Finite Group Databasebeta

← #27393610

Subgroups — #27393610

Order 13568.

1054 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811179C211yesmaximal, characteristic
2C2×#16810678C211yesmaximal, characteristic
3#20277120C211yesmaximal, characteristic
4#20275685C211yesmaximal, characteristic, complemented
5#20275684C211yesmaximal, characteristic, complemented
6#20275683C211yesmaximal, characteristic, complemented
7#20275665C212yesmaximal, complemented
8#20275661C212yesmaximal, complemented
9#20274809C212yesmaximal
10#20274808C212yesmaximal
11C53×#1992C211yesmaximal, characteristic, complemented
12#16811179—24no—
13#16811179C418yescomplemented
14C2×#13133216C2211yescharacteristic
15C2×#13133216C2211yescharacteristic
16C2×#13133209C2211yescharacteristic
17C2×#13133195—21no—
18C2×#13133195—21no—
19C2×#13133182—21no—
20C2×#13133182—21no—
21C2×#13133175—21no—
22C2×#13133175C2211yescharacteristic
23C2×#13133175C2211yescharacteristic
24C2×C42×D53C2211yescharacteristic
25#16810678—24no—
26C2×#13133146C2212yes—
27C2×#13133129C2212yes—
28C2×#13133128C2211yescharacteristic
29#16810493C2211yescharacteristic
30#16810488—21no—
31#16810480—21no—
32#16810480C2212yes—
33C4⋊C4×Dic53—21no—
34C4⋊C4×Dic53C2212yes—
35#16810386C2212yes—
36#16810386C2212yes—
37#16810385C2212yes—
38#16810385C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×#125×C53C2211yescharacteristic
42C2×#97×C53C2211yescharacteristic
43C53×#384C2211yescharacteristic
44C53×#340C2212yes—
45C53×#340C2212yes—
46C22×Dic106—21no—
47C22×Dic106—21no—
48C22×C4×D53D411yescharacteristic
49C22×C4×D53D411yescharacteristic
50C22×C4×D53C2311yescharacteristic
51C22×#88322C2312yes—
52#13133216—22no—
53#13133216—22no—
54#13133216C2×C414yes—
55#13133216C2×C414yes—
56#13133209—22no—
57#13133209C2×C414yes—
58#13133195—28no—
59#13133195—28no—
60#13133182—28no—
61#13133182—28no—
62#13133175—22no—
63#13133175—22no—
64#13133175—44no—
65#13133175C2×C414yes—
66#13133175C2×C414yes—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53C2312yes—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×#11719921—22no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—22no—
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85C42×D53C2×C414yes—
86C42×D53D414yes—
87#13133146—24no—
88#13133129—24no—
89#13133128—22no—
90#13133124—22no—
91#13133124—22no—
92C22×C8×C53C2312yes—
93#125×C53—22no—
94#125×C53C2×C414yes—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53C2312yes—
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C42×C53C2311yescharacteristic
103#97×C53—22no—
104C23×D53C2×D411yescharacteristic
105C22×C4×C53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—22no—
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2411yescharacteristic
111C22×Dic53—21no—
112C22×Dic53—21no—
113C22×Dic53—22no—
