Group #21422811 — subgroups

Subgroups — #21422811

Order 7552.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₄⋊Q₈×D₅₉	C₂	1	1	yes	maximal, characteristic
2	#16837458	C₂	1	1	yes	maximal, characteristic
3	#16837458	C₂	1	1	yes	maximal, characteristic
4	#16837149	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837148	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837147	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₅₉	—	2	1	no	—
9	#14273727	C₂²	1	1	yes	characteristic
10	#14273726	—	2	1	no	—
11	#14273726	C₄	1	2	yes	—
12	#14273725	—	2	1	no	—
13	C₄⋊C₄×D₅₉	—	2	1	no	—
14	C₄⋊C₄×D₅₉	C₄	1	2	yes	—
15	C₄²×D₅₉	C₂²	1	1	yes	characteristic
16	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
17	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
18	(C₂×D₅₉)⋊C₈	—	2	1	no	—
19	(C₂×D₅₉)⋊C₈	—	2	1	no	—
20	#14273637	—	2	1	no	—
21	#14273637	—	2	1	no	—
22	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₅₉	—	2	1	no	—
26	Q₈×D₅₉	—	4	2	no	—
27	Q₈×D₅₉	—	4	2	no	—
28	C₂×C₄×D₅₉	—	2	2	no	—
29	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
31	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
32	C₂×Dic₁₁₈	—	2	1	no	—
33	C₂×Dic₁₁₈	—	2	2	no	—
34	C₂×C₈×C₅₉	—	2	1	no	—
35	C₂×C₈×C₅₉	—	2	1	no	—
36	C₄⋊C₄×C₅₉	—	2	1	no	—
37	C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
38	C₄²×C₅₉	C₂³	1	1	yes	characteristic
39	C₄⋊Dic₅₉	—	2	1	no	—
40	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
41	Dic₅₉⋊C₄	—	2	2	no	—
42	Dic₅₉⋊C₄	—	2	2	no	—
43	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₅₉	D₄	1	1	yes	characteristic
45	C₄×Dic₅₉	D₄	1	1	yes	characteristic
46	C₂×C₅₉⋊C₈	—	2	1	no	—
47	C₂×C₅₉⋊C₈	—	2	1	no	—
48	C₂²×D₅₉	C₂²⋊C₄	1	1	yes	characteristic
49	Q₈×C₅₉	—	4	1	no	—
50	Q₈×C₅₉	—	4	1	no	—
51	C₂×C₄×C₅₉	—	2	2	no	—
52	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₅₉	—	2	2	no	—
56	C₂×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₄×D₅₉	—	2	2	no	—
60	C₄×D₅₉	—	2	1	no	—
61	C₄×D₅₉	—	2	1	no	—
62	C₄×D₅₉	—	2	1	no	—
63	C₄×D₅₉	—	2	1	no	—
64	C₄×D₅₉	—	4	4	no	—
65	Dic₁₁₈	—	4	4	no	—
66	Dic₁₁₈	—	4	1	no	—
67	Dic₁₁₈	—	4	1	no	—
68	C₈×C₅₉	—	4	1	no	—
69	C₈×C₅₉	—	4	1	no	—
70	C₅₉⋊C₈	—	4	1	no	—
71	C₅₉⋊C₈	—	4	1	no	—
72	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×D₅₉	—	2	1	no	—
74	C₂×D₅₉	—	2	1	no	—
75	C₂×D₅₉	#101	1	2	yes	—
76	C₄×C₅₉	—	2	1	no	—
77	C₄×C₅₉	—	4	2	no	—
78	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₄×C₅₉	#137	1	1	yes	characteristic
81	C₄×C₅₉	#137	1	1	yes	characteristic
82	Dic₅₉	—	2	1	no	—
83	Dic₅₉	—	2	1	no	—
84	Dic₅₉	—	2	1	no	—
85	Dic₅₉	—	4	2	no	—
86	#1556	—	59	1	no	maximal, Sylow
