Group #20290437 — subgroups

Subgroups — #20290437

Order 6784.

200 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 78 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	#16811586	C₂	1	2	yes	maximal
2	#16811492	C₂	1	1	yes	maximal, characteristic
3	#16811491	C₂	1	1	yes	maximal, characteristic
4	#16811490	C₂	1	2	yes	maximal
5	#16810580	C₂	1	1	yes	maximal, characteristic
6	D₅₃⋊M₁₆	C₂²	1	2	yes	—
7	#13133296	C₂²	1	2	yes	—
8	#13133271	C₄	1	4	yes	complemented
9	C₈×#6157	C₄	1	4	yes	complemented
10	C₂×#11719912	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#13133174	C₂²	1	1	yes	characteristic
13	#13133127	C₄	1	1	yes	characteristic, complemented
14	C₄×#88322	C₄	1	1	yes	characteristic, complemented
15	C₈×Dic₅₃	C₄	1	1	yes	characteristic, complemented
16	#97×C₅₃	C₄	1	1	yes	characteristic, complemented
17	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
18	C₂²⋊#6157	—	2	2	no	—
19	#11719940	—	2	2	no	—
20	C₄⋊#6157	—	2	2	no	—
21	C₄×#6157	C₂×C₄	1	4	yes	—
22	#11719936	—	2	2	no	—
23	D₅₃⋊C₈	C₂×C₄	1	4	yes	—
24	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	#11719921	—	2	1	no	—
28	#11719919	—	2	1	no	—
29	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×#88322	C₂×C₄	1	1	yes	characteristic
31	C₂×#88322	C₂×C₄	1	1	yes	characteristic
32	#11719912	—	2	2	no	—
33	C₈×D₅₃	C₂×C₄	1	2	yes	—
34	C₈×D₅₃	C₂×C₄	1	2	yes	—
35	C₂²×D₅₃	#41	1	1	yes	characteristic
36	C₂×#6157	—	2	4	no	—
37	C₂×C₄×C₅₃	—	2	1	no	—
38	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×Dic₅₃	—	2	1	no	—
40	C₂×Dic₅₃	#41	1	1	yes	characteristic
41	C₄×D₅₃	C₂²×C₄	1	2	yes	—
42	C₄×D₅₃	#41	1	2	yes	—
43	#88324	—	2	4	no	—
44	C₈×C₅₃	—	2	1	no	—
45	C₈×C₅₃	C₄²	1	2	yes	—
46	#88322	—	2	1	no	—
47	#88322	C₄²	1	1	yes	characteristic
48	#88322	C₄²	1	1	yes	characteristic
49	C₂²×C₅₃	#117	1	1	yes	characteristic
50	C₂×D₅₃	—	4	1	no	—
51	C₂×D₅₃	#117	1	2	yes	—
52	#6157	—	4	4	no	—
53	C₄×C₅₃	—	4	1	no	—
54	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
55	C₄×C₅₃	#117	1	1	yes	characteristic
56	Dic₅₃	—	4	1	no	—
57	Dic₅₃	#117	1	2	yes	—
58	#1747	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	—	2	1	no	—
60	C₂×C₅₃	#376	1	1	yes	characteristic
61	D₅₃	—	2	2	no	—
62	#405	—	53	2	no	—
63	#405	—	53	1	no	—
64	#338	—	53	1	no	—
65	H(3,ℤ/4)	—	53	2	no	—
66	H(3,ℤ/4)	—	53	1	no	—
67	C₅₃	#1747	1	1	yes	characteristic, complemented, Sylow
68	C₂×M₁₆	—	53	2	no	—
69	C₂×M₁₆	—	53	1	no	—
70	C₂²×C₈	—	53	1	no	—
71	#117	—	53	2	no	—
72	#117	—	53	1	no	—
73	#97	—	53	4	no	—
74	#97	—	53	1	no	—
75	#97	#6157	1	1	yes	characteristic, complemented
76	C₄×C₈	—	53	4	no	—
77	C₄×C₈	—	53	1	no	—
78	C₄×C₈	—	53	1	no	—
79	C₂²×C₄	—	53	1	no	—
80	M₁₆	—	106	2	no	—
81	M₁₆	—	106	2	no	—
82	M₁₆	—	106	2	no	—
83	C₂×C₈	—	53	1	no	—
84	C₂×C₈	—	53	2	no	—
85	C₂×C₈	—	53	2	no	—
86	C₂×C₈	—	53	4	no	—
87	C₂×C₈	—	53	1	no	—
88	C₂×C₈	C₂×#6157	1	1	yes	characteristic
89	C₂×C₈	C₂×#6157	1	1	yes	characteristic
90	C₄⋊C₄	—	106	1	no	—
91	C₄⋊C₄	—	106	2	no	—
92	C₂²⋊C₄	—	106	1	no	—
93	C₂²⋊C₄	—	106	2	no	—
94	C₄²	—	53	4	no	—
95	C₄²	—	53	1	no	—
96	C₄²	C₂×#6157	1	1	yes	characteristic
97	C₂³	—	53	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	53	1	no	—
100	C₂×C₄	—	53	2	no	—
101	C₂×C₄	—	53	2	no	—
102	C₂×C₄	—	106	4	no	—
103	C₂×C₄	—	106	1	no	—
104	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	—	53	1	no	—
107	C₈	—	53	1	no	—
108	C₈	—	106	4	no	—
109	C₈	—	106	1	no	—
110	C₈	C₄×#6157	1	2	yes	—
111	C₂²	—	53	2	no	—
112	C₂²	—	212	1	no	—
113	C₂²	#13133296	1	1	yes	characteristic
114	C₄	—	4	1	no	—
115	C₄	—	53	2	no	—
116	C₄	—	212	1	no	—
117	C₄	—	212	4	no	—
118	C₄	C₂×C₄×#6157	1	1	yes	characteristic
119	C₄	#13133296	1	1	yes	characteristic
120	C₂	—	2	1	no	—
121	C₂	—	106	2	no	—
122	C₂	#16811559	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811586

