Group #21433166 — subgroups

Subgroups — #21433166

Order 7552.

394 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 157 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	#16837952	C₂	1	1	yes	maximal, characteristic
2	#16837475	C₂	1	4	yes	maximal, complemented
3	#16837472	C₂	1	4	yes	maximal, complemented
4	#16837459	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837367	C₂	1	4	yes	maximal, complemented
6	C₅₉×#432	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11745685	C₂²	1	2	yes	—
8	C₂×#11745659	C₂²	1	4	yes	—
9	C₂×C₈×D₅₉	C₂²	1	4	yes	—
10	#14273727	C₄	1	1	yes	characteristic
11	#14273724	C₄	1	1	yes	characteristic
12	#14273715	C₄	1	2	yes	—
13	#14273701	C₄	1	2	yes	—
14	#14273701	C₄	1	2	yes	—
15	C₄×D₂₃₆	C₂²	1	2	yes	—
16	C₄×D₂₃₆	C₂²	1	2	yes	—
17	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
18	C₂²⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
19	(C₂×D₅₉)⋊C₈	C₂²	1	8	yes	—
20	#14273638	C₂²	1	2	yes	—
21	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
23	C₂²⋊C₈×C₅₉	C₂²	1	2	yes	—
24	#97×C₅₉	C₂²	1	2	yes	—
25	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊D₅₉	C₂×C₄	1	4	yes	—
29	#11745685	—	4	2	no	—
30	#11745685	—	4	2	no	—
31	C₂×D₂₃₆	C₂×C₄	1	2	yes	—
32	C₂×D₂₃₆	C₂×C₄	1	2	yes	—
33	C₂×D₂₃₆	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₅₉	C₂³	1	4	yes	—
35	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
36	C₂×C₈×C₅₉	C₂³	1	4	yes	—
37	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₅₉	C₂³	1	1	yes	characteristic
40	C₄²×C₅₉	C₂³	1	1	yes	characteristic
41	(C₂×D₅₉)⋊C₄	C₂×C₄	1	4	yes	—
42	(C₂×D₅₉)⋊C₄	C₂×C₄	1	4	yes	—
43	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
44	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
45	C₂×C₅₉⋊C₈	C₂³	1	4	yes	—
46	#11745659	—	4	4	no	—
47	C₈×D₅₉	—	4	4	no	—
48	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×D₅₉	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₅₉	—	2	2	no	—
51	C₂×C₄×C₅₉	—	4	1	no	—
52	C₂×C₄×C₅₉	—	4	1	no	—
53	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
59	C₂²⋊D₅₉	—	4	4	no	—
60	C₂×Dic₅₉	C₂²×C₄	1	4	yes	—
61	D₂₃₆	—	4	2	no	—
62	D₂₃₆	—	4	2	no	—
63	D₂₃₆	—	4	2	no	—
64	C₄×D₅₉	—	2	4	no	—
65	C₄×D₅₉	—	2	4	no	—
66	Dic₁₁₈	—	4	2	no	—
67	C₈×C₅₉	—	2	4	no	—
68	C₅₉⋊C₈	—	2	4	no	—
69	C₂²×C₅₉	—	2	1	no	—
70	C₂²×C₅₉	—	2	1	no	—
71	C₂²×C₅₉	—	2	1	no	—
72	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
73	C₂×D₅₉	—	2	4	no	—
74	C₂×D₅₉	—	2	4	no	—
75	C₂×D₅₉	—	2	4	no	—
76	C₄×C₅₉	—	2	1	no	—
77	C₄×C₅₉	—	2	1	no	—
