Group #20290245 — subgroups

Subgroups — #20290245

Order 6784.

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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	#16811646	C₂	1	1	yes	maximal, characteristic
2	#16811646	C₂	1	1	yes	maximal, characteristic
3	#16811482	C₂	1	4	yes	maximal, complemented
4	C₄⋊Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	#13133284	C₂²	1	2	yes	—
6	#13133284	C₂²	1	2	yes	—
7	#13133274	—	2	1	no	—
8	#13133274	—	2	1	no	—
9	#13133272	—	2	1	no	—
10	#13133272	—	2	1	no	—
11	C₂×Q₈×D₅₃	—	2	1	no	—
12	#13133217	C₄	1	1	yes	characteristic
13	#13133216	—	2	1	no	—
14	#13133216	C₂²	1	2	yes	—
15	#13133215	—	2	1	no	—
16	C₄⋊C₄×D₅₃	—	2	1	no	—
17	C₄⋊C₄×D₅₃	C₄	1	2	yes	—
18	C₄²×D₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
20	C₂×Q₈×C₅₃	—	2	1	no	—
21	Q₈×D₅₃	—	4	4	no	—
22	C₂×C₄×D₅₃	—	2	2	no	—
23	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
26	C₂×Dic₁₀₆	—	2	1	no	—
27	C₂×Dic₁₀₆	—	2	2	no	—
28	C₂×#88324	—	2	2	no	—
29	C₂×#88324	—	2	2	no	—
30	C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
32	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Dic₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
35	#11719919	—	2	2	no	—
36	#11719919	—	2	2	no	—
37	C₄×Dic₅₃	D₄	1	1	yes	characteristic
38	C₄×Dic₅₃	D₄	1	1	yes	characteristic
39	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
40	C₂²×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	Q₈×C₅₃	—	4	2	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₅₃	—	2	2	no	—
47	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
50	C₄×D₅₃	—	2	2	no	—
51	C₄×D₅₃	—	2	2	no	—
52	C₄×D₅₃	—	2	2	no	—
53	C₄×D₅₃	—	4	4	no	—
54	Dic₁₀₆	—	4	2	no	—
55	Dic₁₀₆	—	4	4	no	—
56	#88324	—	4	2	no	—
57	#88324	—	4	2	no	—
58	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×D₅₃	—	2	1	no	—
60	C₂×D₅₃	—	2	1	no	—
61	C₂×D₅₃	#101	1	2	yes	—
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	—	4	2	no	—
66	Dic₅₃	—	2	1	no	—
67	Dic₅₃	—	4	2	no	—
68	Dic₅₃	AGL(1,ℤ/8)	1	2	yes	—
69	Dic₅₃	#137	1	2	yes	—
70	#1556	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	C₂×#101	1	1	yes	characteristic
72	C₂×C₅₃	#418	1	1	yes	characteristic
73	C₂×C₅₃	#419	1	1	yes	characteristic
74	D₅₃	—	2	2	no	—
75	C₂×C₄⋊Q₈	—	53	1	no	—
76	#432	—	53	1	no	—
77	#432	—	53	1	no	—
78	#332	—	53	4	no	—
79	C₅₃	#1556	1	1	yes	characteristic, complemented, Sylow
80	C₂²×Q₈	—	106	1	no	—
81	C₄⋊Q₈	—	53	2	no	—
82	C₄⋊Q₈	—	53	1	no	—
83	C₄⋊Q₈	—	106	1	no	—
84	C₄⋊Q₈	—	106	1	no	—
85	C₄⋊Q₈	#6157	1	1	yes	characteristic
86	C₂×C₄⋊C₄	—	53	2	no	—
87	C₂×C₄⋊C₄	—	106	1	no	—
88	C₂×C₄²	—	53	1	no	—
89	C₄⋊C₈	—	53	2	no	—
90	C₄⋊C₈	—	53	2	no	—
91	C₂²⋊C₈	—	106	1	no	—
92	C₂²⋊C₈	—	106	1	no	—
93	#97	—	106	1	no	—
94	#97	—	106	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	106	2	no	—
97	C₂×Q₈	—	106	1	no	—
98	C₂×Q₈	—	212	4	no	—
99	C₂²×C₄	—	53	1	no	—
100	C₂²×C₄	—	53	1	no	—
101	C₂²×C₄	—	53	1	no	—
102	C₂²×C₄	—	106	2	no	—
103	C₂×C₈	—	106	2	no	—
104	C₂×C₈	—	106	2	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	53	2	no	—
107	C₄⋊C₄	—	106	2	no	—
108	C₄⋊C₄	—	106	2	no	—
109	C₄⋊C₄	—	106	1	no	—
110	C₄⋊C₄	C₂×#6157	1	2	yes	—
111	C₄²	—	53	1	no	—
112	C₄²	—	53	1	no	—
113	C₄²	—	53	1	no	—
114	C₄²	C₂×#6157	1	1	yes	characteristic
115	C₂³	—	53	1	no	—
116	Q₈	—	4	2	no	—
117	Q₈	—	212	2	no	—
118	Q₈	—	212	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	53	1	no	—
121	C₂×C₄	—	53	1	no	—
122	C₂×C₄	—	53	1	no	—
123	C₂×C₄	—	106	2	no	—
124	C₂×C₄	—	106	2	no	—
125	C₂×C₄	—	106	2	no	—
126	C₂×C₄	—	106	2	no	—
127	C₂×C₄	—	212	4	no	—
128	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
129	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
130	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
131	C₈	—	212	2	no	—
132	C₈	—	212	2	no	—
133	C₂²	—	53	2	no	—
134	C₂²	—	106	1	no	—
135	C₂²	—	106	1	no	—
136	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	2	no	—
141	C₄	—	53	2	no	—
142	C₄	—	53	2	no	—
143	C₄	—	106	1	no	—
144	C₄	—	212	2	no	—
145	C₂	—	106	2	no	—
146	C₂	D₅₃⋊#101	1	1	yes	characteristic
147	C₂	#16811615	1	1	yes	characteristic
148	C₂	#16811616	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811646

