Group #20276156 — subgroups

Subgroups — #20276156

Order 6784.

198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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	#16810622	C₂	1	2	yes	maximal
2	#16810589	C₂	1	1	yes	maximal, characteristic
3	#16810392	C₂	1	2	yes	maximal, complemented
4	#16810388	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#107×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719945	C₂²	1	1	yes	characteristic
7	C₂×#11719917	C₂²	1	1	yes	characteristic
8	C₂×#11719917	C₂²	1	2	yes	—
9	#13133140	—	2	2	no	—
10	#13133137	—	2	2	no	—
11	#13133129	—	2	2	no	—
12	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
14	#107×C₅₃	—	2	2	no	—
15	C₂²×C₄×C₅₃	—	2	2	no	—
16	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂×C₂²⋊D₅₃	—	2	1	no	—
18	#11719945	—	2	2	no	—
19	#11719945	—	2	2	no	—
20	C₂×D₂₁₂	D₄	1	1	yes	characteristic
21	C₂×C₄×D₅₃	—	2	1	no	—
22	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	—	2	2	no	—
24	C₂×C₈×C₅₃	D₄	1	2	yes	—
25	C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₄⋊C₄×C₅₃	—	2	2	no	—
27	#11719917	—	2	2	no	—
28	#11719917	—	2	2	no	—
29	#11719917	—	2	2	no	—
30	C₂×#88322	—	2	1	no	—
31	C₂×#88322	—	2	2	no	—
32	C₂³×C₅₃	#41	1	1	yes	characteristic
33	C₂²×D₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	—	4	2	no	—
35	C₂×C₄×C₅₃	—	8	2	no	—
36	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	#41	1	2	yes	—
39	C₂×C₄×C₅₃	#41	1	2	yes	—
40	C₂²⋊D₅₃	—	4	2	no	—
41	C₂×Dic₅₃	—	2	1	no	—
42	D₂₁₂	—	2	2	no	—
43	C₄×D₅₃	—	4	1	no	—
44	C₄×D₅₃	—	4	1	no	—
45	Dic₁₀₆	—	2	2	no	—
46	C₈×C₅₃	—	2	2	no	—
47	#88322	—	4	2	no	—
48	#88322	—	4	1	no	—
49	C₂²×C₅₃	—	2	1	no	—
50	C₂²×C₅₃	—	2	1	no	—
51	C₂²×C₅₃	#121	1	1	yes	characteristic
52	C₂²×C₅₃	#126	1	2	yes	—
53	C₂×D₅₃	—	4	1	no	—
54	C₂×D₅₃	—	4	1	no	—
55	C₂×D₅₃	—	4	1	no	—
56	C₄×C₅₃	—	8	2	no	—
57	C₄×C₅₃	#123	1	1	yes	characteristic
58	C₄×C₅₃	#123	1	2	yes	—
59	C₄×C₅₃	#124	1	1	yes	characteristic
60	Dic₅₃	—	4	1	no	—
61	#2078	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	—	2	2	no	—
63	C₂×C₅₃	#397	1	1	yes	characteristic
64	C₂×C₅₃	#472	1	1	yes	characteristic
65	C₂×C₅₃	#473	1	1	yes	characteristic
66	D₅₃	—	8	1	no	—
67	C₂×#107	D₅₃	1	1	yes	characteristic, complemented
68	#417	—	53	2	no	—
69	#408	—	53	1	no	—
70	#343	—	53	2	no	—
71	#341	—	53	1	no	—
72	C₅₃	#2078	1	1	yes	characteristic, complemented, Sylow
73	C₂×#41	—	53	1	no	—
74	C₂×M₁₆	—	53	2	no	—
75	C₂×M₁₆	—	53	1	no	—
76	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
77	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
78	#107	—	2	2	no	—
79	#103	—	106	2	no	—
80	#102	—	106	2	no	—
81	C₂²⋊C₈	—	106	2	no	—
82	#41	—	106	2	no	—
83	#41	—	106	2	no	—
84	C₂×Q₈	—	53	1	no	—
85	C₂×D₄	—	53	1	no	—
86	C₂×D₄	—	106	1	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	106	1	no	—
89	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
90	M₁₆	—	106	2	no	—
91	M₁₆	—	106	2	no	—
92	M₁₆	—	106	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	106	2	no	—
95	C₂×C₈	—	106	1	no	—
96	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₂³	—	106	1	no	—
100	C₂³	#11719947	1	1	yes	characteristic
101	Q₈	—	106	2	no	—
102	D₄	—	106	2	no	—
103	D₄	—	212	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	—	106	1	no	—
107	C₂×C₄	—	212	1	no	—
108	C₂×C₄	—	212	1	no	—
109	C₂×C₄	#11719945	1	2	yes	—
110	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
111	C₂×C₄	#11719949	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	212	2	no	—
115	C₈	—	212	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	212	1	no	—
119	C₂²	—	212	1	no	—
120	C₂²	—	212	1	no	—
121	C₂²	#13133188	1	1	yes	characteristic
122	C₂²	#13133212	1	2	yes	—
123	C₄	—	8	2	no	—
124	C₄	—	212	1	no	—
125	C₄	#13133200	1	1	yes	characteristic
126	C₄	#13133202	1	2	yes	—
127	C₄	#13133206	1	1	yes	characteristic
128	C₂	—	2	2	no	—
129	C₂	—	424	1	no	—
130	C₂	#16810549	1	1	yes	characteristic
131	C₂	#16810890	1	1	yes	characteristic
132	C₂	#16810895	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810622

