Group #28309872 — subgroups

Subgroups — #28309872

Order 13568.

220 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 96 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	D₅₃⋊#364	C₂	1	2	yes	maximal
2	#20289751	C₂	1	2	yes	maximal, complemented
3	#20289750	C₂	1	2	yes	maximal, complemented
4	#20276308	C₂	1	1	yes	maximal, characteristic
5	D₅₃⋊M₃₂	—	2	2	no	—
6	#16811546	—	2	2	no	—
7	#16811545	C₂²	1	4	yes	—
8	C₂×#13133156	C₄	1	1	yes	characteristic
9	C₂×#13133156	C₄	1	1	yes	characteristic
10	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
11	#16810399	C₄	1	1	yes	characteristic
12	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#346	C₄	1	1	yes	characteristic
15	#13133292	—	4	2	no	—
16	#13133291	—	2	4	no	—
17	D₅₃⋊C₁₆	—	4	2	no	—
18	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
20	C₄²×D₅₃	Q₈	1	1	yes	characteristic
21	#13133156	—	2	2	no	—
22	#13133156	C₈	1	4	yes	—
23	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
24	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
25	C₄×#88322	D₄	1	1	yes	characteristic
26	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
27	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₅₃	—	2	1	no	—
32	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₄×Dic₅₃	—	2	1	no	—
37	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
39	C₂×#88322	C₂×D₄	1	1	yes	characteristic
40	C₈×D₅₃	C₂×C₈	1	2	yes	—
41	C₈×D₅₃	M₁₆	1	2	yes	—
42	C₈×D₅₃	D₈	1	2	yes	—
43	C₈×D₅₃	Q₁₆	1	2	yes	—
44	#11719910	—	4	4	no	—
45	C₁₆×C₅₃	—	2	1	no	—
46	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
47	#11719908	—	2	1	no	—
48	#11719908	C₂×C₈	1	2	yes	—
49	C₂²×D₅₃	#108	1	1	yes	characteristic
50	C₂×C₄×C₅₃	—	2	1	no	—
51	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×Dic₅₃	—	2	1	no	—
53	C₂×Dic₅₃	#108	1	1	yes	characteristic
54	C₄×D₅₃	—	4	2	no	—
55	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
56	C₄×D₅₃	#107	1	2	yes	—
57	C₈×C₅₃	#107	1	2	yes	—
58	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
59	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
60	#88322	C₄⋊C₈	1	2	yes	—
61	#88322	C₂×D₈	1	1	yes	characteristic
62	#88322	C₂×Q₁₆	1	1	yes	characteristic
63	#12011	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
65	C₂×D₅₃	—	4	1	no	—
66	C₂×D₅₃	#335	1	2	yes	—
67	C₄×C₅₃	—	4	1	no	—
68	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
69	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
70	Dic₅₃	—	2	2	no	—
71	Dic₅₃	#335	1	2	yes	—
72	C₂×#364	—	53	2	no	—
73	C₄×M₃₂	—	53	1	no	—
74	#1373	—	53	2	no	—
75	#1373	—	53	2	no	—
76	C₂×C₅₃	—	2	1	no	—
77	C₂×C₅₃	C₂×#335	1	1	yes	characteristic
78	D₅₃	—	2	2	no	—
79	C₂×M₃₂	—	53	1	no	—
80	C₂×M₃₂	—	53	1	no	—
81	C₂×M₃₂	—	106	2	no	—
82	C₂×C₄×C₈	—	53	1	no	—
83	#364	—	53	4	no	—
84	#364	—	106	2	no	—
85	#346	—	53	1	no	—
86	#346	#6157	1	1	yes	characteristic
87	C₄×C₁₆	—	53	1	no	—
88	C₄×C₁₆	—	53	1	no	—
89	C₅₃	#12011	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	—	53	1	no	—
91	C₂²×C₈	—	53	1	no	—
92	C₂×C₄²	—	53	1	no	—
93	M₃₂	—	53	4	no	—
94	M₃₂	—	106	2	no	—
95	M₃₂	—	106	4	no	—
96	M₃₂	—	212	2	no	—
97	C₂×C₁₆	—	53	1	no	—
98	C₂×C₁₆	—	53	1	no	—
99	C₂×C₁₆	—	212	2	no	—
100	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
101	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
102	C₄×C₈	—	53	1	no	—
103	C₄×C₈	—	53	1	no	—
104	C₄×C₈	—	53	1	no	—
105	C₄×C₈	C₂×#6157	1	1	yes	characteristic
106	C₂²×C₄	—	53	1	no	—
107	C₂²×C₄	—	106	1	no	—
108	C₂×C₈	—	53	1	no	—
109	C₂×C₈	—	53	1	no	—
110	C₂×C₈	—	53	2	no	—
111	C₂×C₈	—	53	2	no	—
112	C₂×C₈	—	53	2	no	—
113	C₂×C₈	—	53	2	no	—
114	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
115	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
116	C₄²	—	53	1	no	—
117	C₄²	—	106	1	no	—
118	C₄²	C₄⋊#6157	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	53	2	no	—
121	C₁₆	—	106	1	no	—
122	C₁₆	—	212	4	no	—
123	C₁₆	D₅₃⋊C₈	1	2	yes	—
124	C₂³	—	53	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	53	1	no	—
127	C₂×C₄	—	53	2	no	—
128	C₂×C₄	—	53	2	no	—
129	C₂×C₄	—	106	1	no	—
130	C₂×C₄	—	212	2	no	—
131	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
132	C₈	—	53	2	no	—
133	C₈	—	53	1	no	—
134	C₈	—	53	1	no	—
135	C₈	#13133286	1	2	yes	—
136	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
137	C₈	D₅₃⋊M₁₆	1	1	yes	characteristic
138	C₂²	—	53	2	no	—
139	C₂²	—	212	1	no	—
140	C₂²	D₅₃⋊#108	1	1	yes	characteristic
141	C₄	—	4	1	no	—
142	C₄	—	53	2	no	—
143	C₄	—	106	2	no	—
144	C₄	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
145	C₄	C₂×#13133286	1	1	yes	characteristic
146	C₂	—	2	1	no	—
147	C₂	—	106	2	no	—
148	C₂	D₅₃⋊#335	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊#364

