Group #27298470 — subgroups

Subgroups — #27298470

Order 13568.

188 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20276623	C₂	1	1	yes	maximal, characteristic
2	#20274116	C₂	1	2	yes	maximal
3	#20272789	C₂	1	1	yes	maximal, characteristic
4	#20272773	C₂	1	2	yes	maximal
5	C₅₃×#2252	C₂	1	1	yes	maximal, characteristic
6	#16810915	—	2	2	no	—
7	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
8	C₄⋊#11719908	C₂²	1	1	yes	characteristic
9	#16810441	—	2	1	no	—
10	#16810398	—	2	1	no	—
11	#16810376	C₂²	1	2	yes	—
12	#16810372	—	2	2	no	—
13	C₅₃×#500	C₂²	1	2	yes	—
14	C₅₃×#366	—	2	1	no	—
15	C₅₃×#366	—	2	1	no	—
16	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
18	C₂×#11719908	—	2	1	no	—
19	#13133150	—	4	2	no	—
20	C₄⋊#88322	—	2	2	no	—
21	C₄⋊#88322	D₄	1	1	yes	characteristic
22	#13133146	—	2	1	no	—
23	#13133140	—	4	2	no	—
24	C₈×Dic₅₃	—	2	1	no	—
25	C₄⋊Q₈×C₅₃	—	2	2	no	—
26	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
28	#107×C₅₃	—	2	2	no	—
29	#107×C₅₃	—	2	2	no	—
30	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×Q₈×C₅₃	—	4	2	no	—
32	C₂×Dic₁₀₆	D₈	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	—	4	2	no	—
36	C₄⋊C₄×C₅₃	—	4	2	no	—
37	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
39	#11719919	—	2	1	no	—
40	C₄×Dic₅₃	—	2	1	no	—
41	C₂×#88322	—	2	1	no	—
42	C₂×#88322	—	4	2	no	—
43	C₁₆×C₅₃	—	2	1	no	—
44	C₁₆×C₅₃	Q₁₆	1	2	yes	—
45	#11719908	—	4	1	no	—
46	Q₈×C₅₃	—	16	2	no	—
47	C₂×C₄×C₅₃	—	8	2	no	—
48	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
51	C₂×Dic₅₃	—	2	1	no	—
52	C₂×Dic₅₃	—	2	1	no	—
53	Dic₁₀₆	—	2	2	no	—
54	C₈×C₅₃	—	2	1	no	—
55	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
56	C₈×C₅₃	#137	1	1	yes	characteristic
57	#88322	—	4	1	no	—
58	#88322	—	8	2	no	—
59	#12461	—	53	1	no	maximal, Sylow
60	C₂²×C₅₃	#466	1	1	yes	characteristic
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	—	16	2	no	—
63	C₄×C₅₃	#462	1	1	yes	characteristic
64	C₄×C₅₃	#473	1	1	yes	characteristic
65	C₄×C₅₃	#510	1	2	yes	—
66	Dic₅₃	—	4	1	no	—
67	Dic₅₃	—	4	1	no	—
68	#2252	D₅₃	1	1	yes	characteristic
69	#2183	—	53	1	no	—
70	#1676	—	53	2	no	—
71	#1372	—	53	1	no	—
72	#1365	—	53	2	no	—
73	C₂×C₅₃	#1674	1	1	yes	characteristic
74	C₂×C₅₃	#2219	1	1	yes	characteristic
75	C₂×C₅₃	#2221	1	1	yes	characteristic
76	#500	C₂×D₅₃	1	2	yes	—
77	#477	—	106	2	no	—
78	C₈×Q₈	—	53	1	no	—
79	#366	—	2	1	no	—
80	#366	—	2	1	no	—
81	C₄⋊C₁₆	—	53	1	no	—
82	C₄⋊C₁₆	—	106	1	no	—
83	#346	—	106	1	no	—
84	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
85	#335	—	53	2	no	—
86	#333	—	106	2	no	—
87	C₅₃	#12461	1	1	yes	characteristic, complemented, Sylow
88	C₄⋊Q₈	—	2	2	no	—
89	C₄×Q₈	—	53	1	no	—
90	C₂×C₁₆	—	106	1	no	—
91	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
92	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
93	#107	—	2	2	no	—
94	#107	—	2	2	no	—
95	#107	—	212	2	no	—
96	C₄⋊C₈	—	53	1	no	—
97	C₄⋊C₈	—	106	1	no	—
98	C₄⋊C₈	—	106	2	no	—
99	#103	—	212	2	no	—
100	C₄×C₈	—	106	1	no	—
101	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
102	C₂×Q₈	—	4	2	no	—
103	C₂×Q₈	—	53	1	no	—
104	C₂×C₈	—	106	1	no	—
105	C₂×C₈	—	212	2	no	—
106	C₂×C₈	#11719947	1	1	yes	characteristic
107	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	53	1	no	—
111	C₄⋊C₄	—	106	1	no	—
112	C₄²	—	106	1	no	—
113	C₄²	D₄×D₅₃	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	212	1	no	—
116	C₁₆	#11719925	1	2	yes	—
117	Q₈	—	16	2	no	—
118	Q₈	—	106	2	no	—
119	C₂×C₄	—	8	2	no	—
120	C₂×C₄	—	106	1	no	—
121	C₂×C₄	—	106	1	no	—
122	C₂×C₄	#13133188	1	1	yes	characteristic
123	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
124	C₂×C₄	#13133236	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	424	2	no	—
127	C₈	—	212	1	no	—
128	C₈	C₂×#11719925	1	1	yes	characteristic
129	C₈	#13133248	1	1	yes	characteristic
130	C₂²	#16810856	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	16	2	no	—
133	C₄	—	212	1	no	—
134	C₄	—	212	1	no	—
135	C₄	#16810830	1	1	yes	characteristic
136	C₄	#16810895	1	1	yes	characteristic
137	C₄	#16811065	1	2	yes	—
138	C₂	#20274106	1	1	yes	characteristic
139	C₂	#20276820	1	1	yes	characteristic
140	C₂	#20276832	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276623

