Group #20291093 — subgroups

Subgroups — #20291093

Order 6784.

174 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 74 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	C₂×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#11719910	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811544	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811518	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊#11719910	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811509	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810686	C₂	1	1	yes	maximal, characteristic
8	D₅₃⋊C₁₆	—	2	2	no	—
9	C₂×#11719910	—	2	2	no	—
10	C₂×#11719910	C₂²	1	1	yes	characteristic
11	C₂×#11719910	C₂²	1	1	yes	characteristic
12	C₂×#11719910	C₂²	1	1	yes	characteristic
13	C₂×#11719910	C₂²	1	1	yes	characteristic
14	C₂²×#88322	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊D₅₃	C₄	1	1	yes	characteristic
17	#13133146	C₄	1	1	yes	characteristic
18	#13133129	C₄	1	1	yes	characteristic
19	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
20	C₂²⋊C₈×C₅₃	C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₂²⋊D₅₃	C₈	1	1	yes	characteristic
23	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²⋊Dic₅₃	C₈	1	1	yes	characteristic
27	#11719921	C₈	1	1	yes	characteristic
28	#11719919	C₈	1	1	yes	characteristic
29	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×#88322	—	2	1	no	—
31	C₂×#88322	—	2	1	no	—
32	C₂×#88322	C₂×C₄	1	1	yes	characteristic
33	C₂×#88322	C₂³	1	1	yes	characteristic
34	C₈×D₅₃	—	2	1	no	—
35	C₈×D₅₃	—	2	1	no	—
36	#11719910	—	2	1	no	—
37	#11719910	—	2	1	no	—
38	#11719910	—	2	1	no	—
39	#11719910	D₄	1	2	yes	complemented
40	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×D₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₅₃	—	2	1	no	—
43	C₂×C₄×C₅₃	—	2	1	no	—
44	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²⋊D₅₃	C₁₆	1	4	yes	complemented
47	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
49	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
50	C₄×D₅₃	—	2	1	no	—
51	C₄×D₅₃	—	2	1	no	—
52	C₈×C₅₃	—	2	1	no	—
53	C₈×C₅₃	—	2	1	no	—
54	#88322	—	2	1	no	—
55	#88322	C₂×D₄	1	1	yes	characteristic
56	#88322	#41	1	1	yes	characteristic
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	—	2	1	no	—
59	C₂²×C₅₃	C₂×C₁₆	1	2	yes	complemented
60	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×D₅₃	—	2	1	no	—
62	C₂×D₅₃	—	2	1	no	—
63	C₂×D₅₃	C₂×C₁₆	1	2	yes	complemented
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
66	C₄×C₅₃	#131	1	1	yes	characteristic
67	Dic₅₃	—	2	1	no	—
68	Dic₅₃	—	2	1	no	—
69	Dic₅₃	C₂×C₁₆	1	2	yes	complemented
70	D₄×C₁₆	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	—	2	2	no	—
72	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
73	C₂×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
74	C₂×C₅₃	#504	1	1	yes	characteristic
75	D₅₃	—	2	2	no	—
76	C₂²×C₁₆	—	53	1	no	—
77	C₂²×C₁₆	—	53	1	no	—
78	C₈×D₄	—	53	1	no	—
79	C₄⋊C₁₆	—	53	1	no	—
80	C₂²⋊C₁₆	—	53	1	no	—
81	C₂²⋊C₁₆	—	53	1	no	—
82	C₄×C₁₆	—	53	1	no	—
83	C₅₃	D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	53	1	no	—
85	C₂²×C₈	—	53	1	no	—
86	C₄×D₄	—	53	1	no	—
87	C₂×C₁₆	—	53	1	no	—
88	C₂×C₁₆	—	53	1	no	—
89	C₂×C₁₆	—	53	1	no	—
90	C₂×C₁₆	—	53	1	no	—
91	C₂×C₁₆	—	106	2	no	—
92	C₂×C₁₆	—	106	2	no	—
93	C₄⋊C₈	—	53	1	no	—
94	C₂²⋊C₈	—	53	1	no	—
95	C₂²⋊C₈	#6157	1	1	yes	characteristic
96	C₄×C₈	—	53	1	no	—
97	C₂×D₄	—	53	1	no	—
98	C₂²×C₄	—	53	1	no	—
99	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
100	C₂×C₈	—	53	1	no	—
101	C₂×C₈	—	53	1	no	—
102	C₂×C₈	—	106	1	no	—
103	C₂×C₈	—	106	1	no	—
104	C₂×C₈	—	106	1	no	—
105	C₂×C₈	—	106	1	no	—
106	C₂×C₈	C₂×#6157	1	1	yes	characteristic
107	C₂×C₈	C₂×#6157	1	1	yes	characteristic
108	C₄⋊C₄	—	53	1	no	—
109	C₂²⋊C₄	—	53	1	no	—
110	C₂²⋊C₄	—	53	1	no	—
111	C₄²	—	53	1	no	—
112	C₁₆	—	53	2	no	—
113	C₁₆	—	106	1	no	—
114	C₁₆	—	106	1	no	—
115	C₁₆	—	106	1	no	—
116	C₂³	—	53	1	no	—
117	C₂³	D₅₃⋊C₈	1	1	yes	characteristic
118	D₄	—	53	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	53	1	no	—
122	C₂×C₄	—	53	1	no	—
123	C₂×C₄	—	53	1	no	—
124	C₂×C₄	—	106	1	no	—
125	C₂×C₄	—	106	1	no	—
126	C₂×C₄	D₅₃⋊C₈	1	1	yes	characteristic
127	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	2	1	no	—
130	C₈	—	53	1	no	—
131	C₈	—	53	1	no	—
132	C₈	—	106	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	53	2	no	—
136	C₂²	—	106	1	no	—
137	C₂²	—	106	1	no	—
138	C₂²	D₅₃⋊C₁₆	1	2	yes	complemented
139	C₂²	C₂×D₅₃⋊C₈	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	53	2	no	—
142	C₄	—	106	1	no	—
143	C₄	—	106	1	no	—
144	C₄	D₄×#6157	1	1	yes	characteristic
145	C₄	#13133304	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	—	106	2	no	—
148	C₂	#16811650	1	1	yes	characteristic
149	C₂	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
150	C₂	#16811682	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₅₃⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂²×#11719910