C₂

yes

maximal

#16811492

C₂

yes

maximal, characteristic

#16811491

C₂

yes

maximal, characteristic

#16811490

C₂

yes

maximal

#16810580

C₂

yes

maximal, characteristic

D₅₃⋊M₁₆

C₂²

yes

—

#13133296

C₂²

yes

—

#13133271

C₄

yes

complemented

C₈×#6157

C₄

yes

complemented

C₂×#11719912

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133174

C₂²

yes

characteristic

#13133127

C₄

yes

characteristic, complemented

C₄×#88322

C₄

yes

characteristic, complemented

C₈×Dic₅₃

C₄

yes

characteristic, complemented

#97×C₅₃

C₄

yes

characteristic, complemented

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂²⋊#6157

—

#11719940

—

C₄⋊#6157

—

C₄×#6157

C₂×C₄

yes

—

#11719936

—

D₅₃⋊C₈

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

#11719921

—

#11719919

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

#11719912

—

C₈×D₅₃

C₂×C₄

yes

—

C₈×D₅₃

C₂×C₄

yes

—

C₂²×D₅₃

#41

yes

characteristic

C₂×#6157

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

#41

yes

characteristic

C₄×D₅₃

C₂²×C₄

yes

—

C₄×D₅₃

#41

yes

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

C₄²

yes

—

#88322

—

#88322

C₄²

yes

characteristic

#88322

C₄²

yes

characteristic

C₂²×C₅₃

#117

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

#117

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

#117

yes

characteristic

Dic₅₃

—

Dic₅₃

#117

yes

—

#1747

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#376

yes

characteristic

D₅₃

—

—

—

—

—

—

C₅₃

#1747

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

—

—

—

—

yes

characteristic, complemented

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

106

—

103

C₂×C₄

—

106

—

104

C₂×C₄

C₂²×#6157

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

106

—

109

C₈

—

106

—

110

C₈

C₄×#6157

yes

—

111

C₂²

—

112

C₂²

—

212

—

113

C₂²

#13133296

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

212

—

117

C₄

—

212

—

118

C₄

C₂×C₄×#6157

yes

characteristic

119

C₄

#13133296

yes

characteristic

120

C₂

—

121

C₂

—

106

—

122

C₂

#16811559

yes

characteristic

	subgroup	G/H
center	C₄	C₂×C₄×#6157
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄	C₂²×#6157
Fitting	#97×C₅₃	C₄
socle	C₂×C₅₃	#376
hypercenter	#97	#6157
hypocenter	C₅₃	#1747
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ #97

subgroup

G/H

center

C₄

C₂×C₄×#6157

commutator

C₄×C₅₃

C₂×C₄²

Frattini

C₂×C₄

C₂²×#6157

Fitting

#97×C₅₃

C₄

C₂×C₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₄ ◁ #97

N	H ≅ G/N	type
8. #13133271	C₄	semidirect
9. C₈×#6157	C₄	semidirect
13. #13133127	C₄	semidirect
14. C₄×#88322	C₄	semidirect
15. C₈×Dic₅₃	C₄	semidirect
16. #97×C₅₃	C₄	semidirect
67. C₅₃	#1747	semidirect
75. #97	#6157	semidirect

H ≅ G/N

type

8. #13133271

C₄

semidirect

9. C₈×#6157

C₄

semidirect

13. #13133127

C₄

semidirect

14. C₄×#88322

C₄

semidirect

15. C₈×Dic₅₃

C₄

semidirect

16. #97×C₅₃

C₄

semidirect

67. C₅₃

#1747

semidirect

75. #97

#6157

semidirect

Subgroups — #20290437

Conjugacy classes of subgroups

Complements