Finite Group Databasebeta

← #27441616

Subgroups — #27441616

Order 13568.

920 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810928C211yesmaximal, characteristic, complemented
2C2×#16810632C211yesmaximal, characteristic
3#20278429C211yesmaximal, characteristic
4Q8×C22⋊Dic53C211yesmaximal, characteristic
5#20276251C211yesmaximal, characteristic, complemented
6#20276094C211yesmaximal, characteristic, complemented
7#20276092C211yesmaximal, characteristic, complemented
8#20275864C211yesmaximal, characteristic, complemented
9#20274981C211yesmaximal, characteristic, complemented
10#20274971C211yesmaximal, characteristic, complemented
11#20274916C211yesmaximal, characteristic
12#20274897C211yesmaximal, characteristic
13#20274746C211yesmaximal, characteristic
14#20274718C211yesmaximal, characteristic, complemented
15C53×#1787C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×#13133208C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19C2×Q8×Dic53C2211yescharacteristic
20C2×#13133176C2211yescharacteristic
21#16810928—24no—
22C2×#13133151C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133146C2211yescharacteristic
27C2×#13133146C2211yescharacteristic
28#16810632—24no—
29C2×#13133141C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×#13133141C2211yescharacteristic
32C2×#13133141C2211yescharacteristic
33#16810559—21no—
34#16810546—21no—
35#16810530—21no—
36#16810530—21no—
37#16810530C2211yescharacteristic
38#16810530C2211yescharacteristic
39#16810522C2211yescharacteristic
40#16810520—21no—
41#16810519C2211yescharacteristic
42#16810510—21no—
43#16810492C2211yescharacteristic
44#16810483—21no—
45#16810483C2211yescharacteristic
46C4×C22⋊Dic53—21no—
47C4×C22⋊Dic53C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C23×Q8×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×#103×C53C2211yescharacteristic
59C2×C22⋊C8×C53C2211yescharacteristic
60C53×#380C2211yescharacteristic
61C2×#11719917—22no—
62C2×#11719917—22no—
63C2×#11719917—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133208—22no—
67#13133208—41no—
68Q8×Dic53—22no—
69Q8×Dic53—22no—
70Q8×Dic53—41no—
71Q8×Dic53—41no—
72#13133176—22no—
73#13133176—41no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53D411yescharacteristic
89C2×C22⋊Dic53D411yescharacteristic
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53C2311yescharacteristic
94C2×C4×Dic53C2311yescharacteristic
95#13133151—41no—
96#13133150—22no—
97#13133149—22no—
98#13133148—41no—
99#13133146—22no—
100#13133146—41no—
101#13133141—22no—
102#13133141—22no—
103#13133141—22no—
104#13133141D414yes—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124D411yescharacteristic
113#13133124D411yescharacteristic
114C22×Q8×C53—21no—
115C22×Q8×C53—22no—
116C22×Q8×C53—22no—
117C22×Q8×C53—22no—
118C22×Q8×C53—41no—
119C22×Q8×C53—41no—
120C22×Q8×C53Q814yes—
121C22×Q8×C53C2311yescharacteristic
122C22×Q8×C53C2311yescharacteristic
123C23×C4×C53—21no—
124C23×C4×C53C2311yescharacteristic
125C22×C8×C53C2311yescharacteristic
126C22×C8×C53C2311yescharacteristic
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C22⋊C4×C53—21no—
130#103×C53—22no—
131#103×C53—22no—
132#103×C53—41no—
133#103×C53—41no—
134C22⋊C8×C53—22no—
135#95×C53—21no—
136C24×C53C2×D411yescharacteristic
137C2×Q8×C53—24no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—28no—
