Finite Group Databasebeta

← #27281557

Subgroups — #27281557

Order 13568.

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

normal subgroups only

This group has 720 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#20279646C211yesmaximal, characteristic
2#20273230C212yesmaximal, complemented
3#20273042C211yesmaximal, characteristic, complemented
4#20272601C212yesmaximal, complemented
5C53×#2312C211yesmaximal, characteristic, complemented
6C2×#13133200—21no—
7C2×#13133191C2211yescharacteristic
8C2×#13133186—21no—
9#16810587C2212yes—
10#16810559C412yescomplemented
11#16810532C412yescomplemented
12#16810523—22no—
13C4×C22⋊Dic53C2211yescharacteristic
14#16810356—24no—
15#16810347—22no—
16C22×C22⋊C4×C53C2211yescharacteristic
17C53×#388—24no—
18C53×#380—21no—
19C53×#379—21no—
20C4×C22⋊C4×C53C2212yes—
21C22×D212C2×C411yescharacteristic
22C22×C22⋊D53—21no—
23C22×C22⋊D53—22no—
24C2×#11719917—22no—
25#13133200—44no—
26#13133191—22no—
27#13133191—22no—
28#13133186—44no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—22no—
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53—21no—
33C2×#11719921—22no—
34C2×#11719921—21no—
35C2×C4×Dic53—21no—
36C22⋊#88322—22no—
37C22⋊#88322—22no—
38#13133124C2×C412yes—
39C25×C53D411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C23×C4×C53C2311yescharacteristic
42C2×C4⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53C2×C412yes—
49C2×C22⋊C4×C53D412yes—
50C2×C42×C53—22no—
51#95×C53—22no—
52#95×C53D414yescomplemented
53C24×C53—21no—
54C24×C53—21no—
55C24×C53—42no—
56C24×C53—42no—
57C24×C53—81no—
58C24×C53C22⋊C412yes—
59C24×C53C2×D411yescharacteristic
60C23×D53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—24no—
64C22×C4×C53—42no—
65C22×C4×C53—81no—
66C22×C4×C53C22⋊C412yes—
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D412yes—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53#4112yes—
71C2×C22⋊D53—44no—
72C2×C22⋊D53—42no—
73C2×C22⋊D53—42no—
74C2×C22⋊D53—44no—
75C2×C22⋊D53—41no—
76C2×C22⋊D53—41no—
77C2×C22⋊D53—82no—
78C22×Dic53—22no—
79C22×Dic53—21no—
80C2×D212—22no—
81C2×D212—81no—
82C4⋊C4×C53—82no—
83C22⋊C4×C53—24no—
84C22⋊C4×C53—24no—
85C22⋊C4×C53—44no—
86C22⋊C4×C53—82no—
87C22⋊C4×C53—82no—
88C42×C53—82no—
89C22⋊Dic53—24no—
90C22⋊Dic53—44no—
91#11719921—44no—
92#11719921—42no—
93C4⋊Dic53—22no—
94#11719919—44no—
95C4×Dic53—81no—
96#11719917—82no—
97C2×#88322—42no—
98C23×C53—22no—
99C23×C53—22no—
100C23×C53—22no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53—41no—
104C23×C53—42no—
105C23×C53—42no—
106C23×C53—42no—
107C23×C53—42no—
108C23×C53—42no—
109C23×C53—42no—
110C23×C53—81no—
111C23×C53—81no—
112C23×C53—88no—
113C23×C53C2×C22⋊C411yescharacteristic
114C23×C53#12011yescharacteristic
115C23×C53#12311yescharacteristic
116C22×D53—42no—
117C22×D53—41no—
118C22×D53—41no—
119C22×D53—41no—
120C22×D53—42no—
121C2×C4×C53—22no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—44no—
129C2×C4×C53—44no—
130C2×C4×C53—44no—
131C2×C4×C53—81no—
132C2×C4×C53C4×D412yes—
133C2×C4×C53#12112yes—
134C22⋊D53—88no—
135C22⋊D53—82no—
136C22⋊D53—82no—
137C2×Dic53—24no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143D212—82no—
144#88322—82no—
145#9413—531nomaximal, Sylow
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—44no—
154C22×C53—44no—
155C22×C53—42no—
156C22×C53—42no—
157C22×C53—42no—
158C22×C53—82no—
159C22×C53—84no—
160C22×C53—84no—
161C22×C53#38611yescharacteristic
162C22×C53#42112yes—
163C22×C53#45312yes—
164C2×D53—82no—
165C2×D53—82no—
166C2×D53—81no—
167C2×D53—81no—
168C2×D53—81no—
169C4×C53—22no—
170C4×C53—22no—
171C4×C53—42no—
172C4×C53—84no—
173Dic53—44no—
174Dic53—81no—
175#2539—531no—
176#2312D5311yescharacteristic, complemented
