Finite Group Databasebeta

← #27439849

Subgroups — #27439849

Order 13568.

1078 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811176C211yesmaximal, characteristic
2C2×#16811027C211yesmaximal, characteristic, complemented
3C2×#16811020C211yesmaximal, characteristic, complemented
4#20283065C211yesmaximal, characteristic
5#20276056C211yesmaximal, characteristic, complemented
6#20276055C211yesmaximal, characteristic, complemented
7#20276052C211yesmaximal, characteristic, complemented
8#20276051C211yesmaximal, characteristic, complemented
9#20275674C211yesmaximal, characteristic, complemented
10#20275674C211yesmaximal, characteristic, complemented
11#20274916C211yesmaximal, characteristic
12#20274914C211yesmaximal, characteristic
13#20274912C211yesmaximal, characteristic
14C22⋊C8×Dic53C211yesmaximal, characteristic
15C53×#1854C211yesmaximal, characteristic
16#16811176—24no—
17#16811176—24no—
18C2×#13133216C2211yescharacteristic
19C2×#13133216C2211yescharacteristic
20C2×#13133216C2211yescharacteristic
21C2×#13133208C2211yescharacteristic
22C2×#13133208C2211yescharacteristic
23C2×#13133194—21no—
24C2×#13133194—21no—
25C2×#13133194—21no—
26C2×C4×Dic106—21no—
27C2×C4×Dic106—21no—
28C2×#13133176C2211yescharacteristic
29C22×C4×Dic53C2211yescharacteristic
30#16811027—24no—
31#16811020—24no—
32C2×#13133151C2211yescharacteristic
33C2×#13133150C2211yescharacteristic
34C2×#13133150C2211yescharacteristic
35C2×#13133149C2211yescharacteristic
36C2×#13133148C2211yescharacteristic
37C2×#13133148C2211yescharacteristic
38C2×C22⋊#88322C2211yescharacteristic
39C2×C8×Dic53C2211yescharacteristic
40C2×C8×Dic53C2211yescharacteristic
41#16810527—21no—
42#16810522C2211yescharacteristic
43#16810520—21no—
44#16810519C2211yescharacteristic
45#16810492C2211yescharacteristic
46#16810483C2211yescharacteristic
47#16810481—21no—
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810377C2211yescharacteristic
55#16810377C2211yescharacteristic
56C2×#117×C53C2211yescharacteristic
57C2×#107×C53C2211yescharacteristic
58C2×#106×C53C2211yescharacteristic
59C2×C22⋊C8×C53C2211yescharacteristic
60C53×#380C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C53×#340C2211yescharacteristic
63C22×Dic106—21no—
64C22×Dic106—21no—
65C22×Dic106—21no—
66C23×Dic53D411yescharacteristic
67C23×Dic53D411yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70#13133216—22no—
71#13133216—22no—
72#13133216—22no—
73#13133216—22no—
74#13133216D414yes—
75#13133216D414yes—
76#13133208—22no—
77#13133208—22no—
78#13133208—22no—
79#13133208—22no—
80#13133194—28no—
81#13133194—28no—
82#13133194—44no—
83C4×Dic106—28no—
84C4×Dic106—28no—
85#13133176—22no—
86#13133176—22no—
87C2×C4⋊Dic53—21no—
88C2×C4⋊Dic53—21no—
89C2×C4⋊Dic53—21no—
90C2×C4⋊Dic53C2311yescharacteristic
91C2×C4⋊Dic53C2311yescharacteristic
92C2×C4⋊Dic53C2311yescharacteristic
93C2×C4⋊Dic53C2311yescharacteristic
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×#11719919—21no—
97C2×#11719919—21no—
98C2×#11719919—21no—
99C2×#11719919—21no—
100C2×#11719919—21no—
101C2×#11719919—21no—
102C2×C22⋊Dic53—21no—
103C2×C22⋊Dic53—21no—
104C2×C4×Dic53—22no—
105C2×C4×Dic53—22no—
106C2×C4×Dic53—21no—
107C2×C4×Dic53D411yescharacteristic
108C2×C4×Dic53D411yescharacteristic
109C2×C4×Dic53C2311yescharacteristic
110C2×C4×Dic53C2311yescharacteristic
111#13133151—22no—
112#13133150—22no—
