Group #20273573 — subgroups

Subgroups — #20273573

Order 6784.

168 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 71 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	#16810754	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#16810376	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊#88322	C₂	1	1	yes	maximal, characteristic
6	#16810345	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×SD₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#88322	—	2	1	no	—
9	C₄⋊#88322	—	2	1	no	—
10	C₄⋊#88322	C₂²	1	1	yes	characteristic
11	C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₂²⋊#88322	—	2	1	no	—
13	#13133128	—	2	1	no	—
14	C₄×#88322	C₂²	1	1	yes	characteristic
15	C₄×#88322	C₂²	1	1	yes	characteristic
16	C₂×SD₁₆×C₅₃	C₄	1	1	yes	characteristic
17	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
19	#106×C₅₃	C₄	1	1	yes	characteristic
20	#103×C₅₃	C₄	1	1	yes	characteristic
21	#102×C₅₃	C₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	—	2	1	no	—
26	SD₁₆×C₅₃	—	2	2	no	—
27	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²⋊C₄×C₅₃	—	2	1	no	—
33	C₄²×C₅₃	—	2	1	no	—
34	C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×#88322	—	2	1	no	—
36	C₂×#88322	—	2	1	no	—
37	C₂×#88322	—	2	1	no	—
38	C₂×#88322	—	4	2	no	—
39	C₂×#88322	D₄	1	1	yes	characteristic
40	C₂×#88322	D₄	1	1	yes	characteristic
41	C₂³×C₅₃	—	2	1	no	—
42	Q₈×C₅₃	—	2	1	no	—
43	Q₈×C₅₃	—	2	1	no	—
44	D₄×C₅₃	—	2	1	no	—
45	D₄×C₅₃	M₁₆	1	2	yes	—
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	—	4	1	no	—
50	C₂×C₄×C₅₃	—	4	1	no	—
51	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
54	C₈×C₅₃	—	2	1	no	—
55	C₈×C₅₃	—	2	1	no	—
56	#88322	—	2	2	no	—
57	#88322	—	2	1	no	—
58	#88322	—	2	1	no	—
59	#88322	—	4	1	no	—
60	#88322	—	4	1	no	—
61	C₂²×C₅₃	—	2	2	no	—
62	C₂²×C₅₃	—	4	1	no	—
63	C₂²×C₅₃	—	4	1	no	—
64	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	4	1	no	—
67	C₄×C₅₃	—	4	1	no	—
68	C₄×C₅₃	—	4	1	no	—
69	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
70	C₄×C₅₃	#131	1	1	yes	characteristic
71	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₄×C₅₃	#137	1	1	yes	characteristic
73	#1585	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	—	4	2	no	—
75	C₂×C₅₃	#435	1	1	yes	characteristic
76	C₂×C₅₃	#440	1	1	yes	characteristic
77	C₂×C₅₃	#443	1	1	yes	characteristic
78	#446	—	53	1	no	—
79	C₄×SD₁₆	D₅₃	1	1	yes	characteristic
80	C₈×D₄	—	53	1	no	—
81	#335	—	53	1	no	—
82	Q₈⋊C₈	—	53	1	no	—
83	D₄⋊C₈	—	53	1	no	—
84	#322	—	53	1	no	—
85	C₅₃	#1585	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	Dic₅₃	1	1	yes	characteristic
87	C₂²×C₈	—	106	1	no	—
88	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
89	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
90	#106	Dic₅₃	1	1	yes	characteristic
91	C₄⋊C₈	—	53	1	no	—
92	C₄⋊C₈	—	53	1	no	—
93	C₄⋊C₈	—	106	1	no	—
94	#103	Dic₅₃	1	1	yes	characteristic
95	#102	Dic₅₃	1	1	yes	characteristic
96	C₂²⋊C₈	—	106	1	no	—
97	#97	—	106	1	no	—
98	C₄×C₈	—	53	1	no	—
99	C₄×C₈	—	53	1	no	—
100	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
101	C₂×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
102	C₂×D₄	C₂×Dic₅₃	1	1	yes	characteristic
103	C₂²×C₄	—	2	1	no	—
104	SD₁₆	—	2	2	no	—
105	C₂×C₈	—	53	1	no	—
106	C₂×C₈	—	53	1	no	—
107	C₂×C₈	—	106	1	no	—
108	C₂×C₈	—	106	1	no	—
109	C₂×C₈	—	106	1	no	—
110	C₂×C₈	—	212	2	no	—
111	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
112	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
115	C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
116	C₂²⋊C₄	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	C₂²×D₅₃	1	1	yes	characteristic
119	C₂³	—	2	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	—	2	1	no	—
122	D₄	—	2	1	no	—
123	D₄	#11719917	1	2	yes	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
130	C₂×C₄	#11719947	1	1	yes	characteristic
131	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	106	2	no	—
135	C₈	—	106	1	no	—
136	C₈	—	106	1	no	—
137	C₈	—	212	1	no	—
138	C₈	—	212	1	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	D₄×Dic₅₃	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	4	1	no	—
147	C₄	C₂×#11719917	1	1	yes	characteristic
148	C₄	#13133224	1	1	yes	characteristic
149	C₄	#13133244	1	1	yes	characteristic
150	C₄	#13133246	1	1	yes	characteristic
151	C₂	—	4	2	no	—
152	C₂	#16810695	1	1	yes	characteristic
153	C₂	#16810724	1	1	yes	characteristic
154	C₂	#16810740	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810754

