Group #21425458 — subgroups

Subgroups — #21425458

Order 7552.

208 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 69 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	#16837444	C₂	1	1	yes	maximal, characteristic
2	#16837428	C₂	1	1	yes	maximal, characteristic
3	#16837412	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273652	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273645	C₂	1	1	yes	maximal, characteristic
6	#16837170	C₂	1	1	yes	maximal, characteristic
7	C₂×#108×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
9	C₂×#11745664	C₂²	1	1	yes	characteristic
10	#14273686	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
12	#14273661	—	2	1	no	—
13	#14273659	—	2	1	no	—
14	Dic₅₉⋊C₈	—	2	2	no	—
15	#14273652	—	2	2	no	—
16	#14273652	—	2	1	no	—
17	#14273652	—	2	1	no	—
18	#14273645	—	2	2	no	—
19	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	#108×C₅₉	—	2	2	no	—
23	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
24	C₂²×Dic₅₉	—	2	1	no	—
25	C₂²×Dic₅₉	—	2	1	no	—
26	C₂×Dic₁₁₈	—	2	1	no	—
27	C₂×Dic₁₁₈	—	2	1	no	—
28	C₂×Dic₁₁₈	—	2	1	no	—
29	C₂×Dic₁₁₈	—	2	1	no	—
30	C₂×Dic₁₁₈	—	2	1	no	—
31	C₂×Dic₁₁₈	Q₈	1	2	yes	—
32	M₁₆×C₅₉	—	2	1	no	—
33	M₁₆×C₅₉	—	2	1	no	—
34	M₁₆×C₅₉	—	2	1	no	—
35	M₁₆×C₅₉	D₄	1	2	yes	—
36	C₂×C₈×C₅₉	—	2	1	no	—
37	C₂×C₈×C₅₉	—	2	1	no	—
38	C₂×C₈×C₅₉	—	2	1	no	—
39	C₂×C₈×C₅₉	—	2	1	no	—
40	C₂×C₈×C₅₉	D₄	1	2	yes	—
41	C₂²⋊Dic₅₉	—	2	1	no	—
42	C₄⋊Dic₅₉	—	2	1	no	—
43	C₄⋊Dic₅₉	—	2	1	no	—
44	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
45	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
46	C₄×Dic₅₉	—	2	1	no	—
47	#11745664	—	2	1	no	—
48	#11745664	—	2	1	no	—
49	C₂×C₅₉⋊C₈	—	2	1	no	—
50	C₂³×C₅₉	#41	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
56	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
57	C₂×Dic₅₉	—	2	1	no	—
58	C₂×Dic₅₉	—	2	1	no	—
59	C₂×Dic₅₉	—	2	2	no	—
60	C₂×Dic₅₉	—	4	1	no	—
61	C₂×Dic₅₉	—	4	1	no	—
62	C₂×Dic₅₉	—	4	1	no	—
63	C₂×Dic₅₉	—	8	1	no	—
64	Dic₁₁₈	—	2	2	no	—
65	Dic₁₁₈	—	2	2	no	—
66	Dic₁₁₈	—	4	1	no	—
67	Dic₁₁₈	—	4	1	no	—
68	C₈×C₅₉	—	2	2	no	—
69	C₈×C₅₉	—	2	2	no	—
70	C₈×C₅₉	—	4	1	no	—
71	C₅₉⋊C₈	—	4	1	no	—
72	C₂²×C₅₉	—	2	1	no	—
73	C₂²×C₅₉	—	2	1	no	—
74	C₂²×C₅₉	#121	1	1	yes	characteristic
75	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₅₉	#126	1	1	yes	characteristic
77	C₄×C₅₉	#121	1	1	yes	characteristic
78	C₄×C₅₉	#121	1	1	yes	characteristic
79	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
80	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
81	Dic₅₉	—	4	2	no	—
82	Dic₅₉	—	4	1	no	—
83	Dic₅₉	—	4	1	no	—
84	Dic₅₉	—	8	1	no	—
85	#2064	—	59	1	no	maximal, Sylow
86	C₂×C₅₉	—	2	1	no	—
87	C₂×C₅₉	—	2	1	no	—
