Group #20272725 — subgroups

Subgroups — #20272725

Order 6784.

126 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 39 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	#16810859	C₂	1	1	yes	maximal, characteristic
2	#16810386	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄₂₄	C₄	1	1	yes	characteristic, complemented
5	#13133191	—	2	1	no	—
6	C₂×C₄⋊Dic₅₃	—	2	1	no	—
7	#13133151	C₄	1	1	yes	characteristic, complemented
8	#13133139	—	2	1	no	—
9	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×C₅₃	—	2	1	no	—
11	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
12	C₂×C₂²⋊D₅₃	—	4	1	no	—
13	C₂²×Dic₅₃	—	4	1	no	—
14	C₂×D₂₁₂	—	2	1	no	—
15	C₂×C₈×C₅₃	—	4	1	no	—
16	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
18	#11719921	—	4	1	no	—
19	C₄⋊Dic₅₃	—	2	2	no	—
20	C₄⋊Dic₅₃	—	2	1	no	—
21	C₄⋊Dic₅₃	—	4	1	no	—
22	D₄₂₄	—	4	1	no	—
23	C₁₆×C₅₃	—	4	1	no	—
24	C₂³×C₅₃	D₈	1	1	yes	characteristic
25	C₂²×D₅₃	—	4	1	no	—
26	C₂×C₄×C₅₃	—	2	1	no	—
27	C₂×C₄×C₅₃	—	2	1	no	—
28	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
30	C₂²⋊D₅₃	—	8	1	no	—
31	C₂²⋊D₅₃	—	8	1	no	—
32	C₂×Dic₅₃	—	4	2	no	—
33	C₂×Dic₅₃	—	4	1	no	—
34	C₂×Dic₅₃	—	8	1	no	—
35	C₂×Dic₅₃	—	8	1	no	—
36	D₂₁₂	—	4	1	no	—
37	D₂₁₂	—	4	1	no	—
38	C₈×C₅₃	—	2	1	no	—
39	C₈×C₅₃	—	2	1	no	—
40	C₂²×C₅₃	—	2	1	no	—
41	C₂²×C₅₃	—	2	1	no	—
42	C₂²×C₅₃	#102	1	1	yes	characteristic
43	C₂²×C₅₃	D₁₆	1	1	yes	characteristic
44	C₂²×C₅₃	SD₃₂	1	1	yes	characteristic
45	C₂×D₅₃	—	8	1	no	—
46	C₂×D₅₃	—	8	1	no	—
47	C₂×D₅₃	—	8	1	no	—
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	#99	1	1	yes	characteristic
50	C₄×C₅₃	#104	1	1	yes	characteristic
51	Dic₅₃	—	8	2	no	—
52	Dic₅₃	—	8	1	no	—
53	#1347	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	—	2	1	no	—
55	C₂×C₅₃	—	2	1	no	—
56	C₂×C₅₃	#327	1	1	yes	characteristic
57	C₂×C₅₃	#357	1	1	yes	characteristic
58	C₂×C₅₃	#360	1	1	yes	characteristic
59	D₅₃	—	16	1	no	—
60	#466	—	53	1	no	—
61	C₂²⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
62	#340	—	53	1	no	—
63	C₅₃	#1347	1	1	yes	characteristic, complemented, Sylow
64	C₂×D₈	—	53	1	no	—
65	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
66	#121	—	106	1	no	—
67	C₂×C₄⋊C₄	—	106	1	no	—
68	C₂×C₁₆	—	2	1	no	—
69	#107	—	53	1	no	—
70	#102	—	106	1	no	—
71	C₂×D₄	—	106	1	no	—
72	C₂×D₄	—	212	1	no	—
73	C₂²×C₄	—	212	1	no	—
74	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
75	D₈	—	212	1	no	—
76	C₂×C₈	—	4	1	no	—
77	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
78	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
79	C₄⋊C₄	—	106	2	no	—
80	C₄⋊C₄	—	106	1	no	—
81	C₄⋊C₄	—	212	1	no	—
82	C₂²⋊C₄	—	212	1	no	—
83	C₁₆	—	4	1	no	—
84	C₂³	—	212	1	no	—
85	C₂³	D₄₂₄	1	1	yes	characteristic
86	D₄	—	424	1	no	—
87	D₄	—	424	1	no	—
88	D₄	—	212	1	no	—
89	D₄	—	212	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	424	1	no	—
93	C₂×C₄	—	424	1	no	—
94	C₂×C₄	—	212	2	no	—
95	C₂×C₄	—	212	1	no	—
96	C₂×C₄	#11719913	1	1	yes	characteristic
97	C₂×C₄	#11719921	1	1	yes	characteristic
98	C₈	—	2	1	no	—
99	C₈	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	424	1	no	—
103	C₂²	—	424	1	no	—
104	C₂²	—	424	1	no	—
105	C₂²	#13133139	1	1	yes	characteristic
106	C₂²	D₈₄₈	1	1	yes	characteristic
107	C₂²	#13133162	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	424	2	no	—
110	C₄	—	424	1	no	—
111	C₄	#13133131	1	1	yes	characteristic
112	C₄	#13133143	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	—	848	1	no	—
116	C₂	#16810358	1	1	yes	characteristic
117	C₂	#16810425	1	1	yes	characteristic
118	C₂	#16810434	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810859

