Finite Group Databasebeta

← #27389249

Subgroups — #27389249

Order 13568.

1086 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811236C211yesmaximal, characteristic
2C2×#16810683C211yesmaximal, characteristic
3#20277069C211yesmaximal, characteristic
4#20275673C211yesmaximal, characteristic, complemented
5#20275638C211yesmaximal, characteristic, complemented
6#20275637C211yesmaximal, characteristic, complemented
7#20275026C211yesmaximal, characteristic
8#20275026C211yesmaximal, characteristic
9#20275021C211yesmaximal, characteristic
10#20275021C211yesmaximal, characteristic
11#20274544C211yesmaximal, characteristic, complemented
12#20274544C211yesmaximal, characteristic, complemented
13#20274534C211yesmaximal, characteristic, complemented
14#20274534C211yesmaximal, characteristic, complemented
15C53×#1989C211yesmaximal, characteristic, complemented
16#16811236—24no—
17#16811236C418yescomplemented
18C2×#13133217C2211yescharacteristic
19C2×#13133212—21no—
20C2×#13133212—21no—
21C2×#13133208—21no—
22C2×#13133208—21no—
23C2×#13133197C2211yescharacteristic
24C2×#13133197C2211yescharacteristic
25C2×#13133182C2211yescharacteristic
26C2×#13133182C2211yescharacteristic
27C2×#13133175C2211yescharacteristic
28C2×#13133174C2211yescharacteristic
29C2×C4⋊C4×D53C2211yescharacteristic
30#16810683—24no—
31C2×C4⋊#88322C2211yescharacteristic
32C2×#13133142C2211yescharacteristic
33C2×#13133142C2211yescharacteristic
34C2×#13133142C2211yescharacteristic
35C2×#13133142C2211yescharacteristic
36C2×#13133140C2211yescharacteristic
37C2×#13133140C2211yescharacteristic
38C2×#13133140C2211yescharacteristic
39C2×#13133140C2211yescharacteristic
40C2×#13133129C2211yescharacteristic
41C2×#13133129C2211yescharacteristic
42C2×#13133127C2211yescharacteristic
43C2×C8×Dic53C2211yescharacteristic
44C4⋊C4×Dic53C2211yescharacteristic
45C4⋊C4×Dic53C2211yescharacteristic
46C4×#11719919—21no—
47C4×C4⋊Dic53C2211yescharacteristic
48#16810474—21no—
49C4×#11719921—21no—
50#16810470—21no—
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810385C2211yescharacteristic
55C2×#125×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C22×Dic106C2311yescharacteristic
63C22×Dic106C2311yescharacteristic
64C22×C4×D53—21no—
65C22×C4×D53C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133217—22no—
69#13133217C2×C414yescomplemented
70#13133212—28no—
71#13133212—28no—
72#13133208—28no—
73#13133208—28no—
74#13133197—22no—
75#13133197—22no—
76#13133197C2×C414yes—
77#13133197C2×C414yes—
78#13133182—22no—
79#13133182—22no—
80#13133182C2×C414yescomplemented
81#13133182C2×C414yescomplemented
82#13133175—22no—
83#13133175—22no—
84#13133174C2×C414yes—
85#13133174D414yes—
86C2×C4⋊Dic53—21no—
87C2×C4⋊Dic53—21no—
88C2×C4⋊Dic53—21no—
89C2×C4⋊Dic53—21no—
90C2×C4⋊Dic53C2311yescharacteristic
91C2×C4⋊Dic53C2311yescharacteristic
92C2×#11719919—21no—
