Finite Group Databasebeta

← #27435895

Subgroups — #27435895

Order 13568.

756 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811033C211yesmaximal, characteristic, complemented
2#20279888C211yesmaximal, characteristic
3#20277815C211yesmaximal, characteristic, complemented
4#20276200C211yesmaximal, characteristic, complemented
5#20276199C211yesmaximal, characteristic, complemented
6#20276099C211yesmaximal, characteristic
7#20276085C211yesmaximal, characteristic
8#20275666C211yesmaximal, characteristic, complemented
9#20275484C211yesmaximal, characteristic, complemented
10#20275480C211yesmaximal, characteristic, complemented
11#20275436C211yesmaximal, characteristic, complemented
12#20275345C211yesmaximal, characteristic
13#20275338C211yesmaximal, characteristic
14#20274540C211yesmaximal, characteristic
15C2×C53×#478C211yesmaximal, characteristic
16C2×#13133217C2211yescharacteristic
17#16811033—24no—
18C2×#13133150C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133128C2211yescharacteristic
27#16810535—21no—
28#16810535—21no—
29#16810526C2211yescharacteristic
30#16810526C2211yescharacteristic
31#16810522—21no—
32#16810522C2211yescharacteristic
33#16810522C2211yescharacteristic
34#16810509—21no—
35#16810509—21no—
36#16810488C2211yescharacteristic
37#16810488C2211yescharacteristic
38#16810483—21no—
39#16810483—21no—
40C4⋊C4×Dic53—21no—
41C4⋊C4×Dic53C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#125×C53C2211yescharacteristic
54C2×C4×D4×C53C2211yescharacteristic
55C53×#478—24no—
56C2×#107×C53C2211yescharacteristic
57C2×#106×C53C2211yescharacteristic
58C2×C4⋊C8×C53C2211yescharacteristic
59C2×#102×C53C2211yescharacteristic
60C2×#102×C53C2211yescharacteristic
61C22×Dic106—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133217—22no—
65#13133217—22no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53D411yescharacteristic
70C2×C4⋊Dic53D411yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53C2311yescharacteristic
83C2×C4×Dic53C2311yescharacteristic
84#13133150—22no—
85#13133150—22no—
86#13133148—22no—
87#13133148—22no—
88#13133146—22no—
89#13133146—22no—
90#13133138—22no—
91#13133138Q814yescomplemented
92#13133128—22no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124D411yescharacteristic
103#13133124D411yescharacteristic
104C22×D4×C53C2311yescharacteristic
105C23×C4×C53—21no—
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108#125×C53—22no—
109#125×C53D414yes—
110C4×D4×C53—22no—
111C4×D4×C53—22no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C42×C53C2311yescharacteristic
119#107×C53—22no—
120#106×C53—22no—
121C4⋊C8×C53—22no—
122#102×C53—22no—
123#102×C53—22no—
124C24×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53C2×Q812yes—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—42no—
135C22×C4×C53—42no—
136C22×C4×C53—42no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C2×Dic106—42no—
149C2×Dic106—81no—
150C2×Dic106—81no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C4⋊C4×C53—24no—
158C4⋊C4×C53—24no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53C2×D412yes—
164C4⋊C4×C53C2×D412yes—
165C22⋊C4×C53—42no—
166C42×C53C2×D412yes—
167C42×C53#4112yes—
168C22⋊Dic53—81no—
169C22⋊Dic53—81no—
170C22⋊Dic53—81no—
171C22⋊Dic53—81no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—41no—
180C4⋊Dic53—41no—
181C4⋊Dic53—42no—
182C4⋊Dic53—42no—
183C4⋊Dic53—81no—
184C4⋊Dic53C2×Q812yes—
185C4×Dic53—21no—
186C4×Dic53—21no—
187C4×Dic53—21no—
188C4×Dic53—21no—
189C4×Dic53—42no—
190C4×Dic53—81no—
