Group #20273509 — subgroups

Subgroups — #20273509

Order 6784.

196 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic
2	#16810688	C₂	1	2	yes	maximal, complemented
3	#16810350	C₂	1	2	yes	maximal, complemented
4	C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₂₄	C₄	1	1	yes	characteristic
7	C₂×C₈×D₅₃	—	2	2	no	—
8	C₄×D₂₁₂	C₂²	1	2	yes	—
9	#13133151	C₄	1	1	yes	characteristic
10	#13133139	C₄	1	2	yes	—
11	#13133129	—	2	2	no	—
12	C₄×#88322	C₂²	1	1	yes	characteristic
13	C₄×#88322	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
17	C₂×C₄×D₅₃	—	2	2	no	—
18	C₂×C₈×C₅₃	—	2	2	no	—
19	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₅₃	C₂³	1	1	yes	characteristic
22	#11719921	—	2	2	no	—
23	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
24	C₂×#88322	—	2	1	no	—
25	C₂×#88322	D₄	1	1	yes	characteristic
26	C₂×#88322	D₄	1	1	yes	characteristic
27	D₄₂₄	C₈	1	4	yes	complemented
28	C₈×D₅₃	—	4	4	no	—
29	C₂²×D₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
33	C₂×Dic₅₃	—	2	2	no	—
34	D₂₁₂	—	2	2	no	—
35	D₂₁₂	C₂×C₈	1	4	yes	complemented
36	C₄×D₅₃	—	4	2	no	—
37	C₄×D₅₃	—	4	2	no	—
38	C₈×C₅₃	—	2	1	no	—
39	C₈×C₅₃	—	4	2	no	—
40	C₈×C₅₃	C₂×C₈	1	2	yes	complemented
41	#88322	—	2	2	no	—
42	#88322	—	2	1	no	—
43	#88322	D₈	1	2	yes	complemented
44	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
45	C₂×D₅₃	—	2	4	no	—
46	C₂×D₅₃	—	4	2	no	—
47	C₂×D₅₃	—	4	2	no	—
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
50	C₄×C₅₃	#131	1	1	yes	characteristic
51	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
52	C₄×C₅₃	#135	1	1	yes	characteristic
53	Dic₅₃	—	4	2	no	—
54	C₈×D₈	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
56	C₂×C₅₃	C₄×D₈	1	1	yes	characteristic
57	C₂×C₅₃	#443	1	1	yes	characteristic
58	D₅₃	—	4	4	no	—
59	C₄×D₈	—	53	1	no	—
60	C₈×D₄	—	53	2	no	—
61	#335	D₅₃	1	1	yes	characteristic, complemented
62	D₄⋊C₈	—	53	2	no	—
63	C₈²	—	53	1	no	—
64	C₅₃	C₈×D₈	1	1	yes	characteristic, complemented, Sylow
65	C₂×D₈	—	53	1	no	—
66	C₂²×C₈	—	106	2	no	—
67	C₄×D₄	—	53	2	no	—
68	#107	—	53	1	no	—
69	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
70	#102	—	53	2	no	—
71	C₂²⋊C₈	—	106	2	no	—
72	C₄×C₈	—	53	1	no	—
73	C₄×C₈	—	53	1	no	—
74	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
75	C₂×D₄	—	53	2	no	—
76	C₂²×C₄	—	106	2	no	—
77	D₈	—	53	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	53	1	no	—
80	C₂×C₈	—	53	1	no	—
81	C₂×C₈	—	106	1	no	—
82	C₂×C₈	—	212	4	no	—
83	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
84	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
85	C₄⋊C₄	—	53	2	no	—
86	C₂²⋊C₄	—	106	2	no	—
87	C₄²	C₂²×D₅₃	1	1	yes	characteristic
88	C₂³	—	106	2	no	—
89	D₄	—	53	4	no	—
90	D₄	—	106	2	no	—
91	C₂×C₄	—	106	2	no	—
92	C₂×C₄	—	212	2	no	—
93	C₂×C₄	—	212	2	no	—
94	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
95	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
96	C₂×C₄	#11719949	1	1	yes	characteristic
97	C₈	—	2	1	no	—
98	C₈	—	4	2	no	—
99	C₈	—	53	2	no	—
100	C₈	—	106	2	no	—
101	C₈	—	106	1	no	—
102	C₈	C₈×D₅₃	1	2	yes	complemented
103	C₂²	—	106	4	no	—
104	C₂²	—	212	2	no	—
105	C₂²	—	212	2	no	—
106	C₂²	#13133179	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	212	2	no	—
109	C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
110	C₄	#13133225	1	1	yes	characteristic
111	C₄	D₈×D₅₃	1	1	yes	characteristic
112	C₄	#13133240	1	1	yes	characteristic
113	C₂	—	212	4	no	—
114	C₂	#16810688	1	1	yes	characteristic
115	C₂	#16810705	1	1	yes	characteristic
116	C₂	#16810739	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₄₂₄