114C22×Dic53C2×D411yescharacteristic
115C22×Dic53C2×D411yescharacteristic
116C22×Dic53C2×D411yescharacteristic
117C2×C4×D53—22no—
118C2×C4×D53—22no—
119C2×C4×D53—22no—
120C2×C4×D53—22no—
121C2×C4×D53—41no—
122C2×C4×D53—41no—
123C2×C4×D53C22⋊C414yes—
124C2×C4×D53C22⋊C414yes—
125C2×C4×D53C22×C412yes—
126C2×C4×D53C2×D412yes—
127C2×Dic106—24no—
128C2×Dic106—24no—
129C2×Dic106—42no—
130C2×Dic106—42no—
131C2×Dic106—42no—
132C2×Dic106—42no—
133C2×C8×C53—22no—
134C2×C8×C53—22no—
135C2×C8×C53—22no—
136C4⋊C4×C53—24no—
137C4⋊C4×C53—24no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—24no—
141C4⋊C4×C53—42no—
142C4⋊C4×C53—42no—
143C4⋊C4×C53C22×C412yes—
144C4⋊C4×C53C22×C412yes—
145C42×C53C22×C412yes—
146C42×C53C2×D412yes—
147#11719921—24no—
148#11719921—24no—
149#11719921—44no—
150C4⋊Dic53—21no—
151C4⋊Dic53—21no—
152C4⋊Dic53—24no—
153C4⋊Dic53—42no—
154C4⋊Dic53—42no—
155C4⋊Dic53C22×C412yes—
156C4⋊Dic53C22×C412yes—
157#11719919—24no—
158#11719919—24no—
159#11719919—24no—
160#11719919—24no—
161C4×Dic53—24no—
162C4×Dic53—24no—
163C4×Dic53—44no—
164C4×Dic53C22⋊C414yes—
165C4×Dic53C22⋊C414yes—
166C4×Dic53C22×C412yes—
167C4×Dic53C2×D412yes—
168C2×#88322—22no—
169C2×#88322—22no—
170C2×#88322—22no—
171C23×C53C22×D411yescharacteristic
172C22×D53—21no—
173C22×D53—21no—
174C22×D53—21no—
175C22×D53—21no—
176C22×D53—21no—
177C22×D53—21no—
178C22×D53C2×C22⋊C412yes—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—22no—
184C2×C4×C53—22no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53—42no—
192C2×C4×C53C2×C22⋊C412yes—
193C2×C4×C53C2×C22⋊C412yes—
194C2×C4×C53C23×C411yescharacteristic
195C2×C4×C53C22×D411yescharacteristic
196C2×C4×C532−1+412yes—
197C2×C4×C532−1+412yes—
198C2×Dic53—21no—
199C2×Dic53—21no—
200C2×Dic53—21no—
201C2×Dic53—21no—
202C2×Dic53—22no—
203C2×Dic53—22no—
204C2×Dic53—41no—
205C2×Dic53—41no—
206C2×Dic53—41no—
207C2×Dic53—41no—
208C2×Dic53—42no—
209C2×Dic53—42no—
210C2×Dic53—42no—
211C2×Dic53C2×C22⋊C412yes—
212C2×Dic53C2×C22⋊C412yes—
213C2×Dic53C2×C22⋊C412yes—
214C2×Dic53AGL(1,ℤ/8)12yes—
215C2×Dic53#13712yes—
216C4×D53—24no—
217C4×D53—24no—
218C4×D53—28no—
219C4×D53—44no—
220Dic106—48no—
221Dic106—48no—
222C8×C53—42no—
223#88322—42no—
224#21839—531nomaximal, Sylow
225C22×C53C22×C22⋊C411yescharacteristic
226C22×C53#53611yescharacteristic
227C22×C53#53611yescharacteristic
228C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
229C22×C53C2×#13711yescharacteristic
230C22×C53#57511yescharacteristic
231C22×C53#57511yescharacteristic
232C2×D53—22no—
233C2×D53—22no—
234C2×D53—22no—
235C2×D53—41no—
236C2×D53—41no—
237C2×D53—41no—
238C2×D53—41no—
239C4×C53—22no—
240C4×C53—22no—
241C4×C53—42no—
242C4×C53—42no—
243C4×C53—82no—
244C4×C53#51912yes—
245C4×C53#53612yes—
246Dic53—22no—
247Dic53—22no—
248Dic53—42no—
249Dic53—42no—
250Dic53—82no—
251Dic53#41714yes—
