Finite Group Databasebeta

← #27430134

Subgroups — #27430134

Order 13568.

766 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279636C211yesmaximal, characteristic
2#20277627C211yesmaximal, characteristic, complemented
3#20276249C211yesmaximal, characteristic
4#20276245C211yesmaximal, characteristic
5#20276195C211yesmaximal, characteristic, complemented
6#20275649C211yesmaximal, characteristic, complemented
7#20275475C211yesmaximal, characteristic, complemented
8#20275343C211yesmaximal, characteristic
9#20275341C211yesmaximal, characteristic
10#20275291C211yesmaximal, characteristic, complemented
11#20275290C211yesmaximal, characteristic, complemented
12#20274870C211yesmaximal, characteristic, complemented
13#20274714C211yesmaximal, characteristic
14#20274529C211yesmaximal, characteristic
15C53×#2084C211yesmaximal, characteristic, complemented
16C2×#13133195—21no—
17C2×C4×Dic106C2211yescharacteristic
18C2×#13133182—21no—
19C2×#13133179C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×C8×Dic53C2211yescharacteristic
30#16810547—21no—
31#16810545—21no—
32#16810525C2211yescharacteristic
33#16810524C2211yescharacteristic
34#16810519—21no—
35#16810519C2211yescharacteristic
36#16810518—21no—
37#16810518—21no—
38#16810492C2211yescharacteristic
39#16810492C2211yescharacteristic
40#16810492C2211yescharacteristic
41#16810482—21no—
42C4⋊C4×Dic53C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810378C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#106×C53C2211yescharacteristic
53C2×#106×C53C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#384C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106D411yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C4×D53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133195—44no—
65C4×Dic106—22no—
66C4×Dic106—22no—
67#13133182—44no—
68#13133179—22no—
69#13133179—22no—
70C2×C4⋊Dic53D411yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919D411yescharacteristic
82C2×#11719919D411yescharacteristic
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53C2311yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93#13133149—41no—
94#13133148—22no—
95#13133148—22no—
96#13133148—41no—
97#13133146—22no—
98#13133139—41no—
99#13133139Q814yescomplemented
100#13133137—22no—
101#13133137—41no—
102C8×Dic53—22no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117#106×C53—41no—
118#106×C53—41no—
119#95×C53—21no—
120C23×D53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2411yescharacteristic
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53C2×D411yescharacteristic
132C22×Dic53C2×D411yescharacteristic
133C22×Dic53C2×D411yescharacteristic
134C22×Dic53C2×D411yescharacteristic
135C2×D212—21no—
136C2×D212—21no—
137C2×D212—21no—
138C2×D212—21no—
139C2×D212—21no—
140C2×D212C2×Q812yes—
141C2×C4×D53—42no—
142C2×C4×D53—42no—
143C2×C4×D53—42no—
144C2×Dic106—22no—
145C2×Dic106—22no—
146C2×Dic106SD1614yes—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53C2×Q812yes—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—42no—
162C4⋊C4×C53—42no—
163C4⋊C4×C53—81no—
164C42×C53—21no—
165C42×C53—21no—
166#11719921—42no—
167#11719921—42no—
168#11719921—42no—
169#11719921—81no—
170#11719921—81no—
171C4⋊Dic53—21no—
172C4⋊Dic53—22no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53C2×Q812yes—
178#11719919—22no—
179#11719919—22no—
180#11719919—42no—
181#11719919—42no—
182#11719919—42no—
183#11719919—42no—
184#11719919—81no—
185#11719919—81no—
186#11719919—81no—
187C4×Dic53—21no—
188C4×Dic53—21no—
189C4×Dic53—42no—
190C4×Dic53—42no—
191C4×Dic53—42no—
192C4×Dic53#4112yes—
193C4×Dic53#4112yes—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322#4112yes—
200C23×C53C22×D411yescharacteristic
