Finite Group Databasebeta

← #27446953

Subgroups — #27446953

Order 13568.

812 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810933C211yesmaximal, characteristic, complemented
2#20278422C211yesmaximal, characteristic
3#20277720C211yesmaximal, characteristic
4#20276250C211yesmaximal, characteristic, complemented
5#20276204C211yesmaximal, characteristic, complemented
6#20276091C211yesmaximal, characteristic, complemented
7#20276055C211yesmaximal, characteristic, complemented
8#20275470C211yesmaximal, characteristic, complemented
9#20275295C211yesmaximal, characteristic, complemented
10#20275148C211yesmaximal, characteristic
11#20275027C211yesmaximal, characteristic
12#20274966C211yesmaximal, characteristic, complemented
13#20274903C211yesmaximal, characteristic
14#20274898C211yesmaximal, characteristic
15C2×C53×#428C211yesmaximal, characteristic
16C2×#13133208C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133182C2211yescharacteristic
19#16810933—24no—
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29#16810559—21no—
30#16810546—21no—
31#16810528C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810521—21no—
34#16810519C2211yescharacteristic
35#16810519C2211yescharacteristic
36#16810518—21no—
37#16810507—21no—
38#16810504—21no—
39#16810496—21no—
40#16810492C2211yescharacteristic
41#16810483C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4×C22⋊Dic53—21no—
44C4×C22⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×M16×C53C2211yescharacteristic
54C2×#117×C53C2211yescharacteristic
55C22×C4⋊C4×C53C2211yescharacteristic
56C53×#428—24no—
57C2×#107×C53C2211yescharacteristic
58C2×#107×C53C2211yescharacteristic
59C2×#106×C53C2211yescharacteristic
60C2×#106×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133208—22no—
65#13133208—41no—
66#13133196—22no—
67#13133196Q814yes—
68#13133182—22no—
69#13133182—41no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53D411yescharacteristic
85C2×C22⋊Dic53D411yescharacteristic
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53C2311yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90#13133150—22no—
91#13133150—41no—
92#13133146—22no—
93#13133146—22no—
94#13133142—22no—
95#13133142—22no—
96#13133140—22no—
97#13133140—41no—
98C22⋊#88322—22no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124D411yescharacteristic
107#13133124D411yescharacteristic
108C23×C4×C53—21no—
109C23×C4×C53C2311yescharacteristic
110C2×M16×C53—22no—
111C2×M16×C53—22no—
112C2×M16×C53—22no—
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115#117×C53—22no—
116#117×C53—41no—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53—22no—
119C2×C4⋊C4×C53—41no—
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C22⋊C4×C53—21no—
125C2×C42×C53—21no—
126#107×C53—22no—
127#107×C53—22no—
128#106×C53—22no—
129#106×C53D414yes—
130C24×C53C2×D411yescharacteristic
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—21no—
138C22×C4×C53—21no—
139C22×C4×C53—42no—
140C22×C4×C53—42no—
141C22×C4×C53—81no—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×Q812yes—
144C22×C4×C53#4112yes—
145C22×C4×C53C2411yescharacteristic
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106C2×Q812yes—
160M16×C53—28no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53C2×D412yes—
166C2×C8×C53#4112yes—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—42no—
174C4⋊C4×C53C2×D412yes—
175C4⋊C4×C53C2×D412yes—
176C22⋊C4×C53—42no—
177C42×C53—42no—
178C22⋊Dic53—24no—
179C22⋊Dic53—22no—
180C22⋊Dic53—41no—
181C22⋊Dic53—81no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—42no—
188C4⋊Dic53—42no—
189C4⋊Dic53C2×Q812yes—
190#11719919—24no—
191#11719919—42no—
