Group #21423455 — subgroups

Subgroups — #21423455

Order 7552.

188 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16837778	C₂	1	1	yes	maximal, characteristic
2	#16837603	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837599	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837526	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837154	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837145	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745660	—	2	1	no	—
9	C₂×#11745670	—	2	1	no	—
10	C₂×#11745669	—	2	1	no	—
11	C₂×#11745669	—	2	1	no	—
12	#14273724	C₂²	1	1	yes	characteristic
13	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
14	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₅₉⋊C₈	—	2	1	no	—
16	#14273649	—	2	1	no	—
17	#14273648	—	2	1	no	—
18	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×Dic₅₉	—	2	1	no	—
20	#127×C₅₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₅₉	—	2	1	no	—
24	C₂×D₄×C₅₉	—	2	1	no	—
25	C₂×D₄×C₅₉	—	2	1	no	—
26	C₂×D₂₃₆	—	2	1	no	—
27	C₂×D₂₃₆	—	2	1	no	—
28	C₂×D₂₃₆	—	2	1	no	—
29	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
30	C₂×C₈×C₅₉	—	2	1	no	—
31	C₂×C₈×C₅₉	—	2	1	no	—
32	C₄²×C₅₉	C₂³	1	1	yes	characteristic
33	#11745670	—	4	2	no	—
34	#11745669	—	4	2	no	—
35	#11745669	—	4	2	no	—
36	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
37	Dic₅₉⋊C₄	—	2	1	no	—
38	C₄×Dic₅₉	—	2	1	no	—
39	C₂×C₅₉⋊C₈	—	2	1	no	—
40	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
42	#11745660	—	4	2	no	—
43	C₂³×C₅₉	—	4	1	no	—
44	C₂²×D₅₉	—	4	1	no	—
45	D₄×C₅₉	—	4	1	no	—
46	D₄×C₅₉	—	4	1	no	—
47	D₄×C₅₉	—	4	1	no	—
48	D₄×C₅₉	—	4	1	no	—
49	D₄×C₅₉	—	8	1	no	—
50	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
53	C₂×Dic₅₉	—	2	1	no	—
54	C₂×Dic₅₉	—	2	1	no	—
55	D₂₃₆	—	4	1	no	—
56	D₂₃₆	—	4	1	no	—
57	D₂₃₆	—	4	1	no	—
58	D₂₃₆	—	4	1	no	—
59	D₂₃₆	—	8	1	no	—
60	Dic₁₁₈	—	2	1	no	—
61	Dic₁₁₈	D₈	1	2	yes	complemented
62	C₈×C₅₉	—	4	1	no	—
63	C₈×C₅₉	—	4	1	no	—
64	C₅₉⋊C₈	—	2	1	no	—
65	C₅₉⋊C₈	—	4	1	no	—
66	C₅₉⋊C₈	SD₁₆	1	2	yes	complemented
67	C₂²×C₅₉	—	8	1	no	—
68	C₂²×C₅₉	—	8	1	no	—
69	C₂²×C₅₉	—	8	1	no	—
70	C₂²×C₅₉	#121	1	1	yes	characteristic
71	C₂×D₅₉	—	8	1	no	—
72	C₂×D₅₉	—	8	1	no	—
73	C₂×D₅₉	—	8	1	no	—
74	C₄×C₅₉	—	2	1	no	—
75	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
76	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
77	C₄×C₅₉	#135	1	1	yes	characteristic
78	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
79	Dic₅₉	—	2	2	no	—
80	Dic₅₉	—	4	1	no	—
81	#1668	—	59	1	no	maximal, Sylow
82	C₂×C₅₉	—	16	1	no	—
83	C₂×C₅₉	#460	1	1	yes	characteristic
84	C₂×C₅₉	#466	1	1	yes	characteristic
85	C₂×C₅₉	#472	1	1	yes	characteristic
86	D₅₉	—	16	1	no	—
87	#493	—	59	1	no	—
88	#460	—	59	1	no	—
89	#460	—	59	1	no	—
90	C₈×Q₈	—	59	1	no	—
91	#335	—	59	1	no	—
92	#331	—	59	1	no	—
93	#331	D₅₉	1	1	yes	characteristic, complemented
94	C₅₉	#1668	1	1	yes	characteristic, complemented, Sylow
95	C₂×SD₁₆	—	118	1	no	—
96	C₂×SD₁₆	—	118	1	no	—
97	C₂×D₈	—	118	1	no	—
98	C₂×D₈	—	118	1	no	—
99	#127	—	59	1	no	—
100	#127	C₂×D₅₉	1	1	yes	characteristic
101	C₄×Q₈	—	59	1	no	—
102	C₄⋊C₈	—	59	1	no	—
103	C₄⋊C₈	—	118	1	no	—
104	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
105	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
106	#102	—	118	1	no	—
107	#102	—	118	1	no	—
108	C₄×C₈	—	59	1	no	—
109	C₄×C₈	—	118	1	no	—
110	C₂×Q₈	—	59	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	—	118	1	no	—
115	C₂×D₄	—	118	1	no	—
116	C₂×D₄	—	118	1	no	—
117	SD₁₆	—	236	2	no	—
118	SD₁₆	—	236	2	no	—
119	D₈	—	236	2	no	—
120	D₈	—	236	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	59	1	no	—
124	C₂×C₈	—	59	1	no	—
125	C₂×C₈	—	118	1	no	—
126	C₄⋊C₄	—	59	1	no	—
127	C₄⋊C₄	—	118	1	no	—
128	C₄²	—	118	1	no	—
129	C₄²	C₂²×D₅₉	1	1	yes	characteristic
130	C₂³	—	4	1	no	—
131	C₂³	—	236	1	no	—
132	Q₈	—	59	2	no	—
133	Q₈	—	118	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	4	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	4	1	no	—
138	D₄	—	8	1	no	—
139	D₄	—	472	1	no	—
140	D₄	—	236	1	no	—
141	D₄	—	236	1	no	—
142	D₄	—	236	1	no	—
143	D₄	—	236	1	no	—
144	C₂×C₄	—	118	1	no	—
145	C₂×C₄	—	118	1	no	—
146	C₂×C₄	#11745685	1	1	yes	characteristic
147	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
148	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	59	2	no	—
152	C₈	—	118	1	no	—
153	C₈	—	236	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	8	1	no	—
156	C₂²	—	8	1	no	—
157	C₂²	—	472	1	no	—
158	C₂²	—	472	1	no	—
159	C₂²	—	472	1	no	—
160	C₂²	#14273700	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	118	2	no	—
163	C₄	—	236	1	no	—
164	C₄	D₈×D₅₉	1	1	yes	characteristic
165	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
166	C₄	#14273748	1	1	yes	characteristic
167	C₄	#14273751	1	1	yes	characteristic
168	C₂	—	16	1	no	—
169	C₂	—	944	1	no	—
170	C₂	#16837601	1	1	yes	characteristic
171	C₂	#16837637	1	1	yes	characteristic
172	C₂	#16837671	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837778