78	C₄×C₅₉	—	2	1	no	—
79	C₄×C₅₉	—	2	1	no	—
80	C₄×C₅₉	—	2	1	no	—
81	C₄×C₅₉	H(5,𝔽₂)	1	2	yes	—
82	Dic₅₉	—	2	4	no	—
83	#2980	—	59	1	no	maximal, Sylow
84	C₂×C₅₉	—	4	1	no	—
85	C₂×C₅₉	#518	1	1	yes	characteristic
86	C₂×C₅₉	#568	1	1	yes	characteristic
87	C₂×C₅₉	#568	1	1	yes	characteristic
88	D₅₉	—	4	4	no	—
89	#532	—	59	1	no	—
90	#435	—	59	4	no	—
91	#435	—	59	4	no	—
92	#432	—	59	1	no	—
93	#432	D₅₉	1	1	yes	characteristic, complemented
94	#408	—	59	4	no	—
95	C₅₉	#2980	1	1	yes	characteristic, complemented, Sylow
96	C₂×#41	—	59	2	no	—
97	C₂×M₁₆	—	59	4	no	—
98	C₂²×C₈	—	59	4	no	—
99	C₄⋊Q₈	—	59	1	no	—
100	#127	—	59	1	no	—
101	#124	—	59	2	no	—
102	#121	—	59	2	no	—
103	#121	—	59	2	no	—
104	C₄×D₄	—	59	2	no	—
105	C₄×D₄	—	59	2	no	—
106	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
107	C₄⋊C₈	—	59	2	no	—
108	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
109	C₂²⋊C₈	—	59	2	no	—
110	C₂²⋊C₈	—	59	8	no	—
111	C₂²⋊C₈	C₂×D₅₉	1	2	yes	—
112	#97	—	59	2	no	—
113	#97	C₂×D₅₉	1	2	yes	—
114	#41	—	236	2	no	—
115	#41	—	236	2	no	—
116	C₂×Q₈	—	59	2	no	—
117	C₂×D₄	—	59	4	no	—
118	C₂×D₄	—	59	2	no	—
119	C₂×D₄	—	59	2	no	—
120	C₂×D₄	—	59	2	no	—
121	C₂²×C₄	—	59	4	no	—
122	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
123	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
125	M₁₆	—	236	4	no	—
126	C₂×C₈	—	59	4	no	—
127	C₂×C₈	—	236	4	no	—
128	C₂×C₈	C₂²×D₅₉	1	4	yes	—
129	C₄⋊C₄	—	59	2	no	—
130	C₄⋊C₄	—	59	2	no	—
131	C₂²⋊C₄	—	59	4	no	—
132	C₂²⋊C₄	—	59	4	no	—
133	C₄²	C₄×D₅₉	1	1	yes	characteristic
134	C₄²	C₄×D₅₉	1	1	yes	characteristic
135	C₄²	C₂²×D₅₉	1	1	yes	characteristic
136	C₄²	C₂²×D₅₉	1	1	yes	characteristic
137	C₂³	—	59	4	no	—
138	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
139	Q₈	—	236	2	no	—
140	D₄	—	236	2	no	—
141	D₄	—	236	2	no	—
142	D₄	—	236	2	no	—
143	D₄	—	236	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	59	4	no	—
148	C₂×C₄	—	118	4	no	—
149	C₂×C₄	—	118	4	no	—
150	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
155	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
156	C₈	—	2	4	no	—
157	C₈	—	118	4	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	118	4	no	—
162	C₂²	—	118	4	no	—
163	C₂²	—	118	4	no	—
164	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	118	4	no	—
171	C₄	#14273773	1	2	yes	—
172	C₂	—	4	1	no	—
173	C₂	—	236	4	no	—
174	C₂	#16837876	1	1	yes	characteristic
175	C₂	#16838168	1	1	yes	characteristic
176	C₂	#16838168	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837952

