Group #20287351 — subgroups

Subgroups — #20287351

Order 6784.

426 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 139 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	#16811295	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810970	C₂	1	3	yes	maximal, complemented
4	#16810904	C₂	1	3	yes	maximal, complemented
5	#16810819	C₂	1	3	yes	maximal, complemented
6	C₄×D₄₂₄	C₂	1	3	yes	maximal, complemented
7	C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719949	—	2	1	no	—
9	C₂×Q₈×D₅₃	—	2	1	no	—
10	C₂×D₄₂₄	C₂²	1	3	yes	—
11	#13133217	C₂²	1	3	yes	—
12	#13133214	C₂²	1	3	yes	—
13	#13133197	—	2	3	no	—
14	#13133192	—	2	3	no	—
15	C₄×D₂₁₂	C₂²	1	3	yes	—
16	C₄×D₂₁₂	C₂²	1	3	yes	—
17	#13133151	C₂²	1	3	yes	—
18	#13133139	C₂²	1	6	yes	—
19	#13133139	C₂²	1	1	yes	characteristic
20	#13133139	C₂²	1	3	yes	—
21	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	C₂²	1	3	yes	—
23	C₄×C₈×C₅₃	C₂²	1	3	yes	—
24	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
25	#11719949	—	4	4	no	—
26	Q₈×D₅₃	—	4	4	no	—
27	C₂×D₂₁₂	—	2	3	no	—
28	C₂×D₂₁₂	—	2	3	no	—
29	C₂×D₂₁₂	C₂³	1	3	yes	—
30	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
31	C₂×C₄×D₅₃	—	2	3	no	—
32	C₂×C₄×D₅₃	—	2	3	no	—
33	C₂×Dic₁₀₆	—	2	3	no	—
34	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂³	1	3	yes	—
36	C₄⋊C₄×C₅₃	D₄	1	3	yes	—
37	C₄²×C₅₃	C₂³	1	3	yes	—
38	#11719921	—	2	3	no	—
39	#11719921	—	2	3	no	—
40	C₄⋊Dic₅₃	—	2	3	no	—
41	C₄⋊Dic₅₃	C₂³	1	3	yes	—
42	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
43	D₄₂₄	—	2	6	no	—
44	C₂²×D₅₃	—	2	1	no	—
45	C₂²×D₅₃	—	2	3	no	—
46	Q₈×C₅₃	D₈	1	4	yes	complemented
47	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
48	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
49	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₅₃	—	2	3	no	—
51	C₂×Dic₅₃	—	2	1	no	—
52	D₂₁₂	—	2	3	no	—
53	D₂₁₂	—	2	1	no	—
54	D₂₁₂	—	2	3	no	—
55	D₂₁₂	—	4	6	no	—
56	D₂₁₂	—	8	3	no	—
57	D₂₁₂	#41	1	2	yes	complemented
58	C₄×D₅₃	—	4	6	no	—
59	C₄×D₅₃	—	4	6	no	—
60	Dic₁₀₆	—	4	6	no	—
61	C₈×C₅₃	—	2	3	no	—
62	C₈×C₅₃	—	2	1	no	—
63	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
64	C₂×D₅₃	—	2	2	no	—
65	C₂×D₅₃	—	4	1	no	—
66	C₂×D₅₃	—	4	1	no	—
67	C₂×D₅₃	—	4	3	no	—
68	C₂×D₅₃	—	4	3	no	—
69	C₂×D₅₃	—	4	3	no	—
70	C₄×C₅₃	—	2	3	no	—
71	C₄×C₅₃	C₂×D₈	1	6	yes	—
72	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
73	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
