Group #20290268 — subgroups

Subgroups — #20290268

Order 6784.

180 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 84 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	#16811639	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#88324	C₂	1	1	yes	maximal, characteristic
3	#16811487	C₂	1	2	yes	maximal, complemented
4	#16811485	C₂	1	2	yes	maximal, complemented
5	M₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88324	—	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	C₂²⋊#88324	—	2	1	no	—
12	C₄×#88324	—	2	1	no	—
13	C₂×#11719917	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
15	#13133127	C₄	1	2	yes	—
16	C₈×Dic₅₃	C₂²	1	2	yes	—
17	C₂×M₁₆×C₅₃	C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
20	C₂²×Dic₅₃	Q₈	1	1	yes	characteristic
21	C₂×#88324	—	2	1	no	—
22	C₂×#88324	—	2	1	no	—
23	C₂×#88324	—	2	2	no	—
24	C₂×#88324	—	4	2	no	—
25	M₁₆×C₅₃	—	2	2	no	—
26	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₄×Dic₅₃	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	#11719917	C₈	1	4	yes	complemented
32	C₂×#88322	C₂×C₄	1	2	yes	—
33	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
37	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₅₃	—	2	1	no	—
39	C₂×Dic₅₃	—	2	1	no	—
40	C₂×Dic₅₃	—	4	1	no	—
41	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
45	#88324	—	4	1	no	—
46	#88324	—	4	2	no	—
47	#88324	—	4	1	no	—
48	C₈×C₅₃	—	2	2	no	—
49	#88322	C₂×C₈	1	4	yes	complemented
50	C₂²×C₅₃	—	2	1	no	—
51	C₂²×C₅₃	—	2	1	no	—
52	C₂²×C₅₃	C₄⋊C₈	1	2	yes	—
53	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₅₃	C₄⋊C₈	1	2	yes	—
55	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
56	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
57	Dic₅₃	—	2	1	no	—
58	Dic₅₃	—	2	1	no	—
59	Dic₅₃	—	2	1	no	—
60	Dic₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
61	Dic₅₃	#137	1	1	yes	characteristic
62	#1565	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	—	2	2	no	—
64	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₅₃	#428	1	1	yes	characteristic
66	C₂×C₅₃	#430	1	1	yes	characteristic
67	#431	—	53	1	no	—
68	C₂×C₄⋊C₈	—	53	1	no	—
69	C₄×M₁₆	—	53	1	no	—
70	#335	—	53	2	no	—
71	#334	—	53	2	no	—
72	C₅₃	#1565	1	1	yes	characteristic, complemented, Sylow
73	C₂×M₁₆	—	53	1	no	—
74	C₂×M₁₆	#6157	1	1	yes	characteristic
75	C₂²×C₈	—	106	1	no	—
76	C₂×C₄²	—	53	1	no	—
77	C₄⋊C₈	—	53	1	no	—
78	C₄⋊C₈	—	53	2	no	—
79	C₄⋊C₈	—	53	1	no	—
80	C₄⋊C₈	—	106	1	no	—
81	C₂²⋊C₈	—	106	1	no	—
82	#97	—	53	2	no	—
83	C₄×C₈	—	53	2	no	—
84	C₄×C₈	—	106	1	no	—
85	C₂²×C₄	—	53	1	no	—
86	C₂²×C₄	—	53	1	no	—
87	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	—	53	4	no	—
90	C₂×C₈	—	53	2	no	—
91	C₂×C₈	—	106	1	no	—
92	C₂×C₈	—	106	1	no	—
93	C₂×C₈	—	106	2	no	—
94	C₂×C₈	—	212	2	no	—
95	C₂×C₈	C₂×#6157	1	2	yes	—
96	C₄²	—	53	1	no	—
97	C₄²	—	53	1	no	—
98	C₄²	—	53	1	no	—
99	C₄²	—	53	1	no	—
100	C₂³	C₄⋊#6157	1	1	yes	characteristic
101	C₂×C₄	—	53	1	no	—
102	C₂×C₄	—	53	1	no	—
103	C₂×C₄	—	53	1	no	—
104	C₂×C₄	—	53	1	no	—
105	C₂×C₄	—	106	1	no	—
106	C₂×C₄	—	106	1	no	—
107	C₂×C₄	—	212	1	no	—
108	C₂×C₄	D₅₃⋊C₈	1	2	yes	—
109	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
110	C₂×C₄	#11719940	1	2	yes	—
111	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	53	4	no	—
114	C₈	—	212	1	no	—
115	C₈	—	212	1	no	—
116	C₈	—	212	2	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#13133284	1	2	yes	—
120	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
121	C₄	—	53	1	no	—
122	C₄	—	53	1	no	—
123	C₄	—	106	1	no	—
124	C₄	—	106	1	no	—
125	C₄	—	106	1	no	—
126	C₄	#13133284	1	2	yes	—
127	C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
128	C₄	C₂×#11719940	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	C₂×#13133284	1	1	yes	characteristic
131	C₂	#16811634	1	1	yes	characteristic
132	C₂	#16811638	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811639

