Group #20290306 — subgroups

Subgroups — #20290306

Order 6784.

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

normal subgroups only

This group has 176 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	#16811672	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#88324	C₂	1	1	yes	maximal, characteristic
3	#16811487	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811473	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊#88324	C₂	1	1	yes	maximal, characteristic
6	#16811464	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810714	C₂	1	1	yes	maximal, characteristic
8	C₂²×#88324	—	2	1	no	—
9	#13133284	—	2	1	no	—
10	C₄⋊#88324	C₂²	1	1	yes	characteristic
11	C₄⋊#88324	C₂²	1	1	yes	characteristic
12	C₂²⋊#88324	—	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	C₂×#11719923	C₄	1	1	yes	characteristic
17	#13133182	C₂²	1	1	yes	characteristic
18	D₄×Dic₅₃	C₂²	1	1	yes	characteristic
19	#13133148	C₄	1	1	yes	characteristic
20	#13133142	C₄	1	1	yes	characteristic
21	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
22	#102×C₅₃	C₄	1	1	yes	characteristic
23	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₅₃	—	2	1	no	—
25	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
26	C₂×#88324	—	2	1	no	—
27	C₂×#88324	—	2	1	no	—
28	C₂×#88324	—	2	1	no	—
29	C₂×#88324	—	4	2	no	—
30	C₂×#88324	D₄	1	1	yes	characteristic
31	C₂×#88324	D₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²⋊Dic₅₃	—	2	1	no	—
35	#11719923	C₈	1	4	yes	complemented
36	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
37	#11719919	—	2	1	no	—
38	C₄×Dic₅₃	—	2	1	no	—
39	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
40	C₂×#88322	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₅₃	—	2	1	no	—
42	D₄×C₅₃	—	2	1	no	—
43	D₄×C₅₃	C₂×C₈	1	2	yes	—
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₅₃	—	2	1	no	—
47	C₂×Dic₅₃	—	2	1	no	—
48	C₂×Dic₅₃	—	4	1	no	—
49	C₂×Dic₅₃	—	4	1	no	—
50	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₅₃	#41	1	1	yes	characteristic
52	Dic₁₀₆	—	2	1	no	—
53	Dic₁₀₆	C₂×C₈	1	2	yes	complemented
54	#88324	—	2	1	no	—
55	#88324	—	2	1	no	—
56	#88324	—	4	1	no	—
57	#88324	—	4	1	no	—
58	#88324	—	4	1	no	—
59	C₈×C₅₃	—	2	1	no	—
60	#88322	C₂×C₈	1	2	yes	complemented
61	C₂²×C₅₃	—	2	2	no	—
62	C₂²×C₅₃	—	4	1	no	—
63	C₂²×C₅₃	—	4	1	no	—
64	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
65	C₄×C₅₃	—	4	1	no	—
66	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
67	C₄×C₅₃	#131	1	1	yes	characteristic
68	Dic₅₃	—	2	2	no	—
69	Dic₅₃	—	2	1	no	—
70	Dic₅₃	—	4	1	no	—
71	Dic₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
72	Dic₅₃	#137	1	1	yes	characteristic
73	SD₁₆⋊C₈	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	—	4	2	no	—
75	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
76	C₂×C₅₃	#440	1	1	yes	characteristic
77	C₂×C₅₃	#444	1	1	yes	characteristic
78	C₈×Q₈	—	53	1	no	—
79	C₄×SD₁₆	—	53	1	no	—
80	C₈×D₄	—	53	1	no	—
81	#335	—	53	1	no	—
82	Q₈⋊C₈	—	53	1	no	—
83	D₄⋊C₈	—	53	1	no	—
84	#322	—	53	1	no	—
85	C₅₃	SD₁₆⋊C₈	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	53	1	no	—
87	C₂²×C₈	—	106	1	no	—
88	C₄×Q₈	—	53	1	no	—
89	C₄×D₄	—	53	1	no	—
90	#106	—	53	1	no	—
91	C₄⋊C₈	—	53	1	no	—
92	C₄⋊C₈	—	53	1	no	—
93	C₄⋊C₈	—	106	1	no	—
94	#103	—	53	1	no	—
95	#102	#6157	1	1	yes	characteristic
96	C₂²⋊C₈	—	106	1	no	—
97	C₄×C₈	—	53	1	no	—
98	C₄×C₈	—	53	1	no	—
99	C₄×C₈	—	53	1	no	—
100	C₄×C₈	—	106	1	no	—
101	C₂×Q₈	—	53	1	no	—
102	C₂×D₄	C₂×#6157	1	1	yes	characteristic
103	C₂²×C₄	—	106	1	no	—
104	SD₁₆	—	53	4	no	—
105	C₂×C₈	—	53	1	no	—
106	C₂×C₈	—	53	1	no	—
107	C₂×C₈	—	53	1	no	—
108	C₂×C₈	—	106	1	no	—
109	C₂×C₈	—	106	1	no	—
110	C₂×C₈	—	106	1	no	—
111	C₂×C₈	—	212	2	no	—
112	C₂×C₈	C₂×#6157	1	1	yes	characteristic
113	C₄⋊C₄	—	53	1	no	—
114	C₄⋊C₄	—	106	1	no	—
115	C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
116	C₂²⋊C₄	—	106	1	no	—
117	C₄²	—	53	1	no	—
118	C₄²	—	106	1	no	—
119	C₂³	—	2	1	no	—
120	Q₈	—	53	2	no	—
121	Q₈	—	106	1	no	—
122	D₄	—	2	1	no	—
123	D₄	D₅₃⋊C₈	1	2	yes	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	53	1	no	—
126	C₂×C₄	—	53	1	no	—
127	C₂×C₄	—	106	1	no	—
128	C₂×C₄	—	106	1	no	—
129	C₂×C₄	—	212	1	no	—
130	C₂×C₄	—	212	1	no	—
131	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	53	2	no	—
134	C₈	—	106	1	no	—
135	C₈	—	106	1	no	—
136	C₈	—	212	1	no	—
137	C₈	—	212	1	no	—
138	C₈	—	212	1	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	D₄×#6157	1	1	yes	characteristic
143	C₄	—	4	1	no	—
144	C₄	—	53	1	no	—
145	C₄	—	53	1	no	—
146	C₄	—	106	2	no	—
147	C₄	—	106	1	no	—
148	C₄	—	212	1	no	—
149	C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
150	C₄	#13133304	1	1	yes	characteristic
151	C₂	—	4	2	no	—
152	C₂	#16811650	1	1	yes	characteristic
153	C₂	#16811662	1	1	yes	characteristic
154	C₂	#16811670	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811672

