Group #20276752 — subgroups

Subgroups — #20276752

Order 6784.

174 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 55 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×#13133161	C₂	1	2	yes	maximal, complemented
2	C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810424	C₂	1	2	yes	maximal, complemented
5	C₅₃×#493	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
7	C₄×Dic₁₀₆	—	2	1	no	—
8	#13133161	—	2	4	no	—
9	C₂×#11719908	C₂²	1	2	yes	—
10	#13133151	C₂²	1	1	yes	characteristic
11	#13133142	—	2	1	no	—
12	C₂×D₈×C₅₃	—	2	1	no	—
13	C₂×D₈×C₅₃	C₂²	1	2	yes	—
14	#127×C₅₃	—	2	1	no	—
15	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₅₃	—	2	2	no	—
17	C₂×D₄×C₅₃	—	4	1	no	—
18	C₂×D₄×C₅₃	—	4	1	no	—
19	C₂×Dic₁₀₆	—	2	1	no	—
20	D₈×C₅₃	—	2	2	no	—
21	D₈×C₅₃	—	2	2	no	—
22	D₈×C₅₃	—	4	2	no	—
23	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₄²×C₅₃	D₄	1	1	yes	characteristic
26	C₄⋊Dic₅₃	—	2	1	no	—
27	#11719919	—	4	1	no	—
28	C₄×Dic₅₃	—	4	1	no	—
29	Dic₂₁₂	—	2	1	no	—
30	Dic₂₁₂	D₄	1	2	yes	complemented
31	#11719908	—	2	2	no	—
32	C₂³×C₅₃	—	4	2	no	—
33	D₄×C₅₃	—	4	2	no	—
34	D₄×C₅₃	—	4	2	no	—
35	D₄×C₅₃	—	8	2	no	—
36	D₄×C₅₃	—	8	2	no	—
37	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
38	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₅₃	—	4	1	no	—
41	C₂×Dic₅₃	—	4	1	no	—
42	Dic₁₀₆	—	2	2	no	—
43	Dic₁₀₆	—	4	1	no	—
44	C₈×C₅₃	—	2	1	no	—
45	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₈×C₅₃	#41	1	1	yes	characteristic
47	C₂²×C₅₃	—	8	2	no	—
48	C₂²×C₅₃	—	8	2	no	—
49	C₂²×C₅₃	—	8	2	no	—
50	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	SD₃₂	1	2	yes	complemented
53	C₄×C₅₃	#121	1	1	yes	characteristic
54	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
55	Dic₅₃	—	4	2	no	—
56	Dic₅₃	—	8	1	no	—
57	#2207	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	—	16	2	no	—
59	C₂×C₅₃	#459	1	1	yes	characteristic
60	C₂×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
61	C₂×C₅₃	#509	1	1	yes	characteristic
62	C₂×SD₃₂	—	53	2	no	—
63	#493	D₅₃	1	1	yes	characteristic
64	C₄×Q₁₆	—	53	1	no	—
65	C₄⋊C₁₆	—	53	1	no	—
66	#357	—	53	2	no	—
67	C₅₃	#2207	1	1	yes	characteristic, complemented, Sylow
68	C₂×Q₁₆	—	53	1	no	—
69	C₂×D₈	—	2	1	no	—
70	C₂×D₈	C₂×D₅₃	1	2	yes	—
71	#127	—	2	1	no	—
72	C₄×Q₈	—	106	1	no	—
73	SD₃₂	—	106	4	no	—
74	C₂×C₁₆	—	53	2	no	—
75	#107	—	53	1	no	—
76	#103	—	106	1	no	—
77	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
78	C₂×Q₈	—	106	1	no	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	—	4	1	no	—
81	C₂×D₄	—	4	1	no	—
82	Q₁₆	—	53	2	no	—
83	Q₁₆	—	106	1	no	—
84	D₈	—	2	2	no	—
85	D₈	—	2	2	no	—
86	D₈	—	4	2	no	—
87	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
88	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
89	C₄⋊C₄	—	106	1	no	—
90	C₄⋊C₄	—	212	1	no	—
91	C₄²	—	212	1	no	—
92	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
93	C₁₆	—	106	2	no	—
94	C₂³	—	4	2	no	—
95	Q₈	—	106	2	no	—
96	Q₈	—	212	1	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	D₄	—	8	2	no	—
100	D₄	—	8	2	no	—
101	C₂×C₄	—	212	1	no	—
102	C₂×C₄	—	212	1	no	—
103	C₂×C₄	#11719922	1	1	yes	characteristic
104	C₂×C₄	#11719922	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
106	C₈	—	2	1	no	—
107	C₈	D₄×D₅₃	1	1	yes	characteristic
108	C₈	#11719947	1	1	yes	characteristic
109	C₂²	—	8	2	no	—
110	C₂²	—	8	2	no	—
111	C₂²	—	8	2	no	—
112	C₂²	C₂×#11719922	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	424	1	no	—
115	C₄	—	212	2	no	—
116	C₄	#13133161	1	2	yes	complemented
117	C₄	#13133188	1	1	yes	characteristic
118	C₄	#13133243	1	1	yes	characteristic
119	C₂	—	16	2	no	—
120	C₂	#16810815	1	1	yes	characteristic
121	C₂	C₂×#13133161	1	1	yes	characteristic
122	C₂	#16811059	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133161

