C₂×D₂₃₆

yes

characteristic

163

C₂×C₄

C₂×D₂₃₆

yes

characteristic

164

C₂×C₄

C₂³×D₅₉

yes

characteristic

165

C₈

—

166

C₂²

—

167

C₂²

—

168

C₂²

C₂×#11745660

yes

—

169

C₂²

C₂²×D₂₃₆

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

236

—

173

C₄

—

236

—

174

C₄

—

236

—

175

C₄

C₂×#11745660

yes

—

176

C₄

C₂×#11745660

yes

—

177

C₄

#14273760

yes

—

178

C₄

#14273774

yes

—

179

C₂

—

180

C₂

#16837978

yes

characteristic

181

C₂

C₂²×#11745660

yes

characteristic

182

C₂

C₂×#14273760

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#431	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#431	D₅₉
hypocenter	C₅₉	#3229
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂²×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#431

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #16837800	C₂	semidirect
4. #16837753	C₂	semidirect
6. #16837688	C₂	semidirect
99. C₅₉	#3229	semidirect

H ≅ G/N

type

3. #16837800

C₂

semidirect

4. #16837753

C₂

semidirect

6. #16837688

C₂

semidirect

99. C₅₉

#3229

semidirect

Subgroups — #21435242

Conjugacy classes of subgroups

Complements