C₂

yes

maximal, characteristic, complemented

D₄×#88324

C₂

yes

maximal, characteristic

#16811487

C₂

yes

maximal, characteristic, complemented

#16811473

C₂

yes

maximal, characteristic, complemented

D₄⋊#88324

C₂

yes

maximal, characteristic

#16811464

C₂

yes

maximal, characteristic, complemented

#16810714

C₂

yes

maximal, characteristic

C₂²×#88324

—

#13133284

—

C₄⋊#88324

C₂²

yes

characteristic

C₄⋊#88324

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×#88324

C₂²

yes

characteristic

C₂×#11719923

C₄

yes

characteristic

#13133182

C₂²

yes

characteristic

D₄×Dic₅₃

C₂²

yes

characteristic

#13133148

C₄

yes

characteristic

#13133142

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

#102×C₅₃

C₄

yes

characteristic

C₂×D₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

—

—

—

D₄

yes

characteristic

C₂×#88324

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²⋊Dic₅₃

—

#11719923

C₈

yes

complemented

C₄⋊Dic₅₃

C₂×C₄

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

#41

yes

characteristic

Dic₁₀₆

—

Dic₁₀₆

C₂×C₈

yes

—

—

—

—

—

C₈×C₅₃

—

#88322

C₂×C₈

yes

complemented

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

#131

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

SD₁₆⋊C₈

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₈×D₄

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₈×Q₈

—

C₄×SD₁₆

—

C₈×D₄

—

#335

—

Q₈⋊C₈

—

D₄⋊C₈

—

#322

—

C₅₃

SD₁₆⋊C₈

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂²×C₈

—

106

—

C₄×Q₈

—

C₄×D₄

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

—

yes

C₂²⋊C₈

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

106

—

101

C₂×Q₈

—

102

C₂×D₄

C₂×#6157

yes

characteristic

103

C₂²×C₄

—

106

—

104

SD₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

106

—

109

C₂×C₈

—

106

—

110

C₂×C₈

—

106

—

111

C₂×C₈

—

212

—

112

C₂×C₈

C₂×#6157

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

106

—

115

C₄⋊C₄

C₂×#6157

yes

characteristic

116

C₂²⋊C₄

—

106

—

117

C₄²

—

118

C₄²

—

106

—

119

C₂³

—

120

Q₈

—

121

Q₈

—

106

—

122

D₄

—

123

D₄

D₅₃⋊C₈

yes

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

106

—

128

C₂×C₄

—

106

—

129

C₂×C₄

—

212

—

130

C₂×C₄

—

212

—

131

C₂×C₄

C₂²×#6157

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

106

—

135

C₈

—

106

—

136

C₈

—

212

—

137

C₈

—

212

—

138

C₈

—

212

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

D₄×#6157

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

106

—

147

C₄

—

106

—

148

C₄

—

212

—

149

C₄

C₂×D₅₃⋊C₈

yes

characteristic

150

C₄

#13133304

yes

characteristic

151

C₂

—

152

C₂

#16811650

yes

characteristic

153

C₂

#16811662

yes

characteristic

154

C₂

#16811670

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#6157
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6157
Fitting	#102×C₅₃	C₄
socle	C₂²×C₅₃	C₄×D₄
hypercenter	#102	#6157
hypocenter	C₅₃	SD₁₆⋊C₈
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

subgroup

G/H

center

C₂²

D₄×#6157

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₂×C₄

C₂²×#6157

Fitting

#102×C₅₃

C₄

socle

C₂²×C₅₃

C₄×D₄

C₅₃

SD₁₆⋊C₈

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

N	H ≅ G/N	type
1. #16811672	C₂	semidirect
3. #16811487	C₂	semidirect
4. #16811473	C₂	semidirect
6. #16811464	C₂	semidirect
35. #11719923	C₈	semidirect
53. Dic₁₀₆	C₂×C₈	semidirect
60. #88322	C₂×C₈	semidirect
85. C₅₃	SD₁₆⋊C₈	semidirect

H ≅ G/N

type

1. #16811672

C₂

semidirect

3. #16811487

C₂

semidirect

4. #16811473

C₂

semidirect

6. #16811464

C₂

semidirect

35. #11719923

C₈

semidirect

53. Dic₁₀₆

C₂×C₈

semidirect

60. #88322

C₂×C₈

semidirect

85. C₅₃

SD₁₆⋊C₈

semidirect

Subgroups — #20290306

Conjugacy classes of subgroups

Complements