Finite Group Databasebeta

← #27381541

Subgroups — #27381541

Order 13568.

2046 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811390C211yesmaximal, characteristic
2C2×#16811106C211yesmaximal, characteristic
3#20274453C216yesmaximal, complemented
4C2×#16810391C216yesmaximal, complemented
5C2×C53×#568C211yesmaximal, characteristic
6#16811390—24no—
7#16811390C418yescomplemented
8C2×#13133224—24no—
9C22×#11719917C2216yes—
10#16811106—24no—
11#16811106—24no—
12C2×Q8×Dic53C412yes—
13C2×D4×Dic53C416yes—
14C2×#13133176C2213yes—
15C22×C4⋊Dic53C2213yes—
16#16810391C22112yes—
17#16810391C2214yes—
18C22×#41×C53C2211yescharacteristic
19C53×#568—24no—
20C53×#568C418yescomplemented
21C2×#131×C53—24no—
22C22×M16×C53C2216yes—
23C23×Dic53C2×C416yes—
24#13133224—232no—
25C2×#11719917—26no—
26C2×#11719917—212no—
27C2×#11719917C2×C4124yescomplemented
28C2×#11719917C2314yes—
29C22×#88322—26no—
30Q8×Dic53—28no—
31D4×Dic53—224no—
32#13133176—26no—
33#13133176—26no—
34C2×C4⋊Dic53—26no—
35C2×C4⋊Dic53—26no—
36C2×C4⋊Dic53C2×C416yes—
37C2×C4⋊Dic53C2313yes—
38C2×C4⋊Dic53C2311yescharacteristic
39C2×C22⋊Dic53C2×C416yes—
40C2×C4×Dic53C2×C416yes—
41C2×#41×C53C2×C414yes—
42C2×#41×C53C2×C414yes—
43C2×#41×C53D414yes—
44C2×#41×C53D414yes—
45C2×#41×C53D414yes—
46C2×#41×C53Q814yes—
47C22×Q8×C53C2×C411yescharacteristic
48C22×D4×C53C2×C413yes—
49C23×C4×C53C2313yes—
50#131×C53—232no—
51C2×M16×C53—26no—
52C2×M16×C53—212no—
53C2×M16×C53C2×C4124yescomplemented
54C2×M16×C53C2314yes—
55C22×C8×C53—26no—
56C24×C53C22×C413yes—
57#41×C53C22⋊C4116yes—
58#41×C53C22⋊C4116yes—
59#41×C53C4⋊C4116yes—
60#41×C53C4⋊C4116yes—
61C2×Q8×C53C4214yes—
62C2×Q8×C53C22⋊C414yes—
63C2×Q8×C53C4⋊C414yes—
64C2×D4×C53C42112yes—
65C2×D4×C53C22⋊C4112yes—
66C2×D4×C53C4⋊C4112yes—
67C22×C4×C53C22×C413yes—
68C22×C4×C53C22×C416yes—
69C22×C4×C53C22×C416yes—
70C22×C4×C53C2×D416yes—
71C22×C4×C53C2×D416yes—
72C22×C4×C53C2×D416yes—
73C22×C4×C53C2×Q816yes—
74C22×C4×C53C2411yescharacteristic
75C22×Dic53—212no—
76C22×Dic53—86no—
77C22×Dic53C22×C412yes—
78C22×Dic53C22×C416yes—
79M16×C53—224no—
80M16×C53—24no—
81M16×C53C22×C418yes—
82C2×C8×C53—224no—
83C2×C8×C53—24no—
84C22⋊Dic53—212no—
85C4⋊Dic53—212no—
86C4⋊Dic53—23no—
87C4⋊Dic53—23no—
88C4⋊Dic53—21no—
89C4⋊Dic53—21no—
90C4×Dic53—212no—
91#11719917—224no—
92#11719917—24no—
93#11719917C22×C418yes—
94C2×#88322—224no—
95C2×#88322—24no—
96C23×C53—43no—
97C23×C53—43no—
98C23×C53C2×C4216yes—
99C23×C53C2×C22⋊C416yes—
100C23×C53C2×C4⋊C416yes—
101C23×C53C23×C411yescharacteristic
102Q8×C53#95116yes—
103D4×C53#95148yes—
104C2×C4×C53C2×C4216yes—