252C2×#533—531no—
253C2×#432—531no—
254#2285—531no—
255#1992—531no—
256#1992—531no—
257#1992—531no—
258#1992D5311yescharacteristic, complemented
259#1988—532no—
260#1987—532no—
261#1825—532no—
262#1825—532no—
263C2×C53#231011yescharacteristic
264C2×C53C2×#41711yescharacteristic
265C2×C53#289611yescharacteristic
266C2×C53#323511yescharacteristic
267C2×C53#323511yescharacteristic
268C2×C53#323711yescharacteristic
269C2×C53#323711yescharacteristic
270D53—42no—
271#533—538no—
272#533—1064no—
273C2×C4⋊Q8—531no—
274C2×C4⋊Q8—531no—
275C2×#125—531no—
276C2×#125C2×D5311yescharacteristic
277C2×C22⋊Q8—1061no—
278C2×C22⋊Q8—1061no—
279C2×C4×Q8—1061no—
280C2×C4×Q8—1061no—
281C2×#117—531no—
282C2×#117—531no—
283C2×#117—1061no—
284C22×C42—531no—
285#432—1064no—
286C2×C4⋊C8—532no—
287C2×C22⋊C8—532no—
288C2×#97—531no—
289C2×#97C2×D5311yescharacteristic
290#384—531no—
291#384C2×D5311yescharacteristic
292#382—1061no—
293#380—532no—
294#380—1061no—
295C4×C4⋊C4—532no—
296C4×C4⋊C4—1061no—
297#340—532no—
298#340—532no—
299#340—532no—
300#340—532no—
301#340—532no—
302#340—532no—
303#340C2×D5312yes—
304#340C2×D5312yes—
305C53#2183911yescharacteristic, complemented, Sylow
306C22×Q8—1061no—
307C22×Q8—1061no—
308C23×C4—531no—
309C23×C4—531no—
310C23×C4—531no—
311C22×C8—532no—
312C22×C8C22×D5312yes—
313C4⋊Q8—534no—
314C4⋊Q8—534no—
315C4⋊Q8—1062no—
316C4⋊Q8—1062no—
317#125—22no—
318#125—534no—
319#125—1062no—
320#125C4×D5314yes—
321C22⋊Q8—1068no—
322C22⋊Q8—1068no—
323C4×Q8—1068no—
324C4×Q8—1068no—
325#117—534no—
326#117—534no—
327#117—1062no—
328#117—1062no—
329#117—2124no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—531no—
335C2×C4⋊C4—531no—
336C2×C4⋊C4—532no—
337C2×C4⋊C4—1061no—
338C2×C4⋊C4—1061no—
339C2×C4⋊C4—1061no—
340C2×C4⋊C4—1061no—
341C2×C4⋊C4—1061no—
342C2×C4⋊C4—1061no—
343C2×C4⋊C4C22×D5312yes—
344C2×C4⋊C4C22×D5311yescharacteristic
345C2×C4⋊C4C22×D5311yescharacteristic
346C2×C22⋊C4—1061no—
347C2×C22⋊C4—1061no—
348C2×C22⋊C4—1062no—
349C2×C42—531no—
350C2×C42—531no—
351C2×C42—531no—
352C2×C42—534no—
353C2×C42—534no—
354C2×C42—1061no—
355C2×C42—1061no—
356C2×C42—1062no—
357C2×C42C22×D5311yescharacteristic
358C4⋊C8—1064no—
359C22⋊C8—1064no—
360#97—22no—
361#97—1062no—
362#95—1062no—
363#95—1062no—
364C24—531no—
365C2×Q8—1064no—
366C2×Q8—1064no—
367C2×Q8—2122no—
368C2×Q8—2122no—
369C2×Q8—2122no—
370C2×Q8—2122no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—531no—
375C22×C4—531no—
376C22×C4—531no—
377C22×C4—534no—
378C22×C4—532no—
379C22×C4—532no—
380C22×C4—534no—
381C22×C4—1062no—
382C22×C4—1062no—
383C22×C4—1062no—
384C22×C4—1062no—
385C22×C4—1061no—
386C22×C4—1061no—
387C22×C4—1062no—
388C22×C4—2121no—
389C22×C4—2121no—
390C22×C4D4×D5311yescharacteristic
391C22×C4D4×D5311yescharacteristic
392C22×C4C23×D5311yescharacteristic
