Finite Group Databasebeta

← #27351525

Subgroups — #27351525

Order 13568.

992 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811175C211yesmaximal, characteristic
2#20283344C211yesmaximal, characteristic
3#20277109C211yesmaximal, characteristic
4#20273277C214yesmaximal, complemented
5#20273264C214yesmaximal, complemented
6#20273236C211yesmaximal, characteristic
7#20273235C211yesmaximal, characteristic
8#20273234C211yesmaximal, characteristic
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—
27C22×C4×Dic53C2211yescharacteristic
28#16810677—21no—
29#16810677C2212yes—
30#16810673—21no—
31#16810673C2212yes—
32#16810646—21no—
33#16810587C2211yescharacteristic
34M16×Dic53—21no—
35#16810490C2212yes—
36#16810487C2212yes—
37#16810484—21no—
38#16810481—21no—
39C4×#11719919—21no—
40C4×C22⋊Dic53C2211yescharacteristic
41#16810469—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—22no—
87#13133173—22no—
88#13133173—22no—
89#13133173—22no—
90C2×C4⋊Dic53—21no—
91C2×C4⋊Dic53C2×C411yescharacteristic
92C2×#11719919—21no—
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×#11719919—21no—
97C2×#11719919—22no—
98C2×C22⋊Dic53—21no—
99C2×C22⋊Dic53—21no—
100C2×C4×Dic53—21no—
101C2×C4×Dic53—21no—
102C2×C4×Dic53—21no—
103C2×C4×Dic53C2×C411yescharacteristic
104C2×C4×Dic53C2×C411yescharacteristic
105C2×C4×Dic53C2×C411yescharacteristic
106C2×C4×Dic53C2×C411yescharacteristic
107C2×C4×Dic53D411yescharacteristic
108C2×C4×Dic53D411yescharacteristic
109C2×C4×Dic53D411yescharacteristic
110C2×C4×Dic53D411yescharacteristic
111C2×C4×Dic53D411yescharacteristic
112C2×C4×Dic53D411yescharacteristic
113C2×C4×Dic53C2311yescharacteristic
114C2×C4×Dic53C2311yescharacteristic
115#13133146—22no—
116#13133146—22no—
117C22⋊#88322—21no—
118C22⋊#88322C2312yes—
119#13133127—22no—
120C8×Dic53—22no—
121#13133124—22no—
122#13133124—22no—
123#13133124—22no—
124#13133124C2312yes—
125C22×Q8×C53C2×C411yescharacteristic
126C23×C4×C53C2311yescharacteristic
127C2×M16×C53—21no—
128C22⋊Q8×C53—21no—
129C22⋊Q8×C53—21no—
130C22⋊Q8×C53C2×C412yes—
131C22⋊Q8×C53C2312yes—
132C2×C4⋊C4×C53—21no—
133C2×C4⋊C4×C53C2×C411yescharacteristic
134C2×C22⋊C4×C53—21no—
135C2×C22⋊C4×C53—21no—
136C2×C42×C53—21no—
137C22⋊C8×C53—21no—
138C22⋊C8×C53C2312yes—
139#95×C53C2312yes—
140C24×C53C2×D411yescharacteristic
141C2×Q8×C53—21no—
142C2×Q8×C53—42no—
143C2×Q8×C53C22×C412yes—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—22no—
147C22×C4×C53—42no—
148C22×C4×C53C22×C411yescharacteristic
149C22×C4×C53C22×C411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2411yescharacteristic
154C22×Dic53—24no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—22no—
158C22×Dic53—42no—
159C22×Dic53C22⋊C411yescharacteristic
160C22×Dic53C22⋊C411yescharacteristic
161C22×Dic53C22⋊C411yescharacteristic
162C22×Dic53C22⋊C411yescharacteristic
163C22×Dic53C22⋊C411yescharacteristic
