Group #20274764 — subgroups

Subgroups — #20274764

Order 6784.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₂×#13133225	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133152	C₂	1	2	yes	maximal, complemented
3	#16810605	C₂	1	2	yes	maximal
4	#16810381	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#338	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719945	C₂²	1	1	yes	characteristic
7	#13133225	—	2	4	no	—
8	C₂×#11719917	C₂²	1	2	yes	—
9	C₂×#11719912	C₄	1	2	yes	—
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₅₃	C₄	1	2	yes	—
12	#13133152	—	2	4	no	—
13	#13133135	—	2	2	no	—
14	#13133133	—	2	2	no	—
15	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×C₂²⋊D₅₃	Q₈	1	2	yes	—
19	#11719945	—	2	2	no	—
20	#11719945	—	2	2	no	—
21	C₂×D₂₁₂	D₄	1	1	yes	characteristic
22	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
23	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
24	M₁₆×C₅₃	—	2	2	no	—
25	M₁₆×C₅₃	—	2	2	no	—
26	M₁₆×C₅₃	—	2	2	no	—
27	C₂×C₈×C₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	#11719917	—	2	2	no	—
30	#11719917	—	2	2	no	—
31	C₂×#88322	—	2	2	no	—
32	C₂×#88322	C₂×C₄	1	1	yes	characteristic
33	C₂×#88322	C₂×C₄	1	1	yes	characteristic
34	C₂×#88322	D₄	1	4	yes	—
35	#11719912	—	2	4	no	—
36	C₈×D₅₃	—	2	4	no	—
37	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₂²×D₅₃	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	#41	1	2	yes	—
43	C₂²⋊D₅₃	—	2	4	no	—
44	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
45	D₂₁₂	—	2	2	no	—
46	C₄×D₅₃	—	2	2	no	—
47	C₄×D₅₃	—	2	2	no	—
48	Dic₁₀₆	—	2	2	no	—
49	C₈×C₅₃	—	2	2	no	—
50	C₈×C₅₃	—	4	2	no	—
51	#88322	—	4	2	no	—
52	#88322	C₂²⋊C₄	1	2	yes	—
53	#88322	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₅₃	—	4	1	no	—
55	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
57	C₂×D₅₃	—	2	2	no	—
58	C₂×D₅₃	—	4	2	no	—
59	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₅₃	#117	1	1	yes	characteristic
61	C₄×C₅₃	#121	1	2	yes	—
62	Dic₅₃	—	2	2	no	—
63	#1813	—	53	1	no	maximal, Sylow
64	C₂×C₅₃	—	2	1	no	—
65	C₂×C₅₃	—	2	2	no	—
66	C₂×C₅₃	#380	1	1	yes	characteristic
67	D₅₃	—	4	2	no	—
68	C₂×#131	—	53	1	no	—
69	C₂×#108	—	53	2	no	—
70	#413	—	53	2	no	—
71	#338	—	53	1	no	—
72	#338	D₅₃	1	1	yes	characteristic, complemented
73	C₅₃	#1813	1	1	yes	characteristic, complemented, Sylow
74	C₂×#41	—	53	1	no	—
75	#131	—	106	4	no	—
76	C₂×M₁₆	—	53	2	no	—
77	C₂×M₁₆	—	53	2	no	—
78	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
79	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
80	C₂²×C₈	—	53	1	no	—
81	C₂²×C₈	—	53	2	no	—
82	#108	—	106	4	no	—
83	#101	—	106	2	no	—
84	#100	—	106	2	no	—
85	#41	—	106	2	no	—
86	#41	—	106	2	no	—
87	C₂×Q₈	—	53	1	no	—
88	C₂×D₄	—	53	2	no	—
89	C₂×D₄	—	53	1	no	—
90	C₂²×C₄	—	53	2	no	—
91	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	106	4	no	—
96	M₁₆	—	106	2	no	—
97	M₁₆	—	106	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	53	1	no	—
100	C₂×C₈	—	53	1	no	—
101	C₂×C₈	—	53	4	no	—
102	C₂×C₈	—	106	4	no	—
103	C₂×C₈	—	106	2	no	—
104	C₂×C₈	C₄×D₅₃	1	2	yes	—
105	C₂³	—	53	2	no	—
106	C₂³	Q₈×D₅₃	1	1	yes	characteristic
107	Q₈	—	106	2	no	—
108	D₄	—	106	4	no	—
109	D₄	—	106	2	no	—
110	C₂×C₄	—	53	2	no	—
111	C₂×C₄	—	106	2	no	—
112	C₂×C₄	—	106	2	no	—
113	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
114	C₂×C₄	#11719945	1	2	yes	—
115	C₂×C₄	D₄×D₅₃	1	2	yes	—
116	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	53	2	no	—
120	C₈	—	53	2	no	—
121	C₈	—	212	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	106	2	no	—
124	C₂²	—	212	2	no	—
125	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
126	C₂²	#13133195	1	2	yes	—
127	C₄	—	106	2	no	—
128	C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
129	C₄	#13133174	1	1	yes	characteristic
130	C₄	#13133190	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	212	2	no	—
134	C₂	#16810482	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133225

