Group #21423189 — subgroups

Subgroups — #21423189

Order 7552.

210 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16838127	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837599	C₂	1	1	yes	maximal, characteristic
3	#16837599	C₂	1	1	yes	maximal, characteristic
4	#16837148	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837134	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837134	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#486	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745670	—	2	1	no	—
9	C₂×#11745670	—	2	1	no	—
10	#14273725	—	2	1	no	—
11	#14273718	—	2	1	no	—
12	#14273717	—	2	2	no	—
13	Q₈×Dic₅₉	—	2	1	no	—
14	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
15	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
16	#14273692	—	2	1	no	—
17	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
18	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
19	#14273648	—	2	1	no	—
20	#14273648	—	2	1	no	—
21	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
22	#127×C₅₉	C₂²	1	1	yes	characteristic
23	#102×C₅₉	—	2	2	no	—
24	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₅₉	—	2	1	no	—
26	C₂×D₄×C₅₉	—	2	1	no	—
27	C₂×D₄×C₅₉	—	2	1	no	—
28	C₂×D₄×C₅₉	—	2	1	no	—
29	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
30	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
31	C₂×C₈×C₅₉	—	2	1	no	—
32	C₄⋊C₄×C₅₉	—	2	1	no	—
33	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
34	C₄²×C₅₉	C₂³	1	1	yes	characteristic
35	#11745670	—	4	2	no	—
36	#11745670	—	4	2	no	—
37	C₄⋊Dic₅₉	—	2	2	no	—
38	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
39	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
40	Dic₅₉⋊C₄	—	2	2	no	—
41	Dic₅₉⋊C₄	—	2	2	no	—
42	Dic₅₉⋊C₄	—	2	1	no	—
43	Dic₅₉⋊C₄	—	2	1	no	—
44	C₄×Dic₅₉	—	2	2	no	—
45	C₄×Dic₅₉	—	2	1	no	—
46	C₄×Dic₅₉	—	2	1	no	—
47	C₂×C₅₉⋊C₈	—	2	1	no	—
48	C₂×C₅₉⋊C₈	—	2	1	no	—
49	C₂³×C₅₉	—	4	1	no	—
50	Q₈×C₅₉	—	4	1	no	—
51	Q₈×C₅₉	—	4	1	no	—
52	D₄×C₅₉	—	4	2	no	—
53	D₄×C₅₉	—	4	1	no	—
54	D₄×C₅₉	—	4	1	no	—
55	D₄×C₅₉	—	4	1	no	—
56	D₄×C₅₉	—	4	1	no	—
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₅₉	—	2	1	no	—
62	C₂×Dic₅₉	—	2	1	no	—
63	C₂×Dic₅₉	—	2	1	no	—
64	C₂×Dic₅₉	—	2	1	no	—
65	Dic₁₁₈	—	2	1	no	—
66	Dic₁₁₈	—	2	1	no	—
67	Dic₁₁₈	—	2	1	no	—
68	Dic₁₁₈	SD₁₆	1	2	yes	complemented
69	C₈×C₅₉	—	4	1	no	—
70	C₅₉⋊C₈	—	4	1	no	—
71	C₅₉⋊C₈	—	4	1	no	—
72	C₂²×C₅₉	—	8	1	no	—
73	C₂²×C₅₉	—	8	1	no	—
74	C₂²×C₅₉	—	8	1	no	—
75	C₂²×C₅₉	#120	1	1	yes	characteristic
76	C₄×C₅₉	—	2	1	no	—
77	C₄×C₅₉	—	4	2	no	—
78	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
79	C₄×C₅₉	#135	1	1	yes	characteristic
80	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₄×C₅₉	#137	1	1	yes	characteristic
