Group #21434747 — subgroups

Subgroups — #21434747

Order 7552.

386 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16838008	C₂	1	2	yes	maximal, complemented
2	#16837779	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837768	C₂	1	2	yes	maximal, complemented
4	#16837656	C₂	1	2	yes	maximal, complemented
5	#16837639	C₂	1	2	yes	maximal, complemented
6	#16837513	C₂	1	2	yes	maximal, complemented
7	#16837484	C₂	1	2	yes	maximal, complemented
8	#16837356	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#499	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₄₇₂	C₂²	1	2	yes	—
11	C₂×D₄₇₂	C₂²	1	2	yes	—
12	C₂×#11745659	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
14	#14273724	C₂²	1	2	yes	—
15	#14273712	—	2	4	no	—
16	#14273702	—	2	4	no	—
17	#14273702	C₂²	1	4	yes	—
18	#14273689	C₂²	1	4	yes	—
19	#14273684	C₂²	1	1	yes	characteristic
20	#14273647	C₂²	1	4	yes	—
21	#14273647	C₂²	1	4	yes	—
22	#14273638	C₂²	1	1	yes	characteristic
23	#14273637	C₂²	1	1	yes	characteristic
24	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
25	#125×C₅₉	C₂²	1	2	yes	—
26	#107×C₅₉	C₂²	1	2	yes	—
27	#106×C₅₉	C₂²	1	2	yes	—
28	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
29	C₂×D₂₃₆	—	2	4	no	—
30	C₂×D₂₃₆	—	2	4	no	—
31	C₂×D₂₃₆	C₂³	1	4	yes	—
32	C₂×C₄×D₅₉	—	2	4	no	—
33	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₅₉	—	2	4	no	—
37	C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
38	C₄²×C₅₉	C₂³	1	1	yes	characteristic
39	(C₂×D₅₉)⋊C₄	—	2	4	no	—
40	(C₂×D₅₉)⋊C₄	—	2	4	no	—
41	(C₂×D₅₉)⋊C₄	D₄	1	2	yes	—
42	Dic₅₉⋊C₄	D₄	1	2	yes	—
43	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
44	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
45	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
46	D₄₇₂	—	2	4	no	—
47	D₄₇₂	—	2	4	no	—
48	#11745659	—	2	2	no	—
49	C₈×D₅₉	—	2	2	no	—
50	C₂²×D₅₉	—	2	4	no	—
51	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	—	2	4	no	—
53	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₅₉	C₂×D₄	1	2	yes	—
56	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
57	D₂₃₆	—	2	4	no	—
58	D₂₃₆	—	2	4	no	—
59	D₂₃₆	—	8	4	no	—
60	D₂₃₆	—	8	4	no	—
61	C₄×D₅₉	—	2	1	no	—
62	C₄×D₅₉	—	2	1	no	—
63	C₄×D₅₉	—	8	4	no	—
64	C₈×C₅₉	#41	1	2	yes	—
65	C₈×C₅₉	#41	1	2	yes	—
66	C₅₉⋊C₈	—	2	1	no	—
67	C₅₉⋊C₈	—	2	1	no	—
68	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
69	C₂×D₅₉	—	2	1	no	—
70	C₂×D₅₉	—	2	2	no	—
71	C₂×D₅₉	—	4	4	no	—
72	C₂×D₅₉	—	4	4	no	—
73	C₂×D₅₉	—	4	4	no	—
74	C₄×C₅₉	—	2	2	no	—
75	C₄×C₅₉	—	4	4	no	—
