Finite Group Databasebeta

← #27393805

Subgroups — #27393805

Order 13568.

1106 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811177C211yesmaximal, characteristic
2C2×#16810675C211yesmaximal, characteristic
3#20277121C211yesmaximal, characteristic
4#20275674C212yesmaximal, complemented
5#20275515C212yesmaximal, complemented
6#20275026C212yesmaximal
7#20275024C212yesmaximal
8#20274731C211yesmaximal, characteristic, complemented
9#20274730C211yesmaximal, characteristic, complemented
10#20274727C211yesmaximal, characteristic, complemented
11C53×#1805C211yesmaximal, characteristic, complemented
12#16811177—24no—
13#16811177C418yescomplemented
14C2×#13133216C2211yescharacteristic
15C2×#13133215C2211yescharacteristic
16C2×#13133207—21no—
17C2×#13133197C2211yescharacteristic
18C2×#13133195—21no—
19C2×#13133193C2211yescharacteristic
20C2×C4×Dic106—21no—
21C2×C4×Dic106C2211yescharacteristic
22C2×#13133175C2212yes—
23C2×#13133174—21no—
24C2×C42×D53C2211yescharacteristic
25#16810675—24no—
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133142C2212yes—
28C2×#13133141C2212yes—
29C2×#13133140C2212yes—
30C2×#13133129C2212yes—
31C2×C8×Dic53C2212yes—
32#16810492C2212yes—
33#16810489C2211yescharacteristic
34#16810482—21no—
35#16810482—21no—
36#16810474—21no—
37#16810474—21no—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×C4×Q8×C53C2211yescharacteristic
42C2×C4⋊C8×C53C2211yescharacteristic
43C2×#103×C53C2212yes—
44C53×#382C2211yescharacteristic
45C53×#340C2212yes—
46C22×Dic106—21no—
47C22×Dic106C2311yescharacteristic
48C22×C4×D53D411yescharacteristic
49C22×C4×D53D411yescharacteristic
50C22×C4×D53C2311yescharacteristic
51C22×#88322C2312yes—
52#13133216—22no—
53#13133216C2×C414yes—
54#13133215—22no—
55#13133215C2×C414yes—
56#13133207—28no—
57#13133197—22no—
58#13133197C2×C414yes—
59#13133195—28no—
60#13133193—22no—
61#13133193C2×C414yes—
62C4×Dic106—22no—
63C4×Dic106—28no—
64C4×Dic106C2×C414yes—
65#13133175—42no—
66#13133175—42no—
67#13133174—28no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53C2312yes—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—22no—
77C2×#11719919—22no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—22no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—22no—
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53D411yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86C42×D53C2×C414yes—
87C42×D53D414yes—
88C4⋊#88322—22no—
89#13133142—42no—
90#13133141—42no—
91#13133140—42no—
92#13133129—24no—
93C8×Dic53—24no—
94#13133124—21no—
95#13133124—21no—
96C22×Q8×C53C2311yescharacteristic
97C22×C8×C53C2312yes—
98C4×Q8×C53—22no—
99C4×Q8×C53C2×C414yes—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53C2312yes—
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C42×C53—21no—
104C2×C42×C53C2311yescharacteristic
105C4⋊C8×C53—22no—
106#103×C53—42no—
107#95×C53—21no—
108#95×C53—21no—
109C23×D53C2×D411yescharacteristic
