Finite Group Databasebeta

← #27446957

Subgroups — #27446957

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×#16810932C211yesmaximal, characteristic, complemented
2C2×#16810666C211yesmaximal, characteristic
3#20278422C211yesmaximal, characteristic
4#20277724C211yesmaximal, characteristic
5#20276251C211yesmaximal, characteristic, complemented
6#20276207C211yesmaximal, characteristic, complemented
7#20276092C211yesmaximal, characteristic, complemented
8#20276056C211yesmaximal, characteristic, complemented
9#20275472C211yesmaximal, characteristic, complemented
10#20275296C211yesmaximal, characteristic, complemented
11#20275149C211yesmaximal, characteristic
12#20274967C211yesmaximal, characteristic, complemented
13#20274905C211yesmaximal, characteristic
14#20274897C211yesmaximal, characteristic
15C53×#1873C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×#13133208C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20#16810932—24no—
21#16810666—24no—
22C2×#13133151C2211yescharacteristic
23C2×#13133151C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×#13133146C2211yescharacteristic
29C2×#13133146C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×#13133141C2211yescharacteristic
32#16810559—21no—
33#16810546—21no—
34#16810530C2211yescharacteristic
35#16810522—21no—
36#16810522C2211yescharacteristic
37#16810522C2211yescharacteristic
38#16810521—21no—
39#16810519C2211yescharacteristic
40#16810509—21no—
41#16810507—21no—
42#16810498—21no—
43#16810492C2211yescharacteristic
44#16810483C2211yescharacteristic
45C4⋊C4×Dic53C2211yescharacteristic
46C22⋊C4×Dic53—21no—
47C4×C22⋊Dic53C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C2×C22⋊Q8×C53C2211yescharacteristic
55C2×#117×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C23×Dic53—21no—
62C2×#11719917—22no—
63C2×#11719917—22no—
64C2×#11719917—22no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133208—22no—
68#13133208—41no—
69Q8×Dic53—22no—
70Q8×Dic53—41no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53—41no—
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⋊Dic53D411yescharacteristic
88C2×C22⋊Dic53D411yescharacteristic
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53C2311yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93#13133151—22no—
94#13133151—41no—
95#13133150—22no—
96#13133150—22no—
97#13133148—22no—
98#13133148D414yes—
99#13133146—22no—
100#13133146—22no—
101#13133141—22no—
102#13133141—22no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124D411yescharacteristic
111#13133124D411yescharacteristic
112C22×Q8×C53C2311yescharacteristic
113C23×C4×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C22⋊Q8×C53—22no—
117C22⋊Q8×C53Q814yes—
118#117×C53—22no—
119#117×C53—41no—
120C2×C4⋊C4×C53—21no—
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C22⋊C4×C53—21no—
125C2×C22⋊C4×C53—21no—
126C2×C42×C53—21no—
127#103×C53—22no—
128#103×C53—41no—
129C22⋊C8×C53—22no—
130C24×C53C2×D411yescharacteristic
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53C2×Q812yes—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×Q812yes—
147C22×C4×C53#4112yes—
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C22×Dic53—81no—
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×D411yescharacteristic
159C22×Dic53C2×D411yescharacteristic
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C4⋊C4×C53—24no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53C2×D412yes—
172C4⋊C4×C53C2×Q812yes—
173C22⋊C4×C53—24no—
174C22⋊C4×C53—42no—
175C42×C53—42no—
176C22⋊Dic53—22no—
177C22⋊Dic53—41no—
178C22⋊Dic53—81no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—42no—
187C4⋊Dic53—42no—
188C4⋊Dic53—42no—
189C4⋊Dic53—42no—
190C4⋊Dic53—81no—
191C4⋊Dic53C2×D412yes—
192#11719919—42no—
193#11719919—42no—
194#11719919—81no—
195#11719919—81no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—42no—
201C4×Dic53—81no—
202#11719917—28no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322C2×D412yes—
208C2×#88322#4112yes—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53C22⋊Q812yes—
216C23×C53C22×D411yescharacteristic
217Q8×C53—22no—
218Q8×C53—22no—
219Q8×C53—41no—
220Q8×C53—41no—
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—22no—
229C2×C4×C53—22no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53C22⋊Q812yes—
240C2×C4×C53C22×D411yescharacteristic
241C2×C4×C53C22×Q811yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53H(5,𝔽2)11yescharacteristic
245C2×C4×C532−1+411yescharacteristic
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—22no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
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—41no—
268C2×Dic53—41no—
269C2×Dic53—84no—
270C2×Dic53#13712yes—
271C8×C53—41no—
272C8×C53—41no—
273#88322—22no—
274#88322—22no—
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—42no—
294C4×C53—81no—
295C4×C53#52511yescharacteristic
296C4×C53#53911yescharacteristic
297C4×C53D4×Q811yescharacteristic
298C4×C53#55411yescharacteristic
299Dic53—22no—
300Dic53—41no—
301Dic53—41no—
302Dic53—42no—
