Group #20273405 — subgroups

Subgroups — #20273405

Order 6784.

216 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16810647	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#88322	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	C₂²×#88322	—	2	1	no	—
8	C₄⋊#88322	—	2	1	no	—
9	C₄⋊#88322	—	2	1	no	—
10	C₄⋊#88322	C₂²	1	2	yes	—
11	C₄⋊#88322	C₂²	1	2	yes	—
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	C₂×#88322	—	4	2	no	—
42	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	Q₈×C₅₃	—	4	2	no	—
44	Q₈×C₅₃	—	4	1	no	—
45	Q₈×C₅₃	—	4	1	no	—
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	—	2	1	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	SD₁₆	1	2	yes	—
55	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
56	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	#88322	—	4	2	no	—
60	#88322	—	4	2	no	—
61	C₂²×C₅₃	—	2	1	no	—
62	C₂²×C₅₃	—	2	1	no	—
63	C₂²×C₅₃	#103	1	2	yes	—
64	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	2	1	no	—
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	—	4	2	no	—
69	C₄×C₅₃	—	4	2	no	—
70	C₄×C₅₃	#103	1	2	yes	—
71	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
73	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₄×C₅₃	#137	1	1	yes	characteristic
75	#1550	—	53	1	no	maximal, Sylow
76	C₂×C₅₃	—	2	2	no	—
77	C₂×C₅₃	#413	1	1	yes	characteristic
78	C₂×C₅₃	C₂×#103	1	1	yes	characteristic
79	C₂×C₅₃	#417	1	1	yes	characteristic
80	#533	D₅₃	1	1	yes	characteristic
81	#423	—	53	1	no	—
82	C₂×C₄⋊C₈	—	53	1	no	—
83	#333	—	53	2	no	—
84	#332	—	53	2	no	—
85	C₅₃	#1550	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₁₆	—	106	1	no	—
87	C₂²×C₈	—	106	1	no	—
88	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
89	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
90	#125	—	2	1	no	—
91	C₂²⋊Q₈	—	2	1	no	—
92	C₂²⋊Q₈	Dic₅₃	1	2	yes	—
93	C₄×Q₈	—	2	1	no	—
94	C₄×Q₈	Dic₅₃	1	2	yes	—
95	#117	—	2	1	no	—
96	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
97	C₄⋊C₈	—	53	2	no	—
98	C₄⋊C₈	—	53	2	no	—
99	C₄⋊C₈	—	106	1	no	—
100	C₄⋊C₈	—	106	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
103	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
104	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
106	M₁₆	—	212	2	no	—
107	C₂×C₈	—	106	2	no	—
108	C₂×C₈	—	106	2	no	—
109	C₂×C₈	—	212	2	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
116	C₂²⋊C₄	—	2	2	no	—
117	C₄²	—	2	2	no	—
118	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
119	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
120	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
121	C₄²	C₂²×D₅₃	1	1	yes	characteristic
122	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
123	Q₈	—	4	2	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	4	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	#11719924	1	2	yes	—
132	C₂×C₄	#11719925	1	2	yes	—
133	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
134	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
135	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
136	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
139	C₈	—	212	2	no	—
140	C₈	—	212	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#13133141	1	2	yes	—
144	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	4	2	no	—
149	C₄	—	4	2	no	—
150	C₄	#13133141	1	2	yes	—
151	C₄	C₂×#11719924	1	1	yes	characteristic
152	C₄	C₂×#11719925	1	1	yes	characteristic
153	C₄	#13133242	1	1	yes	characteristic
154	C₄	#13133247	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	#16810607	1	1	yes	characteristic
157	C₂	C₂×#13133141	1	1	yes	characteristic
158	C₂	#16810624	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810647

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic

#16810372

C₂

yes

maximal, complemented

#16810369

C₂

yes

maximal, complemented

C₅₃×#533

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₂²×#88322

—

C₄⋊#88322

—

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

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

#103

yes

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#103

yes

—

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

characteristic

#423

—

C₂×C₄⋊C₈

—

#333

—

#332

—

C₅₃

#1550

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

—

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

—

100

C₄⋊C₈

—

106

—

101

C₂×Q₈

—

102

C₂×Q₈

C₂×Dic₅₃

yes

—

103

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

104

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

105

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

106

M₁₆

—

212

—

107

C₂×C₈

—

106

—

108

C₂×C₈

—

106

—

109

C₂×C₈

—

212

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂×Dic₅₃

yes

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂×Dic₅₃

yes

characteristic

119

C₄²

C₂²⋊D₅₃

yes

characteristic

120

C₄²

C₂²⋊D₅₃

yes

characteristic

121

C₄²

C₂²×D₅₃

yes

characteristic

122

C₂³

C₂²⋊Dic₅₃

yes

characteristic

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#11719924

yes

—

132

C₂×C₄

#11719925

yes

—

133

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

134

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

135

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

136

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

139

C₈

—

212

—

140

C₈

—

212

—

141

C₂²

—

142

C₂²

—

143

C₂²

#13133141

yes

—

144

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#13133141

yes

—

151

C₄

C₂×#11719924

yes

characteristic

152

C₄

C₂×#11719925

yes

characteristic

153

C₄

#13133242

yes

characteristic

154

C₄

#13133247

yes

characteristic

155

C₂

—

156

C₂

#16810607

yes

characteristic

157

C₂

C₂×#13133141

yes

characteristic

158

C₂

#16810624

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	#1550
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
85. C₅₃	#1550	semidirect

H ≅ G/N

type

3. #16810372

C₂

semidirect

4. #16810369

C₂

semidirect

85. C₅₃

#1550

semidirect

Subgroups — #20273405

Conjugacy classes of subgroups

Complements