Group #21426020 — subgroups

Subgroups — #21426020

Order 7552.

230 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 51 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×#14273746	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14273672	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14273668	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837639	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837205	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₅₉)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#357	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745687	—	2	1	no	—
9	#14273746	—	2	2	no	—
10	#14273746	—	2	2	no	—
11	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
12	C₂×#11745670	—	2	1	no	—
13	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
15	#14273702	—	2	1	no	—
16	#14273672	—	2	2	no	—
17	#14273668	—	2	2	no	—
18	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
19	#14273647	—	2	1	no	—
20	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
22	#107×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₅₉	—	2	1	no	—
24	C₂×C₂²⋊D₅₉	—	4	1	no	—
25	C₂²×Dic₅₉	—	4	1	no	—
26	#11745687	—	2	4	no	—
27	#11745687	—	4	2	no	—
28	C₂×D₂₃₆	—	2	1	no	—
29	C₂×D₂₃₆	—	4	1	no	—
30	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
31	C₂×Dic₁₁₈	—	2	1	no	—
32	D₈×C₅₉	—	2	1	no	—
33	D₈×C₅₉	D₄	1	2	yes	complemented
34	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₅₉	—	2	1	no	—
36	#11745670	—	2	4	no	—
37	(C₂×D₅₉)⋊C₄	—	4	1	no	—
38	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
39	Dic₂₃₆	—	2	1	no	—
40	Dic₂₃₆	D₄	1	2	yes	complemented
41	D₄₇₂	—	2	1	no	—
42	D₄₇₂	—	2	1	no	—
43	C₈×D₅₉	—	2	1	no	—
44	C₈×D₅₉	—	2	1	no	—
45	C₁₆×C₅₉	—	2	1	no	—
46	C₅₉⋊C₁₆	—	2	1	no	—
47	C₂³×C₅₉	—	4	1	no	—
48	C₂²×D₅₉	—	4	1	no	—
49	C₂²×D₅₉	D₈	1	1	yes	characteristic
50	D₄×C₅₉	—	2	2	no	—
51	D₄×C₅₉	—	4	1	no	—
52	C₂×C₄×C₅₉	—	4	1	no	—
53	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂²⋊D₅₉	—	4	4	no	—
55	C₂×Dic₅₉	—	4	4	no	—
56	C₂×Dic₅₉	—	4	1	no	—
57	C₂×Dic₅₉	D₈	1	1	yes	characteristic
58	D₂₃₆	—	4	1	no	—
59	D₂₃₆	—	4	1	no	—
60	D₂₃₆	—	8	2	no	—
61	C₄×D₅₉	—	2	1	no	—
62	C₄×D₅₉	—	2	1	no	—
63	Dic₁₁₈	—	2	2	no	—
64	Dic₁₁₈	—	4	1	no	—
65	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
66	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
67	C₅₉⋊C₈	—	2	1	no	—
68	C₂²×C₅₉	—	4	2	no	—
69	C₂²×C₅₉	—	8	1	no	—
70	C₂²×C₅₉	—	8	1	no	—
71	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
72	C₂×D₅₉	—	2	1	no	—
73	C₂×D₅₉	—	2	1	no	—
74	C₂×D₅₉	—	2	1	no	—
75	C₂×D₅₉	—	8	1	no	—
76	C₂×D₅₉	—	8	1	no	—
77	C₂×D₅₉	—	8	1	no	—
78	C₄×C₅₉	—	8	1	no	—
79	C₄×C₅₉	#120	1	1	yes	characteristic
