Group #20273473 — subgroups

Subgroups — #20273473

Order 6784.

184 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16810647	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#16810375	C₂	1	4	yes	maximal, complemented
4	C₅₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	—	2	1	no	—
6	C₂²×#88322	C₄	1	2	yes	—
7	C₄⋊#88322	—	2	1	no	—
8	C₄⋊#88322	C₂²	1	2	yes	—
9	C₄×#88322	C₄	1	1	yes	characteristic
10	C₄×#88322	C₄	1	1	yes	characteristic
11	C₄×#88322	C₂²	1	2	yes	—
12	C₂×M₁₆×C₅₃	—	2	1	no	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₅₃	—	2	1	no	—
15	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
19	M₁₆×C₅₃	—	4	2	no	—
20	C₂×C₈×C₅₃	—	2	2	no	—
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	4	no	—
27	C₂×#88322	—	2	2	no	—
28	C₂×#88322	C₂×C₄	1	2	yes	—
29	C₂×#88322	C₂×C₄	1	2	yes	—
30	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	—	2	2	no	—
33	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	D₈	1	2	yes	—
37	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
38	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₈×C₅₃	—	4	2	no	—
42	#88322	—	2	2	no	—
43	#88322	—	4	2	no	—
44	#88322	M₁₆	1	4	yes	complemented
45	C₂²×C₅₃	—	2	1	no	—
46	C₂²×C₅₃	—	2	1	no	—
47	C₂²×C₅₃	#107	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	#107	1	2	yes	—
52	C₄×C₅₃	#108	1	2	yes	—
53	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
54	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
55	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
56	#1567	—	53	1	no	maximal, Sylow
57	C₂×C₅₃	—	2	2	no	—
58	C₂×C₅₃	#423	1	1	yes	characteristic
59	C₂×C₅₃	C₂×#107	1	1	yes	characteristic
60	C₂×C₅₃	C₂×#108	1	1	yes	characteristic
61	#423	—	53	1	no	—
62	#423	D₅₃	1	1	yes	characteristic
63	C₂×C₄×C₈	—	53	1	no	—
64	#335	—	53	4	no	—
65	C₅₃	#1567	1	1	yes	characteristic, complemented, Sylow
66	C₂×M₁₆	—	2	1	no	—
67	C₂×M₁₆	—	106	1	no	—
68	C₂²×C₈	—	53	2	no	—
69	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	53	2	no	—
72	C₄⋊C₈	—	106	1	no	—
73	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
74	C₄×C₈	—	53	1	no	—
75	C₄×C₈	—	53	1	no	—
76	C₄×C₈	—	53	2	no	—
77	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
78	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
80	M₁₆	—	4	2	no	—
81	M₁₆	—	212	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	53	2	no	—
84	C₂×C₈	—	53	2	no	—
85	C₂×C₈	—	106	4	no	—
86	C₂×C₈	—	106	2	no	—
87	C₄²	Dic₁₀₆	1	1	yes	characteristic
88	C₄²	C₄×D₅₃	1	1	yes	characteristic
89	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
90	C₄²	C₂²×D₅₃	1	1	yes	characteristic
91	C₂³	#11719919	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	#11719919	1	1	yes	characteristic
95	C₂×C₄	#11719919	1	1	yes	characteristic
96	C₂×C₄	#11719919	1	1	yes	characteristic
97	C₂×C₄	#11719922	1	2	yes	—
98	C₂×C₄	#11719925	1	2	yes	—
99	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
102	C₈	—	4	2	no	—
103	C₈	—	53	4	no	—
104	C₈	—	106	2	no	—
105	C₈	—	212	2	no	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#13133150	1	2	yes	—
109	C₂²	C₂×#11719919	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	#13133150	1	2	yes	—
113	C₄	#13133152	1	2	yes	—
114	C₄	M₁₆×D₅₃	1	2	yes	—
115	C₄	C₂×#11719922	1	1	yes	characteristic
116	C₄	C₂×#11719925	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	#16810646	1	1	yes	characteristic
119	C₂	C₂×#13133150	1	1	yes	characteristic
120	C₂	C₂×#13133152	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810647

C₂

yes

maximal, characteristic

C₂×C₄×#88322

C₂

yes

maximal, characteristic

#16810375

C₂

yes

maximal, complemented

C₅₃×#423

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₂²×#88322

C₄

yes

—

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

—

C₄×#88322

C₄

yes

characteristic

C₄×#88322

C₄

yes

characteristic

C₄×#88322

C₂²

yes

—

C₂×M₁₆×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

Q₈

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

Q₈

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

—

—

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

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

D₈

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

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

M₁₆

yes

complemented

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#107

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#107

yes

—

C₄×C₅₃

#108

yes

—

C₄×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

characteristic

#1567

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

characteristic

C₂×C₄×C₈

—

#335

—

C₅₃

#1567

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

C₂×D₅₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

Dic₁₀₆

yes

characteristic

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

characteristic

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

#11719919

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#11719925

yes

—

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

100

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

106

—

105

C₈

—

212

—

106

C₂²

—

107

C₂²

—

108

C₂²

#13133150

yes

—

109

C₂²

C₂×#11719919

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

#13133150

yes

—

113

C₄

#13133152

yes

—

114

C₄

M₁₆×D₅₃

yes

—

115

C₄

C₂×#11719922

yes

characteristic

116

C₄

C₂×#11719925

yes

characteristic

117

C₂

—

118

C₂

#16810646

yes

characteristic

119

C₂

C₂×#13133150

yes

characteristic

120

C₂

C₂×#13133152

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#423	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#423	D₅₃
hypocenter	C₅₃	#1567
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#423

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #16810375	C₂	semidirect
44. #88322	M₁₆	semidirect
65. C₅₃	#1567	semidirect

H ≅ G/N

type

3. #16810375

C₂

semidirect

44. #88322

M₁₆

semidirect

65. C₅₃

#1567

semidirect

Subgroups — #20273473

Conjugacy classes of subgroups

Complements