Finite Group Databasebeta

← C22×#16810392 #27381534

Subgroups — C22×#16810392 #27381534

Order 13568.

2494 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11719917C212yesmaximal
2C2×#16810392C2112yesmaximal, direct product factor
3C23×C4⋊C4×C53C211yesmaximal, characteristic
4C22×#11719917—28no—
5C22×#11719917—26no—
6C22×#11719917C4116yescomplemented
7C22×#11719917C22112yes—
8C23×#88322—22no—
9#16810392C22116yesdirect product factor
10C24×C4×C53C412yes—
11C24×C4×C53C2211yescharacteristic
12C22×C4⋊C4×C53—22no—
13C22×C4⋊C4×C53—26no—
14C22×C4⋊C4×C53C416yes—
15C22×C4⋊C4×C53C2216yes—
16C2×#11719917—248no—
17C2×#11719917—224no—
18C2×#11719917—212no—
19C2×#11719917C2×C4148yescomplemented
20C2×#11719917C2318yes—
21C22×#88322—216no—
22C22×#88322—212no—
23C25×C53C2×C411yescharacteristic
24C23×C4×C53—26no—
25C23×C4×C53—22no—
26C23×C4×C53—48no—
27C23×C4×C53C2×C4112yes—
28C23×C4×C53C2×C413yes—
29C23×C4×C53C2×C418yes—
30C23×C4×C53D414yes—
31C23×C4×C53D412yes—
32C23×C4×C53D416yes—
33C23×C4×C53Q814yes—
34C23×C4×C53C2313yes—
35C2×C4⋊C4×C53—212no—
36C2×C4⋊C4×C53—212no—
37C2×C4⋊C4×C53—212no—
38C2×C4⋊C4×C53—212no—
39C2×C4⋊C4×C53C2×C4112yes—
40C2×C4⋊C4×C53C2314yes—
41C24×C53—24no—
42C24×C53—24no—
43C24×C53C4214yes—
44C24×C53C22⋊C413yes—
45C24×C53C22⋊C411yescharacteristic
46C24×C53C4⋊C414yes—
47C24×C53C22×C413yes—
48C22×C4×C53—212no—
49C22×C4×C53—212no—
50C22×C4×C53—424no—
51C22×C4×C53—48no—
52C22×C4×C53—424no—
53C22×C4×C53C42112yes—
54C22×C4×C53C22⋊C416yes—
55C22×C4×C53C22⋊C416yes—
56C22×C4×C53C22⋊C418yes—
57C22×C4×C53C22⋊C4124yes—
58C22×C4×C53C4⋊C4112yes—
59C22×C4×C53C4⋊C4124yes—
60C22×C4×C53C4⋊C418yes—
61C22×C4×C53C22×C418yes—
62C22×C4×C53C22×C413yes—
63C22×C4×C53C22×C4112yes—
64C22×C4×C53C2×D416yes—
65C22×C4×C53C2×D416yes—
66C22×C4×C53C2×D416yes—
67C22×C4×C53C2×Q816yes—
68C22×C4×C53C2411yescharacteristic
69C4⋊C4×C53—224no—
70C4⋊C4×C53—28no—
71#11719917—248no—
72#11719917—28no—
73#11719917C22×C4116yescomplemented
74C2×#88322—248no—
75C2×#88322—28no—
76C23×C53—216no—
77C23×C53—216no—
78C23×C53—26no—
79C23×C53—26no—
80C23×C53—26no—
81C23×C53—26no—
82C23×C53#95112yes—
83C23×C53#9514yes—
84C23×C53#10014yes—
85C23×C53#10114yes—
86C23×C53C2×C4216yes—
87C23×C53C2×C22⋊C413yes—
88C23×C53C2×C22⋊C413yes—
89C23×C53C2×C4⋊C416yes—
90C23×C53C23×C411yescharacteristic
91C2×C4×C53—28no—
92C2×C4×C53—424no—
93C2×C4×C53—424no—
94C2×C4×C53—48no—
95C2×C4×C53#95124yes—
96C2×C4×C53#95124yes—