110C2×Q8×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×Q8×C53C22×C412yes—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2411yescharacteristic
122C22×Dic53—21no—
123C22×Dic53—21no—
124C22×Dic53—22no—
125C22×Dic53C2×D411yescharacteristic
126C22×Dic53C2×D411yescharacteristic
127C22×Dic53C2×D411yescharacteristic
128C2×C4×D53—21no—
129C2×C4×D53—21no—
130C2×C4×D53—21no—
131C2×C4×D53—21no—
132C2×C4×D53—24no—
133C2×C4×D53—24no—
134C2×C4×D53C22⋊C414yes—
135C2×C4×D53C22⋊C414yes—
136C2×C4×D53C22×C412yes—
137C2×C4×D53C2×D412yes—
138C2×Dic106—21no—
139C2×Dic106—21no—
140C2×Dic106—21no—
141C2×Dic106—21no—
142C2×Dic106—21no—
143C2×Dic106—24no—
144C2×Dic106—44no—
145C2×Dic106C22×C412yes—
146C2×C8×C53—22no—
147C2×C8×C53—22no—
148C2×C8×C53—22no—
149C4⋊C4×C53—24no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53C22×C412yes—
154C42×C53—24no—
155C42×C53C22×C412yes—
156C42×C53C2×D412yes—
157#11719921—24no—
158#11719921—24no—
159#11719921—44no—
160C4⋊Dic53—21no—
161C4⋊Dic53—24no—
162C4⋊Dic53—22no—
163C4⋊Dic53—22no—
164C4⋊Dic53C22×C412yes—
165#11719919—24no—
166#11719919—24no—
167#11719919—24no—
168#11719919—24no—
169#11719919—24no—
170#11719919—44no—
171#11719919—44no—
172C4×Dic53—24no—
173C4×Dic53—44no—
174C4×Dic53C22⋊C414yes—
175C4×Dic53C22⋊C414yes—
176C4×Dic53C22×C412yes—
177C4×Dic53C2×D412yes—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C23×C53C22×D411yescharacteristic
182C22×D53—21no—
183C22×D53—21no—
184C22×D53—22no—
185C22×D53—22no—
186C22×D53C2×C22⋊C412yes—
187Q8×C53—22no—
188Q8×C53—22no—
189Q8×C53—41no—
190Q8×C53—41no—
191C2×C4×C53—22no—
192C2×C4×C53—22no—
193C2×C4×C53—22no—
194C2×C4×C53—22no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53—42no—
200C2×C4×C53—42no—
201C2×C4×C53C2×C22⋊C412yes—
202C2×C4×C53C2×C22⋊C412yes—
203C2×C4×C53C23×C411yescharacteristic
204C2×C4×C53C22×D411yescharacteristic
205C2×C4×C53H(5,𝔽2)11yescharacteristic
206C2×C4×C53H(5,𝔽2)11yescharacteristic
207C2×C4×C532−1+411yescharacteristic
208C2×C4×C532−1+411yescharacteristic
209C2×Dic53—21no—
210C2×Dic53—21no—
211C2×Dic53—22no—
212C2×Dic53—22no—
213C2×Dic53—41no—
214C2×Dic53—41no—
215C2×Dic53—42no—
216C2×Dic53—42no—
217C2×Dic53—42no—
218C2×Dic53—42no—
219C2×Dic53C2×C22⋊C412yes—
220C2×Dic53C2×C22⋊C412yes—
221C2×Dic53C2×C22⋊C412yes—
222C2×Dic53#13514yes—
223C2×Dic53#13714yes—
224C4×D53—22no—
225C4×D53—22no—
226C4×D53—22no—
227C4×D53—22no—
228C4×D53—28no—
229C4×D53—28no—
230Dic106—22no—
231Dic106—22no—
232Dic106—41no—
233Dic106—41no—
234Dic106—48no—
235C8×C53—42no—
236#88322—42no—
237#21864—531nomaximal, Sylow
238C22×C53C22×C22⋊C411yescharacteristic
239C22×C53#53511yescharacteristic
240C22×C53#53611yescharacteristic
241C22×C53C2×#13512yes—
242C22×C53C2×#13712yes—
243C2×D53—22no—
244C2×D53—22no—
245C2×D53—22no—
246C2×D53—42no—
247C2×D53—42no—
248C4×C53—22no—
249C4×C53—22no—
250C4×C53—42no—
251C4×C53—42no—
252C4×C53—82no—
253C4×C53#52011yescharacteristic
254C4×C53#52011yescharacteristic
255C4×C53#53711yescharacteristic
