Group #20273396 — subgroups

Subgroups — #20273396

Order 6784.

240 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 101 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	#16810676	C₂	1	1	yes	maximal, characteristic
2	#16810647	C₂	1	1	yes	maximal, characteristic
3	#16810372	C₂	1	4	yes	maximal, complemented
4	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	—	2	1	no	—
6	C₄⋊#88322	—	2	1	no	—
7	C₄⋊#88322	C₂²	1	2	yes	—
8	C₄⋊#88322	C₂²	1	2	yes	—
9	C₂²⋊#88322	—	2	1	no	—
10	C₄×#88322	—	2	1	no	—
11	C₂²×Q₈×C₅₃	C₄	1	2	yes	—
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	—	2	1	no	—
18	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₅₃	—	2	2	no	—
20	C₂×Q₈×C₅₃	—	4	4	no	—
21	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	—
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	—	2	2	no	—
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	#11719917	—	4	2	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂×#88322	—	2	2	no	—
37	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	Q₈×C₅₃	—	4	4	no	—
39	Q₈×C₅₃	—	4	4	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	SD₁₆	1	2	yes	—
48	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
49	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	#88322	—	4	2	no	—
53	#88322	—	4	2	no	—
54	C₂²×C₅₃	—	2	1	no	—
55	C₂²×C₅₃	—	2	1	no	—
56	C₂²×C₅₃	#103	1	2	yes	—
57	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₅₃	—	2	1	no	—
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	—	4	4	no	—
61	C₄×C₅₃	#100	1	2	yes	—
62	C₄×C₅₃	#103	1	2	yes	—
63	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₄×C₅₃	#137	1	2	yes	—
66	#1548	—	53	1	no	maximal, Sylow
67	C₂×C₅₃	—	2	2	no	—
68	C₂×C₅₃	C₂×#100	1	1	yes	characteristic
69	C₂×C₅₃	C₂×#103	1	1	yes	characteristic
70	C₂×C₅₃	#419	1	1	yes	characteristic
71	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
72	#431	—	53	1	no	—
73	#423	—	53	1	no	—
74	#333	—	53	4	no	—
75	C₅₃	#1548	1	1	yes	characteristic, complemented, Sylow
76	C₂²×Q₈	Dic₅₃	1	2	yes	—
77	C₂×M₁₆	—	106	1	no	—
78	C₄⋊Q₈	—	2	1	no	—
79	C₄⋊Q₈	—	2	1	no	—
80	C₄⋊Q₈	Dic₅₃	1	2	yes	—
81	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
85	C₄⋊C₈	—	53	2	no	—
86	C₄⋊C₈	—	53	2	no	—
87	C₄⋊C₈	—	106	1	no	—
88	C₂²⋊C₈	—	106	1	no	—
89	C₄×C₈	—	106	1	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	4	4	no	—
92	C₂×Q₈	C₂×Dic₅₃	1	4	yes	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
95	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
97	M₁₆	—	212	2	no	—
98	C₂×C₈	—	106	2	no	—
99	C₂×C₈	—	106	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
105	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
106	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
107	C₄²	C₂²×D₅₃	1	1	yes	characteristic
108	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
109	Q₈	—	4	4	no	—
110	Q₈	—	4	4	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	#11719924	1	2	yes	—
116	C₂×C₄	#11719925	1	2	yes	—
117	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
118	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
119	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
120	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
123	C₈	—	212	2	no	—
124	C₈	—	212	2	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#13133141	1	2	yes	—
128	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	4	4	no	—
132	C₄	#13133134	1	2	yes	—
133	C₄	#13133141	1	2	yes	—
134	C₄	C₂×#11719924	1	1	yes	characteristic
135	C₄	C₂×#11719925	1	1	yes	characteristic
136	C₄	#13133248	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂×#13133134	1	1	yes	characteristic
139	C₂	C₂×#13133141	1	1	yes	characteristic
140	C₂	#16810632	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810676

C₂

yes

maximal, characteristic

#16810647

C₂

yes

maximal, characteristic

#16810372

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

C₂²⋊#88322

—

C₄×#88322

—

C₂²×Q₈×C₅₃

C₄

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₄

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

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

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

—

—

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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

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

#103

yes

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#100

yes

—

C₄×C₅₃

#103

yes

—

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₄×C₅₃

#137

yes

—

#1548

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊Q₈

D₅₃

yes

—

—

—

C₅₃

#1548

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₅₃

yes

—

C₂×M₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₅₃

yes

—

C₄⋊Q₈

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

106

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

M₁₆

—

212

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

C₂×Dic₅₃

yes

characteristic

105

C₄²

C₂²⋊D₅₃

yes

characteristic

106

C₄²

C₂²⋊D₅₃

yes

characteristic

107

C₄²

C₂²×D₅₃

yes

characteristic

108

C₂³

C₂²⋊Dic₅₃

yes

characteristic

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

#11719924

yes

—

116

C₂×C₄

#11719925

yes

—

117

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

118

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

119

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

120

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

123

C₈

—

212

—

124

C₈

—

212

—

125

C₂²

—

126

C₂²

—

127

C₂²

#13133141

yes

—

128

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#13133134

yes

—

133

C₄

#13133141

yes

—

134

C₄

C₂×#11719924

yes

characteristic

135

C₄

C₂×#11719925

yes

characteristic

136

C₄

#13133248

yes

—

137

C₂

—

138

C₂

C₂×#13133134

yes

characteristic

139

C₂

C₂×#13133141

yes

characteristic

140

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	#1548
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

#1548

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #16810372	C₂	semidirect
75. C₅₃	#1548	semidirect

H ≅ G/N

type

3. #16810372

C₂

semidirect

75. C₅₃

#1548

semidirect

Subgroups — #20273396

Conjugacy classes of subgroups

Complements