143C2×Q8×C53—21no—
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53—21no—
147C2×Q8×C53—42no—
148C2×Q8×C53—42no—
149C2×Q8×C53—44no—
150C2×Q8×C53—44no—
151C2×Q8×C53—44no—
152C2×Q8×C53C2×D412yes—
153C2×Q8×C53C2×Q812yes—
154C2×Q8×C53C2×Q812yes—
155C2×Q8×C53#4112yes—
156C22×C4×C53—21no—
157C22×C4×C53—21no—
158C22×C4×C53—21no—
159C22×C4×C53—21no—
160C22×C4×C53—21no—
161C22×C4×C53—24no—
162C22×C4×C53—21no—
163C22×C4×C53—21no—
164C22×C4×C53—21no—
165C22×C4×C53—42no—
166C22×C4×C53—42no—
167C22×C4×C53C2×D411yescharacteristic
168C22×C4×C53C2×Q812yes—
169C22×C4×C53#4112yes—
170C22×C4×C53C2411yescharacteristic
171C22×Dic53—21no—
172C22×Dic53—21no—
173C22×Dic53—21no—
174C22×Dic53—21no—
175C22×Dic53C2×D411yescharacteristic
176C22×Dic53C2×D411yescharacteristic
177C22×Dic53C2×D411yescharacteristic
178C22×Dic53C2×D411yescharacteristic
179C2×C8×C53—21no—
180C2×C8×C53—21no—
181C2×C8×C53—21no—
182C2×C8×C53—21no—
183C2×C8×C53—21no—
184C2×C8×C53—21no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C22⋊C4×C53—81no—
190C22⋊Dic53—22no—
191C22⋊Dic53—41no—
192C22⋊Dic53—42no—
193C22⋊Dic53—42no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—42no—
202C4⋊Dic53—42no—
203C4⋊Dic53—42no—
204C4⋊Dic53—81no—
205C4⋊Dic53C2×D412yes—
206#11719919—42no—
207#11719919—42no—
208#11719919—42no—
209#11719919—81no—
210C4×Dic53—21no—
211C4×Dic53—21no—
212C4×Dic53—21no—
213C4×Dic53—21no—
214C4×Dic53—42no—
215C4×Dic53—42no—
216C4×Dic53—42no—
217#11719917—44no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322—21no—
223C2×#88322C2×D412yes—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—21no—
227C23×C53—21no—
228C23×C53—21no—
229C23×C53—21no—
230C23×C53C22⋊Q812yes—
231C23×C53C22×D411yescharacteristic
232Q8×C53—22no—
233Q8×C53—22no—
234Q8×C53—22no—
235Q8×C53—22no—
236Q8×C53—22no—
237Q8×C53—48no—
238Q8×C53—41no—
239Q8×C53—41no—
240Q8×C53C22⋊Q814yes—
241C2×C4×C53—22no—
242C2×C4×C53—21no—
243C2×C4×C53—22no—
244C2×C4×C53—21no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—22no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
251C2×C4×C53—48no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53C22⋊Q812yes—
261C2×C4×C53C22×D411yescharacteristic
262C2×C4×C53C22×Q811yescharacteristic
263C2×C4×C53C2×#4111yescharacteristic
264C2×C4×C53C2×#4111yescharacteristic
265C2×C4×C53H(5,𝔽2)11yescharacteristic
266C2×C4×C53H(5,𝔽2)11yescharacteristic
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—22no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—21no—
275C2×Dic53—21no—
276C2×Dic53—21no—
277C2×Dic53—21no—
278C2×Dic53—21no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287C2×Dic53—41no—
288C2×Dic53—41no—
289C2×Dic53—41no—
290C2×Dic53#13712yes—
291C8×C53—41no—
292C8×C53—41no—
293#88322—22no—
294#88322—41no—
295#25981—531nomaximal, Sylow
296C22×C53—22no—
297C22×C53—22no—
298C22×C53—22no—
299C22×C53—41no—
300C22×C53—41no—
301C22×C53—41no—
302C22×C53—41no—
303C22×C53C2×C22⋊Q811yescharacteristic
304C22×C53D4211yescharacteristic
305C22×C53#54611yescharacteristic
306C22×C53C2×#13711yescharacteristic
307C22×C53#57511yescharacteristic
308C22×C53#57711yescharacteristic
309C22×C53#57811yescharacteristic
310C4×C53—22no—
311C4×C53—24no—
312C4×C53—42no—
313C4×C53—81no—
314C4×C53—81no—
315C4×C53C2×C22⋊Q811yescharacteristic
316C4×C53#54311yescharacteristic