88	C₂×C₅₉	#396	1	1	yes	characteristic
89	C₂×C₅₉	#471	1	1	yes	characteristic
90	C₂×C₅₉	#473	1	1	yes	characteristic
91	C₂×#108	D₅₉	1	1	yes	characteristic
92	#428	—	59	1	no	—
93	#424	—	59	1	no	—
94	#419	—	59	1	no	—
95	C₂×#103	—	59	1	no	—
96	C₂×#101	—	59	1	no	—
97	#343	—	59	1	no	—
98	C₅₉	#2064	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	—	59	1	no	—
100	C₂×M₁₆	—	59	1	no	—
101	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
102	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
103	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
104	#117	—	59	1	no	—
105	C₂×C₄⋊C₄	—	59	1	no	—
106	#108	—	2	2	no	—
107	#107	—	118	1	no	—
108	#106	—	118	1	no	—
109	C₄⋊C₈	—	118	2	no	—
110	#103	—	118	1	no	—
111	#103	—	118	1	no	—
112	#103	—	118	2	no	—
113	#101	—	118	2	no	—
114	C₂×Q₈	—	59	2	no	—
115	C₂×Q₈	—	118	1	no	—
116	C₂×Q₈	—	118	1	no	—
117	C₂×Q₈	—	118	1	no	—
118	C₂×Q₈	—	118	1	no	—
119	C₂×Q₈	—	118	1	no	—
120	C₂²×C₄	—	118	1	no	—
121	C₂²×C₄	—	118	1	no	—
122	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
123	M₁₆	—	2	1	no	—
124	M₁₆	—	2	1	no	—
125	M₁₆	—	2	1	no	—
126	M₁₆	—	118	1	no	—
127	M₁₆	—	118	1	no	—
128	M₁₆	C₂²⋊D₅₉	1	2	yes	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	118	1	no	—
134	C₂×C₈	D₂₃₆	1	2	yes	—
135	C₄⋊C₄	—	59	1	no	—
136	C₄⋊C₄	—	59	1	no	—
137	C₄⋊C₄	—	118	1	no	—
138	C₄⋊C₄	—	118	1	no	—
139	C₂²⋊C₄	—	118	1	no	—
140	C₄²	—	118	1	no	—
141	C₂³	#11745689	1	1	yes	characteristic
142	Q₈	—	118	2	no	—
143	Q₈	—	118	2	no	—
144	Q₈	—	236	1	no	—
145	Q₈	—	236	1	no	—
146	C₂×C₄	—	118	1	no	—
147	C₂×C₄	—	118	1	no	—
148	C₂×C₄	—	118	2	no	—
149	C₂×C₄	—	472	1	no	—
150	C₂×C₄	—	236	1	no	—
151	C₂×C₄	—	236	1	no	—
152	C₂×C₄	—	236	1	no	—
153	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
154	C₂×C₄	#11745685	1	1	yes	characteristic
155	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
156	C₂×C₄	#11745687	1	1	yes	characteristic
157	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
158	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	2	2	no	—
161	C₈	—	4	1	no	—
162	C₈	—	236	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	#14273702	1	1	yes	characteristic
166	C₂²	#14273703	1	1	yes	characteristic
167	C₂²	#14273722	1	1	yes	characteristic
168	C₄	—	472	1	no	—
169	C₄	—	236	1	no	—
170	C₄	—	236	1	no	—
171	C₄	—	236	2	no	—
172	C₄	#14273699	1	1	yes	characteristic
173	C₄	#14273701	1	1	yes	characteristic
174	C₄	#14273705	1	1	yes	characteristic
175	C₄	#14273707	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#16837324	1	1	yes	characteristic
179	C₂	#16837666	1	1	yes	characteristic
180	C₂	#16837678	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837444