303Dic53—41no—
304Dic53—81no—
305Dic53—81no—
306Dic53—81no—
307C2×#479—531no—
308C2×#428—531no—
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#1873D5311yescharacteristic
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×M16—531no—
331C2×#125—531no—
332C2×C22⋊Q8C2×D5311yescharacteristic
333C2×C4×Q8—531no—
334C2×#117C2×D5311yescharacteristic
335C22×C4⋊C4—531no—
336#479—1064no—
337#428—1064no—
338C2×#107—531no—
339C2×#107—531no—
340C2×#107—531no—
341C2×#107—531no—
342C2×#106—531no—
343C2×#106—531no—
344C2×C4⋊C8—531no—
345C2×C4⋊C8—531no—
346C2×#103—531no—
347C2×#103—531no—
348C2×#103C2×D5311yescharacteristic
349C2×#103C2×D5311yescharacteristic
350C2×C22⋊C8C2×D5311yescharacteristic
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#340C2×D5311yescharacteristic
372#340C2×D5311yescharacteristic
373#336—531no—
374#336—531no—
375C53#2636211yescharacteristic, complemented, Sylow
376C22×Q8C22×D5311yescharacteristic
377C23×C4—1061no—
378C23×C4C22×D5311yescharacteristic
379C2×M16—1062no—
380C2×M16—1062no—
381C2×M16—1062no—
382C22×C8—531no—
383C22×C8—531no—
384C22×C8C22×D5311yescharacteristic
385C22×C8C22×D5311yescharacteristic
386#125—1062no—
387#125—2121no—
388C22⋊Q8—22no—
389C22⋊Q8Dic10614yes—
390C4×Q8—1062no—
391C4×Q8—2121no—
392#117—22no—
393#117—41no—
394C2×C4⋊C4—21no—
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×C4⋊C4C22×D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C4⋊C4C22×D5311yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—531no—
416C2×C22⋊C4—531no—
417C2×C22⋊C4—1061no—
418C2×C42—21no—
419C2×C42—531no—
420C2×C42—531no—
421C2×C42—1061no—
422C2×C42—1061no—
423#107—1062no—
424#107—1062no—
425#107—1062no—
426#107—2121no—
427#106—534no—
428#106—1062no—
429C4⋊C8—1062no—
430C4⋊C8—1062no—
431#103—22no—
432#103—41no—
433#103—1062no—
434#103—1062no—
435C22⋊C8—22no—
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—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8C2×Dic10612yes—
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—531no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—531no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—4241no—
469C22×C4—2122no—
470C22×C4—2122no—
471C22×C4C2×Dic10612yes—
472C22×C4D4×D5311yescharacteristic
473C22×C4#1171994712yes—
474C22×C4C23×D5311yescharacteristic
475M16—1068no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—532no—
483C2×C8—532no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C4⋊C4—24no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—532no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—4241no—
502C4⋊C4—4241no—
503C4⋊C4—4241no—
504C4⋊C4—2122no—
505C4⋊C4—2122no—
506C4⋊C4—2122no—
507C4⋊C4—2122no—
508C4⋊C4—2122no—
509C4⋊C4—2122no—
510C4⋊C4C2×Dic10612yes—
511C4⋊C4D4×D5312yes—
512C22⋊C4—24no—
513C22⋊C4—42no—
514C22⋊C4—1062no—
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#1313319412yes—
531C23C2×D4×D5311yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
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—22no—
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—532no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1062no—
567C2×C4—4244no—
568C2×C4—2121no—
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#1313319412yes—
580C2×C4C2×D4×D5311yescharacteristic
581C2×C4C22×Dic10611yescharacteristic
582C2×C4#41×D5311yescharacteristic
583C2×C4C2×#1171994711yescharacteristic
584C2×C4#1313326211yescharacteristic
585C2×C4#1313326411yescharacteristic
586C8—41no—
587C8—41no—
588C8—1062no—
589C8—1062no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22C2×#1313319411yescharacteristic
598C22#1681125511yescharacteristic
599C22#1681127111yescharacteristic
600C22#137×D5311yescharacteristic
601C22#1681142211yescharacteristic
602C22#1681142511yescharacteristic
603C22#1681143711yescharacteristic
604C4—22no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—81no—
609C4—1062no—
610C4—4241no—
611C4—4241no—
612C4—4241no—
613C4—2121no—
614C4—2122no—
615C4—2121no—
616C4—2121no—
617C4#1681113311yescharacteristic
618C4#1681121811yescharacteristic
619C4D4×Dic10611yescharacteristic
620C4#1681133211yescharacteristic
621C2—42no—
622C2#2027852811yescharacteristic
623C2#2028451711yescharacteristic
624C2#2028456411yescharacteristic
625C2#2028625811yescharacteristic
626C2#2028635911yescharacteristic
627C2#2028719311yescharacteristic
628C2#2028728311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1873C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1873D53
hypocenterC53#26362
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
1. C2×#16810932C2semidirect
5. #20276251C2semidirect
6. #20276207C2semidirect
7. #20276092C2semidirect
8. #20276056C2semidirect
9. #20275472C2semidirect
10. #20275296C2semidirect
12. #20274967C2semidirect
375. C53#26362semidirect

Maximal subgroups

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

subgroupindex
C2×#168109322
C2×#168106662
#202784222
#202777242
#202762512
#202762072
#202760922
#202760562
#202754722
#202752962
#202751492
#202749672
#202749052
#202748972
C53×#18732
#2636253