C₂

yes

maximal, characteristic

#16837475

C₂

yes

maximal, complemented

#16837472

C₂

yes

maximal, complemented

#16837459

C₂

yes

maximal, characteristic, complemented

#16837367

C₂

yes

maximal, complemented

C₅₉×#432

C₂

yes

maximal, characteristic, complemented

C₂×#11745685

C₂²

yes

—

C₂×#11745659

C₂²

yes

—

C₂×C₈×D₅₉

C₂²

yes

—

#14273727

C₄

yes

characteristic

#14273724

C₄

yes

characteristic

#14273715

C₄

yes

—

#14273701

C₄

yes

—

#14273701

C₄

yes

—

C₄×D₂₃₆

C₂²

yes

—

C₄×D₂₃₆

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₅₉⋊C₈)

C₂²

yes

—

(C₂×D₅₉)⋊C₈

C₂²

yes

—

#14273638

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₂²⋊C₈×C₅₉

C₂²

yes

—

#97×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×C₂²⋊D₅₉

C₂×C₄

yes

—

#11745685

—

#11745685

—

C₂×D₂₃₆

C₂×C₄

yes

—

C₂×D₂₃₆

C₂×C₄

yes

—

C₂×D₂₃₆

C₂×C₄

yes

—

C₂×C₄×D₅₉

C₂³

yes

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

(C₂×D₅₉)⋊C₄

C₂×C₄

yes

—

(C₂×D₅₉)⋊C₄

C₂×C₄

yes

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

—

#11745659

—

C₈×D₅₉

—

C₂³×C₅₉

C₂²×C₄

yes

characteristic

C₂²×D₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

C₂²×C₄

yes

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂³×C₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

H(5,𝔽₂)

yes

—

Dic₅₉

—

#2980

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

—

—

—

—

D₅₉

yes

characteristic, complemented

#408

—

C₅₉

#2980

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

100

#127

—

101

#124

—

102

#121

—

103

#121

—

104

C₄×D₄

—

105

C₄×D₄

—

106

C₂×C₄²

C₂×D₅₉

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

C₂×D₅₉

yes

—

109

C₂²⋊C₈

—

110

C₂²⋊C₈

—

111

C₂²⋊C₈

C₂×D₅₉

yes

—

112

#97

—

113

#97

C₂×D₅₉

yes

—

114

#41

—

236

—

115

#41

—

236

—

116

C₂×Q₈

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₄×D₅₉

yes

characteristic

123

C₂²×C₄

C₄×D₅₉

yes

characteristic

124

C₂²×C₄

C₂²×D₅₉

yes

characteristic

125

M₁₆

—

236

—

126

C₂×C₈

—

127

C₂×C₈

—

236

—

128

C₂×C₈

C₂²×D₅₉

yes

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

C₄×D₅₉

yes

characteristic

134

C₄²

C₄×D₅₉

yes

characteristic

135

C₄²

C₂²×D₅₉

yes

characteristic

136

C₄²

C₂²×D₅₉

yes

characteristic

137

C₂³

—

138

C₂³

C₂×C₄×D₅₉

yes

characteristic

139

Q₈

—

236

—

140

D₄

—

236

—

141

D₄

—

236

—

142

D₄

—

236

—

143

D₄

—

236

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

118

—

149

C₂×C₄

—

118

—

150

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

151

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

152

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

153

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

154

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

155

C₂×C₄

C₂³×D₅₉

yes

characteristic

156

C₈

—

157

C₈

—

118

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

118

—

162

C₂²

—

118

—

163

C₂²

—

118

—

164

C₂²

C₂²×C₄×D₅₉

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

118

—

171

C₄

#14273773

yes

—

172

C₂

—

173

C₂

—

236

—

174

C₂

#16837876

yes

characteristic

175

C₂

#16838168

yes

characteristic

176

C₂

#16838168

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₄
Fitting	C₅₉×#432	C₂
hypercenter	#432	D₅₉
hypocenter	C₅₉	#2980
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #432

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₉

commutator, socle

C₂²×C₅₉

C₂³×C₄

Fitting

C₅₉×#432

C₂

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₄ ◁ #432

N	H ≅ G/N	type
2. #16837475	C₂	semidirect
3. #16837472	C₂	semidirect
4. #16837459	C₂	semidirect
5. #16837367	C₂	semidirect
6. C₅₉×#432	C₂	semidirect
93. #432	D₅₉	semidirect
95. C₅₉	#2980	semidirect

H ≅ G/N

type

2. #16837475

C₂

semidirect

3. #16837472

C₂

semidirect

4. #16837459

C₂

semidirect

5. #16837367

C₂

semidirect

6. C₅₉×#432

C₂

semidirect

93. #432

D₅₉

semidirect

95. C₅₉

#2980

semidirect

Subgroups — #21433166

Conjugacy classes of subgroups

Complements