192#11719919—42no—
193#11719919—42no—
194#11719919—81no—
195#11719919—81no—
196#11719919—81no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—42no—
202C4×Dic53—81no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53C22⋊Q812yes—
216C23×C53C22×D411yescharacteristic
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—84no—
236C2×C4×C53C22⋊Q812yes—
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C22×Q811yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53H(5,𝔽2)11yescharacteristic
242C2×C4×C532−1+411yescharacteristic
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—41no—
257C2×Dic53—41no—
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#13712yes—
267Dic106—22no—
268Dic106—22no—
269Dic106—41no—
270Dic106—41no—
271C8×C53—22no—
272C8×C53—22no—
273#88322—41no—
274#88322—41no—
275#26362—531nomaximal, Sylow
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53C2×C22⋊Q811yescharacteristic
284C22×C53D4211yescharacteristic
285C22×C53#54711yescharacteristic
286C22×C53C2×#13711yescharacteristic
287C22×C53#57511yescharacteristic
288C22×C53#57611yescharacteristic
289C22×C53#57811yescharacteristic
290C4×C53—22no—
291C4×C53—42no—
292C4×C53—42no—
293C4×C53—81no—
294C4×C53—81no—
295C4×C53#52511yescharacteristic
296C4×C53#53911yescharacteristic
297C4×C53D4×Q811yescharacteristic
298C4×C53#55411yescharacteristic
299Dic53—22no—
300Dic53—42no—
301Dic53—41no—
302Dic53—42no—
303Dic53—41no—
304Dic53—41no—
305Dic53—81no—
306Dic53—81no—
307C2×#479—531no—
308C2×#428D5311yescharacteristic
309#2441—531no—
310#2360—531no—
311#2095—531no—
312#2086—531no—
313#2066—531no—
314#2059—531no—
315#1947—531no—
316#1919—531no—
317#1894—531no—
318#1873—531no—
319#1863—531no—
320#1852—531no—
321#1851—531no—
322C2×C53—42no—
323C2×C53#244911yescharacteristic
324C2×C53#307811yescharacteristic
325C2×C53#308311yescharacteristic
326C2×C53#326811yescharacteristic
327C2×C53#327611yescharacteristic
328C2×C53#334611yescharacteristic
329C2×C53#335211yescharacteristic
330C22×M16C2×D5311yescharacteristic
331C2×#125—531no—
332C2×C22⋊Q8—531no—
333C2×C4×Q8—531no—
334C2×#117C2×D5311yescharacteristic
335C22×C4⋊C4C2×D5311yescharacteristic
336#479—1064no—
337#428—24no—
338C2×#107—531no—
339C2×#107—531no—
340C2×#107C2×D5311yescharacteristic
341C2×#107C2×D5311yescharacteristic
342C2×#106C2×D5311yescharacteristic
343C2×#106C2×D5311yescharacteristic
344C2×C4⋊C8—531no—
345C2×C4⋊C8—531no—
346C2×#103—531no—
347C2×#103—531no—
348C2×#103—531no—
349C2×#103—531no—
350C2×C22⋊C8—531no—
351#399—1061no—
352#396—1061no—
353#391—531no—
354#389—531no—
355#389—531no—
356#389—531no—
357#389—1061no—
358#389—1061no—
359#387—1061no—
360#387—1061no—
361#385—1061no—
362#384—531no—
363#380—531no—
364C4×C4⋊C4—531no—
365C4×C22⋊C4—531no—
366C4×C22⋊C4—1061no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#336—531no—
374#336—531no—
375C53#2636211yescharacteristic, complemented, Sylow
376C22×Q8—531no—
377C23×C4—21no—
378C23×C4C22×D5311yescharacteristic
379C2×M16—22no—
380C2×M16—22no—
381C2×M16—22no—
382C22×C8—531no—
383C22×C8—531no—
384C22×C8C22×D5311yescharacteristic
385C22×C8C22×D5311yescharacteristic
386#125—1062no—
387#125—2121no—
388C22⋊Q8—534no—
389C22⋊Q8—1062no—
390C4×Q8—1062no—
391C4×Q8—2121no—
392#117—22no—
393#117—41no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—41no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
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⋊C4C22×D5311yescharacteristic
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C4⋊C4C22×D5311yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—531no—
415C2×C22⋊C4—531no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1061no—
418C2×C42—21no—
