Group #20274168 — subgroups

Subgroups — #20274168

Order 6784.

160 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16810975	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810903	C₂	1	1	yes	maximal, characteristic
3	D₄×#88322	C₂	1	1	yes	maximal, characteristic
4	#16810375	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810372	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810368	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#476	C₂	1	1	yes	maximal, characteristic
8	C₂²×#88322	—	2	1	no	—
9	#13133217	C₂²	1	1	yes	characteristic
10	#13133150	—	2	1	no	—
11	C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	#13133140	—	2	1	no	—
14	#13133140	—	2	1	no	—
15	#13133138	—	2	1	no	—
16	C₂²⋊#88322	—	2	1	no	—
17	C₄×#88322	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
20	#106×C₅₃	—	2	1	no	—
21	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	#102×C₅₃	—	2	1	no	—
23	C₂×Q₈×C₅₃	—	2	1	no	—
24	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	—	2	1	no	—
26	C₂×Dic₁₀₆	—	2	1	no	—
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂²⋊C₄×C₅₃	—	2	1	no	—
32	C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	—	2	1	no	—
35	C₂×#88322	—	2	1	no	—
36	C₂×#88322	—	2	1	no	—
37	C₂×#88322	—	4	2	no	—
38	C₂×#88322	D₄	1	1	yes	characteristic
39	C₂×#88322	D₄	1	1	yes	characteristic
40	C₂³×C₅₃	—	2	1	no	—
41	Q₈×C₅₃	—	8	1	no	—
42	D₄×C₅₃	—	2	1	no	—
43	D₄×C₅₃	Q₁₆	1	2	yes	—
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	—	4	1	no	—
46	C₂×C₄×C₅₃	—	4	1	no	—
47	C₂×C₄×C₅₃	—	4	1	no	—
48	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
51	C₂×Dic₅₃	—	4	1	no	—
52	Dic₁₀₆	—	4	1	no	—
53	Dic₁₀₆	—	4	1	no	—
54	C₈×C₅₃	—	4	1	no	—
55	#88322	—	2	1	no	—
56	#88322	—	2	1	no	—
57	#88322	—	4	1	no	—
58	#88322	—	4	1	no	—
59	C₂²×C₅₃	—	2	2	no	—
60	C₂²×C₅₃	—	4	1	no	—
61	C₂²×C₅₃	—	4	1	no	—
62	C₂²×C₅₃	#121	1	1	yes	characteristic
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	—	4	1	no	—
65	C₄×C₅₃	—	8	1	no	—
66	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
67	C₄×C₅₃	#135	1	1	yes	characteristic
68	C₄×C₅₃	#135	1	1	yes	characteristic
69	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
70	Dic₅₃	—	8	1	no	—
71	#1683	—	53	1	no	maximal, Sylow
72	C₂×C₅₃	—	4	2	no	—
73	C₂×C₅₃	#463	1	1	yes	characteristic
74	C₂×C₅₃	#467	1	1	yes	characteristic
75	C₂×C₅₃	#471	1	1	yes	characteristic
76	#487	—	53	1	no	—
77	#476	D₅₃	1	1	yes	characteristic
78	#474	—	53	1	no	—
79	C₈×D₄	—	53	1	no	—
80	#335	—	53	1	no	—
81	#333	—	53	1	no	—
82	#332	—	53	1	no	—
83	C₅₃	#1683	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	106	1	no	—
85	C₄⋊Q₈	—	53	1	no	—
86	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
87	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
88	#107	—	106	1	no	—
89	#106	—	2	1	no	—
90	C₄⋊C₈	—	53	1	no	—
91	C₄⋊C₈	—	53	1	no	—
92	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
93	#103	—	106	1	no	—
94	#103	—	106	1	no	—
95	#102	—	2	1	no	—
96	#102	—	106	1	no	—
97	C₂²⋊C₈	—	106	1	no	—
98	C₄×C₈	—	53	1	no	—
99	C₂×Q₈	—	2	1	no	—
100	C₂×Q₈	—	106	1	no	—
101	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
102	C₂²×C₄	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	53	1	no	—
105	C₂×C₈	—	53	1	no	—
106	C₂×C₈	—	106	1	no	—
107	C₂×C₈	—	106	1	no	—
108	C₂×C₈	—	212	2	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	106	1	no	—
112	C₄⋊C₄	—	106	1	no	—
113	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
114	C₂²⋊C₄	—	2	1	no	—
115	C₄²	C₂²×D₅₃	1	1	yes	characteristic
116	C₂³	—	2	1	no	—
117	Q₈	—	8	1	no	—
118	Q₈	—	212	1	no	—
119	Q₈	—	212	1	no	—
120	D₄	—	2	1	no	—
121	D₄	#11719925	1	2	yes	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	212	1	no	—
127	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
128	C₂×C₄	#11719947	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
130	C₈	—	4	1	no	—
131	C₈	—	106	1	no	—
132	C₈	—	106	1	no	—
133	C₈	—	212	1	no	—
134	C₈	—	212	1	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	4	1	no	—
137	C₂²	—	4	1	no	—
138	C₂²	#13133189	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	4	1	no	—
141	C₄	—	8	1	no	—
142	C₄	—	424	1	no	—
143	C₄	C₂×#11719925	1	1	yes	characteristic
144	C₄	#13133237	1	1	yes	characteristic
145	C₄	#13133238	1	1	yes	characteristic
146	C₄	#13133244	1	1	yes	characteristic
147	C₂	—	4	2	no	—
148	C₂	#16810840	1	1	yes	characteristic
149	C₂	#16810862	1	1	yes	characteristic
150	C₂	#16810884	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810975

