Finite Group Databasebeta

← C22×C53×#568 #27270586

Subgroups — C22×C53×#568 #27270586

Order 13568.

11118 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#41×C53C211yesmaximal, characteristic
2C2×C53×#568C2148yesmaximal, direct product factor
3C22×#131×C53C218yesmaximal, complemented
4C23×M16×C53C216yesmaximal, complemented
5C22×#41×C53C414yes—
6C22×#41×C53C4124yes—
7C22×#41×C53C2214yes—
8C22×#41×C53C22124yes—
9C23×Q8×C53C411yescharacteristic
10C23×D4×C53C413yes—
11C24×C4×C53C2213yes—
12C53×#568C221256yesdirect product factor
13C2×#131×C53C221192yescomplemented
14C22×M16×C53C2214yes—
15C22×M16×C53C22112yescomplemented
16C22×M16×C53C221144yescomplemented
17C23×C8×C53C22112yescomplemented
18C25×C53C2×C413yes—
19C2×#41×C53C2×C4148yes—
20C2×#41×C53C2×C4196yes—
21C2×#41×C53C2×C41192yes—
22C2×#41×C53C23148yes—
23C2×#41×C53C23164yes—
24C22×Q8×C53C2×C414yes—
25C22×Q8×C53C2×C4124yes—
26C22×D4×C53C2×C4112yes—
27C22×D4×C53C2×C4172yes—
28C23×C4×C53C2×C413yes—
29C23×C4×C53C2×C416yes—
30C23×C4×C53C2×C4136yes—
31C23×C4×C53C23136yes—
32C23×C4×C53C2316yes—
33C23×C4×C53C2311yescharacteristic
34#131×C53C231512yescomplemented
35C2×M16×C53C23148yes—
36C2×M16×C53C231144yescomplemented
37C2×M16×C53C231384yescomplemented
38C22×C8×C53C2318yes—
39C22×C8×C53C231144yescomplemented
40C24×C53—412no—
41C24×C53C22×C411yescharacteristic
42C24×C53C22×C416yes—
43C24×C53C22×C4136yes—
44#41×C53C22×C41192yes—
45#41×C53C22×C41256yes—
46#41×C53C24164yes—
47C2×Q8×C53C22×C4148yes—
48C2×Q8×C53C22×C4164yes—
49C2×D4×C53C22×C41144yes—
50C2×D4×C53C22×C41192yes—
51C22×C4×C53C22×C411yescharacteristic
52C22×C4×C53C22×C416yes—
53C22×C4×C53C22×C4136yes—
54C22×C4×C53C22×C416yes—
55C22×C4×C53C22×C4172yes—
56C22×C4×C53C22×C41144yes—
57C22×C4×C53C22×C4136yes—
58C22×C4×C53C24136yes—
59C22×C4×C53C2416yes—
60C22×C4×C53C24148yes—
61C22×C4×C53C2411yescharacteristic
62M16×C53C24164yes—
63M16×C53C241192yescomplemented
64C2×C8×C53C24148yes—
65C2×C8×C53C241192yescomplemented
66C23×C53—24no—
67C23×C53—224no—
68C23×C53—472no—
69C23×C53C23×C416yes—
70C23×C53C23×C4136yes—
71C23×C53C23×C4148yes—
72C23×C53C23×C411yescharacteristic
73Q8×C53C23×C4164yes—
74D4×C53C23×C41192yes—
75C2×C4×C53C23×C4112yes—
76C2×C4×C53C23×C416yes—
77C2×C4×C53C23×C413yes—
78C2×C4×C53C23×C4148yes—
79C2×C4×C53C23×C4136yes—
80C2×C4×C53C23×C4172yes—
81C2×C4×C53C23×C4196yes—
82C2×C4×C53C2514yes—
83C2×C4×C53C2513yes—
84C2×C4×C53C25124yes—
85C8×C53C25132yes—
86C22×#568C5311yesmaximal, characteristic, direct product factor, Sylow
87C22×C53—212no—
88C22×C53—272no—
89C22×C53—448no—
90C22×C53#56814yesdirect product factor
91C22×C53C24×C414yes—
92C22×C53C24×C4124yes—
93C22×C53C24×C413yes—
94C4×C53C24×C414yes—
95C4×C53C24×C413yes—
96C4×C53C24×C4124yes—
97C4×C53C2611yescharacteristic
98C23×#41C2×C5311yescharacteristic
99C2×#568C2×C53148yesdirect product factor
100C22×#131C2×C5318yescomplemented
101C23×M16C2×C5316yescomplemented
102C2×C53—24no—
103C2×C53—224no—
104C2×C53C2×#56816yesdirect product factor
105C2×C53C25×C411yescharacteristic
106C22×#41C4×C5314yes—
107C22×#41C4×C53124yes—
108C22×#41C22×C5314yes—
109C22×#41C22×C53124yes—
110C23×Q8C4×C5311yescharacteristic
111C23×D4C4×C5313yes—
112C24×C4C22×C5313yes—
113#568C22×C531256yesdirect product factor
114C2×#131C22×C531192yescomplemented
115C22×M16C22×C5314yes—
116C22×M16C22×C53112yescomplemented
117C22×M16C22×C531144yescomplemented
118C23×C8C22×C53112yescomplemented
119C53C22×#56811yescharacteristic, direct product factor, Sylow