317C4×C53#54611yescharacteristic
318C4×C53#55111yescharacteristic
319Dic53—22no—
320Dic53—41no—
321Dic53—41no—
322Dic53—42no—
323Dic53—41no—
324Dic53—81no—
325Dic53—81no—
326Dic53—81no—
327C2×#479—531no—
328C2×#419—531no—
329#2441—531no—
330Q8×C22⋊C4—531no—
331#2095—531no—
332#2066—531no—
333#2066—531no—
334#2026—531no—
335#1865—531no—
336#1863—531no—
337#1854—531no—
338#1851—531no—
339#1809—531no—
340#1803—531no—
341#1787D5311yescharacteristic
342C2×C53—42no—
343C2×C53#243711yescharacteristic
344C2×C53#307811yescharacteristic
345C2×C53#308211yescharacteristic
346C2×C53#327111yescharacteristic
347C2×C53#327611yescharacteristic
348C2×C53#332011yescharacteristic
349C2×C53#332711yescharacteristic
350C23×Q8C2×D5311yescharacteristic
351C22×M16—531no—
352C2×#125—531no—
353C2×C4×Q8—531no—
354C2×C4×Q8—531no—
355C2×#117—531no—
356#479—1064no—
357C2×#107—531no—
358C2×#107—531no—
359C2×#106—531no—
360C2×#106—531no—
361C2×C4⋊C8—531no—
362C2×C4⋊C8—531no—
363#419—1064no—
364C2×#103—531no—
365C2×#103—531no—
366C2×#103—531no—
367C2×#103—531no—
368C2×#103C2×D5311yescharacteristic
369C2×#103C2×D5311yescharacteristic
370C2×#103C2×D5311yescharacteristic
371C2×#103C2×D5311yescharacteristic
372C2×C22⋊C8C2×D5311yescharacteristic
373#399—1061no—
374#396—1061no—
375#391—531no—
376#391—531no—
377#391—1061no—
378#391—1061no—
379#389—531no—
380#389—531no—
381#389—1061no—
382#387—1061no—
383#384—531no—
384#380—531no—
385#380—1061no—
386#380C2×D5311yescharacteristic
387C4×C22⋊C4—531no—
388C4×C22⋊C4—1061no—
389#340—531no—
390#340—531no—
391#340—531no—
392#340—531no—
393#336—531no—
394#336—531no—
395C53#2598111yescharacteristic, complemented, Sylow
396C22×Q8—21no—
397C22×Q8—22no—
398C22×Q8—22no—
399C22×Q8—22no—
400C22×Q8—41no—
401C22×Q8—41no—
402C22×Q8Dic10614yes—
403C22×Q8C22×D5311yescharacteristic
404C22×Q8C22×D5311yescharacteristic
405C23×C4—21no—
406C23×C4C22×D5311yescharacteristic
407C2×M16—1062no—
408C2×M16—1062no—
409C2×M16—2121no—
410C22×C8—531no—
411C22×C8—531no—
412C22×C8C22×D5311yescharacteristic
413C22×C8C22×D5311yescharacteristic
414#125—1062no—
415#125—2121no—
416C4×Q8—1062no—
417C4×Q8—1062no—
418C4×Q8—2121no—
419C4×Q8—2121no—
420#117—1062no—
421#117—2121no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4C22×D5311yescharacteristic
435C2×C4⋊C4C22×D5311yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—531no—
438C2×C22⋊C4—531no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1061no—
441C2×C42—531no—
442C2×C42—531no—
443C2×C42—1061no—
444C2×C42—1061no—
445C2×C42—1061no—
446#107—1062no—
447#107—2121no—
448#106—1062no—
449#106—2121no—
450C4⋊C8—1062no—
451C4⋊C8—2121no—
452#103—22no—
453#103—22no—
454#103—41no—
455#103—41no—
456#103—534no—
457#103—1062no—
458#103—1062no—
459#103—1062no—
460C22⋊C8—22no—
461#95—21no—
462#95—531no—
463#95—531no—
464#95—1061no—
465#95—1061no—
466#95—1061no—
467#95—1061no—
468#95—1061no—
469#95—1061no—
470#95—1061no—
471C24D4×D5311yescharacteristic
472C2×Q8—24no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—28no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—42no—
483C2×Q8—42no—
484C2×Q8—44no—
485C2×Q8—44no—
486C2×Q8—44no—
487C2×Q8C2×Dic10612yes—
488C2×Q8C2×Dic10612yes—
489C2×Q8D4×D5312yes—
490C2×Q8#1171994712yes—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—24no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—531no—
