Group #20290291 — subgroups

Subgroups — #20290291

Order 6784.

158 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 62 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	#16811639	C₂	1	1	yes	maximal, characteristic
2	#16811626	C₂	1	1	yes	maximal, characteristic
3	#16811487	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811486	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811485	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811484	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810673	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719940	—	2	1	no	—
9	#13133284	—	2	1	no	—
10	C₄⋊#88324	C₂²	1	1	yes	characteristic
11	C₄⋊#88324	C₂²	1	1	yes	characteristic
12	C₄⋊#88324	C₂²	1	1	yes	characteristic
13	C₄⋊#88324	C₂²	1	1	yes	characteristic
14	C₂²⋊#88324	—	2	1	no	—
15	C₄×#88324	—	2	1	no	—
16	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
17	#13133146	C₄	1	1	yes	characteristic
18	#13133146	C₄	1	1	yes	characteristic
19	C₂²⋊#88322	C₄	1	1	yes	characteristic
20	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
21	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
22	C₂²⋊C₈×C₅₃	C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
25	C₂²×Dic₅₃	Q₈	1	1	yes	characteristic
26	#11719940	—	4	2	no	—
27	C₂×#88324	—	2	1	no	—
28	C₂×#88324	—	2	1	no	—
29	C₂×#88324	—	2	1	no	—
30	C₂×#88324	—	2	1	no	—
31	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₅₃	D₄	1	1	yes	characteristic
35	C₄×Dic₅₃	Q₈	1	1	yes	characteristic
36	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
37	C₂×#88322	C₂×C₄	1	1	yes	characteristic
38	C₂×#88322	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	—	2	1	no	—
42	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₅₃	—	2	2	no	—
45	C₂×Dic₅₃	—	2	1	no	—
46	C₂×Dic₅₃	—	2	1	no	—
47	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
51	#88324	—	4	1	no	—
52	#88324	—	4	1	no	—
53	#88324	—	4	1	no	—
54	#88324	—	4	1	no	—
55	C₈×C₅₃	—	2	1	no	—
56	C₈×C₅₃	—	2	1	no	—
57	#88322	—	2	1	no	—
58	#88322	—	2	1	no	—
59	C₂²×C₅₃	—	2	1	no	—
60	C₂²×C₅₃	—	2	1	no	—
61	C₂²×C₅₃	—	2	1	no	—
62	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	#131	1	1	yes	characteristic
65	C₄×C₅₃	#131	1	1	yes	characteristic
66	Dic₅₃	—	2	1	no	—
67	Dic₅₃	—	2	1	no	—
68	Dic₅₃	#108	1	2	yes	—
69	Dic₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
70	Dic₅₃	#137	1	1	yes	characteristic
71	#1574	—	53	1	no	maximal, Sylow
72	C₂×C₅₃	—	4	1	no	—
73	C₂×C₅₃	#424	1	1	yes	characteristic
74	C₂×C₅₃	#428	1	1	yes	characteristic
75	C₂×C₅₃	C₂×#108	1	1	yes	characteristic
76	#431	—	53	1	no	—
77	#431	—	53	1	no	—
78	#423	—	53	1	no	—
79	#335	—	53	1	no	—
80	#335	—	53	1	no	—
81	#334	—	53	1	no	—
82	#334	—	53	1	no	—
83	C₅₃	#1574	1	1	yes	characteristic, complemented, Sylow
84	C₂×M₁₆	—	106	1	no	—
85	C₂×C₄²	—	53	1	no	—
86	C₄⋊C₈	—	53	1	no	—
87	C₄⋊C₈	—	53	1	no	—
88	C₄⋊C₈	—	53	1	no	—
89	C₄⋊C₈	—	53	1	no	—
90	C₄⋊C₈	—	53	1	no	—
91	C₄⋊C₈	—	53	1	no	—
92	C₄⋊C₈	—	106	1	no	—
93	C₂²⋊C₈	—	53	1	no	—
94	C₂²⋊C₈	—	106	1	no	—
95	C₂²⋊C₈	#6157	1	1	yes	characteristic
96	C₄×C₈	—	53	1	no	—
97	C₄×C₈	—	53	1	no	—
98	C₄×C₈	—	106	1	no	—
99	C₂²×C₄	—	53	1	no	—
100	C₂²×C₄	—	53	1	no	—
101	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
102	M₁₆	—	212	2	no	—
103	C₂×C₈	—	53	1	no	—
104	C₂×C₈	—	53	1	no	—
105	C₂×C₈	—	106	1	no	—
106	C₂×C₈	—	106	1	no	—
107	C₂×C₈	—	106	1	no	—
108	C₂×C₈	—	106	1	no	—
109	C₂×C₈	C₂×#6157	1	1	yes	characteristic
110	C₂×C₈	C₂×#6157	1	1	yes	characteristic
111	C₄²	—	53	1	no	—
112	C₄²	—	53	1	no	—
113	C₄²	—	53	1	no	—
114	C₄²	—	53	1	no	—
115	C₂³	C₄⋊#6157	1	1	yes	characteristic
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	53	1	no	—
119	C₂×C₄	—	53	1	no	—
120	C₂×C₄	—	53	1	no	—
121	C₂×C₄	—	53	1	no	—
122	C₂×C₄	—	106	1	no	—
123	C₂×C₄	—	106	1	no	—
124	C₂×C₄	—	106	2	no	—
125	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
126	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
127	C₈	—	2	1	no	—
128	C₈	—	2	1	no	—
129	C₈	—	106	1	no	—
130	C₈	—	106	1	no	—
131	C₈	—	212	1	no	—
132	C₈	—	212	1	no	—
133	C₈	—	212	1	no	—
134	C₈	—	212	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	53	1	no	—
141	C₄	—	53	1	no	—
142	C₄	—	53	2	no	—
143	C₄	—	106	1	no	—
144	C₄	—	106	1	no	—
145	C₄	#13133304	1	1	yes	characteristic
146	C₄	#13133304	1	1	yes	characteristic
147	C₂	—	4	1	no	—
148	C₂	#16811629	1	1	yes	characteristic
149	C₂	#16811634	1	1	yes	characteristic
150	C₂	D₅₃⋊#108	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811639

