Group #27300490 — subgroups

Subgroups — #27300490

Order 13568.

234 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20276314	C₂	1	1	yes	maximal, characteristic
2	#20274840	C₂	1	2	yes	maximal
3	#20272817	C₂	1	1	yes	maximal, characteristic
4	#20272812	C₂	1	2	yes	maximal
5	C₅₃×#1385	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	1	yes	characteristic
7	#16810661	C₂²	1	2	yes	—
8	#16810578	C₂²	1	1	yes	characteristic
9	#16810398	C₄	1	2	yes	complemented
10	C₁₆×Dic₅₃	C₄	1	2	yes	complemented
11	#16810395	—	2	2	no	—
12	#16810380	—	2	2	no	—
13	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#108×C₅₃	C₂²	1	2	yes	—
15	C₂×#11719917	D₄	1	1	yes	characteristic
16	#13133176	—	2	2	no	—
17	#13133176	D₄	1	1	yes	characteristic
18	C₂×#11719908	—	2	2	no	—
19	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
20	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
21	#13133151	—	2	2	no	—
22	#13133149	—	2	2	no	—
23	#13133127	Q₈	1	2	yes	—
24	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₅₃	—	2	2	no	—
26	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	M₃₂×C₅₃	—	2	2	no	—
28	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
29	#108×C₅₃	—	2	2	no	—
30	#108×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	M₁₆×C₅₃	—	4	2	no	—
33	M₁₆×C₅₃	—	4	2	no	—
34	C₂×C₈×C₅₃	—	4	2	no	—
35	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₅₃	#41	1	2	yes	—
38	C₂×C₈×C₅₃	#41	1	2	yes	—
39	C₂²⋊Dic₅₃	—	2	1	no	—
40	C₂²⋊Dic₅₃	—	4	2	no	—
41	C₄⋊Dic₅₃	—	2	1	no	—
42	C₄⋊Dic₅₃	—	4	2	no	—
43	C₄×Dic₅₃	—	4	2	no	—
44	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
45	#11719917	—	2	2	no	—
46	C₂×#88322	C₄⋊C₄	1	2	yes	—
47	C₁₆×C₅₃	—	2	2	no	—
48	#11719908	—	2	1	no	—
49	#11719908	—	2	1	no	—
50	C₂³×C₅₃	#135	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#124	1	2	yes	—
53	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#135	1	2	yes	—
55	C₂×Dic₅₃	—	2	2	no	—
56	C₂×Dic₅₃	—	8	2	no	—
57	C₈×C₅₃	—	8	2	no	—
58	C₈×C₅₃	#117	1	1	yes	characteristic
59	C₈×C₅₃	#117	1	1	yes	characteristic
60	C₈×C₅₃	#125	1	2	yes	—
61	#88322	—	2	2	no	—
62	#12893	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	—	4	1	no	—
64	C₂²×C₅₃	#427	1	1	yes	characteristic
65	C₂²×C₅₃	#488	1	2	yes	—
66	C₄×C₅₃	#382	1	1	yes	characteristic
67	C₄×C₅₃	#427	1	1	yes	characteristic
68	C₄×C₅₃	#488	1	2	yes	—
69	Dic₅₃	—	4	2	no	—
70	Dic₅₃	—	16	2	no	—
71	#2108	—	53	1	no	—
72	#1833	—	53	2	no	—
73	#1385	D₅₃	1	1	yes	characteristic
74	#1384	—	53	1	no	—
75	#1382	—	53	2	no	—
76	C₂×C₅₃	—	2	1	no	—
77	C₂×C₅₃	—	2	2	no	—
78	C₂×C₅₃	#1835	1	1	yes	characteristic
79	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
80	C₂²×C₁₆	—	53	1	no	—
81	C₂×#108	C₂×D₅₃	1	2	yes	—
82	#427	—	53	2	no	—
83	#405	—	53	1	no	—
84	#346	—	53	2	no	—
85	C₄×C₁₆	—	53	2	no	—
86	#344	—	106	2	no	—
87	H(3,ℤ/4)	—	106	2	no	—
88	C₅₃	#12893	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	—	53	1	no	—
91	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
92	#117	—	53	1	no	—
93	#117	—	106	2	no	—
94	M₃₂	—	2	2	no	—
95	C₂×C₁₆	—	53	1	no	—
96	C₂×C₁₆	—	53	1	no	—
97	C₂×C₁₆	—	106	2	no	—
98	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
99	#108	—	2	2	no	—
100	#108	—	2	2	no	—
101	#107	—	106	2	no	—
102	#106	—	106	2	no	—
103	#97	—	53	2	no	—
104	C₄×C₈	—	53	2	no	—
105	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
106	M₁₆	—	4	2	no	—
107	M₁₆	—	4	2	no	—
108	M₁₆	—	106	2	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	—	53	2	no	—
111	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
112	C₂×C₈	#11719945	1	2	yes	—
113	C₂×C₈	#11719947	1	2	yes	—
114	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
115	C₄⋊C₄	—	106	1	no	—
116	C₄⋊C₄	—	212	2	no	—
117	C₂²⋊C₄	—	106	1	no	—
118	C₂²⋊C₄	—	212	2	no	—
119	C₄²	—	53	2	no	—
120	C₄²	—	212	2	no	—
121	C₁₆	—	2	2	no	—
122	C₁₆	—	106	1	no	—
123	C₁₆	—	106	1	no	—
124	C₂³	#13133240	1	1	yes	characteristic
125	C₂×C₄	—	106	2	no	—
126	C₂×C₄	—	424	2	no	—
127	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
128	C₂×C₄	#13133203	1	2	yes	—
129	C₂×C₄	#13133236	1	1	yes	characteristic
130	C₂×C₄	#13133238	1	2	yes	—
131	C₈	—	8	2	no	—
132	C₈	—	106	2	no	—
133	C₈	#13133173	1	1	yes	characteristic
134	C₈	#13133174	1	1	yes	characteristic
135	C₈	#13133207	1	2	yes	—
136	C₂²	—	4	1	no	—
137	C₂²	#16810660	1	1	yes	characteristic
138	C₂²	#16810979	1	2	yes	—
139	C₄	—	848	2	no	—
140	C₄	—	212	2	no	—
141	C₄	#16810487	1	1	yes	characteristic
142	C₄	#16810658	1	1	yes	characteristic
143	C₄	#16810976	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	—	2	1	no	—
146	C₂	#20274844	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276314

