Finite Group Databasebeta

← #27393548

Subgroups — #27393548

Order 13568.

990 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811251C211yesmaximal, characteristic
2#20277316C212yesmaximal
3#20275698C218yesmaximal
4#20274792C211yesmaximal, characteristic
5#20274790C212yesmaximal
6C53×#1820C211yesmaximal, characteristic
7#16811251—24no—
8#16811251—24no—
9C2×#13133217C412yescomplemented
10C2×#13133216C412yescomplemented
11C2×#13133209C412yescomplemented
12C2×#13133208C412yescomplemented
13C2×#13133194C2214yes—
14C2×#13133176C2211yescharacteristic
15C22×C4⋊Dic53C2211yescharacteristic
16#16810508—24no—
17#16810506—24no—
18#16810486C2212yes—
19#16810481C2214yes—
20C22⋊C4×Dic53C2214yes—
21#16810390C2214yes—
22#16810389C2218yes—
23C2×#117×C53C2211yescharacteristic
24C53×#381C2212yes—
25C53×#342C2214yes—
26C22×Dic106C2×C414yes—
27C23×Dic53C2312yes—
28#13133217—24no—
29#13133217—24no—
30#13133216—28no—
31#13133209—24no—
32#13133209—24no—
33#13133208—28no—
34#13133194—28no—
35#13133194—28no—
36#13133176—22no—
37#13133176—22no—
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×C4⋊Dic53C2×C414yes—
41C2×C4⋊Dic53C2×C412yes—
42C2×C4⋊Dic53C2×C412yes—
43C2×C4⋊Dic53D414yes—
44C2×C4⋊Dic53D414yes—
45C2×#11719919—24no—
46C2×#11719919C2×C414yes—
47C2×#11719919C2×C414yes—
48C2×C22⋊Dic53C2314yes—
49C2×C22⋊Dic53C2312yes—
50C2×C4×Dic53—24no—
51C2×C4×Dic53C2×C412yes—
52#13133124—22no—
53#13133124—22no—
54#13133124—24no—
55#13133124—24no—
56C23×C4×C53C2311yescharacteristic
57#117×C53—22no—
58#117×C53—22no—
59C2×C4⋊C4×C53C2×C411yescharacteristic
60C2×C4⋊C4×C53C2×C411yescharacteristic
61C2×C22⋊C4×C53C2314yes—
62C2×C22⋊C4×C53C2312yes—
63C2×C42×C53C2×C412yes—
64#95×C53—22no—
65#95×C53—22no—
66C24×C53C2411yescharacteristic
67C22×C4×C53—24no—
68C22×C4×C53—24no—
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C22×C412yes—
72C22×C4×C53C22×C412yes—
73C22×C4×C53C2×D412yes—
74C22×C4×C53C2×D412yes—
75C22×Dic53—24no—
76C22×Dic53—82no—
77C22×Dic53C22×C412yes—
78C22×Dic53C22×C412yes—
79C22×Dic53C22×C414yes—
80C22×Dic53C2×D414yes—
81C22×Dic53C2×D414yes—
82C2×Dic106—28no—
83C2×Dic106—84no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—22no—
86C22⋊C4×C53—24no—
87C22⋊C4×C53—28no—
88C42×C53—22no—
89C42×C53—22no—
90C22⋊Dic53—24no—
91C22⋊Dic53—28no—
92C4⋊Dic53—21no—
93C4⋊Dic53—21no—
94C4⋊Dic53—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53—28no—
97C4⋊Dic53C22⋊C418yes—
98C4⋊Dic53C22⋊C414yes—
99C4⋊Dic53C22⋊C414yes—
100#11719919—28no—
101#11719919—28no—
102#11719919—84no—
103C4×Dic53—24no—
104C4×Dic53—84no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53C23×C411yescharacteristic
108C23×C53C22×D412yes—
109C23×C532−1+414yes—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—24no—
113C2×C4×C53—24no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—44no—
119C2×C4×C53—44no—
120C2×C4×C53—44no—
121C2×C4×C53C2×C22⋊C412yes—
122C2×C4×C53C2×C22⋊C411yescharacteristic
123C2×C4×C53C2×C22⋊C411yescharacteristic
124C2×Dic53—22no—
125C2×Dic53—22no—
126C2×Dic53—44no—
127C2×Dic53—44no—
128C2×Dic53—44no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131C2×Dic53—44no—
132C2×Dic53—44no—
133C2×Dic53—44no—
134C2×Dic53C2×C22⋊C418yes—
135Dic106—88no—
136#21828—531nomaximal, Sylow
137C22×C53—21no—
138C22×C53—22no—
139C22×C53—21no—
140C22×C53—22no—
141C22×C53—44no—
142C22×C53C22×C22⋊C411yescharacteristic
143C22×C53#51912yes—
144C22×C53#53614yes—
145C4×C53—22no—
146C4×C53—21no—
147C4×C53—21no—
148C4×C53—42no—
149C4×C53—42no—
150C4×C53—84no—
151Dic53—28no—
152Dic53—42no—
153Dic53—42no—
154Dic53—84no—
