Finite Group Databasebeta

← #27447956

Subgroups — #27447956

Order 13568.

784 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
1#20278417C211yesmaximal, characteristic
2#20277740C211yesmaximal, characteristic
3#20276279C211yesmaximal, characteristic, complemented
4#20276213C211yesmaximal, characteristic, complemented
5#20276095C211yesmaximal, characteristic, complemented
6#20275857C211yesmaximal, characteristic, complemented
7#20275666C211yesmaximal, characteristic, complemented
8#20275287C211yesmaximal, characteristic, complemented
9#20275148C211yesmaximal, characteristic
10#20275020C211yesmaximal, characteristic
11#20274972C211yesmaximal, characteristic, complemented
12#20274916C211yesmaximal, characteristic
13#20274721C211yesmaximal, characteristic, complemented
14#20274435C211yesmaximal, characteristic
15C53×#1851C211yesmaximal, characteristic
16C2×#13133217C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133194—21no—
19C2×C4×Dic106C2211yescharacteristic
20C2×#13133176C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28#16810555—21no—
29#16810544—21no—
30#16810529C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810517—21no—
35#16810511—21no—
36#16810505—21no—
37#16810498—21no—
38#16810493C2211yescharacteristic
39#16810483C2211yescharacteristic
40C4×C4⋊Dic53C2211yescharacteristic
41#16810474—21no—
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C22×M16×C53C2211yescharacteristic
53C2×C4⋊C8×C53C2211yescharacteristic
54C2×C4⋊C8×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C4×C22⋊C4×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106—21no—
60C22×Dic106C2311yescharacteristic
61C23×Dic53—21no—
62#13133217—22no—
63#13133217—41no—
64#13133196—22no—
65#13133196Q814yescomplemented
66#13133194—44no—
67C4×Dic106—22no—
68C4×Dic106—41no—
69#13133176—22no—
70#13133176—41no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—22no—
76C2×C4⋊Dic53—41no—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—21no—
89C2×C22⋊Dic53—21no—
90C2×C22⋊Dic53—21no—
91C2×C22⋊Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53—21no—
95#13133151—22no—
96#13133149—41no—
97#13133142—22no—
98#13133142—41no—
99#13133142—41no—
100#13133142—41no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108C23×C4×C53C2311yescharacteristic
109C2×M16×C53—22no—
110C2×M16×C53—41no—
111C2×M16×C53—41no—
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C4⋊C8×C53—41no—
121C4⋊C8×C53—41no—
122C22⋊C8×C53—22no—
123#95×C53D411yescharacteristic
124#95×C53D411yescharacteristic
125C24×C53C2×D411yescharacteristic
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×Q812yes—
136C22×C4×C53#4112yes—
137C22×C4×C53C2411yescharacteristic
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—42no—
147C22×Dic53—42no—
148C22×Dic53—81no—
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×Dic106C2×Q812yes—
158M16×C53—44no—
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—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—22no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—21no—
176C42×C53—21no—
177C22⋊Dic53—24no—
178C22⋊Dic53—42no—
179C22⋊Dic53—42no—
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—81no—
195C4⋊Dic53C2×Q812yes—
196C4⋊Dic53#4112yes—
197C4⋊Dic53#4112yes—
198#11719919—24no—
199#11719919—42no—
200#11719919—42no—
201#11719919—42no—
202#11719919—42no—
203C4×Dic53—42no—
204C4×Dic53—42no—
205C4×Dic53—42no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53C22⋊Q812yes—
213C23×C53C22×D411yescharacteristic
