Finite Group Databasebeta

← #27351271

Subgroups — #27351271

Order 13568.

1084 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811175C211yesmaximal, characteristic
2#20283344C211yesmaximal, characteristic
3C4×#13133173C211yesmaximal, characteristic
4#20273236C211yesmaximal, characteristic
5#20273235C211yesmaximal, characteristic
6#20273234C211yesmaximal, characteristic
7#20273225C214yesmaximal, complemented
8#20273212C214yesmaximal, complemented
9C53×#1508C211yesmaximal, characteristic
10#16811175—21no—
11#16811175—21no—
12#16811175—21no—
13#16811175—21no—
14#16811175C412yescomplemented
15#16811175C412yescomplemented
16#16811175C2212yes—
17#16811175C2212yes—
18C2×#13133216C411yescharacteristic, complemented
19C2×#13133215C411yescharacteristic, complemented
20C2×#13133207—21no—
21C2×#13133196C2211yescharacteristic
22C2×#13133194—21no—
23C2×Q8×Dic53C411yescharacteristic, complemented
24C2×#13133182—21no—
25C2×#13133182C411yescharacteristic, complemented
26C2×#13133173—21no—
27C2×#13133173—21no—
28C22×C4×Dic53C2211yescharacteristic
29#16810677—21no—
30#16810677C2212yes—
31#16810673—21no—
32#16810673C2212yes—
33#16810646—21no—
34#16810587C2211yescharacteristic
35M16×Dic53—21no—
36C4×#11719919—21no—
37#16810474C2212yes—
38C22⋊C4×Dic53—21no—
39C4×C22⋊Dic53C2211yescharacteristic
40#16810468C2212yes—
41C42×Dic53—21no—
42#16810361C2214yes—
43#16810360C2214yes—
44#16810359C2214yes—
45C2×C22⋊Q8×C53C2211yescharacteristic
46C53×#407C2211yescharacteristic
47C4×C22⋊C4×C53C2211yescharacteristic
48C53×#328C2214yes—
49C22×Dic106—21no—
50C22×Dic106C2×C411yescharacteristic
51C23×Dic53D411yescharacteristic
52C23×Dic53D411yescharacteristic
53C2×#11719917—21no—
54#13133216—21no—
55#13133216—21no—
56#13133216C2×C412yes—
57#13133216C2×C412yes—
58#13133215—21no—
59#13133215—21no—
60#13133215C2×C412yes—
61#13133215C2×C412yes—
62#13133207—22no—
63#13133207—22no—
64#13133207—22no—
65#13133207—22no—
66#13133196—21no—
67#13133196—21no—
68#13133196C2×C412yes—
69#13133196C2312yes—
70#13133194—22no—
71#13133194—22no—
72#13133194—22no—
73#13133194—22no—
74Q8×Dic53—21no—
75Q8×Dic53—21no—
76Q8×Dic53C2×C412yes—
77Q8×Dic53C2×C412yes—
78#13133182—22no—
79#13133182—22no—
80#13133182—21no—
81#13133182—21no—
82#13133182—22no—
83#13133182—22no—
84#13133182C2×C412yes—
85#13133182C2×C412yes—
86#13133173—28no—
87#13133173—22no—
88#13133173—22no—
89#13133173—22no—
90#13133173—22no—
91C2×C4⋊Dic53—21no—
92C2×C4⋊Dic53C2×C411yescharacteristic
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×#11719919—21no—
97C2×#11719919—21no—
98C2×#11719919—22no—
99C2×C22⋊Dic53—21no—
100C2×C22⋊Dic53—21no—
101C2×C4×Dic53—21no—
102C2×C4×Dic53—21no—
103C2×C4×Dic53—21no—
104C2×C4×Dic53—22no—
105C2×C4×Dic53—22no—
106C2×C4×Dic53C2×C411yescharacteristic
107C2×C4×Dic53C2×C411yescharacteristic