256C4×C53#53711yescharacteristic
257Dic53—22no—
258Dic53—42no—
259Dic53—42no—
260Dic53—82no—
261Dic53#41614yes—
262Dic53#41914yes—
263C2×#533—531no—
264C2×#431—531no—
265#2285—531no—
266#1990—532no—
267#1956—532no—
268#1873—532no—
269#1873—532no—
270#1805—531no—
271#1805—531no—
272#1805—531no—
273#1805D5311yescharacteristic, complemented
274C2×C53#231111yescharacteristic
275C2×C53C2×#41611yescharacteristic
276C2×C53C2×#41911yescharacteristic
277C2×C53#321012yes—
278C2×C53#323112yes—
279D53—42no—
280#533—538no—
281#533—1064no—
282C2×C4⋊Q8—531no—
283C2×C4⋊Q8—531no—
284C2×#125—1061no—
285C2×C22⋊Q8—531no—
286C2×C22⋊Q8—531no—
287C2×C22⋊Q8—1061no—
288C2×C4×Q8—531no—
289C2×C4×Q8—1061no—
290C2×C4×Q8C2×D5311yescharacteristic
291C2×#117—532no—
292C2×#117—1061no—
293C22×C42—531no—
294#431—1064no—
295C2×C4⋊C8—531no—
296C2×C4⋊C8C2×D5311yescharacteristic
297C2×#103—532no—
298C2×#103—532no—
299C2×#103—532no—
300C2×#103C2×D5312yes—
301C2×C22⋊C8—532no—
302C2×C4×C8—532no—
303#384—532no—
304#382—531no—
305#382C2×D5311yescharacteristic
306#380—1061no—
307#380—1061no—
308C4×C4⋊C4—1061no—
309C4×C4⋊C4—1061no—
310#340—532no—
311#340—532no—
312#340—532no—
313#340C2×D5312yes—
314C53#2186411yescharacteristic, complemented, Sylow
315C22×Q8—531no—
316C22×Q8—1061no—
317C22×Q8C22×D5311yescharacteristic
318C23×C4—531no—
319C23×C4—531no—
320C23×C4—531no—
321C22×C8—532no—
322C22×C8C22×D5312yes—
323C4⋊Q8—534no—
324C4⋊Q8—534no—
325C4⋊Q8—1062no—
326C4⋊Q8—1062no—
327#125—1068no—
328C22⋊Q8—534no—
329C22⋊Q8—534no—
330C22⋊Q8—1062no—
331C22⋊Q8—1068no—
332C22⋊Q8—1062no—
333C4×Q8—22no—
334C4×Q8—534no—
335C4×Q8—1068no—
336C4×Q8—1062no—
337C4×Q8C4×D5314yes—
338#117—1068no—
339#117—2122no—
340#117—2122no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—531no—
343C2×C4⋊C4—532no—
344C2×C4⋊C4—1061no—
345C2×C4⋊C4—1061no—
346C2×C4⋊C4—1061no—
347C2×C4⋊C4—1061no—
348C2×C4⋊C4—1061no—
349C2×C4⋊C4—1061no—
350C2×C4⋊C4—1062no—
351C2×C4⋊C4—1062no—
352C2×C4⋊C4C22×D5312yes—
353C2×C4⋊C4C22×D5311yescharacteristic
354C2×C22⋊C4—1061no—
355C2×C22⋊C4—1061no—
356C2×C22⋊C4—1062no—
357C2×C42—21no—
358C2×C42—531no—
359C2×C42—531no—
360C2×C42—531no—
361C2×C42—534no—
362C2×C42—534no—
363C2×C42—1061no—
364C2×C42—1062no—
365C2×C42C22×D5311yescharacteristic
366C4⋊C8—22no—
367C4⋊C8—1062no—
368#103—42no—
369#103—2122no—
370#103—2122no—
371#103—2122no—
372C22⋊C8—1064no—
373C4×C8—1064no—
374#95—21no—
375#95—21no—
376#95—1061no—
377#95—1061no—
378C24—531no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—532no—
385C2×Q8—1064no—
386C2×Q8—1061no—
387C2×Q8—1061no—
388C2×Q8—1061no—
389C2×Q8—1061no—
390C2×Q8—1061no—
391C2×Q8—2124no—
392C2×Q8C2×C4×D5312yes—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—531no—
397C22×C4—531no—
398C22×C4—531no—
399C22×C4—534no—
400C22×C4—534no—
401C22×C4—532no—
402C22×C4—532no—
403C22×C4—1064no—
404C22×C4—1064no—
405C22×C4—1061no—
406C22×C4—1061no—
407C22×C4—1061no—
408C22×C4—1061no—
409C22×C4—1061no—
