Group #20274711 — subgroups

Subgroups — #20274711

Order 6784.

272 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 113 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	#16810647	C₂	1	2	yes	maximal
2	C₂×#13133136	C₂	1	2	yes	maximal
3	#16810392	C₂	1	2	yes	maximal
4	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	C₂²	1	4	yes	—
6	C₄⋊#88322	—	2	2	no	—
7	C₄⋊#88322	—	2	2	no	—
8	#13133136	—	2	4	no	—
9	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₅₃	—	2	2	no	—
11	C₄×Q₈×C₅₃	—	2	2	no	—
12	C₂×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	C₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₅₃	—	2	2	no	—
17	C₂×Q₈×C₅₃	—	2	2	no	—
18	C₂×Q₈×C₅₃	D₄	1	2	yes	—
19	C₂×Q₈×C₅₃	Q₈	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₅₃	—	2	4	no	—
26	C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₄²×C₅₃	—	2	2	no	—
28	C₄²×C₅₃	—	2	2	no	—
29	C₄²×C₅₃	D₄	1	2	yes	—
30	C₄²×C₅₃	D₄	1	2	yes	—
31	#11719917	—	2	4	no	—
32	#11719917	—	2	4	no	—
33	C₂×#88322	—	2	4	no	—
34	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₅₃	—	2	4	no	—
36	Q₈×C₅₃	—	2	4	no	—
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	4	1	no	—
42	C₂×C₄×C₅₃	—	4	2	no	—
43	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#41	1	2	yes	—
51	#88322	—	4	4	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	—	2	1	no	—
54	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₅₃	#117	1	1	yes	characteristic
57	C₄×C₅₃	—	2	4	no	—
58	C₄×C₅₃	—	2	1	no	—
59	C₄×C₅₃	—	2	2	no	—
60	C₄×C₅₃	—	4	2	no	—
61	C₄×C₅₃	#101	1	2	yes	complemented
62	C₄×C₅₃	#121	1	2	yes	—
63	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
64	#1801	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	—	2	1	no	—
66	C₂×C₅₃	—	2	1	no	—
67	C₂×C₅₃	#380	1	1	yes	characteristic
68	C₂×C₅₃	C₂×#101	1	1	yes	characteristic
69	C₂×C₅₃	#413	1	1	yes	characteristic
70	C₂×C₄×Q₈	D₅₃	1	1	yes	characteristic
71	#423	—	53	2	no	—
72	C₂×#101	—	53	2	no	—
73	#343	—	53	2	no	—
74	C₅₃	#1801	1	1	yes	characteristic, complemented, Sylow
75	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
76	C₂×M₁₆	—	53	4	no	—
77	C₄×Q₈	—	2	2	no	—
78	C₄×Q₈	—	2	2	no	—
79	C₂×C₄⋊C₄	Dic₅₃	1	2	yes	—
80	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
81	C₂×C₄²	Dic₅₃	1	2	yes	—
82	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
83	C₄⋊C₈	—	106	2	no	—
84	C₄⋊C₈	—	106	2	no	—
85	#101	—	106	4	no	—
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	Dic₁₀₆	1	2	yes	—
89	C₂×Q₈	D₂₁₂	1	2	yes	—
90	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
91	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
93	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
95	M₁₆	—	106	4	no	—
96	M₁₆	—	106	4	no	—
97	C₂×C₈	—	106	4	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	C₂²⋊D₅₃	1	2	yes	—
103	C₄²	C₂²⋊D₅₃	1	2	yes	—
104	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
105	Q₈	—	2	4	no	—
106	Q₈	—	2	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₄⋊Dic₅₃	1	4	yes	—
114	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
115	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
116	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
117	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
118	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
119	C₂×C₄	#11719945	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
121	C₈	—	212	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
125	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
126	C₂²	#13133176	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	2	2	no	—
129	C₄	—	2	4	no	—
130	C₄	—	4	2	no	—
131	C₄	#13133136	1	2	yes	complemented
132	C₄	#13133191	1	2	yes	—
133	C₄	#13133196	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#16810483	1	1	yes	characteristic
137	C₂	C₂×#13133136	1	1	yes	characteristic
138	C₂	#16810607	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810647

C₂

yes

maximal

C₂×#13133136

C₂

yes

maximal

#16810392

C₂

yes

maximal

C₂×C₄×Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₄⋊#88322

—

C₄⋊#88322

—

#13133136

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₄

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

D₄

yes

—

#11719917

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#117

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#101

yes

complemented

C₄×C₅₃

#121

yes

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

#1801

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×Q₈

D₅₃

yes

—

—

—

C₅₃

#1801

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₅₃

yes

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₄²

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

#101

—

106

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₁₀₆

yes

—

C₂×Q₈

D₂₁₂

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²⋊D₅₃

yes

—

103

C₄²

C₂²⋊D₅₃

yes

—

104

C₂³

C₂²×Dic₅₃

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₄⋊Dic₅₃

yes

—

114

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

115

C₂×C₄

C₂²⋊Dic₅₃

yes

—

116

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

117

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

118

C₂×C₄

C₂×D₂₁₂

yes

characteristic

119

C₂×C₄

#11719945

yes

—

120

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

121

C₈

—

212

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

125

C₂²

C₂×C₄⋊Dic₅₃

yes

characteristic

126

C₂²

#13133176

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#13133136

yes

complemented

132

C₄

#13133191

yes

—

133

C₄

#13133196

yes

—

134

C₂

—

135

C₂

—

136

C₂

#16810483

yes

characteristic

137

C₂

C₂×#13133136

yes

characteristic

138

C₂

#16810607

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₃
commutator	C₂³×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	#1801
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₃

commutator

C₂³×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄×Q₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

hypercenter

C₂×C₄×Q₈

D₅₃

hypocenter

C₅₃

#1801

LCS

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
61. C₄×C₅₃	#101	semidirect
74. C₅₃	#1801	semidirect
131. C₄	#13133136	semidirect

H ≅ G/N

type

61. C₄×C₅₃

#101

semidirect

74. C₅₃

#1801

semidirect

131. C₄

#13133136

semidirect

Subgroups — #20274711

Conjugacy classes of subgroups

Complements