Group #27296083 — subgroups

Subgroups — #27296083

Order 13568.

210 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 85 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	#20273612	C₂	1	2	yes	maximal
2	#20272787	C₂	1	1	yes	maximal, characteristic
3	#20272679	C₂	1	2	yes	maximal
4	C₁₆×#88322	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16810754	—	2	2	no	—
7	C₄×#11719908	C₂²	1	1	yes	characteristic
8	#16810376	C₂²	1	2	yes	—
9	Q₈⋊#88322	—	2	2	no	—
10	C₈×#88322	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₅₃	C₄	1	1	yes	characteristic
12	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#366	C₄	1	1	yes	characteristic
14	C₅₃×#358	C₄	1	2	yes	—
15	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#11719908	—	2	1	no	—
17	C₄⋊#88322	—	2	2	no	—
18	C₄⋊#88322	—	4	2	no	—
19	#13133128	—	4	2	no	—
20	C₄×#88322	D₄	1	1	yes	characteristic
21	C₄×#88322	D₄	1	1	yes	characteristic
22	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
23	C₄×Q₈×C₅₃	—	2	2	no	—
24	Q₃₂×C₅₃	—	2	2	no	—
25	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
27	#107×C₅₃	C₂×C₄	1	2	yes	—
28	#103×C₅₃	—	2	2	no	—
29	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₅₃	—	2	2	no	—
31	Q₁₆×C₅₃	—	2	2	no	—
32	Q₁₆×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₄⋊C₄×C₅₃	—	4	2	no	—
37	C₄²×C₅₃	—	4	2	no	—
38	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×#88322	—	2	1	no	—
40	C₂×#88322	—	4	2	no	—
41	C₂×#88322	D₈	1	1	yes	characteristic
42	C₂×#88322	D₈	1	1	yes	characteristic
43	C₁₆×C₅₃	—	2	1	no	—
44	C₁₆×C₅₃	M₁₆	1	2	yes	—
45	#11719908	—	2	2	no	—
46	Q₈×C₅₃	—	4	2	no	—
47	Q₈×C₅₃	—	4	2	no	—
48	C₂×C₄×C₅₃	—	4	2	no	—
49	C₂×C₄×C₅₃	—	4	2	no	—
50	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
53	C₈×C₅₃	—	2	1	no	—
54	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
55	C₈×C₅₃	#131	1	1	yes	characteristic
56	#88322	—	2	2	no	—
57	#88322	—	2	1	no	—
58	#88322	—	8	2	no	—
59	#88322	Q₃₂	1	2	yes	complemented
60	#12083	—	53	1	no	maximal, Sylow
61	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	—	8	2	no	—
64	C₄×C₅₃	—	8	2	no	—
65	C₄×C₅₃	#436	1	1	yes	characteristic
66	C₄×C₅₃	#443	1	1	yes	characteristic
67	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
68	C₄×C₅₃	#508	1	1	yes	characteristic
69	C₄×Q₃₂	D₅₃	1	1	yes	characteristic
70	#1591	—	53	2	no	—
71	#1371	—	53	1	no	—
72	#1332	—	53	2	no	—
73	C₈×C₁₆	—	53	1	no	—
74	C₂×C₅₃	#1589	1	1	yes	characteristic
75	C₂×C₅₃	C₄×Q₃₂	1	1	yes	characteristic
76	C₂×C₅₃	#2178	1	1	yes	characteristic
77	C₂×Q₃₂	Dic₅₃	1	1	yes	characteristic
78	#446	—	106	2	no	—
79	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
80	#366	Dic₅₃	1	1	yes	characteristic
81	#358	Dic₅₃	1	2	yes	—
82	C₄×C₁₆	—	53	1	no	—
83	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
84	#335	—	53	2	no	—
85	Q₈⋊C₈	—	106	2	no	—
86	C₈²	—	53	1	no	—
87	C₅₃	#12083	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
89	C₄×Q₈	—	2	2	no	—
90	Q₃₂	—	2	2	no	—
91	C₂×C₁₆	—	106	1	no	—
92	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
93	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
94	#107	C₂×Dic₅₃	1	2	yes	—
95	C₄⋊C₈	—	106	2	no	—
96	C₄⋊C₈	—	212	2	no	—
97	#103	—	2	2	no	—
98	#97	—	212	2	no	—
99	C₄×C₈	—	53	1	no	—
100	C₄×C₈	—	53	1	no	—
101	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
102	C₂×Q₈	—	2	2	no	—
103	Q₁₆	—	2	2	no	—
104	Q₁₆	—	2	2	no	—
105	C₂×C₈	—	53	1	no	—
106	C₂×C₈	—	53	1	no	—
107	C₂×C₈	—	106	1	no	—
108	C₂×C₈	—	212	2	no	—
109	C₂×C₈	#11719947	1	1	yes	characteristic
110	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄²	—	4	2	no	—
114	C₄²	D₄×D₅₃	1	1	yes	characteristic
115	C₁₆	—	2	1	no	—
116	C₁₆	—	106	2	no	—
117	C₁₆	#11719917	1	2	yes	—
118	Q₈	—	4	2	no	—
119	Q₈	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	D₄×Dic₅₃	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	—	53	2	no	—
127	C₈	—	106	2	no	—
128	C₈	—	106	1	no	—
129	C₈	—	424	2	no	—
130	C₈	C₂×#11719917	1	1	yes	characteristic
131	C₈	#13133224	1	1	yes	characteristic
132	C₂²	D₈×Dic₅₃	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	8	2	no	—
135	C₄	—	8	2	no	—
136	C₄	#16810701	1	1	yes	characteristic
137	C₄	#16810740	1	1	yes	characteristic
138	C₄	Q₃₂×D₅₃	1	1	yes	characteristic
139	C₄	#16811056	1	1	yes	characteristic
140	C₂	#20273600	1	1	yes	characteristic
141	C₂	Q₃₂×Dic₅₃	1	1	yes	characteristic
142	C₂	#20276596	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20273612