C₂

yes

maximal

#16810589

C₂

yes

maximal, characteristic

#16810392

C₂

yes

maximal, complemented

#16810388

C₂

yes

maximal, characteristic, complemented

C₂×#107×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

—

#13133140

—

#13133137

—

#13133129

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

#107×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

—

#11719945

—

#11719945

—

C₂×D₂₁₂

D₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

D₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

—

—

—

—

—

C₂³×C₅₃

#41

yes

characteristic

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

#126

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#123

yes

characteristic

C₄×C₅₃

#123

yes

—

C₄×C₅₃

#124

yes

characteristic

Dic₅₃

—

#2078

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂×#107

D₅₃

yes

characteristic, complemented

—

—

—

—

C₅₃

#2078

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

#107

—

#103

—

106

—

#102

—

106

—

C₂²⋊C₈

—

106

—

#41

—

106

—

#41

—

106

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂²⋊D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂³

—

106

—

100

C₂³

#11719947

yes

characteristic

101

Q₈

—

106

—

102

D₄

—

106

—

103

D₄

—

212

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

106

—

107

C₂×C₄

—

212

—

108

C₂×C₄

—

212

—

109

C₂×C₄

#11719945

yes

—

110

C₂×C₄

D₄×D₅₃

yes

characteristic

111

C₂×C₄

#11719949

yes

—

112

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

113

C₈

—

114

C₈

—

212

—

115

C₈

—

212

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

212

—

119

C₂²

—

212

—

120

C₂²

—

212

—

121

C₂²

#13133188

yes

characteristic

122

C₂²

#13133212

yes

—

123

C₄

—

124

C₄

—

212

—

125

C₄

#13133200

yes

characteristic

126

C₄

#13133202

yes

—

127

C₄

#13133206

yes

characteristic

128

C₂

—

129

C₂

—

424

—

130

C₂

#16810549

yes

characteristic

131

C₂

#16810890

yes

characteristic

132

C₂

#16810895

yes

characteristic

	subgroup	G/H
center	C₂²	#13133188
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#107×C₅₃	C₂
socle	C₂²×C₅₃	#121
hypercenter	C₂×#107	D₅₃
hypocenter	C₅₃	#2078
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂²

#13133188

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#107×C₅₃

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
3. #16810392	C₂	semidirect
4. #16810388	C₂	semidirect
5. C₂×#107×C₅₃	C₂	semidirect
67. C₂×#107	D₅₃	semidirect
72. C₅₃	#2078	semidirect

H ≅ G/N

type

3. #16810392

C₂

semidirect

4. #16810388

C₂

semidirect

5. C₂×#107×C₅₃

C₂

semidirect

67. C₂×#107

D₅₃

semidirect

72. C₅₃

#2078

semidirect

Subgroups — #20276156

Conjugacy classes of subgroups

Complements