164C22×Dic53C22⋊C411yescharacteristic
165C22×Dic53C22⋊C411yescharacteristic
166C22×Dic53C22⋊C411yescharacteristic
167C22×Dic53C2×D411yescharacteristic
168C22×Dic53C2×D411yescharacteristic
169C22×Dic53C2×D411yescharacteristic
170C22×Dic53C2×D411yescharacteristic
171C2×Dic106—21no—
172C2×Dic106—22no—
173C2×Dic106—22no—
174C2×Dic106—44no—
175C2×Dic106—42no—
176C2×Dic106C22×C412yes—
177M16×C53—42no—
178C2×C8×C53—22no—
179C4⋊C4×C53—22no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—22no—
182C4⋊C4×C53C22×C412yes—
183C22⋊C4×C53—22no—
184C22⋊C4×C53—22no—
185C22⋊C4×C53—42no—
186C42×C53—42no—
187C22⋊Dic53—22no—
188C22⋊Dic53—22no—
189C22⋊Dic53—42no—
190C4⋊Dic53—21no—
191C4⋊Dic53—22no—
192C4⋊Dic53—22no—
193C4⋊Dic53C22×C412yes—
194#11719919—22no—
195#11719919—22no—
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—44no—
205C4×Dic53—22no—
206C4×Dic53—22no—
207C4×Dic53—22no—
208C4×Dic53—22no—
209C4×Dic53—42no—
210C4×Dic53C22⋊C411yescharacteristic
211C4×Dic53C22⋊C411yescharacteristic
212C4×Dic53C22⋊C411yescharacteristic
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×Dic53C2×D411yescharacteristic
223C4×Dic53C2×D411yescharacteristic
224C4×Dic53C2×D411yescharacteristic
225C4×Dic53C2×D411yescharacteristic
226#11719917—42no—
227C2×#88322—22no—
228C23×C53—22no—
229C23×C53—22no—
230C23×C53—41no—
231C23×C53C2×C22⋊C411yescharacteristic
232C23×C53C2×C22⋊C411yescharacteristic
233C23×C53C22×D411yescharacteristic
234Q8×C53—42no—
235Q8×C53—42no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53C2×C22⋊C411yescharacteristic
250C2×C4×C53C2×C22⋊C411yescharacteristic
251C2×C4×C53C23×C411yescharacteristic
252C2×C4×C53C22×D411yescharacteristic
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53C2×C22⋊C411yescharacteristic
272C2×Dic53C2×C22⋊C411yescharacteristic
273C2×Dic53C2×C22⋊C411yescharacteristic
274C2×Dic53C2×C22⋊C411yescharacteristic
275C2×Dic53C2×C22⋊C411yescharacteristic
276C2×Dic53C2×C22⋊C411yescharacteristic
277C2×Dic53C2×C22⋊C411yescharacteristic
278C2×Dic53C2×C22⋊C411yescharacteristic
279C2×Dic53#13714yes—
280Dic106—44no—
281Dic106—44no—
282Dic106—42no—
283Dic106—42no—
284C8×C53—42no—
285#88322—42no—
286#17584—531nomaximal, Sylow
287C22×C53—21no—
288C22×C53—21no—
289C22×C53—21no—
290C22×C53—21no—
291C22×C53—21no—
292C22×C53—21no—
293C22×C53—21no—
294C22×C53—44no—
295C22×C53#42112yes—
296C22×C53C22×C22⋊C411yescharacteristic
297C22×C53C2×#13712yes—
298C4×C53—21no—
299C4×C53—21no—
300C4×C53—21no—
301C4×C53—42no—
302C4×C53—42no—
303C4×C53—82no—
304C4×C53#41012yes—
305Dic53—21no—
306Dic53—21no—
307Dic53—21no—
308Dic53—21no—
309Dic53—21no—
310Dic53—21no—
311Dic53—42no—
312Dic53—42no—
313Dic53—82no—
314Dic53#41912yes—
315Dic53#41912yes—
316C2×#533—531no—
317#2920—531no—
318#2283—531no—
319#1518—534no—
320#1515—534no—
321#1508—531no—
322#1508—531no—
323#1508—531no—
324#1508D5311yescharacteristic