C₂

yes

maximal, characteristic

#16837428

C₂

yes

maximal, characteristic

#16837412

C₂

yes

maximal, characteristic

C₂×#14273652

C₂

yes

maximal, characteristic

C₂×#14273645

C₂

yes

maximal, characteristic

#16837170

C₂

yes

maximal, characteristic

C₂×#108×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

characteristic

#14273686

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

#14273661

—

#14273659

—

Dic₅₉⋊C₈

—

—

—

—

—

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#108×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

Q₈

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

D₄

yes

characteristic

C₄×Dic₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#41

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×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#121

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

C₂²×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#2064

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

yes

—

—

—

—

—

—

C₅₉

#2064

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

100

C₂×M₁₆

—

101

C₂×M₁₆

C₂×D₅₉

yes

characteristic

102

C₂×M₁₆

C₂×D₅₉

yes

characteristic

103

C₂²×C₈

C₂×D₅₉

yes

characteristic

104

#117

—

105

C₂×C₄⋊C₄

—

106

#108

—

107

#107

—

118

—

108

#106

—

118

—

109

C₄⋊C₈

—

118

—

110

#103

—

118

—

111

#103

—

118

—

112

#103

—

118

—

113

#101

—

118

—

114

C₂×Q₈

—

115

C₂×Q₈

—

118

—

116

C₂×Q₈

—

118

—

117

C₂×Q₈

—

118

—

118

C₂×Q₈

—

118

—

119

C₂×Q₈

—

118

—

120

C₂²×C₄

—

118

—

121

C₂²×C₄

—

118

—

122

C₂²×C₄

C₂²×D₅₉

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

126

M₁₆

—

118

—

127

M₁₆

—

118

—

128

M₁₆

C₂²⋊D₅₉

yes

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

118

—

134

C₂×C₈

D₂₃₆

yes

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

118

—

138

C₄⋊C₄

—

118

—

139

C₂²⋊C₄

—

118

—

140

C₄²

—

118

—

141

C₂³

#11745689

yes

characteristic

142

Q₈

—

118

—

143

Q₈

—

118

—

144

Q₈

—

236

—

145

Q₈

—

236

—

146

C₂×C₄

—

118

—

147

C₂×C₄

—

118

—

148

C₂×C₄

—

118

—

149

C₂×C₄

—

472

—

150

C₂×C₄

—

236

—

151

C₂×C₄

—

236

—

152

C₂×C₄

—

236

—

153

C₂×C₄

C₂×D₂₃₆

yes

characteristic

154

C₂×C₄

#11745685

yes

characteristic

155

C₂×C₄

D₄×D₅₉

yes

characteristic

156

C₂×C₄

#11745687

yes

characteristic

157

C₂×C₄

Q₈×D₅₉

yes

characteristic

158

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

236

—

163

C₂²

—

164

C₂²

—

165

C₂²

#14273702

yes

characteristic

166

C₂²

#14273703

yes

characteristic

167

C₂²

#14273722

yes

characteristic

168

C₄

—

472

—

169

C₄

—

236

—

170

C₄

—

236

—

171

C₄

—

236

—

172

C₄

#14273699

yes

characteristic

173

C₄

#14273701

yes

characteristic

174

C₄

#14273705

yes

characteristic

175

C₄

#14273707

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#16837324

yes

characteristic

179

C₂

#16837666

yes

characteristic

180

C₂

#16837678

yes

characteristic

	subgroup	G/H
center	C₂²	#14273702
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#108×C₅₉	C₂
socle	C₂²×C₅₉	#121
hypercenter	C₂×#108	D₅₉
hypocenter	C₅₉	#2064
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#14273702

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#108×C₅₉

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
98. C₅₉	#2064	semidirect

H ≅ G/N

type

98. C₅₉

#2064

semidirect

Subgroups — #21425458

Conjugacy classes of subgroups

Complements