Finite Group Databasebeta

← #27447699

Subgroups — #27447699

Order 13568.

818 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277718C211yesmaximal, characteristic
2#20276272C211yesmaximal, characteristic, complemented
3#20276216C211yesmaximal, characteristic, complemented
4#20276093C211yesmaximal, characteristic, complemented
5#20275853C211yesmaximal, characteristic, complemented
6#20275668C211yesmaximal, characteristic, complemented
7#20275290C211yesmaximal, characteristic, complemented
8#20275136C211yesmaximal, characteristic
9#20275022C211yesmaximal, characteristic
10#20274982C211yesmaximal, characteristic, complemented
11#20274907C211yesmaximal, characteristic
12#20274895C211yesmaximal, characteristic
13#20274717C211yesmaximal, characteristic, complemented
14C2×#16810385C211yesmaximal, characteristic
15C53×#2440C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133140C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29#16810529C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810520—21no—
32#16810510—21no—
33#16810495—21no—
34C4×C4⋊Dic53C2211yescharacteristic
35C22⋊C4×Dic53—21no—
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C4⋊Q8×C53C2211yescharacteristic
49C2×C22⋊Q8×C53—21no—
50C22×C4⋊C4×C53C2211yescharacteristic
51C53×#398—21no—
52C53×#395—21no—
53C53×#389C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#385—21no—
56C53×#384C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C23×Dic53—21no—
61C22×#88322—22no—
62C22×#88322—41no—
63C22×#88322—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133196—22no—
69#13133196Q814yescomplemented
70C4×Dic106—22no—
71C4×Dic106—41no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—22no—
74C2×C4⋊Dic53—41no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×C22⋊Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84#13133148—22no—
85#13133148—41no—
86C4⋊#88322—41no—
87C4⋊#88322—41no—
88#13133140—22no—
89#13133140—41no—
90#13133140—41no—
91#13133140—41no—
92C22⋊#88322—22no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C22×Q8×C53—21no—
98C23×C4×C53—21no—
99C23×C4×C53C2311yescharacteristic
100C4⋊Q8×C53—22no—
101C4⋊Q8×C53—41no—
102C22⋊Q8×C53—44no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—22no—
110C2×C4⋊C4×C53—41no—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120#95×C53—21no—
121#95×C53—21no—
122#95×C53—21no—
123#95×C53D411yescharacteristic
124#95×C53D411yescharacteristic
125C24×C53C2×D411yescharacteristic
126C2×Q8×C53—42no—
127C2×Q8×C53—42no—
128C2×Q8×C53—81no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—42no—
138C22×C4×C53—42no—
139C22×C4×C53—81no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×Q812yes—
146C22×C4×C53#4112yes—
147C22×C4×C53C2411yescharacteristic
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C22×Dic53—81no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106C2×Q812yes—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—42no—
169C4⋊C4×C53—42no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53—42no—
173C4⋊C4×C53—81no—
174C4⋊C4×C53#4112yes—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—42no—
177C22⋊C4×C53SD1614yes—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C22⋊Dic53—24no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—42no—
188C4⋊Dic53C2×Q812yes—
189C4⋊Dic53#4112yes—
190#11719919—24no—
191#11719919—42no—
192#11719919—81no—
193C4×Dic53—42no—
194C4×Dic53—81no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—44no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53C22⋊Q812yes—
213C23×C53C2×SD1612yes—
214C23×C53#13712yes—
215C23×C53C22×D411yescharacteristic
216Q8×C53—84no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—84no—
246C2×C4×C53C22⋊Q812yes—
247C2×C4×C53C2×SD1612yes—
248C2×C4×C53C2×SD1612yes—
249C2×C4×C53C22×Q811yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53H(5,𝔽2)11yescharacteristic
254C2×C4×C53H(5,𝔽2)11yescharacteristic
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—84no—
268Dic106—22no—
269Dic106—22no—
270Dic106—41no—
271Dic106—41no—
272#88322—41no—
273#88322—41no—
274#88322—41no—
275#88322—41no—
276#26412—531nomaximal, Sylow
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53#47514yes—
284C22×C53C2×C22⋊Q811yescharacteristic
285C22×C53#54611yescharacteristic
286C22×C53#54611yescharacteristic
287C22×C53C22×SD1611yescharacteristic
288C22×C53C2×#13711yescharacteristic
289C22×C53#57611yescharacteristic
290C22×C53#57811yescharacteristic
291C4×C53—22no—
292C4×C53—22no—
293C4×C53—22no—
294C4×C53—42no—
295C4×C53—41no—
296C4×C53—41no—
297C4×C53—81no—
298C4×C53—81no—
299C4×C53—81no—
300C4×C53#52911yescharacteristic
301C4×C53#53811yescharacteristic
302C4×C53#55411yescharacteristic
303C4×C53#55411yescharacteristic
304Dic53—42no—
305Dic53—42no—
306Dic53—42no—
307Dic53—81no—
308#2440D5311yescharacteristic
309#2360—531no—
310#2099—531no—
311#2088—531no—
312#2066—531no—
313#2024—531no—
314#1988—531no—
315#1919—531no—
316#1892—531no—
317#1872—531no—
318#1865—531no—
319#1853—531no—
320#1850—531no—
321#1803—531no—
322C2×#340—531no—
323C2×C53—24no—
324C2×C53#245011yescharacteristic
325C2×C53C2×#47511yescharacteristic
326C2×C53#308311yescharacteristic
327C2×C53#329511yescharacteristic
328C2×C53#329811yescharacteristic
329C2×C53#331211yescharacteristic
330C2×C53#331411yescharacteristic
331C23×C8—531no—