191C4×Dic53—81no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322C2×Q812yes—
197C2×#88322#4112yes—
198C23×C53—22no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53C22×D411yescharacteristic
206D4×C53—22no—
207D4×C53—22no—
208D4×C53—41no—
209D4×C53—41no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—84no—
224C2×C4×C53#12112yes—
225C2×C4×C53#12112yes—
226C2×C4×C53C22×D411yescharacteristic
227C2×C4×C53C22×Q811yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C532−1+411yescharacteristic
231C2×C4×C532−1+411yescharacteristic
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—22no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—41no—
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—
256Dic106—84no—
257C8×C53—41no—
258C8×C53—41no—
259#88322—22no—
260#88322—22no—
261#25466—531nomaximal, Sylow
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—81no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53—81no—
269C22×C53C2×#12111yescharacteristic
270C22×C53#54711yescharacteristic
271C22×C53D4×Q811yescharacteristic
272C22×C53#57511yescharacteristic
273C22×C53#57511yescharacteristic
274C22×C53#57611yescharacteristic
275C22×C53#57711yescharacteristic
276C4×C53—22no—
277C4×C53—22no—
278C4×C53—42no—
279C4×C53—42no—
280C4×C53—81no—
281C4×C53#53311yescharacteristic
282C4×C53#54111yescharacteristic
283C4×C53#54811yescharacteristic
284C4×C53D4×Q811yescharacteristic
285Dic53—41no—
286Dic53—41no—
287Dic53—41no—
288Dic53—41no—
289Dic53—42no—
290Dic53—81no—
291Dic53—81no—
292Dic53—81no—
293C2×#501—531no—
294C2×#478D5311yescharacteristic
295#2563—531no—
296#2370—531no—
297#2085—531no—
298#2085—531no—
299#2067—531no—
300#2065—531no—
301#1988—531no—
302#1949—531no—
303#1949—531no—
304#1940—531no—
305#1926—531no—
306#1925—531no—
307#1762—531no—
308C2×C53—82no—
309C2×C53#242911yescharacteristic
310C2×C53#304011yescharacteristic
311C2×C53#304511yescharacteristic
312C2×C53#334711yescharacteristic
313C2×C53#334811yescharacteristic
314C2×C53#338811yescharacteristic
315C2×C53#338911yescharacteristic
316C2×C4⋊Q8—531no—
317C2×#125C2×D5311yescharacteristic
318C2×C4×D4C2×D5311yescharacteristic
319#501—1064no—
320#478—24no—
321C2×#107—531no—
322C2×#107—531no—
323C2×#107C2×D5311yescharacteristic
324C2×#106—531no—
325C2×#106—531no—
326C2×#106C2×D5311yescharacteristic
327C2×C4⋊C8—531no—
328C2×C4⋊C8—531no—
329C2×C4⋊C8C2×D5311yescharacteristic
330C2×#102—531no—
331C2×#102—531no—
332C2×#102C2×D5311yescharacteristic
333C2×#102C2×D5311yescharacteristic
334C2×#97—531no—
335#393—1061no—
336#393—1061no—
337#390—531no—
338#390—531no—
339#389—531no—
340#389—531no—
341#389—1061no—
342#387—1061no—
343#387—1061no—
344#382—531no—
345#382—531no—
346#380—1061no—
347#380—1061no—
348C4×C4⋊C4—531no—
349C4×C4⋊C4—531no—
350C4×C4⋊C4—531no—
351C4×C4⋊C4—531no—
352C4×C4⋊C4—1061no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#336—531no—
360#336—531no—
361C53#2546611yescharacteristic, complemented, Sylow
362C22×Q8—1061no—
363C22×D4C22×D5311yescharacteristic
364C23×C4—21no—
365C22×C8—531no—
366C22×C8—531no—
367C22×C8C22×D5311yescharacteristic
368C22×C8C22×D5311yescharacteristic
369C4⋊Q8—1062no—
370C4⋊Q8—1062no—
371#125—22no—
372#125C22⋊D5314yes—
373C4×D4—22no—
374C4×D4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4C22×D5311yescharacteristic
387C2×C4⋊C4C22×D5311yescharacteristic
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1061no—
391C2×C22⋊C4—1061no—
392C2×C22⋊C4—1061no—
393C2×C22⋊C4—1061no—
394C2×C42—531no—
395C2×C42—531no—
396C2×C42—1061no—
397C2×C42—1061no—
