Group #20290290 — subgroups

Subgroups — #20290290

Order 6784.

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

normal subgroups only

This group has 160 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	#16811644	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#88324	C₂	1	1	yes	maximal, characteristic
3	#16811486	C₂	1	2	yes	maximal, complemented
4	#16811485	C₂	1	2	yes	maximal, complemented
5	#16810673	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	#13133272	—	2	1	no	—
13	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	#13133146	C₄	1	2	yes	—
15	C₂²⋊#88322	C₄	1	1	yes	characteristic
16	C₈×Dic₅₃	C₂²	1	2	yes	—
17	C₂²⋊C₈×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	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₄×Dic₅₃	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	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	—	2	2	no	—
33	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₅₃	—	2	2	no	—
36	C₂×Dic₅₃	—	2	1	no	—
37	C₂×Dic₅₃	—	2	1	no	—
38	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
42	#88324	—	4	1	no	—
43	#88324	—	4	2	no	—
44	#88324	—	4	1	no	—
45	C₈×C₅₃	—	2	2	no	—
46	#88322	—	2	2	no	—
47	C₂²×C₅₃	—	2	1	no	—
48	C₂²×C₅₃	—	2	1	no	—
49	C₂²×C₅₃	#108	1	2	yes	—
50	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	#131	1	2	yes	—
53	Dic₅₃	—	2	1	no	—
54	Dic₅₃	—	2	1	no	—
55	Dic₅₃	—	2	1	no	—
56	Dic₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
57	Dic₅₃	#137	1	1	yes	characteristic
58	#1573	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	—	2	2	no	—
60	C₂×C₅₃	#424	1	1	yes	characteristic
61	C₂×C₅₃	#428	1	1	yes	characteristic
62	C₂×C₅₃	C₂×#108	1	1	yes	characteristic
63	#432	—	53	1	no	—
64	#431	—	53	1	no	—
65	C₂×C₄⋊C₈	—	53	1	no	—
66	#335	—	53	2	no	—
67	#334	—	53	2	no	—
68	C₅₃	#1573	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	—	106	1	no	—
70	C₂×C₄²	—	53	1	no	—
71	C₄⋊C₈	—	53	2	no	—
72	C₄⋊C₈	—	53	1	no	—
73	C₄⋊C₈	—	53	2	no	—
74	C₄⋊C₈	—	53	1	no	—
75	C₄⋊C₈	—	106	1	no	—
76	C₂²⋊C₈	—	53	1	no	—
77	C₂²⋊C₈	—	106	1	no	—
78	C₂²⋊C₈	#6157	1	1	yes	characteristic
79	#97	—	106	1	no	—
80	C₄×C₈	—	53	2	no	—
81	C₂²×C₄	—	53	1	no	—
82	C₂²×C₄	—	53	1	no	—
83	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
84	C₂×C₈	—	53	2	no	—
85	C₂×C₈	—	106	1	no	—
86	C₂×C₈	—	106	1	no	—
87	C₂×C₈	—	106	2	no	—
88	C₂×C₈	—	212	2	no	—
89	C₂×C₈	C₂×#6157	1	2	yes	—
90	C₄²	—	53	1	no	—
91	C₄²	—	53	1	no	—
92	C₄²	—	53	1	no	—
93	C₄²	—	53	1	no	—
94	C₂³	C₄⋊#6157	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	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	C₄⋊#6157	1	1	yes	characteristic
104	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	106	2	no	—
107	C₈	—	212	1	no	—
108	C₈	—	212	1	no	—
109	C₈	—	212	2	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	#13133288	1	2	yes	—
113	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	53	1	no	—
116	C₄	—	53	1	no	—
117	C₄	—	106	1	no	—
118	C₄	—	106	1	no	—
119	C₄	—	106	1	no	—
120	C₄	#13133304	1	2	yes	—
121	C₂	—	2	2	no	—
122	C₂	#16811629	1	1	yes	characteristic
123	C₂	#16811634	1	1	yes	characteristic
124	C₂	C₂×#13133288	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811644

C₂

yes

maximal, characteristic

C₂×C₄⋊#88324

C₂

yes

maximal, characteristic

#16811486

C₂

yes

maximal, complemented

#16811485

C₂

yes

maximal, complemented

#16810673

C₂

yes

maximal, characteristic

C₂²×#88324

—

C₄⋊#88324

—

C₄⋊#88324

C₂²

yes

—

C₄⋊#88324

C₂²

yes

characteristic

C₄⋊#88324

C₂²

yes

characteristic

C₂²⋊#88324

—

#13133272

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133146

C₄

yes

—

C₂²⋊#88322

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

—

C₂²⋊C₈×C₅₃

C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

Q₈

yes

—

—

—

—

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

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

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

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#108

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#131

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

C₂×C₄⋊C₈

—

#335

—

#334

—

C₅₃

#1573

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

C₂²⋊C₈

#6157

yes

characteristic

#97

—

106

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#6157

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

C₂×#6157

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₄⋊#6157

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

106

—

101

C₂×C₄

—

106

—

102

C₂×C₄

—

106

—

103

C₂×C₄

C₄⋊#6157

yes

characteristic

104

C₂×C₄

C₂²×#6157

yes

characteristic

105

C₈

—

106

C₈

—

106

—

107

C₈

—

212

—

108

C₈

—

212

—

109

C₈

—

212

—

110

C₂²

—

111

C₂²

—

112

C₂²

#13133288

yes

—

113

C₂²

C₂×C₄⋊#6157

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

106

—

118

C₄

—

106

—

119

C₄

—

106

—

120

C₄

#13133304

yes

—

121

C₂

—

122

C₂

#16811629

yes

characteristic

123

C₂

#16811634

yes

characteristic

124

C₂

C₂×#13133288

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₂²⋊C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₂²⋊C₈	#6157
hypocenter	C₅₃	#1573
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₂²⋊C₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₄⋊C₄

C₂²⋊C₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
3. #16811486	C₂	semidirect
4. #16811485	C₂	semidirect
68. C₅₃	#1573	semidirect

H ≅ G/N

type

3. #16811486

C₂

semidirect

4. #16811485

C₂

semidirect

68. C₅₃

#1573

semidirect

Subgroups — #20290290

Conjugacy classes of subgroups

Complements