Group #27295930 — subgroups

Subgroups — #27295930

Order 13568.

212 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20276617	C₂	1	2	yes	maximal
2	C₈×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#20272693	C₂	1	2	yes	maximal
4	#20272633	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1367	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₀₆	C₂²	1	2	yes	—
7	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic
8	#16810441	—	2	2	no	—
9	C₄×#11719908	C₂²	1	1	yes	characteristic
10	C₄×#11719908	C₂²	1	1	yes	characteristic
11	C₁₆×Dic₅₃	—	2	2	no	—
12	#16810376	C₄	1	1	yes	characteristic
13	#16810355	C₄	1	2	yes	—
14	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
15	C₈²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₁₂	C₈	1	1	yes	characteristic
17	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
18	C₂×#11719908	—	2	1	no	—
19	C₂×#11719908	D₄	1	2	yes	—
20	#13133151	C₈	1	1	yes	characteristic
21	C₄⋊#88322	C₂×C₄	1	2	yes	—
22	#13133146	—	2	2	no	—
23	#13133142	C₈	1	2	yes	—
24	C₈×Dic₅₃	—	2	2	no	—
25	C₂×C₁₆×C₅₃	—	2	2	no	—
26	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
30	C₂×C₈×C₅₃	—	2	1	no	—
31	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
35	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
37	#11719919	—	2	2	no	—
38	C₄×Dic₅₃	—	2	2	no	—
39	C₂×#88322	—	2	2	no	—
40	Dic₂₁₂	C₁₆	1	4	yes	complemented
41	C₁₆×C₅₃	—	4	2	no	—
42	#11719908	—	2	2	no	—
43	#11719908	—	4	1	no	—
44	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
47	C₂×Dic₅₃	—	2	2	no	—
48	C₂×Dic₅₃	—	2	2	no	—
49	Dic₁₀₆	—	2	2	no	—
50	Dic₁₀₆	C₂×C₁₆	1	4	yes	—
51	C₈×C₅₃	—	2	1	no	—
52	C₈×C₅₃	—	2	1	no	—
53	C₈×C₅₃	—	4	1	no	—
54	C₈×C₅₃	C₂×C₁₆	1	2	yes	—
55	C₈×C₅₃	#137	1	2	yes	—
56	#88322	—	4	2	no	—
57	#12059	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	#441	1	1	yes	characteristic
61	C₄×C₅₃	#444	1	1	yes	characteristic
62	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
63	C₄×C₅₃	#504	1	1	yes	characteristic
64	Dic₅₃	—	2	4	no	—
65	Dic₅₃	—	4	2	no	—
66	Q₈×C₁₆	—	53	2	no	—
67	C₈×Q₁₆	—	53	1	no	—
68	#1367	D₅₃	1	1	yes	characteristic
69	Q₈⋊C₁₆	—	53	2	no	—
70	#1312	—	53	1	no	—
71	C₂×C₅₃	#1579	1	1	yes	characteristic
72	C₂×C₅₃	D₄×C₁₆	1	1	yes	characteristic
73	C₂×C₅₃	#2171	1	1	yes	characteristic
74	C₈×Q₈	—	53	2	no	—
75	C₄×Q₁₆	—	53	1	no	—
76	C₄⋊C₁₆	—	106	2	no	—
77	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
78	C₄×C₁₆	—	53	1	no	—
79	C₄×C₁₆	—	53	1	no	—
80	C₄×C₁₆	—	106	2	no	—
81	#335	—	53	1	no	—
82	Q₈⋊C₈	—	53	2	no	—
83	C₈²	C₂×D₅₃	1	1	yes	characteristic
84	C₅₃	#12059	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	—	53	1	no	—
86	C₄×Q₈	—	53	2	no	—
87	C₂×C₁₆	—	2	2	no	—
88	C₂×C₁₆	—	53	2	no	—
89	C₂×C₁₆	—	106	1	no	—
90	#107	—	53	1	no	—
91	C₄⋊C₈	—	53	2	no	—
92	C₄⋊C₈	—	106	2	no	—
93	#103	—	53	2	no	—
94	C₄×C₈	—	106	2	no	—
95	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
96	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
98	C₂×Q₈	—	53	2	no	—
99	Q₁₆	—	53	4	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	106	2	no	—
102	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
103	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
104	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
105	C₂×C₈	#11719949	1	1	yes	characteristic
106	C₄⋊C₄	—	53	2	no	—
107	C₄⋊C₄	—	106	2	no	—
108	C₄²	—	106	2	no	—
109	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
110	C₁₆	—	4	2	no	—
111	C₁₆	—	106	2	no	—
112	C₁₆	—	212	1	no	—
113	Q₈	—	53	4	no	—
114	Q₈	—	106	2	no	—
115	C₂×C₄	—	106	2	no	—
116	C₂×C₄	—	106	2	no	—
117	C₂×C₄	#13133179	1	1	yes	characteristic
118	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
119	C₂×C₄	#13133225	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	212	2	no	—
124	C₈	C₁₆×D₅₃	1	2	yes	—
125	C₈	#13133246	1	2	yes	—
126	C₂²	#16810688	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	106	4	no	—
129	C₄	—	212	2	no	—
130	C₄	#16810729	1	1	yes	characteristic
131	C₄	#16810744	1	1	yes	characteristic
132	C₄	C₂×C₁₆×D₅₃	1	1	yes	characteristic
133	C₄	#16811041	1	1	yes	characteristic
134	C₂	#20273533	1	1	yes	characteristic
135	C₂	#20276534	1	1	yes	characteristic
136	C₂	#20276556	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276617

