Group #20290822 — subgroups

Subgroups — #20290822

Order 6784.

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

normal subgroups only

This group has 208 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	C₂×#13133287	C₂	1	1	yes	maximal, characteristic
2	#16811630	C₂	1	2	yes	maximal
3	C₂×C₈×#6157	C₂	1	1	yes	maximal, characteristic
4	#16811493	C₂	1	2	yes	maximal
5	#16810658	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719936	C₂²	1	2	yes	—
7	C₂×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×#6157	C₂²	1	1	yes	characteristic
9	#13133287	—	2	2	no	—
10	#13133284	—	2	2	no	—
11	C₄⋊#88324	—	2	2	no	—
12	C₈×#6157	—	2	2	no	—
13	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
14	#13133175	C₂²	1	2	yes	—
15	#13133150	C₄	1	2	yes	complemented
16	#13133149	C₄	1	1	yes	characteristic, complemented
17	#106×C₅₃	C₄	1	1	yes	characteristic, complemented
18	C₂²×#6157	—	2	1	no	—
19	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
20	C₂×#88324	—	2	2	no	—
21	C₂×#88324	D₄	1	2	yes	—
22	C₄×#6157	—	2	1	no	—
23	C₄×#6157	Q₈	1	2	yes	—
24	#11719936	—	2	2	no	—
25	#11719936	—	2	2	no	—
26	D₅₃⋊C₈	—	2	1	no	—
27	D₅₃⋊C₈	—	2	1	no	—
28	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
30	#11719921	—	2	2	no	—
31	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
32	C₄×Dic₅₃	—	2	2	no	—
33	C₂×#88322	C₂×C₄	1	1	yes	characteristic
34	C₈×D₅₃	D₄	1	2	yes	—
35	C₈×D₅₃	Q₈	1	2	yes	—
36	C₂²×D₅₃	#41	1	1	yes	characteristic
37	C₂×#6157	—	2	4	no	—
38	C₂×#6157	—	2	2	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₅₃	—	2	2	no	—
42	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
43	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
44	C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₄×D₅₃	#41	1	1	yes	characteristic
47	#88324	—	2	2	no	—
48	#88324	—	4	2	no	—
49	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
50	#88322	C₄⋊C₄	1	2	yes	—
51	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
52	C₂×D₅₃	—	2	2	no	—
53	C₂×D₅₃	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂×D₅₃	#125	1	1	yes	characteristic
55	#6157	—	2	4	no	—
56	C₄×C₅₃	—	4	2	no	—
57	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
59	Dic₅₃	—	4	2	no	—
60	Dic₅₃	#121	1	1	yes	characteristic
61	Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
62	#1945	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	#389	1	1	yes	characteristic
64	C₂×C₅₃	#443	1	2	yes	—
65	D₅₃	—	2	1	no	—
66	D₅₃	—	2	1	no	—
67	C₂×#108	—	53	1	no	—
68	#427	—	53	1	no	—
69	#424	—	53	2	no	—
70	C₂×C₄×C₈	—	53	1	no	—
71	#339	—	53	2	no	—
72	C₅₃	#1945	1	1	yes	characteristic, complemented, Sylow
73	C₂×M₁₆	—	53	2	no	—
74	C₂²×C₈	—	53	1	no	—
75	C₂²×C₈	—	53	1	no	—
76	#117	—	53	2	no	—
77	C₂×C₄²	—	53	1	no	—
78	#108	—	106	2	no	—
79	#107	—	53	2	no	—
80	#106	—	53	1	no	—
81	#106	#6157	1	1	yes	characteristic, complemented
82	C₄⋊C₈	—	106	2	no	—
83	C₄⋊C₈	—	106	2	no	—
84	C₄×C₈	—	106	2	no	—
85	C₂²×C₄	—	53	1	no	—
86	C₂²×C₄	—	106	1	no	—
87	M₁₆	—	106	2	no	—
88	M₁₆	—	106	2	no	—
89	C₂×C₈	—	53	1	no	—
90	C₂×C₈	—	53	2	no	—
91	C₂×C₈	—	53	2	no	—
92	C₂×C₈	—	53	2	no	—
93	C₂×C₈	—	106	1	no	—
94	C₂×C₈	—	106	1	no	—
95	C₂×C₈	—	106	2	no	—
96	C₂×C₈	C₂×#6157	1	1	yes	characteristic
97	C₄⋊C₄	—	53	2	no	—
98	C₄⋊C₄	C₂×#6157	1	2	yes	—
99	C₂²⋊C₄	—	106	2	no	—
100	C₄²	—	53	2	no	—
101	C₄²	—	106	2	no	—
102	C₄²	—	106	1	no	—
103	C₂³	—	53	1	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	53	1	no	—
106	C₂×C₄	—	53	1	no	—
107	C₂×C₄	—	53	1	no	—
108	C₂×C₄	—	53	1	no	—
109	C₂×C₄	—	53	1	no	—
110	C₂×C₄	—	106	2	no	—
111	C₂×C₄	—	106	4	no	—
112	C₂×C₄	—	106	2	no	—
113	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
114	C₈	—	53	2	no	—
115	C₈	—	106	2	no	—
116	C₈	—	212	2	no	—
117	C₈	C₄⋊#6157	1	2	yes	—
118	C₂²	—	53	1	no	—
119	C₂²	—	53	1	no	—
120	C₂²	—	106	2	no	—
121	C₂²	D₄×#6157	1	1	yes	characteristic
122	C₄	—	4	2	no	—
123	C₄	—	53	1	no	—
124	C₄	—	53	1	no	—
125	C₄	—	106	4	no	—
126	C₄	—	212	2	no	—
127	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
128	C₄	Q₈×#6157	1	1	yes	characteristic
129	C₂	—	106	1	no	—
130	C₂	—	106	1	no	—
131	C₂	#16811574	1	1	yes	characteristic
132	C₂	#16811666	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133287

