Finite Group Databasebeta

← #27281547

Subgroups — #27281547

Order 13568.

606 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 103 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#20273234C212yesmaximal, complemented
3#20273042C211yesmaximal, characteristic
4#20272601C212yesmaximal, complemented
5C53×#2312C211yesmaximal, characteristic
6C2×#13133196C2211yescharacteristic
7#16810587C2212yes—
8#16810548C412yescomplemented
9#16810535C412yescomplemented
10#16810517—22no—
11#16810486—21no—
12#16810479—21no—
13C4×C22⋊Dic53C2211yescharacteristic
14#16810359—24no—
15#16810347—22no—
16C22×C22⋊C4×C53C2211yescharacteristic
17C53×#388—24no—
18C53×#380—21no—
19C53×#379—21no—
20C4×C22⋊C4×C53C2212yes—
21C22×Dic106C2×C411yescharacteristic
22C2×#11719917—22no—
23#13133196—22no—
24#13133196—22no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×#11719919—21no—
27C2×C22⋊Dic53—22no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—21no—
31C2×C4×Dic53—21no—
32C22⋊#88322—22no—
33C22⋊#88322—22no—
34#13133124—22no—
35#13133124—22no—
36#13133124C2×C412yes—
37C25×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53C2311yescharacteristic
40C2×C4⋊C4×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53C2×C412yes—
47C2×C22⋊C4×C53D412yes—
48C2×C42×C53—22no—
49#95×C53—22no—
50#95×C53D414yes—
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—24no—
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—22no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C2×Dic106—22no—
72C2×Dic106—81no—
73C4⋊C4×C53—82no—
74C22⋊C4×C53—24no—
75C22⋊C4×C53—24no—
76C22⋊C4×C53—44no—
77C22⋊C4×C53—82no—
78C22⋊C4×C53—82no—
79C42×C53—82no—
80C22⋊Dic53—42no—
81C22⋊Dic53—44no—
82C22⋊Dic53—41no—
83C22⋊Dic53—41no—
84C22⋊Dic53—82no—
85C4⋊Dic53—22no—
86#11719919—42no—
87C4×Dic53—81no—
88#11719917—82no—
89C2×#88322—42no—
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—41no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—44no—
116C2×C4×C53—44no—
117C2×C4×C53—44no—
118C2×C4×C53—81no—
119C2×C4×C53C4×D412yes—
120C2×C4×C53#12112yes—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—42no—
127C2×Dic53—42no—
128Dic106—82no—
129#88322—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—
152C4×C53—84no—
153Dic53—82no—
154Dic53—81no—
155Dic53—81no—
156#2561—531no—
157#2312D5311yescharacteristic
158#1508—532no—
159#1462—531no—
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#407—532no—
171#397—532no—
172#393—532no—
173#388—24no—
174#388—1062no—
175#381—1061no—
176#380—21no—
177#379—21no—
178#379—1061no—
179C4×C22⋊C4—531no—
180C4×C22⋊C4C2×D5312yes—
181#328—1064no—
182#323—1062no—
183C53#941111yescharacteristic, complemented, Sylow
184C25C22⋊D5311yescharacteristic
185C22×Q8—531no—
186C23×C4C22⋊D5311yescharacteristic
187C23×C4C22×D5311yescharacteristic
188C2×M16—1062no—
189C22⋊Q8—1062no—
190C22⋊Q8—1062no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—531no—
193C2×C4⋊C4—1061no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—1062no—
200C2×C22⋊C4—1061no—
201C2×C22⋊C4—1061no—
202C2×C22⋊C4—1062no—
203C2×C22⋊C4C4×D5312yes—
204C2×C22⋊C4D21212yes—
205C2×C42—22no—
206C2×C42—1061no—
207C22⋊C8—1062no—
208C22⋊C8—1062no—
209#95—22no—
210#95—532no—
211#95—1062no—
212#95—1062no—
213#95C22⋊D5314yes—
214C24—21no—
215C24—21no—
216C24—42no—
217C24—42no—
218C24—81no—
219C24#1171992112yes—
220C24C2×C22⋊D5311yescharacteristic
221C2×Q8—1062no—
222C2×Q8—4241no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—24no—
226C22×C4—42no—
227C22×C4—81no—
228C22×C4—1062no—
229C22×C4—1061no—
230C22×C4—1061no—
231C22×C4#1171992112yes—
232C22×C4C2×C4×D5311yescharacteristic
233C22×C4C2×D21211yescharacteristic
234C22×C4#1171994512yes—
235C22×C4C2×C22⋊D5312yes—
236M16—4242no—
237C2×C8—2122no—
238C4⋊C4—82no—
239C4⋊C4—1062no—
240C4⋊C4—2122no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—44no—
244C22⋊C4—82no—
245C22⋊C4—82no—
246C22⋊C4—4242no—
247C22⋊C4—2122no—
248C22⋊C4—2124no—
249C22⋊C4—2121no—
250C22⋊C4—2121no—
251C42—82no—
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#1313318611yescharacteristic
270C23#1313320011yescharacteristic
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—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—81no—
283C2×C4—2122no—
284C2×C4—2122no—
285C2×C4—2122no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4—2122no—
289C2×C4—2122no—
290C2×C4C4×C22⋊D5312yes—
291C2×C4#1313319112yes—
292C8—4242no—
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#1681050211yescharacteristic
309C22#1681064112yes—
310C22#1681080112yes—
311C4—22no—
312C4—22no—
313C4—42no—
314C4—84no—
315C4—4242no—
316C4—4241no—
317C4—4241no—
318C2—22no—
319C2—22no—
320C2—41no—
321C2—82no—
322C2#2027510311yescharacteristic
323C2#2027516311yescharacteristic
324C2#2027525711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2312C2
socleC22×C53#386
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenter#2312D53
hypocenterC53#9411
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #20273234C2semidirect
4. #20272601C2semidirect
8. #16810548C4semidirect
9. #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
#202732342
#202730422
#202726012
C53×#23122
#941153