Group #20273478 — subgroups

Subgroups — #20273478

Order 6784.

168 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 93 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	#16810680	C₂	1	2	yes	maximal
2	#16810376	C₂	1	2	yes	maximal, complemented
3	#16810374	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#88322	C₂²	1	4	yes	—
6	C₂²⋊#88322	—	2	2	no	—
7	#13133128	—	2	2	no	—
8	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
13	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
18	C₂×C₈×C₅₃	—	2	2	no	—
19	C₂×C₈×C₅₃	—	2	2	no	—
20	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₅₃	D₄	1	1	yes	characteristic
26	C₄²×C₅₃	Q₈	1	1	yes	characteristic
27	C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×#88322	—	2	4	no	—
29	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	2	1	no	—
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	—	2	2	no	—
34	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₅₃	—	2	1	no	—
41	C₈×C₅₃	—	2	1	no	—
42	C₈×C₅₃	M₁₆	1	2	yes	—
43	C₈×C₅₃	M₁₆	1	2	yes	—
44	#88322	—	4	4	no	—
45	C₂²×C₅₃	—	2	1	no	—
46	C₂²×C₅₃	—	2	1	no	—
47	C₂²×C₅₃	#108	1	2	yes	—
48	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	—	2	1	no	—
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	—	2	1	no	—
53	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
54	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
55	C₄×C₅₃	#135	1	2	yes	—
56	#1568	—	53	1	no	maximal, Sylow
57	C₂×C₅₃	—	2	2	no	—
58	C₂×C₅₃	#423	1	1	yes	characteristic
59	C₂×C₅₃	#427	1	1	yes	characteristic
60	C₂×C₅₃	C₂×#108	1	1	yes	characteristic
61	#432	—	53	2	no	—
62	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
63	#335	—	53	2	no	—
64	#334	—	53	2	no	—
65	C₅₃	#1568	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₈	Dic₅₃	1	1	yes	characteristic
67	C₂²×C₈	Dic₅₃	1	1	yes	characteristic
68	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
69	C₄⋊C₈	—	53	4	no	—
70	C₂²⋊C₈	—	106	2	no	—
71	#97	—	106	2	no	—
72	C₄×C₈	Dic₅₃	1	1	yes	characteristic
73	C₄×C₈	Dic₅₃	1	1	yes	characteristic
74	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
75	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
76	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
77	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	106	4	no	—
82	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
83	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
84	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
85	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
86	C₄²	Dic₁₀₆	1	1	yes	characteristic
87	C₄²	D₂₁₂	1	1	yes	characteristic
88	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
89	C₄²	C₂²×D₅₃	1	1	yes	characteristic
90	C₂³	C₄⋊Dic₅₃	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
97	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
98	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
99	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
100	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	2	1	no	—
103	C₈	—	212	4	no	—
104	C₈	#11719917	1	2	yes	—
105	C₈	#11719917	1	2	yes	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#13133153	1	2	yes	—
109	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	C₂×#11719917	1	1	yes	characteristic
115	C₄	C₂×#11719917	1	1	yes	characteristic
116	C₄	#13133239	1	2	yes	—
117	C₂	—	2	2	no	—
118	C₂	#16810647	1	1	yes	characteristic
119	C₂	#16810661	1	1	yes	characteristic
120	C₂	C₂×#13133153	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810680

C₂

yes

maximal

#16810376

C₂

yes

maximal, complemented

#16810374

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic

C₄⋊#88322

C₂²

yes

—

C₂²⋊#88322

—

#13133128

—

C₂²×C₈×C₅₃

C₄

yes

characteristic

C₂²×C₈×C₅₃

C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₄

yes

characteristic

C₄×C₈×C₅₃

C₄

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

Q₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

Q₈

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂×#88322

—

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

M₁₆

yes

—

C₈×C₅₃

M₁₆

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#108

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

#135

yes

—

#1568

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₄×C₈

D₅₃

yes

characteristic

#335

—

#334

—

C₅₃

#1568

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₅₃

yes

characteristic

C₂²×C₈

Dic₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

106

—

#97

—

106

—

C₄×C₈

Dic₅₃

yes

characteristic

C₄×C₈

Dic₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₄²

Dic₁₀₆

yes

characteristic

C₄²

D₂₁₂

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

characteristic

100

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

212

—

104

C₈

#11719917

yes

—

105

C₈

#11719917

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

#13133153

yes

—

109

C₂²

C₂×C₄⋊Dic₅₃

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

C₂×#11719917

yes

characteristic

115

C₄

C₂×#11719917

yes

characteristic

116

C₄

#13133239

yes

—

117

C₂

—

118

C₂

#16810647

yes

characteristic

119

C₂

#16810661

yes

characteristic

120

C₂

C₂×#13133153

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#1568
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₂×C₄×C₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₄

hypercenter

C₂×C₄×C₈

D₅₃

hypocenter

C₅₃

#1568

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16810376	C₂	semidirect
3. #16810374	C₂	semidirect
65. C₅₃	#1568	semidirect

H ≅ G/N

type

2. #16810376

C₂

semidirect

3. #16810374

C₂

semidirect

65. C₅₃

#1568

semidirect

Subgroups — #20273478

Conjugacy classes of subgroups

Complements