Finite Group Databasebeta

← #27306648

Subgroups — #27306648

Order 13568.

1766 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic106C211yesmaximal, characteristic
2C2×#16810527C212yesmaximal
3C2×#16810487C211yesmaximal, characteristic
4C2×#16810474C211yesmaximal, characteristic
5#20272967C212yesmaximal
6#20272966C212yesmaximal
7#20272943C212yesmaximal
8#20272942C211yesmaximal, characteristic
9#20272939C211yesmaximal, characteristic
10#20272939C211yesmaximal, characteristic
11C53×#1434C211yesmaximal, characteristic
12C23×Dic106C2211yescharacteristic
13C2×C4×Dic106—24no—
14C2×C4×Dic106—24no—
15C2×C4×Dic106C418yescomplemented
16C22×C4⋊Dic53C2211yescharacteristic
17C22×#11719919C2212yes—
18C22×#11719919C2212yes—
19C22×#11719919C2211yescharacteristic
20C22×#11719919C2211yescharacteristic
21C22×C4×Dic53C2212yes—
22C22×C4×Dic53C2211yescharacteristic
23C22×C4×Dic53C2211yescharacteristic
24#16810527—28no—
25#16810487—24no—
26#16810474—24no—
27C2×#13133124C2211yescharacteristic
28C2×#13133124C2211yescharacteristic
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C2×#13133124C2212yes—
36C22×C42×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C2×#95×C53C2212yes—
41C22×Dic106—22no—
42C22×Dic106—22no—
43C22×Dic106—22no—
44C22×Dic106—22no—
45C22×Dic106C2×C414yes—
46C22×Dic106D414yes—
47C22×Dic106Q814yes—
48C23×Dic53C2312yes—
49C23×Dic53C2311yescharacteristic
50C23×Dic53C2312yes—
51C23×Dic53C2311yescharacteristic
52C23×Dic53C2311yescharacteristic
53C23×Dic53C2311yescharacteristic
54C4×Dic106—232no—
55C2×C4⋊Dic53—22no—
56C2×C4⋊Dic53—22no—
57C2×C4⋊Dic53C2×C414yes—
58C2×#11719919—24no—
59C2×#11719919—22no—
60C2×#11719919—22no—
61C2×#11719919—22no—
62C2×#11719919—22no—
63C2×#11719919—24no—
64C2×#11719919—24no—
65C2×#11719919—24no—
66C2×#11719919—24no—
67C2×#11719919—24no—
68C2×#11719919C2×C414yes—
69C2×#11719919C2×C414yes—
70C2×C4×Dic53—22no—
71C2×C4×Dic53—22no—
72C2×C4×Dic53—24no—
73C2×C4×Dic53—24no—
74C2×C4×Dic53—24no—
75C2×C4×Dic53C2×C414yes—
76C2×C4×Dic53C2×C414yes—
77C2×C4×Dic53D414yes—
78C2×C4×Dic53D414yes—
79#13133124—24no—
80#13133124—22no—
81#13133124—22no—
82#13133124—24no—
83#13133124—24no—
84#13133124—24no—
85#13133124—24no—
86#13133124—24no—
87#13133124—24no—
88C23×C4×C53C2312yes—
89C23×C4×C53C2311yescharacteristic
90C23×C4×C53C2312yes—
91C23×C4×C53C2311yescharacteristic
92C23×C4×C53C2311yescharacteristic
93C2×C42×C53—22no—
94C2×C42×C53—22no—
95C2×C42×C53C2×C414yes—
96#95×C53—24no—
97#95×C53—22no—
98#95×C53—22no—
99#95×C53—24no—
100C24×C53C2411yescharacteristic
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—21no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53C22×C412yes—
125C22×C4×C53C22×C412yes—
126C22×C4×C53C22×C412yes—
127C22×C4×C53C2×D412yes—
128C22×C4×C53C2×D412yes—
129C22×C4×C53C2×D412yes—
130C22×C4×C53C2×Q812yes—