108C2×C4×Dic53C2×C411yescharacteristic
109C2×C4×Dic53C2×C411yescharacteristic
110C2×C4×Dic53D411yescharacteristic
111C2×C4×Dic53D411yescharacteristic
112C2×C4×Dic53D411yescharacteristic
113C2×C4×Dic53D411yescharacteristic
114C2×C4×Dic53D411yescharacteristic
115C2×C4×Dic53D411yescharacteristic
116C2×C4×Dic53C2311yescharacteristic
117C2×C4×Dic53C2311yescharacteristic
118#13133146—22no—
119#13133146—22no—
120C22⋊#88322—21no—
121C22⋊#88322C2312yes—
122#13133127—22no—
123C8×Dic53—22no—
124#13133124—22no—
125#13133124C2312yes—
126C22×Q8×C53C2×C411yescharacteristic
127C23×C4×C53C2311yescharacteristic
128C2×M16×C53—21no—
129C22⋊Q8×C53—21no—
130C22⋊Q8×C53—21no—
131C22⋊Q8×C53C2×C412yes—
132C22⋊Q8×C53C2312yes—
133C2×C4⋊C4×C53—21no—
134C2×C4⋊C4×C53C2×C411yescharacteristic
135C2×C22⋊C4×C53—21no—
136C2×C22⋊C4×C53—21no—
137C2×C42×C53—21no—
138C22⋊C8×C53—21no—
139C22⋊C8×C53C2312yes—
140#95×C53C2312yes—
141C24×C53C2×D411yescharacteristic
142C2×Q8×C53—21no—
143C2×Q8×C53—42no—
144C2×Q8×C53C22×C412yes—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—22no—
148C22×C4×C53—42no—
149C22×C4×C53C22×C411yescharacteristic
150C22×C4×C53C22×C411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2411yescharacteristic
155C22×Dic53—24no—
156C22×Dic53—24no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—22no—
160C22×Dic53C22⋊C411yescharacteristic
161C22×Dic53C22⋊C411yescharacteristic
162C22×Dic53C22⋊C411yescharacteristic
163C22×Dic53C22⋊C411yescharacteristic
164C22×Dic53C22⋊C411yescharacteristic
165C22×Dic53C22⋊C411yescharacteristic
166C22×Dic53C22⋊C411yescharacteristic
167C22×Dic53C22⋊C411yescharacteristic
168C22×Dic53C2×D411yescharacteristic
169C22×Dic53C2×D411yescharacteristic
170C22×Dic53C2×D411yescharacteristic
171C22×Dic53C2×D411yescharacteristic
172C2×Dic106—21no—
173C2×Dic106—22no—
174C2×Dic106—22no—
175C2×Dic106—44no—
176C2×Dic106—42no—
177C2×Dic106C22×C412yes—
178M16×C53—42no—
179C2×C8×C53—22no—
180C4⋊C4×C53—22no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—22no—
183C4⋊C4×C53C22×C412yes—
184C22⋊C4×C53—24no—
185C22⋊C4×C53—22no—
186C22⋊C4×C53—22no—
187C42×C53—42no—
188C22⋊Dic53—24no—
189C22⋊Dic53—22no—
190C22⋊Dic53—22no—
191C4⋊Dic53—21no—
192C4⋊Dic53—22no—
193C4⋊Dic53—22no—
194C4⋊Dic53C22×C412yes—
195#11719919—28no—
196#11719919—22no—
197#11719919—22no—
198#11719919—22no—
199#11719919—22no—
200#11719919—22no—
201#11719919—22no—
202#11719919—22no—
203#11719919—22no—
204#11719919—22no—
205#11719919—22no—
206C4×Dic53—28no—
207C4×Dic53—22no—
208C4×Dic53—22no—
209C4×Dic53—22no—
210C4×Dic53—22no—
211C4×Dic53—42no—
212C4×Dic53—44no—
213C4×Dic53C22⋊C411yescharacteristic
214C4×Dic53C22⋊C411yescharacteristic
215C4×Dic53C22⋊C411yescharacteristic
216C4×Dic53C22⋊C411yescharacteristic
217C4×Dic53C22⋊C411yescharacteristic
218C4×Dic53C22⋊C411yescharacteristic
