Finite Group Databasebeta

← #27466322

Subgroups — #27466322

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279193C211yesmaximal, characteristic
2#20278417C211yesmaximal, characteristic
3#20276279C211yesmaximal, characteristic, complemented
4#20276279C211yesmaximal, characteristic, complemented
5#20276125C211yesmaximal, characteristic, complemented
6#20275857C211yesmaximal, characteristic, complemented
7#20275676C211yesmaximal, characteristic, complemented
8#20275391C211yesmaximal, characteristic, complemented
9#20275391C211yesmaximal, characteristic, complemented
10#20275138C211yesmaximal, characteristic
11#20275020C211yesmaximal, characteristic
12#20274843C211yesmaximal, characteristic, complemented
13#20274806C211yesmaximal, characteristic
14C2×#16810386C211yesmaximal, characteristic
15C8×C22⋊C4×C53C211yesmaximal, characteristic
16C2×#13133217C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133194—21no—
19C22×C4⋊Dic53C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21C2×#13133142C2211yescharacteristic
22C2×#13133142C2211yescharacteristic
23C2×#13133142C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25#16810555—21no—
26#16810555—21no—
27#16810555—21no—
28#16810544—21no—
29#16810529C2211yescharacteristic
30#16810529C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810498—21no—
34#16810498—21no—
35#16810498—21no—
36#16810493C2211yescharacteristic
37#16810493C2211yescharacteristic
38#16810493C2211yescharacteristic
39#16810483C2211yescharacteristic
40#16810386C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C23×C8×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C2×C4×C8×C53C2211yescharacteristic
55C2×C4×C8×C53C2211yescharacteristic
56C4×C22⋊C4×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106—21no—
60C22×Dic106C2311yescharacteristic
61C23×Dic53—21no—
62C23×Dic53—21no—
63#13133217—22no—
64#13133217—41no—
65#13133196—22no—
66#13133196—22no—
67#13133194—44no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—22no—
78C2×C4⋊Dic53—22no—
79C2×C4⋊Dic53—22no—
80C2×C4⋊Dic53—41no—
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×C4⋊Dic53C2311yescharacteristic
87C2×C4⋊Dic53C2311yescharacteristic
88C2×#11719919—21no—
89C2×#11719919—21no—
90C2×#11719919—21no—
91C2×C22⋊Dic53—21no—
92C2×C22⋊Dic53—21no—
93#13133142—22no—
94#13133142—22no—
95#13133142—41no—
96#13133142—41no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104C23×C4×C53C2311yescharacteristic
105C22×C8×C53—22no—
106C22×C8×C53—22no—
107C22×C8×C53—41no—
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C2×C22⋊C4×C53D411yescharacteristic
113C2×C22⋊C4×C53D411yescharacteristic
114C2×C42×C53C2311yescharacteristic
115C2×C42×C53C2311yescharacteristic
116C22⋊C8×C53—22no—
117C4×C8×C53—22no—
118C4×C8×C53—41no—
119#95×C53D411yescharacteristic
120#95×C53D411yescharacteristic
121C24×C53C2×D411yescharacteristic
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53#4112yes—
132C22×C4×C53#4112yes—
133C22×C4×C53C2411yescharacteristic
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—42no—
143C22×Dic53—42no—
144C22×Dic53—42no—
145C22×Dic53—81no—
146C22×Dic53—81no—
147C22×Dic53—81no—
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—42no—
155C2×Dic106—42no—
156C2×Dic106—81no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—44no—
169C2×C8×C53#4112yes—
170C22⋊C4×C53SD1614yes—
171C22⋊C4×C53Q1614yes—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—21no—
176C22⋊Dic53—42no—
177C22⋊Dic53—42no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—42no—
192C4⋊Dic53—42no—
193C4⋊Dic53—42no—
