Group #21423658 — subgroups

Subgroups — #21423658

Order 7552.

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

normal subgroups only

This group has 176 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	#16837749	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837608	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837598	C₂	1	1	yes	maximal, characteristic
4	#16837481	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837149	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837145	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#334	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745660	—	2	1	no	—
9	C₂×D₄₇₂	—	2	1	no	—
10	C₂×#11745659	—	2	1	no	—
11	#14273727	C₂²	1	1	yes	characteristic
12	#14273724	C₂²	1	1	yes	characteristic
13	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
14	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
15	#14273652	—	2	1	no	—
16	#14273649	—	2	1	no	—
17	#14273649	—	2	1	no	—
18	#14273649	—	2	1	no	—
19	(C₂×D₅₉)⋊C₈	—	2	1	no	—
20	C₄⋊C₈×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×D₂₃₆	—	2	1	no	—
24	C₂×D₂₃₆	—	2	1	no	—
25	C₂×D₂₃₆	—	2	1	no	—
26	C₂×D₂₃₆	D₄	1	1	yes	characteristic
27	C₂×C₄×D₅₉	—	2	1	no	—
28	C₂×Dic₁₁₈	—	2	1	no	—
29	C₂×C₈×C₅₉	—	2	1	no	—
30	C₂×C₈×C₅₉	—	2	1	no	—
31	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₄²×C₅₉	C₂³	1	1	yes	characteristic
34	(C₂×D₅₉)⋊C₄	—	2	1	no	—
35	C₄⋊Dic₅₉	—	2	1	no	—
36	C₄⋊Dic₅₉	—	2	1	no	—
37	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
38	C₂×C₅₉⋊C₈	—	2	1	no	—
39	D₄₇₂	—	4	2	no	—
40	#11745660	—	4	2	no	—
41	#11745659	—	4	2	no	—
42	C₂²×D₅₉	—	2	1	no	—
43	C₂²×D₅₉	—	4	1	no	—
44	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
47	C₂×Dic₅₉	—	2	1	no	—
48	C₂×Dic₅₉	—	4	1	no	—
49	D₂₃₆	—	2	1	no	—
50	D₂₃₆	—	2	1	no	—
51	D₂₃₆	—	4	1	no	—
52	D₂₃₆	—	4	1	no	—
53	D₂₃₆	—	4	1	no	—
54	D₂₃₆	—	4	1	no	—
55	D₂₃₆	—	8	1	no	—
56	C₄×D₅₉	—	4	1	no	—
57	C₄×D₅₉	—	4	1	no	—
58	Dic₁₁₈	—	4	1	no	—
59	Dic₁₁₈	—	4	1	no	—
60	C₈×C₅₉	—	2	1	no	—
61	C₈×C₅₉	—	2	1	no	—
62	C₈×C₅₉	—	4	1	no	—
63	C₈×C₅₉	—	4	1	no	—
64	C₅₉⋊C₈	—	4	1	no	—
65	C₂²×C₅₉	#121	1	1	yes	characteristic
66	C₂×D₅₉	—	4	1	no	—
67	C₂×D₅₉	—	4	1	no	—
68	C₂×D₅₉	—	4	1	no	—
69	C₂×D₅₉	—	8	1	no	—
70	C₂×D₅₉	—	8	1	no	—
71	C₂×D₅₉	—	8	1	no	—
72	C₄×C₅₉	—	2	1	no	—
73	C₄×C₅₉	#135	1	1	yes	characteristic
74	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
77	Dic₅₉	—	4	1	no	—
78	Dic₅₉	—	8	1	no	—
79	#1701	—	59	1	no	maximal, Sylow
