Group #21424970 — subgroups

Subgroups — #21424970

Order 7552.

376 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 105 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×#14273637	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837303	C₂	1	2	yes	maximal, complemented
5	C₂×#106×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	#14273637	—	2	2	no	—
15	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
17	#106×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	4	no	—
56	D₂₃₆	—	2	4	no	—
57	D₂₃₆	—	2	4	no	—
58	D₂₃₆	—	4	2	no	—
59	D₂₃₆	—	4	2	no	—
60	C₈×C₅₉	C₂²⋊C₄	1	4	yes	complemented
61	C₅₉⋊C₈	—	2	2	no	—
62	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
63	C₂²×C₅₉	#121	1	1	yes	characteristic
64	C₂²×C₅₉	#127	1	1	yes	characteristic
65	C₂²×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
66	C₂×D₅₉	—	4	4	no	—
67	C₂×D₅₉	—	4	8	no	—
68	C₂×D₅₉	—	8	4	no	—
69	C₂×D₅₉	—	8	2	no	—
70	C₂×D₅₉	—	8	2	no	—
71	C₄×C₅₉	—	8	2	no	—
72	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
74	C₄×C₅₉	#121	1	1	yes	characteristic
75	C₄×C₅₉	#124	1	1	yes	characteristic
76	Dic₅₉	—	4	2	no	—
77	#1972	—	59	1	no	maximal, Sylow
78	C₂×C₅₉	#390	1	1	yes	characteristic
79	C₂×C₅₉	#442	1	2	yes	—
80	C₂×C₅₉	#469	1	2	yes	—
81	C₂×C₅₉	#496	1	2	yes	—
82	D₅₉	—	8	4	no	—
83	C₂²×D₈	—	59	1	no	—
84	C₂×#106	D₅₉	1	1	yes	characteristic, complemented
85	C₂×#102	—	59	2	no	—
86	C₂×#97	—	59	1	no	—
87	#390	—	59	2	no	—
88	C₅₉	#1972	1	1	yes	characteristic, complemented, Sylow
89	C₂²×D₄	—	59	2	no	—
90	C₂×D₈	—	59	4	no	—
91	C₂×D₈	—	118	2	no	—
92	C₂×D₈	—	118	2	no	—
93	C₂²×C₈	—	59	1	no	—
94	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
95	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
96	C₂×C₂²⋊C₄	—	118	4	no	—
97	C₂×C₄²	—	59	1	no	—
98	#106	—	2	2	no	—
99	#102	—	118	4	no	—
100	#97	—	118	2	no	—
101	C₂⁴	—	118	2	no	—
102	C₂×D₄	—	59	4	no	—
103	C₂×D₄	—	118	2	no	—
104	C₂×D₄	—	118	2	no	—
105	C₂×D₄	—	118	2	no	—
106	C₂×D₄	—	118	2	no	—
107	C₂×D₄	—	118	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	59	2	no	—
110	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
111	D₈	—	118	8	no	—
112	C₂×C₈	—	59	2	no	—
113	C₂×C₈	—	118	1	no	—
114	C₂×C₈	—	118	1	no	—
115	C₂×C₈	C₄×D₅₉	1	2	yes	—
116	C₂×C₈	D₂₃₆	1	2	yes	—
117	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	236	8	no	—
121	C₄²	—	118	1	no	—
122	C₄²	—	118	1	no	—
123	C₂³	—	118	4	no	—
124	C₂³	—	236	2	no	—
125	C₂³	—	236	2	no	—
126	C₂³	—	236	4	no	—
127	C₂³	—	236	4	no	—
128	C₂³	D₄×D₅₉	1	1	yes	characteristic
129	D₄	—	118	4	no	—
130	D₄	—	118	4	no	—
131	D₄	—	236	2	no	—
132	D₄	—	236	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	118	2	no	—
136	C₂×C₄	—	118	4	no	—
137	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
138	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
139	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
140	C₂×C₄	#11745689	1	1	yes	characteristic
141	C₂×C₄	#11745689	1	1	yes	characteristic
142	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
143	C₈	—	118	2	no	—
144	C₈	(C₂×D₅₉)⋊C₄	1	4	yes	complemented
145	C₂²	—	472	4	no	—
146	C₂²	—	472	2	no	—
147	C₂²	—	472	2	no	—
148	C₂²	—	236	8	no	—
149	C₂²	—	236	4	no	—
150	C₂²	#14273689	1	1	yes	characteristic
151	C₂²	#14273702	1	1	yes	characteristic
152	C₂²	#14273723	1	1	yes	characteristic
153	C₂²	#14273754	1	4	yes	—
154	C₄	—	8	2	no	—
155	C₄	—	236	2	no	—
156	C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
157	C₄	#14273689	1	1	yes	characteristic
158	C₄	#14273702	1	1	yes	characteristic
159	C₄	#14273716	1	1	yes	characteristic
160	C₂	—	472	4	no	—
161	C₂	#16837303	1	1	yes	characteristic
162	C₂	#16837513	1	2	yes	—
163	C₂	#16837656	1	2	yes	—
164	C₂	#16837788	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₄₇₂

