Group #20273504 — subgroups

Subgroups — #20273504

Order 6784.

158 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 85 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	#16810676	C₂	1	1	yes	maximal, characteristic
2	#16810647	C₂	1	1	yes	maximal, characteristic
3	#16810376	C₂	1	2	yes	maximal, complemented
4	#16810374	C₂	1	2	yes	maximal, complemented
5	C₅₃×#431	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	—	2	1	no	—
7	C₄⋊#88322	—	2	1	no	—
8	C₄⋊#88322	C₂²	1	1	yes	characteristic
9	C₄⋊#88322	C₂²	1	2	yes	—
10	C₄⋊#88322	C₂²	1	1	yes	characteristic
11	C₂²⋊#88322	—	2	1	no	—
12	C₄×#88322	—	2	1	no	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₅₃	C₄	1	2	yes	—
15	C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
16	C₄×C₈×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
21	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₃	D₄	1	1	yes	characteristic
23	C₄²×C₅₃	Q₈	1	1	yes	characteristic
24	C₄²×C₅₃	C₂³	1	1	yes	characteristic
25	#11719917	—	4	2	no	—
26	C₂×#88322	—	2	1	no	—
27	C₂×#88322	—	2	1	no	—
28	C₂×#88322	—	2	2	no	—
29	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	—	2	1	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	4	no	—
41	#88322	—	4	1	no	—
42	#88322	—	4	2	no	—
43	#88322	—	4	1	no	—
44	C₂²×C₅₃	—	2	1	no	—
45	C₂²×C₅₃	—	2	1	no	—
46	C₂²×C₅₃	—	2	1	no	—
47	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	—	2	1	no	—
51	C₄×C₅₃	#108	1	2	yes	—
52	C₄×C₅₃	#131	1	2	yes	—
53	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₄×C₅₃	#137	1	1	yes	characteristic
55	#1574	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	—	4	1	no	—
57	C₂×C₅₃	#424	1	1	yes	characteristic
58	C₂×C₅₃	#428	1	1	yes	characteristic
59	C₂×C₅₃	C₂×#108	1	1	yes	characteristic
60	#431	—	53	1	no	—
61	#431	D₅₃	1	1	yes	characteristic
62	#423	—	53	1	no	—
63	#335	—	53	2	no	—
64	#334	—	53	2	no	—
65	C₅₃	#1574	1	1	yes	characteristic, complemented, Sylow
66	C₂×M₁₆	—	106	1	no	—
67	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
68	C₄⋊C₈	—	53	1	no	—
69	C₄⋊C₈	—	53	2	no	—
70	C₄⋊C₈	—	53	1	no	—
71	C₄⋊C₈	—	106	1	no	—
72	C₄⋊C₈	Dic₅₃	1	2	yes	—
73	C₂²⋊C₈	—	106	1	no	—
74	C₂²⋊C₈	Dic₅₃	1	2	yes	—
75	C₄×C₈	—	106	1	no	—
76	C₄×C₈	C₂×D₅₃	1	2	yes	—
77	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
78	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
79	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
80	M₁₆	—	212	2	no	—
81	C₂×C₈	—	106	1	no	—
82	C₂×C₈	—	106	1	no	—
83	C₂×C₈	—	106	2	no	—
84	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
85	C₄²	Dic₁₀₆	1	1	yes	characteristic
86	C₄²	D₂₁₂	1	1	yes	characteristic
87	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
88	C₄²	C₂²×D₅₃	1	1	yes	characteristic
89	C₂³	C₄⋊Dic₅₃	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
95	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
97	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
98	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
99	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
100	C₈	—	2	4	no	—
101	C₈	—	212	1	no	—
102	C₈	—	212	2	no	—
103	C₈	—	212	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	#13133153	1	2	yes	—
112	C₄	#13133224	1	2	yes	—
113	C₄	#13133245	1	1	yes	characteristic
114	C₄	#13133250	1	1	yes	characteristic
115	C₂	—	4	1	no	—
116	C₂	#16810650	1	1	yes	characteristic
117	C₂	#16810665	1	1	yes	characteristic
118	C₂	C₂×#13133153	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810676

C₂

yes

maximal, characteristic

#16810647

C₂

yes

maximal, characteristic

#16810376

C₂

yes

maximal, complemented

#16810374

C₂

yes

maximal, complemented

C₅₃×#431

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

characteristic

C₂²⋊#88322

—

C₄×#88322

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₄

yes

—

C₂²⋊C₈×C₅₃

C₄

yes

—

C₄×C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

Q₈

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

Q₈

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

—

—

—

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

—

#88322

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#108

yes

—

C₄×C₅₃

#131

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

—

—

—

C₅₃

#1574

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

Dic₅₃

yes

—

C₂²⋊C₈

—

106

—

C₂²⋊C₈

Dic₅₃

yes

—

C₄×C₈

—

106

—

C₄×C₈

C₂×D₅₃

yes

—

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

M₁₆

—

212

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×Dic₅₃

yes

—

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

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

100

C₈

—

101

C₈

—

212

—

102

C₈

—

212

—

103

C₈

—

212

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×C₄⋊Dic₅₃

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#13133153

yes

—

112

C₄

#13133224

yes

—

113

C₄

#13133245

yes

characteristic

114

C₄

#13133250

yes

characteristic

115

C₂

—

116

C₂

#16810650

yes

characteristic

117

C₂

#16810665

yes

characteristic

118

C₂

C₂×#13133153

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#431	D₅₃
hypocenter	C₅₃	#1574
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#431

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #16810376	C₂	semidirect
4. #16810374	C₂	semidirect
65. C₅₃	#1574	semidirect

H ≅ G/N

type

3. #16810376

C₂

semidirect

4. #16810374

C₂

semidirect

65. C₅₃

#1574

semidirect

Subgroups — #20273504

Conjugacy classes of subgroups

Complements