82	Dic₅₉	—	2	2	no	—
83	Dic₅₉	—	4	1	no	—
84	Dic₅₉	—	4	1	no	—
85	Dic₅₉	—	4	1	no	—
86	#1626	—	59	1	no	maximal, Sylow
87	C₂×C₅₉	—	16	1	no	—
88	C₂×C₅₉	#448	1	1	yes	characteristic
89	C₂×C₅₉	#449	1	1	yes	characteristic
90	C₂×C₅₉	#454	1	1	yes	characteristic
91	#557	—	59	1	no	—
92	#486	D₅₉	1	1	yes	characteristic
93	#460	—	59	1	no	—
94	#460	—	59	1	no	—
95	#332	—	59	1	no	—
96	Q₈⋊C₈	—	59	1	no	—
97	Q₈⋊C₈	—	59	1	no	—
98	C₅₉	#1626	1	1	yes	characteristic, complemented, Sylow
99	C₂×SD₁₆	—	118	1	no	—
100	C₂×SD₁₆	—	118	1	no	—
101	C₄⋊Q₈	—	118	1	no	—
102	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
103	#127	C₂×D₅₉	1	1	yes	characteristic
104	#125	—	118	1	no	—
105	#125	—	118	2	no	—
106	C₄×Q₈	—	59	1	no	—
107	C₄×Q₈	—	59	1	no	—
108	C₄×Q₈	—	118	1	no	—
109	C₄×Q₈	—	118	1	no	—
110	C₄⋊C₈	—	59	1	no	—
111	C₄⋊C₈	—	59	1	no	—
112	#102	—	2	2	no	—
113	#102	—	118	1	no	—
114	#102	—	118	1	no	—
115	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	59	1	no	—
118	C₂×Q₈	—	59	1	no	—
119	C₂×D₄	—	2	1	no	—
120	C₂×D₄	—	2	1	no	—
121	C₂×D₄	—	2	1	no	—
122	SD₁₆	—	236	2	no	—
123	SD₁₆	—	236	2	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	118	1	no	—
126	C₂×C₈	—	118	1	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	59	1	no	—
129	C₄⋊C₄	—	59	1	no	—
130	C₄⋊C₄	—	118	2	no	—
131	C₄⋊C₄	—	118	2	no	—
132	C₄⋊C₄	—	118	2	no	—
133	C₄⋊C₄	—	118	1	no	—
134	C₄⋊C₄	—	118	1	no	—
135	C₄⋊C₄	C₂²⋊D₅₉	1	2	yes	—
136	C₄²	—	118	2	no	—
137	C₄²	—	118	1	no	—
138	C₄²	—	118	1	no	—
139	C₄²	C₂²×D₅₉	1	1	yes	characteristic
140	C₂³	—	4	1	no	—
141	Q₈	—	4	1	no	—
142	Q₈	—	4	1	no	—
143	Q₈	—	59	2	no	—
144	Q₈	—	118	1	no	—
145	Q₈	—	118	1	no	—
146	Q₈	—	118	1	no	—
147	D₄	—	4	2	no	—
148	D₄	—	4	1	no	—
149	D₄	—	4	1	no	—
150	D₄	—	4	1	no	—
151	D₄	—	4	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	118	1	no	—
154	C₂×C₄	—	118	1	no	—
155	C₂×C₄	—	118	1	no	—
156	C₂×C₄	—	118	1	no	—
157	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
158	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
159	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
160	C₈	—	4	1	no	—
161	C₈	—	236	1	no	—
162	C₈	—	236	1	no	—
163	C₂²	—	8	1	no	—
164	C₂²	—	8	1	no	—
165	C₂²	—	8	1	no	—
166	C₂²	#14273695	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	4	2	no	—
169	C₄	—	118	2	no	—
170	C₄	—	236	1	no	—
171	C₄	—	236	1	no	—
172	C₄	—	236	1	no	—
173	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
174	C₄	#14273746	1	1	yes	characteristic
175	C₄	#14273751	1	1	yes	characteristic
176	C₄	#14273756	1	1	yes	characteristic
177	C₂	—	16	1	no	—
178	C₂	#16837539	1	1	yes	characteristic
179	C₂	#16837544	1	1	yes	characteristic
180	C₂	#16837574	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838127

