Finite Group Databasebeta

← #27446681

Subgroups — #27446681

Order 13568.

838 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
1C2×#16810931C211yesmaximal, characteristic, complemented
2C22×#13133151C211yesmaximal, characteristic
3#20278439C211yesmaximal, characteristic
4#20277718C211yesmaximal, characteristic
5#20276262C211yesmaximal, characteristic, complemented
6#20276203C211yesmaximal, characteristic, complemented
7#20276109C211yesmaximal, characteristic, complemented
8#20276054C211yesmaximal, characteristic, complemented
9#20275485C211yesmaximal, characteristic, complemented
10#20275287C211yesmaximal, characteristic, complemented
11#20275138C211yesmaximal, characteristic
12#20275020C211yesmaximal, characteristic
13#20274972C211yesmaximal, characteristic, complemented
14#20274910C211yesmaximal, characteristic
15C53×#1850C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21#16810931—24no—
22C2×#13133151—24no—
23C2×#13133151C2211yescharacteristic
24C2×#13133151C2211yescharacteristic
25C2×#13133151C2211yescharacteristic
26C2×#13133151C2211yescharacteristic
27C2×#13133149C2211yescharacteristic
28C2×#13133149C2211yescharacteristic
29C2×#13133142C2211yescharacteristic
30C2×#13133142C2211yescharacteristic
31C2×#13133142C2211yescharacteristic
32C2×#13133142C2211yescharacteristic
33#16810565—21no—
34#16810557—21no—
35#16810529C2211yescharacteristic
36#16810521C2211yescharacteristic
37#16810521C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810520—21no—
40#16810510—21no—
41#16810495—21no—
42#16810495—21no—
43#16810489C2211yescharacteristic
44#16810483C2211yescharacteristic
45#16810481—21no—
46C4×C4⋊Dic53C2211yescharacteristic
47C22⋊C4×Dic53—21no—
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53#16810386C2211yescharacteristic
54C23×C8×C53C2211yescharacteristic
55C2×C4⋊C8×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C53×#336C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C23×Dic53—21no—
63C23×Dic53—21no—
64#13133209—22no—
65#13133209—41no—
66#13133196—22no—
67#13133196Q814yes—
68C4×Dic106—22no—
69C4×Dic106—41no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53—22no—
74C2×C4⋊Dic53—41no—
75C2×C4⋊Dic53—41no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—21no—
89C2×#11719919—21no—
90C2×C22⋊Dic53—21no—
91C2×C22⋊Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94#13133151—22no—
95#13133151—22no—
96#13133151—22no—
97#13133151D414yes—
98#13133149—22no—
99#13133149—41no—
100#13133142—22no—
101#13133142—22no—
102#13133142—22no—
103#13133142—41no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113C23×C4×C53C2311yescharacteristic
114C22×C8×C53—22no—
115C22×C8×C53—22no—
116C22×C8×C53—22no—
117C22×C8×C53C2311yescharacteristic
118C22×C8×C53C2311yescharacteristic
119C22×C8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C2×C22⋊C4×C53D411yescharacteristic
122C2×C22⋊C4×C53D411yescharacteristic
123C2×C42×C53C2311yescharacteristic
124C2×C42×C53C2311yescharacteristic
125C4⋊C8×C53—22no—
126C4⋊C8×C53—22no—
127C22⋊C8×C53—22no—
128#95×C53D411yescharacteristic
129#95×C53D411yescharacteristic
130C24×C53C2×D411yescharacteristic
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×Q812yes—
141C22×C4×C53#4112yes—
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—42no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C22×Dic53—81no—
156C22×Dic53—81no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106C2×Q812yes—
163C2×C8×C53—28no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53C2×D412yes—
175C2×C8×C53#4112yes—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—22no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C22⋊Dic53—24no—
183C22⋊Dic53—42no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—42no—
196C4⋊Dic53—42no—
197C4⋊Dic53C2×D412yes—
198C4⋊Dic53C2×D412yes—
199C4⋊Dic53C2×Q812yes—