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719919D411yescharacteristic
96C2×#11719919D411yescharacteristic
97C4⋊C4×D53—22no—
98C4⋊C4×D53—22no—
99C2×#11719921—21no—
100C2×#11719921—21no—
101C2×#11719921—21no—
102C2×#11719921D411yescharacteristic
103C2×#11719921D411yescharacteristic
104C2×C4×Dic53—21no—
105C2×C4×Dic53—21no—
106C2×C4×Dic53—21no—
107C2×C4×Dic53—21no—
108C2×C4×Dic53—21no—
109C2×C4×Dic53—21no—
110C2×C4×Dic53C2311yescharacteristic
111C4⋊#88322—22no—
112#13133142—22no—
113#13133142—22no—
114#13133142—22no—
115#13133142—22no—
116#13133140—22no—
117#13133140—22no—
118#13133140—22no—
119#13133140—22no—
120#13133129—22no—
121#13133129—22no—
122#13133127—22no—
123C8×Dic53—22no—
124#13133124—21no—
125#13133124—21no—
126#13133124—21no—
127C22×C8×C53C2311yescharacteristic
128C22×C8×C53C2311yescharacteristic
129#125×C53—22no—
130#125×C53C2×C414yescomplemented
131C2×C4⋊C4×C53—21no—
132C2×C4⋊C4×C53—21no—
133C2×C4⋊C4×C53C2311yescharacteristic
134C2×C4⋊C4×C53C2311yescharacteristic
135C2×C4⋊C4×C53C2311yescharacteristic
136C2×C4⋊C4×C53C2311yescharacteristic
137C2×C42×C53—21no—
138C2×C42×C53C2311yescharacteristic
139C4⋊C8×C53—22no—
140#95×C53—21no—
141C23×D53C2×D411yescharacteristic
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C2×C4×D53—21no—
157C2×C4×D53—21no—
158C2×C4×D53—21no—
159C2×C4×D53—21no—
160C2×C4×D53—24no—
161C2×C4×D53—81no—
162C2×C4×D53—81no—
163C2×C4×D53C22×C412yes—
164C2×C4×D53C2×D412yes—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106—21no—
171C2×Dic106—21no—
172C2×Dic106—21no—
173C2×Dic106—21no—
174C2×Dic106—21no—
175C2×Dic106C22×C412yes—
176C2×Dic106C22×C412yes—
177C2×C8×C53—21no—
178C2×C8×C53—21no—
179C2×C8×C53—21no—
180C2×C8×C53—21no—
181C2×C8×C53—21no—
182C2×C8×C53—21no—
183C4⋊C4×C53—24no—
184C4⋊C4×C53—24no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53C22×C412yes—
190C4⋊C4×C53C22×C412yes—
191C4⋊C4×C53#4112yes—
192C4⋊C4×C53#4112yes—
193C42×C53—81no—
194C42×C53C22×C412yes—
195C42×C53C2×D412yes—
196#11719921—24no—
197#11719921—24no—
198#11719921—24no—
199#11719921C22⋊C414yes—
200#11719921C22⋊C414yes—
201C4⋊Dic53—24no—
202C4⋊Dic53—24no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—24no—
206C4⋊Dic53—24no—
207C4⋊Dic53#4112yes—
208C4⋊Dic53#4112yes—
209#11719919—24no—
210#11719919—24no—
211#11719919—24no—
212#11719919C22⋊C414yes—
213#11719919C22⋊C414yes—
214C4×Dic53—24no—
215C4×Dic53—24no—
216C4×Dic53—24no—
217C4×Dic53—81no—
218C4×Dic53—81no—
219C4×Dic53—81no—
220C4×Dic53C22×C412yes—
221C4×Dic53C2×D412yes—
222C2×#88322—21no—
223C2×#88322—21no—
224C2×#88322—21no—
225C2×#88322—21no—
226C2×#88322—21no—
227C2×#88322—21no—
228C23×C53C22×D411yescharacteristic
229C22×D53—21no—
230C22×D53—21no—
231C22×D53—21no—
232C22×D53—21no—
233C22×D53—21no—
234C22×D53—21no—
235C22×D53C2×C22⋊C412yes—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53C2×C22⋊C412yes—