503C22×C4—531no—
504C22×C4—531no—
505C22×C4—531no—
506C22×C4—1061no—
507C22×C4—1061no—
508C22×C4—1061no—
509C22×C4—1061no—
510C22×C4C2×Dic10612yes—
511C22×C4D4×D5311yescharacteristic
512C22×C4#1171994712yes—
513C22×C4C23×D5311yescharacteristic
514M16—2124no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—532no—
522C2×C8—1061no—
523C2×C8—1061no—
524C2×C8—1061no—
525C2×C8—1061no—
526C2×C8—1061no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—532no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—1061no—
536C4⋊C4—1061no—
537C4⋊C4—1061no—
538C4⋊C4—1061no—
539C4⋊C4—4241no—
540C4⋊C4—4241no—
541C4⋊C4—2122no—
542C4⋊C4—2122no—
543C4⋊C4—2122no—
544C4⋊C4—2122no—
545C4⋊C4—2122no—
546C4⋊C4—2122no—
547C22⋊C4—81no—
548C22⋊C4—1062no—
549C22⋊C4—2121no—
550C22⋊C4—2122no—
551C22⋊C4—2122no—
552C42—1061no—
553C42—1061no—
554C42—1061no—
555C42—1061no—
556C42—2122no—
557C42—2122no—
558C42—2122no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23#1313319412yes—
566C23C2×D4×D5311yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—22no—
572Q8—48no—
573Q8—41no—
574Q8—41no—
575Q8#1313319414yes—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—48no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—532no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—1061no—
602C2×C4—1061no—
603C2×C4—1061no—
604C2×C4—1061no—
605C2×C4—1061no—
606C2×C4—1062no—
607C2×C4—1061no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—2121no—
619C2×C4#1313319412yes—
620C2×C4C2×D4×D5311yescharacteristic
621C2×C4C22×Dic10611yescharacteristic
622C2×C4C2×#1171994711yescharacteristic
623C2×C4C2×#1171994711yescharacteristic
624C2×C4#1313326211yescharacteristic
625C2×C4#1313326211yescharacteristic
626C8—41no—
627C8—41no—
628C8—1062no—
629C8—2121no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22C2×#1313319411yescharacteristic
638C22#1681125511yescharacteristic
639C22#1681126011yescharacteristic
640C22#137×D5311yescharacteristic
641C22#1681142211yescharacteristic
642C22#1681142911yescharacteristic
643C22#1681143611yescharacteristic
644C4—22no—
645C4—24no—
646C4—42no—
647C4—81no—
648C4—81no—
649C4—1062no—
650C4—4241no—
651C4—4241no—
652C4—4241no—
653C4—2121no—
654C4—2121no—
655C4—2121no—
656C4—2122no—
657C4C2×#1313319411yescharacteristic
658C4#1681124911yescharacteristic
659C4#1681126011yescharacteristic
660C4#1681130411yescharacteristic
661C2—42no—
662C2#2027838111yescharacteristic
663C2#2028451711yescharacteristic
664C2#2028455311yescharacteristic
665C2#2028630311yescharacteristic
666C2#2028635411yescharacteristic
667C2#2028687911yescharacteristic
668C2#2028698711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1787C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1787D53
hypocenterC53#25981
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
1. C2×#16810928C2semidirect
5. #20276251C2semidirect
6. #20276094C2semidirect
7. #20276092C2semidirect
8. #20275864C2semidirect
9. #20274981C2semidirect
10. #20274971C2semidirect
14. #20274718C2semidirect
395. C53#25981semidirect

Maximal subgroups

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

subgroupindex
C2×#168109282
C2×#168106322
#202784292
Q8×C22⋊Dic532
#202762512
#202760942
#202760922
#202758642
#202749812
#202749712
#202749162
#202748972
#202747462
#202747182
C53×#17872
#2598153