200#11719919—24no—
201#11719919—42no—
202#11719919—42no—
203#11719919—42no—
204#11719919—81no—
205#11719919—81no—
206#11719919—81no—
207C4×Dic53—42no—
208C4×Dic53—81no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53C22⋊Q812yes—
214C23×C53#13512yes—
215C23×C53#13712yes—
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—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
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—21no—
235C2×C4×C53C22⋊Q812yes—
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C22×Q811yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53H(5,𝔽2)11yescharacteristic
241C2×C4×C532−1+411yescharacteristic
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
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—84no—
267C2×Dic53—84no—
268Dic106—22no—
269Dic106—22no—
270Dic106—41no—
271Dic106—41no—
272C8×C53—22no—
273C8×C53—22no—
274C8×C53—41no—
275C8×C53—41no—
276#26344—531nomaximal, Sylow
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53#47914yescomplemented
284C22×C53C2×C22⋊Q811yescharacteristic
285C22×C53D4211yescharacteristic
286C22×C53#54711yescharacteristic
287C22×C53C2×#13511yescharacteristic
288C22×C53C2×#13711yescharacteristic
289C22×C53#57611yescharacteristic
290C22×C53#57811yescharacteristic
291C4×C53—22no—
292C4×C53—41no—
293C4×C53—41no—
294C4×C53—41no—
295C4×C53—41no—
296C4×C53#52511yescharacteristic
297C4×C53#53911yescharacteristic
298C4×C53D4×Q811yescharacteristic
299C4×C53#55411yescharacteristic
300Dic53—42no—
301Dic53—42no—
302Dic53—42no—
303Dic53—42no—
304Dic53—81no—
305Dic53—81no—
306Dic53—81no—
307Dic53—81no—
308C2×#479—531no—
309C22×#107—531no—
310#2442—531no—
311#2360—531no—
312#2097—531no—
313#2086—531no—
314#2068—531no—
315#2059—531no—
316#1949—531no—
317#1918—531no—
318#1892—531no—
319#1872—531no—
320#1863—531no—
321#1853—531no—
322#1850D5311yescharacteristic
323C2×C53—24no—
324C2×C53#244911yescharacteristic
325C2×C53C2×#47911yescharacteristic
326C2×C53#308311yescharacteristic
327C2×C53#327011yescharacteristic
328C2×C53#328011yescharacteristic
329C2×C53#333611yescharacteristic
330C2×C53#334611yescharacteristic
331C23×C8C2×D5311yescharacteristic
332C2×#125—531no—
333C2×C22⋊Q8—531no—
334C2×C4×Q8—531no—
335C22×C4⋊C4—531no—
336C22×C4⋊C4—531no—
337#479—1064no—
338C2×#107—531no—
339C2×#107—531no—
340C2×#107—531no—
341C2×#107—531no—
342C2×#107—1064no—
343C2×#106—531no—
344C2×#106—531no—
345C2×C4⋊C8C2×D5311yescharacteristic
346C2×C4⋊C8C2×D5311yescharacteristic
347C2×#103—531no—
348C2×#103—531no—
349C2×#103—531no—
350C2×#103—531no—
351C2×C22⋊C8C2×D5311yescharacteristic
352#400—1061no—
353#398—1061no—
354#391—531no—
355#389—531no—
356#389—531no—
357#389—531no—
358#389—1061no—
359#387—1061no—
360#385—1061no—
361#385—1061no—
362#382—531no—
363#380—531no—
364#380—1061no—
365C4×C4⋊C4—531no—
366C4×C22⋊C4—1061no—
367C4×C22⋊C4C2×D5311yescharacteristic
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#336C2×D5311yescharacteristic
375#336C2×D5311yescharacteristic
376C53#2634411yescharacteristic, complemented, Sylow
377C22×Q8—531no—
378C23×C4—1061no—
379C23×C4—1061no—
380C23×C4C22×D5311yescharacteristic
381C22×C8—22no—
382C22×C8—22no—
383C22×C8—22no—
384C22×C8C22×D5311yescharacteristic
385C22×C8C22×D5311yescharacteristic
386C22×C8C22×D5311yescharacteristic
387C22×C8C22×D5311yescharacteristic
388#125—1062no—
389#125—2121no—
390C22⋊Q8—534no—
391C22⋊Q8—1062no—
392C4×Q8—1062no—
393C4×Q8—2121no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—1062no—
402C2×C4⋊C4—1062no—
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—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—2121no—
413C2×C4⋊C4—2121no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4—1061no—
416C2×C22⋊C4D21211yescharacteristic
417C2×C22⋊C4D21211yescharacteristic
418C2×C42—1061no—
