Group #20274131 — subgroups

Subgroups — #20274131

Order 6784.

160 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16810970	C₂	1	1	yes	maximal, characteristic
2	#16810918	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810823	C₂	1	1	yes	maximal, characteristic
4	#16810374	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810371	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810366	C₂	1	1	yes	maximal, characteristic
7	C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719924	—	2	1	no	—
9	#13133217	C₂²	1	1	yes	characteristic
10	#13133214	C₂²	1	1	yes	characteristic
11	#13133150	—	2	1	no	—
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	C₄⋊#88322	C₂²	1	1	yes	characteristic
14	#13133141	—	2	1	no	—
15	#13133139	—	2	1	no	—
16	#13133139	—	2	1	no	—
17	#13133137	—	2	1	no	—
18	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₅₃	—	2	1	no	—
20	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₅₃	—	2	1	no	—
22	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×D₂₁₂	—	2	1	no	—
25	C₂×D₂₁₂	—	2	1	no	—
26	C₂×D₂₁₂	—	2	1	no	—
27	C₂×Dic₁₀₆	—	2	1	no	—
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
33	C₄²×C₅₃	—	2	1	no	—
34	C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	#11719924	—	4	2	no	—
36	C₄⋊Dic₅₃	—	2	1	no	—
37	C₄⋊Dic₅₃	—	2	1	no	—
38	C₂×#88322	—	2	1	no	—
39	C₂×#88322	—	2	1	no	—
40	C₂²×D₅₃	—	4	1	no	—
41	Q₈×C₅₃	—	2	1	no	—
42	Q₈×C₅₃	SD₁₆	1	2	yes	—
43	C₂×C₄×C₅₃	—	2	1	no	—
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
48	C₂×Dic₅₃	—	4	1	no	—
49	D₂₁₂	—	4	1	no	—
50	D₂₁₂	—	4	1	no	—
51	D₂₁₂	—	4	1	no	—
52	D₂₁₂	—	4	1	no	—
53	D₂₁₂	—	8	1	no	—
54	Dic₁₀₆	—	8	1	no	—
55	C₈×C₅₃	—	2	1	no	—
56	C₈×C₅₃	—	2	1	no	—
57	C₈×C₅₃	—	4	1	no	—
58	#88322	—	4	1	no	—
59	#88322	—	4	1	no	—
60	C₂²×C₅₃	#121	1	1	yes	characteristic
61	C₂×D₅₃	—	8	1	no	—
62	C₂×D₅₃	—	8	1	no	—
63	C₂×D₅₃	—	8	1	no	—
64	C₄×C₅₃	—	2	2	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	4	1	no	—
67	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₅₃	#135	1	1	yes	characteristic
69	C₄×C₅₃	#135	1	1	yes	characteristic
70	C₄×C₅₃	#137	1	1	yes	characteristic
71	Dic₅₃	—	8	1	no	—
72	#1678	—	53	1	no	maximal, Sylow
73	C₂×C₅₃	#464	1	1	yes	characteristic
74	C₂×C₅₃	#465	1	1	yes	characteristic
75	C₂×C₅₃	#472	1	1	yes	characteristic
76	D₅₃	—	16	1	no	—
77	#486	—	53	1	no	—
78	#477	—	53	1	no	—
79	#460	—	53	1	no	—
80	C₈×Q₈	D₅₃	1	1	yes	characteristic, complemented
81	#334	—	53	1	no	—
82	#333	—	53	1	no	—
83	#331	—	53	1	no	—
84	C₅₃	#1678	1	1	yes	characteristic, complemented, Sylow
85	C₂×SD₁₆	—	106	1	no	—
86	C₄⋊Q₈	—	53	1	no	—
87	#127	—	53	1	no	—
88	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
89	#107	—	106	1	no	—
90	C₄⋊C₈	—	2	1	no	—
91	C₄⋊C₈	—	53	1	no	—
92	C₄⋊C₈	—	53	1	no	—
93	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
94	#103	—	106	1	no	—
95	#102	—	106	1	no	—
96	#102	—	106	1	no	—
97	#102	—	106	1	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
100	C₂×Q₈	—	106	1	no	—
101	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
102	C₂×D₄	—	106	1	no	—
103	C₂×D₄	—	106	1	no	—
104	C₂×D₄	—	106	1	no	—
105	SD₁₆	—	212	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	106	1	no	—
108	C₂×C₈	—	106	1	no	—
109	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
110	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	106	1	no	—
113	C₄⋊C₄	—	106	1	no	—
114	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
115	C₄²	—	2	1	no	—
116	C₄²	C₂²×D₅₃	1	1	yes	characteristic
117	C₂³	—	212	1	no	—
118	Q₈	—	2	1	no	—
119	Q₈	—	424	1	no	—
120	Q₈	#11719913	1	2	yes	—
121	D₄	—	424	1	no	—
122	D₄	—	212	1	no	—
123	D₄	—	212	1	no	—
124	D₄	—	212	1	no	—
125	D₄	—	212	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	212	1	no	—
129	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
130	C₂×C₄	#11719945	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	4	1	no	—
135	C₈	—	212	1	no	—
136	C₈	—	212	1	no	—
137	C₂²	—	424	1	no	—
138	C₂²	—	424	1	no	—
139	C₂²	—	424	1	no	—
140	C₂²	#13133191	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	2	1	no	—
143	C₄	—	4	1	no	—
144	C₄	—	424	1	no	—
145	C₄	C₂×#11719913	1	1	yes	characteristic
146	C₄	#13133237	1	1	yes	characteristic
147	C₄	#13133239	1	1	yes	characteristic
148	C₄	#13133248	1	1	yes	characteristic
149	C₂	—	848	1	no	—
150	C₂	#16810845	1	1	yes	characteristic
151	C₂	#16810852	1	1	yes	characteristic
152	C₂	#16810893	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810970

