Finite Group Databasebeta

← #27389455

Subgroups — #27389455

Order 13568.

1162 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811222C211yesmaximal, characteristic
2#20277317C211yesmaximal, characteristic, complemented
3#20277070C211yesmaximal, characteristic, complemented
4#20275701C211yesmaximal, characteristic, complemented
5#20275551C211yesmaximal, characteristic
6#20275550C211yesmaximal, characteristic
7#20275172C214yesmaximal, complemented
8#20274502C214yesmaximal, complemented
9C53×#1996C211yesmaximal, characteristic, complemented
10#16811222—22no—
11#16811222—22no—
12#16811222—22no—
13#16811222—22no—
14C2×#13133212C412yescomplemented
15C2×#13133208C412yescomplemented
16C2×#13133202C412yescomplemented
17C2×#13133201C2212yes—
18C2×#13133193C412yescomplemented
19C2×#13133191C2211yescharacteristic
20C2×#13133177C2212yes—
21C2×#13133173C2212yes—
22C22×#11719919C2211yescharacteristic
23C2×#13133131C2214yes—
24C2×#13133131C2214yes—
25#16810510—22no—
26#16810504—22no—
27#16810484C2212yes—
28#16810483C2211yescharacteristic
29C4×#11719919—22no—
30#16810474—22no—
31C22⋊C4×Dic53C2212yes—
32C22⋊C4×Dic53C2212yes—
33C4×C22⋊Dic53C2211yescharacteristic
34#16810390C2214yes—
35C2×C22⋊Q8×C53C2211yescharacteristic
36C53×#380C2211yescharacteristic
37C4×C22⋊C4×C53C2211yescharacteristic
38C53×#342C2214yes—
39C22×D212C2×C411yescharacteristic
40C22×C22⋊D53C2312yes—
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2311yescharacteristic
43C22×C4×D53C2×C412yes—
44#13133212—24no—
45#13133212—24no—
46#13133208—24no—
47#13133208—24no—
48#13133202—24no—
49#13133202—24no—
50#13133201—24no—
51#13133201—24no—
52#13133193—24no—
53#13133193—24no—
54#13133191—21no—
55#13133191—21no—
56#13133191—21no—
57#13133191—21no—
58#13133177—24no—
59#13133177—24no—
60#13133173—24no—
61#13133173—24no—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53C2×C411yescharacteristic
64C2×C4⋊Dic53C2×C412yes—
65C2×C4⋊Dic53C2×C412yes—
66C2×#11719919—22no—
67C2×#11719919C2×C412yes—
68C2×#11719919C2×C412yes—
69C2×#11719919C2×C412yes—
70C2×#11719919D412yes—
71C2×#11719919D412yes—
72C2×#11719919D412yes—
73C2×#11719919D412yes—
74C2×C22⋊Dic53C2312yes—
75C2×C22⋊Dic53C2312yes—
76C2×#11719921C2×C412yes—
77C2×#11719921C2×C412yes—
78C2×#11719921C2×C412yes—
79C2×#11719921C2×C412yes—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—22no—
82C2×C4×Dic53—22no—
83C2×C4×Dic53—22no—
84C2×C4×Dic53C2×C412yes—
85#13133131—28no—
86#13133131—28no—
87#13133124—21no—
88#13133124—21no—
89#13133124—22no—
90#13133124—24no—
91#13133124—22no—
92C22×Q8×C53C2×C411yescharacteristic
93C23×C4×C53C2311yescharacteristic
94C22⋊Q8×C53—21no—
95C22⋊Q8×C53—21no—
96C22⋊Q8×C53—21no—
97C22⋊Q8×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53C2×C412yes—
100C2×C4⋊C4×C53C2×C411yescharacteristic
101C2×C22⋊C4×C53C2312yes—
102C2×C22⋊C4×C53C2312yes—
103C2×C22⋊C4×C53C2312yes—
104C2×C42×C53—21no—
105#95×C53—21no—
106#95×C53—21no—
107C24×C53C2411yescharacteristic