C₂

yes

maximal, characteristic

#16810386

C₂

yes

maximal, characteristic, complemented

C₅₃×C₂²⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

C₄

yes

characteristic, complemented

#13133191

—

C₂×C₄⋊Dic₅₃

—

#13133151

C₄

yes

characteristic, complemented

#13133139

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂×C₂²⋊D₅₃

—

C₂²×Dic₅₃

—

C₂×D₂₁₂

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

D₄₂₄

—

C₁₆×C₅₃

—

C₂³×C₅₃

D₈

yes

characteristic

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

characteristic

C₂²⋊D₅₃

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#102

yes

characteristic

C₂²×C₅₃

D₁₆

yes

characteristic

C₂²×C₅₃

SD₃₂

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

#1347

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#466

—

C₂²⋊C₁₆

D₅₃

yes

characteristic, complemented

#340

—

C₅₃

#1347

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#121

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₁₆

—

#107

—

#102

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂²×C₄

—

212

—

C₂²×C₄

D₂₁₂

yes

characteristic

D₈

—

212

—

C₂×C₈

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

C₁₆

—

C₂³

—

212

—

C₂³

D₄₂₄

yes

characteristic

D₄

—

424

—

D₄

—

424

—

D₄

—

212

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

424

—

C₂×C₄

—

424

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

424

—

103

C₂²

—

424

—

104

C₂²

—

424

—

105

C₂²

#13133139

yes

characteristic

106

C₂²

D₈₄₈

yes

characteristic

107

C₂²

#13133162

yes

characteristic

108

C₄

—

109

C₄

—

424

—

110

C₄

—

424

—

111

C₄

#13133131

yes

characteristic

112

C₄

#13133143

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

—

848

—

116

C₂

#16810358

yes

characteristic

117

C₂

#16810425

yes

characteristic

118

C₂

#16810434

yes

characteristic

	subgroup	G/H
center	C₂²	#13133139
commutator	C₂×C₈×C₅₃	C₂×C₄
Frattini	C₂²×C₈	C₂×D₅₃
Fitting	C₅₃×C₂²⋊C₁₆	C₂
socle	C₂²×C₅₃	#102
hypercenter	C₂²⋊C₁₆	D₅₃
hypocenter	C₅₃	#1347
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center

C₂²

#13133139

commutator

C₂×C₈×C₅₃

C₂×C₄

Frattini

C₂²×C₈

C₂×D₅₃

Fitting

C₅₃×C₂²⋊C₁₆

C₂

socle

C₂²×C₅₃

#102

hypercenter

C₂²⋊C₁₆

D₅₃

hypocenter

C₅₃

#1347

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
2. #16810386	C₂	semidirect
3. C₅₃×C₂²⋊C₁₆	C₂	semidirect
4. C₂×D₄₂₄	C₄	semidirect
7. #13133151	C₄	semidirect
61. C₂²⋊C₁₆	D₅₃	semidirect
63. C₅₃	#1347	semidirect

H ≅ G/N

type

2. #16810386

C₂

semidirect

3. C₅₃×C₂²⋊C₁₆

C₂

semidirect

4. C₂×D₄₂₄

C₄

semidirect

7. #13133151

C₄

semidirect

61. C₂²⋊C₁₆

D₅₃

semidirect

63. C₅₃

#1347

semidirect

Subgroups — #20272725

Conjugacy classes of subgroups

Complements