Group #28299199 — subgroups

Subgroups — #28299199

Order 13568.

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

normal subgroups only

This group has 162 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	C₂×#16811552	C₂	1	1	yes	maximal, characteristic
2	D₅₃⋊#368	C₂	1	1	yes	maximal, characteristic
3	M₆₄×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133288	—	2	1	no	—
5	D₅₃⋊#108	—	2	1	no	—
6	#16811552	—	2	1	no	—
7	#16811552	—	2	1	no	—
8	#16811552	C₄	1	2	yes	—
9	#16811551	C₄	1	2	yes	—
10	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
11	#16810457	C₄	1	1	yes	characteristic
12	#16810456	—	2	1	no	—
13	C₃₂×D₅₃	—	2	1	no	—
14	C₅₃×M₆₄	C₄	1	1	yes	characteristic
15	C₂×#11719936	—	4	1	no	—
16	D₅₃⋊M₁₆	—	4	1	no	—
17	#13133288	—	2	2	no	—
18	#13133288	—	2	2	no	—
19	#13133288	—	4	1	no	—
20	#13133287	—	4	1	no	—
21	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
22	C₂×#11719908	Q₈	1	1	yes	characteristic
23	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic
25	C₁₆×D₅₃	D₄	1	1	yes	characteristic
26	C₁₆×D₅₃	D₄	1	1	yes	characteristic
27	#13133123	—	2	1	no	—
28	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₃₂×C₅₃	—	2	1	no	—
30	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
31	#11719940	—	8	1	no	—
32	C₂×#88324	—	8	1	no	—
33	#11719936	—	4	2	no	—
34	#11719936	—	4	2	no	—
35	#11719936	—	8	1	no	—
36	D₅₃⋊C₈	—	8	1	no	—
37	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×#88322	Q₁₆	1	1	yes	characteristic
39	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₈×D₅₃	SD₁₆	1	1	yes	characteristic
42	C₈×D₅₃	SD₁₆	1	1	yes	characteristic
43	C₁₆×C₅₃	C₄²	1	1	yes	characteristic
44	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	#11719908	C₄⋊C₄	1	1	yes	characteristic
46	#11719908	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×D₅₃	Q₃₂	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
49	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
50	C₄×D₅₃	#102	1	1	yes	characteristic
51	C₄×D₅₃	#102	1	1	yes	characteristic
52	C₄×D₅₃	SD₃₂	1	1	yes	characteristic
53	C₄×D₅₃	SD₃₂	1	1	yes	characteristic
54	#88324	—	8	2	no	—
55	#88324	—	8	2	no	—
56	C₈×C₅₃	#95	1	1	yes	characteristic
57	C₈×C₅₃	#106	1	1	yes	characteristic
58	#88322	#103	1	1	yes	characteristic
59	#88322	#103	1	1	yes	characteristic
60	#7479	—	53	1	no	maximal, Sylow