80	C₂×C₅₉	#463	1	1	yes	characteristic
81	C₂×C₅₉	#469	1	1	yes	characteristic
82	C₂×C₅₉	#471	1	1	yes	characteristic
83	D₅₉	—	8	1	no	—
84	D₅₉	—	16	1	no	—
85	#486	—	59	1	no	—
86	#461	—	59	1	no	—
87	#459	—	59	1	no	—
88	#436	—	59	1	no	—
89	#334	D₅₉	1	1	yes	characteristic, complemented
90	#332	—	59	1	no	—
91	#331	—	59	1	no	—
92	C₅₉	#1701	1	1	yes	characteristic, complemented, Sylow
93	C₂×SD₁₆	—	118	1	no	—
94	C₂×D₈	—	118	1	no	—
95	C₂×M₁₆	—	118	1	no	—
96	C₄⋊Q₈	—	59	1	no	—
97	#127	—	59	1	no	—
98	C₄×D₄	—	59	1	no	—
99	C₄⋊C₈	—	59	1	no	—
100	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
101	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
102	#103	—	118	1	no	—
103	#102	—	118	1	no	—
104	#102	—	118	1	no	—
105	#102	—	118	1	no	—
106	C₂²⋊C₈	—	118	1	no	—
107	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
108	C₂×Q₈	—	118	1	no	—
109	C₂×D₄	—	59	1	no	—
110	C₂×D₄	—	118	1	no	—
111	C₂×D₄	—	118	1	no	—
112	C₂×D₄	—	118	1	no	—
113	C₂²×C₄	—	118	1	no	—
114	SD₁₆	—	236	2	no	—
115	D₈	—	236	2	no	—
116	M₁₆	—	236	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	118	1	no	—
120	C₂×C₈	D₂₃₆	1	1	yes	characteristic
121	C₂×C₈	D₂₃₆	1	1	yes	characteristic
122	C₄⋊C₄	—	59	1	no	—
123	C₄⋊C₄	—	118	1	no	—
124	C₄⋊C₄	—	118	1	no	—
125	C₂²⋊C₄	—	118	1	no	—
126	C₄²	C₂²×D₅₉	1	1	yes	characteristic
127	C₂³	—	118	1	no	—
128	C₂³	—	236	1	no	—
129	Q₈	—	236	1	no	—
130	Q₈	—	236	1	no	—
131	D₄	—	118	1	no	—
132	D₄	—	118	1	no	—
133	D₄	—	472	1	no	—
134	D₄	—	236	1	no	—
135	D₄	—	236	1	no	—
136	D₄	—	236	1	no	—
137	D₄	—	236	1	no	—
138	C₂×C₄	—	118	1	no	—
139	C₂×C₄	—	236	1	no	—
140	C₂×C₄	—	236	1	no	—
141	C₂×C₄	—	236	1	no	—
142	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
143	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
144	C₂×C₄	#11745689	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	4	1	no	—
148	C₈	—	4	1	no	—
149	C₈	—	236	1	no	—
150	C₂²	—	472	1	no	—
151	C₂²	—	472	1	no	—
152	C₂²	—	472	1	no	—
153	C₂²	—	236	1	no	—
154	C₂²	—	236	1	no	—
155	C₂²	—	236	1	no	—
156	C₂²	#14273702	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	472	1	no	—
159	C₄	—	236	1	no	—
160	C₄	#14273749	1	1	yes	characteristic
161	C₄	#14273754	1	1	yes	characteristic
162	C₄	#14273754	1	1	yes	characteristic
163	C₄	#14273755	1	1	yes	characteristic
164	C₂	—	944	1	no	—
165	C₂	—	472	1	no	—
166	C₂	#16837618	1	1	yes	characteristic
167	C₂	#16837656	1	1	yes	characteristic
168	C₂	#16837666	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837749

