Finite Group Databasebeta

← #27281525

Subgroups — #27281525

Order 13568.

726 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279873C211yesmaximal, characteristic
2#20277068C211yesmaximal, characteristic
3#20273236C212yesmaximal, complemented
4#20272600C212yesmaximal, complemented
5C53×#1462C211yesmaximal, characteristic
6C2×#13133196C2211yescharacteristic
7C22×C22⋊Dic53—21no—
8#16810548C412yescomplemented
9#16810535C412yescomplemented
10#16810517—22no—
11#16810486—21no—
12#16810479—21no—
13C22⋊C4×Dic53—24no—
14C4×C22⋊Dic53—21no—
15C4×C22⋊Dic53C2211yescharacteristic
16C4×C22⋊Dic53C2212yes—
17#16810361—24no—
18C22×C22⋊C4×C53C2211yescharacteristic
19C53×#407C2212yes—
20C53×#323—22no—
21C22×Dic106C2×C411yescharacteristic, complemented
22C23×Dic53—22no—
23#13133196—22no—
24#13133196—22no—
25C2×C4⋊Dic53C2×C411yescharacteristic, complemented
26C2×#11719919—21no—
27C2×C22⋊Dic53—28no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C4×Dic53—22no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—22no—
37#13133124—22no—
38#13133124—22no—
39#13133124—22no—
40#13133124C2×C412yescomplemented
41#13133124D414yes—
42C25×C53D411yescharacteristic
43C23×C4×C53D411yescharacteristic
44C23×C4×C53C2311yescharacteristic
45C2×M16×C53—22no—
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53C2×C412yes—
49C2×C22⋊C4×C53D412yes—
50C22⋊C8×C53—22no—
51C22⋊C8×C53—22no—
52C24×C53—21no—
53C24×C53—24no—
54C24×C53—21no—
55C24×C53—42no—
56C24×C53—81no—
57C24×C53C22⋊C412yes—
58C24×C53C2×D411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—42no—
62C22×C4×C53—81no—
63C22×C4×C53C22⋊C412yes—
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D412yes—
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53#4112yes—
68C22×Dic53—24no—
69C22×Dic53—28no—
70C22×Dic53—22no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—82no—
74C22×Dic53—82no—
75C2×Dic106—22no—
76C2×Dic106—81no—
77M16×C53—82no—
78C2×C8×C53—42no—
79C22⋊C4×C53—24no—
80C22⋊C4×C53—24no—
81C22⋊Dic53—28no—
82C22⋊Dic53—28no—
83C22⋊Dic53—42no—
84C22⋊Dic53—44no—
85C22⋊Dic53—41no—
86C22⋊Dic53—41no—
87C22⋊Dic53—82no—
88C4⋊Dic53—22no—
89#11719919—42no—
90C4×Dic53—82no—
91C4×Dic53—82no—
92C4×Dic53—81no—
93C23×C53—22no—
94C23×C53—22no—
95C23×C53—24no—
96C23×C53—22no—
97C23×C53—22no—
98C23×C53—22no—
99C23×C53—22no—
100C23×C53—41no—
101C23×C53—42no—
102C23×C53—42no—
103C23×C53—42no—
104C23×C53—42no—
105C23×C53—81no—
106C23×C53—81no—
107C23×C53—88no—
108C23×C53#10414yes—
109C23×C53C2×C22⋊C411yescharacteristic
110C23×C53#12011yescharacteristic
111C23×C53#12311yescharacteristic
112C2×C4×C53—22no—
113C2×C4×C53—41no—
114C2×C4×C53—42no—
115C2×C4×C53—41no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—81no—
120C2×C4×C53C4×D412yes—
121C2×C4×C53#12112yes—
122C2×Dic53—28no—
123C2×Dic53—44no—
124C2×Dic53—44no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53—42no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—88no—
133Dic106—82no—
134C8×C53—82no—
135#9407—531nomaximal, Sylow
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—24no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—44no—
145C22×C53—44no—
146C22×C53—44no—
147C22×C53—44no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53—82no—
151C22×C53—84no—
152C22×C53#38611yescharacteristic
153C22×C53C2×#10412yes—
154C22×C53#45512yes—
155C4×C53—22no—
156C4×C53—22no—
157C4×C53—42no—
158Dic53—48no—
159Dic53—82no—
160Dic53—81no—
161Dic53—81no—
162#2561—531no—
163#2273—531no—
164#1508—532no—
165#1462D5311yescharacteristic
166#1297—532no—
167C2×C53—22no—
168C2×C53—22no—
169C2×C53—41no—
170C2×C53—44no—