61	C₂²×C₅₃	#366	1	1	yes	characteristic
62	C₂×D₅₃	—	4	1	no	—
63	C₂×D₅₃	#358	1	1	yes	characteristic
64	C₂×D₅₃	#358	1	1	yes	characteristic
65	C₄×C₅₃	#340	1	1	yes	characteristic
66	C₄×C₅₃	#367	1	1	yes	characteristic
67	Dic₅₃	#357	1	1	yes	characteristic
68	Dic₅₃	#357	1	1	yes	characteristic
69	C₂×M₆₄	—	53	1	no	—
70	C₂×#368	—	53	1	no	—
71	C₂×#368	—	53	1	no	—
72	C₂×C₅₃	—	2	1	no	—
73	C₂×C₅₃	#1380	1	1	yes	characteristic
74	D₅₃	—	2	1	no	—
75	D₅₃	—	2	1	no	—
76	C₂²×C₁₆	—	53	1	no	—
77	C₂×#108	—	106	1	no	—
78	C₂×#108	—	106	1	no	—
79	M₆₄	—	53	1	no	—
80	M₆₄	—	106	1	no	—
81	M₆₄	#6157	1	1	yes	characteristic
82	C₂×C₃₂	—	106	1	no	—
83	#368	—	53	2	no	—
84	#368	—	53	2	no	—
85	#368	—	106	1	no	—
86	#368	—	106	1	no	—
87	C₅₃	#7479	1	1	yes	characteristic, complemented, Sylow
88	C₂×M₁₆	—	212	1	no	—
89	C₂×M₁₆	—	212	1	no	—
90	C₂²×C₈	—	53	1	no	—
91	C₂×C₁₆	—	53	1	no	—
92	C₂×C₁₆	—	53	1	no	—
93	C₂×C₁₆	—	53	1	no	—
94	C₂×C₁₆	—	53	1	no	—
95	C₂×C₁₆	—	53	1	no	—
96	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
97	#108	—	106	2	no	—
98	#108	—	106	2	no	—
99	#108	—	212	1	no	—
100	#108	—	212	1	no	—
101	C₃₂	—	2	1	no	—
102	C₃₂	—	106	1	no	—
103	C₂²×C₄	—	53	1	no	—
104	M₁₆	—	424	1	no	—
105	M₁₆	—	424	1	no	—
106	M₁₆	—	212	2	no	—
107	M₁₆	—	212	2	no	—
108	C₂×C₈	—	53	1	no	—
109	C₂×C₈	—	53	1	no	—
110	C₂×C₈	—	53	1	no	—
111	C₂×C₈	—	53	1	no	—
112	C₂×C₈	—	53	1	no	—
113	C₂×C₈	—	424	1	no	—
114	C₂×C₈	—	424	1	no	—
115	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
116	C₁₆	—	53	1	no	—
117	C₁₆	—	53	1	no	—
118	C₁₆	C₄×#6157	1	1	yes	characteristic
119	C₁₆	C₂²⋊#6157	1	1	yes	characteristic
120	C₂³	—	53	1	no	—
121	C₂×C₄	—	53	1	no	—
122	C₂×C₄	—	53	1	no	—
123	C₂×C₄	—	53	1	no	—
124	C₂×C₄	—	53	1	no	—
125	C₂×C₄	—	53	1	no	—
126	C₂×C₄	#13133286	1	1	yes	characteristic
127	C₈	—	53	1	no	—
128	C₈	—	53	1	no	—
129	C₈	—	424	2	no	—
130	C₈	—	424	2	no	—
131	C₈	(C₂×#6157)⋊C₄	1	1	yes	characteristic
132	C₈	#13133285	1	1	yes	characteristic
133	C₂²	—	53	1	no	—
134	C₂²	—	53	1	no	—
135	C₂²	—	212	1	no	—
136	C₂²	#16811549	1	1	yes	characteristic
137	C₄	—	53	1	no	—
138	C₄	—	53	1	no	—
139	C₄	#16811497	1	1	yes	characteristic
140	C₄	#16811550	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	106	1	no	—
143	C₂	—	106	1	no	—
144	C₂	#20289773	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811552

