Group #20272603 — subgroups

Subgroups — #20272603

Order 6784.

190 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16810587	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	C₄	1	2	yes	—
5	C₂²×#88322	—	2	1	no	—
6	C₂²⋊#88322	—	2	1	no	—
7	C₂²⋊#88322	—	2	1	no	—
8	C₂²⋊#88322	—	2	1	no	—
9	C₂²⋊#88322	C₄	1	2	yes	—
10	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₅₃	—	2	1	no	—
12	C₂²⋊C₈×C₅₃	—	2	1	no	—
13	C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
14	C₂⁴×C₅₃	Q₈	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	—	4	2	no	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	—	2	2	no	—
23	C₂×C₈×C₅₃	—	4	2	no	—
24	#11719917	—	4	2	no	—
25	C₂×#88322	—	2	2	no	—
26	C₂×#88322	—	2	2	no	—
27	C₂×#88322	—	4	2	no	—
28	C₂³×C₅₃	—	2	2	no	—
29	C₂³×C₅₃	—	2	2	no	—
30	C₂³×C₅₃	—	4	1	no	—
31	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	—	2	2	no	—
35	C₂×C₄×C₅₃	—	2	2	no	—
36	C₂×C₄×C₅₃	—	2	2	no	—
37	C₂×C₄×C₅₃	—	4	1	no	—
38	C₂×C₄×C₅₃	—	4	1	no	—
39	C₂×C₄×C₅₃	—	4	1	no	—
40	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₈×C₅₃	—	4	2	no	—
45	#88322	—	4	2	no	—
46	#88322	—	4	2	no	—
47	C₂²×C₅₃	—	2	1	no	—
48	C₂²×C₅₃	—	2	1	no	—
49	C₂²×C₅₃	—	2	1	no	—
50	C₂²×C₅₃	—	2	1	no	—
51	C₂²×C₅₃	—	2	1	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	—	2	1	no	—
54	C₂²×C₅₃	—	4	4	no	—
55	C₂²×C₅₃	#95	1	1	yes	characteristic
56	C₂²×C₅₃	#108	1	2	yes	—
57	C₂²×C₅₃	#108	1	2	yes	—
58	C₄×C₅₃	—	2	1	no	—
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	—	2	1	no	—
61	C₄×C₅₃	#99	1	2	yes	—
62	#1298	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	—	2	2	no	—
64	C₂×C₅₃	—	2	2	no	—
65	C₂×C₅₃	—	4	1	no	—
66	C₂×C₅₃	#341	1	1	yes	characteristic
67	C₂×C₅₃	#341	1	1	yes	characteristic
68	C₂×C₅₃	#342	1	1	yes	characteristic
69	#407	—	53	1	no	—
70	C₂×C₂²⋊C₈	—	53	1	no	—
71	C₂×C₂²⋊C₈	D₅₃	1	1	yes	characteristic
72	C₅₃	#1298	1	1	yes	characteristic, complemented, Sylow
73	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
74	C₂×M₁₆	—	53	2	no	—
75	C₂²×C₈	—	2	1	no	—
76	C₂²×C₈	—	106	1	no	—
77	C₂²⋊C₈	—	2	1	no	—
78	C₂²⋊C₈	—	53	2	no	—
79	C₂²⋊C₈	—	106	1	no	—
80	C₂²⋊C₈	—	106	1	no	—
81	C₂²⋊C₈	—	106	1	no	—
82	C₂²⋊C₈	Dic₅₃	1	2	yes	—
83	C₂⁴	Dic₁₀₆	1	1	yes	characteristic
84	C₂²×C₄	—	4	2	no	—
85	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
86	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
87	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
89	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
90	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
91	M₁₆	—	212	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	4	2	no	—
94	C₂×C₈	—	106	2	no	—
95	C₂×C₈	—	106	2	no	—
96	C₂×C₈	—	212	2	no	—
97	C₂³	—	2	2	no	—
98	C₂³	—	2	2	no	—
99	C₂³	—	4	1	no	—
100	C₂³	#11719919	1	1	yes	characteristic
101	C₂³	#11719919	1	1	yes	characteristic
102	C₂³	C₄⋊Dic₅₃	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
110	C₂×C₄	#11719921	1	1	yes	characteristic
111	C₂×C₄	#11719921	1	1	yes	characteristic
112	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
113	C₈	—	4	2	no	—
114	C₈	—	212	2	no	—
115	C₈	—	212	2	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	4	4	no	—
124	C₂²	#13133124	1	1	yes	characteristic
125	C₂²	#13133152	1	2	yes	—
126	C₂²	#13133153	1	2	yes	—
127	C₄	—	2	1	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	#13133131	1	2	yes	—
131	C₂	—	2	2	no	—
132	C₂	—	2	2	no	—
133	C₂	—	4	1	no	—
134	C₂	#16810387	1	1	yes	characteristic
135	C₂	#16810388	1	1	yes	characteristic
136	C₂	#16810389	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810587

C₂

yes

maximal, characteristic

C₂×C₂²⋊#88322

C₂

yes

maximal, characteristic

C₂×C₂²⋊C₈×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₄

yes

—

C₂²×#88322

—

—

—

—

C₄

yes

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

C₄

yes

—

C₂⁴×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

—

—

—

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#95

yes

characteristic

C₂²×C₅₃

#108

yes

—

C₂²×C₅₃

#108

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#99

yes

—

#1298

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

D₅₃

yes

characteristic

C₅₃

#1298

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

C₂²⋊C₈

Dic₅₃

yes

—

C₂⁴

Dic₁₀₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

#11719919

yes

characteristic

101

C₂³

#11719919

yes

characteristic

102

C₂³

C₄⋊Dic₅₃

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄×Dic₅₃

yes

characteristic

110

C₂×C₄

#11719921

yes

characteristic

111

C₂×C₄

#11719921

yes

characteristic

112

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

113

C₈

—

114

C₈

—

212

—

115

C₈

—

212

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

#13133124

yes

characteristic

125

C₂²

#13133152

yes

—

126

C₂²

#13133153

yes

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#13133131

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

134

C₂

#16810387

yes

characteristic

135

C₂

#16810388

yes

characteristic

136

C₂

#16810389

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#11719921
commutator	C₂×C₄×C₅₃	C₄²
Frattini	C₂³×C₄	C₂×D₅₃
Fitting	C₂×C₂²⋊C₈×C₅₃	C₂
socle	C₂²×C₅₃	#95
hypercenter	C₂×C₂²⋊C₈	D₅₃
hypocenter	C₅₃	#1298
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂×C₄

#11719921

commutator

C₂×C₄×C₅₃

C₄²

Frattini

C₂³×C₄

C₂×D₅₃

Fitting

C₂×C₂²⋊C₈×C₅₃

C₂

socle

C₂²×C₅₃

#95

hypercenter

C₂×C₂²⋊C₈

D₅₃

hypocenter

C₅₃

#1298

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
72. C₅₃	#1298	semidirect

H ≅ G/N

type

72. C₅₃

#1298

semidirect

Subgroups — #20272603

Conjugacy classes of subgroups

Complements