C₂

yes

maximal, characteristic

C₂×C₄⋊#88324

C₂

yes

maximal, characteristic

#16811487

C₂

yes

maximal, complemented

#16811485

C₂

yes

maximal, complemented

M₁₆×Dic₅₃

C₂

yes

maximal, characteristic

C₂²×#88324

—

C₄⋊#88324

—

C₄⋊#88324

C₂²

yes

—

C₄⋊#88324

C₂²

yes

characteristic

C₄⋊#88324

C₂²

yes

—

—

C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133127

C₄

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₂×M₁₆×C₅₃

C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

Q₈

yes

—

—

—

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

Q₈

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

#11719917

C₈

yes

complemented

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×Q₈

yes

—

—

—

C₈×C₅₃

—

#88322

C₂×C₈

yes

complemented

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄⋊C₈

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₄⋊C₈

yes

—

C₄×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄⋊C₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

#335

—

#334

—

C₅₃

#1565

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

#6157

yes

characteristic

C₂²×C₈

—

106

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

#97

—

C₄×C₈

—

C₄×C₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#6157

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

C₂×#6157

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

100

C₂³

C₄⋊#6157

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

—

106

C₂×C₄

—

106

—

107

C₂×C₄

—

212

—

108

C₂×C₄

D₅₃⋊C₈

yes

—

109

C₂×C₄

C₄⋊#6157

yes

characteristic

110

C₂×C₄

#11719940

yes

—

111

C₂×C₄

C₂²×#6157

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

212

—

115

C₈

—

212

—

116

C₈

—

212

—

117

C₂²

—

118

C₂²

—

119

C₂²

#13133284

yes

—

120

C₂²

C₂×C₄⋊#6157

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

106

—

124

C₄

—

106

—

125

C₄

—

106

—

126

C₄

#13133284

yes

—

127

C₄

C₂×D₅₃⋊C₈

yes

characteristic

128

C₄

C₂×#11719940

yes

characteristic

129

C₂

—

130

C₂

C₂×#13133284

yes

characteristic

131

C₂

#16811634

yes

characteristic

132

C₂

#16811638

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6157
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₂×M₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₂×M₁₆	#6157
hypocenter	C₅₃	#1565
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₂²

C₂×C₄⋊#6157

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₂×M₁₆×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₄⋊C₄

C₂×M₁₆

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
3. #16811487	C₂	semidirect
4. #16811485	C₂	semidirect
31. #11719917	C₈	semidirect
49. #88322	C₂×C₈	semidirect
72. C₅₃	#1565	semidirect

H ≅ G/N

type

3. #16811487

C₂

semidirect

4. #16811485

C₂

semidirect

31. #11719917

C₈

semidirect

49. #88322

C₂×C₈

semidirect

72. C₅₃

#1565

semidirect

Subgroups — #20290268

Conjugacy classes of subgroups

Complements