253C2×C4×C53C2×C22⋊C412yes—
254C2×C4×C53C23×C411yescharacteristic
255C2×C4×C53C22×D411yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—22no—
267C2×Dic53—22no—
268C2×Dic53—22no—
269C2×Dic53—22no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53C2×C22⋊C412yes—
279C2×Dic53C2×C22⋊C412yes—
280C2×Dic53C2×C22⋊C412yes—
281C4×D53—22no—
282C4×D53—22no—
283C4×D53—22no—
284C4×D53—22no—
285C4×D53—84no—
286Dic106—22no—
287Dic106—22no—
288Dic106—22no—
289Dic106—22no—
290Dic106—41no—
291Dic106—41no—
292Dic106—41no—
293Dic106—41no—
294C8×C53—41no—
295C8×C53—41no—
296#88322—41no—
297#88322—41no—
298#21398—531nomaximal, Sylow
299C22×C53C22×C22⋊C411yescharacteristic
300C22×C53#52511yescharacteristic
301C22×C53#52511yescharacteristic
302C22×C53#57711yescharacteristic
303C22×C53#57711yescharacteristic
304C22×C53#57811yescharacteristic
305C22×C53#57811yescharacteristic
306C2×D53—22no—
307C2×D53—22no—
308C2×D53—22no—
309C2×D53—41no—
310C2×D53—41no—
311C2×D53—41no—
312C2×D53—41no—
313C4×C53—22no—
314C4×C53—22no—
315C4×C53—42no—
316C4×C53—42no—
317C4×C53—42no—
318C4×C53—42no—
319C4×C53C4×#4111yescharacteristic
320C4×C53C4×#4111yescharacteristic
321C4×C53#53211yescharacteristic
322C4×C53#53211yescharacteristic
323Dic53—22no—
324Dic53—22no—
325Dic53—22no—
326Dic53—42no—
327Dic53—42no—
328Dic53—42no—
329Dic53—42no—
330C2×#541—531no—
331C2×#433—531no—
332#2274—531no—
333#1989—531no—
334#1989D5311yescharacteristic, complemented
335#1981—531no—
336#1981—531no—
337#1873—531no—
338#1873—531no—
339#1872—531no—
340#1872—531no—
341#1763—531no—
342#1763—531no—
343C4×#103—531no—
344C4×#103—531no—
345C2×C53#229711yescharacteristic
346C2×C53#289811yescharacteristic
347C2×C53#289811yescharacteristic
348C2×C53#312011yescharacteristic
349C2×C53#312011yescharacteristic
350C2×C53#312311yescharacteristic
351C2×C53#312711yescharacteristic
352D53—42no—
353#541—538no—
354#541—1064no—
355C2×C4⋊Q8—531no—
356C2×#126—1061no—
357C2×#126—1061no—
358C2×#125—1061no—
359C2×#125—1061no—
360C2×#125C2×D5311yescharacteristic
361C2×C22⋊Q8—531no—
362C2×C22⋊Q8—531no—
363C2×C4×Q8—531no—
364C2×C4×Q8—531no—
365C2×#117—531no—
366C2×#117—531no—
367C22×C4⋊C4—531no—
368#433—1064no—
369C2×C4⋊C8—531no—
370C2×C4⋊C8C2×D5311yescharacteristic
371C2×#103—531no—
372C2×#103—531no—
373C2×#103—531no—
374C2×#103—531no—
375C2×#103—531no—
376C2×#103—531no—
377C2×#103—531no—
378C2×#103—531no—
379C2×C22⋊C8—531no—
380C2×C22⋊C8—531no—
381C2×#97—531no—
382C2×C4×C8—531no—
383C4×C4⋊C4—531no—
384C4×C4⋊C4—531no—
385C4×C4⋊C4—531no—
386C4×C4⋊C4—1061no—
387C4×C4⋊C4—1061no—
388C4×C4⋊C4C2×D5311yescharacteristic
389C4×C22⋊C4—1061no—
390C4×C22⋊C4—1061no—
391#340—531no—
392#340—531no—
393#340—531no—
394#340—531no—
395#340C2×D5311yescharacteristic
396#340C2×D5311yescharacteristic
397#340C2×D5311yescharacteristic
398#340C2×D5311yescharacteristic
399C53#2139811yescharacteristic, complemented, Sylow