87	C₂×C₅₉	C₂×#101	1	1	yes	characteristic
88	C₂×C₅₉	#418	1	1	yes	characteristic
89	C₂×C₅₉	#419	1	1	yes	characteristic
90	D₅₉	—	2	2	no	—
91	C₂×C₄⋊Q₈	—	59	1	no	—
92	#432	—	59	1	no	—
93	#432	—	59	1	no	—
94	#332	—	59	1	no	—
95	#332	—	59	1	no	—
96	#332	—	59	1	no	—
97	#332	D₅₉	1	1	yes	characteristic, complemented
98	C₅₉	#1556	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	—	118	1	no	—
100	C₄⋊Q₈	—	59	2	no	—
101	C₄⋊Q₈	—	59	1	no	—
102	C₄⋊Q₈	—	118	1	no	—
103	C₄⋊Q₈	—	118	1	no	—
104	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	59	2	no	—
106	C₂×C₄⋊C₄	—	118	1	no	—
107	C₂×C₄²	—	59	1	no	—
108	C₄⋊C₈	—	59	1	no	—
109	C₄⋊C₈	—	59	1	no	—
110	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
111	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
112	C₂²⋊C₈	—	118	1	no	—
113	C₂²⋊C₈	—	118	1	no	—
114	#97	—	118	1	no	—
115	#97	—	118	1	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	118	2	no	—
118	C₂×Q₈	—	118	1	no	—
119	C₂×Q₈	—	236	2	no	—
120	C₂×Q₈	—	236	2	no	—
121	C₂²×C₄	—	59	1	no	—
122	C₂²×C₄	—	59	1	no	—
123	C₂²×C₄	—	59	1	no	—
124	C₂²×C₄	—	118	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	118	1	no	—
128	C₂×C₈	—	118	1	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	59	2	no	—
131	C₄⋊C₄	—	118	2	no	—
132	C₄⋊C₄	—	118	2	no	—
133	C₄⋊C₄	—	118	1	no	—
134	C₄⋊C₄	C₄×D₅₉	1	2	yes	—
135	C₄²	—	59	1	no	—
136	C₄²	—	59	1	no	—
137	C₄²	—	59	1	no	—
138	C₄²	C₂²×D₅₉	1	1	yes	characteristic
139	C₂³	—	59	1	no	—
140	Q₈	—	4	1	no	—
141	Q₈	—	4	1	no	—
142	Q₈	—	236	4	no	—
143	Q₈	—	236	1	no	—
144	Q₈	—	236	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	59	1	no	—
147	C₂×C₄	—	59	1	no	—
148	C₂×C₄	—	59	1	no	—
149	C₂×C₄	—	118	2	no	—
150	C₂×C₄	—	118	1	no	—
151	C₂×C₄	—	118	1	no	—
152	C₂×C₄	—	118	1	no	—
153	C₂×C₄	—	118	1	no	—
154	C₂×C₄	—	118	2	no	—
155	C₂×C₄	—	236	4	no	—
156	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
157	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
158	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
159	C₈	—	4	1	no	—
160	C₈	—	4	1	no	—
161	C₈	—	236	1	no	—
162	C₈	—	236	1	no	—
163	C₂²	—	59	2	no	—
164	C₂²	—	118	1	no	—
165	C₂²	—	118	1	no	—
166	C₂²	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	4	2	no	—
169	C₄	—	118	1	no	—
170	C₄	—	118	1	no	—
171	C₄	—	118	1	no	—
172	C₄	—	236	2	no	—
173	C₄	#14273751	1	1	yes	characteristic
174	C₄	#14273754	1	1	yes	characteristic
175	C₄	#14273756	1	1	yes	characteristic
176	C₄	#14273759	1	1	yes	characteristic
177	C₂	—	118	2	no	—
178	C₂	#101×D₅₉	1	1	yes	characteristic
179	C₂	#16837405	1	1	yes	characteristic
180	C₂	#16837409	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₅₉

