Group #20290342 — subgroups

Subgroups — #20290342

Order 6784.

166 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 56 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	Q₈×#88324	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811657	C₂	1	1	yes	maximal, characteristic
3	#16811486	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#88324	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811471	C₂	1	1	yes	maximal, characteristic
6	#88324⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810708	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719936	—	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	#13133274	—	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×#11719924	C₄	1	1	yes	characteristic
17	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	#13133179	C₂²	1	1	yes	characteristic
19	#13133148	C₄	1	1	yes	characteristic
20	#13133139	C₄	1	1	yes	characteristic
21	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
22	#103×C₅₃	C₄	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	—	2	1	no	—
26	C₂×#88324	—	2	1	no	—
27	C₂×#88324	—	2	1	no	—
28	C₂×#88324	—	2	1	no	—
29	C₂×#88324	D₄	1	1	yes	characteristic
30	C₂×#88324	D₄	1	1	yes	characteristic
31	#11719936	—	4	2	no	—
32	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	#11719924	—	2	2	no	—
35	#11719921	—	2	1	no	—
36	C₄⋊Dic₅₃	—	2	1	no	—
37	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₅₃	—	2	1	no	—
39	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
40	C₂×#88322	C₂×C₄	1	1	yes	characteristic
41	C₂²×D₅₃	—	2	1	no	—
42	Q₈×C₅₃	—	2	1	no	—
43	Q₈×C₅₃	M₁₆	1	2	yes	complemented
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	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₅₃	#41	1	1	yes	characteristic
50	D₂₁₂	—	2	1	no	—
51	D₂₁₂	—	2	1	no	—
52	C₄×D₅₃	—	4	1	no	—
53	C₄×D₅₃	—	4	1	no	—
54	#88324	—	2	2	no	—
55	#88324	—	2	1	no	—
56	#88324	—	2	1	no	—
57	#88324	—	4	1	no	—
58	#88324	—	4	1	no	—
59	C₈×C₅₃	—	2	1	no	—
60	#88322	—	2	1	no	—
61	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
62	C₂×D₅₃	—	4	1	no	—
63	C₂×D₅₃	—	4	1	no	—
64	C₂×D₅₃	—	4	1	no	—
65	C₄×C₅₃	—	2	2	no	—
66	C₄×C₅₃	—	4	1	no	—
67	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
68	C₄×C₅₃	#131	1	1	yes	characteristic
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	#1588	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	#435	1	1	yes	characteristic
75	C₂×C₅₃	#440	1	1	yes	characteristic
76	C₂×C₅₃	#443	1	1	yes	characteristic
77	D₅₃	—	8	1	no	—
78	C₈×Q₈	—	53	1	no	—
79	C₄×SD₁₆	—	53	1	no	—
80	#436	—	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	#1588	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	53	1	no	—
87	C₂×M₁₆	—	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	#6157	1	1	yes	characteristic
95	#102	—	53	1	no	—
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	C₂×#6157	1	1	yes	characteristic
102	C₂×D₄	—	53	1	no	—
103	C₂²×C₄	—	106	1	no	—
104	SD₁₆	—	106	2	no	—
105	M₁₆	—	212	2	no	—
106	C₂×C₈	—	53	1	no	—
107	C₂×C₈	—	53	1	no	—
108	C₂×C₈	—	53	1	no	—
109	C₂×C₈	—	106	1	no	—
110	C₂×C₈	—	106	1	no	—
111	C₂×C₈	—	106	1	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	—	106	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	#11719936	1	2	yes	complemented
122	D₄	—	106	1	no	—
123	D₄	—	106	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	53	1	no	—
127	C₂×C₄	—	53	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	—	106	1	no	—
134	C₈	—	106	1	no	—
135	C₈	—	106	2	no	—
136	C₈	—	106	1	no	—
137	C₈	—	212	1	no	—
138	C₈	—	212	1	no	—
139	C₂²	—	212	1	no	—
140	C₂²	—	212	1	no	—
141	C₂²	—	212	1	no	—
142	C₂²	D₄×#6157	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	4	1	no	—
145	C₄	—	53	1	no	—
146	C₄	—	53	1	no	—
147	C₄	—	106	1	no	—
148	C₄	—	212	1	no	—
149	C₄	C₂×#11719936	1	1	yes	characteristic
150	C₄	#13133304	1	1	yes	characteristic
151	C₂	—	424	1	no	—
152	C₂	#16811652	1	1	yes	characteristic
153	C₂	#16811662	1	1	yes	characteristic
154	C₂	#16811667	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#88324