177#1507—532no—
178#1462—531no—
179#1297—532no—
180C2×C53—22no—
181C2×C53—22no—
182C2×C53—41no—
183C2×C53—82no—
184C2×C53#188711yescharacteristic
185C2×C53#189711yescharacteristic
186C2×C53#191311yescharacteristic
187D53—161no—
188C2×#123—1061no—
189C2×#121—531no—
190C2×#120—1061no—
191C22×C22⋊C4C2×D5311yescharacteristic
192#407—532no—
193#399—532no—
194#392—532no—
195#390—1062no—
196#388—24no—
197#380—21no—
198#379—21no—
199C4×C22⋊C4—531no—
200C4×C22⋊C4C2×D5312yes—
201#327—1064no—
202#323—1062no—
203C53#941311yescharacteristic, complemented, Sylow
204C25C22⋊D5311yescharacteristic
205C22×D4—531no—
206C22×D4—1062no—
207C22×D4—1061no—
208C23×C4C22⋊D5311yescharacteristic
209C23×C4C22×D5311yescharacteristic
210C2×M16—1062no—
211#123—2124no—
212#121—1062no—
213#121—1062no—
214#120—2124no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—1062no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—1061no—
224C2×C22⋊C4—1062no—
225C2×C22⋊C4—1061no—
226C2×C22⋊C4—1062no—
227C2×C22⋊C4C4×D5312yes—
228C2×C22⋊C4D21212yes—
229C2×C42—22no—
230C2×C42—1061no—
231C22⋊C8—1062no—
232C22⋊C8—1062no—
233#95—22no—
234#95—532no—
235#95C22⋊D5314yescomplemented
236C24—21no—
237C24—21no—
238C24—42no—
239C24—42no—
240C24—81no—
241C24—1061no—
242C24#1171992112yes—
243C24C2×C22⋊D5311yescharacteristic
244C2×D4—1062no—
245C2×D4—4241no—
246C2×D4—4242no—
247C2×D4—2124no—
248C2×D4—2122no—
249C2×D4—2122no—
250C2×D4—2124no—
251C2×D4—2121no—
252C2×D4—2121no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—42no—
257C22×C4—81no—
258C22×C4—1062no—
259C22×C4—1061no—
260C22×C4#1171992112yes—
261C22×C4C2×C4×D5311yescharacteristic
262C22×C4C2×D21211yescharacteristic
263C22×C4#1171994512yes—
264C22×C4C2×C22⋊D5312yes—
265M16—4242no—
266C2×C8—2122no—
267C4⋊C4—82no—
268C4⋊C4—1062no—
269C4⋊C4—2124no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—44no—
273C22⋊C4—82no—
274C22⋊C4—82no—
275C22⋊C4—1064no—
276C22⋊C4—2124no—
277C22⋊C4—2124no—
278C22⋊C4—2122no—
279C42—82no—
280C42—4241no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—41no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—81no—
294C23—81no—
295C23—88no—
296C23—2121no—
297C23—2122no—
298C23—2121no—
299C23—2121no—
300C23—2122no—
301C23C2×#1171992111yescharacteristic
302C23#1313318611yescharacteristic
303C23#1313320011yescharacteristic
304D4—4242no—
305D4—4242no—
306D4—4242no—
307D4—4248no—
308C2×C4—22no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—81no—
319C2×C4—1064no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2122no—
324C2×C4—2122no—
325C2×C4C4×C22⋊D5312yes—
326C2×C4#1313319112yes—
327C8—4242no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—44no—
336C22—44no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—82no—
341C22—84no—
342C22—84no—
343C22—4242no—
344C22—4242no—
345C22—4241no—
346C22—4241no—
347C22—4241no—
348C22#1681050211yescharacteristic
349C22#1681064112yes—
350C22#1681079112yes—
351C4—22no—
352C4—22no—
353C4—42no—
354C4—84no—
355C4—4241no—
356C4—2124no—
357C2—22no—
358C2—22no—
359C2—41no—
360C2—82no—
361C2—8481no—
362C2#2027511311yescharacteristic
363C2#2027516311yescharacteristic
364C2#2027525711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2312C2
socleC22×C53#386
prime radical#20279646C2
cosocleC23×C4×C53C23
hypercenter#2312D53
hypocenterC53#9413
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. #20273230C2semidirect
3. #20273042C2semidirect
4. #20272601C2semidirect
5. C53×#2312C2semidirect
10. #16810559C4semidirect
11. #16810532C4semidirect
52. #95×C53D4semidirect
176. #2312D53semidirect
203. C53#9413semidirect
235. #95C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202796462
#202732302
#202730422
#202726012
C53×#23122
#941353