325C2×C53—22no—
326C2×C53—22no—
327C2×C53—41no—
328C2×C53C2×#41011yescharacteristic
329C2×C53C2×#41911yescharacteristic
330C2×C53C2×#42111yescharacteristic
331#533—532no—
332#533—532no—
333#533—532no—
334#533—532no—
335#533—1061no—
336#533—1061no—
337#533—1061no—
338#533—1061no—
339C2×C4⋊Q8—531no—
340C2×C4⋊Q8—531no—
341C2×#125—1061no—
342C2×C22⋊Q8—531no—
343C2×C22⋊Q8—1061no—
344C2×C22⋊Q8C2×D5311yescharacteristic
345C2×C4×Q8—531no—
346C2×C4×Q8—531no—
347C2×C4×Q8—1061no—
348C2×#117—1061no—
349C22×C42—531no—
350#432—532no—
351#432—1061no—
352#431—532no—
353#431—1061no—
354#423—1061no—
355#407—531no—
356#407C2×D5311yescharacteristic
357C4×M16—1061no—
358#383—532no—
359#382—532no—
360#381—1061no—
361#380—1061no—
362C4×C4⋊C4—1061no—
363C4×C22⋊C4—531no—
364C4×C22⋊C4C2×D5311yescharacteristic
365#376—1061no—
366#328—534no—
367#328—534no—
368#328—534no—
369#328C2×D5314yes—
370C53#1758411yescharacteristic, complemented, Sylow
371C22×Q8—531no—
372C22×Q8—1061no—
373C22×Q8C4×D5311yescharacteristic
374C23×C4—531no—
375C23×C4—531no—
376C23×C4C22×D5311yescharacteristic
377C2×M16—21no—
378C2×M16—1061no—
379C4⋊Q8—532no—
380C4⋊Q8—532no—
381C4⋊Q8—532no—
382C4⋊Q8—532no—
383C4⋊Q8—1061no—
384C4⋊Q8—1061no—
385C4⋊Q8—1061no—
386C4⋊Q8—1061no—
387#125—1062no—
388#125—1062no—
389#125—1062no—
390#125—1062no—
391C22⋊Q8—21no—
392C22⋊Q8—21no—
393C22⋊Q8—532no—
394C22⋊Q8—532no—
395C22⋊Q8—1062no—
396C22⋊Q8—1062no—
397C22⋊Q8—1062no—
398C22⋊Q8—1062no—
399C22⋊Q8—1061no—
400C22⋊Q8—1061no—
401C22⋊Q8C4×D5312yes—
402C22⋊Q8C22×D5312yes—
403C4×Q8—532no—
404C4×Q8—532no—
405C4×Q8—532no—
406C4×Q8—532no—
407C4×Q8—1061no—
408C4×Q8—1061no—
409C4×Q8—1062no—
410C4×Q8—1062no—
411C4×Q8—1062no—
412C4×Q8—1062no—
413C4×Q8—1061no—
414C4×Q8—1061no—
415#117—1062no—
416#117—1062no—
417#117—1062no—
418#117—1062no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1062no—
428C2×C4⋊C4C4×D5311yescharacteristic
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4—1061no—
433C2×C42—21no—
434C2×C42—531no—
435C2×C42—531no—
436C2×C42—531no—
437C2×C42—531no—
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—1061no—
447C2×C42—1061no—
448C2×C42—1061no—
449C4⋊C8—1062no—
450C4⋊C8—1062no—
451C22⋊C8—21no—
452C22⋊C8—532no—
453C22⋊C8—1061no—
454C22⋊C8C22×D5312yes—
455#97—1062no—
456C4×C8—1062no—
457#95—532no—
458#95—1062no—
459#95—1062no—
460#95—1062no—
461#95C22×D5312yes—
462C24D4×D5311yescharacteristic
463C2×Q8—21no—
464C2×Q8—42no—
465C2×Q8—532no—
466C2×Q8—1062no—
467C2×Q8—1062no—
468C2×Q8—1061no—
469C2×Q8—2122no—
470C2×Q8—2124no—
471C2×Q8C2×C4×D5312yes—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—42no—
476C22×C4—531no—
477C22×C4—531no—
478C22×C4—531no—
479C22×C4—531no—
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—1064no—
489C22×C4—1061no—
490C22×C4—1061no—
491C22×C4—1062no—
492C22×C4—2122no—
493C22×C4C2×C4×D5311yescharacteristic