80	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
81	Dic₅₉	—	2	1	no	—
82	Dic₅₉	—	4	2	no	—
83	#2186	—	59	1	no	maximal, Sylow
84	C₂×C₅₉	—	8	2	no	—
85	C₂×C₅₉	#447	1	1	yes	characteristic
86	C₂×C₅₉	#508	1	1	yes	characteristic
87	C₂×C₅₉	#509	1	1	yes	characteristic
88	D₅₉	—	4	1	no	—
89	D₅₉	—	16	1	no	—
90	C₂×#135	—	59	1	no	—
91	C₂×SD₃₂	—	59	1	no	—
92	C₂×D₁₆	—	59	1	no	—
93	#466	—	59	1	no	—
94	#358	—	59	1	no	—
95	#357	D₅₉	1	1	yes	characteristic, complemented
96	C₂²⋊C₁₆	—	59	1	no	—
97	C₅₉	#2186	1	1	yes	characteristic, complemented, Sylow
98	C₂×#41	—	118	1	no	—
99	#135	—	118	2	no	—
100	#135	—	118	2	no	—
101	C₂×Q₁₆	—	59	1	no	—
102	C₂×SD₁₆	—	118	1	no	—
103	C₂×D₈	—	59	1	no	—
104	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
105	C₂²×C₈	—	59	1	no	—
106	#121	—	118	1	no	—
107	SD₃₂	—	118	2	no	—
108	D₁₆	—	118	2	no	—
109	C₂×C₁₆	—	59	1	no	—
110	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
111	#107	C₂×D₅₉	1	1	yes	characteristic
112	#102	—	118	1	no	—
113	#41	—	118	4	no	—
114	#41	—	236	2	no	—
115	C₂×Q₈	—	118	1	no	—
116	C₂×D₄	—	2	1	no	—
117	C₂×D₄	—	118	1	no	—
118	C₂×D₄	—	236	1	no	—
119	C₂×D₄	—	236	1	no	—
120	C₂²×C₄	—	59	1	no	—
121	C₂²×C₄	—	236	1	no	—
122	Q₁₆	—	59	2	no	—
123	Q₁₆	—	118	1	no	—
124	SD₁₆	—	118	4	no	—
125	D₈	—	2	1	no	—
126	D₈	—	118	1	no	—
127	D₈	—	118	1	no	—
128	D₈	D₂₃₆	1	2	yes	complemented
129	C₂×C₈	—	59	1	no	—
130	C₂×C₈	—	118	1	no	—
131	C₂×C₈	—	118	1	no	—
132	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
133	C₄⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	236	1	no	—
135	C₁₆	—	2	1	no	—
136	C₁₆	—	118	1	no	—
137	C₂³	—	4	1	no	—
138	C₂³	—	59	1	no	—
139	C₂³	—	236	1	no	—
140	Q₈	—	118	2	no	—
141	Q₈	—	236	1	no	—
142	D₄	—	2	2	no	—
143	D₄	—	4	1	no	—
144	D₄	—	472	2	no	—
145	D₄	—	236	4	no	—
146	D₄	—	236	1	no	—
147	D₄	—	236	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	59	1	no	—
150	C₂×C₄	—	118	1	no	—
151	C₂×C₄	—	118	1	no	—
152	C₂×C₄	—	236	4	no	—
153	C₂×C₄	—	236	1	no	—
154	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
155	C₈	—	118	1	no	—
156	C₈	C₂×D₂₃₆	1	1	yes	characteristic
157	C₈	D₄×D₅₉	1	1	yes	characteristic
158	C₂²	—	4	2	no	—
159	C₂²	—	8	1	no	—
160	C₂²	—	8	1	no	—
161	C₂²	—	118	1	no	—
162	C₂²	—	118	1	no	—
163	C₂²	—	118	1	no	—
164	C₂²	—	472	1	no	—
165	C₂²	—	472	1	no	—
166	C₂²	—	472	1	no	—
167	C₂²	D₈×D₅₉	1	1	yes	characteristic
168	C₄	—	8	1	no	—
169	C₄	—	118	1	no	—
170	C₄	—	236	2	no	—
171	C₄	#14273697	1	1	yes	characteristic
172	C₄	#14273754	1	1	yes	characteristic
173	C₂	—	8	2	no	—
174	C₂	—	944	1	no	—
175	C₂	—	236	1	no	—
176	C₂	#16837537	1	1	yes	characteristic
177	C₂	#16837833	1	1	yes	characteristic
178	C₂	#16837836	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273746

