Group #21424960 — subgroups

Subgroups — #21424960

Order 7552.

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

normal subgroups only

This group has 384 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	C₂²×D₄₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273647	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837303	C₂	1	2	yes	maximal, complemented
5	C₂×#107×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₃₆	C₂²	1	2	yes	—
7	C₂×D₄₇₂	—	2	2	no	—
8	C₂×D₄₇₂	—	2	2	no	—
9	C₂×D₄₇₂	C₄	1	4	yes	complemented
10	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×(C₂×D₅₉)⋊C₄	—	2	4	no	—
12	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
13	#14273647	—	2	4	no	—
14	C₈×Dic₅₉	—	2	2	no	—
15	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
17	#107×C₅₉	—	2	2	no	—
18	C₂³×D₅₉	—	2	2	no	—
19	C₂²×C₄×C₅₉	—	2	2	no	—
20	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
21	C₂²×Dic₅₉	D₄	1	2	yes	—
22	C₂×D₂₃₆	—	2	2	no	—
23	C₂×D₂₃₆	—	2	2	no	—
24	C₂×D₂₃₆	—	2	2	no	—
25	C₂×D₂₃₆	—	2	2	no	—
26	C₂×D₂₃₆	—	2	2	no	—
27	C₂×D₂₃₆	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₅₉	D₄	1	2	yes	—
30	C₂×C₈×C₅₉	D₄	1	2	yes	—
31	C₄⋊C₄×C₅₉	—	2	2	no	—
32	C₄⋊C₄×C₅₉	—	2	2	no	—
33	(C₂×D₅₉)⋊C₄	—	4	8	no	—
34	C₄×Dic₅₉	—	2	1	no	—
35	C₄×Dic₅₉	—	2	1	no	—
36	C₂×C₅₉⋊C₈	—	2	1	no	—
37	C₂×C₅₉⋊C₈	—	2	1	no	—
38	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
39	D₄₇₂	—	2	8	no	—
40	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₅₉	—	2	4	no	—
42	C₂²×D₅₉	—	4	2	no	—
43	C₂²×D₅₉	—	4	2	no	—
44	C₂²×D₅₉	—	4	4	no	—
45	C₂²×D₅₉	—	4	4	no	—
46	C₂×C₄×C₅₉	—	4	2	no	—
47	C₂×C₄×C₅₉	—	4	4	no	—
48	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
53	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
54	C₂×Dic₅₉	—	2	2	no	—
55	C₂×Dic₅₉	—	2	2	no	—
56	C₂×Dic₅₉	D₈	1	4	yes	complemented
57	D₂₃₆	—	2	4	no	—
58	D₂₃₆	—	2	4	no	—
59	D₂₃₆	—	4	2	no	—
60	D₂₃₆	—	4	2	no	—
61	C₈×C₅₉	C₂²⋊C₄	1	4	yes	complemented
62	C₅₉⋊C₈	—	2	2	no	—
63	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
64	C₂²×C₅₉	#121	1	1	yes	characteristic
65	C₂²×C₅₉	#127	1	1	yes	characteristic
66	C₂²×C₅₉	C₂×D₈	1	2	yes	—
67	C₂²×C₅₉	#135	1	2	yes	—
68	C₂×D₅₉	—	4	4	no	—
69	C₂×D₅₉	—	4	8	no	—
70	C₂×D₅₉	—	8	4	no	—
71	C₂×D₅₉	—	8	2	no	—
72	C₂×D₅₉	—	8	2	no	—
73	C₄×C₅₉	—	8	2	no	—
74	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
76	C₄×C₅₉	#121	1	1	yes	characteristic
77	C₄×C₅₉	#124	1	1	yes	characteristic
78	Dic₅₉	—	2	4	no	—
79	#1970	—	59	1	no	maximal, Sylow
80	C₂×C₅₉	#390	1	1	yes	characteristic
81	C₂×C₅₉	C₄×D₈	1	2	yes	—