494C22×C4C2×C4×D5311yescharacteristic
495C22×C4D4×D5311yescharacteristic
496C22×C4D4×D5311yescharacteristic
497C22×C4D4×D5311yescharacteristic
498C22×C4C23×D5311yescharacteristic
499M16—42no—
500M16—2122no—
501C2×C8—22no—
502C2×C8—1062no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—532no—
507C4⋊C4—1062no—
508C4⋊C4—1062no—
509C4⋊C4—1062no—
510C4⋊C4—1062no—
511C4⋊C4—1062no—
512C4⋊C4—1062no—
513C4⋊C4—1062no—
514C4⋊C4—1062no—
515C4⋊C4—1062no—
516C4⋊C4—1062no—
517C4⋊C4—1061no—
518C4⋊C4—1062no—
519C4⋊C4—1062no—
520C4⋊C4—2124no—
521C4⋊C4C2×C4×D5312yes—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—42no—
525C22⋊C4—1062no—
526C22⋊C4—1062no—
527C22⋊C4—2122no—
528C42—42no—
529C42—531no—
530C42—531no—
531C42—531no—
532C42—531no—
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—1062no—
546C42—1062no—
547C42—1062no—
548C42—1062no—
549C42—2122no—
550C23—22no—
551C23—22no—
552C23—41no—
553C23C22⋊C4×D5311yescharacteristic
554C23C22⋊C4×D5311yescharacteristic
555C23C2×D4×D5311yescharacteristic
556Q8—42no—
557Q8—42no—
558Q8—2122no—
559Q8—2124no—
560Q8—2124no—
561Q8—2122no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—531no—
576C2×C4—531no—
577C2×C4—534no—
578C2×C4—531no—
579C2×C4—531no—
580C2×C4—531no—
581C2×C4—531no—
582C2×C4—531no—
583C2×C4—531no—
584C2×C4—1062no—
585C2×C4—1062no—
586C2×C4—1062no—
587C2×C4—1062no—
588C2×C4—1062no—
589C2×C4—1062no—
590C2×C4—1062no—
591C2×C4—1062no—
592C2×C4—1062no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2122no—
598C2×C4—2122no—
599C2×C4—2122no—
600C2×C4—2122no—
601C2×C4—2122no—
602C2×C4C22⋊C4×D5311yescharacteristic
603C2×C4C22⋊C4×D5311yescharacteristic
604C2×C4C22×C4×D5311yescharacteristic
605C2×C4C2×D4×D5311yescharacteristic
606C8—42no—
607C8—2122no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—44no—
616C22#1681063812yes—
617C22C2×C22⋊C4×D5311yescharacteristic
618C22#137×D5312yes—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—42no—
623C4—42no—
624C4—82no—
625C4—532no—
626C4—532no—
627C4—1061no—
628C4—1061no—
629C4—1061no—
630C4—1061no—
631C4—1061no—
632C4—1061no—
633C4—4242no—
634C4—2122no—
635C4—2122no—
636C4#1681059512yes—
637C2—22no—
638C2—22no—
639C2—41no—
640C2C2×#1681059511yescharacteristic
641C2#419×D5311yescharacteristic
642C2C2×#1681063811yescharacteristic
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#17584
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
4. #20273277C2semidirect
5. #20273264C2semidirect
14. #16811175C4semidirect
15. #16811175C4semidirect
18. C2×#13133216C4semidirect
19. C2×#13133215C4semidirect
23. C2×Q8×Dic53C4semidirect
25. C2×#13133182C4semidirect
370. C53#17584semidirect

Maximal subgroups

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

subgroupindex
C2×#168111752
#202833442
#202771092
#202732772
#202732642
#202732362
#202732352
#202732342
C53×#15082
#1758453