C₂

yes

maximal, characteristic, complemented

#16810903

C₂

yes

maximal, characteristic

D₄×#88322

C₂

yes

maximal, characteristic

#16810375

C₂

yes

maximal, characteristic, complemented

#16810372

C₂

yes

maximal, characteristic, complemented

#16810368

C₂

yes

maximal, characteristic, complemented

C₅₃×#476

C₂

yes

maximal, characteristic

C₂²×#88322

—

#13133217

C₂²

yes

characteristic

#13133150

—

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

C₂²

yes

characteristic

#106×C₅₃

—

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

#102×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

—

—

—

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

C₂³×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

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

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1683

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

characteristic

#474

—

C₈×D₄

—

#335

—

#333

—

#332

—

C₅₃

#1683

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

C₄×D₄

C₂×D₅₃

yes

characteristic

#107

—

106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

#103

—

106

—

#103

—

106

—

#102

—

#102

—

106

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

C₂×Q₈

—

100

C₂×Q₈

—

106

—

101

C₂×D₄

C₂²⋊D₅₃

yes

characteristic

102

C₂²×C₄

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

106

—

107

C₂×C₈

—

106

—

108

C₂×C₈

—

212

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

106

—

112

C₄⋊C₄

—

106

—

113

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

114

C₂²⋊C₄

—

115

C₄²

C₂²×D₅₃

yes

characteristic

116

C₂³

—

117

Q₈

—

118

Q₈

—

212

—

119

Q₈

—

212

—

120

D₄

—

121

D₄

#11719925

yes

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

212

—

127

C₂×C₄

D₄×D₅₃

yes

characteristic

128

C₂×C₄

#11719947

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

130

C₈

—

131

C₈

—

106

—

132

C₈

—

106

—

133

C₈

—

212

—

134

C₈

—

212

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#13133189

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

424

—

143

C₄

C₂×#11719925

yes

characteristic

144

C₄

#13133237

yes

characteristic

145

C₄

#13133238

yes

characteristic

146

C₄

#13133244

yes

characteristic

147

C₂

—

148

C₂

#16810840

yes

characteristic

149

C₂

#16810862

yes

characteristic

150

C₂

#16810884

yes

characteristic

	subgroup	G/H
center	C₂²	#13133189
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#476	C₂
socle	C₂²×C₅₃	#121
hypercenter	#476	D₅₃
hypocenter	C₅₃	#1683
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

subgroup

G/H

center

C₂²

#13133189

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#476

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #476

N	H ≅ G/N	type
1. #16810975	C₂	semidirect
4. #16810375	C₂	semidirect
5. #16810372	C₂	semidirect
6. #16810368	C₂	semidirect
83. C₅₃	#1683	semidirect

H ≅ G/N

type

1. #16810975

C₂

semidirect

4. #16810375

C₂

semidirect

5. #16810372

C₂

semidirect

6. #16810368

C₂

semidirect

83. C₅₃

#1683

semidirect

Subgroups — #20274168

Conjugacy classes of subgroups

Complements