74	Dic₅₃	—	4	3	no	—
75	Dic₅₃	—	4	1	no	—
76	#3358	—	53	1	no	maximal, Sylow
77	C₂×C₅₃	#548	1	1	yes	characteristic
78	C₂×C₅₃	C₂²×D₈	1	1	yes	characteristic
79	C₂×C₅₃	#577	1	1	yes	characteristic
80	D₅₃	—	4	2	no	—
81	D₅₃	—	8	3	no	—
82	#550	—	53	1	no	—
83	D₄×Q₈	—	53	1	no	—
84	#486	—	53	3	no	—
85	#474	—	53	3	no	—
86	#459	—	53	3	no	—
87	C₈×Q₈	D₅₃	1	1	yes	characteristic, complemented
88	C₄×D₈	—	53	3	no	—
89	C₅₃	#3358	1	1	yes	characteristic, complemented, Sylow
90	C₂×#41	—	106	1	no	—
91	C₂²×Q₈	—	106	1	no	—
92	C₂×D₈	—	53	3	no	—
93	C₄⋊Q₈	—	53	3	no	—
94	#127	—	53	3	no	—
95	C₂²⋊Q₈	—	106	3	no	—
96	#121	—	106	3	no	—
97	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
98	C₄×D₄	—	53	3	no	—
99	C₄×D₄	—	53	3	no	—
100	#107	—	53	3	no	—
101	C₄⋊C₈	C₂×D₅₃	1	3	yes	—
102	#102	—	53	1	no	—
103	#102	—	53	3	no	—
104	#102	—	53	6	no	—
105	C₄×C₈	C₂×D₅₃	1	3	yes	—
106	#41	—	212	4	no	—
107	C₂×Q₈	—	106	3	no	—
108	C₂×Q₈	—	212	4	no	—
109	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
110	C₂×D₄	—	53	1	no	—
111	C₂×D₄	—	53	3	no	—
112	C₂×D₄	—	106	3	no	—
113	C₂×D₄	—	106	3	no	—
114	C₂²×C₄	—	106	3	no	—
115	C₂²×C₄	—	106	3	no	—
116	D₈	—	106	6	no	—
117	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
118	C₂×C₈	C₂²×D₅₃	1	3	yes	—
119	C₄⋊C₄	—	53	3	no	—
120	C₄⋊C₄	—	53	1	no	—
121	C₄⋊C₄	—	106	3	no	—
122	C₄⋊C₄	D₂₁₂	1	3	yes	—
123	C₂²⋊C₄	—	106	3	no	—
124	C₂²⋊C₄	—	106	3	no	—
125	C₄²	C₂²×D₅₃	1	3	yes	—
126	C₂³	—	106	1	no	—
127	C₂³	—	106	3	no	—
128	Q₈	—	212	6	no	—
129	Q₈	D₄₂₄	1	4	yes	complemented
130	D₄	—	53	2	no	—
131	D₄	—	106	3	no	—
132	D₄	—	106	1	no	—
133	D₄	—	106	3	no	—
134	D₄	—	424	3	no	—
135	D₄	—	212	6	no	—
136	C₂×C₄	—	106	1	no	—
137	C₂×C₄	—	106	3	no	—
138	C₂×C₄	—	212	6	no	—
139	C₂×C₄	—	212	6	no	—
140	C₂×C₄	C₂×D₂₁₂	1	3	yes	—
141	C₂×C₄	C₂×D₂₁₂	1	3	yes	—
142	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
143	C₈	—	2	3	no	—
144	C₈	—	2	1	no	—
145	C₂²	—	106	2	no	—
146	C₂²	—	212	3	no	—
147	C₂²	—	212	3	no	—
148	C₂²	—	212	3	no	—
149	C₂²	—	212	1	no	—
150	C₂²	—	212	1	no	—
151	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
152	C₄	—	2	3	no	—
153	C₄	—	212	1	no	—
154	C₄	—	212	3	no	—
155	C₄	C₂×D₄₂₄	1	6	yes	—
156	C₄	#41×D₅₃	1	1	yes	characteristic
157	C₄	#13133265	1	1	yes	characteristic
158	C₂	—	424	3	no	—
159	C₂	—	212	2	no	—
160	C₂	#16811285	1	1	yes	characteristic
161	C₂	C₂²×D₄₂₄	1	1	yes	characteristic
162	C₂	#16811435	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811295