219C4×Dic53C22⋊C411yescharacteristic
220C4×Dic53C22⋊C411yescharacteristic
221C4×Dic53C22×C411yescharacteristic
222C4×Dic53C22×C411yescharacteristic
223C4×Dic53C22×C411yescharacteristic
224C4×Dic53C22×C411yescharacteristic
225C4×Dic53C2×D411yescharacteristic
226C4×Dic53C2×D411yescharacteristic
227C4×Dic53C2×D411yescharacteristic
228C4×Dic53C2×D411yescharacteristic
229#11719917—42no—
230C2×#88322—22no—
231C23×C53—22no—
232C23×C53—22no—
233C23×C53—41no—
234C23×C53C2×C22⋊C411yescharacteristic
235C23×C53C2×C22⋊C411yescharacteristic
236C23×C53C22×D411yescharacteristic
237Q8×C53—42no—
238Q8×C53—42no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—24no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53C2×C22⋊C411yescharacteristic
253C2×C4×C53C2×C22⋊C411yescharacteristic
254C2×C4×C53C23×C411yescharacteristic
255C2×C4×C53C22×D411yescharacteristic
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—24no—
267C2×Dic53—22no—
268C2×Dic53—22no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53#10414yes—
274C2×Dic53C2×C22⋊C411yescharacteristic
275C2×Dic53C2×C22⋊C411yescharacteristic
276C2×Dic53C2×C22⋊C411yescharacteristic
277C2×Dic53C2×C22⋊C411yescharacteristic
278C2×Dic53C2×C22⋊C411yescharacteristic
279C2×Dic53C2×C22⋊C411yescharacteristic
280C2×Dic53C2×C22⋊C411yescharacteristic
281C2×Dic53C2×C22⋊C411yescharacteristic
282C2×Dic53#13714yes—
283Dic106—42no—
284Dic106—44no—
285Dic106—44no—
286Dic106—42no—
287C8×C53—42no—
288#88322—42no—
289#17545—531nomaximal, Sylow
290C22×C53—21no—
291C22×C53—21no—
292C22×C53—21no—
293C22×C53—21no—
294C22×C53—21no—
295C22×C53—21no—
296C22×C53—21no—
297C22×C53—44no—
298C22×C53C2×#10412yes—
299C22×C53C22×C22⋊C411yescharacteristic
300C22×C53C2×#13712yes—
301C4×C53—21no—
302C4×C53—21no—
303C4×C53—21no—
304C4×C53—44no—
305C4×C53—42no—
306C4×C53—42no—
307C4×C53#41012yes—
308Dic53—21no—
309Dic53—21no—
310Dic53—21no—
311Dic53—21no—
312Dic53—44no—
313Dic53—42no—
314Dic53—42no—
315Dic53#41912yes—
316Dic53#41912yes—
317Dic53C2×#10412yes—
318Dic53C2×#10412yes—
319C2×#533—531no—
320#2920—531no—
321C4×#117—531no—
322#1508—531no—
323#1508—531no—
324#1508—531no—
325#1508D5311yescharacteristic
326#1506—534no—
327#1503—534no—
328C2×C53—22no—
329C2×C53—22no—
330C2×C53—41no—
331C2×C53C2×#41011yescharacteristic
332C2×C53C2×#41911yescharacteristic
333C2×C53C22×#10411yescharacteristic
334#533—532no—
335#533—532no—
336#533—532no—
337#533—532no—
338#533—1061no—
339#533—1061no—
340#533—1061no—
341#533—1061no—
342C2×C4⋊Q8—531no—
343C2×C4⋊Q8—531no—
344C2×#125—1061no—
345C2×C22⋊Q8—531no—
346C2×C22⋊Q8—1061no—
347C2×C22⋊Q8C2×D5311yescharacteristic
348C2×C4×Q8—531no—
349C2×C4×Q8—531no—
350C2×C4×Q8—1061no—
351C2×#117—1061no—
352C2×#117—1061no—
353C22×C42—531no—
354#432—532no—
355#432—1061no—
356#431—532no—
357#431—1061no—
358#423—1061no—
359#407—531no—