108C23×D53C22×C412yes—
109C2×Q8×C53—21no—
110C2×Q8×C53—21no—
111C2×Q8×C53—81no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—81no—
115C22×C4×C53C22×C412yes—
116C22×C4×C53C22×C412yes—
117C22×C4×C53C22×C411yescharacteristic
118C22×C4×C53C22×C411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C2×C22⋊D53—22no—
124C2×C22⋊D53—22no—
125C2×C22⋊D53—24no—
126C2×C22⋊D53—24no—
127C22×Dic53—22no—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—81no—
132C22×Dic53C22×C412yes—
133C22×Dic53C22×C412yes—
134C22×Dic53C22×C412yes—
135C22×Dic53C2×D412yes—
136C22×Dic53C2×D412yes—
137C22×Dic53C2×D412yes—
138C22×Dic53C2×D412yes—
139C2×D212—21no—
140C2×D212—21no—
141C2×D212—81no—
142C2×C4×D53—24no—
143C2×C4×D53—82no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—81no—
149C22⋊C4×C53—22no—
150C22⋊C4×C53—22no—
151C22⋊C4×C53—22no—
152C22⋊C4×C53—24no—
153C22⋊C4×C53#4114yes—
154C42×C53—81no—
155C22⋊Dic53—22no—
156C22⋊Dic53—24no—
157C22⋊Dic53#4114yes—
158#11719921—22no—
159#11719921—22no—
160#11719921—24no—
161#11719921—24no—
162#11719921—24no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—24no—
166C4⋊Dic53—24no—
167C4⋊Dic53—81no—
168#11719919—28no—
169#11719919—24no—
170#11719919C22⋊C414yes—
171#11719919C22⋊C414yes—
172#11719919C22⋊C414yes—
173#11719919C22⋊C414yes—
174C4×Dic53—28no—
175C4×Dic53—24no—
176C4×Dic53—82no—
177C4×Dic53—82no—
178C4×Dic53—81no—
179C23×C53—41no—
180C23×C53—41no—
181C23×C53C23×C411yescharacteristic
182C23×C53C22×D411yescharacteristic
183C23×C53C22×D411yescharacteristic
184C23×C53C2×#4112yes—
185C23×C53C2×#4112yes—
186C22×D53—22no—
187C22×D53—24no—
188C22×D53—42no—
189C22×D53—42no—
190Q8×C53—81no—
191Q8×C53—81no—
192C2×C4×C53—22no—
193C2×C4×C53—22no—
194C2×C4×C53—24no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53—42no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53C2×C22⋊C411yescharacteristic
209C2×C4×C53C2×C22⋊C411yescharacteristic
210C2×C4×C53C2×C22⋊C411yescharacteristic
211C2×C4×C53C2×C22⋊C411yescharacteristic
212C22⋊D53—44no—
213C22⋊D53—42no—
214C22⋊D53—42no—
215C2×Dic53—24no—
216C2×Dic53—22no—
217C2×Dic53—22no—
218C2×Dic53—22no—
219C2×Dic53—22no—
220C2×Dic53—22no—
221C2×Dic53—24no—
222C2×Dic53—24no—
223C2×Dic53—42no—
224C2×Dic53—42no—
225C2×Dic53—42no—
226C2×Dic53—42no—
227C2×Dic53—42no—
228C2×Dic53—42no—
229C2×Dic53—42no—
230C2×Dic53—42no—
231C2×Dic53C2×C22⋊C414yes—
232C2×Dic53C2×C22⋊C414yes—
233D212—81no—
234D212—81no—
235C4×D53—84no—
236#21423—531nomaximal, Sylow
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—21no—
243C22×C53—42no—
244C22×C53—42no—
245C22×C53C22×C22⋊C411yescharacteristic
246C22×C53C4×#4112yes—
247C22×C53#52512yes—
248C22×C53#53211yescharacteristic
249C22×C53#53211yescharacteristic
250C2×D53—42no—
251C2×D53—42no—
252C2×D53—42no—
253C2×D53—84no—
254C4×C53—21no—
255C4×C53—21no—
256C4×C53—21no—
257C4×C53—21no—
258C4×C53—44no—
259C4×C53—42no—
260C4×C53—42no—
261C4×C53—82no—
262Dic53—24no—
263Dic53—24no—