C₂

yes

maximal, characteristic, complemented

Q₈×D₂₁₂

C₂

yes

maximal, characteristic, complemented

#16810970

C₂

yes

maximal, complemented

#16810904

C₂

yes

maximal, complemented

#16810819

C₂

yes

maximal, complemented

C₄×D₄₂₄

C₂

yes

maximal, complemented

C₈×Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#11719949

—

C₂×Q₈×D₅₃

—

C₂×D₄₂₄

C₂²

yes

—

#13133217

C₂²

yes

—

#13133214

C₂²

yes

—

#13133197

—

#13133192

—

C₄×D₂₁₂

C₂²

yes

—

C₄×D₂₁₂

C₂²

yes

—

#13133151

C₂²

yes

—

#13133139

C₂²

yes

—

#13133139

C₂²

yes

characteristic

#13133139

C₂²

yes

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

#11719949

—

Q₈×D₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂³

yes

—

C₂×D₂₁₂

C₂³

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

#11719921

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

characteristic

D₄₂₄

—

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

D₈

yes

complemented

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

#41

yes

complemented

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×D₈

yes

—

C₄×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂²×D₈

yes

characteristic

C₂×C₅₃

#577

yes

characteristic

D₅₃

—

D₅₃

—

#550

—

D₄×Q₈

—

#486

—

#474

—

#459

—

C₈×Q₈

D₅₃

yes

characteristic, complemented

C₄×D₈

—

C₅₃

#3358

yes

characteristic, complemented, Sylow

C₂×#41

—

106

—

C₂²×Q₈

—

106

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂²⋊Q₈

—

106

—

#121

—

106

—

C₄×Q₈

C₂×D₅₃

yes

characteristic

C₄×D₄

—

C₄×D₄

—

100

#107

—

101

C₄⋊C₈

C₂×D₅₃

yes

—

102

#102

—

103

#102

—

104

#102

—

105

C₄×C₈

C₂×D₅₃

yes

—

106

#41

—

212

—

107

C₂×Q₈

—

106

—

108

C₂×Q₈

—

212

—

109

C₂×Q₈

D₂₁₂

yes

characteristic

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

106

—

113

C₂×D₄

—

106

—

114

C₂²×C₄

—

106

—

115

C₂²×C₄

—

106

—

116

D₈

—

106

—

117

C₂×C₈

C₂²×D₅₃

yes

characteristic

118

C₂×C₈

C₂²×D₅₃

yes

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

106

—

122

C₄⋊C₄

D₂₁₂

yes

—

123

C₂²⋊C₄

—

106

—

124

C₂²⋊C₄

—

106

—

125

C₄²

C₂²×D₅₃

yes

—

126

C₂³

—

106

—

127

C₂³

—

106

—

128

Q₈

—

212

—

129

Q₈

D₄₂₄

yes

complemented

130

D₄

—

131

D₄

—

106

—

132

D₄

—

106

—

133

D₄

—

106

—

134

D₄

—

424

—

135

D₄

—

212

—

136

C₂×C₄

—

106

—

137

C₂×C₄

—

106

—

138

C₂×C₄

—

212

—

139

C₂×C₄

—

212

—

140

C₂×C₄

C₂×D₂₁₂

yes

—

141

C₂×C₄

C₂×D₂₁₂

yes

—

142

C₂×C₄

C₂³×D₅₃

yes

characteristic

143

C₈

—

144

C₈

—

145

C₂²

—

106

—

146

C₂²

—

212

—

147

C₂²

—

212

—

148

C₂²

—

212

—

149

C₂²

—

212

—

150

C₂²

—

212

—

151

C₂²

C₂²×D₂₁₂

yes

characteristic

152

C₄

—

153

C₄

—

212

—

154

C₄

—

212

—

155

C₄

C₂×D₄₂₄

yes

—

156

C₄

#41×D₅₃

yes

characteristic

157

C₄

#13133265

yes

characteristic

158

C₂

—

424

—

159

C₂

—

212

—

160

C₂

#16811285

yes

characteristic

161

C₂

C₂²×D₄₂₄

yes

characteristic

162

C₂

#16811435

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₈×Q₈	D₅₃
hypocenter	C₅₃	#3358
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₈×Q₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×D₄

hypercenter

C₈×Q₈

D₅₃

hypocenter

C₅₃

#3358

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. #16811295	C₂	semidirect
2. Q₈×D₂₁₂	C₂	semidirect
3. #16810970	C₂	semidirect
4. #16810904	C₂	semidirect
5. #16810819	C₂	semidirect
6. C₄×D₄₂₄	C₂	semidirect
7. C₈×Q₈×C₅₃	C₂	semidirect
46. Q₈×C₅₃	D₈	semidirect
57. D₂₁₂	#41	semidirect
87. C₈×Q₈	D₅₃	semidirect
89. C₅₃	#3358	semidirect
129. Q₈	D₄₂₄	semidirect

H ≅ G/N

type

1. #16811295

C₂

semidirect

2. Q₈×D₂₁₂

C₂

semidirect

3. #16810970

C₂

semidirect

4. #16810904

C₂

semidirect

5. #16810819

C₂

semidirect

6. C₄×D₄₂₄

C₂

semidirect

7. C₈×Q₈×C₅₃

C₂

semidirect

46. Q₈×C₅₃

D₈

semidirect

57. D₂₁₂

#41

semidirect

87. C₈×Q₈

D₅₃

semidirect

89. C₅₃

#3358

semidirect

129. Q₈

D₄₂₄

semidirect

Subgroups — #20287351

Conjugacy classes of subgroups

Complements