105C2×C4×C53C2×C22⋊C4112yes—
106C2×C4×C53C2×C22⋊C416yes—
107C2×C4×C53C2×C22⋊C4112yes—
108C2×C4×C53C2×C4⋊C4112yes—
109C2×C4×C53C2×C4⋊C416yes—
110C2×C4×C53C2×C4⋊C4112yes—
111C2×C4×C53C23×C411yescharacteristic
112C2×C4×C53C23×C411yescharacteristic
113C2×C4×C53C22×D411yescharacteristic
114C2×C4×C53C22×D411yescharacteristic
115C2×C4×C53C22×D411yescharacteristic
116C2×C4×C53C22×Q811yescharacteristic
117C2×Dic53—26no—
118C2×Dic53—22no—
119C2×Dic53—46no—
120C2×Dic53—42no—
121C2×Dic53—812no—
122C8×C53—28no—
123#88322—28no—
124#20358—531nomaximal, Sylow
125C22×C53—21no—
126C22×C53—26no—
127C22×C53—21no—
128C22×C53—26no—
129C22×C53—46no—
130C22×C53—46no—
131C22×C53C2×#95112yes—
132C22×C53C22×C4211yescharacteristic
133C22×C53C22×C22⋊C411yescharacteristic
134C22×C53C22×C4⋊C411yescharacteristic
135C4×C53C2×#95112yes—
136C4×C53C22×C22⋊C411yescharacteristic
137C4×C53C22×C22⋊C411yescharacteristic
138C4×C53C22×C4⋊C411yescharacteristic
139C4×C53C22×C4⋊C411yescharacteristic
140Dic53—46no—
141Dic53—42no—
142C2×#568—531no—
143C2×#568D5311yescharacteristic
144C2×#520—531no—
145#1742—536no—
146C2×#343—536no—
147C2×C53—21no—
148C2×C53—21no—
149C2×C53—212no—
150C2×C53C22×#9511yescharacteristic
151C2×C53#288811yescharacteristic
152C2×C53#288911yescharacteristic
153C22×#41C2×D5311yescharacteristic
154#568—24no—
155#568—538no—
156#568—1064no—
157#568Dic5318yescomplemented
158C2×#131—24no—
159C2×#131—1064no—
160C22×M16—536no—
161C22×M16C2×D5316yes—
162#520—1064no—
163#520—1064no—
164C2×C4×Q8—532no—
165C2×C4×D4—536no—
166C2×#117—533no—
167C22×C4⋊C4—533no—
168#343—5312no—
169#343—534no—
170C53#2035811yescharacteristic, complemented, Sylow
171C2×#41Dic10614yes—
172C2×#41C4×D5314yes—
173C2×#41D21214yes—
174C2×#41C2×Dic5314yes—
175C2×#41C22⋊D5314yes—
176C2×#41C22⋊D5314yes—
177C22×Q8C4×D5311yescharacteristic
178C22×D4C4×D5313yes—
179C23×C4—536no—
180C23×C4C22×D5313yes—
181#131—232no—
182#131—10632no—
183C2×M16—26no—
184C2×M16—212no—
185C2×M16—5324no—
186C2×M16—534no—
187C2×M16—1066no—
188C2×M16—10612no—
189C2×M16C2×Dic53124yescomplemented
190C2×M16C22×D5314yes—
191C22×C8—26no—
192C22×C8—1066no—
193C4×Q8—1068no—
194C4×D4—10624no—
195#117—1066no—
196#117—1066no—
197C2×C4⋊C4—536no—
198C2×C4⋊C4—533no—
199C2×C4⋊C4—531no—
200C2×C4⋊C4—1066no—
201C2×C4⋊C4—1066no—
202C2×C22⋊C4—536no—
203C2×C42—536no—
204C24C2×C4×D5313yes—
205#41#11719919116yes—
206#41C4⋊Dic53116yes—
207#41#11719921116yes—
208#41C22⋊Dic53116yes—
209C2×Q8C4×Dic5314yes—
210C2×Q8#1171991914yes—
211C2×Q8#1171992114yes—
212C2×D4C4×Dic53112yes—
213C2×D4#11719919112yes—
214C2×D4#11719921112yes—
