Finite Group Databasebeta

← #27469689

Subgroups — #27469689

Order 13568.

706 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283024C211yesmaximal, characteristic
2#20277120C211yesmaximal, characteristic
3#20276279C211yesmaximal, characteristic, complemented
4#20276278C211yesmaximal, characteristic, complemented
5#20276272C211yesmaximal, characteristic, complemented
6#20275678C212yesmaximal, complemented
7#20274844C212yesmaximal, complemented
8#20274808C212yesmaximal
9#20274808C212yesmaximal
10#20274472C211yesmaximal, characteristic
11C53×#2099C211yesmaximal, characteristic, complemented
12C2×#13133208C2211yescharacteristic
13C2×#13133175C2212yes—
14C2×C42×D53C2211yescharacteristic
15C2×#13133129C2212yes—
16C2×C8×Dic53C2212yes—
17#16810521C2211yescharacteristic
18#16810509—21no—
19#16810505—21no—
20#16810497—21no—
21#16810494—21no—
22#16810493C2211yescharacteristic
23#16810488C2212yes—
24#16810488C2211yescharacteristic
25#16810480—21no—
26#16810480—21no—
27#16810480C2211yescharacteristic
28#16810480C2211yescharacteristic
29C4⋊C4×Dic53—21no—
30C4⋊C4×Dic53—21no—
31#16810386C2212yes—
32#16810386C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810378C2211yescharacteristic
38C53×#389C2211yescharacteristic
39C53×#384C2211yescharacteristic
40C53×#340C2212yes—
41C53×#340C2212yes—
42C53×#336C2211yescharacteristic
43C22×C4×D53D411yescharacteristic
44C22×C4×D53D411yescharacteristic
45C22×C4×D53C2311yescharacteristic
46C22×#88322C2312yes—
47#13133208—24no—
48#13133175—28no—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53C2311yescharacteristic
53C2×C4⋊Dic53C2311yescharacteristic
54C2×C4⋊Dic53C2312yes—
55C2×#11719919—21no—
56C2×#11719919—21no—
57C2×#11719921—21no—
58C2×#11719921—21no—
59C2×#11719921—22no—
60C2×C4×Dic53—22no—
61C2×C4×Dic53D411yescharacteristic
62C2×C4×Dic53D411yescharacteristic
63C2×C4×Dic53C2311yescharacteristic
64C42×D53—24no—
65#13133129—24no—
66C8×Dic53—24no—
67#13133124—21no—
68#13133124—21no—
69#13133124—21no—
70#13133124—21no—
71#13133124—21no—
72#13133124—22no—
73#13133124—22no—
74C22×C8×C53C2312yes—
75C2×C4⋊C4×C53—21no—
76C2×C4⋊C4×C53—21no—
77C2×C4⋊C4×C53—21no—
78C2×C4⋊C4×C53C2312yes—
79C2×C4⋊C4×C53C2311yescharacteristic
80C2×C4⋊C4×C53C2311yescharacteristic
81C2×C42×C53C2311yescharacteristic
82#95×C53—21no—
83C23×D53C2×D411yescharacteristic
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—22no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2411yescharacteristic
90C22×Dic53—21no—
91C22×Dic53—22no—
92C22×Dic53—21no—
93C22×Dic53C2×D411yescharacteristic
94C22×Dic53C2×D411yescharacteristic
95C22×Dic53C2×D411yescharacteristic
96C2×C4×D53—22no—
97C2×C4×D53—21no—
98C2×C4×D53—21no—
99C2×C4×D53—22no—
100C2×C4×D53—22no—
101C2×C4×D53—22no—
102C2×C4×D53—22no—
103C2×C4×D53—41no—
104C2×C4×D53—41no—
105C2×C4×D53#4114yes—
106C2×C8×C53—22no—
107C2×C8×C53—22no—
108C2×C8×C53—22no—
109C4⋊C4×C53—24no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—42no—
114C4⋊C4×C53—42no—
115C4⋊C4×C53—42no—
116C42×C53—22no—
117#11719921—42no—
118#11719921—42no—
119#11719921—44no—
120C4⋊Dic53—24no—
121C4⋊Dic53—22no—
122C4⋊Dic53—21no—
123C4⋊Dic53—21no—
124C4⋊Dic53—42no—
125C4⋊Dic53—42no—
126C4⋊Dic53—42no—
127#11719919—42no—
128#11719919—42no—
129C4×Dic53—22no—
130C4×Dic53—22no—
131C4×Dic53—44no—
132C4×Dic53#4114yes—
