Group #20274596 — subgroups

Subgroups — #20274596

Order 6784.

232 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 117 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×#13133153	C₂	1	1	yes	maximal, characteristic
2	#16810579	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810382	C₂	1	2	yes	maximal
5	C₅₃×#427	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₂²×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
9	#13133153	C₄	1	4	yes	complemented
10	#13133128	—	2	2	no	—
11	C₄×#88322	—	2	2	no	—
12	C₈×Dic₅₃	C₄	1	4	yes	complemented
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	#117×C₅₃	C₂²	1	2	yes	—
15	#107×C₅₃	C₄	1	2	yes	complemented
16	#106×C₅₃	C₄	1	2	yes	complemented
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₃	—	2	1	no	—
19	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
24	C₂²⋊C₄×C₅₃	—	2	2	no	—
25	C₄²×C₅₃	—	2	2	no	—
26	C₄×Dic₅₃	—	2	1	no	—
27	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
28	#11719917	—	2	2	no	—
29	#11719917	C₂×C₄	1	4	yes	—
30	C₂×#88322	—	2	2	no	—
31	C₂×#88322	C₂×C₄	1	2	yes	—
32	C₂×#88322	D₄	1	1	yes	characteristic
33	C₂×#88322	D₄	1	1	yes	characteristic
34	C₂×#88322	Q₈	1	2	yes	—
35	C₂³×C₅₃	#41	1	1	yes	characteristic
36	C₂×C₄×C₅₃	—	2	4	no	—
37	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
43	C₂×Dic₅₃	—	2	4	no	—
44	C₂×Dic₅₃	—	2	2	no	—
45	C₈×C₅₃	C₄²	1	2	yes	—
46	C₈×C₅₃	C₄²	1	2	yes	—
47	#88322	—	2	4	no	—
48	#88322	C₄⋊C₄	1	2	yes	—
49	#88322	C₄⋊C₄	1	2	yes	—
50	C₂²×C₅₃	—	2	2	no	—
51	C₂²×C₅₃	#117	1	1	yes	characteristic
52	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₅₃	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₅₃	—	4	4	no	—
55	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
56	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
58	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
59	Dic₅₃	—	2	4	no	—
60	#1772	—	53	1	no	maximal, Sylow
61	C₂×C₅₃	—	2	1	no	—
62	C₂×C₅₃	—	2	1	no	—
63	C₂×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₅₃	#443	1	2	yes	—
65	C₂×#108	—	53	1	no	—
66	#427	D₅₃	1	1	yes	characteristic
67	#405	—	53	2	no	—
68	C₂×C₄×C₈	—	53	1	no	—
69	#339	—	53	2	no	—
70	C₅₃	#1772	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	—	53	2	no	—
72	C₂²×C₈	—	53	1	no	—
73	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
74	#117	C₂×D₅₃	1	2	yes	—
75	C₂×C₄²	—	53	1	no	—
76	#108	—	53	4	no	—
77	#107	Dic₅₃	1	2	yes	complemented
78	#106	Dic₅₃	1	2	yes	complemented
79	#97	—	106	2	no	—
80	C₄×C₈	—	53	4	no	—
81	C₄×C₈	—	106	2	no	—
82	C₂²×C₄	—	106	1	no	—
83	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
84	M₁₆	—	53	4	no	—
85	M₁₆	—	106	2	no	—
86	C₂×C₈	—	53	2	no	—
87	C₂×C₈	—	53	1	no	—
88	C₂×C₈	—	53	1	no	—
89	C₂×C₈	—	53	2	no	—
90	C₂×C₈	—	106	2	no	—
91	C₂×C₈	C₄×D₅₃	1	2	yes	—
92	C₂×C₈	C₄×D₅₃	1	2	yes	—
93	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
94	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
95	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
96	C₂²⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	—	53	2	no	—
99	C₄²	—	106	1	no	—
100	C₂³	#11719947	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	106	4	no	—
103	C₂×C₄	—	106	2	no	—
104	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
106	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
108	C₂×C₄	#11719949	1	1	yes	characteristic
109	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
110	C₈	—	53	2	no	—
111	C₈	—	53	2	no	—
112	C₈	—	106	4	no	—
113	C₈	C₄×Dic₅₃	1	2	yes	—
114	C₈	C₄×Dic₅₃	1	2	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	#13133175	1	1	yes	characteristic
117	C₂²	D₄×Dic₅₃	1	1	yes	characteristic
118	C₂²	#13133182	1	1	yes	characteristic
119	C₄	—	4	4	no	—
120	C₄	—	106	4	no	—
121	C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
122	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
123	C₄	#13133179	1	1	yes	characteristic
124	C₄	Q₈×Dic₅₃	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
128	C₂	#16810740	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133153