400C22×Q8—531no—
401C22×Q8—531no—
402C23×C4—531no—
403C23×C4—1061no—
404C22×C8—531no—
405C22×C8—531no—
406C22×C8C22×D5311yescharacteristic
407C22×C8C22×D5311yescharacteristic
408C4⋊Q8—534no—
409C4⋊Q8—1062no—
410#126—1068no—
411#126—1068no—
412#125—22no—
413#125—1068no—
414#125—1068no—
415#125C4×D5314yescomplemented
416C22⋊Q8—534no—
417C22⋊Q8—534no—
418C22⋊Q8—1062no—
419C22⋊Q8—1062no—
420C4×Q8—534no—
421C4×Q8—534no—
422C4×Q8—1062no—
423C4×Q8—1062no—
424#117—534no—
425#117—534no—
426#117—1062no—
427#117—1062no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—531no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—531no—
434C2×C4⋊C4—1062no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4—1061no—
439C2×C4⋊C4—1061no—
440C2×C4⋊C4—1061no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4—1062no—
443C2×C4⋊C4C22×D5311yescharacteristic
444C2×C4⋊C4C22×D5311yescharacteristic
445C2×C4⋊C4C22×D5311yescharacteristic
446C2×C4⋊C4C22×D5311yescharacteristic
447C2×C22⋊C4—531no—
448C2×C22⋊C4—531no—
449C2×C22⋊C4—1061no—
450C2×C22⋊C4—1061no—
451C2×C22⋊C4—1061no—
452C2×C42—21no—
453C2×C42—531no—
454C2×C42—1061no—
455C2×C42—1061no—
456C2×C42—1061no—
457C2×C42—1061no—
458C2×C42—1061no—
459C2×C42—1061no—
460C2×C42C22×D5311yescharacteristic
461C4⋊C8—22no—
462C4⋊C8—1062no—
463#103—1062no—
464#103—1062no—
465#103—1062no—
466#103—1062no—
467#103—1062no—
468#103—1062no—
469#103—1062no—
470#103—1062no—
471C22⋊C8—1062no—
472C22⋊C8—1062no—
473#97—1062no—
474C4×C8—1062no—
475#95—21no—
476#95—1061no—
477#95—1061no—
478#95—1061no—
479C24—531no—
480C2×Q8—532no—
481C2×Q8—532no—
482C2×Q8—1061no—
483C2×Q8—1061no—
484C2×Q8—1061no—
485C2×Q8—1061no—
486C2×Q8—1061no—
487C2×Q8—1061no—
488C2×Q8—1061no—
489C2×Q8—1061no—
490C2×Q8—1061no—
491C2×Q8—1061no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—531no—
497C22×C4—531no—
498C22×C4—531no—
499C22×C4—532no—
500C22×C4—532no—
501C22×C4—1064no—
502C22×C4—1061no—
503C22×C4—1061no—
504C22×C4—1061no—
505C22×C4—1061no—
506C22×C4—1061no—
507C22×C4—1061no—
508C22×C4—1061no—
509C22×C4—1061no—
510C22×C4—4241no—
511C22×C4—4241no—
512C22×C4D4×D5311yescharacteristic
513C22×C4D4×D5311yescharacteristic
514C22×C4C23×D5311yescharacteristic
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—1061no—
522C2×C8—1061no—
523C2×C8—1061no—
524C2×C8—1061no—
525C2×C8—1061no—
526C2×C8—1061no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—534no—
534C4⋊C4—534no—
535C4⋊C4—532no—
536C4⋊C4—532no—
537C4⋊C4—1064no—
538C4⋊C4—1064no—
539C4⋊C4—1064no—
540C4⋊C4—1064no—
541C4⋊C4—1064no—
542C4⋊C4—1064no—
543C4⋊C4—1064no—
544C4⋊C4—1061no—
545C4⋊C4—1061no—
546C4⋊C4C2×C4×D5312yes—
547C4⋊C4C2×C4×D5312yes—
548C4⋊C4#1171994912yes—
549C4⋊C4#1171994912yes—
550C22⋊C4—534no—
551C22⋊C4—534no—
552C22⋊C4—1064no—
553C22⋊C4—1064no—
554C22⋊C4—1064no—
555C42—81no—
556C42—532no—
557C42—532no—
558C42—1064no—
559C42—1064no—