264Dic53—44no—
265Dic53—42no—
266Dic53—82no—
267C2×#539—531no—
268#2313—531no—
269#2274—531no—
270#1996—531no—
271#1996D5311yescharacteristic, complemented
272#1964—531no—
273#1964—531no—
274#1899—534no—
275C4×#99—534no—
276C2×C53—22no—
277C2×C53—21no—
278C2×C53—21no—
279C2×C53—22no—
280C2×C53#229711yescharacteristic
281C2×C53#288311yescharacteristic
282C2×C53#288411yescharacteristic
283D53—82no—
284#539—1062no—
285#539—1062no—
286#539—1062no—
287#539—1062no—
288C2×#126—532no—
289C2×#125—532no—
290C2×#123—532no—
291C2×#123—532no—
292C2×C22⋊Q8—532no—
293C2×C22⋊Q8C2×D5311yescharacteristic
294C2×#121—531no—
295C2×C4×D4—532no—
296C2×#117—532no—
297C22×C4⋊C4—531no—
298C2×#99—534no—
299C2×#99—534no—
300#387—1062no—
301#387—1062no—
302#381—532no—
303#380—531no—
304#380C2×D5311yescharacteristic
305C4×C4⋊C4—1062no—
306C4×C4⋊C4—1062no—
307C4×C22⋊C4—531no—
308C4×C22⋊C4—532no—
309C4×C22⋊C4—532no—
310C4×C22⋊C4C2×D5311yescharacteristic
311#342—534no—
312#342C2×D5314yes—
313C53#2142311yescharacteristic, complemented, Sylow
314C22×Q8C4×D5311yescharacteristic
315C22×D4—532no—
316C22×D4—531no—
317C23×C4—531no—
318C23×C4—531no—
319C23×C4—532no—
320C23×C4C22×D5311yescharacteristic
321#126—1064no—
322#126—1064no—
323#125—1064no—
324#125—1064no—
325#123—1064no—
326#123—1064no—
327#123—1064no—
328#123—1064no—
329C22⋊Q8—21no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—21no—
333C22⋊Q8—1064no—
334C22⋊Q8—1064no—
335#121—1061no—
336#121—1061no—
337#121—1061no—
338#121—1061no—
339C4×D4—1064no—
340C4×D4—1064no—
341#117—1064no—
342#117—1064no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—532no—
345C2×C4⋊C4—532no—
346C2×C4⋊C4—532no—
347C2×C4⋊C4—532no—
348C2×C4⋊C4—532no—
349C2×C4⋊C4—532no—
350C2×C4⋊C4—532no—
351C2×C4⋊C4—531no—
352C2×C4⋊C4—532no—
353C2×C4⋊C4—532no—
354C2×C4⋊C4—1062no—
355C2×C4⋊C4—1061no—
356C2×C4⋊C4C4×D5312yes—
357C2×C4⋊C4C4×D5311yescharacteristic
358C2×C22⋊C4—532no—
359C2×C22⋊C4—532no—
360C2×C22⋊C4—532no—
361C2×C22⋊C4—532no—
362C2×C22⋊C4—532no—
363C2×C22⋊C4—532no—
364C2×C22⋊C4C22×D5312yes—
365C2×C22⋊C4C22×D5312yes—
366C2×C22⋊C4C22×D5312yes—
367C2×C42—21no—
368C2×C42—532no—
369C2×C42—1062no—
370C2×C42—1062no—
371C2×C42—1062no—
372C2×C42—1061no—
373#99—1068no—
374#99—1068no—
375#95—21no—
376#95—21no—
377#95—1062no—
378#95—1062no—
379#95—1064no—
380#95—1061no—
381#95—1061no—
382C24—532no—
383C24C23×D5311yescharacteristic
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—81no—
387C2×D4—1062no—
388C2×D4—1062no—
389C2×D4—1064no—
390C2×D4—1064no—
391C2×D4—1061no—
392C2×D4—1061no—
393C2×D4—4241no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—81no—
397C22×C4—532no—
398C22×C4—532no—
399C22×C4—532no—
400C22×C4—532no—
401C22×C4—532no—
402C22×C4—532no—
403C22×C4—532no—
404C22×C4—1064no—
405C22×C4—1062no—
406C22×C4—1062no—
407C22×C4—1062no—
408C22×C4—1062no—
409C22×C4—4242no—
410C22×C4—4241no—
411C22×C4C2×C4×D5312yes—
412C22×C4C2×C4×D5312yes—
413C22×C4C2×C4×D5311yescharacteristic