360#407C2×D5311yescharacteristic
361C4×M16—1061no—
362C4×C4⋊C4—532no—
363C4×C4⋊C4—1061no—
364C4×C22⋊C4—531no—
365C4×C22⋊C4—1061no—
366C4×C22⋊C4C2×D5311yescharacteristic
367#376—532no—
368C43—1061no—
369#328—534no—
370#328—534no—
371#328—534no—
372#328C2×D5314yes—
373C53#1754511yescharacteristic, complemented, Sylow
374C22×Q8—531no—
375C22×Q8—1061no—
376C22×Q8C4×D5311yescharacteristic
377C23×C4—531no—
378C23×C4—531no—
379C23×C4C22×D5311yescharacteristic
380C2×M16—21no—
381C2×M16—1061no—
382C4⋊Q8—532no—
383C4⋊Q8—532no—
384C4⋊Q8—532no—
385C4⋊Q8—532no—
386C4⋊Q8—1061no—
387C4⋊Q8—1061no—
388C4⋊Q8—1061no—
389C4⋊Q8—1061no—
390#125—1062no—
391#125—1062no—
392#125—1062no—
393#125—1062no—
394C22⋊Q8—21no—
395C22⋊Q8—21no—
396C22⋊Q8—532no—
397C22⋊Q8—532no—
398C22⋊Q8—1062no—
399C22⋊Q8—1062no—
400C22⋊Q8—1062no—
401C22⋊Q8—1062no—
402C22⋊Q8—1061no—
403C22⋊Q8—1061no—
404C22⋊Q8C4×D5312yes—
405C22⋊Q8C22×D5312yes—
406C4×Q8—532no—
407C4×Q8—532no—
408C4×Q8—532no—
409C4×Q8—532no—
410C4×Q8—1061no—
411C4×Q8—1061no—
412C4×Q8—1062no—
413C4×Q8—1062no—
414C4×Q8—1062no—
415C4×Q8—1062no—
416C4×Q8—1061no—
417C4×Q8—1061no—
418#117—1062no—
419#117—1062no—
420#117—1062no—
421#117—1062no—
422#117—1068no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1062no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4C4×D5311yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—1061no—
436C2×C22⋊C4—1061no—
437C2×C42—21no—
438C2×C42—531no—
439C2×C42—531no—
440C2×C42—531no—
441C2×C42—531no—
442C2×C42—531no—
443C2×C42—531no—
444C2×C42—531no—
445C2×C42—531no—
446C2×C42—531no—
447C2×C42—531no—
448C2×C42—531no—
449C2×C42—531no—
450C2×C42—1061no—
451C2×C42—1061no—
452C2×C42—1062no—
453C2×C42—1062no—
454C2×C42—1061no—
455C4⋊C8—1062no—
456C4⋊C8—1062no—
457C22⋊C8—21no—
458C22⋊C8—532no—
459C22⋊C8—1061no—
460C22⋊C8C22×D5312yes—
461#97—1062no—
462C4×C8—1062no—
463#95—532no—
464#95—1062no—
465#95C22×D5312yes—
466C24D4×D5311yescharacteristic
467C2×Q8—21no—
468C2×Q8—42no—
469C2×Q8—532no—
470C2×Q8—1062no—
471C2×Q8—1062no—
472C2×Q8—1061no—
473C2×Q8—2122no—
474C2×Q8—2124no—
475C2×Q8C2×C4×D5312yes—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—42no—
480C22×C4—531no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—531no—
484C22×C4—531no—
485C22×C4—531no—
486C22×C4—531no—
487C22×C4—531no—
488C22×C4—531no—
489C22×C4—531no—
490C22×C4—531no—
491C22×C4—531no—
492C22×C4—1064no—
493C22×C4—1064no—
494C22×C4—1061no—
495C22×C4—1061no—
496C22×C4—1062no—
497C22×C4C2×C4×D5311yescharacteristic
498C22×C4C2×C4×D5311yescharacteristic
499C22×C4D4×D5311yescharacteristic
500C22×C4D4×D5311yescharacteristic
501C22×C4D4×D5311yescharacteristic
502C22×C4C23×D5311yescharacteristic
503M16—42no—
504M16—2122no—
505C2×C8—22no—
506C2×C8—1062no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—532no—
511C4⋊C4—1062no—
