Group #20290281 — subgroups

Subgroups — #20290281

Order 6784.

166 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 64 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	#16811645	C₂	1	1	yes	maximal, characteristic
2	#16811627	C₂	1	1	yes	maximal, characteristic
3	#16811487	C₂	1	2	yes	maximal, complemented
4	#16811485	C₂	1	2	yes	maximal, complemented
5	#16810575	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719936	—	2	1	no	—
7	C₄⋊#88324	—	2	1	no	—
8	C₄⋊#88324	C₂²	1	2	yes	—
9	C₄⋊#88324	C₂²	1	1	yes	characteristic
10	C₄⋊#88324	C₂²	1	1	yes	characteristic
11	#13133274	—	2	1	no	—
12	#13133272	—	2	1	no	—
13	C₂×#11719912	C₄	1	2	yes	—
14	C₄²×D₅₃	C₂²	1	1	yes	characteristic
15	#13133128	C₄	1	1	yes	characteristic
16	C₈×Dic₅₃	C₂²	1	2	yes	—
17	#97×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
20	C₂×C₄×D₅₃	Q₈	1	1	yes	characteristic
21	C₂×#88324	—	2	1	no	—
22	C₂×#88324	—	2	1	no	—
23	C₂×#88324	—	2	2	no	—
24	#11719936	—	4	2	no	—
25	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₅₃	D₄	1	1	yes	characteristic
28	C₄×Dic₅₃	Q₈	1	1	yes	characteristic
29	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
30	C₂×#88322	C₂×C₄	1	2	yes	—
31	#11719912	—	2	4	no	—
32	C₂²×D₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₄×D₅₃	—	2	2	no	—
40	C₄×D₅₃	—	2	1	no	—
41	C₄×D₅₃	—	2	1	no	—
42	C₄×D₅₃	—	4	1	no	—
43	#88324	—	4	2	no	—
44	#88324	—	4	1	no	—
45	#88324	—	4	1	no	—
46	C₈×C₅₃	—	2	2	no	—
47	#88322	M₁₆	1	4	yes	complemented
48	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×D₅₃	—	2	1	no	—
50	C₂×D₅₃	—	2	1	no	—
51	C₂×D₅₃	—	2	1	no	—
52	C₄×C₅₃	—	2	1	no	—
53	C₄×C₅₃	—	2	1	no	—
54	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
55	Dic₅₃	—	2	1	no	—
56	Dic₅₃	—	2	1	no	—
57	Dic₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
58	Dic₅₃	#137	1	1	yes	characteristic
59	#1569	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	#423	1	1	yes	characteristic
61	C₂×C₅₃	#428	1	1	yes	characteristic
62	C₂×C₅₃	#430	1	1	yes	characteristic
63	D₅₃	—	4	1	no	—
64	#432	—	53	1	no	—
65	#423	—	53	1	no	—
66	C₄×M₁₆	—	53	1	no	—
67	#335	—	53	2	no	—
68	#334	—	53	2	no	—
69	C₅₃	#1569	1	1	yes	characteristic, complemented, Sylow
70	C₂×M₁₆	—	53	2	no	—
71	C₂×M₁₆	—	106	1	no	—
72	C₂×C₄²	—	53	1	no	—
73	C₄⋊C₈	—	53	1	no	—
74	C₄⋊C₈	—	53	2	no	—
75	C₄⋊C₈	—	53	1	no	—
76	C₄⋊C₈	—	106	1	no	—
77	C₂²⋊C₈	—	106	1	no	—
78	#97	—	53	1	no	—
79	#97	—	106	1	no	—
80	#97	#6157	1	1	yes	characteristic
81	C₄×C₈	—	53	2	no	—
82	C₂²×C₄	—	53	1	no	—
83	C₂²×C₄	—	53	1	no	—
84	C₂²×C₄	—	53	1	no	—
85	M₁₆	—	106	4	no	—
86	M₁₆	—	212	2	no	—
87	C₂×C₈	—	53	2	no	—
88	C₂×C₈	—	106	1	no	—
89	C₂×C₈	—	106	1	no	—
90	C₂×C₈	—	106	2	no	—
91	C₂×C₈	C₂×#6157	1	2	yes	—
92	C₄²	—	53	1	no	—
93	C₄²	—	53	1	no	—
94	C₄²	—	53	1	no	—
95	C₄²	C₂×#6157	1	1	yes	characteristic
96	C₂³	—	53	1	no	—
97	C₂×C₄	—	53	1	no	—
98	C₂×C₄	—	53	1	no	—
99	C₂×C₄	—	53	1	no	—
100	C₂×C₄	—	106	1	no	—
101	C₂×C₄	—	106	1	no	—
102	C₂×C₄	—	106	2	no	—
103	C₂×C₄	—	212	1	no	—
104	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
105	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
106	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	53	4	no	—
109	C₈	—	212	1	no	—
110	C₈	—	212	1	no	—
111	C₈	—	212	2	no	—
112	C₂²	—	106	1	no	—
113	C₂²	—	106	1	no	—
114	C₂²	—	106	1	no	—
115	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	53	1	no	—
119	C₄	—	53	1	no	—
120	C₄	—	106	1	no	—
121	C₄	—	106	1	no	—
122	C₄	D₅₃⋊M₁₆	1	2	yes	—
123	C₂	—	212	1	no	—
124	C₂	#16811628	1	1	yes	characteristic
125	C₂	#16811634	1	1	yes	characteristic
126	C₂	#16811638	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811645