C₂

yes

maximal

#20289751

C₂

yes

maximal, complemented

#20289750

C₂

yes

maximal, complemented

#20276308

C₂

yes

maximal, characteristic

D₅₃⋊M₃₂

—

#16811546

—

#16811545

C₂²

yes

—

C₂×#13133156

C₄

yes

characteristic

C₂×#13133156

C₄

yes

characteristic

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810399

C₄

yes

characteristic

C₁₆×Dic₅₃

C₂²

yes

characteristic

C₁₆×Dic₅₃

C₂²

yes

characteristic

C₅₃×#346

C₄

yes

characteristic

#13133292

—

#13133291

—

D₅₃⋊C₁₆

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

D₄

yes

characteristic

C₄²×D₅₃

Q₈

yes

characteristic

#13133156

—

#13133156

C₈

yes

—

C₂×#11719908

C₂×C₄

yes

characteristic

C₂×#11719908

C₂×C₄

yes

characteristic

C₄×#88322

D₄

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₈×Dic₅₃

C₂³

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×Q₈

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂×#88322

C₂×D₄

yes

characteristic

C₈×D₅₃

C₂×C₈

yes

—

C₈×D₅₃

M₁₆

yes

—

C₈×D₅₃

D₈

yes

—

C₈×D₅₃

Q₁₆

yes

—

#11719910

—

C₁₆×C₅₃

—

C₁₆×C₅₃

C₂×C₈

yes

—

#11719908

—

#11719908

C₂×C₈

yes

—

C₂²×D₅₃

#108

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

#108

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

C₄⋊C₈

yes

—

C₄×D₅₃

#107

yes

—

C₈×C₅₃

#107

yes

—

C₈×C₅₃

C₂²×C₈

yes

characteristic

C₈×C₅₃

C₂×M₁₆

yes

characteristic

#88322

C₄⋊C₈

yes

—

#88322

C₂×D₈

yes

characteristic

#88322

C₂×Q₁₆

yes

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

#335

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₅₃

C₂×#107

yes

characteristic

Dic₅₃

—

Dic₅₃

#335

yes

—

C₂×#364

—

C₄×M₃₂

—

#1373

—

#1373

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#335

yes

characteristic

D₅₃

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

—

106

—

C₂×C₄×C₈

—

#364

—

#364

—

106

—

—

yes

C₄×C₁₆

—

C₄×C₁₆

—

C₅₃

#12011

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

M₃₂

—

106

—

M₃₂

—

106

—

M₃₂

—

212

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

212

—

100

C₂×C₁₆

C₂×#6157

yes

characteristic

101

C₂×C₁₆

C₂×#6157

yes

characteristic

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

C₂×#6157

yes

characteristic

106

C₂²×C₄

—

107

C₂²×C₄

—

106

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₄⋊#6157

yes

characteristic

115

C₂×C₈

C₂²×#6157

yes

characteristic

116

C₄²

—

117

C₄²

—

106

—

118

C₄²

C₄⋊#6157

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

—

106

—

122

C₁₆

—

212

—

123

C₁₆

D₅₃⋊C₈

yes

—

124

C₂³

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

106

—

130

C₂×C₄

—

212

—

131

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

#13133286

yes

—

136

C₈

C₂×D₅₃⋊C₈

yes

characteristic

137

C₈

D₅₃⋊M₁₆

yes

characteristic

138

C₂²

—

139

C₂²

—

212

—

140

C₂²

D₅₃⋊#108

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

106

—

144

C₄

D₅₃⋊(C₄⋊C₈)

yes

characteristic

145

C₄

C₂×#13133286

yes

characteristic

146

C₂

—

147

C₂

—

106

—

148

C₂

D₅₃⋊#335

yes

characteristic

	subgroup	G/H
center	C₄	C₂×#13133286
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₅₃×#346	C₄
socle	C₂×C₅₃	C₂×#335
prime radical	C₂²×D₅₃	#108
cosocle	C₈×Dic₅₃	C₂³
hypercenter	#346	#6157
hypocenter	C₅₃	#12011
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₄

C₂×#13133286

commutator

C₈×C₅₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₅₃×#346

C₄

C₂×C₅₃

C₂²×D₅₃

C₈×Dic₅₃

C₂³

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
2. #20289751	C₂	semidirect
3. #20289750	C₂	semidirect
89. C₅₃	#12011	semidirect

H ≅ G/N

type

2. #20289751

C₂

semidirect

3. #20289750

C₂

semidirect

89. C₅₃

#12011

semidirect

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

subgroup	index
D₅₃⋊#364	2
#20289751	2
#20289750	2
#20276308	2
#12011	53

subgroup

index

D₅₃⋊#364

Subgroups — #28309872

Conjugacy classes of subgroups

Complements

Maximal subgroups