419C2×C42—1061no—
420C2×C42C22×D5311yescharacteristic
421C2×C42C22×D5311yescharacteristic
422#107—534no—
423#107—1062no—
424#107—1062no—
425#107—1062no—
426#106—1062no—
427#106—2121no—
428C4⋊C8—22no—
429C4⋊C8—22no—
430#103—1062no—
431#103—1062no—
432#103—1062no—
433#103—2121no—
434C22⋊C8—22no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444#95D21211yescharacteristic
445#95D21211yescharacteristic
446C24C2×D21211yescharacteristic
447C2×Q8—532no—
448C2×Q8—1061no—
449C2×Q8—1061no—
450C2×Q8—1061no—
451C2×Q8—1061no—
452C2×Q8—1061no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—4241no—
466C22×C4—4241no—
467C22×C4—2122no—
468C22×C4—2122no—
469C22×C4—2122no—
470C22×C4—2122no—
471C22×C4C2×D21211yescharacteristic
472C22×C4C2×D21211yescharacteristic
473C22×C4C2×D21211yescharacteristic
474C22×C4C2×D21211yescharacteristic
475C22×C4C2×D21211yescharacteristic
476C22×C4#1171994712yes—
477C22×C4Q8×D5312yes—
478C22×C4C23×D5311yescharacteristic
479C2×C8—28no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8D4×D5312yes—
491C2×C8#1171994712yes—
492C4⋊C4—532no—
493C4⋊C4—532no—
494C4⋊C4—532no—
495C4⋊C4—1064no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—4241no—
508C4⋊C4—4241no—
509C4⋊C4—4241no—
510C4⋊C4—2122no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C4⋊C4—2122no—
514C4⋊C4—2122no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—1064no—
518C22⋊C4—2122no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—4241no—
524C42—2122no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23#1313319712yes—
530C23#1313323912yes—
531C23#1313325012yes—
532C23C22×D21211yescharacteristic
533Q8—1062no—
534Q8—1062no—
535Q8—2121no—
536Q8—2121no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—4244no—
560C2×C4—4244no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
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—2121no—
580C2×C4—2121no—
581C2×C4#1313319712yes—
582C2×C4C2×D4×D5311yescharacteristic
583C2×C4C2×Q8×D5311yescharacteristic
584C2×C4C2×#1171994711yescharacteristic
585C2×C4C2×#1171994711yescharacteristic
586C2×C4#1313326311yescharacteristic
587C2×C4#1313326411yescharacteristic
588C8—22no—
589C8—22no—
590C8—41no—
591C8—41no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22#1681093114yescomplemented
599C22C2×#1313319711yescharacteristic
600C22D4×D21211yescharacteristic
601C22#1681127711yescharacteristic
602C22C2×#1313323911yescharacteristic
603C22C2×#1313325011yescharacteristic
604C22#1681142811yescharacteristic
605C22#1681144011yescharacteristic
606C4—22no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—4241no—
612C4—4241no—
613C4—4241no—
614C4—4241no—
615C4—2122no—
616C4—2122no—
617C4—2122no—
618C4—2122no—
619C4#1681112911yescharacteristic
620C4#1681121711yescharacteristic
621C4#1681128611yescharacteristic
622C4#1681133111yescharacteristic
623C2—24no—
624C2#2027852211yescharacteristic
625C2C2×#1681093111yescharacteristic
626C2#2028456811yescharacteristic
627C2#2028629011yescharacteristic
628C2#2028640211yescharacteristic
629C2#2028708411yescharacteristic
630C2#2028720211yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1850C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1850D53
hypocenterC53#26344
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
1. C2×#16810931C2semidirect
5. #20276262C2semidirect
6. #20276203C2semidirect
7. #20276109C2semidirect
8. #20276054C2semidirect
9. #20275485C2semidirect
10. #20275287C2semidirect
13. #20274972C2semidirect
283. C22×C53#479semidirect
376. C53#26344semidirect
598. C22#16810931semidirect

Maximal subgroups

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

subgroupindex
C2×#168109312
C22×#131331512
#202784392
#202777182
#202762622
#202762032
#202761092
#202760542
#202754852
#202752872
#202751382
#202750202
#202749722
#202749102
C53×#18502
#2634453