Group #20290285 — subgroups

Subgroups — #20290285

Order 6784.

168 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 70 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	#16811627	C₂	1	1	yes	maximal, characteristic
2	D₅₃⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#16811487	C₂	1	2	yes	maximal, complemented
4	#16811486	C₂	1	2	yes	maximal, complemented
5	#16810675	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719936	—	2	1	no	—
7	C₂×D₅₃⋊C₈	—	2	1	no	—
8	#13133284	—	2	1	no	—
9	C₄⋊#88324	—	2	1	no	—
10	C₄⋊#88324	C₂²	1	1	yes	characteristic
11	C₄⋊#88324	C₂²	1	1	yes	characteristic
12	C₄⋊#88324	C₂²	1	2	yes	—
13	C₄²×D₅₃	C₂²	1	1	yes	characteristic
14	C₄⋊#88322	C₄	1	1	yes	characteristic
15	#13133129	C₄	1	2	yes	—
16	C₈×Dic₅₃	C₂²	1	2	yes	—
17	C₄⋊C₈×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	2	no	—
22	C₂×#88324	—	2	1	no	—
23	C₂×#88324	—	2	1	no	—
24	#11719936	—	4	2	no	—
25	D₅₃⋊C₈	—	4	2	no	—
26	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₅₃	D₄	1	1	yes	characteristic
29	C₄×Dic₅₃	Q₈	1	1	yes	characteristic
30	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
31	C₂×#88322	C₂×C₄	1	2	yes	—
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	1	no	—
40	C₄×D₅₃	—	2	1	no	—
41	C₄×D₅₃	—	4	1	no	—
42	C₄×D₅₃	D₈	1	2	yes	—
43	C₄×D₅₃	Q₁₆	1	2	yes	—
44	#88324	—	4	2	no	—
45	#88324	—	4	1	no	—
46	#88324	—	4	1	no	—
47	C₈×C₅₃	—	2	2	no	—
48	#88322	—	2	2	no	—
49	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×D₅₃	—	2	1	no	—
51	C₂×D₅₃	—	2	1	no	—
52	C₂×D₅₃	#107	1	2	yes	—
53	C₄×C₅₃	—	2	1	no	—
54	C₄×C₅₃	#107	1	2	yes	—
55	C₄×C₅₃	#131	1	1	yes	characteristic
56	C₄×C₅₃	#131	1	1	yes	characteristic
57	Dic₅₃	—	2	1	no	—
58	Dic₅₃	—	2	1	no	—
59	Dic₅₃	C₂×D₈	1	1	yes	characteristic
60	Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
61	#1571	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	#424	1	1	yes	characteristic
63	C₂×C₅₃	C₂×#107	1	1	yes	characteristic
64	C₂×C₅₃	#430	1	1	yes	characteristic
65	D₅₃	—	2	2	no	—
66	#431	—	53	1	no	—
67	#423	—	53	1	no	—
68	C₂×C₄⋊C₈	—	53	1	no	—
69	#335	—	53	2	no	—
70	#335	—	53	2	no	—
71	C₅₃	#1571	1	1	yes	characteristic, complemented, Sylow
72	C₂×M₁₆	—	106	1	no	—
73	C₂²×C₈	—	106	1	no	—
74	C₂×C₄²	—	53	1	no	—
75	C₄⋊C₈	—	53	1	no	—
76	C₄⋊C₈	—	53	1	no	—
77	C₄⋊C₈	—	53	2	no	—
78	C₄⋊C₈	—	53	1	no	—
79	C₄⋊C₈	—	106	1	no	—
80	C₄⋊C₈	—	106	1	no	—
81	C₄⋊C₈	#6157	1	1	yes	characteristic
82	C₂²⋊C₈	—	53	2	no	—
83	C₄×C₈	—	53	2	no	—
84	C₂²×C₄	—	53	1	no	—
85	C₂²×C₄	—	53	1	no	—
86	C₂²×C₄	—	53	1	no	—
87	M₁₆	—	212	2	no	—
88	C₂×C₈	—	53	2	no	—
89	C₂×C₈	—	106	1	no	—
90	C₂×C₈	—	106	1	no	—
91	C₂×C₈	—	106	2	no	—
92	C₂×C₈	—	212	2	no	—
93	C₂×C₈	C₂×#6157	1	2	yes	—
94	C₄²	—	53	1	no	—
95	C₄²	—	53	1	no	—
96	C₄²	—	53	1	no	—
97	C₄²	C₂×#6157	1	1	yes	characteristic
98	C₂³	—	53	1	no	—
99	C₂×C₄	—	53	1	no	—
100	C₂×C₄	—	53	2	no	—
101	C₂×C₄	—	53	2	no	—
102	C₂×C₄	—	53	1	no	—
103	C₂×C₄	—	53	1	no	—
104	C₂×C₄	—	106	1	no	—
105	C₂×C₄	—	106	1	no	—
106	C₂×C₄	—	212	1	no	—
107	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
108	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
109	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	106	2	no	—
112	C₈	—	212	1	no	—
113	C₈	—	212	1	no	—
114	C₈	—	212	2	no	—
115	C₂²	—	53	2	no	—
116	C₂²	—	106	1	no	—
117	C₂²	—	106	1	no	—
118	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	53	1	no	—
121	C₄	—	53	1	no	—
122	C₄	—	106	1	no	—
123	C₄	—	106	1	no	—
124	C₄	#13133286	1	2	yes	—
125	C₄	#13133304	1	1	yes	characteristic
126	C₄	#13133305	1	1	yes	characteristic
127	C₂	—	106	2	no	—
128	C₂	#16811630	1	1	yes	characteristic
129	C₂	C₂×#13133286	1	1	yes	characteristic
130	C₂	#16811638	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811627