C₂

yes

maximal, characteristic, complemented

D₄×#88322

C₂

yes

maximal, characteristic

#16810376

C₂

yes

maximal, characteristic, complemented

Q₈⋊#88322

C₂

yes

maximal, characteristic, complemented

D₄⋊#88322

C₂

yes

maximal, characteristic

#16810345

C₂

yes

maximal, characteristic, complemented

C₄×SD₁₆×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

—

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₂×SD₁₆×C₅₃

C₄

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

C₂²

yes

characteristic

#106×C₅₃

C₄

yes

characteristic

#103×C₅₃

C₄

yes

characteristic

#102×C₅₃

C₄

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

SD₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

—

—

—

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

C₂³×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

M₁₆

yes

—

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

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₄×SD₁₆

D₅₃

yes

characteristic

C₈×D₄

—

#335

—

Q₈⋊C₈

—

D₄⋊C₈

—

#322

—

C₅₃

#1585

yes

characteristic, complemented, Sylow

C₂×SD₁₆

Dic₅₃

yes

characteristic

C₂²×C₈

—

106

—

C₄×Q₈

C₂×D₅₃

yes

characteristic

C₄×D₄

C₂×D₅₃

yes

characteristic

#106

Dic₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

#103

Dic₅₃

yes

characteristic

#102

Dic₅₃

yes

characteristic

C₂²⋊C₈

—

106

—

#97

—

106

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×D₅₃

yes

characteristic

101

C₂×Q₈

C₂×Dic₅₃

yes

characteristic

102

C₂×D₄

C₂×Dic₅₃

yes

characteristic

103

C₂²×C₄

—

104

SD₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

106

—

108

C₂×C₈

—

106

—

109

C₂×C₈

—

106

—

110

C₂×C₈

—

212

—

111

C₂×C₈

C₂×Dic₅₃

yes

characteristic

112

C₂×C₈

C₂×Dic₅₃

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

C₂×Dic₅₃

yes

characteristic

115

C₄⋊C₄

C₂×Dic₅₃

yes

characteristic

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂²×D₅₃

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

—

122

D₄

—

123

D₄

#11719917

yes

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

D₄×D₅₃

yes

characteristic

130

C₂×C₄

#11719947

yes

characteristic

131

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

106

—

135

C₈

—

106

—

136

C₈

—

106

—

137

C₈

—

212

—

138

C₈

—

212

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

D₄×Dic₅₃

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

C₂×#11719917

yes

characteristic

148

C₄

#13133224

yes

characteristic

149

C₄

#13133244

yes

characteristic

150

C₄

#13133246

yes

characteristic

151

C₂

—

152

C₂

#16810695

yes

characteristic

153

C₂

#16810724

yes

characteristic

154

C₂

#16810740

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₄×SD₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	C₄×SD₁₆	D₅₃
hypocenter	C₅₃	#1585
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

subgroup

G/H

center

C₂×C₄

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₄×SD₁₆×C₅₃

C₂

socle

C₂²×C₅₃

C₄×D₄

hypercenter

C₄×SD₁₆

D₅₃

hypocenter

C₅₃

#1585

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

N	H ≅ G/N	type
1. #16810754	C₂	semidirect
3. #16810376	C₂	semidirect
4. Q₈⋊#88322	C₂	semidirect
6. #16810345	C₂	semidirect
85. C₅₃	#1585	semidirect

H ≅ G/N

type

1. #16810754

C₂

semidirect

3. #16810376

C₂

semidirect

4. Q₈⋊#88322

C₂

semidirect

6. #16810345

C₂

semidirect

85. C₅₃

#1585

semidirect

Subgroups — #20273573

Conjugacy classes of subgroups

Complements