Group #20290328 — subgroups

Subgroups — #20290328

Order 6784.

152 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 58 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	#16811674	C₂	1	1	yes	maximal, characteristic
2	Q₈×#88324	C₂	1	1	yes	maximal, characteristic
3	#16811487	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#88324	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#88324	C₂	1	1	yes	maximal, characteristic
6	#88324⋊C₈	C₂	1	1	yes	maximal, characteristic
7	Q₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
8	#13133284	—	2	1	no	—
9	C₄⋊#88324	—	2	1	no	—
10	C₄⋊#88324	C₂²	1	1	yes	characteristic
11	C₄⋊#88324	C₂²	1	1	yes	characteristic
12	#13133272	—	2	1	no	—
13	C₄×#88324	—	2	1	no	—
14	C₄×#88324	C₂²	1	1	yes	characteristic
15	C₄×#88324	C₂²	1	1	yes	characteristic
16	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic
17	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	#13133151	C₄	1	1	yes	characteristic
19	#13133141	C₄	1	1	yes	characteristic
20	#13133141	C₄	1	1	yes	characteristic
21	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
22	C₂×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×#88324	—	2	1	no	—
26	C₂×#88324	—	2	1	no	—
27	C₂×#88324	—	2	1	no	—
28	C₂×#88324	D₄	1	1	yes	characteristic
29	C₂×#88324	D₄	1	1	yes	characteristic
30	Q₁₆×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄⋊Dic₅₃	—	2	1	no	—
33	C₄⋊Dic₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₅₃	—	2	1	no	—
37	C₄×Dic₅₃	—	2	1	no	—
38	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
39	C₂×#88322	C₂×C₄	1	1	yes	characteristic
40	Q₈×C₅₃	—	2	1	no	—
41	Q₈×C₅₃	—	2	1	no	—
42	Q₈×C₅₃	—	2	1	no	—
43	Q₈×C₅₃	M₁₆	1	2	yes	—
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₅₃	—	2	1	no	—
48	C₂×Dic₅₃	—	2	1	no	—
49	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₅₃	#41	1	1	yes	characteristic
51	#88324	—	2	1	no	—
52	#88324	—	4	1	no	—
53	#88324	—	4	1	no	—
54	#88324	—	4	1	no	—
55	#88324	Q₁₆	1	2	yes	complemented
56	C₈×C₅₃	—	2	1	no	—
57	#88322	—	2	1	no	—
58	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
59	C₄×C₅₃	—	2	2	no	—
60	C₄×C₅₃	—	4	1	no	—
61	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
62	C₄×C₅₃	#131	1	1	yes	characteristic
63	Dic₅₃	—	2	1	no	—
64	Dic₅₃	—	4	1	no	—
65	Dic₅₃	—	4	1	no	—
66	Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
67	Dic₅₃	#135	1	1	yes	characteristic
68	#1584	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	#435	1	1	yes	characteristic
70	C₂×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
71	C₂×C₅₃	#444	1	1	yes	characteristic
72	#446	—	53	1	no	—
73	C₈×Q₈	—	53	1	no	—
74	C₄×Q₁₆	—	53	1	no	—
75	#335	—	53	1	no	—
76	Q₈⋊C₈	—	53	1	no	—
77	Q₈⋊C₈	—	53	1	no	—
78	#322	—	53	1	no	—
79	C₅₃	#1584	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₁₆	#6157	1	1	yes	characteristic
81	C₄×Q₈	—	53	1	no	—
82	C₄×Q₈	—	53	1	no	—
83	#107	—	53	1	no	—
84	C₄⋊C₈	—	53	1	no	—
85	C₄⋊C₈	—	53	1	no	—
86	C₄⋊C₈	—	106	1	no	—
87	C₄⋊C₈	—	106	1	no	—
88	#103	—	53	1	no	—
89	#103	—	53	1	no	—
90	#97	—	106	1	no	—
91	C₄×C₈	—	53	1	no	—
92	C₄×C₈	—	53	1	no	—
93	C₄×C₈	—	53	1	no	—
94	C₄×C₈	—	106	1	no	—
95	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
96	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
97	Q₁₆	—	2	2	no	—
98	C₂×C₈	—	53	1	no	—
99	C₂×C₈	—	53	1	no	—
100	C₂×C₈	—	53	1	no	—
101	C₂×C₈	—	106	1	no	—
102	C₂×C₈	—	106	1	no	—
103	C₂×C₈	—	106	1	no	—
104	C₂×C₈	C₂×#6157	1	1	yes	characteristic
105	C₄⋊C₄	—	53	1	no	—
106	C₄⋊C₄	—	53	1	no	—
107	C₄⋊C₄	—	106	1	no	—
108	C₄⋊C₄	—	106	1	no	—
109	C₄²	—	53	1	no	—
110	C₄²	—	106	1	no	—
111	C₄²	—	106	1	no	—
112	Q₈	—	2	1	no	—
113	Q₈	—	2	1	no	—
114	Q₈	—	2	1	no	—
115	Q₈	#11719940	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	53	1	no	—
119	C₂×C₄	—	53	1	no	—
120	C₂×C₄	—	106	1	no	—
121	C₂×C₄	—	106	1	no	—
122	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	53	2	no	—
125	C₈	—	106	1	no	—
126	C₈	—	106	1	no	—
127	C₈	—	212	1	no	—
128	C₈	—	212	1	no	—
129	C₈	—	212	1	no	—
130	C₂²	D₄×#6157	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	4	1	no	—
133	C₄	—	53	1	no	—
134	C₄	—	53	1	no	—
135	C₄	—	106	1	no	—
136	C₄	—	212	1	no	—
137	C₄	—	212	1	no	—
138	C₄	C₂×#11719940	1	1	yes	characteristic
139	C₄	#13133304	1	1	yes	characteristic
140	C₂	#16811654	1	1	yes	characteristic
141	C₂	Q₁₆×#6157	1	1	yes	characteristic
142	C₂	#16811669	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811674