131C22×C4×C53#4114yes—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53C22×C412yes—
156C22×Dic53C22×C412yes—
157C22×Dic53C22×C412yes—
158C22×Dic53C22×C412yes—
159C22×Dic53C2×D412yes—
160C22×Dic53C2×D412yes—
161C22×Dic53C2×D412yes—
162C22×Dic53C2×D412yes—
163C22×Dic53C2×D412yes—
164C22×Dic53C2×D412yes—
165C22×Dic53C2×Q812yes—
166C22×Dic53C2×Q812yes—
167C22×Dic53#4114yes—
168C22×Dic53#4112yes—
169C22×Dic53#4112yes—
170C2×Dic106—28no—
171C2×Dic106—28no—
172C2×Dic106—28no—
173C2×Dic106—28no—
174C2×Dic106—28no—
175C2×Dic106—28no—
176C2×Dic106C4⋊C4116yescomplemented
177C42×C53—28no—
178C4⋊Dic53—28no—
179#11719919—216no—
180#11719919—28no—
181#11719919—28no—
182#11719919—48no—
183C4×Dic53—28no—
184C4×Dic53—48no—
185C4×Dic53C22⋊C4116yes—
186C23×C53C23×C411yescharacteristic
187C23×C53C22×D411yescharacteristic
188C23×C53C22×D411yescharacteristic
189C23×C53C22×D411yescharacteristic
190C23×C53C22×Q811yescharacteristic
191C23×C53C2×#4111yescharacteristic
192C23×C53C2×#4111yescharacteristic
193C23×C53C2×#4112yes—
194C23×C53C2×#4112yes—
195C23×C532−1+412yes—
196C23×C532−1+412yes—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—24no—
211C2×C4×C53—24no—
212C2×C4×C53—24no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53C2×C22⋊C414yes—
227C2×C4×C53C2×C4⋊C414yes—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—24no—
237C2×Dic53—24no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—24no—
242C2×Dic53—24no—
243C2×Dic53—24no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53C2×C22⋊C414yes—
258C2×Dic53C2×C22⋊C414yes—
259C2×Dic53C2×C4⋊C414yes—
260C2×Dic53C2×C4⋊C414yes—
261C2×Dic53#11714yes—
262C2×Dic53#12114yes—
263C2×Dic53#12114yes—
264C2×Dic53C22⋊Q814yes—
265C2×Dic53C22⋊Q814yes—
266Dic106—232no—
267#13942—531nomaximal, Sylow
268C22×C53C22×C22⋊C411yescharacteristic
269C22×C53C22×C4⋊C411yescharacteristic
270C22×C53C2×#11711yescharacteristic
271C22×C53C4×#4111yescharacteristic
272C22×C53C4×#4111yescharacteristic
273C22×C53#51911yescharacteristic
274C22×C53#51911yescharacteristic
275C22×C53C2×#12111yescharacteristic
276C22×C53C2×#12111yescharacteristic
277C22×C53C2×C22⋊Q811yescharacteristic
278C22×C53C2×C22⋊Q811yescharacteristic
279C22×C53#52512yes—
280C22×C53#53212yes—
281C22×C53#53612yes—
282C22×C53#53612yes—
283C22×C53#54812yes—
284C22×C53#54812yes—
285C22×C53#55212yes—
286C22×C53#55212yes—
287C22×C53#55612yes—
288C22×C53#55612yes—
289C22×C53#55712yes—
290C22×C53#55712yes—
291C4×C53—24no—
292C4×C53—24no—
293C4×C53—44no—
294C4×C53—42no—
295C4×C53—82no—
296Dic53—24no—
297Dic53—24no—
298Dic53—44no—
299Dic53—42no—
300Dic53—82no—
301Dic53#38018yes—
302C22×C4×Q8—531no—
303C2×#391—532no—
304C2×#382—531no—
305C2×C4×C4⋊C4—531no—
306#1441—532no—
307#1441—532no—
308C4⋊#95—532no—
309#1434—531no—
310#1434—531no—
311#1434—531no—
312#1434D5311yescharacteristic
