Group #20276651 — subgroups

Subgroups — #20276651

Order 6784.

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

normal subgroups only

This group has 216 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×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₄₂₄	C₂	1	2	yes	maximal, complemented
3	#16810864	C₂	1	1	yes	maximal, characteristic
4	#16810428	C₂	1	2	yes	maximal, complemented
5	C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	—	2	1	no	—
7	C₂×Dic₂₁₂	—	2	1	no	—
8	C₂×Dic₂₁₂	—	2	4	no	—
9	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₁₂	C₂²	1	2	yes	—
11	#13133196	—	2	1	no	—
12	Dic₄₂₄	—	2	4	no	—
13	#13133151	C₂²	1	1	yes	characteristic
14	#13133142	—	2	1	no	—
15	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
18	C₂²×Dic₅₃	—	4	1	no	—
19	C₂×Dic₁₀₆	—	2	4	no	—
20	C₂×Dic₁₀₆	—	2	2	no	—
21	C₂×Dic₁₀₆	—	2	1	no	—
22	C₂×Dic₁₀₆	—	4	2	no	—
23	C₂×Dic₁₀₆	—	4	1	no	—
24	C₂×C₈×C₅₃	—	2	2	no	—
25	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²⋊Dic₅₃	—	4	1	no	—
28	C₄⋊Dic₅₃	—	2	1	no	—
29	#11719919	—	4	1	no	—
30	Dic₂₁₂	—	2	4	no	—
31	Dic₂₁₂	—	2	2	no	—
32	Dic₂₁₂	—	2	2	no	—
33	Dic₂₁₂	D₄	1	4	yes	—
34	C₁₆×C₅₃	—	2	2	no	—
35	C₂³×C₅₃	D₈	1	1	yes	characteristic
36	C₂×C₄×C₅₃	—	2	1	no	—
37	C₂×C₄×C₅₃	—	2	1	no	—
38	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₅₃	—	4	4	no	—
41	C₂×Dic₅₃	—	4	2	no	—
42	C₂×Dic₅₃	—	4	1	no	—
43	C₂×Dic₅₃	—	4	1	no	—
44	Dic₁₀₆	—	2	4	no	—
45	Dic₁₀₆	—	4	4	no	—
46	Dic₁₀₆	—	4	2	no	—
47	Dic₁₀₆	—	4	1	no	—
48	Dic₁₀₆	—	4	1	no	—
49	C₈×C₅₃	—	2	1	no	—
50	C₈×C₅₃	C₂×D₄	1	2	yes	—
51	C₂²×C₅₃	—	2	1	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	Q₃₂	1	2	yes	complemented
54	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
55	C₄×C₅₃	—	2	1	no	—
56	C₄×C₅₃	#120	1	1	yes	characteristic
57	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
58	Dic₅₃	—	4	4	no	—
59	Dic₅₃	—	8	1	no	—
60	Dic₅₃	—	8	1	no	—
61	#2188	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	—	2	2	no	—
63	C₂×C₅₃	#447	1	1	yes	characteristic
64	C₂×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
65	C₂×C₅₃	#510	1	1	yes	characteristic
66	C₂²×Q₁₆	—	53	1	no	—
67	C₂×Q₃₂	—	53	2	no	—
68	#467	—	53	1	no	—
69	#358	—	53	2	no	—
70	C₂²⋊C₁₆	D₅₃	1	1	yes	characteristic
71	C₅₃	#2188	1	1	yes	characteristic, complemented, Sylow
72	C₂²×Q₈	—	106	1	no	—
73	C₂×Q₁₆	—	53	2	no	—
74	C₂×Q₁₆	—	53	1	no	—
75	C₂×Q₁₆	—	106	4	no	—
76	C₂×Q₁₆	—	106	1	no	—
77	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
78	C₂²⋊Q₈	—	106	1	no	—
79	Q₃₂	—	106	4	no	—
80	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
81	#107	—	53	1	no	—
82	#103	—	106	1	no	—
83	C₂×Q₈	—	106	4	no	—
84	C₂×Q₈	—	106	2	no	—
85	C₂×Q₈	—	106	1	no	—
86	C₂×Q₈	—	212	2	no	—
87	C₂×Q₈	—	212	1	no	—
88	C₂²×C₄	—	212	1	no	—
89	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
90	Q₁₆	—	53	4	no	—
91	Q₁₆	—	106	4	no	—
92	Q₁₆	—	106	2	no	—
93	Q₁₆	—	106	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	D₂₁₂	1	1	yes	characteristic
96	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
97	C₄⋊C₄	—	106	1	no	—
98	C₄⋊C₄	—	212	1	no	—
99	C₂²⋊C₄	—	212	1	no	—
100	C₁₆	—	2	2	no	—
101	C₂³	D₄₂₄	1	1	yes	characteristic
102	Q₈	—	106	4	no	—
103	Q₈	—	212	4	no	—
104	Q₈	—	212	1	no	—
105	Q₈	—	212	1	no	—
106	Q₈	—	212	2	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	212	1	no	—
110	C₂×C₄	—	212	1	no	—
111	C₂×C₄	—	212	4	no	—
112	C₂×C₄	—	212	2	no	—
113	C₂×C₄	D₄₂₄	1	1	yes	characteristic
114	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
115	C₈	—	2	1	no	—
116	C₈	D₄×D₅₃	1	2	yes	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	Dic₄₂₄	1	2	yes	complemented
120	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	424	1	no	—
123	C₄	—	424	1	no	—
124	C₄	—	212	4	no	—
125	C₄	#13133187	1	1	yes	characteristic
126	C₄	#13133245	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	#16810759	1	1	yes	characteristic
129	C₂	C₂×Dic₄₂₄	1	1	yes	characteristic
130	C₂	#16811066	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₁₂