414C22×C4C2×C4×D5311yescharacteristic
415C22×C4D4×D5311yescharacteristic
416C22×C4D4×D5311yescharacteristic
417C22×C4D4×D5311yescharacteristic
418C22×C4D4×D5311yescharacteristic
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—81no—
424C4⋊C4—534no—
425C4⋊C4—534no—
426C4⋊C4—534no—
427C4⋊C4—534no—
428C4⋊C4—1064no—
429C4⋊C4—1064no—
430C4⋊C4—1064no—
431C4⋊C4—1068no—
432C4⋊C4—1061no—
433C4⋊C4—1061no—
434C4⋊C4—4241no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—24no—
439C22⋊C4—534no—
440C22⋊C4—1064no—
441C22⋊C4—1064no—
442C22⋊C4—1064no—
443C22⋊C4—1062no—
444C22⋊C4—1062no—
445C22⋊C4—1062no—
446C22⋊C4—1064no—
447C22⋊C4#1171994914yes—
448C42—81no—
449C42—1064no—
450C42—1068no—
451C42—4241no—
452C42—4242no—
453C42—4242no—
454C23—41no—
455C23—41no—
456C23—1064no—
457C23—1062no—
458C23—2122no—
459C23—2122no—
460C23C22×C4×D5311yescharacteristic
461C23C2×D4×D5311yescharacteristic
462C23C2×D4×D5311yescharacteristic
463C23#41×D5312yes—
464C23C2×#1171994912yes—
465Q8—81no—
466Q8—81no—
467D4—4241no—
468D4—4241no—
469D4—2124no—
470D4—2122no—
471D4—2122no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—534no—
489C2×C4—534no—
490C2×C4—1064no—
491C2×C4—1062no—
492C2×C4—1062no—
493C2×C4—1062no—
494C2×C4—1062no—
495C2×C4—1062no—
496C2×C4—1064no—
497C2×C4—1064no—
498C2×C4—4244no—
499C2×C4—2122no—
500C2×C4—2122no—
501C2×C4—2122no—
502C2×C4—2122no—
503C2×C4—2122no—
504C2×C4—2122no—
505C2×C4—2122no—
506C2×C4—2122no—
507C2×C4C22⋊C4×D5311yescharacteristic
508C2×C4C22⋊C4×D5311yescharacteristic
509C2×C4C22⋊C4×D5311yescharacteristic
510C2×C4C22⋊C4×D5311yescharacteristic
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—42no—
518C22—42no—
519C22—4244no—
520C22—2122no—
521C22—2122no—
522C22—2122no—
523C22C2×C22⋊C4×D5311yescharacteristic
524C22C4×#1171994912yes—
525C22#1681113612yes—
526C22#1681117211yescharacteristic
527C22#1681117411yescharacteristic
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—44no—
533C4—42no—
534C4—42no—
535C4—82no—
536C4—1064no—
537C4—1064no—
538C4—4242no—
539C4—2124no—
540C4—2122no—
541C2—22no—
542C2—21no—
543C2—21no—
544C2—22no—
545C2—4242no—
546C2#2027720411yescharacteristic
547C2#2028317311yescharacteristic
548C2#2028317711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC53×#1996C2
socleC22×C53C22×C22⋊C4
prime radicalC2×#13133191C22
cosocleC24×C53C24
hypercenter#1996D53
hypocenterC53#21423
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
2. #20277317C2semidirect
3. #20277070C2semidirect
4. #20275701C2semidirect
7. #20275172C2semidirect
8. #20274502C2semidirect
9. C53×#1996C2semidirect
14. C2×#13133212C4semidirect
15. C2×#13133208C4semidirect
16. C2×#13133202C4semidirect
18. C2×#13133193C4semidirect
271. #1996D53semidirect
313. C53#21423semidirect

Maximal subgroups

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

subgroupindex
C2×#168112222
#202773172
#202770702
#202757012
#202755512
#202755502
#202751722
#202745022
C53×#19962
#2142353