410C22×C4—1061no—
411C22×C4—1062no—
412C22×C4D4×D5311yescharacteristic
413C22×C4D4×D5311yescharacteristic
414C22×C4C23×D5311yescharacteristic
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—1062no—
419C2×C8—1062no—
420C2×C8—1062no—
421C4⋊C4—24no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—532no—
426C4⋊C4—1064no—
427C4⋊C4—1064no—
428C4⋊C4—1064no—
429C4⋊C4—1064no—
430C4⋊C4—1064no—
431C4⋊C4—1064no—
432C4⋊C4—1061no—
433C4⋊C4—1062no—
434C4⋊C4—1062no—
435C4⋊C4—2124no—
436C4⋊C4—2124no—
437C4⋊C4C2×C4×D5312yes—
438C22⋊C4—1064no—
439C22⋊C4—1064no—
440C22⋊C4—2124no—
441C42—24no—
442C42—534no—
443C42—534no—
444C42—532no—
445C42—532no—
446C42—1064no—
447C42—2124no—
448C42C2×C4×D5312yes—
449C42D4×D5312yes—
450C23—532no—
451C23—1062no—
452C23—1062no—
453C23—1061no—
454C23—1061no—
455C23C2×D4×D5311yescharacteristic
456Q8—22no—
457Q8—22no—
458Q8—41no—
459Q8—41no—
460Q8—1062no—
461Q8—1062no—
462Q8—2128no—
463Q8—2121no—
464Q8—2121no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—532no—
476C2×C4—532no—
477C2×C4—534no—
478C2×C4—534no—
479C2×C4—532no—
480C2×C4—1068no—
481C2×C4—1068no—
482C2×C4—1061no—
483C2×C4—1061no—
484C2×C4—1062no—
485C2×C4—1062no—
486C2×C4—1062no—
487C2×C4—1062no—
488C2×C4—1062no—
489C2×C4—1062no—
490C2×C4—2122no—
491C2×C4—2122no—
492C2×C4—2122no—
493C2×C4—2122no—
494C2×C4—2121no—
495C2×C4—2121no—
496C2×C4C22⋊C4×D5312yes—
497C2×C4C22⋊C4×D5312yes—
498C2×C4C22×C4×D5311yescharacteristic
499C2×C4C2×D4×D5311yescharacteristic
500C2×C4#1313326211yescharacteristic
501C2×C4#1313326311yescharacteristic
502C2×C4#1313326411yescharacteristic
503C2×C4#1313326511yescharacteristic
504C8—42no—
505C8—2122no—
506C22—1062no—
507C22—1062no—
508C22—1062no—
509C22—2122no—
510C22—2122no—
511C22C2×C22⋊C4×D5311yescharacteristic
512C22#1681118911yescharacteristic
513C22#1681119611yescharacteristic
514C22#135×D5312yes—
515C22#137×D5312yes—
516C4—22no—
517C4—22no—
518C4—42no—
519C4—42no—
520C4—82no—
521C4—534no—
522C4—534no—
523C4—1062no—
524C4—4242no—
525C4—2122no—
526C4—2122no—
527C4#1681110311yescharacteristic
528C4#1681110411yescharacteristic
529C4#1681120211yescharacteristic
530C4#1681120311yescharacteristic
531C2—2122no—
532C2#2027730411yescharacteristic
533C2#416×D5311yescharacteristic
534C2#419×D5311yescharacteristic
535C2#2028567012yes—
536C2#2028587312yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1805C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1805D53
hypocenterC53#21864
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
4. #20275674C2semidirect
5. #20275515C2semidirect
8. #20274731C2semidirect
9. #20274730C2semidirect
10. #20274727C2semidirect
11. C53×#1805C2semidirect
13. #16811177C4semidirect
273. #1805D53semidirect
314. C53#21864semidirect

Maximal subgroups

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

subgroupindex
C2×#168111772
C2×#168106752
#202771212
#202756742
#202755152
#202750262
#202750242
#202747312
#202747302
#202747272
C53×#18052
#2186453