C₂

yes

maximal, characteristic

#16811626

C₂

yes

maximal, characteristic

#16811487

C₂

yes

maximal, characteristic, complemented

#16811486

C₂

yes

maximal, characteristic, complemented

#16811485

C₂

yes

maximal, characteristic, complemented

#16811484

C₂

yes

maximal, characteristic, complemented

#16810673

C₂

yes

maximal, characteristic

C₂×#11719940

—

#13133284

—

C₄⋊#88324

C₂²

yes

characteristic

C₄⋊#88324

C₂²

yes

characteristic

C₄⋊#88324

C₂²

yes

characteristic

C₄⋊#88324

C₂²

yes

characteristic

C₂²⋊#88324

—

C₄×#88324

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133146

C₄

yes

characteristic

#13133146

C₄

yes

characteristic

C₂²⋊#88322

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

C₂²⋊C₈×C₅₃

C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

Q₈

yes

—

—

—

—

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

Q₈

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×Q₈

yes

—

—

—

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#108

yes

—

Dic₅₃

yes

Dic₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

—

—

—

—

C₅₃

#1574

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

C₂²⋊C₈

#6157

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

106

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂×#6157

yes

characteristic

102

M₁₆

—

212

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

—

106

C₂×C₈

—

106

—

107

C₂×C₈

—

106

—

108

C₂×C₈

—

106

—

109

C₂×C₈

C₂×#6157

yes

characteristic

110

C₂×C₈

C₂×#6157

yes

characteristic

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₂³

C₄⋊#6157

yes

characteristic

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

106

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

106

—

125

C₂×C₄

C₄⋊#6157

yes

characteristic

126

C₂×C₄

C₂²×#6157

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

106

—

130

C₈

—

106

—

131

C₈

—

212

—

132

C₈

—

212

—

133

C₈

—

212

—

134

C₈

—

212

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂×C₄⋊#6157

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

106

—

144

C₄

—

106

—

145

C₄

#13133304

yes

characteristic

146

C₄

#13133304

yes

characteristic

147

C₂

—

148

C₂

#16811629

yes

characteristic

149

C₂

#16811634

yes

characteristic

150

C₂

D₅₃⋊#108

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₂²⋊C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₂²⋊C₈	#6157
hypocenter	C₅₃	#1574
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₂²⋊C₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₄⋊C₄

C₂²⋊C₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
3. #16811487	C₂	semidirect
4. #16811486	C₂	semidirect
5. #16811485	C₂	semidirect
6. #16811484	C₂	semidirect
83. C₅₃	#1574	semidirect

H ≅ G/N

type

3. #16811487

C₂

semidirect

4. #16811486

C₂

semidirect

5. #16811485

C₂

semidirect

6. #16811484

C₂

semidirect

83. C₅₃

#1574

semidirect

Subgroups — #20290291

Conjugacy classes of subgroups

Complements