Group #21426028 — subgroups

Subgroups — #21426028

Order 7552.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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×Q₁₆×D₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273673	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837640	C₂	1	1	yes	maximal, characteristic
4	#16837207	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837206	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₅₉)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₅₉	—	2	1	no	—
9	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
10	C₂×#11745672	—	2	1	no	—
11	Q₁₆×D₅₉	—	2	2	no	—
12	Q₁₆×D₅₉	—	2	2	no	—
13	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
14	#14273703	—	2	1	no	—
15	#14273673	—	2	2	no	—
16	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
17	#14273661	C₂²	1	1	yes	characteristic
18	#14273651	—	2	1	no	—
19	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	Q₃₂×C₅₉	—	2	2	no	—
22	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₉	—	2	1	no	—
24	C₂×Q₈×C₅₉	—	2	1	no	—
25	Q₈×D₅₉	—	2	4	no	—
26	Q₈×D₅₉	—	4	2	no	—
27	C₂×C₄×D₅₉	—	4	1	no	—
28	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
29	C₂×Dic₁₁₈	—	2	1	no	—
30	C₂×Dic₁₁₈	—	4	1	no	—
31	Q₁₆×C₅₉	—	2	1	no	—
32	Q₁₆×C₅₉	—	2	1	no	—
33	Q₁₆×C₅₉	—	2	1	no	—
34	Q₁₆×C₅₉	D₄	1	2	yes	—
35	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
36	#11745672	—	2	4	no	—
37	(C₂×D₅₉)⋊C₄	—	4	1	no	—
38	C₄⋊Dic₅₉	—	2	1	no	—
39	Dic₅₉⋊C₄	—	4	1	no	—
40	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
41	Dic₂₃₆	—	2	1	no	—
42	Dic₂₃₆	D₄	1	2	yes	—
43	C₈×D₅₉	—	2	1	no	—
44	C₈×D₅₉	—	2	1	no	—
45	C₁₆×C₅₉	—	2	1	no	—
46	C₅₉⋊C₁₆	—	2	1	no	—
47	C₂²×D₅₉	D₈	1	1	yes	characteristic
48	Q₈×C₅₉	—	2	2	no	—
49	Q₈×C₅₉	—	4	1	no	—
50	Q₈×C₅₉	—	4	1	no	—
51	Q₈×C₅₉	—	4	1	no	—
52	C₂×C₄×C₅₉	—	4	1	no	—
53	C₂×C₄×C₅₉	—	4	1	no	—
54	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₅₉	—	4	1	no	—
56	C₂×Dic₅₉	—	4	1	no	—
57	C₂×Dic₅₉	D₈	1	1	yes	characteristic
58	C₄×D₅₉	—	2	1	no	—
59	C₄×D₅₉	—	2	1	no	—
60	C₄×D₅₉	—	4	4	no	—
61	Dic₁₁₈	—	2	2	no	—
62	Dic₁₁₈	—	4	4	no	—
63	Dic₁₁₈	—	4	1	no	—
64	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
66	C₅₉⋊C₈	—	2	1	no	—
67	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
68	C₂×D₅₉	—	2	1	no	—
69	C₂×D₅₉	—	2	1	no	—
70	C₂×D₅₉	Q₃₂	1	2	yes	complemented
71	C₄×C₅₉	—	4	2	no	—
72	C₄×C₅₉	—	8	1	no	—
73	C₄×C₅₉	#120	1	1	yes	characteristic
74	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