419C2×C42—531no—
420C2×C42—531no—
421C2×C42—1061no—
422C2×C42—1061no—
423#107—22no—
424#107—22no—
425#107—1062no—
426#107—2121no—
427#106—22no—
428#106C22⋊D5314yes—
429C4⋊C8—1062no—
430C4⋊C8—1062no—
431#103—1062no—
432#103—1062no—
433#103—1062no—
434#103—2121no—
435C22⋊C8—1062no—
436#95—531no—
437#95—531no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445C24D4×D5311yescharacteristic
446C2×Q8—532no—
447C2×Q8—1061no—
448C2×Q8—1061no—
449C2×Q8—1061no—
450C2×Q8—1061no—
451C2×Q8—1061no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—81no—
463C22×C4—531no—
464C22×C4—531no—
465C22×C4—531no—
466C22×C4—531no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4#1171994512yes—
472C22×C4D4×D5311yescharacteristic
473C22×C4Q8×D5312yes—
474C22×C4C23×D5311yescharacteristic
475M16—28no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8#1171994512yes—
487C2×C8C2×C22⋊D5312yes—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—42no—
495C4⋊C4—532no—
496C4⋊C4—1064no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—4241no—
503C4⋊C4—4241no—
504C4⋊C4—4241no—
505C4⋊C4—2122no—
506C4⋊C4—2122no—
507C4⋊C4—2122no—
508C4⋊C4—2122no—
509C4⋊C4—2122no—
510C4⋊C4C2×C22⋊D5312yes—
511C4⋊C4C2×C22⋊D5312yes—
512C22⋊C4—42no—
513C22⋊C4—1062no—
514C22⋊C4—1064no—
515C22⋊C4—4241no—
516C22⋊C4—2121no—
517C42—42no—
518C42—1061no—
519C42—1061no—
520C42—1061no—
521C42—1061no—
522C42—4241no—
523C42—2122no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#1313319512yes—
531C23C2×D4×D5311yescharacteristic
532Q8—1062no—
533Q8—1062no—
534Q8—2121no—
535Q8—2121no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—84no—
555C2×C4—532no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1062no—
560C2×C4—1062no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—2121no—
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#1313319512yes—
580C2×C4C22×C22⋊D5311yescharacteristic
581C2×C4C2×Q8×D5311yescharacteristic
582C2×C4C2×#1171994511yescharacteristic
583C2×C4C2×#1171994511yescharacteristic
584C2×C4#1313326211yescharacteristic
585C2×C4#1313326511yescharacteristic
586C8—22no—
587C8—22no—
588C8—2121no—
589C8—2121no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22C2×#1313319511yescharacteristic
598C22D4×C22⋊D5311yescharacteristic
599C22#1681127411yescharacteristic
600C22#137×D5311yescharacteristic
601C22#1681141911yescharacteristic
602C22#1681142711yescharacteristic
603C22#1681143911yescharacteristic
604C4—22no—
605C4—42no—
606C4—42no—
607C4—81no—
608C4—81no—
609C4—1062no—
610C4—4241no—
611C4—4241no—
612C4—2121no—
613C4—2121no—
614C4—2121no—
615C4—2122no—
616C4—2122no—
617C4#1681113411yescharacteristic
618C4#1681122011yescharacteristic
619C4Q8×C22⋊D5311yescharacteristic
620C4#1681132811yescharacteristic
621C2—42no—
622C2#2027852411yescharacteristic
623C2#2028451911yescharacteristic
624C2#2028456511yescharacteristic
625C2#2028624911yescharacteristic
626C2#2028635511yescharacteristic
627C2#2028720111yescharacteristic
628C2#2028728611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#428C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC2×#428D53
hypocenterC53#26362
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×#16810933C2semidirect
4. #20276250C2semidirect
5. #20276204C2semidirect
6. #20276091C2semidirect
7. #20276055C2semidirect
8. #20275470C2semidirect
9. #20275295C2semidirect
12. #20274966C2semidirect
375. C53#26362semidirect

Maximal subgroups

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

subgroupindex
C2×#168109332
#202784222
#202777202
#202762502
#202762042
#202760912
#202760552
#202754702
#202752952
#202751482
#202750272
#202749662
#202749032
#202748982
C2×C53×#4282
#2636253