Finite Group Databasebeta

← #27281545

Subgroups — #27281545

Order 13568.

606 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 99 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#20277313C211yesmaximal, characteristic
3#20273236C212yesmaximal, complemented
4#20272600C212yesmaximal, complemented
5C53×#1462C211yesmaximal, characteristic
6C2×#13133196C2211yescharacteristic
7#16810548C412yescomplemented
8#16810535C412yescomplemented
9#16810517—22no—
10#16810516—24no—
11#16810486—21no—
12#16810483—21no—
13#16810479—21no—
14#16810479—21no—
15C4×C22⋊Dic53C2211yescharacteristic
16C4×C22⋊Dic53C2212yes—
17#16810361—24no—
18C22×C22⋊C4×C53C2211yescharacteristic
19C53×#407C2212yes—
20C53×#323—22no—
21C22×Dic106C2×C411yescharacteristic
22#13133196—22no—
23#13133196—22no—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×#11719919—21no—
27C2×C22⋊Dic53—22no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53—21no—
34C2×C4×Dic53—22no—
35C2×C4×Dic53—21no—
36#13133124—22no—
37#13133124—22no—
38#13133124—22no—
39#13133124C2×C412yes—
40#13133124D414yes—
41C25×C53D411yescharacteristic
42C23×C4×C53D411yescharacteristic
43C23×C4×C53C2311yescharacteristic
44C2×M16×C53—22no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53C2×C412yes—
48C2×C22⋊C4×C53D412yes—
49C22⋊C8×C53—22no—
50C22⋊C8×C53—22no—
51C24×C53—21no—
52C24×C53—21no—
53C24×C53—42no—
54C24×C53—42no—
55C24×C53—81no—
56C24×C53C22⋊C412yes—
57C24×C53C2×D411yescharacteristic
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—42no—
61C22×C4×C53—81no—
62C22×C4×C53C22⋊C412yes—
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D412yes—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53#4112yes—
67C22×Dic53—22no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—24no—
71C2×Dic106—22no—
72C2×Dic106—81no—
73M16×C53—82no—
74C2×C8×C53—42no—
75C22⋊C4×C53—24no—
76C22⋊C4×C53—24no—
77C22⋊Dic53—44no—
78C22⋊Dic53—42no—
79C22⋊Dic53—44no—
80C22⋊Dic53—41no—
81C22⋊Dic53—41no—
82C22⋊Dic53—82no—
83C22⋊Dic53—82no—
84C22⋊Dic53—82no—
85C4⋊Dic53—22no—
86C4⋊Dic53—82no—
87#11719919—42no—
88C4×Dic53—82no—
89C4×Dic53—81no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—22no—
95C23×C53—41no—
96C23×C53—42no—
97C23×C53—42no—
98C23×C53—42no—
99C23×C53—42no—
100C23×C53—42no—
101C23×C53—42no—
102C23×C53—81no—
103C23×C53—81no—
104C23×C53—88no—
105C23×C53C2×C22⋊C411yescharacteristic
106C23×C53#12011yescharacteristic
107C23×C53#12311yescharacteristic
108C2×C4×C53—22no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—81no—
116C2×C4×C53C4×D412yes—
117C2×C4×C53#12112yes—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—44no—
121C2×Dic53—42no—
122C2×Dic53—44no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53—44no—
128Dic106—82no—
129C8×C53—82no—
130#9411—531nomaximal, Sylow
131C22×C53—21no—
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—44no—
139C22×C53—44no—
140C22×C53—42no—
141C22×C53—42no—
142C22×C53—42no—
143C22×C53—82no—
144C22×C53—84no—
145C22×C53—84no—
146C22×C53#38611yescharacteristic
147C22×C53#42112yes—
148C22×C53#45512yes—
149C4×C53—22no—
150C4×C53—22no—
151C4×C53—42no—
152Dic53—84no—
153Dic53—82no—
154Dic53—81no—
155Dic53—81no—
156#2561—531no—
157#2312—531no—
158#1508—532no—
159#1462D5311yescharacteristic
