Group #27298901 — subgroups

Subgroups — #27298901

Order 13568.

250 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20276995	C₂	1	2	yes	maximal
2	#20276944	C₂	1	2	yes	maximal
3	#20274312	C₂	1	1	yes	maximal, characteristic
4	#20272633	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1713	C₂	1	1	yes	maximal, characteristic
6	#16811028	C₂²	1	1	yes	characteristic
7	#16811021	—	2	1	no	—
8	#16811003	C₂²	1	1	yes	characteristic
9	#16810974	—	2	1	no	—
10	#16810450	—	2	4	no	—
11	#16810426	—	2	4	no	—
12	C₄×#11719908	C₂²	1	2	yes	—
13	C₅₃×#500	—	2	1	no	—
14	C₅₃×#500	C₂²	1	2	yes	—
15	C₅₃×#494	—	2	1	no	—
16	C₈²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₁₂	—	2	2	no	—
18	#13133217	—	2	1	no	—
19	#13133217	—	2	1	no	—
20	C₂×#11719908	D₄	1	4	yes	—
21	#13133151	—	2	2	no	—
22	#13133149	—	4	2	no	—
23	#13133142	—	4	2	no	—
24	C₂×Q₁₆×C₅₃	—	4	2	no	—
25	C₄⋊Q₈×C₅₃	—	2	2	no	—
26	#107×C₅₃	—	2	2	no	—
27	#107×C₅₃	—	2	2	no	—
28	#107×C₅₃	—	4	2	no	—
29	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₅₃	—	4	2	no	—
33	C₂×Dic₁₀₆	—	4	1	no	—
34	C₂×Dic₁₀₆	—	4	1	no	—
35	Q₁₆×C₅₃	—	8	4	no	—
36	C₂×C₈×C₅₃	D₈	1	2	yes	—
37	C₂×C₈×C₅₃	D₈	1	2	yes	—
38	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
39	C₄⋊C₄×C₅₃	—	4	2	no	—
40	C₄⋊C₄×C₅₃	—	4	2	no	—
41	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₅₃	—	4	1	no	—
43	C₄⋊Dic₅₃	—	4	1	no	—
44	C₄⋊Dic₅₃	—	4	1	no	—
45	C₄⋊Dic₅₃	—	4	1	no	—
46	Dic₂₁₂	—	4	4	no	—
47	#11719908	—	2	4	no	—
48	Q₈×C₅₃	—	8	4	no	—
49	C₂×C₄×C₅₃	—	8	2	no	—
50	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₅₃	—	8	1	no	—
54	C₂×Dic₅₃	—	8	1	no	—
55	Dic₁₀₆	—	8	2	no	—
56	Dic₁₀₆	—	8	2	no	—
57	C₈×C₅₃	—	2	2	no	—
58	C₈×C₅₃	SD₃₂	1	4	yes	—
59	C₈×C₅₃	#137	1	4	yes	—
60	#12535	—	53	1	no	maximal, Sylow
61	C₂²×C₅₃	#493	1	1	yes	characteristic
62	C₄×C₅₃	—	16	2	no	—
63	C₄×C₅₃	#497	1	2	yes	—
64	C₄×C₅₃	C₂×SD₃₂	1	2	yes	—
65	C₄×C₅₃	#510	1	2	yes	—
66	Dic₅₃	—	16	1	no	—
67	Dic₅₃	—	16	1	no	—
68	#2254	—	53	2	no	—
69	#2241	—	53	2	no	—
70	#1713	D₅₃	1	1	yes	characteristic
71	#1707	—	53	1	no	—
72	#1312	—	53	1	no	—
73	C₂×C₅₃	#1722	1	1	yes	characteristic
74	C₂×C₅₃	#2248	1	1	yes	characteristic
75	C₂×C₅₃	#2251	1	1	yes	characteristic
76	#500	—	2	1	no	—
77	#500	—	53	1	no	—
78	#500	C₂×D₅₃	1	2	yes	—
79	#498	—	106	1	no	—
80	#494	—	2	1	no	—
81	#494	—	53	1	no	—
82	#487	—	106	1	no	—
83	#367	—	106	4	no	—
84	#358	—	106	4	no	—
85	C₄×C₁₆	—	53	2	no	—
86	C₈²	C₂×D₅₃	1	1	yes	characteristic
87	C₅₃	#12535	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₁₆	—	4	2	no	—
89	C₂×Q₁₆	—	106	2	no	—
90	C₄⋊Q₈	—	2	2	no	—
91	C₄⋊Q₈	—	106	1	no	—
92	C₄⋊Q₈	—	106	1	no	—
93	C₂×C₁₆	—	53	4	no	—
94	#107	—	2	2	no	—
95	#107	—	2	2	no	—
96	#107	—	4	2	no	—
97	#107	—	106	2	no	—
98	#106	—	212	2	no	—
99	#103	—	212	2	no	—
100	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
101	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
102	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
103	C₂×Q₈	—	4	2	no	—
104	C₂×Q₈	—	212	1	no	—
105	C₂×Q₈	—	212	1	no	—
106	Q₁₆	—	8	4	no	—
107	Q₁₆	—	212	4	no	—
108	C₂×C₈	#11719922	1	2	yes	—
109	C₂×C₈	#11719922	1	2	yes	—
110	C₂×C₈	D₄×D₅₃	1	2	yes	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	212	1	no	—
114	C₄⋊C₄	—	212	1	no	—
115	C₄⋊C₄	—	212	1	no	—
116	C₄⋊C₄	—	212	1	no	—
117	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
118	C₁₆	—	106	4	no	—
119	Q₈	—	8	4	no	—
120	Q₈	—	424	2	no	—
121	Q₈	—	424	2	no	—
122	C₂×C₄	—	8	2	no	—
123	C₂×C₄	—	424	1	no	—
124	C₂×C₄	—	424	1	no	—
125	C₂×C₄	#13133213	1	1	yes	characteristic
126	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
127	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	#13133161	1	4	yes	—
130	C₈	#13133249	1	4	yes	—
131	C₂²	#16810999	1	1	yes	characteristic
132	C₄	—	16	2	no	—
133	C₄	—	848	1	no	—
134	C₄	—	848	1	no	—
135	C₄	#16811016	1	2	yes	—
136	C₄	C₂×#13133161	1	2	yes	—
137	C₄	#16811064	1	2	yes	—
138	C₂	#20274387	1	1	yes	characteristic
139	C₂	#20276970	1	1	yes	characteristic
140	C₂	#20276984	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276995

