Group #20276632 — subgroups

Subgroups — #20276632

Order 6784.

226 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 61 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×#13133160	C₂	1	2	yes	maximal, complemented
2	#16810859	C₂	1	1	yes	maximal, characteristic
3	#16810427	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
7	#13133191	—	2	1	no	—
8	#13133160	—	2	4	no	—
9	C₂×#11719908	C₂²	1	2	yes	—
10	#13133151	C₂²	1	1	yes	characteristic
11	#13133139	—	2	1	no	—
12	C₂²×Q₈×C₅₃	—	2	1	no	—
13	C₂×Q₁₆×C₅₃	—	2	4	no	—
14	C₂×Q₁₆×C₅₃	—	2	1	no	—
15	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	2	4	no	—
18	C₂×Q₈×C₅₃	—	2	2	no	—
19	C₂×Q₈×C₅₃	—	4	2	no	—
20	C₂×Q₈×C₅₃	—	4	1	no	—
21	C₂²×C₄×C₅₃	—	4	1	no	—
22	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂×C₂²⋊D₅₃	—	4	1	no	—
24	C₂×D₂₁₂	—	2	1	no	—
25	Q₁₆×C₅₃	—	2	4	no	—
26	Q₁₆×C₅₃	—	2	2	no	—
27	Q₁₆×C₅₃	D₄	1	4	yes	complemented
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	#11719921	—	4	1	no	—
32	C₄⋊Dic₅₃	—	2	1	no	—
33	D₄₂₄	—	2	2	no	—
34	#11719908	—	2	2	no	—
35	C₂³×C₅₃	D₈	1	1	yes	characteristic
36	C₂²×D₅₃	—	4	1	no	—
37	Q₈×C₅₃	—	2	4	no	—
38	Q₈×C₅₃	—	4	4	no	—
39	Q₈×C₅₃	—	4	2	no	—
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	—	2	1	no	—
42	C₂×C₄×C₅₃	—	4	4	no	—
43	C₂×C₄×C₅₃	—	4	2	no	—
44	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²⋊D₅₃	—	8	2	no	—
47	C₂×Dic₅₃	—	4	1	no	—
48	D₂₁₂	—	4	1	no	—
49	D₂₁₂	—	4	1	no	—
50	C₈×C₅₃	—	2	1	no	—
51	C₈×C₅₃	C₂×D₄	1	2	yes	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	—	2	1	no	—
54	C₂²×C₅₃	SD₃₂	1	2	yes	complemented
55	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
56	C₂×D₅₃	—	8	1	no	—
57	C₂×D₅₃	—	8	1	no	—
58	C₂×D₅₃	—	8	1	no	—
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	—	4	4	no	—
61	C₄×C₅₃	#120	1	1	yes	characteristic
62	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
63	Dic₅₃	—	8	1	no	—
64	#2185	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	—	2	2	no	—
66	C₂×C₅₃	#447	1	1	yes	characteristic
67	C₂×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
68	C₂×C₅₃	#510	1	1	yes	characteristic
69	D₅₃	—	16	1	no	—
70	C₂²×Q₁₆	D₅₃	1	1	yes	characteristic, complemented
71	C₂×SD₃₂	—	53	2	no	—
72	#466	—	53	1	no	—
73	#358	—	53	2	no	—
74	C₂²⋊C₁₆	—	53	1	no	—
75	C₅₃	#2185	1	1	yes	characteristic, complemented, Sylow
76	C₂²×Q₈	—	2	1	no	—
77	C₂×Q₁₆	—	2	4	no	—
78	C₂×Q₁₆	—	2	1	no	—
79	C₂×Q₁₆	C₂×D₅₃	1	2	yes	—
80	C₂×D₈	—	53	1	no	—
81	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
82	#121	—	106	1	no	—
83	SD₃₂	—	106	4	no	—
84	C₂×C₁₆	—	53	2	no	—
85	#107	—	53	1	no	—
86	#102	—	106	1	no	—
87	C₂×Q₈	—	2	4	no	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	—	4	2	no	—
90	C₂×Q₈	—	4	1	no	—
91	C₂×D₄	—	106	1	no	—
92	C₂×D₄	—	212	1	no	—
93	C₂²×C₄	—	4	1	no	—
94	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
95	Q₁₆	—	2	4	no	—
96	Q₁₆	—	2	2	no	—
97	Q₁₆	C₂²⋊D₅₃	1	4	yes	complemented
98	D₈	—	106	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
101	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
102	C₄⋊C₄	—	106	1	no	—
103	C₂²⋊C₄	—	212	1	no	—
104	C₁₆	—	106	2	no	—
105	C₂³	—	212	1	no	—
106	C₂³	#11719922	1	1	yes	characteristic
107	Q₈	—	2	4	no	—
108	Q₈	—	4	4	no	—
109	Q₈	—	4	2	no	—
110	D₄	—	424	2	no	—
111	D₄	—	212	1	no	—
112	D₄	—	212	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	212	1	no	—
118	C₂×C₄	#11719922	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	424	1	no	—
125	C₂²	—	424	1	no	—
126	C₂²	—	424	1	no	—
127	C₂²	#13133160	1	2	yes	complemented
128	C₂²	C₂×#11719922	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	4	4	no	—
131	C₄	—	424	1	no	—
132	C₄	#13133186	1	1	yes	characteristic
133	C₄	#13133243	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	—	848	1	no	—
136	C₂	#16810757	1	1	yes	characteristic
137	C₂	C₂×#13133160	1	1	yes	characteristic
138	C₂	#16811064	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133160

