Group #20273445 — subgroups

Subgroups — #20273445

Order 6784.

202 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 81 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	#16810680	C₂	1	1	yes	maximal, characteristic
2	#16810647	C₂	1	1	yes	maximal, characteristic
3	#16810372	C₂	1	2	yes	maximal, complemented
4	#16810369	C₂	1	2	yes	maximal, complemented
5	C₅₃×#533	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	—	2	1	no	—
7	C₄⋊#88322	—	2	1	no	—
8	C₄⋊#88322	C₂²	1	2	yes	—
9	C₄⋊#88322	C₂²	1	2	yes	—
10	C₂²⋊#88322	—	2	1	no	—
11	#13133128	—	2	1	no	—
12	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	#125×C₅₃	—	2	1	no	—
15	C₂²⋊Q₈×C₅₃	—	2	1	no	—
16	C₂²⋊Q₈×C₅₃	C₄	1	2	yes	—
17	C₄×Q₈×C₅₃	—	2	1	no	—
18	C₄×Q₈×C₅₃	C₄	1	2	yes	—
19	#117×C₅₃	—	2	1	no	—
20	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₅₃	—	2	1	no	—
22	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²⋊C₄×C₅₃	—	2	2	no	—
33	C₄²×C₅₃	—	2	2	no	—
34	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₅₃	D₄	1	1	yes	characteristic
36	C₄²×C₅₃	D₄	1	1	yes	characteristic
37	C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	#11719917	—	4	2	no	—
39	C₂×#88322	—	2	2	no	—
40	C₂×#88322	—	2	2	no	—
41	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	Q₈×C₅₃	—	4	2	no	—
43	Q₈×C₅₃	—	4	1	no	—
44	Q₈×C₅₃	—	4	1	no	—
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	4	1	no	—
51	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	#88322	—	4	2	no	—
58	#88322	—	4	2	no	—
59	C₂²×C₅₃	—	2	1	no	—
60	C₂²×C₅₃	—	2	1	no	—
61	C₂²×C₅₃	—	2	1	no	—
62	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	2	1	no	—
67	C₄×C₅₃	—	4	2	no	—
68	C₄×C₅₃	—	4	2	no	—
69	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₄×C₅₃	#137	1	1	yes	characteristic
71	C₄×C₅₃	#137	1	2	yes	—
72	#1559	—	53	1	no	maximal, Sylow
73	C₂×C₅₃	—	4	1	no	—
74	C₂×C₅₃	#413	1	1	yes	characteristic
75	C₂×C₅₃	#417	1	1	yes	characteristic
76	C₂×C₅₃	#419	1	1	yes	characteristic
77	#533	D₅₃	1	1	yes	characteristic
78	#432	—	53	1	no	—
79	#423	—	53	1	no	—
80	#333	—	53	2	no	—
81	#332	—	53	2	no	—
82	C₅₃	#1559	1	1	yes	characteristic, complemented, Sylow
83	C₂×M₁₆	—	106	1	no	—
84	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
85	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
86	#125	—	2	1	no	—
87	C₂²⋊Q₈	—	2	1	no	—
88	C₂²⋊Q₈	Dic₅₃	1	2	yes	—
89	C₄×Q₈	—	2	1	no	—
90	C₄×Q₈	Dic₅₃	1	2	yes	—
91	#117	—	2	1	no	—
92	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
93	C₄⋊C₈	—	53	2	no	—
94	C₄⋊C₈	—	53	2	no	—
95	C₄⋊C₈	—	106	1	no	—
96	C₂²⋊C₈	—	106	1	no	—
97	#97	—	106	1	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
100	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
101	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
102	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
103	M₁₆	—	212	2	no	—
104	C₂×C₈	—	106	2	no	—
105	C₂×C₈	—	106	2	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
115	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
116	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
117	C₄²	C₂²×D₅₃	1	1	yes	characteristic
118	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
119	Q₈	—	4	2	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	4	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
129	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
130	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
131	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
134	C₈	—	212	2	no	—
135	C₈	—	212	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	4	2	no	—
145	C₄	—	4	2	no	—
146	C₄	#13133242	1	1	yes	characteristic
147	C₄	#13133247	1	1	yes	characteristic
148	C₄	#13133248	1	2	yes	—
149	C₂	—	4	1	no	—
150	C₂	#16810607	1	1	yes	characteristic
151	C₂	#16810624	1	1	yes	characteristic
152	C₂	#16810632	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810680

C₂

yes

maximal, characteristic

#16810647

C₂

yes

maximal, characteristic

#16810372

C₂

yes

maximal, complemented

#16810369

C₂

yes

maximal, complemented

C₅₃×#533

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

C₂²⋊#88322

—

#13133128

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#125×C₅₃

—

C₂²⋊Q₈×C₅₃

—

C₂²⋊Q₈×C₅₃

C₄

yes

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

C₄

yes

—

#117×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

—

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#1559

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

—

—

—

—

C₅₃

#1559

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

Dic₅₃

yes

—

C₄×Q₈

—

C₄×Q₈

Dic₅₃

yes

—

#117

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

#97

—

106

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₃

yes

—

100

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

101

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

102

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

103

M₁₆

—

212

—

104

C₂×C₈

—

106

—

105

C₂×C₈

—

106

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂×Dic₅₃

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₂×Dic₅₃

yes

characteristic

115

C₄²

C₂²⋊D₅₃

yes

characteristic

116

C₄²

C₂²⋊D₅₃

yes

characteristic

117

C₄²

C₂²×D₅₃

yes

characteristic

118

C₂³

C₂²⋊Dic₅₃

yes

characteristic

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

129

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

130

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

131

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

134

C₈

—

212

—

135

C₈

—

212

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#13133242

yes

characteristic

147

C₄

#13133247

yes

characteristic

148

C₄

#13133248

yes

—

149

C₂

—

150

C₂

#16810607

yes

characteristic

151

C₂

#16810624

yes

characteristic

152

C₂

#16810632

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#533	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#533	D₅₃
hypocenter	C₅₃	#1559
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#533

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. #16810372	C₂	semidirect
4. #16810369	C₂	semidirect
82. C₅₃	#1559	semidirect

H ≅ G/N

type

3. #16810372

C₂

semidirect

4. #16810369

C₂

semidirect

82. C₅₃

#1559

semidirect

Subgroups — #20273445

Conjugacy classes of subgroups

Complements