C₂

yes

maximal, characteristic, complemented

#16811544

C₂

yes

maximal, characteristic, complemented

#16811518

C₂

yes

maximal, characteristic, complemented

C₂²⋊#11719910

C₂

yes

maximal, characteristic, complemented

#16811509

C₂

yes

maximal, characteristic, complemented

#16810686

C₂

yes

maximal, characteristic

D₅₃⋊C₁₆

—

C₂×#11719910

—

C₂×#11719910

C₂²

yes

characteristic

C₂×#11719910

C₂²

yes

characteristic

C₂×#11719910

C₂²

yes

characteristic

C₂×#11719910

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

C₄×C₂²⋊D₅₃

C₄

yes

characteristic

#13133146

C₄

yes

characteristic

#13133129

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

C₂²⋊C₈×C₅₃

C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₅₃

C₈

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²⋊Dic₅₃

C₈

yes

characteristic

#11719921

C₈

yes

characteristic

#11719919

C₈

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

—

—

C₂×C₄

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

—

—

—

D₄

yes

complemented

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂²×D₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²⋊D₅₃

C₁₆

yes

complemented

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

C₂×D₄

yes

yes

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₁₆

yes

complemented

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂×C₁₆

yes

complemented

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

#131

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×C₁₆

yes

complemented

D₄×C₁₆

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₈×D₄

yes

characteristic

C₂×C₅₃

C₂²×C₁₆

yes

characteristic

C₂×C₅₃

#504

yes

characteristic

D₅₃

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₅₃

D₄×C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₄×D₄

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

—

106

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

#6157

yes

characteristic

C₄×C₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#6157

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

106

—

103

C₂×C₈

—

106

—

104

C₂×C₈

—

106

—

105

C₂×C₈

—

106

—

106

C₂×C₈

C₂×#6157

yes

characteristic

107

C₂×C₈

C₂×#6157

yes

characteristic

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₁₆

—

113

C₁₆

—

106

—

114

C₁₆

—

106

—

115

C₁₆

—

106

—

116

C₂³

—

117

C₂³

D₅₃⋊C₈

yes

characteristic

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

106

—

125

C₂×C₄

—

106

—

126

C₂×C₄

D₅₃⋊C₈

yes

characteristic

127

C₂×C₄

C₂²×#6157

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

106

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

106

—

137

C₂²

—

106

—

138

C₂²

D₅₃⋊C₁₆

yes

complemented

139

C₂²

C₂×D₅₃⋊C₈

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

106

—

143

C₄

—

106

—

144

C₄

D₄×#6157

yes

characteristic

145

C₄

#13133304

yes

characteristic

146

C₂

—

147

C₂

—

106

—

148

C₂

#16811650

yes

characteristic

149

C₂

C₂×D₅₃⋊C₁₆

yes

characteristic

150

C₂

#16811682

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₂²×C₁₆
Fitting	C₂²⋊C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₈
hypercenter	C₂²⋊C₈	#6157
hypocenter	C₅₃	D₄×C₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#6157

commutator

C₂×C₅₃

C₂²×C₁₆

Fitting

C₂²⋊C₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂²×C₈

hypercenter

C₂²⋊C₈

#6157

hypocenter

C₅₃

D₄×C₁₆

LCS

G ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
1. C₂×D₅₃⋊C₁₆	C₂	semidirect
2. C₂²×#11719910	C₂	semidirect
3. #16811544	C₂	semidirect
4. #16811518	C₂	semidirect
5. C₂²⋊#11719910	C₂	semidirect
6. #16811509	C₂	semidirect
39. #11719910	D₄	semidirect
46. C₂²⋊D₅₃	C₁₆	semidirect
59. C₂²×C₅₃	C₂×C₁₆	semidirect
63. C₂×D₅₃	C₂×C₁₆	semidirect
69. Dic₅₃	C₂×C₁₆	semidirect
83. C₅₃	D₄×C₁₆	semidirect
138. C₂²	D₅₃⋊C₁₆	semidirect

H ≅ G/N

type

1. C₂×D₅₃⋊C₁₆

C₂

semidirect

2. C₂²×#11719910

C₂

semidirect

3. #16811544

C₂

semidirect

4. #16811518

C₂

semidirect

5. C₂²⋊#11719910

C₂

semidirect

6. #16811509

C₂

semidirect

39. #11719910

D₄

semidirect

46. C₂²⋊D₅₃

C₁₆

semidirect

59. C₂²×C₅₃

C₂×C₁₆

semidirect

63. C₂×D₅₃

C₂×C₁₆

semidirect

69. Dic₅₃

C₂×C₁₆

semidirect

83. C₅₃

D₄×C₁₆

semidirect

138. C₂²

D₅₃⋊C₁₆

semidirect

Subgroups — #20291093

Conjugacy classes of subgroups

Complements