215C22×C4—532no—
216C22×C4—536no—
217C22×C4—10612no—
218C22×C4—4246no—
219C22×C4C2×Dic10616yes—
220C22×C4C2×C4×D5313yes—
221C22×C4C2×C4×D5316yes—
222C22×C4C2×D21216yes—
223C22×C4C22×Dic5316yes—
224C22×C4C2×C22⋊D5316yes—
225C22×C4C2×C22⋊D5316yes—
226C22×C4C23×D5311yescharacteristic
227M16—224no—
228M16—24no—
229M16—538no—
230M16—10624no—
231M16—1064no—
232M16C22×Dic5318yes—
233C2×C8—224no—
234C2×C8—24no—
235C2×C8—10624no—
236C2×C8—1064no—
237C4⋊C4—10612no—
238C4⋊C4—1063no—
239C4⋊C4—1061no—
240C4⋊C4—1063no—
241C4⋊C4—1061no—
242C22⋊C4—10612no—
243C42—10612no—
244C23—43no—
245C23—43no—
246C23C2×C4×Dic5316yes—
247C23C2×#1171992116yes—
248C23C2×#1171991916yes—
249C23C22×C4×D5311yescharacteristic
250Q8#13133124116yes—
251D4#13133124148yes—
252C2×C4—1062no—
253C2×C4—1066no—
254C2×C4—42412no—
255C2×C4—2122no—
256C2×C4—2126no—
257C2×C4C2×C4×Dic5316yes—
258C2×C4C2×#11719921112yes—
259C2×C4C2×#1171992116yes—
260C2×C4C2×C22⋊Dic53112yes—
261C2×C4C2×#11719919112yes—
262C2×C4C2×#1171991916yes—
263C2×C4C2×C4⋊Dic53112yes—
264C2×C4C22×C4×D5311yescharacteristic
265C2×C4C23×Dic5311yescharacteristic
266C2×C4C22×C22⋊D5311yescharacteristic
267C2×C4C22×C22⋊D5311yescharacteristic
268C2×C4C22×D21211yescharacteristic
269C2×C4C22×Dic10611yescharacteristic
270C8—28no—
271C8—1068no—
272C22—26no—
273C22—21no—
274C22—21no—
275C22—26no—
276C22—46no—
277C22—46no—
278C22C2×#13133124112yes—
279C22C22×C4×Dic5311yescharacteristic
280C22C22×#1171992111yescharacteristic
281C22C22×#1171991911yescharacteristic
282C4—2126no—
283C4—2122no—
284C4C2×#13133124112yes—
285C4C22×#1171992111yescharacteristic
286C4C22×C22⋊Dic5311yescharacteristic
287C4C22×#1171991911yescharacteristic
288C4C22×C4⋊Dic5311yescharacteristic
289C2—21no—
290C2—21no—
291C2—212no—
292C2C22×#1313312411yescharacteristic
293C2#2028319311yescharacteristic
294C2#2028319711yescharacteristic
subgroupG/H
centerC22C22×#11719921
commutatorC22×C53C22×C42
FrattiniC22×C4C23×D53
FittingC2×C53×#568C2
socleC22×C53C22×C22⋊C4
prime radicalC22×#41×C53C22
cosocleC22×C4×C53C24
hypercenterC2×#568D53
hypocenterC53#20358
LCSG ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×#41 ◁ C2×#568

Complements

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

NH ≅ G/Ntype
3. #20274453C2semidirect
4. C2×#16810391C2semidirect
7. #16811390C4semidirect
20. C53×#568C4semidirect
27. C2×#11719917C2×C4semidirect
53. C2×M16×C53C2×C4semidirect
157. #568Dic53semidirect
170. C53#20358semidirect
189. C2×M16C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168113902
C2×#168111062
#202744532
C2×#168103912
C2×C53×#5682
#2035853