Group #20276896 — subgroups

Subgroups — #20276896

Order 6784.

150 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 53 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	C₂×#13133151	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810449	C₂	1	2	yes	maximal, complemented
4	#16810425	C₂	1	2	yes	maximal, complemented
5	C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
7	#13133191	—	2	1	no	—
8	C₂×C₄⋊Dic₅₃	—	2	1	no	—
9	#13133151	—	2	1	no	—
10	#13133151	C₂²	1	1	yes	characteristic
11	#13133151	C₂²	1	2	yes	—
12	#13133139	—	2	1	no	—
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₂²⋊D₅₃	—	4	1	no	—
17	C₂²×Dic₅₃	—	4	1	no	—
18	C₂×D₂₁₂	—	2	1	no	—
19	C₂×C₈×C₅₃	—	2	2	no	—
20	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	#11719921	—	4	1	no	—
23	C₄⋊Dic₅₃	—	2	1	no	—
24	C₄⋊Dic₅₃	—	2	2	no	—
25	C₄⋊Dic₅₃	—	4	1	no	—
26	D₄₂₄	—	2	2	no	—
27	C₁₆×C₅₃	—	2	2	no	—
28	C₂³×C₅₃	D₈	1	1	yes	characteristic
29	C₂²×D₅₃	—	4	1	no	—
30	C₂×C₄×C₅₃	—	2	1	no	—
31	C₂×C₄×C₅₃	—	2	1	no	—
32	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂²⋊D₅₃	—	8	2	no	—
35	C₂×Dic₅₃	—	4	2	no	—
36	C₂×Dic₅₃	—	4	1	no	—
37	C₂×Dic₅₃	—	8	2	no	—
38	D₂₁₂	—	4	1	no	—
39	D₂₁₂	—	4	1	no	—
40	C₈×C₅₃	—	2	1	no	—
41	C₈×C₅₃	#41	1	2	yes	—
42	C₂²×C₅₃	—	2	1	no	—
43	C₂²×C₅₃	—	2	1	no	—
44	C₂²×C₅₃	D₁₆	1	2	yes	—
45	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
46	C₂×D₅₃	—	8	1	no	—
47	C₂×D₅₃	—	8	1	no	—
48	C₂×D₅₃	—	8	1	no	—
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	#123	1	1	yes	characteristic
51	C₄×C₅₃	#137	1	1	yes	characteristic
52	Dic₅₃	—	8	2	no	—
53	Dic₅₃	—	8	1	no	—
54	#2232	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	—	2	2	no	—
56	C₂×C₅₃	#480	1	1	yes	characteristic
57	C₂×C₅₃	C₂×D₁₆	1	1	yes	characteristic
58	C₂×C₅₃	#510	1	1	yes	characteristic
59	D₅₃	—	16	1	no	—
60	#466	—	53	1	no	—
61	C₂×#107	—	53	1	no	—
62	#366	—	53	2	no	—
63	#357	—	53	2	no	—
64	C₂²⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
65	C₅₃	#2232	1	1	yes	characteristic, complemented, Sylow
66	C₂×D₈	—	53	1	no	—
67	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
68	#121	—	106	1	no	—
69	C₂×C₄⋊C₄	—	106	1	no	—
70	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
71	#107	—	53	2	no	—
72	#107	—	53	1	no	—
73	#107	—	106	1	no	—
74	#102	—	106	1	no	—
75	C₂×D₄	—	106	1	no	—
76	C₂×D₄	—	212	1	no	—
77	C₂²×C₄	—	212	1	no	—
78	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
79	D₈	—	106	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	D₂₁₂	1	1	yes	characteristic
82	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
83	C₄⋊C₄	—	106	1	no	—
84	C₄⋊C₄	—	106	2	no	—
85	C₄⋊C₄	—	212	1	no	—
86	C₂²⋊C₄	—	212	1	no	—
87	C₁₆	—	2	2	no	—
88	C₂³	—	212	1	no	—
89	C₂³	D₄₂₄	1	1	yes	characteristic
90	D₄	—	424	2	no	—
91	D₄	—	212	1	no	—
92	D₄	—	212	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	424	2	no	—
96	C₂×C₄	—	212	1	no	—
97	C₂×C₄	—	212	2	no	—
98	C₂×C₄	D₄₂₄	1	1	yes	characteristic
99	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
100	C₈	—	2	1	no	—
101	C₈	#11719947	1	2	yes	—
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	424	1	no	—
105	C₂²	—	424	1	no	—
106	C₂²	—	424	1	no	—
107	C₂²	D₈₄₈	1	2	yes	—
108	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
109	C₄	—	2	1	no	—
110	C₄	—	424	2	no	—
111	C₄	—	424	1	no	—
112	C₄	#13133201	1	1	yes	characteristic
113	C₄	#13133250	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	—	848	1	no	—
116	C₂	#16810937	1	1	yes	characteristic
117	C₂	C₂×D₈₄₈	1	1	yes	characteristic
118	C₂	#16811066	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810859