120C25C2×C4×C5313yes—
121C2×#41C2×C4×C53148yes—
122C2×#41C2×C4×C53196yes—
123C2×#41C2×C4×C531192yes—
124C2×#41C23×C53148yes—
125C2×#41C23×C53164yes—
126C22×Q8C2×C4×C5314yes—
127C22×Q8C2×C4×C53124yes—
128C22×D4C2×C4×C53112yes—
129C22×D4C2×C4×C53172yes—
130C23×C4C2×C4×C5313yes—
131C23×C4C2×C4×C5316yes—
132C23×C4C2×C4×C53136yes—
133C23×C4C23×C5316yes—
134C23×C4C23×C5311yescharacteristic
135C23×C4C23×C53136yes—
136#131C23×C531512yescomplemented
137C2×M16C23×C53148yes—
138C2×M16C23×C531144yescomplemented
139C2×M16C23×C531384yescomplemented
140C22×C8C23×C5318yes—
141C22×C8C23×C531144yescomplemented
142C24—412no—
143C24C22×C4×C5311yescharacteristic
144C24C22×C4×C5316yes—
145C24C22×C4×C53136yes—
146#41C22×C4×C531192yes—
147#41C22×C4×C531256yes—
148#41C24×C53164yes—
149C2×Q8C22×C4×C53148yes—
150C2×Q8C22×C4×C53164yes—
151C2×D4C22×C4×C531144yes—
152C2×D4C22×C4×C531192yes—
153C22×C4C22×C4×C5316yes—
154C22×C4C22×C4×C5311yescharacteristic
155C22×C4C22×C4×C53136yes—
156C22×C4C22×C4×C5316yes—
157C22×C4C22×C4×C53136yes—
158C22×C4C22×C4×C53172yes—
159C22×C4C22×C4×C531144yes—
160C22×C4C24×C53136yes—
161C22×C4C24×C5316yes—
162C22×C4C24×C53148yes—
163C22×C4C24×C5311yescharacteristic
164M16C24×C53164yes—
165M16C24×C531192yescomplemented
166C2×C8C24×C53148yes—
167C2×C8C24×C531192yescomplemented
168C23—24no—
169C23—224no—
170C23—472no—
171C23C23×C4×C5316yes—
172C23C23×C4×C53136yes—
173C23C23×C4×C53148yes—
174C23C23×C4×C5311yescharacteristic
175Q8C23×C4×C53164yes—
176D4C23×C4×C531192yes—
177C2×C4C23×C4×C53112yes—
178C2×C4C23×C4×C5316yes—
179C2×C4C23×C4×C5313yes—
180C2×C4C23×C4×C53148yes—
181C2×C4C23×C4×C53136yes—
182C2×C4C23×C4×C53172yes—
183C2×C4C23×C4×C53196yes—
184C2×C4C25×C5313yes—
185C2×C4C25×C53124yes—
186C2×C4C25×C5314yes—
187C8C25×C53132yes—
188C22—212no—
189C22—272no—
190C22—448no—
191C22C53×#56814yesdirect product factor
192C22C24×C4×C5314yes—
193C22C24×C4×C53124yes—
194C22C24×C4×C5313yes—
195C4C24×C4×C5313yes—
196C4C24×C4×C5314yes—
197C4C24×C4×C53124yes—
198C4C26×C5311yescharacteristic
199C2—24no—
200C2—224no—
201C2C2×C53×#56816yesdirect product factor
202C2C25×C4×C5311yescharacteristic
subgroupG/H
centerC22×C4×C53C24
commutatorC2C25×C4×C53
Frattini, cosocleC4C26×C53
socleC23×C53C23×C4
prime radicalC23×#41×C53C2
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C53 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C53×#568C2direct (N × H)
3. C22×#131×C53C2semidirect
4. C23×M16×C53C2semidirect
12. C53×#568C22direct (N × H)
13. C2×#131×C53C22semidirect
15. C22×M16×C53C22semidirect
16. C22×M16×C53C22semidirect
17. C23×C8×C53C22semidirect
34. #131×C53C23semidirect
36. C2×M16×C53C23semidirect
37. C2×M16×C53C23semidirect
39. C22×C8×C53C23semidirect
63. M16×C53C24semidirect
65. C2×C8×C53C24semidirect
86. C22×#568C53direct (N × H)
90. C22×C53#568direct (N × H)
99. C2×#568C2×C53direct (N × H)
100. C22×#131C2×C53semidirect
101. C23×M16C2×C53semidirect
104. C2×C53C2×#568direct (N × H)
113. #568C22×C53direct (N × H)
114. C2×#131C22×C53semidirect
116. C22×M16C22×C53semidirect
117. C22×M16C22×C53semidirect
118. C23×C8C22×C53semidirect
119. C53C22×#568direct (N × H)
136. #131C23×C53semidirect
138. C2×M16C23×C53semidirect
139. C2×M16C23×C53semidirect
141. C22×C8C23×C53semidirect
165. M16C24×C53semidirect
167. C2×C8C24×C53semidirect
191. C22C53×#568direct (N × H)
201. C2C2×C53×#568direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×#41×C532
C2×C53×#5682
C22×#131×C532
C23×M16×C532
C22×#56853