97C2×C4×C53C2×C4216yes—
98C2×C4×C53C2×C22⋊C4112yes—
99C2×C4×C53C2×C22⋊C4112yes—
100C2×C4×C53C2×C22⋊C416yes—
101C2×C4×C53C2×C4⋊C4112yes—
102C2×C4×C53C2×C4⋊C4112yes—
103C2×C4×C53C2×C4⋊C416yes—
104C2×C4×C53C23×C412yes—
105C2×C4×C53C22×D412yes—
106C2×C4×C53C22×D411yescharacteristic
107C2×C4×C53C22×Q811yescharacteristic
108#88322—216no—
109C22×#343—531nomaximal, Sylow
110C22×C53—23no—
111C22×C53—21no—
112C22×C53—23no—
113C22×C53—21no—
114C22×C53—224no—
115C22×C53—212no—
116C22×C53—212no—
117C22×C53#343116yescomplemented
118C22×C53C2×#9516yes—
119C22×C53C2×#9516yes—
120C22×C53C2×#10016yes—
121C22×C53C2×#10116yes—
122C22×C53C22×C4211yescharacteristic
123C22×C53C22×C22⋊C411yescharacteristic
124C22×C53C22×C4⋊C411yescharacteristic
125C4×C53—48no—
126C4×C53C2×#95112yes—
127C4×C53C22×C22⋊C412yes—
128C4×C53C22×C4⋊C412yes—
129C23×M16—532no—
130C23×C4⋊C4D5311yescharacteristic
131C2×#343—5312no—
132C2×C53—21no—
133C2×C53—26no—
134C2×C53—21no—
135C2×C53C2×#343112yescomplemented
136C2×C53C22×#9511yescharacteristic
137C2×C53C22×#10011yescharacteristic
138C2×C53C22×#10111yescharacteristic
139C24×C4Dic5312yes—
140C24×C4C2×D5311yescharacteristic
141C22×M16—5316no—
142C22×M16—5312no—
143C22×M16—1068no—
144C22×M16—1066no—
145C23×C8—1062no—
146C22×C4⋊C4—26no—
147C22×C4⋊C4—22no—
148C22×C4⋊C4Dic5316yes—
149C22×C4⋊C4C2×D5316yes—
150#343—5316no—
151C53C22×#34311yescharacteristic, complemented, Sylow
152C25C2×Dic5311yescharacteristic
153C23×C4—26no—
154C23×C4—22no—
155C23×C4—48no—
156C23×C4Dic10614yes—
157C23×C4C4×D5318yes—
158C23×C4D21214yes—
159C23×C4C2×Dic53112yes—
160C23×C4C2×Dic5313yes—
161C23×C4C22⋊D5316yes—
162C23×C4C22⋊D5312yes—
163C23×C4C22×D5313yes—
164C2×M16—5348no—
165C2×M16—538no—
166C2×M16—10624no—
167C2×M16—10612no—
168C2×M16—10648no—
169C22×C8—10612no—
170C22×C8—10616no—
171C2×C4⋊C4—212no—
172C2×C4⋊C4—212no—
173C2×C4⋊C4—212no—
174C2×C4⋊C4—212no—
175C2×C4⋊C4C2×Dic53112yes—
176C2×C4⋊C4C22×D5314yes—
177C24—24no—
178C24—24no—
179C24C4×Dic5314yes—
180C24C4⋊Dic5314yes—
181C24C22⋊Dic5313yes—
182C24C22⋊Dic5311yescharacteristic
183C24C22×Dic5313yes—
184C22×C4—212no—
185C22×C4—212no—
186C22×C4—424no—
187C22×C4—48no—
188C22×C4—424no—
189C22×C4C4×Dic53112yes—
190C22×C4#11719919124yes—
191C22×C4#1171991918yes—
192C22×C4C4⋊Dic53112yes—
193C22×C4#1171992118yes—
194C22×C4#11719921124yes—
195C22×C4C22⋊Dic5316yes—
196C22×C4C22⋊Dic5316yes—
197C22×C4C2×Dic10616yes—
