Group #20274200 — subgroups

Subgroups — #20274200

Order 6784.

184 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16811000	C₂	1	1	yes	maximal, characteristic
2	#16810824	C₂	1	2	yes	maximal, complemented
3	#16810753	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810366	C₂	1	2	yes	maximal, complemented
5	C₅₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719913	—	2	2	no	—
7	C₂×D₄₂₄	—	2	1	no	—
8	C₂×D₄₂₄	—	2	1	no	—
9	#13133214	C₂²	1	2	yes	—
10	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
11	C₄⋊#88322	C₂²	1	1	yes	characteristic
12	#13133146	—	2	1	no	—
13	#13133139	—	2	2	no	—
14	#13133127	—	2	1	no	—
15	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×D₂₁₂	—	2	2	no	—
18	C₂×D₂₁₂	—	2	2	no	—
19	C₂×D₂₁₂	—	2	2	no	—
20	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	—	2	2	no	—
22	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₄²×C₅₃	C₂³	1	1	yes	characteristic
25	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
26	#11719919	—	2	1	no	—
27	C₄×Dic₅₃	—	2	1	no	—
28	C₂×#88322	—	2	1	no	—
29	D₄₂₄	—	4	2	no	—
30	D₄₂₄	—	4	2	no	—
31	#11719913	—	4	4	no	—
32	C₂²×D₅₃	—	4	2	no	—
33	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
36	C₂×Dic₅₃	—	2	1	no	—
37	C₂×Dic₅₃	—	2	1	no	—
38	D₂₁₂	—	4	2	no	—
39	D₂₁₂	—	4	2	no	—
40	D₂₁₂	—	4	2	no	—
41	D₂₁₂	—	4	2	no	—
42	D₂₁₂	—	8	2	no	—
43	Dic₁₀₆	—	2	2	no	—
44	C₈×C₅₃	—	2	1	no	—
45	C₈×C₅₃	—	4	2	no	—
46	C₈×C₅₃	SD₁₆	1	2	yes	complemented
47	#88322	—	4	1	no	—
48	C₂²×C₅₃	#121	1	1	yes	characteristic
49	C₂×D₅₃	—	8	2	no	—
50	C₂×D₅₃	—	8	2	no	—
51	C₂×D₅₃	—	8	2	no	—
52	C₄×C₅₃	—	2	1	no	—
53	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
54	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
56	Dic₅₃	—	4	1	no	—
57	Dic₅₃	—	4	1	no	—
58	#1688	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	#460	1	1	yes	characteristic
60	C₂×C₅₃	#469	1	1	yes	characteristic
61	C₂×C₅₃	#472	1	1	yes	characteristic
62	D₅₃	—	16	2	no	—
63	#493	—	53	1	no	—
64	#460	—	53	2	no	—
65	#446	—	53	1	no	—
66	#334	D₅₃	1	1	yes	characteristic, complemented
67	#331	—	53	2	no	—
68	C₅₃	#1688	1	1	yes	characteristic, complemented, Sylow
69	C₂×SD₁₆	—	106	2	no	—
70	C₂×D₈	—	106	1	no	—
71	C₂×D₈	—	106	1	no	—
72	#127	—	53	2	no	—
73	C₄×Q₈	—	53	1	no	—
74	C₄⋊C₈	—	53	1	no	—
75	C₄⋊C₈	—	106	1	no	—
76	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
77	#102	—	106	2	no	—
78	#97	—	106	1	no	—
79	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
80	C₂×Q₈	—	53	1	no	—
81	C₂×D₄	—	106	2	no	—
82	C₂×D₄	—	106	2	no	—
83	C₂×D₄	—	106	2	no	—
84	SD₁₆	—	212	4	no	—
85	D₈	—	212	2	no	—
86	D₈	—	212	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	106	1	no	—
89	C₂×C₈	D₂₁₂	1	1	yes	characteristic
90	C₂×C₈	D₂₁₂	1	1	yes	characteristic
91	C₄⋊C₄	—	53	1	no	—
92	C₄⋊C₄	—	106	1	no	—
93	C₄²	—	106	1	no	—
94	C₄²	C₂²×D₅₃	1	1	yes	characteristic
95	C₂³	—	212	2	no	—
96	Q₈	—	106	2	no	—
97	D₄	—	424	2	no	—
98	D₄	—	212	2	no	—
99	D₄	—	212	2	no	—
100	D₄	—	212	2	no	—
101	D₄	—	212	2	no	—
102	C₂×C₄	—	106	1	no	—
103	C₂×C₄	—	106	1	no	—
104	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
105	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
106	C₂×C₄	#11719949	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	4	2	no	—
109	C₈	—	212	1	no	—
110	C₈	#11719913	1	2	yes	complemented
111	C₂²	—	424	2	no	—
112	C₂²	—	424	2	no	—
113	C₂²	—	424	2	no	—
114	C₂²	#13133192	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	212	1	no	—
117	C₄	—	212	1	no	—
118	C₄	C₂×#11719913	1	1	yes	characteristic
119	C₄	#13133244	1	2	yes	—
120	C₄	#13133245	1	1	yes	characteristic
121	C₂	—	848	2	no	—
122	C₂	#16810824	1	1	yes	characteristic
123	C₂	#16810877	1	1	yes	characteristic
124	C₂	#16810894	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811000