512C4⋊C4—1062no—
513C4⋊C4—1062no—
514C4⋊C4—1062no—
515C4⋊C4—1062no—
516C4⋊C4—1062no—
517C4⋊C4—1062no—
518C4⋊C4—1062no—
519C4⋊C4—1062no—
520C4⋊C4—1062no—
521C4⋊C4—1068no—
522C4⋊C4—1061no—
523C4⋊C4—1062no—
524C4⋊C4—1062no—
525C4⋊C4C2×C4×D5312yes—
526C22⋊C4—24no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—1064no—
530C22⋊C4—1062no—
531C22⋊C4—1062no—
532C42—42no—
533C42—531no—
534C42—531no—
535C42—531no—
536C42—531no—
537C42—531no—
538C42—531no—
539C42—531no—
540C42—531no—
541C42—531no—
542C42—531no—
543C42—531no—
544C42—531no—
545C42—531no—
546C42—531no—
547C42—531no—
548C42—531no—
549C42—1062no—
550C42—1062no—
551C42—1062no—
552C42—1062no—
553C42—1068no—
554C42—2122no—
555C42—2124no—
556C23—22no—
557C23—22no—
558C23—41no—
559C23C22⋊C4×D5311yescharacteristic
560C23C22⋊C4×D5311yescharacteristic
561C23C2×D4×D5311yescharacteristic
562Q8—42no—
563Q8—42no—
564Q8—2122no—
565Q8—2124no—
566Q8—2124no—
567Q8—2122no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—531no—
582C2×C4—531no—
583C2×C4—531no—
584C2×C4—531no—
585C2×C4—534no—
586C2×C4—534no—
587C2×C4—531no—
588C2×C4—531no—
589C2×C4—531no—
590C2×C4—531no—
591C2×C4—1062no—
592C2×C4—1062no—
593C2×C4—1062no—
594C2×C4—1062no—
595C2×C4—1062no—
596C2×C4—1062no—
597C2×C4—1062no—
598C2×C4—1062no—
599C2×C4—1062no—
600C2×C4—1062no—
601C2×C4—1064no—
602C2×C4—1062no—
603C2×C4—1062no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4—2122no—
607C2×C4—2122no—
608C2×C4C22⋊C4×D5311yescharacteristic
609C2×C4C22⋊C4×D5311yescharacteristic
610C2×C4C22×C4×D5311yescharacteristic
611C2×C4C2×D4×D5311yescharacteristic
612C8—42no—
613C8—2122no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—44no—
622C22#104×D5312yes—
623C22C2×C22⋊C4×D5311yescharacteristic
624C22#137×D5312yes—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—44no—
629C4—42no—
630C4—42no—
631C4—532no—
632C4—532no—
633C4—532no—
634C4—532no—
635C4—1061no—
636C4—1061no—
637C4—1061no—
638C4—1061no—
639C4—2124no—
640C4—2122no—
641C4—2122no—
642C4#1681059512yes—
643C2—22no—
644C2—22no—
645C2—41no—
646C2C2×#1681059511yescharacteristic
647C2#419×D5311yescharacteristic
648C2C2×#104×D5311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1508C2
socleC22×C53C22×C22⋊C4
prime radicalC24×C53C2×D4
cosocleC22×C4×C53C24
hypercenter#1508D53
hypocenterC53#17545
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
7. #20273225C2semidirect
8. #20273212C2semidirect
14. #16811175C4semidirect
15. #16811175C4semidirect
18. C2×#13133216C4semidirect
19. C2×#13133215C4semidirect
23. C2×Q8×Dic53C4semidirect
25. C2×#13133182C4semidirect
373. C53#17545semidirect

Maximal subgroups

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

subgroupindex
C2×#168111752
#202833442
C4×#131331732
#202732362
#202732352
#202732342
#202732252
#202732122
C53×#15082
#1754553