C₂

yes

maximal, characteristic, complemented

#16837599

C₂

yes

maximal, characteristic

#16837599

C₂

yes

maximal, characteristic

#16837148

C₂

yes

maximal, characteristic, complemented

#16837134

C₂

yes

maximal, characteristic, complemented

#16837134

C₂

yes

maximal, characteristic, complemented

C₅₉×#486

C₂

yes

maximal, characteristic

—

—

—

—

—

Q₈×Dic₅₉

—

C₄×Dic₁₁₈

C₂²

yes

characteristic

C₄×Dic₁₁₈

C₂²

yes

characteristic

#14273692

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

#14273648

—

#14273648

—

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

#127×C₅₉

C₂²

yes

characteristic

#102×C₅₉

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×Dic₁₁₈

D₄

yes

characteristic

C₂×Dic₁₁₈

D₄

yes

characteristic

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745670

—

#11745670

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

D₄

yes

characteristic

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

SD₁₆

yes

complemented

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#120

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×SD₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#1626

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

yes

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₅₉

#1626

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

118

—

100

C₂×SD₁₆

—

118

—

101

C₄⋊Q₈

—

118

—

102

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

103

#127

C₂×D₅₉

yes

characteristic

104

#125

—

118

—

105

#125

—

118

—

106

C₄×Q₈

—

107

C₄×Q₈

—

108

C₄×Q₈

—

118

—

109

C₄×Q₈

—

118

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

#102

—

113

#102

—

118

—

114

#102

—

118

—

115

C₄×C₈

C₂×D₅₉

yes

characteristic

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

—

122

SD₁₆

—

236

—

123

SD₁₆

—

236

—

124

C₂×C₈

—

125

C₂×C₈

—

118

—

126

C₂×C₈

—

118

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

118

—

131

C₄⋊C₄

—

118

—

132

C₄⋊C₄

—

118

—

133

C₄⋊C₄

—

118

—

134

C₄⋊C₄

—

118

—

135

C₄⋊C₄

C₂²⋊D₅₉

yes

—

136

C₄²

—

118

—

137

C₄²

—

118

—

138

C₄²

—

118

—

139

C₄²

C₂²×D₅₉

yes

characteristic

140

C₂³

—

141

Q₈

—

142

Q₈

—

143

Q₈

—

144

Q₈

—

118

—

145

Q₈

—

118

—

146

Q₈

—

118

—

147

D₄

—

148

D₄

—

149

D₄

—

150

D₄

—

151

D₄

—

152

C₂×C₄

—

153

C₂×C₄

—

118

—

154

C₂×C₄

—

118

—

155

C₂×C₄

—

118

—

156

C₂×C₄

—

118

—

157

C₂×C₄

D₄×D₅₉

yes

characteristic

158

C₂×C₄

D₄×D₅₉

yes

characteristic

159

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

160

C₈

—

161

C₈

—

236

—

162

C₈

—

236

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

#14273695

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

118

—

170

C₄

—

236

—

171

C₄

—

236

—

172

C₄

—

236

—

173

C₄

SD₁₆×D₅₉

yes

characteristic

174

C₄

#14273746

yes

characteristic

175

C₄

#14273751

yes

characteristic

176

C₄

#14273756

yes

characteristic

177

C₂

—

178

C₂

#16837539

yes

characteristic

179

C₂

#16837544

yes

characteristic

180

C₂

#16837574

yes

characteristic

	subgroup	G/H
center	C₂²	#14273695
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#486	C₂
socle	C₂²×C₅₉	#120
hypercenter	#486	D₅₉
hypocenter	C₅₉	#1626
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#14273695

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#486

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #16838127	C₂	semidirect
4. #16837148	C₂	semidirect
5. #16837134	C₂	semidirect
6. #16837134	C₂	semidirect
68. Dic₁₁₈	SD₁₆	semidirect
98. C₅₉	#1626	semidirect

H ≅ G/N

type

1. #16838127

C₂

semidirect

4. #16837148

C₂

semidirect

5. #16837134

C₂

semidirect

6. #16837134

C₂

semidirect

68. Dic₁₁₈

SD₁₆

semidirect

98. C₅₉

#1626

semidirect

Subgroups — #21423189

Conjugacy classes of subgroups

Complements