Group #20287608 — subgroups

Subgroups — #20287608

Order 6784.

324 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16811348	C₂	1	2	yes	maximal
2	#16811024	C₂	1	2	yes	maximal
3	#16810929	C₂	1	4	yes	maximal
4	#16810905	C₂	1	2	yes	maximal
5	C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
7	#16810717	C₂	1	2	yes	maximal
8	C₅₃×#500	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
10	#13133215	—	2	2	no	—
11	#13133208	C₂²	1	4	yes	—
12	#13133207	—	2	4	no	—
13	Q₈×Dic₅₃	—	2	2	no	—
14	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
15	C₄×Dic₁₀₆	C₂²	1	2	yes	—
16	#13133182	C₂²	1	4	yes	—
17	#13133151	C₂²	1	1	yes	characteristic
18	#13133148	C₂²	1	4	yes	—
19	C₄⋊#88322	C₂²	1	1	yes	characteristic
20	#13133146	C₂²	1	2	yes	—
21	#13133142	C₂²	1	2	yes	—
22	#13133140	C₂²	1	4	yes	—
23	C₈×Dic₅₃	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
25	#107×C₅₃	C₂²	1	2	yes	—
26	#107×C₅₃	C₂²	1	2	yes	—
27	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₅₃	—	2	2	no	—
29	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
30	C₂×Dic₁₀₆	C₂³	1	2	yes	—
31	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
35	C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	C₄⋊Dic₅₃	—	2	4	no	—
37	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
38	C₄⋊Dic₅₃	C₂³	1	2	yes	—
39	#11719919	—	2	2	no	—
40	#11719919	—	2	4	no	—
41	#11719919	—	2	4	no	—
42	#11719919	D₄	1	2	yes	—
43	C₄×Dic₅₃	—	2	2	no	—
44	C₄×Dic₅₃	—	2	4	no	—
45	C₄×Dic₅₃	C₂³	1	2	yes	—
46	C₂×#88322	C₂³	1	2	yes	—
47	Dic₂₁₂	Q₈	1	4	yes	complemented
48	Q₈×C₅₃	—	4	4	no	—
49	C₂×C₄×C₅₃	—	2	4	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	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₅₃	—	2	2	no	—
54	C₂×Dic₅₃	—	2	2	no	—
55	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
56	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
57	Dic₁₀₆	—	2	2	no	—
58	Dic₁₀₆	—	2	2	no	—
59	Dic₁₀₆	C₂×Q₈	1	4	yes	—
60	C₈×C₅₃	—	2	1	no	—
61	C₈×C₅₃	C₂×Q₈	1	2	yes	—
62	#88322	—	2	2	no	—
63	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	—	4	4	no	—
66	C₄×C₅₃	#135	1	2	yes	—
67	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
68	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
69	Dic₅₃	—	2	4	no	—
70	Dic₅₃	—	2	2	no	—
71	Dic₅₃	—	2	2	no	—
72	Dic₅₃	—	4	2	no	—
73	#3383	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	D₄×Q₈	1	1	yes	characteristic
75	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
76	C₂×C₅₃	#578	1	1	yes	characteristic
77	#557	—	53	2	no	—
78	#500	D₅₃	1	1	yes	characteristic
79	#499	—	53	2	no	—
80	#479	—	53	4	no	—
81	#475	—	53	2	no	—
82	C₈×Q₈	—	53	1	no	—
83	C₄×Q₁₆	—	53	2	no	—
84	C₄×Q₁₆	—	53	1	no	—
85	C₅₃	#3383	1	1	yes	characteristic, complemented, Sylow
86	C₂×Q₁₆	—	53	1	no	—
87	C₄⋊Q₈	—	106	2	no	—
88	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
89	#125	—	53	4	no	—
90	#125	—	106	4	no	—
91	C₄×Q₈	—	53	1	no	—
92	C₄×Q₈	—	53	4	no	—
93	C₄×Q₈	—	53	2	no	—
94	C₄×Q₈	—	106	2	no	—
95	#107	—	53	1	no	—
96	#107	C₂×D₅₃	1	2	yes	—
97	#107	C₂×D₅₃	1	2	yes	—
98	#106	—	53	4	no	—
99	C₄⋊C₈	—	53	2	no	—
100	C₄⋊C₈	—	53	1	no	—
101	#103	—	53	2	no	—
102	#103	—	53	4	no	—
103	C₄×C₈	—	53	2	no	—
104	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	53	1	no	—
107	C₂×Q₈	—	53	2	no	—
108	Q₁₆	—	53	4	no	—
109	C₂×C₈	—	53	2	no	—
110	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
111	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	53	1	no	—
114	C₄⋊C₄	—	53	2	no	—
115	C₄⋊C₄	—	53	2	no	—
116	C₄⋊C₄	—	106	4	no	—
117	C₄⋊C₄	—	106	4	no	—
118	C₄⋊C₄	—	106	4	no	—
119	C₄⋊C₄	—	106	2	no	—
120	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
121	C₄²	—	53	2	no	—
122	C₄²	—	106	4	no	—
123	C₄²	—	106	2	no	—
124	C₄²	C₂²×D₅₃	1	1	yes	characteristic
125	Q₈	—	4	4	no	—
126	Q₈	—	53	4	no	—
127	Q₈	—	106	2	no	—
128	Q₈	—	106	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	53	2	no	—
131	C₂×C₄	—	53	2	no	—
132	C₂×C₄	—	106	2	no	—
133	C₂×C₄	—	106	2	no	—
134	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
135	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
136	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	106	2	no	—
139	C₈	Q₈×D₅₃	1	2	yes	—
140	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	4	4	no	—
143	C₄	—	106	4	no	—
144	C₄	—	106	2	no	—
145	C₄	—	106	2	no	—
146	C₄	—	212	2	no	—
147	C₄	#13133236	1	2	yes	—
148	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
149	C₄	#13133265	1	1	yes	characteristic
150	C₂	#16811288	1	1	yes	characteristic
151	C₂	C₂×#13133236	1	1	yes	characteristic
152	C₂	#16811436	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811348