C₂

yes

maximal, characteristic

#16810579

C₂

yes

maximal

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic

#16810382

C₂

yes

maximal

C₅₃×#427

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133153

C₄

yes

complemented

#13133128

—

C₄×#88322

—

C₈×Dic₅₃

C₄

yes

complemented

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

—

#107×C₅₃

C₄

yes

complemented

#106×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

—

#11719917

—

#11719917

C₂×C₄

yes

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

C₂×#88322

Q₈

yes

—

C₂³×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

C₄²

yes

—

C₈×C₅₃

C₄²

yes

—

#88322

—

#88322

C₄⋊C₄

yes

—

#88322

C₄⋊C₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

#117

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

C₄×Q₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

Dic₅₃

—

#1772

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₅₃

#443

yes

—

C₂×#108

—

#427

D₅₃

yes

characteristic

#405

—

C₂×C₄×C₈

—

#339

—

C₅₃

#1772

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

C₂×D₅₃

yes

—

C₂×C₄²

—

#108

—

#107

Dic₅₃

yes

complemented

#106

Dic₅₃

yes

complemented

#97

—

106

—

C₄×C₈

—

C₄×C₈

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

106

—

100

C₂³

#11719947

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

106

—

103

C₂×C₄

—

106

—

104

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

105

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

106

C₂×C₄

D₄×D₅₃

yes

characteristic

107

C₂×C₄

Q₈×D₅₃

yes

characteristic

108

C₂×C₄

#11719949

yes

characteristic

109

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

106

—

113

C₈

C₄×Dic₅₃

yes

—

114

C₈

C₄×Dic₅₃

yes

—

115

C₂²

—

116

C₂²

#13133175

yes

characteristic

117

C₂²

D₄×Dic₅₃

yes

characteristic

118

C₂²

#13133182

yes

characteristic

119

C₄

—

120

C₄

—

106

—

121

C₄

C₂×C₄×Dic₅₃

yes

characteristic

122

C₄

C₄⋊C₄×D₅₃

yes

characteristic

123

C₄

#13133179

yes

characteristic

124

C₄

Q₈×Dic₅₃

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₄⋊C₄×Dic₅₃

yes

characteristic

128

C₂

#16810740

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#427	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	#427	D₅₃
hypocenter	C₅₃	#1772
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₅₃

commutator

C₄×C₅₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#427

C₂

socle

C₂²×C₅₃

C₄×D₄

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
9. #13133153	C₄	semidirect
12. C₈×Dic₅₃	C₄	semidirect
15. #107×C₅₃	C₄	semidirect
16. #106×C₅₃	C₄	semidirect
70. C₅₃	#1772	semidirect
77. #107	Dic₅₃	semidirect
78. #106	Dic₅₃	semidirect

H ≅ G/N

type

9. #13133153

C₄

semidirect

12. C₈×Dic₅₃

C₄

semidirect

15. #107×C₅₃

C₄

semidirect

16. #106×C₅₃

C₄

semidirect

70. C₅₃

#1772

semidirect

77. #107

Dic₅₃

semidirect

78. #106

Dic₅₃

semidirect

Subgroups — #20274596

Conjugacy classes of subgroups

Complements