Group #20275999 — subgroups

Subgroups — #20275999

Order 6784.

266 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 67 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	#16810646	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133144	C₂	1	2	yes	maximal, complemented
3	#16810607	C₂	1	2	yes	maximal, complemented
4	#16810381	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#575	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	1	yes	characteristic
7	C₂×#11719917	C₂²	1	2	yes	—
8	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
9	#13133146	—	2	2	no	—
10	#13133144	—	2	2	no	—
11	#13133144	—	2	2	no	—
12	#13133136	—	2	2	no	—
13	#13133134	—	2	2	no	—
14	C₂×#41×C₅₃	C₂²	1	2	yes	—
15	#137×C₅₃	—	2	2	no	—
16	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
17	#135×C₅₃	—	2	2	no	—
18	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	#41×C₅₃	—	2	2	no	—
20	#41×C₅₃	—	2	2	no	—
21	#41×C₅₃	—	2	2	no	—
22	#41×C₅₃	—	2	2	no	—
23	C₂×Q₈×C₅₃	D₄	1	2	yes	—
24	C₂×D₄×C₅₃	—	2	2	no	—
25	C₂×D₄×C₅₃	D₄	1	2	yes	—
26	C₂²×C₄×C₅₃	—	2	2	no	—
27	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×Dic₅₃	—	2	1	no	—
29	Q₁₆×C₅₃	—	2	2	no	—
30	SD₁₆×C₅₃	—	2	4	no	—
31	D₈×C₅₃	—	2	2	no	—
32	M₁₆×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	D₄	1	2	yes	—
34	C₄×Dic₅₃	—	2	1	no	—
35	C₄×Dic₅₃	D₄	1	1	yes	characteristic
36	C₄×Dic₅₃	D₄	1	1	yes	characteristic
37	#11719917	—	2	2	no	—
38	#11719917	—	2	2	no	—
39	#11719917	—	2	2	no	—
40	C₂×#88322	—	2	1	no	—
41	C₂×#88322	—	2	2	no	—
42	C₂³×C₅₃	—	2	2	no	—
43	C₂³×C₅₃	#41	1	1	yes	characteristic
44	Q₈×C₅₃	—	2	2	no	—
45	Q₈×C₅₃	—	2	2	no	—
46	D₄×C₅₃	—	2	2	no	—
47	D₄×C₅₃	—	2	2	no	—
48	D₄×C₅₃	—	4	2	no	—
49	D₄×C₅₃	—	4	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	4	2	no	—
52	C₂×C₄×C₅₃	—	4	2	no	—
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#41	1	2	yes	—
57	C₂×Dic₅₃	—	2	1	no	—
58	C₂×Dic₅₃	—	2	1	no	—
59	C₂×Dic₅₃	—	4	2	no	—
60	C₈×C₅₃	—	2	2	no	—
61	#88322	—	4	1	no	—
62	#88322	—	4	2	no	—
63	C₂²×C₅₃	—	4	1	no	—
64	C₂²×C₅₃	—	4	2	no	—
65	C₂²×C₅₃	—	8	2	no	—
66	C₂²×C₅₃	#121	1	1	yes	characteristic
67	C₂²×C₅₃	#123	1	2	yes	—
68	C₄×C₅₃	—	4	2	no	—
69	C₄×C₅₃	#120	1	1	yes	characteristic
70	C₄×C₅₃	#121	1	2	yes	—
71	C₄×C₅₃	#124	1	1	yes	characteristic
72	Dic₅₃	—	2	2	no	—
73	Dic₅₃	—	4	1	no	—
74	#2050	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	—	2	1	no	—
76	C₂×C₅₃	—	2	2	no	—
77	C₂×C₅₃	—	8	2	no	—
78	C₂×C₅₃	#394	1	1	yes	characteristic
79	#575	D₅₃	1	1	yes	characteristic
80	#423	—	53	1	no	—
81	C₂×#104	—	53	2	no	—
82	#413	—	53	2	no	—
83	#338	—	53	1	no	—
84	C₅₃	#2050	1	1	yes	characteristic, complemented, Sylow
85	C₂×#41	C₂×D₅₃	1	2	yes	—
86	#137	—	2	2	no	—
87	AGL(1,ℤ/8)	—	2	2	no	—
88	#135	—	2	2	no	—
89	C₂×M₁₆	—	53	2	no	—
90	C₂×M₁₆	—	53	1	no	—
91	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
92	C₂×C₄²	—	53	1	no	—
93	C₄⋊C₈	—	106	2	no	—
94	#104	—	106	2	no	—
95	#104	—	106	2	no	—
96	#101	—	106	2	no	—
97	#100	—	106	2	no	—
98	#41	—	2	2	no	—
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	#41	—	2	2	no	—
102	C₂×Q₈	C₂²⋊D₅₃	1	2	yes	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	C₂²⋊D₅₃	1	2	yes	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	106	1	no	—
107	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
108	Q₁₆	—	2	2	no	—
109	SD₁₆	—	2	4	no	—
110	D₈	—	2	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	106	2	no	—
113	M₁₆	—	106	2	no	—
114	M₁₆	—	106	2	no	—
115	C₂×C₈	—	106	2	no	—
116	C₂×C₈	—	106	1	no	—
117	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
118	C₄²	—	53	1	no	—
119	C₄²	—	53	1	no	—
120	C₄²	—	106	1	no	—
121	C₂³	—	2	2	no	—
122	C₂³	#11719947	1	1	yes	characteristic
123	Q₈	—	2	2	no	—
124	Q₈	—	2	2	no	—
125	D₄	—	2	2	no	—
126	D₄	—	2	2	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	106	1	no	—
133	C₂×C₄	—	106	1	no	—
134	C₂×C₄	—	212	2	no	—
135	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
136	C₂×C₄	#11719947	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	212	1	no	—
141	C₈	—	212	2	no	—
142	C₂²	—	4	1	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	8	2	no	—
145	C₂²	#13133188	1	1	yes	characteristic
146	C₂²	#13133198	1	2	yes	—
147	C₄	—	4	2	no	—
148	C₄	—	106	2	no	—
149	C₄	—	212	1	no	—
150	C₄	#13133186	1	1	yes	characteristic
151	C₄	#13133189	1	2	yes	—
152	C₄	#13133204	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	2	no	—
155	C₂	—	8	2	no	—
156	C₂	#16810537	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810646