198C22×C4C2×C4×D53112yes—
199C22×C4C2×D21216yes—
200C22×C4C22×Dic5318yes—
201C22×C4C22×Dic5313yes—
202C22×C4C2×C22⋊D5316yes—
203C22×C4C2×C22⋊D5316yes—
204C22×C4C23×D5311yescharacteristic
205M16—5316no—
206M16—10648no—
207M16—1068no—
208C2×C8—10648no—
209C2×C8—1068no—
210C4⋊C4—224no—
211C4⋊C4—28no—
212C23—26no—
213C23—26no—
214C23—26no—
215C23—26no—
216C23—216no—
217C23—216no—
218C23#13133124112yes—
219C23#1313312414yes—
220C23#1313313414yes—
221C23#1313313614yes—
222C23C2×C4×Dic5316yes—
223C23C2×C22⋊Dic5313yes—
224C23C2×C22⋊Dic5313yes—
225C23C2×C4⋊Dic5316yes—
226C23C23×Dic5311yescharacteristic
227C2×C4—28no—
228C2×C4—424no—
229C2×C4—424no—
230C2×C4—48no—
231C2×C4#13133124124yes—
232C2×C4#13133124124yes—
233C2×C4C2×C4×Dic5316yes—
234C2×C4C2×#11719921112yes—
235C2×C4C2×#11719921112yes—
236C2×C4C2×C22⋊Dic5316yes—
237C2×C4C2×#11719919112yes—
238C2×C4C2×#11719919112yes—
239C2×C4C2×C4⋊Dic5316yes—
240C2×C4C22×C4×D5312yes—
241C2×C4C22×C22⋊D5312yes—
242C2×C4C22×D21211yescharacteristic
243C2×C4C22×Dic10611yescharacteristic
244C8—10616no—
245C22—224no—
246C22—212no—
247C22—212no—
248C22—23no—
249C22—21no—
250C22—23no—
251C22—21no—
252C22#16810392116yesdirect product factor
253C22C2×#1313312416yes—
254C22C2×#1313312416yes—
255C22C2×#1313313416yes—
256C22C2×#1313313616yes—
257C22C22×C4×Dic5311yescharacteristic
258C22C22×C22⋊Dic5311yescharacteristic
259C22C22×C4⋊Dic5311yescharacteristic
260C4—48no—
261C4C2×#13133124112yes—
262C4C22×#1171992112yes—
263C4C22×#1171991912yes—
264C2—21no—
265C2—21no—
266C2—26no—
267C2C2×#16810392112yesdirect product factor
268C2C22×#1313312411yescharacteristic
269C2C22×#1313313411yescharacteristic
270C2C22×#1313313611yescharacteristic
subgroupG/H
centerC24C22⋊Dic53
commutatorC22×C53C22×C42
FrattiniC22×C4C23×D53
FittingC23×C4⋊C4×C53C2
socleC24×C53C22⋊C4
prime radicalC25×C53C2×C4
cosocleC22×C4×C53C24
hypercenterC23×C4⋊C4D53
hypocenterC53C22×#343
LCSG ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C24×C4 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16810392C2direct (N × H)
6. C22×#11719917C4semidirect
9. #16810392C22direct (N × H)
19. C2×#11719917C2×C4semidirect
73. #11719917C22×C4semidirect
117. C22×C53#343semidirect
135. C2×C53C2×#343semidirect
151. C53C22×#343semidirect
252. C22#16810392direct (N × H)
267. C2C2×#16810392direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117199172
C2×#168103922
C23×C4⋊C4×C532
C22×#34353