C₂

yes

maximal, characteristic

#16837458

C₂

yes

maximal, characteristic

#16837458

C₂

yes

maximal, characteristic

#16837149

C₂

yes

maximal, characteristic, complemented

#16837148

C₂

yes

maximal, characteristic, complemented

#16837147

C₂

yes

maximal, characteristic, complemented

C₅₉×#332

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₅₉

—

#14273727

C₂²

yes

characteristic

#14273726

—

#14273726

C₄

yes

—

#14273725

—

C₄⋊C₄×D₅₉

—

C₄⋊C₄×D₅₉

C₄

yes

—

C₄²×D₅₉

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

(C₂×D₅₉)⋊C₈

—

(C₂×D₅₉)⋊C₈

—

#14273637

—

#14273637

—

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

Q₈×D₅₉

—

Q₈×D₅₉

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

C₂×C₄

yes

characteristic

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₄×Dic₅₉

D₄

yes

characteristic

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂²×D₅₉

C₂²⋊C₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₉

C₂²⋊C₄

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

#101

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

C₂×C₄⋊Q₈

—

—

—

—

—

—

D₅₉

yes

characteristic, complemented

C₅₉

#1556

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

118

—

100

C₄⋊Q₈

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

118

—

103

C₄⋊Q₈

—

118

—

104

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

118

—

107

C₂×C₄²

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄⋊C₈

C₂×D₅₉

yes

characteristic

111

C₄⋊C₈

C₂×D₅₉

yes

characteristic

112

C₂²⋊C₈

—

118

—

113

C₂²⋊C₈

—

118

—

114

#97

—

118

—

115

#97

—

118

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

—

118

C₂×Q₈

—

118

—

119

C₂×Q₈

—

236

—

120

C₂×Q₈

—

236

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

118

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

118

—

128

C₂×C₈

—

118

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

118

—

132

C₄⋊C₄

—

118

—

133

C₄⋊C₄

—

118

—

134

C₄⋊C₄

C₄×D₅₉

yes

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

C₂²×D₅₉

yes

characteristic

139

C₂³

—

140

Q₈

—

141

Q₈

—

142

Q₈

—

236

—

143

Q₈

—

236

—

144

Q₈

—

236

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

118

—

150

C₂×C₄

—

118

—

151

C₂×C₄

—

118

—

152

C₂×C₄

—

118

—

153

C₂×C₄

—

118

—

154

C₂×C₄

—

118

—

155

C₂×C₄

—

236

—

156

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

157

C₂×C₄

D₄×D₅₉

yes

characteristic

158

C₂×C₄

D₄×D₅₉

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

236

—

162

C₈

—

236

—

163

C₂²

—

164

C₂²

—

118

—

165

C₂²

—

118

—

166

C₂²

C₂²⋊C₄×D₅₉

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

118

—

170

C₄

—

118

—

171

C₄

—

118

—

172

C₄

—

236

—

173

C₄

#14273751

yes

characteristic

174

C₄

#14273754

yes

characteristic

175

C₄

#14273756

yes

characteristic

176

C₄

#14273759

yes

characteristic

177

C₂

—

118

—

178

C₂

#101×D₅₉

yes

characteristic

179

C₂

#16837405

yes

characteristic

180

C₂

#16837409

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#332	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	#332	D₅₉
hypocenter	C₅₉	#1556
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#332

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
4. #16837149	C₂	semidirect
5. #16837148	C₂	semidirect
6. #16837147	C₂	semidirect
7. C₅₉×#332	C₂	semidirect
97. #332	D₅₉	semidirect
98. C₅₉	#1556	semidirect

H ≅ G/N

type

4. #16837149

C₂

semidirect

5. #16837148

C₂

semidirect

6. #16837147

C₂

semidirect

7. C₅₉×#332

C₂

semidirect

97. #332

D₅₉

semidirect

98. C₅₉

#1556

semidirect

Subgroups — #21422811

Conjugacy classes of subgroups

Complements