C₂

yes

maximal, complemented

#16810859

C₂

yes

maximal, characteristic

#16810427

C₂

yes

maximal, complemented

C₂²⋊#11719908

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

C₂²

yes

—

—

C₂²

yes

—

#13133151

C₂²

yes

characteristic

#13133139

—

C₂²×Q₈×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂×C₂²⋊D₅₃

—

C₂×D₂₁₂

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

D₄

yes

complemented

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

—

D₄₂₄

—

#11719908

—

C₂³×C₅₃

D₈

yes

characteristic

C₂²×D₅₃

—

Q₈×C₅₃

—

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

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

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

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#2185

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#447

yes

characteristic

C₂×C₅₃

C₂×SD₃₂

yes

characteristic

C₂×C₅₃

#510

yes

characteristic

D₅₃

—

C₂²×Q₁₆

D₅₃

yes

characteristic, complemented

C₂×SD₃₂

—

#466

—

#358

—

C₂²⋊C₁₆

—

C₅₃

#2185

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₅₃

yes

—

C₂×D₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#121

—

106

—

SD₃₂

—

106

—

C₂×C₁₆

—

#107

—

#102

—

106

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

C₂²⋊D₅₃

yes

complemented

D₈

—

106

—

C₂×C₈

—

100

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

101

C₂×C₈

C₂²×D₅₃

yes

characteristic

102

C₄⋊C₄

—

106

—

103

C₂²⋊C₄

—

212

—

104

C₁₆

—

106

—

105

C₂³

—

212

—

106

C₂³

#11719922

yes

characteristic

107

Q₈

—

108

Q₈

—

109

Q₈

—

110

D₄

—

424

—

111

D₄

—

212

—

112

D₄

—

212

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

212

—

118

C₂×C₄

#11719922

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

120

C₈

—

121

C₈

C₂×C₂²⋊D₅₃

yes

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

424

—

125

C₂²

—

424

—

126

C₂²

—

424

—

127

C₂²

#13133160

yes

complemented

128

C₂²

C₂×#11719922

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

424

—

132

C₄

#13133186

yes

characteristic

133

C₄

#13133243

yes

characteristic

134

C₂

—

135

C₂

—

848

—

136

C₂

#16810757

yes

characteristic

137

C₂

C₂×#13133160

yes

characteristic

138

C₂

#16811064

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂²×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₂²×Q₁₆	D₅₃
hypocenter	C₅₃	#2185
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

C₂×#11719922

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₂²×Q₁₆×C₅₃

C₂

socle

C₂²×C₅₃

C₂×D₈

hypercenter

C₂²×Q₁₆

D₅₃

hypocenter

C₅₃

#2185

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
1. C₂×#13133160	C₂	semidirect
3. #16810427	C₂	semidirect
4. C₂²⋊#11719908	C₂	semidirect
5. C₂²×Q₁₆×C₅₃	C₂	semidirect
27. Q₁₆×C₅₃	D₄	semidirect
54. C₂²×C₅₃	SD₃₂	semidirect
70. C₂²×Q₁₆	D₅₃	semidirect
75. C₅₃	#2185	semidirect
97. Q₁₆	C₂²⋊D₅₃	semidirect
127. C₂²	#13133160	semidirect

H ≅ G/N

type

1. C₂×#13133160

C₂

semidirect

3. #16810427

C₂

semidirect

4. C₂²⋊#11719908

C₂

semidirect

5. C₂²×Q₁₆×C₅₃

C₂

semidirect

27. Q₁₆×C₅₃

D₄

semidirect

54. C₂²×C₅₃

SD₃₂

semidirect

70. C₂²×Q₁₆

D₅₃

semidirect

75. C₅₃

#2185

semidirect

97. Q₁₆

C₂²⋊D₅₃

semidirect

127. C₂²

#13133160

semidirect

Subgroups — #20276632

Conjugacy classes of subgroups

Complements