C₂

yes

maximal, characteristic

C₂×#13133151

C₂

yes

maximal, characteristic, complemented

#16810449

C₂

yes

maximal, complemented

#16810425

C₂

yes

maximal, complemented

C₅₃×C₂²⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

C₂²

yes

characteristic

#13133191

—

C₂×C₄⋊Dic₅₃

—

#13133151

—

#13133151

C₂²

yes

characteristic

#13133151

C₂²

yes

—

#13133139

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂×C₂²⋊D₅₃

—

C₂²×Dic₅₃

—

C₂×D₂₁₂

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

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

D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

C₈×C₅₃

#41

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

D₁₆

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

#2232

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#480

yes

characteristic

C₂×C₅₃

C₂×D₁₆

yes

characteristic

C₂×C₅₃

#510

yes

characteristic

D₅₃

—

—

—

—

—

C₂²⋊C₁₆

D₅₃

yes

characteristic, complemented

C₅₃

#2232

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#121

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₁₆

C₂×D₅₃

yes

—

#107

—

#107

—

#107

—

106

—

#102

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂²×C₄

—

212

—

C₂²×C₄

D₂₁₂

yes

characteristic

D₈

—

106

—

C₂×C₈

—

C₂×C₈

D₂₁₂

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

—

212

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

424

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

D₄₂₄

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

characteristic

100

C₈

—

101

C₈

#11719947

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

424

—

105

C₂²

—

424

—

106

C₂²

—

424

—

107

C₂²

D₈₄₈

yes

—

108

C₂²

C₂×D₄₂₄

yes

characteristic

109

C₄

—

110

C₄

—

424

—

111

C₄

—

424

—

112

C₄

#13133201

yes

characteristic

113

C₄

#13133250

yes

characteristic

114

C₂

—

115

C₂

—

848

—

116

C₂

#16810937

yes

characteristic

117

C₂

C₂×D₈₄₈

yes

characteristic

118

C₂

#16811066

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×C₂²⋊C₁₆	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₂²⋊C₁₆	D₅₃
hypocenter	C₅₃	#2232
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₄₂₄

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×C₂²⋊C₁₆

C₂

socle

C₂²×C₅₃

C₂×D₈

hypercenter

C₂²⋊C₁₆

D₅₃

hypocenter

C₅₃

#2232

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
2. C₂×#13133151	C₂	semidirect
3. #16810449	C₂	semidirect
4. #16810425	C₂	semidirect
5. C₅₃×C₂²⋊C₁₆	C₂	semidirect
64. C₂²⋊C₁₆	D₅₃	semidirect
65. C₅₃	#2232	semidirect

H ≅ G/N

type

2. C₂×#13133151

C₂

semidirect

3. #16810449

C₂

semidirect

4. #16810425

C₂

semidirect

5. C₅₃×C₂²⋊C₁₆

C₂

semidirect

64. C₂²⋊C₁₆

D₅₃

semidirect

65. C₅₃

#2232

semidirect

Subgroups — #20276896

Conjugacy classes of subgroups

Complements