Group #20273432 — subgroups

Subgroups — #20273432

Order 6784.

228 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 85 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	#16810680	C₂	1	1	yes	maximal, characteristic
2	#16810680	C₂	1	1	yes	maximal, characteristic
3	#16810369	C₂	1	4	yes	maximal, complemented
4	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#88322	C₂²	1	2	yes	—
6	C₄⋊#88322	C₂²	1	2	yes	—
7	C₂²⋊#88322	—	2	1	no	—
8	C₂²⋊#88322	—	2	1	no	—
9	#13133128	—	2	1	no	—
10	#13133128	—	2	1	no	—
11	C₂²×Q₈×C₅₃	—	2	1	no	—
12	C₄⋊Q₈×C₅₃	—	2	1	no	—
13	C₄⋊Q₈×C₅₃	—	2	1	no	—
14	C₄⋊Q₈×C₅₃	C₄	1	2	yes	—
15	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
17	C₂×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₅₃	—	2	2	no	—
20	C₂×Q₈×C₅₃	—	2	2	no	—
21	C₂×Q₈×C₅₃	—	4	4	no	—
22	C₂²×C₄×C₅₃	—	2	2	no	—
23	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
30	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₅₃	D₄	1	1	yes	characteristic
32	C₄²×C₅₃	D₄	1	1	yes	characteristic
33	C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×#88322	—	2	2	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	Q₈×C₅₃	—	4	4	no	—
38	Q₈×C₅₃	—	4	4	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	4	no	—
43	C₂×C₄×C₅₃	—	4	4	no	—
44	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	#88322	—	4	2	no	—
51	#88322	—	4	2	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	—	2	1	no	—
54	C₂²×C₅₃	#101	1	2	yes	—
55	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₄×C₅₃	—	2	1	no	—
57	C₄×C₅₃	—	2	1	no	—
58	C₄×C₅₃	—	2	1	no	—
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	—	4	4	no	—
61	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₅₃	#137	1	2	yes	—
63	#1556	—	53	1	no	maximal, Sylow
64	C₂×C₅₃	—	2	2	no	—
65	C₂×C₅₃	C₂×#101	1	1	yes	characteristic
66	C₂×C₅₃	#418	1	1	yes	characteristic
67	C₂×C₅₃	#419	1	1	yes	characteristic
68	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
69	#432	—	53	1	no	—
70	#432	—	53	1	no	—
71	#332	—	53	4	no	—
72	C₅₃	#1556	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	2	1	no	—
74	C₄⋊Q₈	—	2	1	no	—
75	C₄⋊Q₈	—	2	1	no	—
76	C₄⋊Q₈	Dic₅₃	1	2	yes	—
77	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
78	C₂×C₄⋊C₄	—	2	1	no	—
79	C₂×C₄⋊C₄	Dic₅₃	1	2	yes	—
80	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
81	C₄⋊C₈	—	53	2	no	—
82	C₄⋊C₈	—	53	2	no	—
83	C₂²⋊C₈	—	106	1	no	—
84	C₂²⋊C₈	—	106	1	no	—
85	#97	—	106	1	no	—
86	#97	—	106	1	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	—	4	4	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
92	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
93	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
94	C₂×C₈	—	106	2	no	—
95	C₂×C₈	—	106	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
100	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
101	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
102	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
103	C₄²	C₂²×D₅₃	1	1	yes	characteristic
104	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
105	Q₈	—	4	4	no	—
106	Q₈	—	4	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
113	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
114	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
115	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
116	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
118	C₈	—	212	2	no	—
119	C₈	—	212	2	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	#13133136	1	2	yes	—
123	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	4	4	no	—
129	C₄	#13133243	1	2	yes	—
130	C₄	#13133248	1	2	yes	—
131	C₂	—	2	2	no	—
132	C₂	C₂×#13133136	1	1	yes	characteristic
133	C₂	#16810628	1	1	yes	characteristic
134	C₂	#16810632	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810680

C₂

yes

maximal, characteristic

#16810680

C₂

yes

maximal, characteristic

#16810369

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₄⋊#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

—

—

—

—

C₂²×Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₄

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#101

yes

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

#137

yes

—

#1556

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊Q₈

D₅₃

yes

—

—

—

C₅₃

#1556

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₅₃

yes

—

C₄⋊Q₈

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

#97

—

106

—

#97

—

106

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

100

C₄²

C₂×Dic₅₃

yes

characteristic

101

C₄²

C₂²⋊D₅₃

yes

characteristic

102

C₄²

C₂²⋊D₅₃

yes

characteristic

103

C₄²

C₂²×D₅₃

yes

characteristic

104

C₂³

C₂²⋊Dic₅₃

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

113

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

114

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

115

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

116

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

118

C₈

—

212

—

119

C₈

—

212

—

120

C₂²

—

121

C₂²

—

122

C₂²

#13133136

yes

—

123

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#13133243

yes

—

130

C₄

#13133248

yes

—

131

C₂

—

132

C₂

C₂×#13133136

yes

characteristic

133

C₂

#16810628

yes

characteristic

134

C₂

#16810632

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#1556
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

hypercenter

C₂×C₄⋊Q₈

D₅₃

hypocenter

C₅₃

#1556

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #16810369	C₂	semidirect
72. C₅₃	#1556	semidirect

H ≅ G/N

type

3. #16810369

C₂

semidirect

72. C₅₃

#1556

semidirect

Subgroups — #20273432

Conjugacy classes of subgroups

Complements