160#1297—532no—
161C2×C53—22no—
162C2×C53—22no—
163C2×C53—41no—
164C2×C53—82no—
165C2×C53#188511yescharacteristic
166C2×C53#189711yescharacteristic
167C2×C53#191311yescharacteristic
168C2×C22⋊Q8—531no—
169C22×C22⋊C4C2×D5311yescharacteristic
170#407C2×D5312yes—
171#397—532no—
172#393—532no—
173#388—1064no—
174#388—1062no—
175#381—1061no—
176#380—1061no—
177#379—1061no—
178#379—1061no—
179C4×C22⋊C4—531no—
180C4×C22⋊C4—532no—
181#328—1064no—
182#323—22no—
183C53#941111yescharacteristic, complemented, Sylow
184C25D21211yescharacteristic
185C22×Q8—531no—
186C23×C4D21211yescharacteristic
187C23×C4C22×D5311yescharacteristic
188C2×M16—22no—
189C22⋊Q8—1062no—
190C22⋊Q8—1062no—
191C2×C4⋊C4—531no—
192C2×C4⋊C4—1062no—
193C2×C4⋊C4—1061no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—1062no—
197C2×C22⋊C4—1062no—
198C2×C22⋊C4—1062no—
199C2×C22⋊C4—1061no—
200C2×C22⋊C4—1062no—
201C2×C22⋊C4—1061no—
202C2×C22⋊C4—1062no—
203C2×C22⋊C4C4×D5312yes—
204C2×C22⋊C4C22⋊D5312yes—
205C2×C42—1062no—
206C2×C42—1061no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209#95—532no—
210#95—534no—
211#95—1062no—
212#95—1062no—
213#95—1062no—
214C24—21no—
215C24—21no—
216C24—42no—
217C24—42no—
218C24—81no—
219C24#1171992112yes—
220C24C2×D21211yescharacteristic
221C2×Q8—1062no—
222C2×Q8—4241no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—81no—
227C22×C4—1062no—
228C22×C4—1061no—
229C22×C4—1061no—
230C22×C4—1064no—
231C22×C4#1171992112yes—
232C22×C4C2×C4×D5311yescharacteristic
233C22×C4D4×D5312yes—
234C22×C4#1171994712yes—
235C22×C4C2×C22⋊D5311yescharacteristic
236M16—82no—
237C2×C8—42no—
238C4⋊C4—1062no—
239C4⋊C4—4242no—
240C4⋊C4—2122no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—4242no—
244C22⋊C4—4242no—
245C22⋊C4—4242no—
246C22⋊C4—2122no—
247C22⋊C4—2124no—
248C22⋊C4—2124no—
249C22⋊C4—2121no—
250C22⋊C4—2121no—
251C42—4242no—
252C42—4241no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—41no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—81no—
266C23—81no—
267C23—88no—
268C23C2×#1171992111yescharacteristic
269C23#1313318711yescharacteristic
270C23#1313320111yescharacteristic
271Q8—4242no—
272C2×C4—22no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—81no—
280C2×C4—2122no—
281C2×C4—2124no—
282C2×C4—2122no—
283C2×C4—2122no—
284C2×C4—2124no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2124no—
288C2×C4—2122no—
289C2×C4—2122no—
290C2×C4#1313317712yes—
291C2×C4#1313318912yes—
292C8—82no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—44no—
301C22—44no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22—84no—
307C22—84no—
308C22#1681050011yescharacteristic
309C22#1681063912yes—
310C22#1681080312yes—
311C4—22no—
312C4—22no—
313C4—42no—
314C4—4244no—
315C4—4242no—
316C4—4241no—
317C4—4241no—
318C2—22no—
319C2—22no—
320C2—41no—
321C2—82no—
322C2#2027510111yescharacteristic
323C2#2027516111yescharacteristic
324C2#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#9411
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
7. #16810548C4semidirect
8. #16810535C4semidirect
183. C53#9411semidirect

Maximal subgroups

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

subgroupindex
#202798732
#202773132
#202732362
#202726002
C53×#14622
#941153