Group #20276643 — subgroups

Subgroups — #20276643

Order 6784.

256 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₂²×D₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133160	C₂	1	2	yes	maximal, complemented
3	#16810423	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₁₂	—	2	1	no	—
7	C₂×D₄₂₄	—	2	1	no	—
8	C₂×D₄₂₄	—	2	4	no	—
9	C₂×D₄₂₄	C₂²	1	2	yes	—
10	#13133160	—	2	4	no	—
11	C₂×#11719908	C₂²	1	2	yes	complemented
12	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²⋊Q₈×C₅₃	—	2	1	no	—
15	#107×C₅₃	C₂²	1	1	yes	characteristic, complemented
16	#103×C₅₃	—	2	1	no	—
17	C₂³×D₅₃	—	4	1	no	—
18	C₂×Q₈×C₅₃	—	2	1	no	—
19	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂×D₂₁₂	—	2	4	no	—
21	C₂×D₂₁₂	—	2	2	no	—
22	C₂×D₂₁₂	—	4	2	no	—
23	C₂×D₂₁₂	—	4	1	no	—
24	Q₁₆×C₅₃	—	2	2	no	—
25	C₂×C₈×C₅₃	—	2	2	no	—
26	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₄⋊C₄×C₅₃	—	4	1	no	—
30	C₂²⋊C₄×C₅₃	—	4	1	no	—
31	D₄₂₄	—	2	4	no	—
32	D₄₂₄	—	2	2	no	—
33	D₄₂₄	D₄	1	4	yes	—
34	#11719908	—	2	2	no	—
35	C₂³×C₅₃	D₈	1	1	yes	characteristic
36	C₂²×D₅₃	—	4	4	no	—
37	C₂²×D₅₃	—	4	2	no	—
38	C₂²×D₅₃	—	8	2	no	—
39	C₂²×D₅₃	—	8	2	no	—
40	Q₈×C₅₃	—	4	1	no	—
41	Q₈×C₅₃	—	4	1	no	—
42	C₂×C₄×C₅₃	—	2	1	no	—
43	C₂×C₄×C₅₃	—	2	1	no	—
44	C₂×C₄×C₅₃	—	4	1	no	—
45	C₂×C₄×C₅₃	—	4	1	no	—
46	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	D₂₁₂	—	2	4	no	—
49	D₂₁₂	—	4	4	no	—
50	D₂₁₂	—	4	2	no	—
51	C₈×C₅₃	—	2	1	no	—
52	C₈×C₅₃	C₂×D₄	1	2	yes	—
53	C₂²×C₅₃	—	2	1	no	—
54	C₂²×C₅₃	—	2	1	no	—
55	C₂²×C₅₃	SD₃₂	1	2	yes	complemented
56	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
57	C₂×D₅₃	—	4	4	no	—
58	C₂×D₅₃	—	8	8	no	—
59	C₂×D₅₃	—	8	2	no	—
60	C₂×D₅₃	—	8	2	no	—
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	—	8	1	no	—
63	C₄×C₅₃	—	8	1	no	—
64	C₄×C₅₃	#120	1	1	yes	characteristic
65	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
66	#2187	—	53	1	no	maximal, Sylow
67	C₂×C₅₃	—	2	2	no	—
68	C₂×C₅₃	#447	1	1	yes	characteristic
69	C₂×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
70	C₂×C₅₃	#509	1	1	yes	characteristic
71	D₅₃	—	8	4	no	—
72	C₂²×D₈	—	53	1	no	—
73	C₂×SD₃₂	—	53	2	no	—
74	#467	D₅₃	1	1	yes	characteristic, complemented
75	#357	—	53	2	no	—
76	C₂²⋊C₁₆	—	53	1	no	—
77	C₅₃	#2187	1	1	yes	characteristic, complemented, Sylow
78	C₂²×D₄	—	106	1	no	—
79	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
80	C₂×D₈	—	53	2	no	—
81	C₂×D₈	—	106	4	no	—
82	C₂×D₈	—	106	1	no	—
83	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
84	C₂²⋊Q₈	—	2	1	no	—
85	SD₃₂	—	106	4	no	—
86	C₂×C₁₆	—	53	2	no	—
87	#107	C₂×D₅₃	1	1	yes	characteristic, complemented
88	#103	—	2	1	no	—
89	C₂⁴	—	212	1	no	—
90	C₂×Q₈	—	2	1	no	—
91	C₂×D₄	—	106	4	no	—
92	C₂×D₄	—	106	2	no	—
93	C₂×D₄	—	212	2	no	—
94	C₂×D₄	—	212	1	no	—
95	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
96	Q₁₆	—	2	2	no	—
97	D₈	—	53	4	no	—
98	D₈	—	106	4	no	—
99	D₈	—	106	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₂²⋊C₄	—	4	1	no	—
106	C₁₆	—	106	2	no	—
107	C₂³	—	424	2	no	—
108	C₂³	—	424	2	no	—
109	C₂³	—	212	4	no	—
110	C₂³	—	212	2	no	—
111	C₂³	#11719922	1	1	yes	characteristic
112	Q₈	—	4	1	no	—
113	Q₈	—	4	1	no	—
114	D₄	—	106	4	no	—
115	D₄	—	212	4	no	—
116	D₄	—	212	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	#11719922	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	D₄×D₅₃	1	2	yes	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	424	8	no	—
128	C₂²	—	424	2	no	—
129	C₂²	—	424	2	no	—
130	C₂²	—	212	4	no	—
131	C₂²	#13133160	1	2	yes	complemented
132	C₂²	C₂×#11719922	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	8	1	no	—
136	C₄	#13133185	1	1	yes	characteristic
137	C₄	#13133242	1	1	yes	characteristic
138	C₂	—	2	2	no	—
139	C₂	—	424	4	no	—
140	C₂	#16810756	1	1	yes	characteristic
141	C₂	C₂×#13133160	1	1	yes	characteristic
142	C₂	#16811058	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₄₂₄