171C2×C53#188511yescharacteristic
172C2×C53#189611yescharacteristic
173C2×C53#191311yescharacteristic
174C2×C22⋊Q8—531no—
175C22×C22⋊C4—1061no—
176C22×C22⋊C4C2×D5311yescharacteristic
177#407C2×D5312yes—
178#397—532no—
179#393—532no—
180#388—1062no—
181#381—1061no—
182#379—1061no—
183C4×C22⋊C4—531no—
184C4×C22⋊C4—532no—
185C4×C22⋊C4—1064no—
186C4×C22⋊C4—1061no—
187#328—1064no—
188#323—22no—
189C53#940711yescharacteristic, complemented, Sylow
190C25D21211yescharacteristic
191C22×Q8—531no—
192C23×C4—1062no—
193C23×C4D21211yescharacteristic
194C23×C4C22×D5311yescharacteristic
195C2×M16—22no—
196C22⋊Q8—1062no—
197C22⋊Q8—1062no—
198C2×C4⋊C4—531no—
199C2×C4⋊C4—1061no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—1068no—
203C2×C22⋊C4—1062no—
204C2×C22⋊C4—1062no—
205C2×C22⋊C4—1062no—
206C2×C22⋊C4—1061no—
207C2×C22⋊C4—1061no—
208C2×C22⋊C4—1062no—
209C2×C22⋊C4C4×D5312yes—
210C2×C22⋊C4C22⋊D5312yes—
211C2×C42—1062no—
212C2×C42—1062no—
213C2×C42—1061no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216#95—532no—
217#95—534no—
218#95—1062no—
219#95—1062no—
220#95—1062no—
221C24—21no—
222C24—24no—
223C24—21no—
224C24—42no—
225C24—81no—
226C24#1171992112yes—
227C24C2×D21211yescharacteristic
228C2×Q8—1062no—
229C2×Q8—4241no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—42no—
233C22×C4—81no—
234C22×C4—1068no—
235C22×C4—1064no—
236C22×C4—1062no—
237C22×C4—1061no—
238C22×C4—1061no—
239C22×C4—4242no—
240C22×C4—4242no—
241C22×C4#1171992112yes—
242C22×C4C2×C4×D5311yescharacteristic
243C22×C4D4×D5312yes—
244C22×C4#1171994712yes—
245C22×C4C2×C22⋊D5311yescharacteristic
246M16—82no—
247C2×C8—42no—
248C4⋊C4—1062no—
249C4⋊C4—2122no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—1068no—
253C22⋊C4—1068no—
254C22⋊C4—4242no—
255C22⋊C4—2122no—
256C22⋊C4—2124no—
257C22⋊C4—2121no—
258C22⋊C4—2121no—
259C42—4242no—
260C42—4241no—
261C42—4242no—
262C23—22no—
263C23—22no—
264C23—24no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—41no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—81no—
275C23—81no—
276C23—88no—
277C23#1313314314yes—
278C23C2×#1171992111yescharacteristic
279C23#1313318711yescharacteristic
280C23#1313320111yescharacteristic
281Q8—4242no—
282C2×C4—22no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—81no—
290C2×C4—1068no—
291C2×C4—4248no—
292C2×C4—2122no—
293C2×C4—2122no—
294C2×C4—2122no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2122no—
298C2×C4—2122no—
299C2×C4—2124no—
300C2×C4—2124no—
301C2×C4#1313317712yes—
302C2×C4#1313318912yes—
303C8—82no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—24no—
311C22—21no—
312C22—44no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22#1681050011yescharacteristic
321C22C2×#1313314312yes—
322C22#1681080312yes—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—4242no—
327C4—4241no—
328C4—4241no—
329C4—2128no—
330C2—22no—
331C2—22no—
332C2—41no—
333C2—44no—
334C2#2027510111yescharacteristic
335C2#2027515611yescharacteristic
336C2#2027525511yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1462C2
socleC22×C53#386
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenter#1462D53
hypocenterC53#9407
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
3. #20273236C2semidirect
4. #20272600C2semidirect
8. #16810548C4semidirect
9. #16810535C4semidirect
21. C22×Dic106C2×C4semidirect
25. C2×C4⋊Dic53C2×C4semidirect
40. #13133124C2×C4semidirect
189. C53#9407semidirect

Maximal subgroups

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

subgroupindex
#202798732
#202770682
#202732362
#202726002
C53×#14622
#940753