133C2×#88322—22no—
134C2×#88322—22no—
135C2×#88322—22no—
136C23×C53C22×D411yescharacteristic
137C22×D53—21no—
138C22×D53—21no—
139C22×D53—21no—
140C22×D53—21no—
141C22×D53—21no—
142C22×D53—21no—
143C22×D53#12412yes—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—41no—
149C2×C4×C53—42no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—42no—
153C2×C4×C53—41no—
154C2×C4×C53—42no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53#13512yes—
158C2×C4×C53#13512yes—
159C2×C4×C53C2×#4112yes—
160C2×C4×C53C2×#4112yes—
161C2×C4×C532−1+411yescharacteristic
162C2×C4×C532−1+411yescharacteristic
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—21no—
166C2×Dic53—21no—
167C2×Dic53—21no—
168C2×Dic53—21no—
169C2×Dic53—41no—
170C2×Dic53—42no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—42no—
175C2×Dic53—42no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53#12312yes—
179C2×Dic53#12312yes—
180C2×Dic53#12412yes—
181C4×D53—24no—
182C4×D53—24no—
183C4×D53—44no—
184C4×D53—44no—
185C8×C53—42no—
186#88322—42no—
187#27963—531nomaximal, Sylow
188C22×C53C2×#12311yescharacteristic
189C22×C53C2×#12411yescharacteristic
190C22×C53#53611yescharacteristic
191C22×C53C2×#13511yescharacteristic
192C22×C53C2×#13511yescharacteristic
193C22×C53#57511yescharacteristic
194C22×C53#57511yescharacteristic
195C2×D53—22no—
196C2×D53—22no—
197C2×D53—22no—
198C2×D53—41no—
199C2×D53—41no—
200C2×D53—41no—
201C2×D53—41no—
202C4×C53—22no—
203C4×C53—22no—
204C4×C53—81no—
205C4×C53—82no—
206C4×C53—81no—
207C4×C53#55212yes—
208C4×C53#55212yes—
209Dic53—22no—
210Dic53—22no—
211Dic53—22no—
212Dic53—81no—
213Dic53—82no—
214Dic53—81no—
215#2851—531no—
216#2285—531no—
217#2100—531no—
218#2100—531no—
219#2099—531no—
220#2099D5311yescharacteristic, complemented
221#1991—532no—
222#1835—532no—
223#1825—532no—
224#1825—532no—
225C4×C22⋊C8—531no—
226C2×C53#250311yescharacteristic
227C2×C53#309111yescharacteristic
228C2×C53#309111yescharacteristic
229C2×C53#313111yescharacteristic
230C2×C53#314211yescharacteristic
231C2×C53#323311yescharacteristic
232C2×C53#323311yescharacteristic
233D53—42no—
234C2×#125—531no—
235C2×#117—532no—
236C22×C42—531no—
237C2×C22⋊C8—532no—
238C2×C4×C8—532no—
239#389—531no—
240#389C2×D5311yescharacteristic
241#387—1061no—
242#387—1061no—
243#385—1061no—
244#385—1061no—
245#384—531no—
246#384C2×D5311yescharacteristic
247#382—531no—
248#382—532no—
249#380—531no—
250#380—531no—
251#380—1061no—
252#380—1061no—
253C4×C4⋊C4—1061no—
254C4×C4⋊C4—1061no—
255#340—532no—
256#340—532no—
257#340—532no—
258#340—532no—
259#340—532no—
260#340—532no—
261#340C2×D5312yes—
262#340C2×D5312yes—
263#336—531no—
264#336C2×D5311yescharacteristic
265C53#2796311yescharacteristic, complemented, Sylow
266C23×C4—531no—
267C23×C4—531no—
268C23×C4—531no—
269C22×C8—532no—
270C22×C8C22×D5312yes—
271#125—1064no—
272#117—1068no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—532no—
277C2×C4⋊C4—531no—
278C2×C4⋊C4—531no—
279C2×C4⋊C4—1061no—
280C2×C4⋊C4—1061no—
281C2×C4⋊C4—1061no—
282C2×C4⋊C4—1061no—
283C2×C4⋊C4—1061no—
284C2×C4⋊C4C22×D5312yes—
285C2×C4⋊C4C22×D5311yescharacteristic
286C2×C4⋊C4C22×D5311yescharacteristic
287C2×C22⋊C4—1062no—
288C2×C22⋊C4—1061no—
289C2×C22⋊C4—1061no—
290C2×C42—531no—
291C2×C42—531no—
292C2×C42—531no—
293C2×C42—1064no—