C₂

yes

maximal, characteristic

C₂×#13133160

C₂

yes

maximal, complemented

#16810423

C₂

yes

maximal, complemented

C₂²⋊#11719908

C₂

yes

maximal, characteristic, complemented

C₅₃×#467

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂×D₄₂₄

C₂²

yes

—

#13133160

—

C₂×#11719908

C₂²

yes

complemented

C₂×Q₁₆×C₅₃

C₂²

yes

characteristic, complemented

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²⋊Q₈×C₅₃

—

#107×C₅₃

C₂²

yes

characteristic, complemented

#103×C₅₃

—

C₂³×D₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

D₄₂₄

—

D₄₂₄

—

D₄₂₄

D₄

yes

—

#11719908

—

C₂³×C₅₃

D₈

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

C₈×C₅₃

C₂×D₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

SD₃₂

yes

complemented

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

#447

yes

characteristic

C₂×C₅₃

C₂×SD₃₂

yes

characteristic

C₂×C₅₃

#509

yes

characteristic

D₅₃

—

C₂²×D₈

—

C₂×SD₃₂

—

#467

D₅₃

yes

characteristic, complemented

#357

—

C₂²⋊C₁₆

—

C₅₃

#2187

yes

characteristic, complemented, Sylow

C₂²×D₄

—

106

—

C₂×Q₁₆

C₂×D₅₃

yes

characteristic, complemented

C₂×D₈

—

C₂×D₈

—

106

—

C₂×D₈

—

106

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂²⋊Q₈

—

SD₃₂

—

106

—

C₂×C₁₆

—

#107

C₂×D₅₃

yes

characteristic, complemented

#103

—

C₂⁴

—

212

—

C₂×Q₈

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂×D₄

—

212

—

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

Q₁₆

—

D₈

—

D₈

—

106

—

D₈

—

106

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

102

C₂×C₈

C₂²×D₅₃

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

106

C₁₆

—

106

—

107

C₂³

—

424

—

108

C₂³

—

424

—

109

C₂³

—

212

—

110

C₂³

—

212

—

111

C₂³

#11719922

yes

characteristic

112

Q₈

—

113

Q₈

—

114

D₄

—

106

—

115

D₄

—

212

—

116

D₄

—

212

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

#11719922

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

123

C₈

—

124

C₈

D₄×D₅₃

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

424

—

128

C₂²

—

424

—

129

C₂²

—

424

—

130

C₂²

—

212

—

131

C₂²

#13133160

yes

complemented

132

C₂²

C₂×#11719922

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#13133185

yes

characteristic

137

C₄

#13133242

yes

characteristic

138

C₂

—

139

C₂

—

424

—

140

C₂

#16810756

yes

characteristic

141

C₂

C₂×#13133160

yes

characteristic

142

C₂

#16811058

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#467	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#467	D₅₃
hypocenter	C₅₃	#2187
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #467

subgroup

G/H

center

C₂²

C₂×#11719922

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#467

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #467

N	H ≅ G/N	type
2. C₂×#13133160	C₂	semidirect
3. #16810423	C₂	semidirect
4. C₂²⋊#11719908	C₂	semidirect
5. C₅₃×#467	C₂	semidirect
11. C₂×#11719908	C₂²	semidirect
12. C₂×Q₁₆×C₅₃	C₂²	semidirect
15. #107×C₅₃	C₂²	semidirect
55. C₂²×C₅₃	SD₃₂	semidirect
74. #467	D₅₃	semidirect
77. C₅₃	#2187	semidirect
79. C₂×Q₁₆	C₂×D₅₃	semidirect
87. #107	C₂×D₅₃	semidirect
131. C₂²	#13133160	semidirect

H ≅ G/N

type

2. C₂×#13133160

C₂

semidirect

3. #16810423

C₂

semidirect

4. C₂²⋊#11719908

C₂

semidirect

5. C₅₃×#467

C₂

semidirect

11. C₂×#11719908

C₂²

semidirect

12. C₂×Q₁₆×C₅₃

C₂²

semidirect

15. #107×C₅₃

C₂²

semidirect

55. C₂²×C₅₃

SD₃₂

semidirect

74. #467

D₅₃

semidirect

77. C₅₃

#2187

semidirect

79. C₂×Q₁₆

C₂×D₅₃

semidirect

87. #107

C₂×D₅₃

semidirect

131. C₂²

#13133160

semidirect

Subgroups — #20276643

Conjugacy classes of subgroups

Complements