113#13133150—22no—
114#13133149—22no—
115#13133148—22no—
116#13133148—22no—
117C22⋊#88322Q814yescomplemented
118C8×Dic53—22no—
119C8×Dic53—22no—
120#13133124—21no—
121#13133124—21no—
122#13133124—21no—
123C23×C4×C53C2311yescharacteristic
124C22×C8×C53C2311yescharacteristic
125C22×C8×C53C2311yescharacteristic
126#117×C53—22no—
127#117×C53—22no—
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C4⋊C4×C53C2311yescharacteristic
131C2×C4⋊C4×C53C2311yescharacteristic
132C2×C22⋊C4×C53—21no—
133C2×C22⋊C4×C53—21no—
134C2×C42×C53—21no—
135#107×C53—22no—
136#106×C53—22no—
137C22⋊C8×C53Q814yescomplemented
138#95×C53—21no—
139C24×C53C2×D411yescharacteristic
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×Q812yes—
150C22×C4×C53C2×Q812yes—
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164C22×Dic53C2×D411yescharacteristic
165C22×Dic53C2×D411yescharacteristic
166C2×Dic106—24no—
167C2×Dic106—24no—
168C2×Dic106—42no—
169C2×Dic106—42no—
170C2×Dic106—42no—
171C2×Dic106—42no—
172C2×Dic106—42no—
173C2×Dic106—42no—
174C2×Dic106—42no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C2×C8×C53—21no—
179C2×C8×C53C2×Q812yes—
180C2×C8×C53C2×Q812yes—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53—21no—
187C4⋊C4×C53C2×D412yes—
188C4⋊C4×C53C2×D412yes—
189C22⋊C4×C53—24no—
190C22⋊C4×C53—42no—
191C42×C53—24no—
192C22⋊Dic53—24no—
193C22⋊Dic53—42no—
194C4⋊Dic53—24no—
195C4⋊Dic53—24no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—42no—
203C4⋊Dic53C2×D412yes—
204C4⋊Dic53C2×D412yes—
205#11719919—24no—
206#11719919—24no—
207#11719919—24no—
208#11719919—24no—
209#11719919—42no—
210#11719919—42no—
211#11719919—42no—
212#11719919—42no—
213C4×Dic53—24no—
214C4×Dic53—21no—
215C4×Dic53—21no—
216C4×Dic53—22no—
217C4×Dic53—22no—
218C4×Dic53C2×D412yes—
219C4×Dic53C2×D412yes—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322—21no—
223C2×#88322—21no—
224C2×#88322C2×Q812yes—
225C2×#88322C2×Q812yes—
226C23×C53—21no—
227C23×C53—21no—
228C23×C53—21no—
229C23×C53—21no—
230C23×C53—21no—
231C23×C53—21no—
232C23×C53C4⋊Q812yes—
233C23×C53C22×D411yescharacteristic
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53C4⋊Q812yes—
253C2×C4×C53C22×D411yescharacteristic
254C2×C4×C53C22×D411yescharacteristic
255C2×C4×C53C22×Q811yescharacteristic
256C2×C4×C53C22×Q811yescharacteristic
257C2×C4×C532−1+411yescharacteristic
258C2×C4×C532−1+411yescharacteristic
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—44no—
272C2×Dic53—44no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53#12012yes—
280C2×Dic53#12012yes—
281C2×Dic53#13512yes—
282C2×Dic53#13512yes—
283C2×Dic53AGL(1,ℤ/8)12yes—
284C2×Dic53#13712yes—
285Dic106—48no—
286Dic106—48no—
287Dic106—84no—
288C8×C53—22no—
289C8×C53—22no—
290#88322—22no—
291#88322—22no—
292#25812—531nomaximal, Sylow
293C22×C53—22no—
294C22×C53—22no—
295C22×C53—22no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53—41no—
299C22×C53—41no—
300C22×C53C2×#12011yescharacteristic
301C22×C53C2×C4⋊Q811yescharacteristic
302C22×C53#53611yescharacteristic
303C22×C53C2×#13511yescharacteristic
304C22×C53C2×#13511yescharacteristic
305C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
306C22×C53C2×#13711yescharacteristic
307C4×C53—22no—
308C4×C53—42no—
309C4×C53—42no—
310C4×C53—81no—
311C4×C53—81no—
312C4×C53D4×Q811yescharacteristic
313C4×C53D4×Q811yescharacteristic
314C4×C53D4×Q811yescharacteristic