82	C₂×C₅₉	#466	1	2	yes	—
83	C₂×C₅₉	#493	1	1	yes	characteristic
84	C₂×C₅₉	#495	1	1	yes	characteristic
85	D₅₉	—	8	4	no	—
86	C₂²×D₈	—	59	1	no	—
87	C₂×#107	D₅₉	1	1	yes	characteristic, complemented
88	C₂×#102	—	59	2	no	—
89	C₂×C₄×C₈	—	59	1	no	—
90	#390	—	59	2	no	—
91	C₅₉	#1970	1	1	yes	characteristic, complemented, Sylow
92	C₂²×D₄	—	59	2	no	—
93	C₂×D₈	—	59	4	no	—
94	C₂×D₈	—	118	2	no	—
95	C₂×D₈	—	118	2	no	—
96	C₂²×C₈	—	59	1	no	—
97	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
98	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
99	C₂×C₂²⋊C₄	—	118	4	no	—
100	C₂×C₄²	—	59	1	no	—
101	#107	—	2	2	no	—
102	#102	—	118	4	no	—
103	C₄×C₈	—	118	2	no	—
104	C₂⁴	—	118	2	no	—
105	C₂×D₄	—	59	4	no	—
106	C₂×D₄	—	118	2	no	—
107	C₂×D₄	—	118	2	no	—
108	C₂×D₄	—	118	2	no	—
109	C₂×D₄	—	118	2	no	—
110	C₂×D₄	—	118	2	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	59	2	no	—
113	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
114	D₈	—	118	8	no	—
115	C₂×C₈	—	59	2	no	—
116	C₂×C₈	—	118	1	no	—
117	C₂×C₈	—	118	1	no	—
118	C₂×C₈	C₄×D₅₉	1	2	yes	—
119	C₂×C₈	D₂₃₆	1	2	yes	—
120	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₂²⋊C₄	—	236	8	no	—
124	C₄²	—	118	1	no	—
125	C₄²	—	118	1	no	—
126	C₂³	—	118	4	no	—
127	C₂³	—	236	2	no	—
128	C₂³	—	236	2	no	—
129	C₂³	—	236	4	no	—
130	C₂³	—	236	4	no	—
131	C₂³	D₄×D₅₉	1	1	yes	characteristic
132	D₄	—	118	4	no	—
133	D₄	—	118	4	no	—
134	D₄	—	236	2	no	—
135	D₄	—	236	2	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	4	4	no	—
138	C₂×C₄	—	59	4	no	—
139	C₂×C₄	—	118	2	no	—
140	C₂×C₄	—	118	2	no	—
141	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
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	C₂×C₄	#11745689	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
147	C₈	—	118	2	no	—
148	C₈	(C₂×D₅₉)⋊C₄	1	4	yes	complemented
149	C₂²	—	472	4	no	—
150	C₂²	—	472	2	no	—
151	C₂²	—	472	2	no	—
152	C₂²	—	236	8	no	—
153	C₂²	—	236	4	no	—
154	C₂²	#14273689	1	1	yes	characteristic
155	C₂²	#14273702	1	1	yes	characteristic
156	C₂²	#14273723	1	1	yes	characteristic
157	C₂²	D₈×D₅₉	1	2	yes	—
158	C₂²	#14273750	1	2	yes	—
159	C₄	—	8	2	no	—
160	C₄	—	118	4	no	—
161	C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
162	C₄	#14273689	1	1	yes	characteristic
163	C₄	#14273702	1	1	yes	characteristic
164	C₄	#14273716	1	1	yes	characteristic
165	C₂	—	472	4	no	—
166	C₂	#16837303	1	1	yes	characteristic
167	C₂	#16837484	1	2	yes	—
168	C₂	#16837639	1	2	yes	—
169	C₂	#16837777	1	1	yes	characteristic
170	C₂	#16837787	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₄₇₂