560C42—1064no—
561C42—4241no—
562C42—4241no—
563C42—4241no—
564C42C2×C4×D5312yes—
565C42D4×D5312yes—
566C23—532no—
567C23—1061no—
568C23—1061no—
569C23—1061no—
570C23—1061no—
571C23—1061no—
572C23—1061no—
573C23C2×D4×D5311yescharacteristic
574Q8—1062no—
575Q8—1062no—
576Q8—1062no—
577Q8—1062no—
578Q8—2121no—
579Q8—2121no—
580Q8—2121no—
581Q8—2121no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—532no—
599C2×C4—532no—
600C2×C4—532no—
601C2×C4—1061no—
602C2×C4—1061no—
603C2×C4—1061no—
604C2×C4—1061no—
605C2×C4—1061no—
606C2×C4—1061no—
607C2×C4—1062no—
608C2×C4—1062no—
609C2×C4—1062no—
610C2×C4—1062no—
611C2×C4—1062no—
612C2×C4—1062no—
613C2×C4—1062no—
614C2×C4—1062no—
615C2×C4—4244no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—2121no—
619C2×C4—2121no—
620C2×C4—2121no—
621C2×C4—2121no—
622C2×C4—2121no—
623C2×C4—2121no—
624C2×C4C22⋊C4×D5312yes—
625C2×C4C22⋊C4×D5312yes—
626C2×C4C22×C4×D5311yescharacteristic
627C2×C4C2×D4×D5311yescharacteristic
628C2×C4#41×D5311yescharacteristic
629C2×C4#41×D5311yescharacteristic
630C2×C4C2×#1171994911yescharacteristic
631C2×C4C2×#1171994911yescharacteristic
632C8—41no—
633C8—41no—
634C8—2121no—
635C8—2121no—
636C22—1062no—
637C22—1062no—
638C22—1062no—
639C22—2121no—
640C22—2121no—
641C22—2121no—
642C22—2121no—
643C22C2×C22⋊C4×D5311yescharacteristic
644C22#1681113611yescharacteristic
645C22#1681113611yescharacteristic
646C22#1681143011yescharacteristic
647C22#1681143311yescharacteristic
648C22#1681143711yescharacteristic
649C22#1681143911yescharacteristic
650C4—22no—
651C4—22no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—1062no—
657C4—1062no—
658C4—1062no—
659C4—2122no—
660C4—2122no—
661C4—2122no—
662C4—2122no—
663C4C4×#1171994911yescharacteristic
664C4C4×#1171994911yescharacteristic
665C4#1681117211yescharacteristic
666C4#1681117411yescharacteristic
667C2—2122no—
668C2#2027720411yescharacteristic
669C2#2028324411yescharacteristic
670C2#2028324611yescharacteristic
671C2#2028493411yescharacteristic
672C2#2028493611yescharacteristic
673C2#2028496911yescharacteristic
674C2#2028501711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1989C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1989D53
hypocenterC53#21398
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
4. #20275673C2semidirect
5. #20275638C2semidirect
6. #20275637C2semidirect
11. #20274544C2semidirect
12. #20274544C2semidirect
13. #20274534C2semidirect
14. #20274534C2semidirect
15. C53×#1989C2semidirect
17. #16811236C4semidirect
69. #13133217C2×C4semidirect
80. #13133182C2×C4semidirect
81. #13133182C2×C4semidirect
130. #125×C53C2×C4semidirect
334. #1989D53semidirect
399. C53#21398semidirect
415. #125C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168112362
C2×#168106832
#202770692
#202756732
#202756382
#202756372
#202750262
#202750212
#202745442
#202745342
C53×#19892
#2139853