C₂

yes

maximal

#20272787

C₂

yes

maximal, characteristic

#20272679

C₂

yes

maximal

C₁₆×#88322

C₂

yes

maximal, characteristic

C₄×Q₃₂×C₅₃

C₂

yes

maximal, characteristic

#16810754

—

C₄×#11719908

C₂²

yes

characteristic

#16810376

C₂²

yes

—

Q₈⋊#88322

—

C₈×#88322

C₂²

yes

characteristic

C₂×Q₃₂×C₅₃

C₄

yes

characteristic

C₄×Q₁₆×C₅₃

C₂²

yes

—

C₅₃×#366

C₄

yes

characteristic

C₅₃×#358

C₄

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

—

—

—

—

D₄

yes

characteristic

C₄×#88322

D₄

yes

characteristic

C₂×Q₁₆×C₅₃

C₂×C₄

yes

—

C₄×Q₈×C₅₃

—

Q₃₂×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

#107×C₅₃

C₂×C₄

yes

—

#103×C₅₃

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

—

D₈

yes

characteristic

C₂×#88322

D₈

yes

characteristic

C₁₆×C₅₃

—

C₁₆×C₅₃

M₁₆

yes

—

#11719908

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

C₂×M₁₆

yes

C₈×C₅₃

yes

—

—

—

Q₃₂

yes

complemented

#12083

—

maximal, Sylow

C₂²×C₅₃

C₄×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×Q₃₂

yes

characteristic

C₄×C₅₃

#508

yes

characteristic

C₄×Q₃₂

D₅₃

yes

—

—

—

C₈×C₁₆

—

C₂×C₅₃

#1589

yes

characteristic

C₂×C₅₃

C₄×Q₃₂

yes

characteristic

C₂×C₅₃

#2178

yes

characteristic

C₂×Q₃₂

Dic₅₃

yes

characteristic

#446

—

106

—

C₄×Q₁₆

C₂×D₅₃

yes

—

#366

Dic₅₃

yes

characteristic

#358

Dic₅₃

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

#335

—

Q₈⋊C₈

—

106

—

C₈²

—

C₅₃

#12083

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×Dic₅₃

yes

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₂×Dic₅₃

yes

characteristic

C₂×C₁₆

C₂×Dic₅₃

yes

characteristic

#107

C₂×Dic₅₃

yes

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

212

—

#103

—

#97

—

212

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

C₂²×D₅₃

yes

characteristic

102

C₂×Q₈

—

103

Q₁₆

—

104

Q₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

106

—

108

C₂×C₈

—

212

—

109

C₂×C₈

#11719947

yes

characteristic

110

C₂×C₈

C₂²×Dic₅₃

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

—

114

C₄²

D₄×D₅₃

yes

characteristic

115

C₁₆

—

116

C₁₆

—

106

—

117

C₁₆

#11719917

yes

—

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

D₄×Dic₅₃

yes

characteristic

123

C₂×C₄

D₈×D₅₃

yes

characteristic

124

C₂×C₄

#13133236

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

106

—

128

C₈

—

106

—

129

C₈

—

424

—

130

C₈

C₂×#11719917

yes

characteristic

131

C₈

#13133224

yes

characteristic

132

C₂²

D₈×Dic₅₃

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#16810701

yes

characteristic

137

C₄

#16810740

yes

characteristic

138

C₄

Q₃₂×D₅₃

yes

characteristic

139

C₄

#16811056

yes

characteristic

140

C₂

#20273600

yes

characteristic

141

C₂

Q₃₂×Dic₅₃

yes

characteristic

142

C₂

#20276596

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₄×Q₃₂×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₄×D₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₄×Q₃₂	D₅₃
hypocenter	C₅₃	#12083
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

subgroup

G/H

center

C₂×C₄

D₈×D₅₃

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₄×Q₃₂×C₅₃

C₂

socle, prime radical

C₂²×C₅₃

C₄×D₈

cosocle

C₄×C₈×C₅₃

C₂³

hypercenter

C₄×Q₃₂

D₅₃

hypocenter

C₅₃

#12083

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

N	H ≅ G/N	type
59. #88322	Q₃₂	semidirect
87. C₅₃	#12083	semidirect

H ≅ G/N

type

59. #88322

Q₃₂

semidirect

87. C₅₃

#12083

semidirect

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

subgroup	index
#20273612	2
#20272787	2
#20272679	2
C₁₆×#88322	2
C₄×Q₃₂×C₅₃	2
#12083	53

subgroup

index

#20273612

#20272787

#20272679

C₁₆×#88322

C₄×Q₃₂×C₅₃

#12083

Subgroups — #27296083

Conjugacy classes of subgroups

Complements

Maximal subgroups