C₂

yes

maximal, characteristic

#20274116

C₂

yes

maximal

#20272789

C₂

yes

maximal, characteristic

#20272773

C₂

yes

maximal

C₅₃×#2252

C₂

yes

maximal, characteristic

#16810915

—

C₈×Dic₁₀₆

C₂²

yes

characteristic

C₄⋊#11719908

C₂²

yes

—

—

C₂²

yes

—

#16810372

—

C₅₃×#500

C₂²

yes

—

C₅₃×#366

—

C₅₃×#366

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₄×Dic₁₀₆

D₄

yes

—

—

—

D₄

yes

characteristic

#13133146

—

#13133140

—

C₈×Dic₅₃

—

C₄⋊Q₈×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

D₄

yes

characteristic

#107×C₅₃

—

#107×C₅₃

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Dic₁₀₆

D₈

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

D₈

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₁₆×C₅₃

—

C₁₆×C₅₃

Q₁₆

yes

—

#11719908

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂×Q₁₆

yes

C₈×C₅₃

yes

—

—

—

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

Dic₅₃

—

#2252

D₅₃

yes

—

—

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×D₅₃

yes

—

#477

—

106

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

106

—

#346

—

106

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

#335

—

#333

—

106

—

C₅₃

#12461

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₂²⋊D₅₃

yes

characteristic

C₂×C₁₆

C₂²⋊D₅₃

yes

—

—

—

212

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

#103

—

212

—

100

C₄×C₈

—

106

—

101

C₄×C₈

C₂²×D₅₃

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×C₈

—

106

—

105

C₂×C₈

—

212

—

106

C₂×C₈

#11719947

yes

characteristic

107

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

106

—

112

C₄²

—

106

—

113

C₄²

D₄×D₅₃

yes

characteristic

114

C₁₆

—

115

C₁₆

—

212

—

116

C₁₆

#11719925

yes

—

117

Q₈

—

118

Q₈

—

106

—

119

C₂×C₄

—

120

C₂×C₄

—

106

—

121

C₂×C₄

—

106

—

122

C₂×C₄

#13133188

yes

characteristic

123

C₂×C₄

D₈×D₅₃

yes

characteristic

124

C₂×C₄

#13133236

yes

characteristic

125

C₈

—

126

C₈

—

424

—

127

C₈

—

212

—

128

C₈

C₂×#11719925

yes

characteristic

129

C₈

#13133248

yes

characteristic

130

C₂²

#16810856

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

212

—

134

C₄

—

212

—

135

C₄

#16810830

yes

characteristic

136

C₄

#16810895

yes

characteristic

137

C₄

#16811065

yes

—

138

C₂

#20274106

yes

characteristic

139

C₂

#20276820

yes

characteristic

140

C₂

#20276832

yes

characteristic

	subgroup	G/H
center	C₂²	#16810856
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2252	C₂
socle, prime radical	C₂²×C₅₃	#466
hypercenter	#2252	D₅₃
hypocenter	C₅₃	#12461
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#16810856

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#2252

C₂

C₂²×C₅₃

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
87. C₅₃	#12461	semidirect

H ≅ G/N

type

87. C₅₃

#12461

semidirect

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

subgroup	index
#20276623	2
#20274116	2
#20272789	2
#20272773	2
C₅₃×#2252	2
#12461	53

subgroup

index

C₅₃×#2252

Subgroups — #27298470

Conjugacy classes of subgroups

Complements

Maximal subgroups