C₂

yes

maximal

#16811024

C₂

yes

maximal

#16810929

C₂

yes

maximal

#16810905

C₂

yes

maximal

C₈×Dic₁₀₆

C₂

yes

maximal, characteristic

C₄×Dic₂₁₂

C₂

yes

maximal, characteristic

#16810717

C₂

yes

maximal

C₅₃×#500

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₂²

yes

characteristic

#13133215

—

#13133208

C₂²

yes

—

#13133207

—

Q₈×Dic₅₃

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

C₄×Dic₁₀₆

C₂²

yes

—

#13133182

C₂²

yes

—

#13133151

C₂²

yes

characteristic

#13133148

C₂²

yes

—

C₄⋊#88322

C₂²

yes

characteristic

#13133146

C₂²

yes

—

#13133142

C₂²

yes

—

#13133140

C₂²

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

#107×C₅₃

C₂²

yes

—

#107×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Dic₁₀₆

D₄

yes

characteristic

C₂×Dic₁₀₆

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

C₂³

yes

—

Dic₂₁₂

Q₈

yes

complemented

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

C₂×Q₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₂×Q₈

yes

—

#88322

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#135

yes

—

C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#3383

—

maximal, Sylow

C₂×C₅₃

D₄×Q₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₅₃

#3383

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

—

#125

—

#125

—

106

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

106

—

#107

—

#107

C₂×D₅₃

yes

—

#107

C₂×D₅₃

yes

—

#106

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

#103

—

102

#103

—

103

C₄×C₈

—

104

C₄×C₈

C₂×D₅₃

yes

characteristic

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

Q₁₆

—

109

C₂×C₈

—

110

C₂×C₈

C₂²×D₅₃

yes

characteristic

111

C₂×C₈

C₂²×D₅₃

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

106

—

117

C₄⋊C₄

—

106

—

118

C₄⋊C₄

—

106

—

119

C₄⋊C₄

—

106

—

120

C₄⋊C₄

C₂²×D₅₃

yes

—

121

C₄²

—

122

C₄²

—

106

—

123

C₄²

—

106

—

124

C₄²

C₂²×D₅₃

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

106

—

128

Q₈

—

106

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

106

—

133

C₂×C₄

—

106

—

134

C₂×C₄

D₄×D₅₃

yes

characteristic

135

C₂×C₄

D₄×D₅₃

yes

characteristic

136

C₂×C₄

C₂³×D₅₃

yes

characteristic

137

C₈

—

138

C₈

—

106

—

139

C₈

Q₈×D₅₃

yes

—

140

C₂²

C₂×D₄×D₅₃

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

106

—

144

C₄

—

106

—

145

C₄

—

106

—

146

C₄

—

212

—

147

C₄

#13133236

yes

—

148

C₄

C₂×Q₈×D₅₃

yes

characteristic

149

C₄

#13133265

yes

characteristic

150

C₂

#16811288

yes

characteristic

151

C₂

C₂×#13133236

yes

characteristic

152

C₂

#16811436

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#500	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#500	D₅₃
hypocenter	C₅₃	#3383
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#500

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
47. Dic₂₁₂	Q₈	semidirect
85. C₅₃	#3383	semidirect

H ≅ G/N

type

47. Dic₂₁₂

Q₈

semidirect

85. C₅₃

#3383

semidirect

Subgroups — #20287608

Conjugacy classes of subgroups

Complements