C₂

yes

maximal, characteristic

D₅₃⋊(C₄⋊C₈)

C₂

yes

maximal, characteristic

#16811487

C₂

yes

maximal, complemented

#16811486

C₂

yes

maximal, complemented

#16810675

C₂

yes

maximal, characteristic

C₂×#11719936

—

C₂×D₅₃⋊C₈

—

#13133284

—

C₄⋊#88324

—

C₄⋊#88324

C₂²

yes

characteristic

C₄⋊#88324

C₂²

yes

characteristic

C₄⋊#88324

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊#88322

C₄

yes

characteristic

#13133129

C₄

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₄⋊C₈×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

Q₈

yes

—

—

—

—

D₅₃⋊C₈

—

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

—

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

D₈

yes

—

C₄×D₅₃

Q₁₆

yes

—

#88324

—

#88324

—

#88324

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#107

yes

—

C₄×C₅₃

—

C₄×C₅₃

#107

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×D₈

yes

characteristic

Dic₅₃

C₂×Q₁₆

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#431

—

#423

—

C₂×C₄⋊C₈

—

#335

—

#335

—

C₅₃

#1571

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

—

106

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₄⋊C₈

#6157

yes

characteristic

C₂²⋊C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

C₂×#6157

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

106

—

105

C₂×C₄

—

106

—

106

C₂×C₄

—

212

—

107

C₂×C₄

C₄⋊#6157

yes

characteristic

108

C₂×C₄

C₄⋊#6157

yes

characteristic

109

C₂×C₄

C₂²×#6157

yes

characteristic

110

C₈

—

111

C₈

—

106

—

112

C₈

—

212

—

113

C₈

—

212

—

114

C₈

—

212

—

115

C₂²

—

116

C₂²

—

106

—

117

C₂²

—

106

—

118

C₂²

C₂×C₄⋊#6157

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

106

—

123

C₄

—

106

—

124

C₄

#13133286

yes

—

125

C₄

#13133304

yes

characteristic

126

C₄

#13133305

yes

characteristic

127

C₂

—

106

—

128

C₂

#16811630

yes

characteristic

129

C₂

C₂×#13133286

yes

characteristic

130

C₂

#16811638

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄⋊C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₄⋊C₈	#6157
hypocenter	C₅₃	#1571
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄⋊C₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₄⋊C₄

C₄⋊C₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
3. #16811487	C₂	semidirect
4. #16811486	C₂	semidirect
71. C₅₃	#1571	semidirect

H ≅ G/N

type

3. #16811487

C₂

semidirect

4. #16811486

C₂

semidirect

71. C₅₃

#1571

semidirect

Subgroups — #20290285

Conjugacy classes of subgroups

Complements