C₂

yes

maximal, characteristic

#16811627

C₂

yes

maximal, characteristic

#16811487

C₂

yes

maximal, complemented

#16811485

C₂

yes

maximal, complemented

#16810575

C₂

yes

maximal, characteristic

C₂×#11719936

—

C₄⋊#88324

—

C₄⋊#88324

C₂²

yes

—

C₄⋊#88324

C₂²

yes

characteristic

C₄⋊#88324

C₂²

yes

—

—

C₄

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

#13133128

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

—

#97×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

Q₈

yes

—

—

—

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

Q₈

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂×C₄

yes

—

#11719912

—

C₂²×D₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×Q₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

#88324

—

#88324

—

#88324

—

C₈×C₅₃

—

#88322

M₁₆

yes

complemented

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#432

—

#423

—

C₄×M₁₆

—

#335

—

#334

—

C₅₃

#1569

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

#97

—

#97

—

106

—

#97

#6157

yes

characteristic

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×#6157

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

106

—

101

C₂×C₄

—

106

—

102

C₂×C₄

—

106

—

103

C₂×C₄

—

212

—

104

C₂×C₄

C₄⋊#6157

yes

characteristic

105

C₂×C₄

C₄⋊#6157

yes

characteristic

106

C₂×C₄

C₂²×#6157

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

212

—

110

C₈

—

212

—

111

C₈

—

212

—

112

C₂²

—

106

—

113

C₂²

—

106

—

114

C₂²

—

106

—

115

C₂²

C₂×C₄⋊#6157

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

106

—

121

C₄

—

106

—

122

C₄

D₅₃⋊M₁₆

yes

—

123

C₂

—

212

—

124

C₂

#16811628

yes

characteristic

125

C₂

#16811634

yes

characteristic

126

C₂

#16811638

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	#97×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#97	#6157
hypocenter	C₅₃	#1569
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

#97×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₄⋊C₄

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
3. #16811487	C₂	semidirect
4. #16811485	C₂	semidirect
47. #88322	M₁₆	semidirect
69. C₅₃	#1569	semidirect

H ≅ G/N

type

3. #16811487

C₂

semidirect

4. #16811485

C₂

semidirect

47. #88322

M₁₆

semidirect

69. C₅₃

#1569

semidirect

Subgroups — #20290281

Conjugacy classes of subgroups

Complements