C₂

yes

maximal, characteristic

D₅₃⋊#368

C₂

yes

maximal, characteristic

M₆₄×D₅₃

C₂

yes

maximal, characteristic

C₂×#13133288

—

D₅₃⋊#108

—

#16811552

—

#16811552

—

#16811552

C₄

yes

—

#16811551

C₄

yes

—

C₂×C₁₆×D₅₃

C₂²

yes

characteristic

#16810457

C₄

yes

characteristic

#16810456

—

C₃₂×D₅₃

—

C₅₃×M₆₄

C₄

yes

characteristic

C₂×#11719936

—

D₅₃⋊M₁₆

—

—

—

—

—

C₂×C₈×D₅₃

D₄

yes

characteristic

C₂×#11719908

Q₈

yes

characteristic

C₁₆×D₅₃

C₂×C₄

yes

characteristic

C₁₆×D₅₃

C₂×C₄

yes

characteristic

C₁₆×D₅₃

D₄

yes

characteristic

C₁₆×D₅₃

D₄

yes

characteristic

#13133123

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₃₂×C₅₃

—

C₂×C₄×D₅₃

D₈

yes

—

—

—

—

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×#88322

Q₁₆

yes

characteristic

C₈×D₅₃

C₂²⋊C₄

yes

characteristic

C₈×D₅₃

C₂²⋊C₄

yes

characteristic

C₈×D₅₃

SD₁₆

yes

characteristic

C₈×D₅₃

SD₁₆

yes

characteristic

C₁₆×C₅₃

C₄²

yes

characteristic

C₁₆×C₅₃

C₂²⋊C₄

yes

characteristic

#11719908

C₄⋊C₄

yes

characteristic

#11719908

C₄⋊C₄

yes

characteristic

C₂²×D₅₃

Q₃₂

yes

characteristic

C₂×C₄×C₅₃

#107

yes

characteristic

C₂×Dic₅₃

D₁₆

yes

C₄×D₅₃

yes

C₄×D₅₃

yes

C₄×D₅₃

SD₃₂

yes

characteristic

C₄×D₅₃

SD₃₂

yes

characteristic

#88324

—

#88324

—

C₈×C₅₃

yes

C₈×C₅₃

yes

yes

yes

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

yes

C₂×D₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

yes

Dic₅₃

yes

C₂×M₆₄

—

C₂×#368

—

C₂×#368

—

C₂×C₅₃

—

C₂×C₅₃

#1380

yes

characteristic

D₅₃

—

D₅₃

—

C₂²×C₁₆

—

C₂×#108

—

106

—

C₂×#108

—

106

—

M₆₄

—

M₆₄

—

106

—

M₆₄

#6157

yes

characteristic

C₂×C₃₂

—

106

—

#368

—

#368

—

#368

—

106

—

#368

—

106

—

C₅₃

#7479

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

212

—

C₂×M₁₆

—

212

—

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#6157

yes

characteristic

#108

—

106

—

#108

—

106

—

#108

—

212

—

100

#108

—

212

—

101

C₃₂

—

102

C₃₂

—

106

—

103

C₂²×C₄

—

104

M₁₆

—

424

—

105

M₁₆

—

424

—

106

M₁₆

—

212

—

107

M₁₆

—

212

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

424

—

114

C₂×C₈

—

424

—

115

C₂×C₈

C₄⋊#6157

yes

characteristic

116

C₁₆

—

117

C₁₆

—

118

C₁₆

C₄×#6157

yes

characteristic

119

C₁₆

C₂²⋊#6157

yes

characteristic

120

C₂³

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#13133286

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

424

—

130

C₈

—

424

—

131

C₈

(C₂×#6157)⋊C₄

yes

characteristic

132

C₈

#13133285

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

212

—

136

C₂²

#16811549

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

#16811497

yes

characteristic

140

C₄

#16811550

yes

characteristic

141

C₂

—

142

C₂

—

106

—

143

C₂

—

106

—

144

C₂

#20289773

yes

characteristic

	subgroup	G/H
center	C₂	#20289773
commutator	C₁₆×C₅₃	C₄²
Frattini	C₂×C₁₆	C₂×#6157
Fitting	C₅₃×M₆₄	C₄
socle	C₂×C₅₃	#1380
prime radical	C₂²×D₅₃	Q₃₂
cosocle	C₂×C₁₆×D₅₃	C₂²
hypercenter	M₆₄	#6157
hypocenter	C₅₃	#7479
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

subgroup

G/H

center

C₂

#20289773

commutator

C₁₆×C₅₃

C₄²

Frattini

C₂×C₁₆

C₂×#6157

Fitting

C₅₃×M₆₄

C₄

socle

C₂×C₅₃

#1380

prime radical

C₂²×D₅₃

Q₃₂

cosocle

C₂×C₁₆×D₅₃

C₂²

M₆₄

C₅₃

G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

N	H ≅ G/N	type
87. C₅₃	#7479	semidirect

H ≅ G/N

type

87. C₅₃

#7479

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16811552	2
D₅₃⋊#368	2
M₆₄×D₅₃	2
#7479	53

subgroup

index

C₂×#16811552

D₅₃⋊#368

M₆₄×D₅₃

#7479

Subgroups — #28299199

Conjugacy classes of subgroups

Complements

Maximal subgroups