Group #21433751 — subgroups

Subgroups — #21433751

Order 7552.

460 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 149 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×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837973	C₂	1	2	yes	maximal
3	#16837660	C₂	1	4	yes	maximal, complemented
4	#16837643	C₂	1	4	yes	maximal, complemented
5	#16837440	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837412	C₂	1	2	yes	maximal, complemented
7	C₂×#131×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₁₈	C₂²	1	2	yes	—
9	C₂×Dic₂₃₆	—	2	4	no	—
10	C₂×Dic₂₃₆	C₂²	1	2	yes	—
11	C₂×Dic₂₃₆	C₂²	1	2	yes	—
12	#14273725	—	2	2	no	—
13	#14273714	—	2	2	no	—
14	#14273708	—	2	4	no	—
15	#14273706	C₂²	1	8	yes	—
16	#14273686	C₂²	1	2	yes	—
17	#14273661	C₂²	1	2	yes	—
18	#14273659	C₂²	1	2	yes	—
19	#14273652	C₂²	1	8	yes	—
20	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
21	#131×C₅₉	—	2	4	no	—
22	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
23	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₅₉	C₂²	1	2	yes	—
25	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
26	#41×C₅₉	—	2	2	no	—
27	#41×C₅₉	—	2	2	no	—
28	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
29	C₂×D₄×C₅₉	D₄	1	2	yes	—
30	C₂×D₄×C₅₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
32	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
33	C₂²×Dic₅₉	—	2	2	no	—
34	C₂×Dic₁₁₈	—	2	4	no	—
35	C₂×Dic₁₁₈	—	2	4	no	—
36	C₂×Dic₁₁₈	—	2	2	no	—
37	C₂×Dic₁₁₈	C₂³	1	4	yes	—
38	M₁₆×C₅₉	—	2	2	no	—
39	M₁₆×C₅₉	—	2	4	no	—
40	C₂×C₈×C₅₉	—	2	4	no	—
41	C₂×C₈×C₅₉	D₄	1	4	yes	—
42	C₂×C₈×C₅₉	C₂³	1	2	yes	—
43	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
45	C₂²⋊Dic₅₉	—	2	2	no	—
46	C₂²⋊Dic₅₉	—	2	8	no	—
47	C₄⋊Dic₅₉	C₂³	1	4	yes	—
48	Dic₅₉⋊C₄	—	2	8	no	—
49	C₄×Dic₅₉	—	2	2	no	—
50	Dic₂₃₆	—	2	4	no	—
51	Dic₂₃₆	—	2	4	no	—
52	C₂³×C₅₉	C₂×D₄	1	2	yes	—
53	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
54	Q₈×C₅₉	—	2	2	no	—
55	D₄×C₅₉	—	2	4	no	—
56	D₄×C₅₉	—	2	2	no	—
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	—	2	2	no	—
59	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₅₉	#41	1	2	yes	—
63	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
64	C₂×Dic₅₉	—	2	4	no	—
65	C₂×Dic₅₉	—	2	4	no	—
66	C₂×Dic₅₉	—	4	4	no	—
67	Dic₁₁₈	—	2	4	no	—
68	Dic₁₁₈	—	2	4	no	—
69	Dic₁₁₈	—	4	4	no	—
70	C₈×C₅₉	—	2	2	no	—
71	C₈×C₅₉	C₂×D₄	1	2	yes	—
72	C₈×C₅₉	C₂×D₄	1	2	yes	—
73	C₂²×C₅₉	—	2	2	no	—
74	C₂²×C₅₉	—	2	2	no	—
75	C₂²×C₅₉	—	2	4	no	—
76	C₂²×C₅₉	#121	1	2	yes	—
77	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₅₉	—	2	2	no	—
79	C₄×C₅₉	#121	1	2	yes	—
80	C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
81	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
82	Dic₅₉	—	4	4	no	—
