Finite Group Databasebeta

← #27469751

Subgroups — #27469751

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280545C211yesmaximal, characteristic
2#20277338C211yesmaximal, characteristic
3#20276284C211yesmaximal, characteristic, complemented
4#20276284C211yesmaximal, characteristic, complemented
5#20276279C211yesmaximal, characteristic, complemented
6#20276274C211yesmaximal, characteristic, complemented
7#20275684C211yesmaximal, characteristic, complemented
8#20275680C211yesmaximal, characteristic, complemented
9#20274849C211yesmaximal, characteristic, complemented
10#20274843C211yesmaximal, characteristic, complemented
11#20274811C211yesmaximal, characteristic
12#20274811C211yesmaximal, characteristic
13#20274806C211yesmaximal, characteristic
14#20274802C211yesmaximal, characteristic
15C53×#1750C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19#16810556—21no—
20#16810556—21no—
21#16810546—21no—
22#16810546—21no—
23#16810522—21no—
24#16810522—21no—
25#16810521C2211yescharacteristic
26#16810521C2211yescharacteristic
27#16810498—21no—
28#16810498—21no—
29#16810493C2211yescharacteristic
30#16810493C2211yescharacteristic
31#16810493C2211yescharacteristic
32#16810489C2211yescharacteristic
33#16810489C2211yescharacteristic
34#16810483C2211yescharacteristic
35#16810483C2211yescharacteristic
36#16810386C2211yescharacteristic
37#16810386C2211yescharacteristic
38#16810386C2211yescharacteristic
39#16810386C2211yescharacteristic
40#16810386C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C2×C22⋊C8×C53C2211yescharacteristic
55C2×#97×C53C2211yescharacteristic
56C2×C4×C8×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C23×Dic53—21no—
60#13133209—22no—
61#13133209—22no—
62#13133176—22no—
63#13133176—22no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—22no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C4×Dic53—21no—
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C23×C4×C53C2311yescharacteristic
98C22×C8×C53C2311yescharacteristic
99C22×C8×C53C2311yescharacteristic
100C22×C8×C53C2311yescharacteristic
101C22×C8×C53C2311yescharacteristic
102#117×C53—22no—
103#117×C53—22no—
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C4⋊C4×C53D411yescharacteristic
106C2×C22⋊C4×C53D411yescharacteristic
107C2×C22⋊C4×C53D411yescharacteristic
108C2×C42×C53C2311yescharacteristic
109C2×C42×C53C2311yescharacteristic
110C22⋊C8×C53—22no—
111C22⋊C8×C53—22no—
112#97×C53—22no—
113C4×C8×C53—22no—
114C24×C53C2×D411yescharacteristic
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53#4112yes—
125C22×C4×C53#4112yes—
126C22×C4×C53C2411yescharacteristic
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—42no—
136C22×Dic53—81no—
137C22×Dic53—81no—
138C2×C8×C53—21no—
139C2×C8×C53—21no—
140C2×C8×C53—21no—
141C2×C8×C53—21no—
142C2×C8×C53—21no—
143C2×C8×C53—21no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—22no—
152C22⋊C4×C53—22no—
153C22⋊C4×C53—22no—
154C42×C53—21no—
155C42×C53—21no—
156C42×C53#4112yes—
157C42×C53#4112yes—
158C22⋊Dic53—42no—
159C22⋊Dic53—81no—
160C22⋊Dic53—81no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—42no—
178C4⋊Dic53—81no—
179C4⋊Dic53—81no—
180C4⋊Dic53—81no—
181C4⋊Dic53—81no—
182C4⋊Dic53—81no—
183C4⋊Dic53—81no—
184#11719919—42no—
185#11719919—42no—
186#11719919—81no—
187#11719919—81no—
188C4×Dic53—42no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53#12412yes—
196C23×C53C22×D411yescharacteristic
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53#12312yes—
214C2×C4×C53#12312yes—
215C2×C4×C53#12412yes—