C₂

yes

maximal, complemented

C₄×Dic₂₁₂

C₂

yes

maximal, characteristic, complemented

C₄⋊#11719908

C₂

yes

maximal, characteristic, complemented

#16810424

C₂

yes

maximal, complemented

C₅₃×#493

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₂²

yes

characteristic

C₄×Dic₁₀₆

—

#13133161

—

C₂×#11719908

C₂²

yes

—

#13133151

C₂²

yes

characteristic

#13133142

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

C₂²

yes

—

#127×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×Dic₁₀₆

—

D₈×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

D₄

yes

complemented

#11719908

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

#41

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

SD₃₂

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

#2207

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#459

yes

characteristic

C₂×C₅₃

C₂×SD₃₂

yes

characteristic

C₂×C₅₃

#509

yes

characteristic

C₂×SD₃₂

—

#493

D₅₃

yes

characteristic

C₄×Q₁₆

—

C₄⋊C₁₆

—

#357

—

C₅₃

#2207

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

C₂×D₅₃

yes

—

#127

—

C₄×Q₈

—

106

—

SD₃₂

—

106

—

C₂×C₁₆

—

#107

—

#103

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

106

—

D₈

—

D₈

—

D₈

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄²

—

212

—

C₄²

C₂²⋊D₅₃

yes

characteristic

C₁₆

—

106

—

C₂³

—

Q₈

—

106

—

Q₈

—

212

—

D₄

—

D₄

—

D₄

—

100

D₄

—

101

C₂×C₄

—

212

—

102

C₂×C₄

—

212

—

103

C₂×C₄

#11719922

yes

characteristic

104

C₂×C₄

#11719922

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

106

C₈

—

107

C₈

D₄×D₅₃

yes

characteristic

108

C₈

#11719947

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

C₂×#11719922

yes

characteristic

113

C₄

—

114

C₄

—

424

—

115

C₄

—

212

—

116

C₄

#13133161

yes

complemented

117

C₄

#13133188

yes

characteristic

118

C₄

#13133243

yes

characteristic

119

C₂

—

120

C₂

#16810815

yes

characteristic

121

C₂

C₂×#13133161

yes

characteristic

122

C₂

#16811059

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#493	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#493	D₅₃
hypocenter	C₅₃	#2207
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

subgroup

G/H

center

C₂²

C₂×#11719922

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#493

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
1. C₂×#13133161	C₂	semidirect
2. C₄×Dic₂₁₂	C₂	semidirect
3. C₄⋊#11719908	C₂	semidirect
4. #16810424	C₂	semidirect
30. Dic₂₁₂	D₄	semidirect
52. C₄×C₅₃	SD₃₂	semidirect
67. C₅₃	#2207	semidirect
116. C₄	#13133161	semidirect

H ≅ G/N

type

1. C₂×#13133161

C₂

semidirect

2. C₄×Dic₂₁₂

C₂

semidirect

3. C₄⋊#11719908

C₂

semidirect

4. #16810424

C₂

semidirect

30. Dic₂₁₂

D₄

semidirect

52. C₄×C₅₃

SD₃₂

semidirect

67. C₅₃

#2207

semidirect

116. C₄

#13133161

semidirect

Subgroups — #20276752

Conjugacy classes of subgroups

Complements