C₂

yes

maximal, characteristic, complemented

C₂×#14273672

C₂

yes

maximal, characteristic, complemented

C₂×#14273668

C₂

yes

maximal, characteristic, complemented

#16837639

C₂

yes

maximal, characteristic, complemented

#16837205

C₂

yes

maximal, characteristic, complemented

(C₂×D₅₉)⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₅₉×#357

C₂

yes

maximal, characteristic, complemented

C₂×#11745687

—

#14273746

—

#14273746

—

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×#11745670

—

C₂×D₄₇₂

C₂²

yes

characteristic

C₂×C₈×D₅₉

C₂²

yes

—

—

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273647

—

C₂×D₈×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₂²

yes

characteristic

C₂×D₄×C₅₉

—

C₂×C₂²⋊D₅₉

—

C₂²×Dic₅₉

—

#11745687

—

#11745687

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

D₈×C₅₉

—

D₈×C₅₉

D₄

yes

complemented

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

#11745670

—

(C₂×D₅₉)⋊C₄

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

D₄

yes

complemented

D₄₇₂

—

D₄₇₂

—

C₈×D₅₉

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

D₈

yes

characteristic

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

C₂×D₄

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

#2186

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

D₅₉

—

C₂×#135

—

C₂×SD₃₂

—

C₂×D₁₆

—

#466

—

#358

—

#357

D₅₉

yes

characteristic, complemented

C₂²⋊C₁₆

—

C₅₉

#2186

yes

characteristic, complemented, Sylow

C₂×#41

—

118

—

#135

—

118

—

100

#135

—

118

—

101

C₂×Q₁₆

—

102

C₂×SD₁₆

—

118

—

103

C₂×D₈

—

104

C₂×D₈

C₂×D₅₉

yes

characteristic

105

C₂²×C₈

—

106

#121

—

118

—

107

SD₃₂

—

118

—

108

D₁₆

—

118

—

109

C₂×C₁₆

—

110

C₂×C₁₆

C₂×D₅₉

yes

characteristic

111

#107

C₂×D₅₉

yes

characteristic

112

#102

—

118

—

113

#41

—

118

—

114

#41

—

236

—

115

C₂×Q₈

—

118

—

116

C₂×D₄

—

117

C₂×D₄

—

118

—

118

C₂×D₄

—

236

—

119

C₂×D₄

—

236

—

120

C₂²×C₄

—

121

C₂²×C₄

—

236

—

122

Q₁₆

—

123

Q₁₆

—

118

—

124

SD₁₆

—

118

—

125

D₈

—

126

D₈

—

118

—

127

D₈

—

118

—

128

D₈

D₂₃₆

yes

complemented

129

C₂×C₈

—

130

C₂×C₈

—

118

—

131

C₂×C₈

—

118

—

132

C₂×C₈

C₂²×D₅₉

yes

characteristic

133

C₄⋊C₄

—

134

C₂²⋊C₄

—

236

—

135

C₁₆

—

136

C₁₆

—

118

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

236

—

140

Q₈

—

118

—

141

Q₈

—

236

—

142

D₄

—

143

D₄

—

144

D₄

—

472

—

145

D₄

—

236

—

146

D₄

—

236

—

147

D₄

—

236

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

118

—

151

C₂×C₄

—

118

—

152

C₂×C₄

—

236

—

153

C₂×C₄

—

236

—

154

C₂×C₄

D₄×D₅₉

yes

characteristic

155

C₈

—

118

—

156

C₈

C₂×D₂₃₆

yes

characteristic

157

C₈

D₄×D₅₉

yes

characteristic

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

118

—

162

C₂²

—

118

—

163

C₂²

—

118

—

164

C₂²

—

472

—

165

C₂²

—

472

—

166

C₂²

—

472

—

167

C₂²

D₈×D₅₉

yes

characteristic

168

C₄

—

169

C₄

—

118

—

170

C₄

—

236

—

171

C₄

#14273697

yes

characteristic

172

C₄

#14273754

yes

characteristic

173

C₂

—

174

C₂

—

944

—

175

C₂

—

236

—

176

C₂

#16837537

yes

characteristic

177

C₂

#16837833

yes

characteristic

178

C₂

#16837836

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#357	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#357	D₅₉
hypocenter	C₅₉	#2186
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#357

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. C₂×#14273746	C₂	semidirect
2. C₂×#14273672	C₂	semidirect
3. C₂×#14273668	C₂	semidirect
4. #16837639	C₂	semidirect
5. #16837205	C₂	semidirect
6. (C₂×D₅₉)⋊C₁₆	C₂	semidirect
7. C₅₉×#357	C₂	semidirect
33. D₈×C₅₉	D₄	semidirect
40. Dic₂₃₆	D₄	semidirect
95. #357	D₅₉	semidirect
97. C₅₉	#2186	semidirect
128. D₈	D₂₃₆	semidirect

H ≅ G/N

type

1. C₂×#14273746

C₂

semidirect

2. C₂×#14273672

C₂

semidirect

3. C₂×#14273668

C₂

semidirect

4. #16837639

C₂

semidirect

5. #16837205

C₂

semidirect

6. (C₂×D₅₉)⋊C₁₆

C₂

semidirect

7. C₅₉×#357

C₂

semidirect

33. D₈×C₅₉

D₄

semidirect

40. Dic₂₃₆

D₄

semidirect

95. #357

D₅₉

semidirect

97. C₅₉

#2186

semidirect

128. D₈

D₂₃₆

semidirect

Subgroups — #21426020

Conjugacy classes of subgroups

Complements