C₂

yes

maximal, characteristic

#16811630

C₂

yes

maximal

C₂×C₈×#6157

C₂

yes

maximal, characteristic

#16811493

C₂

yes

maximal

#16810658

C₂

yes

maximal, characteristic

C₂×#11719936

C₂²

yes

—

C₂×D₅₃⋊C₈

C₂²

yes

characteristic

C₂×C₄×#6157

C₂²

yes

—

—

—

—

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133175

C₂²

yes

—

#13133150

C₄

yes

complemented

#13133149

C₄

yes

characteristic, complemented

#106×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×#88324

—

C₂×#88324

D₄

yes

—

C₄×#6157

—

C₄×#6157

Q₈

yes

—

#11719936

—

#11719936

—

D₅₃⋊C₈

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

#11719921

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

—

C₂×#88322

C₂×C₄

yes

characteristic

C₈×D₅₃

D₄

yes

—

C₈×D₅₃

Q₈

yes

—

C₂²×D₅₃

yes

—

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

C₂×Q₈

yes

characteristic

C₄×D₅₃

C₂×Q₈

yes

C₄×D₅₃

yes

—

—

C₈×C₅₃

C₄⋊C₄

yes

—

#88322

C₄⋊C₄

yes

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

C₂²⋊Q₈

yes

C₂×D₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

#121

yes

characteristic

Dic₅₃

C₄⋊Q₈

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

D₅₃

—

C₂×#108

—

#427

—

#424

—

C₂×C₄×C₈

—

#339

—

C₅₃

#1945

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

C₂×C₄²

—

#108

—

106

—

—

—

yes

characteristic, complemented

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×#6157

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×#6157

yes

—

C₂²⋊C₄

—

106

—

100

C₄²

—

101

C₄²

—

106

—

102

C₄²

—

106

—

103

C₂³

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

106

—

111

C₂×C₄

—

106

—

112

C₂×C₄

—

106

—

113

C₂×C₄

C₂²×#6157

yes

characteristic

114

C₈

—

115

C₈

—

106

—

116

C₈

—

212

—

117

C₈

C₄⋊#6157

yes

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

106

—

121

C₂²

D₄×#6157

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

106

—

126

C₄

—

212

—

127

C₄

C₂×C₄⋊#6157

yes

characteristic

128

C₄

Q₈×#6157

yes

characteristic

129

C₂

—

106

—

130

C₂

—

106

—

131

C₂

#16811574

yes

characteristic

132

C₂

#16811666

yes

—

	subgroup	G/H
center	C₂²	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	#106×C₅₃	C₄
socle	C₂²×C₅₃	C₄×D₄
hypercenter	#106	#6157
hypocenter	C₅₃	#1945
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

subgroup

G/H

center

C₂²

D₄×#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

#106×C₅₃

C₄

socle

C₂²×C₅₃

C₄×D₄

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

N	H ≅ G/N	type
15. #13133150	C₄	semidirect
16. #13133149	C₄	semidirect
17. #106×C₅₃	C₄	semidirect
72. C₅₃	#1945	semidirect
81. #106	#6157	semidirect

H ≅ G/N

type

15. #13133150

C₄

semidirect

16. #13133149

C₄

semidirect

17. #106×C₅₃

C₄

semidirect

72. C₅₃

#1945

semidirect

81. #106

#6157

semidirect

Subgroups — #20290822

Conjugacy classes of subgroups

Complements