313C2×C53C2×#38011yescharacteristic
314C2×C53#229711yescharacteristic
315C2×C53#231011yescharacteristic
316C2×C53#235111yescharacteristic
317C2×C53#235111yescharacteristic
318C2×C53#235611yescharacteristic
319C2×C53#235611yescharacteristic
320C2×C53#253612yes—
321C2×C53#254512yes—
322C2×C53#256512yes—
323C2×C53#258112yes—
324C23×Q8—531no—
325C2×C4×Q8—538no—
326C2×C4×Q8—1064no—
327C2×C4×Q8—1064no—
328C22×C4⋊C4—531no—
329C22×C4⋊C4—531no—
330C22×C4⋊C4—531no—
331C22×C4⋊C4—532no—
332C22×C4⋊C4—532no—
333C22×C42—531no—
334C22×C42—531no—
335C22×C42—532no—
336C22×C42C2×D5311yescharacteristic
337#391—1068no—
338#382—1064no—
339C4×C4⋊C4—1064no—
340C2×#95—531no—
341C2×#95—531no—
342C2×#95—532no—
343C2×#95—532no—
344C2×#95—532no—
345C2×#95—532no—
346C2×#95—532no—
347C2×#95—532no—
348C2×#95—532no—
349C2×#95C2×D5311yescharacteristic
350C2×#95C2×D5311yescharacteristic
351C2×#95C2×D5312yes—
352C2×#95C2×D5312yes—
353C53#1394211yescharacteristic, complemented, Sylow
354C22×Q8—534no—
355C22×Q8—534no—
356C22×Q8—534no—
357C22×Q8—1062no—
358C22×Q8—1062no—
359C22×Q8—1062no—
360C22×Q8—1062no—
361C23×C4—531no—
362C23×C4—531no—
363C23×C4—531no—
364C23×C4—531no—
365C23×C4—532no—
366C23×C4—532no—
367C23×C4C22×D5312yes—
368C23×C4C22×D5311yescharacteristic
369C23×C4C22×D5312yes—
370C23×C4C22×D5311yescharacteristic
371C23×C4C22×D5311yescharacteristic
372C4×Q8—10632no—
373C2×C4⋊C4—534no—
374C2×C4⋊C4—534no—
375C2×C4⋊C4—534no—
376C2×C4⋊C4—1062no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1062no—
379C2×C4⋊C4—1062no—
380C2×C4⋊C4—1062no—
381C2×C4⋊C4—1062no—
382C2×C4⋊C4—1064no—
383C2×C4⋊C4—1064no—
384C2×C4⋊C4—1064no—
385C2×C4⋊C4—1064no—
386C2×C4⋊C4—1064no—
387C2×C4⋊C4—1064no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—534no—
391C2×C42—534no—
392C2×C42—534no—
393C2×C42—534no—
394C2×C42—1062no—
395C2×C42—1062no—
396C2×C42—1064no—
397C2×C42—1064no—
398C2×C42—1064no—
399C2×C42C4×D5314yes—
400#95—24no—
401#95—22no—
402#95—22no—
403#95—24no—
404#95—1064no—
405#95—1064no—
406#95—1064no—
407#95—1064no—
408#95—1064no—
409#95—1064no—
410#95—1062no—
411#95—1062no—
412#95—1064no—
413C24C23×D5311yescharacteristic
414C2×Q8—5316no—
415C2×Q8—1068no—
416C2×Q8—1068no—
417C2×Q8—1068no—
418C2×Q8—1068no—
419C2×Q8—1068no—
420C2×Q8—1068no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—532no—
445C22×C4—532no—
446C22×C4—532no—
447C22×C4—532no—
448C22×C4—532no—
449C22×C4—532no—
450C22×C4—532no—
451C22×C4—532no—
452C22×C4—532no—
453C22×C4—532no—
454C22×C4—532no—
455C22×C4—532no—
456C22×C4—532no—
457C22×C4—532no—
458C22×C4—534no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1062no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4C2×C4×D5312yes—
483C22×C4C2×C4×D5312yes—
484C22×C4C2×C4×D5312yes—
485C22×C4#1171994514yes—
486C22×C4D4×D5312yes—
487C22×C4D4×D5312yes—
488C22×C4D4×D5312yes—
489C22×C4Q8×D5312yes—