76	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
77	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
78	Dic₅₉	—	2	1	no	—
79	Dic₅₉	—	2	2	no	—
80	#3176	—	59	1	no	maximal, Sylow
81	C₂×C₅₉	#532	1	1	yes	characteristic
82	C₂×C₅₉	#576	1	2	yes	—
83	D₅₉	—	4	1	no	—
84	D₅₉	—	8	4	no	—
85	#540	—	59	2	no	—
86	#499	D₅₉	1	1	yes	characteristic, complemented
87	#493	—	59	1	no	—
88	#490	—	59	2	no	—
89	#469	—	59	2	no	—
90	#466	—	59	2	no	—
91	#442	—	59	2	no	—
92	C₄×D₈	—	59	2	no	—
93	#405	—	59	1	no	—
94	C₅₉	#3176	1	1	yes	characteristic, complemented, Sylow
95	C₂×D₈	—	59	2	no	—
96	C₂×D₈	—	59	2	no	—
97	C₂×M₁₆	—	59	1	no	—
98	C₂²×C₈	—	59	1	no	—
99	#127	—	59	2	no	—
100	#125	C₂×D₅₉	1	2	yes	—
101	#123	—	118	4	no	—
102	#121	—	59	4	no	—
103	#121	—	118	4	no	—
104	C₄×D₄	—	59	4	no	—
105	#117	—	59	1	no	—
106	#107	C₂×D₅₉	1	2	yes	—
107	#106	C₂×D₅₉	1	2	yes	—
108	#102	—	59	4	no	—
109	#102	—	59	4	no	—
110	#97	—	59	1	no	—
111	#97	—	59	1	no	—
112	C₄×C₈	—	59	1	no	—
113	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
114	C₂×D₄	—	59	4	no	—
115	C₂×D₄	—	118	4	no	—
116	C₂×D₄	—	118	4	no	—
117	C₂²×C₄	—	59	1	no	—
118	C₂²×C₄	—	118	4	no	—
119	D₈	—	118	4	no	—
120	D₈	—	118	4	no	—
121	M₁₆	—	118	2	no	—
122	C₂×C₈	—	59	1	no	—
123	C₂×C₈	—	59	1	no	—
124	C₂×C₈	—	118	2	no	—
125	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
126	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	—	59	2	no	—
129	C₄⋊C₄	C₂²×D₅₉	1	4	yes	—
130	C₂²⋊C₄	—	59	2	no	—
131	C₂²⋊C₄	—	118	4	no	—
132	C₂²⋊C₄	—	118	4	no	—
133	C₄²	—	59	1	no	—
134	C₄²	C₂²×D₅₉	1	1	yes	characteristic
135	C₂³	—	59	1	no	—
136	C₂³	—	118	4	no	—
137	D₄	—	118	4	no	—
138	D₄	—	118	4	no	—
139	D₄	—	472	4	no	—
140	D₄	—	472	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	59	2	no	—
143	C₂×C₄	—	59	1	no	—
144	C₂×C₄	—	118	1	no	—
145	C₂×C₄	—	118	1	no	—
146	C₂×C₄	—	472	4	no	—
147	C₂×C₄	D₄×D₅₉	1	2	yes	—
148	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
149	C₈	—	118	1	no	—
150	C₈	—	118	1	no	—
151	C₈	#11745689	1	2	yes	—
152	C₈	#11745689	1	2	yes	—
153	C₂²	—	118	2	no	—
154	C₂²	—	118	1	no	—
155	C₂²	—	236	4	no	—
156	C₂²	—	236	4	no	—
157	C₂²	—	236	4	no	—
158	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	4	4	no	—
161	C₄	—	118	2	no	—
162	C₄	—	118	1	no	—
163	C₄	C₂×#11745689	1	1	yes	characteristic
164	C₄	C₂×#11745689	1	1	yes	characteristic
165	C₂	—	472	4	no	—
166	C₂	—	236	1	no	—
167	C₂	#16837953	1	1	yes	characteristic
168	C₂	#16838204	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16838008