332C2×C4⋊Q8C2×D5311yescharacteristic
333C2×C22⋊Q8—21no—
334C2×C22⋊Q8—531no—
335C2×C4×Q8—531no—
336C22×C4⋊C4—531no—
337C22×C4⋊C4C2×D5311yescharacteristic
338C2×#106—531no—
339C2×#106—531no—
340C2×C4⋊C8—531no—
341C2×C4⋊C8—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#103—531no—
345C2×#103—531no—
346C2×C22⋊C8—531no—
347#398—21no—
348#395—21no—
349#391—531no—
350#389—531no—
351#389—1061no—
352#389C2×D5311yescharacteristic
353#389C2×D5311yescharacteristic
354#387—1061no—
355#385—21no—
356#385—1061no—
357#384C2×D5311yescharacteristic
358#380C2×D5311yescharacteristic
359C4×C4⋊C4—531no—
360C4×C22⋊C4—1061no—
361C4×C22⋊C4C2×D5311yescharacteristic
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#336—531no—
373#336—531no—
374C53#2641211yescharacteristic, complemented, Sylow
375C22×Q8—21no—
376C22×Q8—531no—
377C23×C4—21no—
378C23×C4—1061no—
379C23×C4C22×D5311yescharacteristic
380C22×C8—531no—
381C22×C8—531no—
382C22×C8—531no—
383C22×C8—531no—
384C22×C8—1062no—
385C22×C8—2121no—
386C22×C8—2121no—
387C4⋊Q8—22no—
388C4⋊Q8—41no—
389C22⋊Q8—44no—
390C22⋊Q8—534no—
391C22⋊Q8—1062no—
392C4×Q8—1062no—
393C4×Q8—2121no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—41no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—1062no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—2121no—
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C4⋊C4C22×D5311yescharacteristic
413C2×C4⋊C4C22×D5311yescharacteristic
414C2×C4⋊C4C22×D5311yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4C22⋊D5311yescharacteristic
418C2×C22⋊C4C22⋊D5311yescharacteristic
419C2×C42—1061no—
420C2×C42—1061no—
421C2×C42C22×D5311yescharacteristic
422C2×C42C22×D5311yescharacteristic
423#106—1062no—
424#106—2121no—
425C4⋊C8—2121no—
426C4⋊C8—2121no—
427#103—1062no—
428#103—2121no—
429#103—2121no—
430#103—2121no—
431C22⋊C8—1062no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95C22⋊D5311yescharacteristic
440#95C22⋊D5311yescharacteristic
441C24C2×C22⋊D5311yescharacteristic
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—81no—
445C2×Q8—532no—
446C2×Q8—1061no—
447C2×Q8—1061no—
448C2×Q8—1061no—
449C2×Q8—1061no—
450C2×Q8—1061no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—81no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—4241no—
467C22×C4—2122no—
468C22×C4—2122no—
469C22×C4Q8×D5312yes—
470C22×C4#1171994912yes—
471C22×C4C2×C22⋊D5311yescharacteristic
472C22×C4C2×C22⋊D5311yescharacteristic
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C23×D5311yescharacteristic
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—2124no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—81no—
503C4⋊C4—532no—
504C4⋊C4—532no—
505C4⋊C4—1064no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—4241no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C4⋊C4#1171994712yes—
514C22⋊C4—22no—
515C22⋊C4—42no—
516C22⋊C4—1064no—
517C22⋊C4#1171992314yes—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—4241no—
523C42—2122no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#1313319312yes—
529C23C2×#1171992312yes—
530C23#1313324812yes—
531C23C22×C22⋊D5311yescharacteristic
532Q8—84no—
533Q8—1062no—
534Q8—1062no—
535Q8—2121no—
536Q8—2121no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—84no—
566C2×C4—1062no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—4244no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4#1313319312yes—
580C2×C4C2×#1171992312yes—
581C2×C4C2×#1171992312yes—
582C2×C4C2×Q8×D5311yescharacteristic
583C2×C4#41×D5311yescharacteristic
584C2×C4C2×#1171994711yescharacteristic
585C2×C4C2×#1171994911yescharacteristic
586C2×C4#1313326211yescharacteristic
587C2×C4#1313326311yescharacteristic
588C8—2121no—
589C8—2121no—
590C8—2121no—
591C8—2121no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22#1681090514yes—
599C22C2×#1313319311yescharacteristic
600C22#1681126111yescharacteristic
601C22#1681126711yescharacteristic
602C22C22×#1171992311yescharacteristic
603C22C2×#1313324811yescharacteristic
604C22#1681142611yescharacteristic
605C22#1681143811yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—41no—
610C4—41no—
611C4—42no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4—4241no—
616C4—2122no—
617C4—2122no—
618C4—2122no—
619C4#1681115911yescharacteristic
620C4#1681121311yescharacteristic
621C4#1681132711yescharacteristic
622C4#1681133111yescharacteristic
623C2—24no—
624C2#2027853511yescharacteristic
625C2C2×#1681090511yescharacteristic
626C2#2028456211yescharacteristic
627C2#2028653911yescharacteristic
628C2#2028657411yescharacteristic
629C2#2028676611yescharacteristic
630C2#2028679011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2440C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2440D53
hypocenterC53#26412
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
2. #20276272C2semidirect
3. #20276216C2semidirect
4. #20276093C2semidirect
5. #20275853C2semidirect
6. #20275668C2semidirect
7. #20275290C2semidirect
10. #20274982C2semidirect
13. #20274717C2semidirect
69. #13133196Q8semidirect
374. C53#26412semidirect

Maximal subgroups

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

subgroupindex
#202777182
#202762722
#202762162
#202760932
#202758532
#202756682
#202752902
#202751362
#202750222
#202749822
#202749072
#202748952
#202747172
C2×#168103852
C53×#24402
#2641253