Group #20273004 — subgroups

Subgroups — #20273004

Order 6784.

208 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	#16810345	C₂	1	2	yes	maximal, complemented
4	#16810344	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₄	1	2	yes	—
7	C₄⋊#88322	C₄	1	2	yes	—
8	C₂²⋊#88322	C₄	1	2	yes	—
9	#13133128	C₄	1	2	yes	—
10	C₄×#88322	C₂²	1	2	yes	—
11	C₄×#88322	C₂²	1	2	yes	—
12	C₂²×C₈×C₅₃	C₄	1	2	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₄	1	2	yes	—
15	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	—	2	4	no	—
21	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	#11719917	—	2	4	no	—
28	C₂×#88322	C₂×C₄	1	4	yes	—
29	C₂×#88322	C₂×C₄	1	4	yes	—
30	C₂³×C₅₃	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	—	2	2	no	—
33	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₅₃	M₁₆	1	4	yes	—
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	C₈×C₅₃	—	2	2	no	—
41	C₈×C₅₃	M₁₆	1	4	yes	complemented
42	#88322	—	2	4	no	—
43	#88322	—	2	4	no	—
44	C₂²×C₅₃	—	2	1	no	—
45	C₂²×C₅₃	—	2	1	no	—
46	C₂²×C₅₃	#97	1	2	yes	—
47	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	#97	1	2	yes	—
51	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
52	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
53	C₄×C₅₃	#131	1	2	yes	—
54	#1451	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	—	2	2	no	—
56	C₂×C₅₃	C₂×#97	1	1	yes	characteristic
57	C₂×C₅₃	C₄×M₁₆	1	1	yes	characteristic
58	C₂×C₅₃	#405	1	1	yes	characteristic
59	#431	—	53	1	no	—
60	C₄×M₁₆	—	53	1	no	—
61	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
62	#322	—	53	2	no	—
63	#322	—	53	2	no	—
64	C₅₃	#1451	1	1	yes	characteristic, complemented, Sylow
65	C₂×M₁₆	—	53	2	no	—
66	C₂²×C₈	Dic₅₃	1	2	yes	—
67	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
68	C₄⋊C₈	—	53	2	no	—
69	C₂²⋊C₈	—	53	2	no	—
70	#97	—	53	2	no	—
71	C₄×C₈	—	53	2	no	—
72	C₄×C₈	—	53	2	no	—
73	C₄×C₈	Dic₅₃	1	2	yes	—
74	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
75	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
76	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
77	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
79	M₁₆	—	106	4	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	—	53	4	no	—
82	C₂×C₈	—	53	4	no	—
83	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
84	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
85	C₄²	C₄×D₅₃	1	1	yes	characteristic
86	C₄²	C₄×D₅₃	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	#11719912	1	4	yes	—
93	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
94	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
95	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
96	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
97	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
98	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	106	4	no	—
101	C₈	—	106	4	no	—
102	C₈	#11719917	1	4	yes	complemented
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	#13133127	1	2	yes	—
106	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	#13133127	1	2	yes	—
110	C₄	C₂×#11719912	1	2	yes	—
111	C₄	C₂×#11719917	1	2	yes	—
112	C₄	#13133223	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	C₂×#13133127	1	1	yes	characteristic
115	C₂	C₄×#11719917	1	1	yes	characteristic
116	C₂	#16810578	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810676

C₂

yes

maximal, characteristic

C₄×#11719917

C₂

yes

maximal, characteristic

#16810345

C₂

yes

maximal, complemented

#16810344

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₄

yes

—

C₄⋊#88322

C₄

yes

—

C₂²⋊#88322

C₄

yes

—

#13133128

C₄

yes

—

C₄×#88322

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₂²×C₈×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₄

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

#11719917

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

M₁₆

yes

complemented

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#97

yes

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#97

yes

—

C₄×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

#131

yes

—

#1451

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#97

yes

characteristic

C₂×C₅₃

C₄×M₁₆

yes

C₂×C₅₃

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

D₅₃

yes

characteristic

#322

—

#322

—

C₅₃

#1451

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₅₃

yes

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

C₄×Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#11719912

yes

—

C₂×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₈

—

100

C₈

—

106

—

101

C₈

—

106

—

102

C₈

#11719917

yes

complemented

103

C₂²

—

104

C₂²

—

105

C₂²

#13133127

yes

—

106

C₂²

C₂×C₄×Dic₅₃

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

#13133127

yes

—

110

C₄

C₂×#11719912

yes

—

111

C₄

C₂×#11719917

yes

—

112

C₄

#13133223

yes

—

113

C₂

—

114

C₂

C₂×#13133127

yes

characteristic

115

C₂

C₄×#11719917

yes

characteristic

116

C₂

#16810578

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂×C₄²
Fitting	C₂×C₄×C₈×C₅₃	C₂
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#1451
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₄²

C₂²×D₅₃

commutator, socle

C₂²×C₅₃

C₂×C₄²

Fitting

C₂×C₄×C₈×C₅₃

C₂

hypercenter

C₂×C₄×C₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
3. #16810345	C₂	semidirect
4. #16810344	C₂	semidirect
41. C₈×C₅₃	M₁₆	semidirect
64. C₅₃	#1451	semidirect
102. C₈	#11719917	semidirect

H ≅ G/N

type

3. #16810345

C₂

semidirect

4. #16810344

C₂

semidirect

41. C₈×C₅₃

M₁₆

semidirect

64. C₅₃

#1451

semidirect

102. C₈

#11719917

semidirect

Subgroups — #20273004

Conjugacy classes of subgroups

Complements