83	Dic₅₉	—	4	4	no	—
84	#3059	—	59	1	no	maximal, Sylow
85	C₂×C₅₉	—	2	2	no	—
86	C₂×C₅₉	—	4	2	no	—
87	C₂×C₅₉	C₂×#121	1	1	yes	characteristic
88	C₂×C₅₉	#578	1	2	yes	—
89	C₂²×Q₁₆	—	59	1	no	—
90	C₂×#131	D₅₉	1	1	yes	characteristic
91	#536	—	59	2	no	—
92	#470	—	59	4	no	—
93	#467	—	59	4	no	—
94	#427	—	59	1	no	—
95	#419	—	59	2	no	—
96	C₅₉	#3059	1	1	yes	characteristic, complemented, Sylow
97	C₂×#41	C₂×D₅₉	1	1	yes	characteristic
98	C₂²×Q₈	—	59	2	no	—
99	C₂×Q₁₆	—	59	2	no	—
100	C₂×Q₁₆	—	59	2	no	—
101	C₂×Q₁₆	—	118	4	no	—
102	#131	—	2	4	no	—
103	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
104	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
105	C₂²×C₈	C₂×D₅₉	1	2	yes	—
106	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
107	C₄⋊Q₈	—	118	2	no	—
108	#124	—	118	2	no	—
109	#123	—	118	4	no	—
110	C₂²⋊Q₈	—	59	8	no	—
111	#117	—	59	2	no	—
112	#107	—	59	2	no	—
113	#106	—	59	2	no	—
114	#103	—	59	8	no	—
115	#41	—	2	2	no	—
116	#41	—	2	2	no	—
117	C₂×Q₈	—	59	4	no	—
118	C₂×Q₈	—	118	2	no	—
119	C₂×Q₈	—	118	4	no	—
120	C₂×Q₈	—	118	4	no	—
121	C₂×Q₈	D₂₃₆	1	1	yes	characteristic
122	C₂×D₄	D₂₃₆	1	2	yes	—
123	C₂×D₄	D₂₃₆	1	1	yes	characteristic
124	C₂²×C₄	—	118	2	no	—
125	C₂²×C₄	C₂²×D₅₉	1	2	yes	—
126	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
127	Q₁₆	—	118	4	no	—
128	Q₁₆	—	118	4	no	—
129	M₁₆	—	2	2	no	—
130	M₁₆	—	2	4	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	C₂²⋊D₅₉	1	4	yes	—
133	C₂×C₈	C₂²×D₅₉	1	2	yes	—
134	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
135	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
136	C₄⋊C₄	—	59	4	no	—
137	C₄⋊C₄	—	118	8	no	—
138	C₂²⋊C₄	—	118	2	no	—
139	C₂²⋊C₄	—	118	8	no	—
140	C₄²	—	118	2	no	—
141	C₂³	C₂×D₂₃₆	1	2	yes	—
142	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
143	Q₈	—	2	2	no	—
144	Q₈	—	118	4	no	—
145	Q₈	—	118	4	no	—
146	Q₈	—	236	4	no	—
147	D₄	—	2	4	no	—
148	D₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	118	4	no	—
152	C₂×C₄	—	118	4	no	—
153	C₂×C₄	—	236	4	no	—
154	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
155	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
156	C₂×C₄	#11745685	1	2	yes	—
157	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
158	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
161	C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	4	no	—
165	C₂²	#14273701	1	2	yes	—
166	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	236	4	no	—
169	C₄	—	236	4	no	—
170	C₄	#14273701	1	2	yes	—
171	C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
172	C₄	C₂×#11745685	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	—	4	2	no	—
175	C₂	C₂×#14273701	1	1	yes	characteristic
176	C₂	#16838219	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₃₆