194C4⋊Dic53—42no—
195C4⋊Dic53—81no—
196C4⋊Dic53—81no—
197C4⋊Dic53—81no—
198C4⋊Dic53—81no—
199C4⋊Dic53—81no—
200C4⋊Dic53#4112yes—
201#11719919—42no—
202#11719919—42no—
203#11719919—42no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53#12412yes—
209C23×C53C2×SD1612yes—
210C23×C53C2×Q1612yes—
211C23×C53C22×D411yescharacteristic
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53#12412yes—
227C2×C4×C53C2×SD1612yes—
228C2×C4×C53C2×SD1612yes—
229C2×C4×C53C2×Q1612yes—
230C2×C4×C53C2×Q1612yes—
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53H(5,𝔽2)11yescharacteristic
236C2×C4×C532−1+411yescharacteristic
237C2×Dic53—22no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—84no—
262C2×Dic53—84no—
263Dic106—41no—
264Dic106—41no—
265Dic106—41no—
266Dic106—41no—
267Dic106—84no—
268C8×C53—22no—
269C8×C53—41no—
270C8×C53—41no—
271C8×C53—41no—
272#27697—531nomaximal, Sylow
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—22no—
279C22×C53#48714yes—
280C22×C53C2×#12411yescharacteristic
281C22×C53#54611yescharacteristic
282C22×C53#54711yescharacteristic
283C22×C53C22×SD1611yescharacteristic
284C22×C53C22×Q1611yescharacteristic
285C22×C53#57611yescharacteristic
286C22×C53#57711yescharacteristic
287C4×C53—22no—
288C4×C53—22no—
289C4×C53—22no—
290C4×C53—22no—
291C4×C53—22no—
292C4×C53#52911yescharacteristic
293C4×C53#53911yescharacteristic
294C4×C53#55111yescharacteristic
295C4×C53#55211yescharacteristic
296Dic53—42no—
297Dic53—81no—
298Dic53—81no—
299Dic53—81no—
300Dic53—81no—
301Dic53—81no—
302Dic53—81no—
303Dic53—81no—
304#2502—531no—
305#2440—531no—
306#2100—531no—
307#2100—531no—
308#2072—531no—
309#2024—531no—
310#1990—531no—
311#1933—531no—
312#1933—531no—
313#1892—531no—
314#1872—531no—
315#1834—531no—
316#1824—531no—
317C8×C22⋊C4D5311yescharacteristic
318C2×#340—531no—
319C2×C53—24no—
320C2×C53#248811yescharacteristic
321C2×C53C2×#48711yescharacteristic
322C2×C53#313611yescharacteristic
323C2×C53#329511yescharacteristic
324C2×C53#329911yescharacteristic
325C2×C53#333511yescharacteristic
326C2×C53#333811yescharacteristic
327C23×C8C2×D5311yescharacteristic
328C2×C4⋊Q8—531no—
329C2×C22⋊Q8—531no—
330C2×C22⋊Q8—1061no—
331C22×C4⋊C4—531no—
332C22×C4⋊C4—531no—
333C2×#103—531no—
334C2×#103—531no—
335C2×#103—531no—
336C2×#103—531no—
337C2×C22⋊C8C2×D5311yescharacteristic
338C2×C4×C8C2×D5311yescharacteristic
339C2×C4×C8C2×D5311yescharacteristic
340#398—1061no—
341#398—1061no—
342#398—1061no—
343#395—1061no—
344#391—531no—
345#391—531no—
346#389—531no—
347#389—531no—
348#385—1061no—
349#385—1061no—
350#385—1061no—
351#384—531no—
352#384—531no—
353#384—531no—
354#380—531no—
355C4×C22⋊C4C2×D5311yescharacteristic
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#336C2×D5311yescharacteristic
369#336C2×D5311yescharacteristic
370C53#2769711yescharacteristic, complemented, Sylow
371C22×Q8—531no—
372C22×Q8—1061no—
373C23×C4—1061no—
374C23×C4—1061no—
375C23×C4C22×D5311yescharacteristic
376C22×C8—22no—
377C22×C8—22no—
378C22×C8—41no—
379C22×C8C22×D5311yescharacteristic
380C22×C8C22×D5311yescharacteristic
381C22×C8C22×D5311yescharacteristic
382C22×C8C22×D5311yescharacteristic
383C4⋊Q8—1062no—
384C4⋊Q8—2121no—
385C22⋊Q8—1062no—
386C22⋊Q8—1062no—
387C22⋊Q8—2124no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—1062no—
396C2×C4⋊C4—1062no—
397C2×C4⋊C4—1062no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—2121no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4D21211yescharacteristic
414C2×C22⋊C4D21211yescharacteristic
415C2×C42C22×D5311yescharacteristic
416C2×C42C22×D5311yescharacteristic
