Group #21424913 — subgroups

Subgroups — #21424913

Order 7552.

344 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 113 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×#14273651	C₂	1	2	yes	maximal, complemented
2	C₂×#14273648	C₂	1	2	yes	maximal
3	C₂×#14273638	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837272	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#124×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₉	—	2	2	no	—
8	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
9	#14273651	—	2	4	no	—
10	#14273648	—	2	4	no	—
11	#14273638	—	2	2	no	—
12	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₅₉	C₂²	1	1	yes	characteristic
14	#124×C₅₉	—	2	2	no	—
15	#124×C₅₉	C₄	1	4	yes	complemented
16	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
17	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
18	C₂⁴×C₅₉	—	2	1	no	—
19	C₂×Q₈×C₅₉	—	2	1	no	—
20	C₂×Q₈×C₅₉	—	2	2	no	—
21	C₂×Q₈×C₅₉	—	2	2	no	—
22	C₂×Q₈×C₅₉	C₂×C₄	1	2	yes	—
23	C₂×D₄×C₅₉	—	2	2	no	—
24	C₂×D₄×C₅₉	—	2	1	no	—
25	C₂×D₄×C₅₉	—	2	2	no	—
26	C₂×D₄×C₅₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₅₉	—	2	1	no	—
28	C₂²×C₄×C₅₉	D₄	1	2	yes	—
29	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
30	C₂²×Dic₅₉	—	2	2	no	—
31	C₂²⋊C₄×C₅₉	—	2	8	no	—
32	C₄²×C₅₉	C₂×C₄	1	2	yes	—
33	C₄²×C₅₉	D₄	1	2	yes	—
34	C₄⋊Dic₅₉	—	2	2	no	—
35	C₄⋊Dic₅₉	—	2	2	no	—
36	C₄⋊Dic₅₉	—	4	4	no	—
37	C₂×C₅₉⋊C₈	—	2	2	no	—
38	C₂×C₅₉⋊C₈	—	2	2	no	—
39	C₂×C₅₉⋊C₈	D₄	1	4	yes	—
40	C₂³×C₅₉	—	2	2	no	—
41	C₂³×C₅₉	—	4	1	no	—
42	C₂³×C₅₉	—	4	1	no	—
43	C₂³×C₅₉	—	4	2	no	—
44	C₂³×C₅₉	—	4	2	no	—
45	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₅₉	—	2	4	no	—
47	Q₈×C₅₉	—	4	2	no	—
48	D₄×C₅₉	—	2	4	no	—
49	D₄×C₅₉	—	4	2	no	—
50	C₂×C₄×C₅₉	—	2	2	no	—
51	C₂×C₄×C₅₉	—	2	2	no	—
52	C₂×C₄×C₅₉	—	2	2	no	—
53	C₂×C₄×C₅₉	—	4	1	no	—
54	C₂×C₄×C₅₉	—	4	1	no	—
55	C₂×C₄×C₅₉	C₂²⋊C₄	1	4	yes	—
56	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₉	#41	1	2	yes	—
60	C₂×Dic₅₉	—	4	2	no	—
61	C₂×Dic₅₉	—	4	4	no	—
62	C₅₉⋊C₈	—	2	4	no	—
63	C₂²×C₅₉	—	4	2	no	—
64	C₂²×C₅₉	—	4	4	no	—
65	C₂²×C₅₉	—	8	2	no	—
66	C₂²×C₅₉	—	8	1	no	—
67	C₂²×C₅₉	—	8	1	no	—
68	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₅₉	#124	1	1	yes	characteristic
70	C₂²×C₅₉	#127	1	1	yes	characteristic
71	C₂²×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
72	C₂²×C₅₉	#137	1	2	yes	—
73	C₄×C₅₉	—	2	4	no	—
74	C₄×C₅₉	—	4	2	no	—
75	C₄×C₅₉	C₄×D₄	1	2	yes	—
76	C₄×C₅₉	#121	1	2	yes	—
77	Dic₅₉	—	8	2	no	—
78	#1959	—	59	1	no	maximal, Sylow
79	C₂×C₅₉	—	8	2	no	—
80	C₂×C₅₉	#390	1	1	yes	characteristic
81	C₂×C₅₉	#417	1	2	yes	—
82	C₂×C₅₉	#489	1	2	yes	—
83	C₂×C₅₉	#496	1	1	yes	characteristic
84	C₂×C₅₉	#497	1	1	yes	characteristic
85	C₂×#124	D₅₉	1	1	yes	characteristic
86	C₂×#103	—	59	2	no	—
87	C₂×#102	—	59	2	no	—
88	C₂×#97	—	59	1	no	—
89	#384	—	59	1	no	—
90	C₅₉	#1959	1	1	yes	characteristic, complemented, Sylow
91	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
92	C₂²×D₄	C₂×D₅₉	1	1	yes	characteristic
93	C₂²×C₈	—	59	2	no	—
94	#124	—	2	2	no	—
95	#124	Dic₅₉	1	4	yes	complemented
96	C₂×C₄⋊C₄	—	59	2	no	—
97	C₂×C₄⋊C₄	—	118	2	no	—
98	C₂×C₂²⋊C₄	—	2	2	no	—
99	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
100	#103	—	118	4	no	—
101	#102	—	118	4	no	—
102	#97	—	118	2	no	—
103	C₂⁴	—	2	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	C₂×Dic₅₉	1	2	yes	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	—	2	2	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	C₂×Dic₅₉	1	2	yes	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	118	2	no	—
114	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
115	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
116	C₂×C₈	—	59	4	no	—
117	C₂×C₈	—	118	2	no	—
118	C₂×C₈	—	118	2	no	—
119	C₄⋊C₄	—	118	2	no	—
120	C₄⋊C₄	—	118	2	no	—
121	C₄⋊C₄	—	236	4	no	—
122	C₂²⋊C₄	—	2	8	no	—
123	C₄²	C₂×Dic₅₉	1	2	yes	—
124	C₄²	C₂²⋊D₅₉	1	2	yes	—
125	C₂³	—	2	2	no	—
126	C₂³	—	4	1	no	—
127	C₂³	—	4	1	no	—
128	C₂³	—	4	2	no	—
129	C₂³	—	4	2	no	—
130	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
131	Q₈	—	2	4	no	—
132	Q₈	—	4	2	no	—
133	D₄	—	2	4	no	—
134	D₄	—	4	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	236	2	no	—
141	C₂×C₄	—	236	4	no	—
142	C₂×C₄	C₂²⋊Dic₅₉	1	4	yes	—
143	C₂×C₄	D₄×D₅₉	1	2	yes	—
144	C₂×C₄	#11745687	1	2	yes	—
145	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
147	C₈	—	118	4	no	—
148	C₂²	—	4	4	no	—
149	C₂²	—	4	2	no	—
150	C₂²	—	8	2	no	—
151	C₂²	—	8	1	no	—
152	C₂²	—	8	1	no	—
153	C₂²	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
154	C₂²	#14273714	1	1	yes	characteristic
155	C₂²	#14273723	1	1	yes	characteristic
156	C₂²	#14273752	1	2	yes	—
157	C₂²	#14273757	1	2	yes	—
158	C₄	—	2	4	no	—
159	C₄	—	4	2	no	—
160	C₄	—	472	2	no	—
161	C₄	D₄×Dic₅₉	1	2	yes	—
162	C₄	#14273698	1	2	yes	—
163	C₂	—	8	2	no	—
164	C₂	#16837305	1	1	yes	characteristic
165	C₂	#16837403	1	2	yes	—
166	C₂	#16837761	1	2	yes	—
167	C₂	#16837792	1	1	yes	characteristic
168	C₂	#16837797	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273651