C₂

yes

maximal, characteristic, complemented

#16837973

C₂

yes

maximal

#16837660

C₂

yes

maximal, complemented

#16837643

C₂

yes

maximal, complemented

#16837440

C₂

yes

maximal, characteristic, complemented

#16837412

C₂

yes

maximal, complemented

C₂×#131×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

C₂²

yes

—

C₂×Dic₂₃₆

C₂²

yes

—

—

—

—

C₂²

yes

—

#14273686

C₂²

yes

—

#14273661

C₂²

yes

—

#14273659

C₂²

yes

—

#14273652

C₂²

yes

—

C₂×#41×C₅₉

C₂²

yes

characteristic

#131×C₅₉

—

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#41×C₅₉

—

#41×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×D₄×C₅₉

D₄

yes

—

C₂×D₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂³

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

Dic₂₃₆

—

Dic₂₃₆

—

C₂³×C₅₉

C₂×D₄

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂×D₄

yes

—

C₈×C₅₉

C₂×D₄

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#121

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

#121

yes

—

C₄×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

C₂×#41

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#3059

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

C₂×#121

yes

characteristic

C₂×C₅₉

#578

yes

—

C₂²×Q₁₆

—

C₂×#131

D₅₉

yes

—

—

—

—

—

C₅₉

#3059

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₉

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

100

C₂×Q₁₆

—

101

C₂×Q₁₆

—

118

—

102

#131

—

103

C₂×M₁₆

C₂×D₅₉

yes

characteristic

104

C₂×M₁₆

C₂×D₅₉

yes

—

105

C₂²×C₈

C₂×D₅₉

yes

—

106

C₂²×C₈

C₂×D₅₉

yes

characteristic

107

C₄⋊Q₈

—

118

—

108

#124

—

118

—

109

#123

—

118

—

110

C₂²⋊Q₈

—

111

#117

—

112

#107

—

113

#106

—

114

#103

—

115

#41

—

116

#41

—

117

C₂×Q₈

—

118

C₂×Q₈

—

118

—

119

C₂×Q₈

—

118

—

120

C₂×Q₈

—

118

—

121

C₂×Q₈

D₂₃₆

yes

characteristic

122

C₂×D₄

D₂₃₆

yes

—

123

C₂×D₄

D₂₃₆

yes

characteristic

124

C₂²×C₄

—

118

—

125

C₂²×C₄

C₂²×D₅₉

yes

—

126

C₂²×C₄

C₂²×D₅₉

yes

characteristic

127

Q₁₆

—

118

—

128

Q₁₆

—

118

—

129

M₁₆

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

C₂²⋊D₅₉

yes

—

133

C₂×C₈

C₂²×D₅₉

yes

—

134

C₂×C₈

C₂²×D₅₉

yes

characteristic

135

C₂×C₈

C₂²×D₅₉

yes

characteristic

136

C₄⋊C₄

—

137

C₄⋊C₄

—

118

—

138

C₂²⋊C₄

—

118

—

139

C₂²⋊C₄

—

118

—

140

C₄²

—

118

—

141

C₂³

C₂×D₂₃₆

yes

—

142

C₂³

C₂×D₂₃₆

yes

characteristic

143

Q₈

—

144

Q₈

—

118

—

145

Q₈

—

118

—

146

Q₈

—

236

—

147

D₄

—

148

D₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

118

—

152

C₂×C₄

—

118

—

153

C₂×C₄

—

236

—

154

C₂×C₄

C₂×D₂₃₆

yes

—

155

C₂×C₄

C₂×D₂₃₆

yes

characteristic

156

C₂×C₄

#11745685

yes

—

157

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

158

C₂×C₄

C₂³×D₅₉

yes

characteristic

159

C₈

—

160

C₈

C₂×C₂²⋊D₅₉

yes

—

161

C₈

C₂×C₂²⋊D₅₉

yes

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#14273701

yes

—

166

C₂²

C₂²×D₂₃₆

yes

characteristic

167

C₄

—

168

C₄

—

236

—

169

C₄

—

236

—

170

C₄

#14273701

yes

—

171

C₄

C₂²×C₂²⋊D₅₉

yes

characteristic

172

C₄

C₂×#11745685

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

C₂×#14273701

yes

characteristic

176

C₂

#16838219

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₂×#131×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	C₂×#131	D₅₉
hypocenter	C₅₉	#3059
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

subgroup

G/H

center

C₂²

C₂²×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₂×#131×C₅₉

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

N	H ≅ G/N	type
1. C₂²×Dic₂₃₆	C₂	semidirect
3. #16837660	C₂	semidirect
4. #16837643	C₂	semidirect
5. #16837440	C₂	semidirect
6. #16837412	C₂	semidirect
96. C₅₉	#3059	semidirect

H ≅ G/N

type

1. C₂²×Dic₂₃₆

C₂

semidirect

3. #16837660

C₂

semidirect

4. #16837643

C₂

semidirect

5. #16837440

C₂

semidirect

6. #16837412

C₂

semidirect

96. C₅₉

#3059

semidirect

Subgroups — #21433751

Conjugacy classes of subgroups

Complements