75	Dic₅₉	—	2	1	no	—
76	Dic₅₉	—	4	2	no	—
77	Dic₅₉	—	8	1	no	—
78	#2188	—	59	1	no	maximal, Sylow
79	C₂×C₅₉	#447	1	1	yes	characteristic
80	C₂×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
81	C₂×C₅₉	#510	1	1	yes	characteristic
82	D₅₉	—	2	2	no	—
83	C₂²×Q₁₆	—	59	1	no	—
84	C₂×Q₃₂	—	59	1	no	—
85	C₂×Q₃₂	D₅₉	1	1	yes	characteristic, complemented
86	#467	—	59	1	no	—
87	#358	—	59	1	no	—
88	#358	—	59	1	no	—
89	C₂²⋊C₁₆	—	59	1	no	—
90	C₅₉	#2188	1	1	yes	characteristic, complemented, Sylow
91	C₂²×Q₈	—	118	1	no	—
92	C₂×Q₁₆	—	59	1	no	—
93	C₂×Q₁₆	—	118	1	no	—
94	C₂×Q₁₆	—	118	2	no	—
95	C₂×Q₁₆	—	118	2	no	—
96	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
97	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
98	C₂²×C₈	—	59	1	no	—
99	C₂²⋊Q₈	—	118	1	no	—
100	Q₃₂	—	2	2	no	—
101	Q₃₂	—	118	2	no	—
102	C₂×C₁₆	—	59	1	no	—
103	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
104	#107	—	59	1	no	—
105	#103	—	118	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	—	118	4	no	—
109	C₂×Q₈	—	118	1	no	—
110	C₂×Q₈	—	236	1	no	—
111	C₂×Q₈	—	236	2	no	—
112	C₂²×C₄	—	59	1	no	—
113	C₂²×C₄	—	236	1	no	—
114	Q₁₆	—	2	1	no	—
115	Q₁₆	—	2	1	no	—
116	Q₁₆	—	2	1	no	—
117	Q₁₆	—	59	2	no	—
118	Q₁₆	—	118	4	no	—
119	Q₁₆	—	118	1	no	—
120	Q₁₆	C₂²⋊D₅₉	1	2	yes	—
121	C₂×C₈	—	59	1	no	—
122	C₂×C₈	—	118	1	no	—
123	C₂×C₈	—	118	1	no	—
124	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
125	C₄⋊C₄	—	118	1	no	—
126	C₄⋊C₄	—	236	1	no	—
127	C₂²⋊C₄	—	236	1	no	—
128	C₁₆	—	2	1	no	—
129	C₁₆	—	118	1	no	—
130	C₂³	—	59	1	no	—
131	Q₈	—	2	2	no	—
132	Q₈	—	4	1	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	118	2	no	—
136	Q₈	—	236	4	no	—
137	Q₈	—	236	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	59	1	no	—
141	C₂×C₄	—	118	1	no	—
142	C₂×C₄	—	118	1	no	—
143	C₂×C₄	—	236	4	no	—
144	C₂×C₄	—	236	1	no	—
145	C₂×C₄	—	236	1	no	—
146	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
147	C₈	—	118	1	no	—
148	C₈	D₄×D₅₉	1	1	yes	characteristic
149	C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
150	C₂²	—	59	2	no	—
151	C₂²	—	118	1	no	—
152	C₂²	—	118	1	no	—
153	C₂²	D₈×D₅₉	1	1	yes	characteristic
154	C₄	—	4	2	no	—
155	C₄	—	8	1	no	—
156	C₄	—	118	1	no	—
157	C₄	—	472	1	no	—
158	C₄	—	236	2	no	—
159	C₄	#14273695	1	1	yes	characteristic
160	C₄	#14273751	1	1	yes	characteristic
161	C₂	—	118	2	no	—
162	C₂	#16837534	1	1	yes	characteristic
163	C₂	Q₃₂×D₅₉	1	1	yes	characteristic
164	C₂	#16837844	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₁₆×D₅₉