490C4⋊C4—1068no—
491C4⋊C4—1068no—
492C4⋊C4—1068no—
493C4⋊C4—10616no—
494C4⋊C4—2128no—
495C42—28no—
496C42—5316no—
497C42—1068no—
498C42—2128no—
499C23C22×C4×D5311yescharacteristic
500C23C2×D4×D5311yescharacteristic
501C23C2×D4×D5311yescharacteristic
502C23C2×D4×D5311yescharacteristic
503C23C2×Q8×D5311yescharacteristic
504C23#41×D5312yes—
505C23#41×D5311yescharacteristic
506C23#41×D5311yescharacteristic
507C23C2×#1171994512yes—
508C23#1313326412yes—
509C23#1313326412yes—
510Q8—10632no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—534no—
541C2×C4—534no—
542C2×C4—534no—
543C2×C4—534no—
544C2×C4—534no—
545C2×C4—534no—
546C2×C4—534no—
547C2×C4—534no—
548C2×C4—534no—
549C2×C4—1062no—
550C2×C4—1062no—
551C2×C4—1062no—
552C2×C4—1062no—
553C2×C4—1062no—
554C2×C4—1062no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—1064no—
558C2×C4—1064no—
559C2×C4—1064no—
560C2×C4—1064no—
561C2×C4—1062no—
562C2×C4—1062no—
563C2×C4—1062no—
564C2×C4—1064no—
565C2×C4—2122no—
566C2×C4—2122no—
567C2×C4—2122no—
568C2×C4—2122no—
569C2×C4—2122no—
570C2×C4—2122no—
571C2×C4—2122no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4—2122no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4C22⋊C4×D5314yes—
579C2×C4C4⋊C4×D5314yes—
580C22C2×C22⋊C4×D5311yescharacteristic
581C22C2×C4⋊C4×D5311yescharacteristic
582C22#117×D5311yescharacteristic
583C22#1681109111yescharacteristic
584C22C4×#1171994511yescharacteristic
585C22#1681110011yescharacteristic
586C22#1681110011yescharacteristic
587C22#121×D5311yescharacteristic
588C22#121×D5311yescharacteristic
589C22C22⋊Q8×D5311yescharacteristic
590C22C22⋊Q8×D5311yescharacteristic
591C22#1681113212yes—
592C22#1681117012yes—
593C22#1681119512yes—
594C22#1681119512yes—
595C22#1681128112yes—
596C22#1681128212yes—
597C22#1681130612yes—
598C22#1681130912yes—
599C22#1681134012yes—
600C22#1681134112yes—
601C22#1681135012yes—
602C22#1681135112yes—
603C4—24no—
604C4—24no—
605C4—44no—
606C4—42no—
607C4—82no—
608C4—538no—
609C4—1064no—
610C4—1064no—
611C4—4242no—
612C4—2124no—
613C4—2122no—
614C2#380×D5311yescharacteristic
615C2#2027720111yescharacteristic
616C2#2027730011yescharacteristic
617C2#2027763211yescharacteristic
618C2#2027763511yescharacteristic
619C2#2027767511yescharacteristic
620C2#2027768111yescharacteristic
621C2#2027962312yes—
622C2#2027969312yes—
623C2#2027990412yes—
624C2#2028008012yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1434C2
socle, prime radical, cosocleC24×C53C24
hypercenter#1434D53
hypocenterC53#13942
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
15. C2×C4×Dic106C4semidirect
176. C2×Dic106C4⋊C4semidirect
353. C53#13942semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1062
C2×#168105272
C2×#168104872
C2×#168104742
#202729672
#202729662
#202729432
#202729422
#202729392
C53×#14342
#1394253