C₂

yes

maximal

C₈×Dic₂₁₂

C₂

yes

maximal, characteristic

#20272693

C₂

yes

maximal

#20272633

C₂

yes

maximal, characteristic

C₅₃×#1367

C₂

yes

maximal, characteristic

C₈×Dic₁₀₆

C₂²

yes

—

C₄×Dic₂₁₂

C₄

yes

characteristic

#16810441

—

C₄×#11719908

C₂²

yes

characteristic

C₄×#11719908

C₂²

yes

characteristic

C₁₆×Dic₅₃

—

#16810376

C₄

yes

characteristic

#16810355

C₄

yes

—

C₅₃×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

C₈

yes

characteristic

C₄×Dic₁₀₆

C₂×C₄

yes

—

C₂×#11719908

—

C₂×#11719908

D₄

yes

—

#13133151

C₈

yes

characteristic

C₄⋊#88322

C₂×C₄

yes

—

#13133146

—

#13133142

C₈

yes

—

C₈×Dic₅₃

—

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

C₂×C₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄⋊Dic₅₃

C₂×C₈

yes

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

Dic₂₁₂

C₁₆

yes

complemented

C₁₆×C₅₃

—

#11719908

—

#11719908

—

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

#131

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

C₂×C₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₁₆

yes

—

C₈×C₅₃

#137

yes

—

#88322

—

#12059

—

maximal, Sylow

C₂²×C₅₃

C₈×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂²×C₁₆

yes

characteristic

C₄×C₅₃

#504

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Q₈×C₁₆

—

C₈×Q₁₆

—

#1367

D₅₃

yes

characteristic

Q₈⋊C₁₆

—

#1312

—

C₂×C₅₃

#1579

yes

characteristic

C₂×C₅₃

D₄×C₁₆

yes

characteristic

C₂×C₅₃

#2171

yes

characteristic

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

106

—

C₄⋊C₁₆

C₂×D₅₃

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

106

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12059

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

#103

—

C₄×C₈

—

106

—

C₄×C₈

C₄×D₅₃

yes

characteristic

C₄×C₈

C₄×D₅₃

yes

characteristic

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

Q₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

106

—

102

C₂×C₈

C₈×D₅₃

yes

characteristic

103

C₂×C₈

C₈×D₅₃

yes

characteristic

104

C₂×C₈

D₄×D₅₃

yes

characteristic

105

C₂×C₈

#11719949

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

106

—

108

C₄²

—

106

—

109

C₄²

C₂×C₄×D₅₃

yes

characteristic

110

C₁₆

—

111

C₁₆

—

106

—

112

C₁₆

—

212

—

113

Q₈

—

114

Q₈

—

106

—

115

C₂×C₄

—

106

—

116

C₂×C₄

—

106

—

117

C₂×C₄

#13133179

yes

characteristic

118

C₂×C₄

C₂×C₈×D₅₃

yes

characteristic

119

C₂×C₄

#13133225

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

212

—

124

C₈

C₁₆×D₅₃

yes

—

125

C₈

#13133246

yes

—

126

C₂²

#16810688

yes

characteristic

127

C₄

—

128

C₄

—

106

—

129

C₄

—

212

—

130

C₄

#16810729

yes

characteristic

131

C₄

#16810744

yes

characteristic

132

C₄

C₂×C₁₆×D₅₃

yes

characteristic

133

C₄

#16811041

yes

characteristic

134

C₂

#20273533

yes

characteristic

135

C₂

#20276534

yes

characteristic

136

C₂

#20276556

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1367	C₂
socle, prime radical	C₂²×C₅₃	C₈×D₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1367	D₅₃
hypocenter	C₅₃	#12059
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

subgroup

G/H

center

C₂×C₈

D₄×D₅₃

commutator

C₄×C₅₃

C₂²×C₁₆

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1367

C₂

socle, prime radical

C₂²×C₅₃

C₈×D₄

cosocle

C₄×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

N	H ≅ G/N	type
40. Dic₂₁₂	C₁₆	semidirect
84. C₅₃	#12059	semidirect

H ≅ G/N

type

40. Dic₂₁₂

C₁₆

semidirect

84. C₅₃

#12059

semidirect

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

subgroup	index
#20276617	2
C₈×Dic₂₁₂	2
#20272693	2
#20272633	2
C₅₃×#1367	2
#12059	53

subgroup

index

C₈×Dic₂₁₂

C₅₃×#1367

Subgroups — #27295930

Conjugacy classes of subgroups

Complements

Maximal subgroups