C₂

yes

maximal, characteristic

#20274840

C₂

yes

maximal

#20272817

C₂

yes

maximal, characteristic

#20272812

C₂

yes

maximal

C₅₃×#1385

C₂

yes

maximal, characteristic

C₂²×#11719908

C₂²

yes

characteristic

#16810661

C₂²

yes

—

#16810578

C₂²

yes

characteristic

#16810398

C₄

yes

complemented

C₁₆×Dic₅₃

C₄

yes

complemented

#16810395

—

#16810380

—

C₂×M₃₂×C₅₃

C₂²

yes

characteristic

C₂×#108×C₅₃

C₂²

yes

—

C₂×#11719917

D₄

yes

characteristic

#13133176

—

#13133176

D₄

yes

characteristic

C₂×#11719908

—

C₂×#11719908

C₂×C₄

yes

characteristic

C₂×#11719908

C₂×C₄

yes

—

—

Q₈

yes

—

C₈×Dic₅₃

C₂×C₄

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

#108×C₅₃

—

#108×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₄⋊C₄

yes

—

#11719917

—

C₂×#88322

C₄⋊C₄

yes

—

C₁₆×C₅₃

—

#11719908

—

#11719908

—

C₂³×C₅₃

#135

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

#135

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

C₈×C₅₃

yes

—

#88322

—

#12893

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#427

yes

characteristic

C₂²×C₅₃

#488

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

Dic₅₃

—

#2108

—

#1833

—

#1385

D₅₃

yes

characteristic

#1384

—

#1382

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1835

yes

characteristic

C₂×M₃₂

C₂×D₅₃

yes

characteristic

C₂²×C₁₆

—

C₂×#108

C₂×D₅₃

yes

—

#427

—

#405

—

#346

—

C₄×C₁₆

—

#344

—

106

—

H(3,ℤ/4)

—

106

—

C₅₃

#12893

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²×D₅₃

yes

characteristic

#117

—

#117

—

106

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₄×D₅₃

yes

—

#108

—

100

#108

—

101

#107

—

106

—

102

#106

—

106

—

103

#97

—

104

C₄×C₈

—

105

C₂²×C₄

D₄×D₅₃

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

106

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

112

C₂×C₈

#11719945

yes

—

113

C₂×C₈

#11719947

yes

—

114

C₂×C₈

Q₈×D₅₃

yes

characteristic

115

C₄⋊C₄

—

106

—

116

C₄⋊C₄

—

212

—

117

C₂²⋊C₄

—

106

—

118

C₂²⋊C₄

—

212

—

119

C₄²

—

120

C₄²

—

212

—

121

C₁₆

—

122

C₁₆

—

106

—

123

C₁₆

—

106

—

124

C₂³

#13133240

yes

characteristic

125

C₂×C₄

—

106

—

126

C₂×C₄

—

424

—

127

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

128

C₂×C₄

#13133203

yes

—

129

C₂×C₄

#13133236

yes

characteristic

130

C₂×C₄

#13133238

yes

—

131

C₈

—

132

C₈

—

106

—

133

C₈

#13133173

yes

characteristic

134

C₈

#13133174

yes

characteristic

135

C₈

#13133207

yes

—

136

C₂²

—

137

C₂²

#16810660

yes

characteristic

138

C₂²

#16810979

yes

—

139

C₄

—

848

—

140

C₄

—

212

—

141

C₄

#16810487

yes

characteristic

142

C₄

#16810658

yes

characteristic

143

C₄

#16810976

yes

—

144

C₂

—

145

C₂

—

146

C₂

#20274844

yes

characteristic

	subgroup	G/H
center	C₄	#16810658
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1385	C₂
socle	C₂×C₅₃	#1835
prime radical	C₂³×C₅₃	#135
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1385	D₅₃
hypocenter	C₅₃	#12893
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#16810658

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1385

C₂

C₂×C₅₃

C₂³×C₅₃

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
9. #16810398	C₄	semidirect
10. C₁₆×Dic₅₃	C₄	semidirect
88. C₅₃	#12893	semidirect

H ≅ G/N

type

9. #16810398

C₄

semidirect

10. C₁₆×Dic₅₃

C₄

semidirect

88. C₅₃

#12893

semidirect

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

subgroup	index
#20276314	2
#20274840	2
#20272817	2
#20272812	2
C₅₃×#1385	2
#12893	53

subgroup

index

C₅₃×#1385

Subgroups — #27300490

Conjugacy classes of subgroups

Complements

Maximal subgroups