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—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53C22⋊Q812yes—
231C2×C4×C53AGL(1,ℤ/8)12yes—
232C2×C4×C53#13712yes—
233C2×C4×C53C22×Q811yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53H(5,𝔽2)11yescharacteristic
238C2×C4×C53H(5,𝔽2)11yescharacteristic
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—22no—
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—41no—
262C2×Dic53—41no—
263C2×Dic53—84no—
264Dic106—22no—
265Dic106—22no—
266Dic106—41no—
267Dic106—41no—
268Dic106—84no—
269C8×C53—41no—
270C8×C53—41no—
271C8×C53—41no—
272C8×C53—41no—
273#26429—531nomaximal, Sylow
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53C2×C22⋊Q811yescharacteristic
282C22×C53#54611yescharacteristic
283C22×C53#54611yescharacteristic
284C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C53C2×#13711yescharacteristic
286C22×C53#57711yescharacteristic
287C22×C53#57811yescharacteristic
288C4×C53—22no—
289C4×C53—22no—
290C4×C53—22no—
291C4×C53—41no—
292C4×C53—41no—
293C4×C53#52911yescharacteristic
294C4×C53#53811yescharacteristic
295C4×C53#55411yescharacteristic
296C4×C53#55411yescharacteristic
297Dic53—42no—
298Dic53—42no—
299Dic53—42no—
300Dic53—42no—
301Dic53—81no—
302Dic53—81no—
303Dic53—81no—
304Dic53—81no—
305#2440—531no—
306#2362—531no—
307#2100—531no—
308#2088—531no—
309#2066—531no—
310#2024—531no—
311#1988—531no—
312#1918—531no—
313#1894—531no—
314#1872—531no—
315#1863—531no—
316#1854—531no—
317#1851D5311yescharacteristic
318#1803—531no—
319#1736—531no—
320C2×C53—42no—
321C2×C53#245011yescharacteristic
322C2×C53#307911yescharacteristic
323C2×C53#308211yescharacteristic
324C2×C53#331011yescharacteristic
325C2×C53#331611yescharacteristic
326C2×C53#332811yescharacteristic
327C2×C53#333011yescharacteristic
328C22×M16C2×D5311yescharacteristic
329C2×C4⋊Q8—531no—
330C2×C22⋊Q8—531no—
331C2×C22⋊Q8—1061no—
332C2×C4×Q8—531no—
333C2×#117—531no—
334C22×C4⋊C4—531no—
335C2×#107—531no—
336C2×#106—531no—
337C2×C4⋊C8C2×D5311yescharacteristic
338C2×C4⋊C8C2×D5311yescharacteristic
339C2×#103—531no—
340C2×#103—531no—
341C2×#103—531no—
342C2×#103—531no—
343C2×C22⋊C8C2×D5311yescharacteristic
344#398—1061no—
345#395—1061no—
346#391—531no—
347#389—531no—
348#389—531no—
349#389—531no—
350#388—1061no—
351#388—1061no—
352#387—1061no—
353#385—1061no—
354#384—531no—
355#380—531no—
356C4×C4⋊C4—531no—
357C4×C4⋊C4—1061no—
358C4×C22⋊C4C2×D5311yescharacteristic
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#340—531no—
369#336C2×D5311yescharacteristic
370#336C2×D5311yescharacteristic
371C53#2642911yescharacteristic, complemented, Sylow
372C22×Q8—531no—
373C22×Q8—1061no—
374C23×C4—1061no—
375C23×C4C22×D5311yescharacteristic
376C2×M16—22no—
377C2×M16—41no—
378C2×M16—41no—
379C22×C8C22×D5311yescharacteristic
380C22×C8C22×D5311yescharacteristic
381C22×C8C22×D5311yescharacteristic
382C22×C8C22×D5311yescharacteristic
383C4⋊Q8—1062no—
384C4⋊Q8—2121no—
385C22⋊Q8—534no—
386C22⋊Q8—1062no—
387C22⋊Q8—2124no—
388C4×Q8—1062no—
389C4×Q8—2121no—
390#117—1062no—
391#117—2121no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—1062no—
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—2121no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4D21211yescharacteristic
414C2×C22⋊C4D21211yescharacteristic
415C2×C42—1061no—
416C2×C42—1061no—
417C2×C42—1061no—
418C2×C42C22×D5311yescharacteristic
419C2×C42C22×D5311yescharacteristic
420#107—1062no—
421#106—2121no—
422C4⋊C8—41no—
423C4⋊C8—41no—
424#103—1062no—
425#103—2121no—
426#103—2121no—
427#103—2121no—