C₂

yes

maximal, characteristic

#16811646

C₂

yes

maximal, characteristic

#16811482

C₂

yes

maximal, complemented

C₄⋊Q₈×D₅₃

C₂

yes

maximal, characteristic

#13133284

C₂²

yes

—

#13133284

C₂²

yes

—

—

—

—

—

C₂×Q₈×D₅₃

—

#13133217

C₄

yes

characteristic

#13133216

—

#13133216

C₂²

yes

—

#13133215

—

C₄⋊C₄×D₅₃

—

C₄⋊C₄×D₅₃

C₄

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

—

Q₈×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×#88324

—

C₂×#88324

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

C₂²×D₅₃

C₂²⋊C₄

yes

characteristic

Q₈×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×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

#88324

—

#88324

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#101

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

AGL(1,ℤ/8)

yes

—

Dic₅₃

#137

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂×C₄⋊Q₈

—

#432

—

#432

—

#332

—

C₅₃

#1556

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

#6157

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

#97

—

106

—

#97

—

106

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

212

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

106

—

103

C₂×C₈

—

106

—

104

C₂×C₈

—

106

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

106

—

108

C₄⋊C₄

—

106

—

109

C₄⋊C₄

—

106

—

110

C₄⋊C₄

C₂×#6157

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂×#6157

yes

characteristic

115

C₂³

—

116

Q₈

—

117

Q₈

—

212

—

118

Q₈

—

212

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

106

—

125

C₂×C₄

—

106

—

126

C₂×C₄

—

106

—

127

C₂×C₄

—

212

—

128

C₂×C₄

C₂²⋊#6157

yes

characteristic

129

C₂×C₄

C₂²⋊#6157

yes

characteristic

130

C₂×C₄

C₂²×#6157

yes

characteristic

131

C₈

—

212

—

132

C₈

—

212

—

133

C₂²

—

134

C₂²

—

106

—

135

C₂²

—

106

—

136

C₂²

C₂×C₂²⋊#6157

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

106

—

144

C₄

—

212

—

145

C₂

—

106

—

146

C₂

D₅₃⋊#101

yes

characteristic

147

C₂

#16811615

yes

characteristic

148

C₂

#16811616

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄⋊Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#6157
hypocenter	C₅₃	#1556
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄⋊Q₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

C₄⋊Q₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
3. #16811482	C₂	semidirect
79. C₅₃	#1556	semidirect

H ≅ G/N

type

3. #16811482

C₂

semidirect

79. C₅₃

#1556

semidirect

Subgroups — #20290245

Conjugacy classes of subgroups

Complements