C₂

yes

maximal, characteristic

C₂×#14273647

C₂

yes

maximal, complemented

C₂×C₈×Dic₅₉

C₂

yes

maximal, characteristic, complemented

#16837303

C₂

yes

maximal, complemented

C₂×#107×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₃₆

C₂²

yes

—

C₂×D₄₇₂

—

C₂×D₄₇₂

—

C₂×D₄₇₂

C₄

yes

complemented

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×(C₂×D₅₉)⋊C₄

—

C₂×C₄×Dic₅₉

C₂²

yes

characteristic

#14273647

—

C₈×Dic₅₉

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

#107×C₅₉

—

C₂³×D₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

D₄

yes

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

(C₂×D₅₉)⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

—

D₄₇₂

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

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×D₄

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

complemented

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

C₂²⋊C₄

yes

complemented

C₅₉⋊C₈

—

C₂²×C₅₉

C₄×D₄

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

C₂×D₈

yes

—

C₂²×C₅₉

#135

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₄×D₈

yes

—

C₂×C₅₉

#466

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

C₂²×D₈

—

C₂×#107

D₅₉

yes

characteristic, complemented

C₂×#102

—

C₂×C₄×C₈

—

#390

—

C₅₉

#1970

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

118

—

C₂×D₈

—

118

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

C₂×C₂²⋊C₄

—

118

—

100

C₂×C₄²

—

101

#107

—

102

#102

—

118

—

103

C₄×C₈

—

118

—

104

C₂⁴

—

118

—

105

C₂×D₄

—

106

C₂×D₄

—

118

—

107

C₂×D₄

—

118

—

108

C₂×D₄

—

118

—

109

C₂×D₄

—

118

—

110

C₂×D₄

—

118

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂²×D₅₉

yes

characteristic

114

D₈

—

118

—

115

C₂×C₈

—

116

C₂×C₈

—

118

—

117

C₂×C₈

—

118

—

118

C₂×C₈

C₄×D₅₉

yes

—

119

C₂×C₈

D₂₃₆

yes

—

120

C₂×C₈

C₂²⋊D₅₉

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

236

—

124

C₄²

—

118

—

125

C₄²

—

118

—

126

C₂³

—

118

—

127

C₂³

—

236

—

128

C₂³

—

236

—

129

C₂³

—

236

—

130

C₂³

—

236

—

131

C₂³

D₄×D₅₉

yes

characteristic

132

D₄

—

118

—

133

D₄

—

118

—

134

D₄

—

236

—

135

D₄

—

236

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

118

—

140

C₂×C₄

—

118

—

141

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

142

C₂×C₄

C₂×D₂₃₆

yes

characteristic

143

C₂×C₄

D₄×D₅₉

yes

characteristic

144

C₂×C₄

#11745689

yes

characteristic

145

C₂×C₄

#11745689

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

147

C₈

—

118

—

148

C₈

(C₂×D₅₉)⋊C₄

yes

complemented

149

C₂²

—

472

—

150

C₂²

—

472

—

151

C₂²

—

472

—

152

C₂²

—

236

—

153

C₂²

—

236

—

154

C₂²

#14273689

yes

characteristic

155

C₂²

#14273702

yes

characteristic

156

C₂²

#14273723

yes

characteristic

157

C₂²

D₈×D₅₉

yes

—

158

C₂²

#14273750

yes

—

159

C₄

—

160

C₄

—

118

—

161

C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

162

C₄

#14273689

yes

characteristic

163

C₄

#14273702

yes

characteristic

164

C₄

#14273716

yes

characteristic

165

C₂

—

472

—

166

C₂

#16837303

yes

characteristic

167

C₂

#16837484

yes

—

168

C₂

#16837639

yes

—

169

C₂

#16837777

yes

characteristic

170

C₂

#16837787

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#107×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#107	D₅₉
hypocenter	C₅₉	#1970
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#107×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
2. C₂×#14273647	C₂	semidirect
3. C₂×C₈×Dic₅₉	C₂	semidirect
4. #16837303	C₂	semidirect
5. C₂×#107×C₅₉	C₂	semidirect
9. C₂×D₄₇₂	C₄	semidirect
56. C₂×Dic₅₉	D₈	semidirect
61. C₈×C₅₉	C₂²⋊C₄	semidirect
87. C₂×#107	D₅₉	semidirect
91. C₅₉	#1970	semidirect
148. C₈	(C₂×D₅₉)⋊C₄	semidirect

H ≅ G/N

type

2. C₂×#14273647

C₂

semidirect

3. C₂×C₈×Dic₅₉

C₂

semidirect

4. #16837303

C₂

semidirect

5. C₂×#107×C₅₉

C₂

semidirect

9. C₂×D₄₇₂

C₄

semidirect

56. C₂×Dic₅₉

D₈

semidirect

61. C₈×C₅₉

C₂²⋊C₄

semidirect

87. C₂×#107

D₅₉

semidirect

91. C₅₉

#1970

semidirect

148. C₈

(C₂×D₅₉)⋊C₄

semidirect

Subgroups — #21424960

Conjugacy classes of subgroups

Complements