C₂

yes

maximal, characteristic

C₂×#14273647

C₂

yes

maximal, complemented

C₂×#14273637

C₂

yes

maximal, characteristic, complemented

#16837303

C₂

yes

maximal, complemented

C₂×#106×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

—

#14273637

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

#106×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

—

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

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

yes

—

C₂×C₅₉

#469

yes

—

C₂×C₅₉

#496

yes

—

D₅₉

—

C₂²×D₈

—

C₂×#106

D₅₉

yes

characteristic, complemented

C₂×#102

—

C₂×#97

—

#390

—

C₅₉

#1972

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

—

C₂×C₄²

—

#106

—

#102

—

118

—

100

#97

—

118

—

101

C₂⁴

—

118

—

102

C₂×D₄

—

103

C₂×D₄

—

118

—

104

C₂×D₄

—

118

—

105

C₂×D₄

—

118

—

106

C₂×D₄

—

118

—

107

C₂×D₄

—

118

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²×D₅₉

yes

characteristic

111

D₈

—

118

—

112

C₂×C₈

—

113

C₂×C₈

—

118

—

114

C₂×C₈

—

118

—

115

C₂×C₈

C₄×D₅₉

yes

—

116

C₂×C₈

D₂₃₆

yes

—

117

C₂×C₈

C₂²⋊D₅₉

yes

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

236

—

121

C₄²

—

118

—

122

C₄²

—

118

—

123

C₂³

—

118

—

124

C₂³

—

236

—

125

C₂³

—

236

—

126

C₂³

—

236

—

127

C₂³

—

236

—

128

C₂³

D₄×D₅₉

yes

characteristic

129

D₄

—

118

—

130

D₄

—

118

—

131

D₄

—

236

—

132

D₄

—

236

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

118

—

136

C₂×C₄

—

118

—

137

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

138

C₂×C₄

C₂×D₂₃₆

yes

characteristic

139

C₂×C₄

D₄×D₅₉

yes

characteristic

140

C₂×C₄

#11745689

yes

characteristic

141

C₂×C₄

#11745689

yes

characteristic

142

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

143

C₈

—

118

—

144

C₈

(C₂×D₅₉)⋊C₄

yes

complemented

145

C₂²

—

472

—

146

C₂²

—

472

—

147

C₂²

—

472

—

148

C₂²

—

236

—

149

C₂²

—

236

—

150

C₂²

#14273689

yes

characteristic

151

C₂²

#14273702

yes

characteristic

152

C₂²

#14273723

yes

characteristic

153

C₂²

#14273754

yes

—

154

C₄

—

155

C₄

—

236

—

156

C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

157

C₄

#14273689

yes

characteristic

158

C₄

#14273702

yes

characteristic

159

C₄

#14273716

yes

characteristic

160

C₂

—

472

—

161

C₂

#16837303

yes

characteristic

162

C₂

#16837513

yes

—

163

C₂

#16837656

yes

—

164

C₂

#16837788

yes

—

	subgroup	G/H
center	C₂³	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#106×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#106	D₅₉
hypocenter	C₅₉	#1972
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#106×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
2. C₂×#14273647	C₂	semidirect
3. C₂×#14273637	C₂	semidirect
4. #16837303	C₂	semidirect
5. C₂×#106×C₅₉	C₂	semidirect
9. C₂×D₄₇₂	C₄	semidirect
60. C₈×C₅₉	C₂²⋊C₄	semidirect
84. C₂×#106	D₅₉	semidirect
88. C₅₉	#1972	semidirect
144. C₈	(C₂×D₅₉)⋊C₄	semidirect

H ≅ G/N

type

2. C₂×#14273647

C₂

semidirect

3. C₂×#14273637

C₂

semidirect

4. #16837303

C₂

semidirect

5. C₂×#106×C₅₉

C₂

semidirect

9. C₂×D₄₇₂

C₄

semidirect

60. C₈×C₅₉

C₂²⋊C₄

semidirect

84. C₂×#106

D₅₉

semidirect

88. C₅₉

#1972

semidirect

144. C₈

(C₂×D₅₉)⋊C₄

semidirect

Subgroups — #21424970

Conjugacy classes of subgroups

Complements