C₂

yes

maximal, characteristic

C₂×Dic₄₂₄

C₂

yes

maximal, complemented

#16810864

C₂

yes

maximal, characteristic

#16810428

C₂

yes

maximal, complemented

C₅₃×C₂²⋊C₁₆

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

C₂²

yes

characteristic

C₂×Dic₂₁₂

C₂²

yes

—

#13133196

—

Dic₄₂₄

—

#13133151

C₂²

yes

characteristic

#13133142

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

D₄

yes

—

C₁₆×C₅₃

—

C₂³×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂×D₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

Q₃₂

yes

complemented

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2188

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#447

yes

characteristic

C₂×C₅₃

C₂×Q₃₂

yes

characteristic

C₂×C₅₃

#510

yes

characteristic

C₂²×Q₁₆

—

C₂×Q₃₂

—

#467

—

#358

—

C₂²⋊C₁₆

D₅₃

yes

characteristic

C₅₃

#2188

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂²⋊Q₈

—

106

—

Q₃₂

—

106

—

C₂×C₁₆

C₂×D₅₃

yes

—

#107

—

#103

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

212

—

C₂×Q₈

—

212

—

C₂²×C₄

—

212

—

C₂²×C₄

D₂₁₂

yes

characteristic

Q₁₆

—

Q₁₆

—

106

—

Q₁₆

—

106

—

Q₁₆

—

106

—

C₂×C₈

—

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

100

C₁₆

—

101

C₂³

D₄₂₄

yes

characteristic

102

Q₈

—

106

—

103

Q₈

—

212

—

104

Q₈

—

212

—

105

Q₈

—

212

—

106

Q₈

—

212

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

212

—

110

C₂×C₄

—

212

—

111

C₂×C₄

—

212

—

112

C₂×C₄

—

212

—

113

C₂×C₄

D₄₂₄

yes

characteristic

114

C₂×C₄

C₂×D₂₁₂

yes

characteristic

115

C₈

—

116

C₈

D₄×D₅₃

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

Dic₄₂₄

yes

complemented

120

C₂²

C₂×D₄₂₄

yes

characteristic

121

C₄

—

122

C₄

—

424

—

123

C₄

—

424

—

124

C₄

—

212

—

125

C₄

#13133187

yes

characteristic

126

C₄

#13133245

yes

characteristic

127

C₂

—

128

C₂

#16810759

yes

characteristic

129

C₂

C₂×Dic₄₂₄

yes

characteristic

130

C₂

#16811066

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×C₂²⋊C₁₆	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₂²⋊C₁₆	D₅₃
hypocenter	C₅₃	#2188
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₄₂₄

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×C₂²⋊C₁₆

C₂

socle

C₂²×C₅₃

C₂×D₈

hypercenter

C₂²⋊C₁₆

D₅₃

hypocenter

C₅₃

#2188

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
2. C₂×Dic₄₂₄	C₂	semidirect
4. #16810428	C₂	semidirect
53. C₂²×C₅₃	Q₃₂	semidirect
71. C₅₃	#2188	semidirect
119. C₂²	Dic₄₂₄	semidirect

H ≅ G/N

type

2. C₂×Dic₄₂₄

C₂

semidirect

4. #16810428

C₂

semidirect

53. C₂²×C₅₃

Q₃₂

semidirect

71. C₅₃

#2188

semidirect

119. C₂²

Dic₄₂₄

semidirect

Subgroups — #20276651

Conjugacy classes of subgroups

Complements