294C2×C42—1062no—
295C2×C42C22×D5311yescharacteristic
296C22⋊C8—1064no—
297C4×C8—1064no—
298#95—21no—
299#95—1062no—
300#95—1062no—
301#95—1061no—
302#95—1061no—
303#95—1061no—
304#95—1061no—
305#95—1061no—
306C24—531no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—531no—
311C22×C4—531no—
312C22×C4—531no—
313C22×C4—534no—
314C22×C4—1062no—
315C22×C4—1062no—
316C22×C4—1062no—
317C22×C4—1061no—
318C22×C4—1061no—
319C22×C4—1062no—
320C22×C4—1062no—
321C22×C4—1061no—
322C22×C4—1062no—
323C22×C4—1061no—
324C22×C4—2121no—
325C22×C4—2121no—
326C22×C4D4×D5311yescharacteristic
327C22×C4D4×D5311yescharacteristic
328C22×C4C23×D5311yescharacteristic
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—1062no—
333C2×C8—1062no—
334C2×C8—1062no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—1064no—
343C4⋊C4—1062no—
344C4⋊C4—1061no—
345C4⋊C4—1061no—
346C4⋊C4—2122no—
347C4⋊C4—2122no—
348C4⋊C4—2122no—
349C4⋊C4—2122no—
350C4⋊C4—2122no—
351C22⋊C4—2124no—
352C22⋊C4—2122no—
353C22⋊C4—2122no—
354C42—22no—
355C42—534no—
356C42—1062no—
357C42—1062no—
358C42—2124no—
359C23—532no—
360C23—1061no—
361C23—1061no—
362C23—1061no—
363C23—1061no—
364C23—1061no—
365C23—1061no—
366C23C2×D4×D5311yescharacteristic
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—532no—
381C2×C4—532no—
382C2×C4—532no—
383C2×C4—1061no—
384C2×C4—1061no—
385C2×C4—1061no—
386C2×C4—1061no—
387C2×C4—1061no—
388C2×C4—1061no—
389C2×C4—1064no—
390C2×C4—1064no—
391C2×C4—2124no—
392C2×C4—2124no—
393C2×C4—2121no—
394C2×C4—2122no—
395C2×C4—2121no—
396C2×C4—2121no—
397C2×C4—2121no—
398C2×C4—2121no—
399C2×C4—2122no—
400C2×C4—2122no—
401C2×C4—2121no—
402C2×C4#1313323612yes—
403C2×C4#1313323812yes—
404C2×C4#41×D5312yes—
405C2×C4#41×D5312yes—
406C2×C4#1313326411yescharacteristic
407C2×C4#1313326511yescharacteristic
408C8—42no—
409C8—2122no—
410C22—1062no—
411C22—1062no—
412C22—1062no—
413C22—2121no—
414C22—2121no—
415C22—2121no—
416C22—2121no—
417C22#123×D5311yescharacteristic
418C22#124×D5311yescharacteristic
419C22#1681119611yescharacteristic
420C22C2×#1313323611yescharacteristic
421C22C2×#1313323811yescharacteristic
422C22#1681142211yescharacteristic
423C22#1681142211yescharacteristic
424C4—22no—
425C4—22no—
426C4—82no—
427C4—81no—
428C4—81no—
429C4—1062no—
430C4—1062no—
431C4—1062no—
432C4—4242no—
433C4—4241no—
434C4—4241no—
435C4#1681130512yes—
436C4#1681130912yes—
437C2—2122no—
438C2#2027919511yescharacteristic
439C2#2028462311yescharacteristic
440C2#2028462911yescharacteristic
441C2#2028503411yescharacteristic
442C2#2028511711yescharacteristic
443C2#2028589211yescharacteristic
444C2#2028589811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2099C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2099D53
hypocenterC53#27963
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2099

Complements

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

NH ≅ G/Ntype
3. #20276279C2semidirect
4. #20276278C2semidirect
5. #20276272C2semidirect
6. #20275678C2semidirect
7. #20274844C2semidirect
11. C53×#2099C2semidirect
220. #2099D53semidirect
265. C53#27963semidirect

Maximal subgroups

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

subgroupindex
#202830242
#202771202
#202762792
#202762782
#202762722
#202756782
#202748442
#202748082
#202744722
C53×#20992
#2796353