C₂

yes

maximal, characteristic, complemented

#16811657

C₂

yes

maximal, characteristic

#16811486

C₂

yes

maximal, characteristic, complemented

Q₈⋊#88324

C₂

yes

maximal, characteristic, complemented

#16811471

C₂

yes

maximal, characteristic

#88324⋊C₈

C₂

yes

maximal, characteristic, complemented

#16810708

C₂

yes

maximal, characteristic

C₂×#11719936

—

C₄⋊#88324

—

C₄⋊#88324

C₂²

yes

characteristic

C₄⋊#88324

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×#88324

C₂²

yes

characteristic

C₂×#11719924

C₄

yes

characteristic

Q₈×Dic₅₃

C₂²

yes

characteristic

#13133179

C₂²

yes

characteristic

#13133148

C₄

yes

characteristic

#13133139

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

#103×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₂₁₂

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

—

—

—

—

D₄

yes

characteristic

C₂×#88324

D₄

yes

characteristic

#11719936

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₃

C₂×C₄

yes

characteristic

#11719924

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

M₁₆

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

#41

yes

characteristic

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

—

—

—

—

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

#131

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₈×Q₈

—

C₄×SD₁₆

—

#436

—

#335

—

Q₈⋊C₈

—

D₄⋊C₈

—

#322

—

C₅₃

#1588

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×M₁₆

—

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

C₂×#6157

yes

characteristic

102

C₂×D₄

—

103

C₂²×C₄

—

106

—

104

SD₁₆

—

106

—

105

M₁₆

—

212

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

106

—

110

C₂×C₈

—

106

—

111

C₂×C₈

—

106

—

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

—

106

—

120

Q₈

—

121

Q₈

#11719936

yes

complemented

122

D₄

—

106

—

123

D₄

—

106

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

106

—

129

C₂×C₄

—

212

—

130

C₂×C₄

—

212

—

131

C₂×C₄

C₂²×#6157

yes

characteristic

132

C₈

—

133

C₈

—

106

—

134

C₈

—

106

—

135

C₈

—

106

—

136

C₈

—

106

—

137

C₈

—

212

—

138

C₈

—

212

—

139

C₂²

—

212

—

140

C₂²

—

212

—

141

C₂²

—

212

—

142

C₂²

D₄×#6157

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

106

—

148

C₄

—

212

—

149

C₄

C₂×#11719936

yes

characteristic

150

C₄

#13133304

yes

characteristic

151

C₂

—

424

—

152

C₂

#16811652

yes

characteristic

153

C₂

#16811662

yes

characteristic

154

C₂

#16811667

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	#103×C₅₃	C₄
socle	C₂²×C₅₃	C₄×D₄
hypercenter	#103	#6157
hypocenter	C₅₃	#1588
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

subgroup

G/H

center

C₂²

D₄×#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

#103×C₅₃

C₄

socle

C₂²×C₅₃

C₄×D₄

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

N	H ≅ G/N	type
1. Q₈×#88324	C₂	semidirect
3. #16811486	C₂	semidirect
4. Q₈⋊#88324	C₂	semidirect
6. #88324⋊C₈	C₂	semidirect
43. Q₈×C₅₃	M₁₆	semidirect
85. C₅₃	#1588	semidirect
121. Q₈	#11719936	semidirect

H ≅ G/N

type

1. Q₈×#88324

C₂

semidirect

3. #16811486

C₂

semidirect

4. Q₈⋊#88324

C₂

semidirect

6. #88324⋊C₈

C₂

semidirect

43. Q₈×C₅₃

M₁₆

semidirect

85. C₅₃

#1588

semidirect

121. Q₈

#11719936

semidirect

Subgroups — #20290342

Conjugacy classes of subgroups

Complements