315C4×C53D4×Q811yescharacteristic
316Dic53—42no—
317Dic53—41no—
318Dic53—42no—
319Dic53—41no—
320Dic53—81no—
321Dic53—81no—
322Dic53#44914yes—
323Dic53#45214yes—
324C2×#533—531no—
325C2×#500—531no—
326C2×#498—531no—
327#2857—531no—
328#2059—531no—
329#2059—531no—
330#2059—531no—
331#2059—531no—
332#1990—531no—
333#1990—531no—
334#1854—531no—
335#1854—531no—
336#1854—531no—
337#1854D5311yescharacteristic
338C4×C22⋊C8—531no—
339C2×C53—42no—
340C2×C53#243411yescharacteristic
341C2×C53C2×#44911yescharacteristic
342C2×C53C2×#45211yescharacteristic
343C2×C53#316011yescharacteristic
344C2×C53#316511yescharacteristic
345C2×C53#322811yescharacteristic
346C2×C53#323111yescharacteristic
347#533—1064no—
348#533—1064no—
349C2×C4⋊Q8—531no—
350C2×C4⋊Q8—531no—
351C2×C4⋊Q8—531no—
352C2×#125—531no—
353C2×#125—531no—
354C2×C22⋊Q8—1061no—
355C2×C22⋊Q8—1061no—
356C2×C22⋊Q8—1061no—
357C2×C4×Q8—1061no—
358C2×C4×Q8—1061no—
359C2×#117—531no—
360C2×#117C2×D5311yescharacteristic
361C22×C42—531no—
362#500—1064no—
363#498—1064no—
364C2×#107—531no—
365C2×#107—531no—
366C2×#107—531no—
367C2×#107C2×D5311yescharacteristic
368C2×#106—531no—
369C2×#106—531no—
370C2×#106—531no—
371C2×#106C2×D5311yescharacteristic
372C2×C22⋊C8—531no—
373C2×C22⋊C8C2×D5311yescharacteristic
374C2×C4×C8—531no—
375C2×C4×C8—531no—
376#391—1061no—
377#389—531no—
378#389—531no—
379#389—1061no—
380#384—531no—
381#380—531no—
382#380—1061no—
383#380C2×D5311yescharacteristic
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340C2×D5311yescharacteristic
391#340C2×D5311yescharacteristic
392#336—531no—
393#336—531no—
394C53#2581211yescharacteristic, complemented, Sylow
395C22×Q8—1061no—
396C22×Q8—1061no—
397C22×Q8—1061no—
398C23×C4—531no—
399C23×C4—531no—
400C23×C4C22×D5311yescharacteristic
401C22×C8—531no—
402C22×C8—531no—
403C22×C8C22×D5311yescharacteristic
404C22×C8C22×D5311yescharacteristic
405C4⋊Q8—534no—
406C4⋊Q8—534no—
407C4⋊Q8—1062no—
408C4⋊Q8—1062no—
409C4⋊Q8—1062no—
410C4⋊Q8—1062no—
411#125—1062no—
412#125—1062no—
413#125—1062no—
414#125—1062no—
415C22⋊Q8—1068no—
416C22⋊Q8—1068no—
417C22⋊Q8—2124no—
418C4×Q8—1068no—
419C4×Q8—1068no—
420#117—22no—
421#117—22no—
422#117—1062no—
423#117—1062no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—531no—
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⋊C4—1061no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4—1061no—
439C2×C4⋊C4C22×D5311yescharacteristic
440C2×C4⋊C4C22×D5311yescharacteristic
441C2×C4⋊C4C22×D5311yescharacteristic
442C2×C4⋊C4C22×D5311yescharacteristic
443C2×C22⋊C4—21no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—1061no—
446C2×C22⋊C4—1061no—
447C2×C42—21no—
448C2×C42—531no—
449C2×C42—531no—
450C2×C42—531no—
451C2×C42—531no—
452C2×C42—1062no—
453C2×C42—1062no—
454C2×C42—1061no—
455#107—22no—
456#107—1062no—
457#107—1062no—
458#107—1062no—
459#106—22no—
460#106—1062no—
461#106—1062no—
462#106—1062no—
463C22⋊C8—534no—
464C22⋊C8Dic10614yescomplemented
465C4×C8—1062no—
466C4×C8—1062no—
467#95—21no—
468#95—1061no—
469#95—1061no—
470#95—1061no—
471C24D4×D5311yescharacteristic
472C2×Q8—1064no—
473C2×Q8—1064no—
474C2×Q8—2122no—
475C2×Q8—2122no—
476C2×Q8—2122no—
477C2×Q8—2122no—
478C2×Q8—2122no—
479C2×Q8—2122no—
480C2×Q8—2122no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—531no—
490C22×C4—531no—
491C22×C4—531no—