C₂

yes

maximal, characteristic

#16810824

C₂

yes

maximal, complemented

#16810753

C₂

yes

maximal, characteristic, complemented

#16810366

C₂

yes

maximal, complemented

C₅₃×#334

C₂

yes

maximal, characteristic, complemented

C₂×#11719913

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

#13133214

C₂²

yes

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

—

C₄⋊C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×Dic₁₀₆

D₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

D₄

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

D₄₂₄

—

D₄₂₄

—

#11719913

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

SD₁₆

yes

complemented

#88322

—

C₂²×C₅₃

#121

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

—

—

—

D₅₃

yes

characteristic, complemented

#331

—

C₅₃

#1688

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

106

—

C₂×D₈

—

106

—

C₂×D₈

—

106

—

#127

—

C₄×Q₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

C₂×D₅₃

yes

—

#102

—

106

—

#97

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

SD₁₆

—

212

—

D₈

—

212

—

D₈

—

212

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄²

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

212

—

Q₈

—

106

—

D₄

—

424

—

D₄

—

212

—

D₄

—

212

—

100

D₄

—

212

—

101

D₄

—

212

—

102

C₂×C₄

—

106

—

103

C₂×C₄

—

106

—

104

C₂×C₄

C₂×D₂₁₂

yes

characteristic

105

C₂×C₄

D₄×D₅₃

yes

characteristic

106

C₂×C₄

#11719949

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

212

—

110

C₈

#11719913

yes

complemented

111

C₂²

—

424

—

112

C₂²

—

424

—

113

C₂²

—

424

—

114

C₂²

#13133192

yes

characteristic

115

C₄

—

116

C₄

—

212

—

117

C₄

—

212

—

118

C₄

C₂×#11719913

yes

characteristic

119

C₄

#13133244

yes

—

120

C₄

#13133245

yes

characteristic

121

C₂

—

848

—

122

C₂

#16810824

yes

characteristic

123

C₂

#16810877

yes

characteristic

124

C₂

#16810894

yes

characteristic

	subgroup	G/H
center	C₂²	#13133192
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#334	C₂
socle	C₂²×C₅₃	#121
hypercenter	#334	D₅₃
hypocenter	C₅₃	#1688
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#13133192

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#334

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
2. #16810824	C₂	semidirect
3. #16810753	C₂	semidirect
4. #16810366	C₂	semidirect
5. C₅₃×#334	C₂	semidirect
46. C₈×C₅₃	SD₁₆	semidirect
66. #334	D₅₃	semidirect
68. C₅₃	#1688	semidirect
110. C₈	#11719913	semidirect

H ≅ G/N

type

2. #16810824

C₂

semidirect

3. #16810753

C₂

semidirect

4. #16810366

C₂

semidirect

5. C₅₃×#334

C₂

semidirect

46. C₈×C₅₃

SD₁₆

semidirect

66. #334

D₅₃

semidirect

68. C₅₃

#1688

semidirect

110. C₈

#11719913

semidirect

Subgroups — #20274200

Conjugacy classes of subgroups

Complements