155C2×#544—531no—
156#2313—532no—
157#1996—538no—
158#1820—532no—
159#1820—531no—
160#1820D5311yescharacteristic
161C2×C53—24no—
162C2×C53—22no—
163C2×C53#231011yescharacteristic
164C2×C53#288311yescharacteristic
165C2×C53#288411yescharacteristic
166#544—1064no—
167#544—1064no—
168C2×C4⋊Q8—532no—
169C2×C4⋊Q8—532no—
170C2×#125—532no—
171C2×#125—532no—
172C2×C22⋊Q8—534no—
173C2×#117—531no—
174C2×#117C2×D5311yescharacteristic
175C22×C4⋊C4—531no—
176#387—1064no—
177#387—1064no—
178#381—532no—
179#381C2×D5312yes—
180#380—534no—
181C4×C22⋊C4—534no—
182#342—538no—
183#342—534no—
184#342C2×D5314yes—
185C53#2182811yescharacteristic, complemented, Sylow
186C22×Q8—534no—
187C23×C4—532no—
188C23×C4C22×D5311yescharacteristic
189C4⋊Q8—1064no—
190C4⋊Q8—1064no—
191C4⋊Q8—1068no—
192#125—1064no—
193#125—1064no—
194#125—1068no—
195C22⋊Q8—1068no—
196C22⋊Q8—1068no—
197#117—22no—
198#117—22no—
199#117—1062no—
200#117—1062no—
201C2×C4⋊C4—532no—
202C2×C4⋊C4—532no—
203C2×C4⋊C4—534no—
204C2×C4⋊C4—534no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—534no—
208C2×C4⋊C4—534no—
209C2×C4⋊C4—534no—
210C2×C4⋊C4—1064no—
211C2×C4⋊C4C4×D5311yescharacteristic
212C2×C4⋊C4C4×D5311yescharacteristic
213C2×C22⋊C4—532no—
214C2×C22⋊C4—534no—
215C2×C22⋊C4C22×D5314yes—
216C2×C22⋊C4C22×D5312yes—
217C2×C42—532no—
218C2×C42—1064no—
219C2×C42C4×D5312yes—
220#95—22no—
221#95—22no—
222#95—1064no—
223#95—1064no—
224#95—1062no—
225#95—1062no—
226C24C23×D5311yescharacteristic
227C2×Q8—1068no—
228C2×Q8—4244no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—534no—
232C22×C4—534no—
233C22×C4—534no—
234C22×C4—532no—
235C22×C4—532no—
236C22×C4—1064no—
237C22×C4—4242no—
238C22×C4C2×C4×D5311yescharacteristic
239C22×C4C2×C4×D5311yescharacteristic
240C22×C4C2×C4×D5312yes—
241C22×C4C2×C4×D5312yes—
242C22×C4D4×D5312yes—
243C22×C4D4×D5312yes—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—534no—
247C4⋊C4—534no—
248C4⋊C4—538no—
249C4⋊C4—1068no—
250C4⋊C4—1068no—
251C4⋊C4—1068no—
252C4⋊C4—1061no—
253C4⋊C4—1061no—
254C4⋊C4—1061no—
255C4⋊C4—1061no—
256C4⋊C4—4244no—
257C22⋊C4—24no—
258C22⋊C4—28no—
259C22⋊C4—1064no—
260C22⋊C4—1068no—
261C42—22no—
262C42—22no—
263C42—1064no—
264C42—4244no—
265C23—41no—
266C23—41no—
267C23C22×C4×D5311yescharacteristic
268C23C2×D4×D5312yes—
269C23#1313326414yes—
270Q8—4248no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—538no—
283C2×C4—1062no—
284C2×C4—1062no—
285C2×C4—2124no—
286C2×C4—2124no—
287C2×C4—2124no—
288C2×C4—2124no—
289C2×C4—2124no—
290C2×C4—2124no—
291C2×C4—2122no—
292C2×C4—2122no—
293C2×C4C22⋊C4×D5311yescharacteristic
294C2×C4C22⋊C4×D5311yescharacteristic
295C2×C4C22⋊C4×D5312yes—
296C22—21no—
297C22—22no—
298C22—21no—
299C22—22no—
300C22—44no—
301C22C2×C22⋊C4×D5311yescharacteristic
302C22#1681110012yes—
303C22#1681119514yes—
304C4—21no—
305C4—21no—
306C4—22no—
307C4—42no—
308C4—42no—
309C4—84no—
310C4—1068no—
311C4—4244no—
312C4—2122no—
313C4—2122no—
314C2—24no—
315C2—22no—
316C2#2027730011yescharacteristic
317C2#2028317411yescharacteristic
318C2#2028317711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC53×#1820C2
socleC22×C53C22×C22⋊C4
prime radical, cosocleC24×C53C24
hypercenter#1820D53
hypocenterC53#21828
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
9. C2×#13133217C4semidirect
10. C2×#13133216C4semidirect
11. C2×#13133209C4semidirect
12. C2×#13133208C4semidirect
185. C53#21828semidirect

Maximal subgroups

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

subgroupindex
C2×#168112512
#202773162
#202756982
#202747922
#202747902
C53×#18202
#2182853