492C22×C4—531no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—1061no—
503C22×C4C2×Dic10612yes—
504C22×C4D4×D5311yescharacteristic
505C22×C4Q8×D5312yes—
506C22×C4C23×D5311yescharacteristic
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—532no—
512C2×C8—532no—
513C2×C8—1061no—
514C2×C8—1061no—
515C2×C8—1061no—
516C2×C8—1061no—
517C2×C8C2×Dic10612yes—
518C2×C8C2×Dic10612yes—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—532no—
526C4⋊C4—532no—
527C4⋊C4—1064no—
528C4⋊C4—1064no—
529C4⋊C4—1064no—
530C4⋊C4—1064no—
531C4⋊C4—1064no—
532C4⋊C4—1064no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—1061no—
536C4⋊C4—1061no—
537C4⋊C4—1061no—
538C4⋊C4—1061no—
539C4⋊C4—2122no—
540C4⋊C4—2122no—
541C4⋊C4—2122no—
542C4⋊C4—2122no—
543C4⋊C4—2122no—
544C4⋊C4D4×D5312yes—
545C4⋊C4D4×D5312yes—
546C22⋊C4—24no—
547C22⋊C4—42no—
548C22⋊C4—1064no—
549C22⋊C4—2122no—
550C42—24no—
551C42—532no—
552C42—532no—
553C42—1062no—
554C42—1062no—
555C42—1064no—
556C42—1061no—
557C42—1061no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23#1313321612yes—
565C23C2×D4×D5311yescharacteristic
566Q8—4244no—
567Q8—2128no—
568Q8—2128no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—532no—
588C2×C4—532no—
589C2×C4—532no—
590C2×C4—532no—
591C2×C4—532no—
592C2×C4—532no—
593C2×C4—1061no—
594C2×C4—1061no—
595C2×C4—1062no—
596C2×C4—1062no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—2124no—
602C2×C4—2124no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4#1313321612yes—
614C2×C4C2×D4×D5311yescharacteristic
615C2×C4C2×D4×D5311yescharacteristic
616C2×C4C2×Q8×D5311yescharacteristic
617C2×C4C22×Dic10611yescharacteristic
618C2×C4#1313326411yescharacteristic
619C2×C4#1313326411yescharacteristic
620C8—22no—
621C8—22no—
622C8—1062no—
623C8—1062no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22#120×D5311yescharacteristic
632C22C2×#1313321611yescharacteristic
633C22#1681119511yescharacteristic
634C22#135×D5311yescharacteristic
635C22#135×D5311yescharacteristic
636C22AGL(1,ℤ/8)×D5311yescharacteristic
637C22#137×D5311yescharacteristic
638C4—22no—
639C4—42no—
640C4—42no—
641C4—81no—
642C4—81no—
643C4—534no—
644C4—534no—
645C4—4241no—
646C4—4241no—
647C4—2121no—
648C4—2122no—
649C4—2121no—
650C4—2122no—
651C4#1681128611yescharacteristic
652C4#1681128611yescharacteristic
653C4D4×Dic10611yescharacteristic
654C4D4×Dic10611yescharacteristic
655C2—42no—
656C2#2027835711yescharacteristic
657C2#449×D5311yescharacteristic
658C2#452×D5311yescharacteristic
659C2#2028522711yescharacteristic
660C2#2028525911yescharacteristic
661C2#2028584911yescharacteristic
662C2#2028587211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1854C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1854D53
hypocenterC53#25812
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
2. C2×#16811027C2semidirect
3. C2×#16811020C2semidirect
5. #20276056C2semidirect
6. #20276055C2semidirect
7. #20276052C2semidirect
8. #20276051C2semidirect
9. #20275674C2semidirect
10. #20275674C2semidirect
117. C22⋊#88322Q8semidirect
137. C22⋊C8×C53Q8semidirect
394. C53#25812semidirect
464. C22⋊C8Dic106semidirect

Maximal subgroups

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

subgroupindex
C2×#168111762
C2×#168110272
C2×#168110202
#202830652
#202760562
#202760552
#202760522
#202760512
#202756742
#202749162
#202749142
#202749122
C22⋊C8×Dic532
C53×#18542
#2581253