C₂

yes

maximal, characteristic

Q₈×#88324

C₂

yes

maximal, characteristic

#16811487

C₂

yes

maximal, characteristic

Q₈⋊#88324

C₂

yes

maximal, characteristic

Q₈⋊#88324

C₂

yes

maximal, characteristic

#88324⋊C₈

C₂

yes

maximal, characteristic

Q₁₆×Dic₅₃

C₂

yes

maximal, characteristic

#13133284

—

C₄⋊#88324

—

C₄⋊#88324

C₂²

yes

characteristic

C₄⋊#88324

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×#88324

C₂²

yes

characteristic

Q₈×Dic₅₃

C₂²

yes

characteristic

Q₈×Dic₅₃

C₂²

yes

characteristic

#13133151

C₄

yes

characteristic

#13133141

C₄

yes

characteristic

#13133141

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

C₂×Q₁₆×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

—

—

—

D₄

yes

characteristic

C₂×#88324

D₄

yes

characteristic

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

characteristic

C₄⋊Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

C₂×Dic₅₃

yes

—

—

—

—

Q₁₆

yes

complemented

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

#131

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×Q₁₆

yes

Dic₅₃

yes

—

C₂×C₅₃

yes

C₂×C₅₃

C₄×Q₁₆

yes

C₂×C₅₃

yes

—

C₈×Q₈

—

C₄×Q₁₆

—

#335

—

Q₈⋊C₈

—

Q₈⋊C₈

—

#322

—

C₅₃

#1584

yes

characteristic, complemented, Sylow

C₂×Q₁₆

#6157

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

#103

—

#103

—

#97

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

106

—

C₂×Q₈

yes

C₂×Q₈

yes

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

106

—

102

C₂×C₈

—

106

—

103

C₂×C₈

—

106

—

104

C₂×C₈

C₂×#6157

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

106

—

108

C₄⋊C₄

—

106

—

109

C₄²

—

110

C₄²

—

106

—

111

C₄²

—

106

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

#11719940

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

106

—

121

C₂×C₄

—

106

—

122

C₂×C₄

C₂²×#6157

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

106

—

126

C₈

—

106

—

127

C₈

—

212

—

128

C₈

—

212

—

129

C₈

—

212

—

130

C₂²

D₄×#6157

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

106

—

136

C₄

—

212

—

137

C₄

—

212

—

138

C₄

C₂×#11719940

yes

characteristic

139

C₄

#13133304

yes

characteristic

140

C₂

#16811654

yes

characteristic

141

C₂

Q₁₆×#6157

yes

characteristic

142

C₂

#16811669

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₂×Q₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₄×D₄
hypercenter	C₂×Q₁₆	#6157
hypocenter	C₅₃	#1584
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₂×Q₁₆×C₅₃

C₄

socle

C₂²×C₅₃

C₄×D₄

C₂×Q₁₆

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
55. #88324	Q₁₆	semidirect
79. C₅₃	#1584	semidirect

H ≅ G/N

type

55. #88324

Q₁₆

semidirect

79. C₅₃

#1584

semidirect

Subgroups — #20290328

Conjugacy classes of subgroups

Complements