Group #20274264 — subgroups

Subgroups — #20274264

Order 6784.

160 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16810701	C₂	1	1	yes	maximal, characteristic
4	#16810375	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810369	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810368	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#461	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719917	—	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	C₂×SD₁₆×C₅₃	—	2	1	no	—
19	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	#103×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	SD₁₆×C₅₃	—	4	2	no	—
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
32	C₂²⋊C₄×C₅₃	—	2	1	no	—
33	C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊Dic₅₃	—	2	1	no	—
35	C₄⋊Dic₅₃	—	2	1	no	—
36	#11719917	—	4	2	no	—
37	C₂×#88322	—	2	1	no	—
38	C₂×#88322	—	2	1	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	—	4	1	no	—
43	Q₈×C₅₃	—	4	1	no	—
44	D₄×C₅₃	—	2	1	no	—
45	D₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	—	4	1	no	—
48	C₂×C₄×C₅₃	—	4	1	no	—
49	C₂×C₄×C₅₃	—	4	1	no	—
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
53	C₂×Dic₅₃	—	4	1	no	—
54	Dic₁₀₆	—	4	1	no	—
55	Dic₁₀₆	—	4	1	no	—
56	C₈×C₅₃	—	4	1	no	—
57	#88322	—	2	1	no	—
58	#88322	—	2	1	no	—
59	#88322	—	4	1	no	—
60	#88322	—	4	1	no	—
61	C₂²×C₅₃	—	4	1	no	—
62	C₂²×C₅₃	—	4	1	no	—
63	C₂²×C₅₃	—	4	1	no	—
64	C₂²×C₅₃	#121	1	1	yes	characteristic
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	4	1	no	—
67	C₄×C₅₃	—	8	1	no	—
68	C₄×C₅₃	#135	1	1	yes	characteristic
69	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₄×C₅₃	#137	1	1	yes	characteristic
72	Dic₅₃	—	8	1	no	—
73	#1699	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	—	8	1	no	—
75	C₂×C₅₃	#463	1	1	yes	characteristic
76	C₂×C₅₃	#468	1	1	yes	characteristic
77	C₂×C₅₃	#473	1	1	yes	characteristic
78	#487	—	53	1	no	—
79	#474	—	53	1	no	—
80	#461	D₅₃	1	1	yes	characteristic
81	#436	—	53	1	no	—
82	#335	—	53	1	no	—
83	#332	—	53	1	no	—
84	#332	—	53	1	no	—
85	C₅₃	#1699	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	2	1	no	—
87	C₂×M₁₆	—	106	1	no	—
88	C₄⋊Q₈	—	53	1	no	—
89	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
90	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
91	#107	—	106	1	no	—
92	C₄⋊C₈	—	53	1	no	—
93	C₄⋊C₈	—	53	1	no	—
94	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
95	#103	—	2	1	no	—
96	#103	—	106	1	no	—
97	#103	—	106	1	no	—
98	#102	—	106	1	no	—
99	C₂²⋊C₈	—	106	1	no	—
100	C₄×C₈	—	53	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	106	1	no	—
103	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
104	C₂²×C₄	—	2	1	no	—
105	SD₁₆	—	4	2	no	—
106	M₁₆	—	212	2	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	53	1	no	—
109	C₂×C₈	—	53	1	no	—
110	C₂×C₈	—	106	1	no	—
111	C₂×C₈	—	106	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	106	1	no	—
115	C₄⋊C₄	—	106	1	no	—
116	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
117	C₂²⋊C₄	—	2	1	no	—
118	C₄²	C₂²×D₅₃	1	1	yes	characteristic
119	C₂³	—	2	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	4	1	no	—
122	Q₈	—	212	1	no	—
123	Q₈	—	212	1	no	—
124	D₄	—	2	1	no	—
125	D₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	212	1	no	—
131	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
132	C₂×C₄	#11719947	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
134	C₈	—	4	1	no	—
135	C₈	—	106	1	no	—
136	C₈	—	106	1	no	—
137	C₈	—	212	1	no	—
138	C₈	—	212	1	no	—
139	C₂²	—	4	1	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	#13133189	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	8	1	no	—
146	C₄	—	424	1	no	—
147	C₄	#13133238	1	1	yes	characteristic
148	C₄	#13133243	1	1	yes	characteristic
149	C₄	#13133244	1	1	yes	characteristic
150	C₄	#13133247	1	1	yes	characteristic
151	C₂	—	8	1	no	—
152	C₂	#16810840	1	1	yes	characteristic
153	C₂	#16810867	1	1	yes	characteristic
154	C₂	#16810896	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810975

C₂

yes

maximal, characteristic, complemented

#16810903

C₂

yes

maximal, characteristic

#16810701

C₂

yes

maximal, characteristic

#16810375

C₂

yes

maximal, characteristic, complemented

#16810369

C₂

yes

maximal, characteristic, complemented

#16810368

C₂

yes

maximal, characteristic, complemented

C₅₃×#461

C₂

yes

maximal, characteristic

C₂×#11719917

—

#13133217

C₂²

yes

characteristic

#13133150

—

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×SD₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

#103×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂×Dic₁₀₆

—

SD₁₆×C₅₃

—

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

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

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

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1699

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

—

—

—

—

C₅₃

#1699

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×M₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

C₄×D₄

C₂×D₅₃

yes

characteristic

#107

—

106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

#103

—

#103

—

106

—

#103

—

106

—

#102

—

106

—

C₂²⋊C₈

—

106

—

100

C₄×C₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

106

—

103

C₂×D₄

C₂²⋊D₅₃

yes

characteristic

104

C₂²×C₄

—

105

SD₁₆

—

106

M₁₆

—

212

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

106

—

111

C₂×C₈

—

106

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

106

—

115

C₄⋊C₄

—

106

—

116

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

117

C₂²⋊C₄

—

118

C₄²

C₂²×D₅₃

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

212

—

123

Q₈

—

212

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

212

—

131

C₂×C₄

D₄×D₅₃

yes

characteristic

132

C₂×C₄

#11719947

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

134

C₈

—

135

C₈

—

106

—

136

C₈

—

106

—

137

C₈

—

212

—

138

C₈

—

212

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

#13133189

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

424

—

147

C₄

#13133238

yes

characteristic

148

C₄

#13133243

yes

characteristic

149

C₄

#13133244

yes

characteristic

150

C₄

#13133247

yes

characteristic

151

C₂

—

152

C₂

#16810840

yes

characteristic

153

C₂

#16810867

yes

characteristic

154

C₂

#16810896

yes

characteristic

	subgroup	G/H
center	C₂²	#13133189
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#461	C₂
socle	C₂²×C₅₃	#121
hypercenter	#461	D₅₃
hypocenter	C₅₃	#1699
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

subgroup

G/H

center

C₂²

#13133189

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#461

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #461

N	H ≅ G/N	type
1. #16810975	C₂	semidirect
4. #16810375	C₂	semidirect
5. #16810369	C₂	semidirect
6. #16810368	C₂	semidirect
85. C₅₃	#1699	semidirect

H ≅ G/N

type

1. #16810975

C₂

semidirect

4. #16810375

C₂

semidirect

5. #16810369

C₂

semidirect

6. #16810368

C₂

semidirect

85. C₅₃

#1699

semidirect

Subgroups — #20274264

Conjugacy classes of subgroups

Complements