C₂

yes

maximal, characteristic

#16810688

C₂

yes

maximal, complemented

#16810350

C₂

yes

maximal, complemented

C₈×#88322

C₂

yes

maximal, characteristic, complemented

C₅₃×#335

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

C₄

yes

characteristic

C₂×C₈×D₅₃

—

C₄×D₂₁₂

C₂²

yes

—

#13133151

C₄

yes

characteristic

#13133139

C₄

yes

—

#13133129

—

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₄×D₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₂×#88322

—

C₂×#88322

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

D₄₂₄

C₈

yes

complemented

C₈×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

C₂×C₈

yes

complemented

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₈

yes

—

—

D₈

yes

complemented

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

#131

yes

characteristic

C₄×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

#135

yes

characteristic

Dic₅₃

—

C₈×D₈

—

maximal, Sylow

C₂×C₅₃

C₈×D₄

yes

characteristic

C₂×C₅₃

C₄×D₈

yes

characteristic

C₂×C₅₃

#443

yes

characteristic

D₅₃

—

C₄×D₈

—

C₈×D₄

—

#335

D₅₃

yes

characteristic, complemented

D₄⋊C₈

—

C₈²

—

C₅₃

C₈×D₈

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

—

106

—

C₄×D₄

—

#107

—

C₄⋊C₈

C₂×D₅₃

yes

—

#102

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

106

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

106

—

D₄

—

D₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

D₄×D₅₃

yes

characteristic

C₂×C₄

#11719949

yes

characteristic

C₈

—

C₈

—

C₈

—

100

C₈

—

106

—

101

C₈

—

106

—

102

C₈

C₈×D₅₃

yes

complemented

103

C₂²

—

106

—

104

C₂²

—

212

—

105

C₂²

—

212

—

106

C₂²

#13133179

yes

characteristic

107

C₄

—

108

C₄

—

212

—

109

C₄

C₂×C₈×D₅₃

yes

characteristic

110

C₄

#13133225

yes

characteristic

111

C₄

D₈×D₅₃

yes

characteristic

112

C₄

#13133240

yes

characteristic

113

C₂

—

212

—

114

C₂

#16810688

yes

characteristic

115

C₂

#16810705

yes

characteristic

116

C₂

#16810739

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#335	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	#335	D₅₃
hypocenter	C₅₃	C₈×D₈
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×D₅₃

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#335

C₂

socle

C₂²×C₅₃

C₄×D₄

hypercenter

#335

D₅₃

hypocenter

C₅₃

C₈×D₈

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
2. #16810688	C₂	semidirect
3. #16810350	C₂	semidirect
4. C₈×#88322	C₂	semidirect
5. C₅₃×#335	C₂	semidirect
27. D₄₂₄	C₈	semidirect
35. D₂₁₂	C₂×C₈	semidirect
40. C₈×C₅₃	C₂×C₈	semidirect
43. #88322	D₈	semidirect
61. #335	D₅₃	semidirect
64. C₅₃	C₈×D₈	semidirect
102. C₈	C₈×D₅₃	semidirect

H ≅ G/N

type

2. #16810688

C₂

semidirect

3. #16810350

C₂

semidirect

4. C₈×#88322

C₂

semidirect

5. C₅₃×#335

C₂

semidirect

27. D₄₂₄

C₈

semidirect

35. D₂₁₂

C₂×C₈

semidirect

40. C₈×C₅₃

C₂×C₈

semidirect

43. #88322

D₈

semidirect

61. #335

D₅₃

semidirect

64. C₅₃

C₈×D₈

semidirect

102. C₈

C₈×D₅₃

semidirect

Subgroups — #20273509

Conjugacy classes of subgroups

Complements