C₂

yes

maximal, complemented

#16837779

C₂

yes

maximal, characteristic, complemented

#16837768

C₂

yes

maximal, complemented

#16837656

C₂

yes

maximal, complemented

#16837639

C₂

yes

maximal, complemented

#16837513

C₂

yes

maximal, complemented

#16837484

C₂

yes

maximal, complemented

#16837356

C₂

yes

maximal, characteristic, complemented

C₅₉×#499

C₂

yes

maximal, characteristic, complemented

C₂×D₄₇₂

C₂²

yes

—

C₂×D₄₇₂

C₂²

yes

—

C₂×#11745659

C₂²

yes

characteristic

C₂×C₈×D₅₉

C₂²

yes

characteristic

#14273724

C₂²

yes

—

#14273712

—

#14273702

—

#14273702

C₂²

yes

—

#14273689

C₂²

yes

—

#14273684

C₂²

yes

characteristic

#14273647

C₂²

yes

—

#14273647

C₂²

yes

—

#14273638

C₂²

yes

characteristic

#14273637

C₂²

yes

characteristic

C₈×Dic₅₉

C₂²

yes

characteristic

#125×C₅₉

C₂²

yes

—

#107×C₅₉

C₂²

yes

—

#106×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

C₂³

yes

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

C₂³

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

D₄

yes

—

Dic₅₉⋊C₄

D₄

yes

—

C₄×Dic₅₉

C₂³

yes

characteristic

C₂×C₅₉⋊C₈

C₂³

yes

characteristic

C₂×C₅₉⋊C₈

C₂³

yes

characteristic

D₄₇₂

—

D₄₇₂

—

#11745659

—

C₈×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

C₂×D₄

yes

—

C₂×Dic₅₉

C₂×D₄

yes

characteristic

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

#41

yes

—

C₈×C₅₉

#41

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#41

yes

characteristic

C₄×C₅₉

C₂×#41

yes

characteristic

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

—

D₅₉

—

#540

—

#499

D₅₉

yes

characteristic, complemented

—

—

—

—

—

C₄×D₈

—

#405

—

C₅₉

#3176

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

—

#127

—

100

#125

C₂×D₅₉

yes

—

101

#123

—

118

—

102

#121

—

103

#121

—

118

—

104

C₄×D₄

—

105

#117

—

106

#107

C₂×D₅₉

yes

—

107

#106

C₂×D₅₉

yes

—

108

#102

—

109

#102

—

110

#97

—

111

#97

—

112

C₄×C₈

—

113

C₄×C₈

C₂×D₅₉

yes

characteristic

114

C₂×D₄

—

115

C₂×D₄

—

118

—

116

C₂×D₄

—

118

—

117

C₂²×C₄

—

118

C₂²×C₄

—

118

—

119

D₈

—

118

—

120

D₈

—

118

—

121

M₁₆

—

118

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

118

—

125

C₂×C₈

C₂²×D₅₉

yes

characteristic

126

C₂×C₈

C₂²×D₅₉

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

C₂²×D₅₉

yes

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

118

—

132

C₂²⋊C₄

—

118

—

133

C₄²

—

134

C₄²

C₂²×D₅₉

yes

characteristic

135

C₂³

—

136

C₂³

—

118

—

137

D₄

—

118

—

138

D₄

—

118

—

139

D₄

—

472

—

140

D₄

—

472

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

118

—

145

C₂×C₄

—

118

—

146

C₂×C₄

—

472

—

147

C₂×C₄

D₄×D₅₉

yes

—

148

C₂×C₄

C₂³×D₅₉

yes

characteristic

149

C₈

—

118

—

150

C₈

—

118

—

151

C₈

#11745689

yes

—

152

C₈

#11745689

yes

—

153

C₂²

—

118

—

154

C₂²

—

118

—

155

C₂²

—

236

—

156

C₂²

—

236

—

157

C₂²

—

236

—

158

C₂²

C₂×D₄×D₅₉

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

118

—

162

C₄

—

118

—

163

C₄

C₂×#11745689

yes

characteristic

164

C₄

C₂×#11745689

yes

characteristic

165

C₂

—

472

—

166

C₂

—

236

—

167

C₂

#16837953

yes

characteristic

168

C₂

#16838204

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#499	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#499	D₅₉
hypocenter	C₅₉	#3176
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

subgroup

G/H

center

C₂²

C₂×D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#499

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #499

N	H ≅ G/N	type
1. #16838008	C₂	semidirect
2. #16837779	C₂	semidirect
3. #16837768	C₂	semidirect
4. #16837656	C₂	semidirect
5. #16837639	C₂	semidirect
6. #16837513	C₂	semidirect
7. #16837484	C₂	semidirect
8. #16837356	C₂	semidirect
9. C₅₉×#499	C₂	semidirect
86. #499	D₅₉	semidirect
94. C₅₉	#3176	semidirect

H ≅ G/N

type

1. #16838008

C₂

semidirect

2. #16837779

C₂

semidirect

3. #16837768

C₂

semidirect

4. #16837656

C₂

semidirect

5. #16837639

C₂

semidirect

6. #16837513

C₂

semidirect

7. #16837484

C₂

semidirect

8. #16837356

C₂

semidirect

9. C₅₉×#499

C₂

semidirect

86. #499

D₅₉

semidirect

94. C₅₉

#3176

semidirect

Subgroups — #21434747

Conjugacy classes of subgroups

Complements