Group #20276827 — subgroups

Subgroups — #20276827

Order 6784.

160 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 47 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×#13133161	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133156	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810865	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810856	C₂	1	1	yes	maximal, characteristic
5	#16810448	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810428	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#357	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	#13133197	—	2	1	no	—
11	#13133188	—	2	1	no	—
12	#13133161	—	2	2	no	—
13	#13133156	—	2	2	no	—
14	C₂×#11719908	C₂²	1	1	yes	characteristic
15	#13133151	C₂²	1	1	yes	characteristic
16	#13133140	—	2	1	no	—
17	#13133138	—	2	1	no	—
18	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	#107×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×D₄×C₅₃	—	2	1	no	—
22	C₂×C₂²⋊D₅₃	—	4	1	no	—
23	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
24	C₂×Dic₁₀₆	—	2	1	no	—
25	D₈×C₅₃	—	2	1	no	—
26	D₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₂²⋊Dic₅₃	—	4	1	no	—
30	#11719921	—	4	1	no	—
31	C₄⋊Dic₅₃	—	2	1	no	—
32	C₄⋊Dic₅₃	—	4	1	no	—
33	C₂×#88322	D₄	1	1	yes	characteristic
34	Dic₂₁₂	—	2	1	no	—
35	Dic₂₁₂	—	2	1	no	—
36	C₈×D₅₃	—	2	1	no	—
37	C₈×D₅₃	—	2	1	no	—
38	C₁₆×C₅₃	—	2	1	no	—
39	#11719908	D₄	1	2	yes	complemented
40	C₂³×C₅₃	—	4	1	no	—
41	C₂²×D₅₃	D₈	1	1	yes	characteristic
42	D₄×C₅₃	—	4	1	no	—
43	D₄×C₅₃	—	4	1	no	—
44	C₂×C₄×C₅₃	—	4	1	no	—
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²⋊D₅₃	—	8	2	no	—
47	C₂×Dic₅₃	—	4	1	no	—
48	C₂×Dic₅₃	—	4	1	no	—
49	C₂×Dic₅₃	D₈	1	1	yes	characteristic
50	C₄×D₅₃	—	2	1	no	—
51	C₄×D₅₃	—	2	1	no	—
52	Dic₁₀₆	—	4	1	no	—
53	Dic₁₀₆	—	4	1	no	—
54	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₈×C₅₃	#41	1	1	yes	characteristic
56	#88322	—	2	1	no	—
57	C₂²×C₅₃	—	8	1	no	—
58	C₂²×C₅₃	—	8	1	no	—
59	C₂²×C₅₃	—	8	1	no	—
60	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
61	C₂×D₅₃	—	2	1	no	—
62	C₂×D₅₃	—	2	1	no	—
63	C₂×D₅₃	—	2	1	no	—
64	C₄×C₅₃	—	8	1	no	—
65	C₄×C₅₃	#121	1	1	yes	characteristic
66	C₄×C₅₃	#135	1	1	yes	characteristic
67	Dic₅₃	—	2	1	no	—
68	Dic₅₃	—	8	1	no	—
69	Dic₅₃	—	8	1	no	—
70	#2220	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	—	16	1	no	—
72	C₂×C₅₃	#466	1	1	yes	characteristic
73	C₂×C₅₃	#509	1	1	yes	characteristic
74	C₂×C₅₃	#510	1	1	yes	characteristic
75	D₅₃	—	4	1	no	—
76	C₂×SD₃₂	—	53	1	no	—
77	C₂×M₃₂	—	53	1	no	—
78	#467	—	53	1	no	—
79	#466	—	53	1	no	—
80	#366	—	53	1	no	—
81	#358	—	53	1	no	—
82	#357	D₅₃	1	1	yes	characteristic, complemented
83	C₅₃	#2220	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₁₆	—	53	1	no	—
85	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic
86	C₂²×C₈	—	53	1	no	—
87	C₂²⋊Q₈	—	106	1	no	—
88	#121	—	106	1	no	—
89	SD₃₂	—	106	2	no	—
90	M₃₂	—	106	2	no	—
91	C₂×C₁₆	—	53	1	no	—
92	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
93	#107	—	53	1	no	—
94	#107	C₂×D₅₃	1	1	yes	characteristic
95	#103	—	106	1	no	—
96	#102	—	106	1	no	—
97	C₂×Q₈	—	106	1	no	—
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	—	212	1	no	—
100	C₂²×C₄	—	53	1	no	—
101	Q₁₆	—	106	1	no	—
102	Q₁₆	—	106	1	no	—
103	D₈	—	2	1	no	—
104	D₈	—	2	1	no	—
105	C₂×C₈	—	53	1	no	—
106	C₂×C₈	—	106	1	no	—
107	C₂×C₈	—	106	1	no	—
108	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	106	1	no	—
111	C₄⋊C₄	—	212	1	no	—
112	C₂²⋊C₄	—	212	1	no	—
113	C₂²⋊C₄	—	212	1	no	—
114	C₁₆	—	2	1	no	—
115	C₁₆	—	53	2	no	—
116	C₂³	—	4	1	no	—
117	C₂³	—	53	1	no	—
118	Q₈	—	212	1	no	—
119	Q₈	—	212	1	no	—
120	D₄	—	4	1	no	—
121	D₄	—	4	1	no	—
122	D₄	—	424	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	53	1	no	—
125	C₂×C₄	—	106	1	no	—
126	C₂×C₄	—	106	1	no	—
127	C₂×C₄	—	212	1	no	—
128	C₂×C₄	—	212	1	no	—
129	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
130	C₈	—	106	1	no	—
131	C₈	#11719945	1	1	yes	characteristic
132	C₈	D₄×D₅₃	1	1	yes	characteristic
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	106	1	no	—
137	C₂²	—	106	1	no	—
138	C₂²	—	106	1	no	—
139	C₂²	D₈×D₅₃	1	1	yes	characteristic
140	C₄	—	8	1	no	—
141	C₄	—	106	1	no	—
142	C₄	—	424	1	no	—
143	C₄	—	424	1	no	—
144	C₄	#13133190	1	1	yes	characteristic
145	C₄	#13133238	1	1	yes	characteristic
146	C₂	—	16	1	no	—
147	C₂	—	212	1	no	—
148	C₂	#16810858	1	1	yes	characteristic
149	C₂	#16811057	1	1	yes	characteristic
150	C₂	#16811062	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133161