C₂

yes

maximal, characteristic

C₂×#13133152

C₂

yes

maximal, complemented

#16810605

C₂

yes

maximal

#16810381

C₂

yes

maximal, characteristic, complemented

C₅₃×#338

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

C₂²

yes

characteristic

#13133225

—

C₂×#11719917

C₂²

yes

—

C₂×#11719912

C₄

yes

—

C₂²×#88322

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₄

yes

—

#13133152

—

#13133135

—

#13133133

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

Q₈

yes

—

#11719945

—

#11719945

—

C₂×D₂₁₂

D₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×Dic₁₀₆

D₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

D₄

yes

—

#11719912

—

C₈×D₅₃

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂²×D₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

C₂²⋊C₄

yes

—

#88322

C₂²⋊C₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

#1813

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#380

yes

characteristic

D₅₃

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#1813

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

106

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#108

—

106

—

#101

—

106

—

#100

—

106

—

#41

—

106

—

#41

—

106

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

106

—

103

C₂×C₈

—

106

—

104

C₂×C₈

C₄×D₅₃

yes

—

105

C₂³

—

106

C₂³

Q₈×D₅₃

yes

characteristic

107

Q₈

—

106

—

108

D₄

—

106

—

109

D₄

—

106

—

110

C₂×C₄

—

111

C₂×C₄

—

106

—

112

C₂×C₄

—

106

—

113

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

114

C₂×C₄

#11719945

yes

—

115

C₂×C₄

D₄×D₅₃

yes

—

116

C₂×C₄

D₄×D₅₃

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

212

—

122

C₂²

—

123

C₂²

—

106

—

124

C₂²

—

212

—

125

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

126

C₂²

#13133195

yes

—

127

C₄

—

106

—

128

C₄

C₂²⋊C₄×D₅₃

yes

characteristic

129

C₄

#13133174

yes

characteristic

130

C₄

#13133190

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

212

—

134

C₂

#16810482

yes

characteristic

	subgroup	G/H
center	C₄	#13133174
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#338	C₂
socle	C₂×C₅₃	#380
hypercenter	#338	D₅₃
hypocenter	C₅₃	#1813
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#13133174

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#338

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#13133152	C₂	semidirect
4. #16810381	C₂	semidirect
5. C₅₃×#338	C₂	semidirect
72. #338	D₅₃	semidirect
73. C₅₃	#1813	semidirect

H ≅ G/N

type

2. C₂×#13133152

C₂

semidirect

4. #16810381

C₂

semidirect

5. C₅₃×#338

C₂

semidirect

72. #338

D₅₃

semidirect

73. C₅₃

#1813

semidirect

Subgroups — #20274764

Conjugacy classes of subgroups

Complements