201C22×D53—22no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53—41no—
207C22×D53—41no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
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—41no—
224C2×C4×C53—41no—
225C2×C4×C53C2×SD1612yes—
226C2×C4×C53C22×Q811yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53H(5,𝔽2)11yescharacteristic
231C2×C4×C532−1+411yescharacteristic
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—22no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53#12312yes—
249C2×Dic53#12312yes—
250C2×Dic53C2×SD1612yes—
251C2×Dic53C2×SD1612yes—
252C2×Dic53AGL(1,ℤ/8)12yes—
253D212—22no—
254D212—22no—
255D212—41no—
256D212—41no—
257C4×D53—84no—
258Dic106—28no—
259C8×C53—22no—
260C8×C53—41no—
261#88322—22no—
262#88322—41no—
263#25058—531nomaximal, Sylow
264C22×C53C2×#12311yescharacteristic
265C22×C53D4×Q811yescharacteristic
266C22×C53#55011yescharacteristic
267C22×C53C22×SD1611yescharacteristic
268C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C53#57711yescharacteristic
270C22×C53#57711yescharacteristic
271C2×D53—42no—
272C2×D53—42no—
273C2×D53—42no—
274C2×D53—81no—
275C2×D53—81no—
276C2×D53—81no—
277C2×D53—81no—
278C4×C53—22no—
279C4×C53—41no—
280C4×C53—81no—
281C4×C53—81no—
282C4×C53—81no—
283C4×C53—81no—
284C4×C53#53311yescharacteristic
285C4×C53#54211yescharacteristic
286C4×C53#55211yescharacteristic
287C4×C53#55411yescharacteristic
288Dic53—22no—
289Dic53—22no—
290Dic53—41no—
291Dic53—42no—
292Dic53—81no—
293Dic53—81no—
294Dic53#48114yes—
295#2538—531no—
296Q8×C4⋊C4—531no—
297#2094—531no—
298#2094—531no—
299#2084—531no—
300#2084D5311yescharacteristic, complemented
301#1984—531no—
302#1948—531no—
303#1926—531no—
304#1926—531no—
305#1919—531no—
306#1919—531no—
307#1844—531no—
308#1802—531no—
309C4×#102—531no—
310C2×C53#242311yescharacteristic
311C2×C53C2×#48111yescharacteristic
312C2×C53#309911yescharacteristic
313C2×C53Q8×SD1611yescharacteristic
314C2×C53#338411yescharacteristic
315C2×C53#339211yescharacteristic
316C2×C53#339711yescharacteristic
317D53—82no—
318C2×C22⋊Q8—1061no—
319C2×C4×Q8—531no—
320C2×C4×Q8—1061no—
321C2×C4×D4—531no—
322C2×#106—531no—
323C2×#106—531no—
324C2×#106—531no—
325C2×#106—531no—
326C2×#106C2×D5311yescharacteristic
327C2×#106C2×D5311yescharacteristic
328C2×C4⋊C8—531no—
329C2×#102—531no—
330C2×#102—531no—
331C2×#102—531no—
332C2×#102—531no—
333C2×C4×C8—531no—
334#396—1061no—
335#396—1061no—
336#390—531no—
337#390—531no—
338#389—531no—
339#389—1061no—
340#389—1061no—
341#389—1061no—
342#389C2×D5311yescharacteristic
343#384—531no—
344#384—531no—
345#384—531no—
346#384C2×D5311yescharacteristic
347#380—1061no—
348C4×C4⋊C4—531no—
349C4×C4⋊C4—531no—
350C4×C4⋊C4—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340C2×D5311yescharacteristic
356#340C2×D5311yescharacteristic
357#336—531no—
358#336—531no—
359#336—531no—
360#336C2×D5311yescharacteristic
361C53#2505811yescharacteristic, complemented, Sylow
362C22×Q8—531no—
363C22×D4—531no—
364C23×C4—1061no—
365C22×C8—531no—
366C22×C8—531no—
367C22×C8C22×D5311yescharacteristic
368C22×C8C22×D5311yescharacteristic
369C22⋊Q8—2124no—
370C4×Q8—1062no—
371C4×Q8—1062no—
372C4×Q8—2124no—
373C4×D4—1062no—
374C4×D4—1062no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C4⋊C4C22×D5311yescharacteristic
393C2×C4⋊C4C22×D5311yescharacteristic
394C2×C4⋊C4C22×D5311yescharacteristic
395C2×C22⋊C4—1061no—
396C2×C22⋊C4—1061no—
397C2×C22⋊C4—1061no—
398C2×C22⋊C4—1061no—
399C2×C22⋊C4—1061no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—1061no—
403C2×C42—1061no—
404C2×C42—1061no—
405C2×C42C22×D5311yescharacteristic
406#106—41no—
407#106—41no—
408#106—1062no—
409#106—1062no—
410#106—2121no—
411#106—2121no—
412C4⋊C8—1062no—