417#103—1062no—
418#103—1062no—
419#103—2121no—
420#103—2121no—
421C22⋊C8—22no—
422C4×C8—22no—
423C4×C8—41no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95D21211yescharacteristic
432#95D21211yescharacteristic
433C24C2×D21211yescharacteristic
434C2×Q8—1061no—
435C2×Q8—1061no—
436C2×Q8—1061no—
437C2×Q8—1061no—
438C2×Q8—1061no—
439C2×Q8—1061no—
440C2×Q8—4241no—
441C2×Q8—2122no—
442C2×Q8—2122no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—4241no—
456C22×C4—4241no—
457C22×C4—4241no—
458C22×C4—2122no—
459C22×C4—2122no—
460C22×C4—2122no—
461C22×C4C2×D21211yescharacteristic
462C22×C4C2×D21211yescharacteristic
463C22×C4C2×D21211yescharacteristic
464C22×C4C2×D21211yescharacteristic
465C22×C4C2×D21211yescharacteristic
466C22×C4#1171994512yes—
467C22×C4#1171994512yes—
468C22×C4C23×D5311yescharacteristic
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—44no—
481C2×C8#1171994712yes—
482C4⋊C4—532no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—4241no—
497C4⋊C4—4241no—
498C4⋊C4—4241no—
499C4⋊C4—4241no—
500C4⋊C4—4241no—
501C4⋊C4—2122no—
502C4⋊C4—2122no—
503C4⋊C4—2122no—
504C4⋊C4—2122no—
505C4⋊C4—2122no—
506C4⋊C4—2122no—
507C4⋊C4—2122no—
508C22⋊C4—2122no—
509C22⋊C4—2122no—
510C22⋊C4#1171991314yes—
511C22⋊C4Dic21214yes—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23#1313320512yes—
521C23C2×#1171991312yes—
522C23C2×Dic21212yes—
523C23C22×D21211yescharacteristic
524Q8—4244no—
525Q8—2121no—
526Q8—2121no—
527Q8—2121no—
528Q8—2121no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—1062no—
544C2×C4—4244no—
545C2×C4—4244no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4#1313320512yes—
570C2×C4C2×#1171991312yes—
571C2×C4C2×#1171991312yes—
572C2×C4C2×Dic21212yes—
573C2×C4C2×Dic21212yes—
574C2×C4#41×D5311yescharacteristic
575C2×C4C2×#1171994711yescharacteristic
576C2×C4C2×#1171994511yescharacteristic
577C2×C4C2×#1171994511yescharacteristic
578C2×C4#1313326211yescharacteristic
579C2×C4#1313326411yescharacteristic
580C8—22no—
581C8—41no—
582C8—41no—
583C8—41no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22#1681097414yes—
591C22C2×#1313320511yescharacteristic
592C22#1681126811yescharacteristic
593C22#1681127711yescharacteristic
594C22C22×#1171991311yescharacteristic
595C22C22×Dic21211yescharacteristic
596C22#1681142811yescharacteristic
597C22#1681143511yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—4241no—
604C4—4241no—
605C4—4241no—
606C4—4241no—
607C4—4241no—
608C4—4241no—
609C4—4241no—
610C4—2122no—
611C4#1681115311yescharacteristic
612C4#1681121511yescharacteristic
613C4#1681129911yescharacteristic
614C4#1681130611yescharacteristic
615C2—24no—
616C2#2027903511yescharacteristic
617C2C2×#1681097411yescharacteristic
618C2#2028507111yescharacteristic
619C2#2028654011yescharacteristic
620C2#2028659211yescharacteristic
621C2#2028707311yescharacteristic
622C2#2028710811yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC8×C22⋊C4×C53C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC8×C22⋊C4D53
hypocenterC53#27697
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #20276279C2semidirect
4. #20276279C2semidirect
5. #20276125C2semidirect
6. #20275857C2semidirect
7. #20275676C2semidirect
8. #20275391C2semidirect
9. #20275391C2semidirect
12. #20274843C2semidirect
370. C53#27697semidirect

Maximal subgroups

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

subgroupindex
#202791932
#202784172
#202762792
#202761252
#202758572
#202756762
#202753912
#202751382
#202750202
#202748432
#202748062
C2×#168103862
C8×C22⋊C4×C532
#2769753