C₂

yes

maximal, complemented

C₂×#14273648

C₂

yes

maximal

C₂×#14273638

C₂

yes

maximal, characteristic, complemented

#16837272

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

#14273651

—

#14273648

—

#14273638

—

C₂²×Q₈×C₅₉

C₂²

yes

characteristic

C₂²×D₄×C₅₉

C₂²

yes

characteristic

#124×C₅₉

—

#124×C₅₉

C₄

yes

complemented

C₂×C₂²⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂⁴×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

C₂×C₄

yes

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#137

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

—

C₄×C₅₉

#121

yes

—

Dic₅₉

—

#1959

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#390

yes

characteristic

C₂×C₅₉

#417

yes

—

C₂×C₅₉

#489

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

yes

characteristic

C₂×#103

—

C₂×#102

—

C₂×#97

—

#384

—

C₅₉

#1959

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₉

yes

characteristic

C₂²×D₄

C₂×D₅₉

yes

characteristic

C₂²×C₈

—

#124

—

#124

Dic₅₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

100

#103

—

118

—

101

#102

—

118

—

102

#97

—

118

—

103

C₂⁴

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

C₂×Dic₅₉

yes

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

C₂×Dic₅₉

yes

—

112

C₂²×C₄

—

113

C₂²×C₄

—

118

—

114

C₂²×C₄

C₂²⋊D₅₉

yes

—

115

C₂²×C₄

C₂²×D₅₉

yes

characteristic

116

C₂×C₈

—

117

C₂×C₈

—

118

—

118

C₂×C₈

—

118

—

119

C₄⋊C₄

—

118

—

120

C₄⋊C₄

—

118

—

121

C₄⋊C₄

—

236

—

122

C₂²⋊C₄

—

123

C₄²

C₂×Dic₅₉

yes

—

124

C₄²

C₂²⋊D₅₉

yes

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

131

Q₈

—

132

Q₈

—

133

D₄

—

134

D₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

236

—

141

C₂×C₄

—

236

—

142

C₂×C₄

C₂²⋊Dic₅₉

yes

—

143

C₂×C₄

D₄×D₅₉

yes

—

144

C₂×C₄

#11745687

yes

—

145

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

147

C₈

—

118

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

C₂×C₂²⋊Dic₅₉

yes

characteristic

154

C₂²

#14273714

yes

characteristic

155

C₂²

#14273723

yes

characteristic

156

C₂²

#14273752

yes

—

157

C₂²

#14273757

yes

—

158

C₄

—

159

C₄

—

160

C₄

—

472

—

161

C₄

D₄×Dic₅₉

yes

—

162

C₄

#14273698

yes

—

163

C₂

—

164

C₂

#16837305

yes

characteristic

165

C₂

#16837403

yes

—

166

C₂

#16837761

yes

—

167

C₂

#16837792

yes

characteristic

168

C₂

#16837797

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#124×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#124	D₅₉
hypocenter	C₅₉	#1959
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#124×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. C₂×#14273651	C₂	semidirect
3. C₂×#14273638	C₂	semidirect
4. #16837272	C₂	semidirect
15. #124×C₅₉	C₄	semidirect
90. C₅₉	#1959	semidirect
95. #124	Dic₅₉	semidirect

H ≅ G/N

type

1. C₂×#14273651

C₂

semidirect

3. C₂×#14273638

C₂

semidirect

4. #16837272

C₂

semidirect

15. #124×C₅₉

C₄

semidirect

90. C₅₉

#1959

semidirect

95. #124

Dic₅₉

semidirect

Subgroups — #21424913

Conjugacy classes of subgroups

Complements