C₂

yes

maximal, characteristic, complemented

#16837608

C₂

yes

maximal, characteristic, complemented

#16837598

C₂

yes

maximal, characteristic

#16837481

C₂

yes

maximal, characteristic, complemented

#16837149

C₂

yes

maximal, characteristic, complemented

#16837145

C₂

yes

maximal, characteristic, complemented

C₅₉×#334

C₂

yes

maximal, characteristic, complemented

C₂×#11745660

—

C₂×D₄₇₂

—

C₂×#11745659

—

#14273727

C₂²

yes

characteristic

#14273724

C₂²

yes

characteristic

C₄×D₂₃₆

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

—

—

—

(C₂×D₅₉)⋊C₈

—

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

D₄

yes

characteristic

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

—

D₄₇₂

—

#11745660

—

#11745659

—

C₂²×D₅₉

—

C₂²×D₅₉

—

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

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

#121

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

D₅₉

—

—

—

—

—

D₅₉

yes

characteristic, complemented

#332

—

#331

—

C₅₉

#1701

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

118

—

C₂×D₈

—

118

—

C₂×M₁₆

—

118

—

C₄⋊Q₈

—

#127

—

C₄×D₄

—

C₄⋊C₈

—

100

C₄⋊C₈

C₂×D₅₉

yes

characteristic

101

C₄⋊C₈

C₂×D₅₉

yes

characteristic

102

#103

—

118

—

103

#102

—

118

—

104

#102

—

118

—

105

#102

—

118

—

106

C₂²⋊C₈

—

118

—

107

C₄×C₈

C₂×D₅₉

yes

characteristic

108

C₂×Q₈

—

118

—

109

C₂×D₄

—

110

C₂×D₄

—

118

—

111

C₂×D₄

—

118

—

112

C₂×D₄

—

118

—

113

C₂²×C₄

—

118

—

114

SD₁₆

—

236

—

115

D₈

—

236

—

116

M₁₆

—

236

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

118

—

120

C₂×C₈

D₂₃₆

yes

characteristic

121

C₂×C₈

D₂₃₆

yes

characteristic

122

C₄⋊C₄

—

123

C₄⋊C₄

—

118

—

124

C₄⋊C₄

—

118

—

125

C₂²⋊C₄

—

118

—

126

C₄²

C₂²×D₅₉

yes

characteristic

127

C₂³

—

118

—

128

C₂³

—

236

—

129

Q₈

—

236

—

130

Q₈

—

236

—

131

D₄

—

118

—

132

D₄

—

118

—

133

D₄

—

472

—

134

D₄

—

236

—

135

D₄

—

236

—

136

D₄

—

236

—

137

D₄

—

236

—

138

C₂×C₄

—

118

—

139

C₂×C₄

—

236

—

140

C₂×C₄

—

236

—

141

C₂×C₄

—

236

—

142

C₂×C₄

C₂×D₂₃₆

yes

characteristic

143

C₂×C₄

D₄×D₅₉

yes

characteristic

144

C₂×C₄

#11745689

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

236

—

150

C₂²

—

472

—

151

C₂²

—

472

—

152

C₂²

—

472

—

153

C₂²

—

236

—

154

C₂²

—

236

—

155

C₂²

—

236

—

156

C₂²

#14273702

yes

characteristic

157

C₄

—

158

C₄

—

472

—

159

C₄

—

236

—

160

C₄

#14273749

yes

characteristic

161

C₄

#14273754

yes

characteristic

162

C₄

#14273754

yes

characteristic

163

C₄

#14273755

yes

characteristic

164

C₂

—

944

—

165

C₂

—

472

—

166

C₂

#16837618

yes

characteristic

167

C₂

#16837656

yes

characteristic

168

C₂

#16837666

yes

characteristic

	subgroup	G/H
center	C₂²	#14273702
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#334	C₂
socle	C₂²×C₅₉	#121
hypercenter	#334	D₅₉
hypocenter	C₅₉	#1701
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#14273702

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#334

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
1. #16837749	C₂	semidirect
2. #16837608	C₂	semidirect
4. #16837481	C₂	semidirect
5. #16837149	C₂	semidirect
6. #16837145	C₂	semidirect
7. C₅₉×#334	C₂	semidirect
89. #334	D₅₉	semidirect
92. C₅₉	#1701	semidirect

H ≅ G/N

type

1. #16837749

C₂

semidirect

2. #16837608

C₂

semidirect

4. #16837481

C₂

semidirect

5. #16837149

C₂

semidirect

6. #16837145

C₂

semidirect

7. C₅₉×#334

C₂

semidirect

89. #334

D₅₉

semidirect

92. C₅₉

#1701

semidirect

Subgroups — #21423658

Conjugacy classes of subgroups

Complements