C₂

yes

maximal, characteristic, complemented

C₂×#13133156

C₂

yes

maximal, characteristic, complemented

#16810865

C₂

yes

maximal, characteristic, complemented

#16810856

C₂

yes

maximal, characteristic

#16810448

C₂

yes

maximal, characteristic, complemented

#16810428

C₂

yes

maximal, characteristic, complemented

C₅₃×#357

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₂

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

#13133151

C₂²

yes

characteristic

#13133140

—

#13133138

—

C₂×D₈×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

#107×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

—

C₂×C₂²⋊D₅₃

—

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊Dic₅₃

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

D₄

yes

characteristic

Dic₂₁₂

—

Dic₂₁₂

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

D₄

yes

complemented

C₂³×C₅₃

—

C₂²×D₅₃

D₈

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

D₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

C₂×D₄

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2220

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂×SD₃₂

—

C₂×M₃₂

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#2220

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

C₂²⋊Q₈

—

106

—

#121

—

106

—

SD₃₂

—

106

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

#107

—

#107

C₂×D₅₃

yes

characteristic

#103

—

106

—

#102

—

106

—

C₂×Q₈

—

106

—

C₂×D₄

—

C₂×D₄

—

212

—

100

C₂²×C₄

—

101

Q₁₆

—

106

—

102

Q₁₆

—

106

—

103

D₈

—

104

D₈

—

105

C₂×C₈

—

106

C₂×C₈

—

106

—

107

C₂×C₈

—

106

—

108

C₂×C₈

C₂²×D₅₃

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

106

—

111

C₄⋊C₄

—

212

—

112

C₂²⋊C₄

—

212

—

113

C₂²⋊C₄

—

212

—

114

C₁₆

—

115

C₁₆

—

116

C₂³

—

117

C₂³

—

118

Q₈

—

212

—

119

Q₈

—

212

—

120

D₄

—

121

D₄

—

122

D₄

—

424

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

106

—

126

C₂×C₄

—

106

—

127

C₂×C₄

—

212

—

128

C₂×C₄

—

212

—

129

C₂×C₄

D₄×D₅₃

yes

characteristic

130

C₈

—

106

—

131

C₈

#11719945

yes

characteristic

132

C₈

D₄×D₅₃

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

106

—

137

C₂²

—

106

—

138

C₂²

—

106

—

139

C₂²

D₈×D₅₃

yes

characteristic

140

C₄

—

141

C₄

—

106

—

142

C₄

—

424

—

143

C₄

—

424

—

144

C₄

#13133190

yes

characteristic

145

C₄

#13133238

yes

characteristic

146

C₂

—

147

C₂

—

212

—

148

C₂

#16810858

yes

characteristic

149

C₂

#16811057

yes

characteristic

150

C₂

#16811062

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#357	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#357	D₅₃
hypocenter	C₅₃	#2220
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₅₃

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#357

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. C₂×#13133161	C₂	semidirect
2. C₂×#13133156	C₂	semidirect
3. #16810865	C₂	semidirect
5. #16810448	C₂	semidirect
6. #16810428	C₂	semidirect
7. C₅₃×#357	C₂	semidirect
39. #11719908	D₄	semidirect
82. #357	D₅₃	semidirect
83. C₅₃	#2220	semidirect

H ≅ G/N

type

1. C₂×#13133161

C₂

semidirect

2. C₂×#13133156

C₂

semidirect

3. #16810865

C₂

semidirect

5. #16810448

C₂

semidirect

6. #16810428

C₂

semidirect

7. C₅₃×#357

C₂

semidirect

39. #11719908

D₄

semidirect

82. #357

D₅₃

semidirect

83. C₅₃

#2220

semidirect

Subgroups — #20276827

Conjugacy classes of subgroups

Complements