C₂

yes

maximal, characteristic

#16810918

C₂

yes

maximal, characteristic, complemented

#16810823

C₂

yes

maximal, characteristic

#16810374

C₂

yes

maximal, characteristic, complemented

#16810371

C₂

yes

maximal, characteristic, complemented

#16810366

C₂

yes

maximal, characteristic

C₈×Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#11719924

—

#13133217

C₂²

yes

characteristic

#13133214

C₂²

yes

characteristic

#13133150

—

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

—

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719924

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

#121

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#486

—

#477

—

#460

—

C₈×Q₈

D₅₃

yes

characteristic, complemented

#334

—

#333

—

#331

—

C₅₃

#1678

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

106

—

C₄⋊Q₈

—

#127

—

C₄×Q₈

C₂×D₅₃

yes

characteristic

#107

—

106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

#103

—

106

—

#102

—

106

—

#102

—

106

—

#102

—

106

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

100

C₂×Q₈

—

106

—

101

C₂×Q₈

D₂₁₂

yes

characteristic

102

C₂×D₄

—

106

—

103

C₂×D₄

—

106

—

104

C₂×D₄

—

106

—

105

SD₁₆

—

212

—

106

C₂×C₈

—

107

C₂×C₈

—

106

—

108

C₂×C₈

—

106

—

109

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

110

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

106

—

113

C₄⋊C₄

—

106

—

114

C₄⋊C₄

D₂₁₂

yes

characteristic

115

C₄²

—

116

C₄²

C₂²×D₅₃

yes

characteristic

117

C₂³

—

212

—

118

Q₈

—

119

Q₈

—

424

—

120

Q₈

#11719913

yes

—

121

D₄

—

424

—

122

D₄

—

212

—

123

D₄

—

212

—

124

D₄

—

212

—

125

D₄

—

212

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

212

—

129

C₂×C₄

C₂×D₂₁₂

yes

characteristic

130

C₂×C₄

#11719945

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

212

—

136

C₈

—

212

—

137

C₂²

—

424

—

138

C₂²

—

424

—

139

C₂²

—

424

—

140

C₂²

#13133191

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

424

—

145

C₄

C₂×#11719913

yes

characteristic

146

C₄

#13133237

yes

characteristic

147

C₄

#13133239

yes

characteristic

148

C₄

#13133248

yes

characteristic

149

C₂

—

848

—

150

C₂

#16810845

yes

characteristic

151

C₂

#16810852

yes

characteristic

152

C₂

#16810893

yes

characteristic

	subgroup	G/H
center	C₂²	#13133191
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	#121
hypercenter	C₈×Q₈	D₅₃
hypocenter	C₅₃	#1678
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#13133191

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₈×Q₈×C₅₃

C₂

socle

C₂²×C₅₃

#121

hypercenter

C₈×Q₈

D₅₃

hypocenter

C₅₃

#1678

LCS

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
2. #16810918	C₂	semidirect
4. #16810374	C₂	semidirect
5. #16810371	C₂	semidirect
7. C₈×Q₈×C₅₃	C₂	semidirect
80. C₈×Q₈	D₅₃	semidirect
84. C₅₃	#1678	semidirect

H ≅ G/N

type

2. #16810918

C₂

semidirect

4. #16810374

C₂

semidirect

5. #16810371

C₂

semidirect

7. C₈×Q₈×C₅₃

C₂

semidirect

80. C₈×Q₈

D₅₃

semidirect

84. C₅₃

#1678

semidirect

Subgroups — #20274131

Conjugacy classes of subgroups

Complements