428C22⋊C8—22no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95D21211yescharacteristic
437#95D21211yescharacteristic
438C24C2×D21211yescharacteristic
439C2×Q8—532no—
440C2×Q8—1061no—
441C2×Q8—1061no—
442C2×Q8—1061no—
443C2×Q8—1061no—
444C2×Q8—1061no—
445C2×Q8—4241no—
446C2×Q8—2122no—
447C2×Q8—2122no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—4241no—
461C22×C4—2122no—
462C22×C4—2122no—
463C22×C4C2×D21211yescharacteristic
464C22×C4C2×D21211yescharacteristic
465C22×C4C2×D21211yescharacteristic
466C22×C4C2×D21211yescharacteristic
467C22×C4C2×D21211yescharacteristic
468C22×C4#1171994712yes—
469C22×C4Q8×D5312yes—
470C22×C4C23×D5311yescharacteristic
471M16—44no—
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—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C4⋊C4—532no—
485C4⋊C4—532no—
486C4⋊C4—532no—
487C4⋊C4—1064no—
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—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—4241no—
500C4⋊C4—2122no—
501C4⋊C4—2122no—
502C4⋊C4—2122no—
503C4⋊C4—2122no—
504C4⋊C4—2122no—
505C4⋊C4—2122no—
506C4⋊C4—2122no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—1064no—
510C22⋊C4—2122no—
511C22⋊C4—2122no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—2122no—
517C42—2122no—
518C42—2122no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23#1313319712yes—
526C23C22×D21211yescharacteristic
527Q8—1062no—
528Q8—1062no—
529Q8—4244no—
530Q8—2121no—
531Q8—2121no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—1062no—
549C2×C4—1062no—
550C2×C4—1062no—
551C2×C4—1062no—
552C2×C4—4244no—
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—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4#1313319712yes—
574C2×C4#1313324512yes—
575C2×C4#1313325012yes—
576C2×C4C2×Q8×D5311yescharacteristic
577C2×C4#41×D5311yescharacteristic
578C2×C4#41×D5311yescharacteristic
579C2×C4C2×#1171994711yescharacteristic
580C2×C4#1313326211yescharacteristic
581C2×C4#1313326211yescharacteristic
582C8—41no—
583C8—41no—
584C8—41no—
585C8—41no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#1313319711yescharacteristic
594C22#1681126811yescharacteristic
595C22#1681126811yescharacteristic
596C22C2×#1313324511yescharacteristic
597C22C2×#1313325011yescharacteristic
598C22#1681143511yescharacteristic
599C22#1681144011yescharacteristic
600C4—22no—
601C4—22no—
602C4—22no—
603C4—41no—
604C4—41no—
605C4—4241no—
606C4—4241no—
607C4—4241no—
608C4—4241no—
609C4—2122no—
610C4—2122no—
611C4—2122no—
612C4—2122no—
613C4#1681115011yescharacteristic
614C4#1681120811yescharacteristic
615C4#1681132711yescharacteristic
616C4#1681132711yescharacteristic
617C2—42no—
618C2#2027853711yescharacteristic
619C2#2028453111yescharacteristic
620C2#2028455811yescharacteristic
621C2#2028673811yescharacteristic
622C2#2028682811yescharacteristic
623C2#2028699911yescharacteristic
624C2#2028702111yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1851C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1851D53
hypocenterC53#26429
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
3. #20276279C2semidirect
4. #20276213C2semidirect
5. #20276095C2semidirect
6. #20275857C2semidirect
7. #20275666C2semidirect
8. #20275287C2semidirect
11. #20274972C2semidirect
13. #20274721C2semidirect
65. #13133196Q8semidirect
371. C53#26429semidirect

Maximal subgroups

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

subgroupindex
#202784172
#202777402
#202762792
#202762132
#202760952
#202758572
#202756662
#202752872
#202751482
#202750202
#202749722
#202749162
#202747212
#202744352
C53×#18512
#2642953