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—1062no—
397C2×C8—1062no—
398C2×C8—1062no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—24no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—532no—
407C4⋊C4—532no—
408C4⋊C4—1064no—
409C4⋊C4—1064no—
410C4⋊C4—1064no—
411C4⋊C4—1064no—
412C4⋊C4—1061no—
413C4⋊C4—1061no—
414C4⋊C4—1064no—
415C4⋊C4—2122no—
416C4⋊C4—2122no—
417C4⋊C4C2×C4×D5312yes—
418C4⋊C4C2×C4×D5312yes—
419C22⋊C4—1064no—
420C22⋊C4—1064no—
421C22⋊C4—2124no—
422C42—534no—
423C42—534no—
424C42—532no—
425C42—532no—
426C42—1064no—
427C42—1064no—
428C42—2124no—
429C42C2×C4×D5312yes—
430C42D4×D5312yes—
431C23—532no—
432C23—1061no—
433C23—1061no—
434C23—1061no—
435C23—1061no—
436C23—1061no—
437C23—1061no—
438C23C2×D4×D5311yescharacteristic
439Q8—2128no—
440Q8—2128no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—532no—
455C2×C4—532no—
456C2×C4—532no—
457C2×C4—532no—
458C2×C4—532no—
459C2×C4—1068no—
460C2×C4—1061no—
461C2×C4—1061no—
462C2×C4—1061no—
463C2×C4—1061no—
464C2×C4—1064no—
465C2×C4—1064no—
466C2×C4—1062no—
467C2×C4—1062no—
468C2×C4—2124no—
469C2×C4—2122no—
470C2×C4—2122no—
471C2×C4—2122no—
472C2×C4—2121no—
473C2×C4—2121no—
474C2×C4—2121no—
475C2×C4—2121no—
476C2×C4C22⋊C4×D5312yes—
477C2×C4C22⋊C4×D5312yes—
478C2×C4C22×C4×D5311yescharacteristic
479C2×C4C2×D4×D5311yescharacteristic
480C2×C4#1313326412yes—
481C2×C4#1313326512yes—
482C8—42no—
483C8—2122no—
484C22—1062no—
485C22—1062no—
486C22—1062no—
487C22—2121no—
488C22—2121no—
489C22—2121no—
490C22—2121no—
491C22C2×C22⋊C4×D5311yescharacteristic
492C22#1681119311yescharacteristic
493C22#1681119811yescharacteristic
494C22AGL(1,ℤ/8)×D5311yescharacteristic
495C22#137×D5311yescharacteristic
496C22#1681141911yescharacteristic
497C22#1681142211yescharacteristic
498C4—22no—
499C4—22no—
500C4—42no—
501C4—42no—
502C4—82no—
503C4—534no—
504C4—1062no—
505C4—1062no—
506C4—4242no—
507C4—2122no—
508C4—2122no—
509C4#1681109912yes—
510C4#1681119612yes—
511C2—2122no—
512C2#2027729811yescharacteristic
513C2#417×D5311yescharacteristic
514C2#2028323111yescharacteristic
515C2#2028590811yescharacteristic
516C2#2028591411yescharacteristic
517C2#2028592411yescharacteristic
518C2#2028593011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1992C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1992D53
hypocenterC53#21839
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
4. #20275685C2semidirect
5. #20275684C2semidirect
6. #20275683C2semidirect
7. #20275665C2semidirect
8. #20275661C2semidirect
11. C53×#1992C2semidirect
13. #16811179C4semidirect
258. #1992D53semidirect
305. C53#21839semidirect

Maximal subgroups

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

subgroupindex
C2×#168111792
C2×#168106782
#202771202
#202756852
#202756842
#202756832
#202756652
#202756612
#202748092
#202748082
C53×#19922
#2183953