C₂

yes

maximal, characteristic

C₂×#14273673

C₂

yes

maximal, characteristic, complemented

#16837640

C₂

yes

maximal, characteristic

#16837207

C₂

yes

maximal, characteristic, complemented

#16837206

C₂

yes

maximal, characteristic, complemented

(C₂×D₅₉)⋊C₁₆

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₅₉

—

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×#11745672

—

Q₁₆×D₅₉

—

Q₁₆×D₅₉

—

C₂×C₈×D₅₉

C₂²

yes

characteristic

#14273703

—

#14273673

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273651

—

C₂×Q₁₆×C₅₉

C₂²

yes

characteristic

C₂×Q₁₆×C₅₉

C₂²

yes

characteristic

Q₃₂×C₅₉

—

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

Q₈×D₅₉

—

Q₈×D₅₉

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

#11745672

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

D₄

yes

—

C₈×D₅₉

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂²×D₅₉

D₈

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

C₂×D₄

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

Q₃₂

yes

complemented

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂×Q₃₂

yes

characteristic

C₂×C₅₉

#510

yes

characteristic

D₅₉

—

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

D₅₉

yes

characteristic, complemented

#467

—

#358

—

#358

—

C₂²⋊C₁₆

—

C₅₉

#2188

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

118

—

C₂×Q₁₆

—

C₂×Q₁₆

—

118

—

C₂×Q₁₆

—

118

—

C₂×Q₁₆

—

118

—

C₂×Q₁₆

C₂×D₅₉

yes

characteristic

C₂×Q₁₆

C₂×D₅₉

yes

characteristic

C₂²×C₈

—

C₂²⋊Q₈

—

118

—

100

Q₃₂

—

101

Q₃₂

—

118

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₂×D₅₉

yes

characteristic

104

#107

—

105

#103

—

118

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

118

—

109

C₂×Q₈

—

118

—

110

C₂×Q₈

—

236

—

111

C₂×Q₈

—

236

—

112

C₂²×C₄

—

113

C₂²×C₄

—

236

—

114

Q₁₆

—

115

Q₁₆

—

116

Q₁₆

—

117

Q₁₆

—

118

Q₁₆

—

118

—

119

Q₁₆

—

118

—

120

Q₁₆

C₂²⋊D₅₉

yes

—

121

C₂×C₈

—

122

C₂×C₈

—

118

—

123

C₂×C₈

—

118

—

124

C₂×C₈

C₂²×D₅₉

yes

characteristic

125

C₄⋊C₄

—

118

—

126

C₄⋊C₄

—

236

—

127

C₂²⋊C₄

—

236

—

128

C₁₆

—

129

C₁₆

—

118

—

130

C₂³

—

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

118

—

136

Q₈

—

236

—

137

Q₈

—

236

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

118

—

142

C₂×C₄

—

118

—

143

C₂×C₄

—

236

—

144

C₂×C₄

—

236

—

145

C₂×C₄

—

236

—

146

C₂×C₄

D₄×D₅₉

yes

characteristic

147

C₈

—

118

—

148

C₈

D₄×D₅₉

yes

characteristic

149

C₈

C₂×C₂²⋊D₅₉

yes

characteristic

150

C₂²

—

151

C₂²

—

118

—

152

C₂²

—

118

—

153

C₂²

D₈×D₅₉

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

118

—

157

C₄

—

472

—

158

C₄

—

236

—

159

C₄

#14273695

yes

characteristic

160

C₄

#14273751

yes

characteristic

161

C₂

—

118

—

162

C₂

#16837534

yes

characteristic

163

C₂

Q₃₂×D₅₉

yes

characteristic

164

C₂

#16837844

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₂×Q₃₂×C₅₉	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	C₂×Q₃₂	D₅₉
hypocenter	C₅₉	#2188
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₂×Q₃₂×C₅₉

C₂

socle

C₂²×C₅₉

C₂×D₈

hypercenter

C₂×Q₃₂

D₅₉

hypocenter

C₅₉

#2188

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
2. C₂×#14273673	C₂	semidirect
4. #16837207	C₂	semidirect
5. #16837206	C₂	semidirect
7. C₂×Q₃₂×C₅₉	C₂	semidirect
70. C₂×D₅₉	Q₃₂	semidirect
85. C₂×Q₃₂	D₅₉	semidirect
90. C₅₉	#2188	semidirect

H ≅ G/N

type

2. C₂×#14273673

C₂

semidirect

4. #16837207

C₂

semidirect

5. #16837206

C₂

semidirect

7. C₂×Q₃₂×C₅₉

C₂

semidirect

70. C₂×D₅₉

Q₃₂

semidirect

85. C₂×Q₃₂

D₅₉

semidirect

90. C₅₉

#2188

semidirect

Subgroups — #21426028

Conjugacy classes of subgroups

Complements