C₂

yes

maximal, characteristic

#16837603

C₂

yes

maximal, characteristic, complemented

#16837599

C₂

yes

maximal, characteristic, complemented

#16837526

C₂

yes

maximal, characteristic, complemented

#16837154

C₂

yes

maximal, characteristic, complemented

#16837145

C₂

yes

maximal, characteristic, complemented

C₅₉×#331

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₁₁₈

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

Dic₅₉⋊C₈

—

#14273649

—

#14273648

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₈×Dic₅₉

—

#127×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×Dic₁₁₈

D₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂³

yes

—

—

—

C₄⋊Dic₅₉

D₄

yes

characteristic

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

#11745660

—

C₂³×C₅₉

—

C₂²×D₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

Dic₁₁₈

—

Dic₁₁₈

D₈

yes

complemented

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

SD₁₆

yes

complemented

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#121

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

#1668

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

#493

—

#460

—

#460

—

C₈×Q₈

—

#335

—

#331

—

#331

D₅₉

yes

characteristic, complemented

C₅₉

#1668

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

118

—

C₂×SD₁₆

—

118

—

C₂×D₈

—

118

—

C₂×D₈

—

118

—

#127

—

100

#127

C₂×D₅₉

yes

characteristic

101

C₄×Q₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

118

—

104

C₄⋊C₈

C₂×D₅₉

yes

characteristic

105

C₄⋊C₈

C₂×D₅₉

yes

characteristic

106

#102

—

118

—

107

#102

—

118

—

108

C₄×C₈

—

109

C₄×C₈

—

118

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

118

—

115

C₂×D₄

—

118

—

116

C₂×D₄

—

118

—

117

SD₁₆

—

236

—

118

SD₁₆

—

236

—

119

D₈

—

236

—

120

D₈

—

236

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

118

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

118

—

128

C₄²

—

118

—

129

C₄²

C₂²×D₅₉

yes

characteristic

130

C₂³

—

131

C₂³

—

236

—

132

Q₈

—

133

Q₈

—

118

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

472

—

140

D₄

—

236

—

141

D₄

—

236

—

142

D₄

—

236

—

143

D₄

—

236

—

144

C₂×C₄

—

118

—

145

C₂×C₄

—

118

—

146

C₂×C₄

#11745685

yes

characteristic

147

C₂×C₄

D₄×D₅₉

yes

characteristic

148

C₂×C₄

D₄×D₅₉

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

118

—

153

C₈

—

236

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

472

—

158

C₂²

—

472

—

159

C₂²

—

472

—

160

C₂²

#14273700

yes

characteristic

161

C₄

—

162

C₄

—

118

—

163

C₄

—

236

—

164

C₄

D₈×D₅₉

yes

characteristic

165

C₄

SD₁₆×D₅₉

yes

characteristic

166

C₄

#14273748

yes

characteristic

167

C₄

#14273751

yes

characteristic

168

C₂

—

169

C₂

—

944

—

170

C₂

#16837601

yes

characteristic

171

C₂

#16837637

yes

characteristic

172

C₂

#16837671

yes

characteristic

	subgroup	G/H
center	C₂²	#14273700
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#331	C₂
socle	C₂²×C₅₉	#121
hypercenter	#331	D₅₉
hypocenter	C₅₉	#1668
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#14273700

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#331

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #16837603	C₂	semidirect
3. #16837599	C₂	semidirect
4. #16837526	C₂	semidirect
5. #16837154	C₂	semidirect
6. #16837145	C₂	semidirect
7. C₅₉×#331	C₂	semidirect
61. Dic₁₁₈	D₈	semidirect
66. C₅₉⋊C₈	SD₁₆	semidirect
93. #331	D₅₉	semidirect
94. C₅₉	#1668	semidirect

H ≅ G/N

type

2. #16837603

C₂

semidirect

3. #16837599

C₂

semidirect

4. #16837526

C₂

semidirect

5. #16837154

C₂

semidirect

6. #16837145

C₂

semidirect

7. C₅₉×#331

C₂

semidirect

61. Dic₁₁₈

D₈

semidirect

66. C₅₉⋊C₈

SD₁₆

semidirect

93. #331

D₅₉

semidirect

94. C₅₉

#1668

semidirect

Subgroups — #21423455

Conjugacy classes of subgroups

Complements