C₂

yes

maximal, characteristic, complemented

C₂×#13133144

C₂

yes

maximal, complemented

#16810607

C₂

yes

maximal, complemented

#16810381

C₂

yes

maximal, characteristic, complemented

C₅₃×#575

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

—

—

—

—

C₂×#41×C₅₃

C₂²

yes

—

#137×C₅₃

—

AGL(1,ℤ/8)×C₅₃

—

#135×C₅₃

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

Q₁₆×C₅₃

—

SD₁₆×C₅₃

—

D₈×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

—

—

—

—

—

C₂³×C₅₃

—

C₂³×C₅₃

#41

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

#123

yes

—

C₄×C₅₃

—

C₄×C₅₃

#120

yes

characteristic

C₄×C₅₃

#121

yes

—

C₄×C₅₃

#124

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#2050

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#394

yes

characteristic

#575

D₅₃

yes

—

—

—

—

C₅₃

#2050

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

—

#137

—

AGL(1,ℤ/8)

—

#135

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

106

—

#104

—

106

—

#104

—

106

—

#101

—

106

—

#100

—

106

—

#41

—

#41

—

100

#41

—

101

#41

—

102

C₂×Q₈

C₂²⋊D₅₃

yes

—

103

C₂×D₄

—

104

C₂×D₄

C₂²⋊D₅₃

yes

—

105

C₂²×C₄

—

106

C₂²×C₄

—

106

—

107

C₂²×C₄

C₂²×D₅₃

yes

characteristic

108

Q₁₆

—

109

SD₁₆

—

110

D₈

—

111

M₁₆

—

112

M₁₆

—

106

—

113

M₁₆

—

106

—

114

M₁₆

—

106

—

115

C₂×C₈

—

106

—

116

C₂×C₈

—

106

—

117

C₂×C₈

C₂²⋊D₅₃

yes

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

106

—

121

C₂³

—

122

C₂³

#11719947

yes

characteristic

123

Q₈

—

124

Q₈

—

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

106

—

133

C₂×C₄

—

106

—

134

C₂×C₄

—

212

—

135

C₂×C₄

D₄×D₅₃

yes

characteristic

136

C₂×C₄

#11719947

yes

—

137

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

138

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

139

C₈

—

140

C₈

—

212

—

141

C₈

—

212

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#13133188

yes

characteristic

146

C₂²

#13133198

yes

—

147

C₄

—

148

C₄

—

106

—

149

C₄

—

212

—

150

C₄

#13133186

yes

characteristic

151

C₄

#13133189

yes

—

152

C₄

#13133204

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

—

156

C₂

#16810537

yes

characteristic

	subgroup	G/H
center	C₄	#13133204
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#575	C₂
socle	C₂×C₅₃	#394
hypercenter	#575	D₅₃
hypocenter	C₅₃	#2050
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

subgroup

G/H

center

C₄

#13133204

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#575

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

N	H ≅ G/N	type
1. #16810646	C₂	semidirect
2. C₂×#13133144	C₂	semidirect
3. #16810607	C₂	semidirect
4. #16810381	C₂	semidirect
84. C₅₃	#2050	semidirect

H ≅ G/N

type

1. #16810646

C₂

semidirect

2. C₂×#13133144

C₂

semidirect

3. #16810607

C₂

semidirect

4. #16810381

C₂

semidirect

84. C₅₃

#2050

semidirect

Subgroups — #20275999

Conjugacy classes of subgroups

Complements