Group #21424112 — subgroups

Subgroups — #21424112

Order 7552.

292 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×C₄×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273652	C₂	1	2	yes	maximal
3	#16837268	C₂	1	1	yes	maximal, characteristic
4	#16837165	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
7	C₄×Dic₁₁₈	—	2	2	no	—
8	C₄×Dic₁₁₈	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₅₉⋊C₄	—	2	1	no	—
12	C₂×C₄×Dic₅₉	—	2	1	no	—
13	#14273652	—	2	4	no	—
14	C₅₉⋊#95	—	2	1	no	—
15	C₅₉⋊#95	—	2	1	no	—
16	C₂²×C₈×C₅₉	C₂²	1	2	yes	—
17	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₅₉	—	2	2	no	—
19	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
22	C₂²×Dic₅₉	—	2	1	no	—
23	C₂²×Dic₅₉	—	2	1	no	—
24	C₂²×Dic₅₉	—	2	2	no	—
25	C₂×Dic₁₁₈	—	2	1	no	—
26	C₂×Dic₁₁₈	—	2	1	no	—
27	C₂×Dic₁₁₈	—	2	1	no	—
28	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₁₈	D₄	1	2	yes	—
30	C₂×Dic₁₁₈	Q₈	1	2	yes	—
31	C₂×C₈×C₅₉	—	2	2	no	—
32	C₂×C₈×C₅₉	—	2	2	no	—
33	C₂×C₈×C₅₉	—	2	2	no	—
34	C₄²×C₅₉	C₂×C₄	1	2	yes	—
35	C₄²×C₅₉	D₄	1	2	yes	—
36	C₄⋊Dic₅₉	—	2	1	no	—
37	C₄⋊Dic₅₉	—	2	2	no	—
38	C₄⋊Dic₅₉	—	2	2	no	—
39	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
40	Dic₅₉⋊C₄	—	2	4	no	—
41	C₄×Dic₅₉	—	2	4	no	—
42	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₅₉	—	2	2	no	—
44	C₂×C₄×C₅₉	—	2	2	no	—
45	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
52	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
53	C₂×Dic₅₉	—	2	2	no	—
54	C₂×Dic₅₉	—	2	2	no	—
55	C₂×Dic₅₉	—	2	2	no	—
56	C₂×Dic₅₉	—	2	2	no	—
57	C₂×Dic₅₉	—	4	2	no	—
58	C₂×Dic₅₉	—	4	2	no	—
59	C₂×Dic₅₉	—	4	2	no	—
60	C₂×Dic₅₉	—	4	2	no	—
61	Dic₁₁₈	—	2	2	no	—
62	Dic₁₁₈	—	2	2	no	—
63	Dic₁₁₈	—	2	2	no	—
64	Dic₁₁₈	C₄⋊C₄	1	4	yes	—
65	C₈×C₅₉	—	4	2	no	—
66	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₅₉	#121	1	1	yes	characteristic
68	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂²×C₅₉	#135	1	2	yes	—
70	C₂²×C₅₉	#137	1	2	yes	—
71	C₄×C₅₉	—	2	2	no	—
72	C₄×C₅₉	—	2	2	no	—
73	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₄×C₅₉	#117	1	1	yes	characteristic
75	C₄×C₅₉	#121	1	1	yes	characteristic
76	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
77	Dic₅₉	—	2	4	no	—
78	Dic₅₉	—	4	2	no	—
79	Dic₅₉	—	8	2	no	—
80	#1805	—	59	1	no	maximal, Sylow
81	C₂×C₅₉	#380	1	1	yes	characteristic
82	C₂×C₅₉	#416	1	1	yes	characteristic
83	C₂×C₅₉	#419	1	1	yes	characteristic
84	C₂×C₅₉	#464	1	2	yes	—
85	C₂×C₅₉	#479	1	2	yes	—
86	C₂×C₄×Q₈	—	59	1	no	—
87	C₂×C₄⋊C₈	D₅₉	1	1	yes	characteristic
88	C₂×#103	—	59	2	no	—
89	#382	—	59	1	no	—
90	#340	—	59	2	no	—
91	C₅₉	#1805	1	1	yes	characteristic, complemented, Sylow
92	C₂²×Q₈	—	59	1	no	—
93	C₂²×C₈	C₂×D₅₉	1	2	yes	—
94	C₄×Q₈	—	59	4	no	—
95	C₄×Q₈	—	118	2	no	—
96	C₂×C₄⋊C₄	—	59	1	no	—
97	C₂×C₄⋊C₄	—	59	2	no	—
98	C₂×C₄⋊C₄	—	118	1	no	—
99	C₂×C₄²	—	118	1	no	—
100	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
101	C₄⋊C₈	—	2	2	no	—
102	#103	—	118	4	no	—
103	#95	—	118	1	no	—
104	#95	—	118	1	no	—
105	C₂×Q₈	—	59	2	no	—
106	C₂×Q₈	—	59	2	no	—
107	C₂×Q₈	—	59	2	no	—
108	C₂×Q₈	—	118	1	no	—
109	C₂×Q₈	—	118	1	no	—
110	C₂×Q₈	—	118	1	no	—
111	C₂²×C₄	—	118	1	no	—
112	C₂²×C₄	—	118	1	no	—
113	C₂²×C₄	—	118	2	no	—
114	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
115	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₄⋊C₄	—	59	2	no	—
121	C₄⋊C₄	—	118	4	no	—
122	C₄⋊C₄	—	118	1	no	—
123	C₄⋊C₄	—	118	2	no	—
124	C₄⋊C₄	—	118	2	no	—
125	C₄²	—	118	4	no	—
126	C₄²	C₄×D₅₉	1	2	yes	—
127	C₄²	C₂²⋊D₅₉	1	2	yes	—
128	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
129	Q₈	—	59	4	no	—
130	Q₈	—	118	2	no	—
131	Q₈	—	118	2	no	—
132	Q₈	—	118	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	118	2	no	—
136	C₂×C₄	—	118	2	no	—
137	C₂×C₄	—	118	2	no	—
138	C₂×C₄	—	118	2	no	—
139	C₂×C₄	—	236	2	no	—
140	C₂×C₄	—	236	2	no	—
141	C₂×C₄	—	236	2	no	—
142	C₂×C₄	—	236	2	no	—
143	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
144	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
145	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
146	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
147	C₂×C₄	#11745687	1	1	yes	characteristic
148	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
149	C₂×C₄	#11745689	1	1	yes	characteristic
150	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
151	C₈	—	4	2	no	—
152	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
153	C₂²	#14273702	1	1	yes	characteristic
154	C₂²	#14273707	1	1	yes	characteristic
155	C₂²	#14273749	1	2	yes	—
156	C₂²	#14273760	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	118	4	no	—
160	C₄	—	472	2	no	—
161	C₄	—	236	2	no	—
162	C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
163	C₄	#14273685	1	1	yes	characteristic
164	C₄	#14273698	1	1	yes	characteristic
165	C₄	#14273703	1	1	yes	characteristic
166	C₂	#16837259	1	1	yes	characteristic
167	C₂	#16837399	1	1	yes	characteristic
168	C₂	#16837412	1	1	yes	characteristic
169	C₂	#16837625	1	2	yes	—
170	C₂	#16837710	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₁₈