C₂

yes

maximal

#20276944

C₂

yes

maximal

#20274312

C₂

yes

maximal, characteristic

#20272633

C₂

yes

maximal, characteristic

C₅₃×#1713

C₂

yes

maximal, characteristic

#16811028

C₂²

yes

characteristic

#16811021

—

#16811003

C₂²

yes

—

—

—

C₂²

yes

—

C₅₃×#500

—

C₅₃×#500

C₂²

yes

—

C₅₃×#494

—

C₈²×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

#13133217

—

#13133217

—

C₂×#11719908

D₄

yes

—

#13133151

—

#13133149

—

#13133142

—

C₂×Q₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

—

#107×C₅₃

—

#107×C₅₃

—

#107×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

Dic₂₁₂

—

#11719908

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

SD₃₂

yes

—

C₈×C₅₃

#137

yes

—

—

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

#497

yes

—

C₄×C₅₃

C₂×SD₃₂

yes

—

C₄×C₅₃

#510

yes

—

Dic₅₃

—

Dic₅₃

—

#2254

—

#2241

—

#1713

D₅₃

yes

characteristic

#1707

—

#1312

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

C₂×D₅₃

yes

—

#498

—

106

—

#494

—

#494

—

#487

—

106

—

#367

—

106

—

#358

—

106

—

C₄×C₁₆

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12535

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

C₂×C₁₆

—

—

—

—

—

106

—

#106

—

212

—

#103

—

212

—

100

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

101

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

102

C₄×C₈

C₂²×D₅₃

yes

characteristic

103

C₂×Q₈

—

104

C₂×Q₈

—

212

—

105

C₂×Q₈

—

212

—

106

Q₁₆

—

107

Q₁₆

—

212

—

108

C₂×C₈

#11719922

yes

—

109

C₂×C₈

#11719922

yes

—

110

C₂×C₈

D₄×D₅₃

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

212

—

114

C₄⋊C₄

—

212

—

115

C₄⋊C₄

—

212

—

116

C₄⋊C₄

—

212

—

117

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

118

C₁₆

—

106

—

119

Q₈

—

120

Q₈

—

424

—

121

Q₈

—

424

—

122

C₂×C₄

—

123

C₂×C₄

—

424

—

124

C₂×C₄

—

424

—

125

C₂×C₄

#13133213

yes

characteristic

126

C₂×C₄

C₂×#11719922

yes

characteristic

127

C₂×C₄

C₂×#11719922

yes

characteristic

128

C₈

—

129

C₈

#13133161

yes

—

130

C₈

#13133249

yes

—

131

C₂²

#16810999

yes

characteristic

132

C₄

—

133

C₄

—

848

—

134

C₄

—

848

—

135

C₄

#16811016

yes

—

136

C₄

C₂×#13133161

yes

—

137

C₄

#16811064

yes

—

138

C₂

#20274387

yes

characteristic

139

C₂

#20276970

yes

characteristic

140

C₂

#20276984

yes

characteristic

	subgroup	G/H
center	C₂²	#16810999
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1713	C₂
socle, prime radical	C₂²×C₅₃	#493
hypercenter	#1713	D₅₃
hypocenter	C₅₃	#12535
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#16810999

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1713

C₂

C₂²×C₅₃

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
87. C₅₃	#12535	semidirect

H ≅ G/N

type

87. C₅₃

#12535

semidirect

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

subgroup	index
#20276995	2
#20276944	2
#20274312	2
#20272633	2
C₅₃×#1713	2
#12535	53

subgroup

index

C₅₃×#1713

Subgroups — #27298901

Conjugacy classes of subgroups

Complements

Maximal subgroups