398C2×C42—1061no—
399C2×C42C22×D5311yescharacteristic
400#107—22no—
401#107—1062no—
402#107—1062no—
403#106—22no—
404#106—1062no—
405#106—1062no—
406C4⋊C8—22no—
407C4⋊C8—1062no—
408C4⋊C8—1062no—
409#102—22no—
410#102—22no—
411#102—534no—
412#102—1062no—
413#97—1062no—
414#95—531no—
415#95—531no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420#95—1061no—
421#95—1061no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425C24—21no—
426C2×Q8—4241no—
427C2×Q8—4241no—
428C2×Q8—2122no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4Q8×D5312yes—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—531no—
442C22×C4—531no—
443C22×C4—531no—
444C22×C4—531no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4D4×D5311yescharacteristic
450C22×C4D4×D5311yescharacteristic
451C22×C4C23×D5311yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—532no—
459C2×C8—532no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—532no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—4241no—
479C4⋊C4—2121no—
480C4⋊C4—2121no—
481C4⋊C4—2122no—
482C4⋊C4—2122no—
483C4⋊C4C2×C22⋊D5312yes—
484C4⋊C4C2×C22⋊D5312yes—
485C22⋊C4—42no—
486C22⋊C4—4241no—
487C22⋊C4—4241no—
488C22⋊C4—4241no—
489C22⋊C4—4241no—
490C42—1061no—
491C42—1061no—
492C42—1061no—
493C42—1061no—
494C42—4241no—
495C42—4241no—
496C42—2122no—
497C42#1171994712yes—
498C42C2×C22⋊D5312yes—
499C23—22no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23C2×D4×D5311yescharacteristic
507Q8—4244no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—84no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1062no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4#1313318912yes—
551C2×C4#1313318912yes—
552C2×C4C22×C22⋊D5311yescharacteristic
553C2×C4C2×Q8×D5311yescharacteristic
554C2×C4#41×D5311yescharacteristic
555C2×C4C2×#1171994711yescharacteristic
556C2×C4#1313326411yescharacteristic
557C2×C4#1313326511yescharacteristic
558C8—41no—
559C8—41no—
560C8—1062no—
561C8—1062no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22C2×#1313318911yescharacteristic
570C22#1681127511yescharacteristic
571C22#1681128611yescharacteristic
572C22#1681141911yescharacteristic
573C22#1681141911yescharacteristic
574C22#1681142511yescharacteristic
575C22#1681143011yescharacteristic
576C4—22no—
577C4—22no—
578C4—42no—
579C4—42no—
580C4—81no—
581C4—4241no—
582C4—4241no—
583C4—4241no—
584C4—2122no—
585C4—2121no—
586C4—2121no—
587C4—2121no—
588C4—2121no—
589C4#1681117511yescharacteristic
590C4#1681123311yescharacteristic
591C4#1681128011yescharacteristic
592C4Q8×C22⋊D5311yescharacteristic
593C2—82no—
594C2#2027831611yescharacteristic
595C2#2028420711yescharacteristic
596C2#2028425211yescharacteristic
597C2#2028721211yescharacteristic
598C2#2028722711yescharacteristic
599C2#2028765711yescharacteristic
600C2#2028767211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#478C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenterC2×#478D53
hypocenterC53#25466
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. C2×#16811033C2semidirect
3. #20277815C2semidirect
4. #20276200C2semidirect
5. #20276199C2semidirect
8. #20275666C2semidirect
9. #20275484C2semidirect
10. #20275480C2semidirect
11. #20275436C2semidirect
91. #13133138Q8semidirect
361. C53#25466semidirect

Maximal subgroups

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

subgroupindex
C2×#168110332
#202798882
#202778152
#202762002
#202761992
#202760992
#202760852
#202756662
#202754842
#202754802
#202754362
#202753452
#202753382
#202745402
C2×C53×#4782
#2546653