C₂

yes

maximal, characteristic

C₂×#14273652

C₂

yes

maximal

#16837268

C₂

yes

maximal, characteristic

#16837165

C₂

yes

maximal

C₂×C₄⋊C₈×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

characteristic

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

C₄

yes

complemented

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×Dic₅₉⋊C₄

—

C₂×C₄×Dic₅₉

—

#14273652

—

C₅₉⋊#95

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×Dic₁₁₈

D₄

yes

—

C₂×Dic₁₁₈

Q₈

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

C₄⋊C₄

yes

—

C₈×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₉

#121

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₉

#135

yes

—

C₂²×C₅₉

#137

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₄⋊C₄

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#479

yes

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

D₅₉

yes

—

—

—

C₅₉

#1805

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

C₂×D₅₉

yes

—

C₄×Q₈

—

C₄×Q₈

—

118

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

—

118

—

100

C₂×C₄²

C₂×D₅₉

yes

characteristic

101

C₄⋊C₈

—

102

#103

—

118

—

103

#95

—

118

—

104

#95

—

118

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

118

—

109

C₂×Q₈

—

118

—

110

C₂×Q₈

—

118

—

111

C₂²×C₄

—

118

—

112

C₂²×C₄

—

118

—

113

C₂²×C₄

—

118

—

114

C₂²×C₄

D₂₃₆

yes

characteristic

115

C₂²×C₄

D₂₃₆

yes

characteristic

116

C₂²×C₄

C₂²×D₅₉

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

118

—

122

C₄⋊C₄

—

118

—

123

C₄⋊C₄

—

118

—

124

C₄⋊C₄

—

118

—

125

C₄²

—

118

—

126

C₄²

C₄×D₅₉

yes

—

127

C₄²

C₂²⋊D₅₉

yes

—

128

C₂³

C₂×D₂₃₆

yes

characteristic

129

Q₈

—

130

Q₈

—

118

—

131

Q₈

—

118

—

132

Q₈

—

118

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

118

—

136

C₂×C₄

—

118

—

137

C₂×C₄

—

118

—

138

C₂×C₄

—

118

—

139

C₂×C₄

—

236

—

140

C₂×C₄

—

236

—

141

C₂×C₄

—

236

—

142

C₂×C₄

—

236

—

143

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

144

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

145

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

146

C₂×C₄

D₄×D₅₉

yes

characteristic

147

C₂×C₄

#11745687

yes

characteristic

148

C₂×C₄

Q₈×D₅₉

yes

characteristic

149

C₂×C₄

#11745689

yes

characteristic

150

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

151

C₈

—

152

C₂²

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

153

C₂²

#14273702

yes

characteristic

154

C₂²

#14273707

yes

characteristic

155

C₂²

#14273749

yes

—

156

C₂²

#14273760

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

118

—

160

C₄

—

472

—

161

C₄

—

236

—

162

C₄

C₄⋊C₄×D₅₉

yes

characteristic

163

C₄

#14273685

yes

characteristic

164

C₄

#14273698

yes

characteristic

165

C₄

#14273703

yes

characteristic

166

C₂

#16837259

yes

characteristic

167

C₂

#16837399

yes

characteristic

168

C₂

#16837412

yes

characteristic

169

C₂

#16837625

yes

—

170

C₂

#16837710

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×C₄⋊C₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₅₉
hypocenter	C₅₉	#1805
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×C₄⋊C₈×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₅₉

hypocenter

C₅₉

#1805

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
8. C₄×Dic₁₁₈	C₄	semidirect
91. C₅₉	#1805	semidirect

H ≅ G/N

type

8. C₄×Dic₁₁₈

C₄

semidirect

91. C₅₉

#1805

semidirect

Subgroups — #21424112

Conjugacy classes of subgroups

Complements