413#102—534no—
414#102—1062no—
415#102—2121no—
416#102—2121no—
417C4×C8—1062no—
418#95—21no—
419#95—1061no—
420#95—1061no—
421#95—1061no—
422#95—1061no—
423C24—1061no—
424C2×Q8—534no—
425C2×Q8—1062no—
426C2×Q8—1062no—
427C2×D4—532no—
428C2×D4—1061no—
429C2×D4—1061no—
430C2×D4—1061no—
431C2×D4—1061no—
432C2×D4—1061no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—531no—
441C22×C4—1061no—
442C22×C4—1061no—
443C22×C4—1061no—
444C22×C4—2122no—
445C22×C4—2122no—
446C22×C4—2122no—
447C22×C4D4×D5311yescharacteristic
448C22×C4D4×D5311yescharacteristic
449C22×C4C23×D5311yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—532no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8Q8×D5312yes—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—81no—
473C4⋊C4—532no—
474C4⋊C4—1062no—
475C4⋊C4—1062no—
476C4⋊C4—1062no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—4241no—
483C4⋊C4—4241no—
484C4⋊C4—4241no—
485C4⋊C4—2122no—
486C4⋊C4—2122no—
487C4⋊C4—2122no—
488C4⋊C4—2122no—
489C22⋊C4—4241no—
490C22⋊C4—4241no—
491C22⋊C4—2122no—
492C22⋊C4—2122no—
493C22⋊C4—2122no—
494C42—21no—
495C42—21no—
496C42—532no—
497C42—532no—
498C42—1061no—
499C42—1061no—
500C42—2122no—
501C42—2122no—
502C42—2122no—
503C23—1062no—
504C23—2121no—
505C23—2121no—
506C23—2121no—
507C23—2121no—
508C23—2121no—
509C23—2121no—
510C23C2×D4×D5311yescharacteristic
511Q8—1068no—
512D4—1062no—
513D4—1062no—
514D4—2121no—
515D4—2121no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—532no—
534C2×C4—532no—
535C2×C4—532no—
536C2×C4—532no—
537C2×C4—532no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1062no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—4244no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4SD16×D5312yes—
556C2×C4C2×Q8×D5311yescharacteristic
557C2×C4#41×D5311yescharacteristic
558C2×C4#41×D5311yescharacteristic
559C2×C4#41×D5311yescharacteristic
560C2×C4#1313326211yescharacteristic
561C2×C4#1313326511yescharacteristic
562C8—22no—
563C8—41no—
564C8—1062no—
565C8—2121no—
566C22—4241no—
567C22—4241no—
568C22—4241no—
569C22—4241no—
570C22—2122no—
571C22—2122no—
572C22—2122no—
573C22#123×D5311yescharacteristic
574C22#1681128811yescharacteristic
575C22#1681129111yescharacteristic
576C22C2×SD16×D5311yescharacteristic
577C22AGL(1,ℤ/8)×D5311yescharacteristic
578C22#1681142911yescharacteristic
579C22#1681143211yescharacteristic
580C4—22no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—534no—
587C4—1062no—
588C4—1062no—
589C4—4241no—
590C4—4241no—
591C4—2121no—
592C4—2122no—
593C4#1681117911yescharacteristic
594C4#1681124311yescharacteristic
595C4#1681130511yescharacteristic
596C4#1681132711yescharacteristic
597C2—4242no—
598C2#2027823511yescharacteristic
599C2#481×D5311yescharacteristic
600C2#2028469911yescharacteristic
601C2#2028757211yescharacteristic
602C2#2028761111yescharacteristic
603C2#2028769111yescharacteristic
604C2#2028773011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2084C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#2084D53
hypocenterC53#25058
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2084

Complements

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

NH ≅ G/Ntype
2. #20277627C2semidirect
5. #20276195C2semidirect
6. #20275649C2semidirect
7. #20275475C2semidirect
10. #20275291C2semidirect
11. #20275290C2semidirect
12. #20274870C2semidirect
15. C53×#2084C2semidirect
99. #13133139Q8semidirect
300. #2084D53semidirect
361. C53#25058semidirect

Maximal subgroups

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

subgroupindex
#202796362
#202776272
#202762492
#202762452
#202761952
#202756492
#202754752
#202753432
#202753412
#202752912
#202752902
#202748702
#202747142
#202745292
C53×#20842
#2505853