216C2×C4×C53C2×#4111yescharacteristic
217C2×C4×C53C2×#4111yescharacteristic
218C2×C4×C53C2×#4111yescharacteristic
219C2×C4×C53C2×#4111yescharacteristic
220C2×C4×C532−1+411yescharacteristic
221C2×C4×C532−1+411yescharacteristic
222C2×Dic53—41no—
223C2×Dic53—41no—
224C2×Dic53—41no—
225C2×Dic53—41no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—84no—
247C8×C53—41no—
248C8×C53—41no—
249C8×C53—41no—
250C8×C53—41no—
251#27970—531nomaximal, Sylow
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53C2×#12311yescharacteristic
260C22×C53C2×#12411yescharacteristic
261C22×C53#53611yescharacteristic
262C22×C53#57611yescharacteristic
263C22×C53#57711yescharacteristic
264C22×C53#57711yescharacteristic
265C22×C53#57811yescharacteristic
266C4×C53—22no—
267C4×C53—22no—
268C4×C53—22no—
269C4×C53—41no—
270C4×C53—41no—
271C4×C53#55211yescharacteristic
272C4×C53#55211yescharacteristic
273C4×C53#55211yescharacteristic
274C4×C53#55211yescharacteristic
275Dic53—81no—
276Dic53—81no—
277Dic53—81no—
278Dic53—81no—
279Dic53—81no—
280Dic53—81no—
281Dic53—81no—
282Dic53—81no—
283#2629—531no—
284#2317—531no—
285#2101—531no—
286#2101—531no—
287#2100—531no—
288#2099—531no—
289#1992—531no—
290#1991—531no—
291#1836—531no—
292#1834—531no—
293#1825—531no—
294#1825—531no—
295#1824—531no—
296#1823—531no—
297#1750D5311yescharacteristic
298C2×C53—42no—
299C2×C53#250311yescharacteristic
300C2×C53#309811yescharacteristic
301C2×C53#309911yescharacteristic
302C2×C53#313611yescharacteristic
303C2×C53#313711yescharacteristic
304C2×C53#324111yescharacteristic
305C2×C53#324211yescharacteristic
306C2×#125—531no—
307C2×#117—531no—
308C2×#117C2×D5311yescharacteristic
309C22×C4⋊C4—531no—
310C2×C22⋊C8C2×D5311yescharacteristic
311C2×C22⋊C8C2×D5311yescharacteristic
312C2×#97C2×D5311yescharacteristic
313C2×C4×C8C2×D5311yescharacteristic
314#398—1061no—
315#398—1061no—
316#396—1061no—
317#396—1061no—
318#389—531no—
319#389—531no—
320#389—1061no—
321#389—1061no—
322#385—1061no—
323#385—1061no—
324#384—531no—
325#384—531no—
326#384—531no—
327#382—531no—
328#382—531no—
329#380—531no—
330#380—531no—
331#340—531no—
332#340—531no—
333#340—531no—
334#340—531no—
335#340—531no—
336#340—531no—
337#340—531no—
338#340—531no—
339#340—531no—
340#340—531no—
341#340—531no—
342#340—531no—
343#340—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#336C2×D5311yescharacteristic
348#336C2×D5311yescharacteristic
349C53#2797011yescharacteristic, complemented, Sylow
350C23×C4—1061no—
351C23×C4C22×D5311yescharacteristic
352C22×C8C22×D5311yescharacteristic
353C22×C8C22×D5311yescharacteristic
354C22×C8C22×D5311yescharacteristic
355C22×C8C22×D5311yescharacteristic
356#125—1062no—
357#125—1062no—
358#117—22no—
359#117—22no—
360#117—1062no—
361#117—1062no—
362C2×C4⋊C4—531no—
363C2×C4⋊C4—531no—
364C2×C4⋊C4—531no—
365C2×C4⋊C4—531no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—1062no—
371C2×C4⋊C4—1062no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4D21211yescharacteristic
384C2×C4⋊C4D21211yescharacteristic
385C2×C22⋊C4—1061no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C22⋊C4D21211yescharacteristic
389C2×C22⋊C4D21211yescharacteristic
390C2×C42—1061no—
391C2×C42C22×D5311yescharacteristic
392C2×C42C22×D5311yescharacteristic
393C22⋊C8—22no—
394C22⋊C8—22no—
395#97—22no—
396C4×C8—22no—
397#95—1061no—
398#95—1061no—
399#95—1061no—
400#95—1061no—
401#95—1061no—
402#95—1061no—
403#95—1061no—
404#95—1061no—
405C24C2×D21211yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—1061no—
411C22×C4—1061no—
412C22×C4—1061no—
413C22×C4—1061no—
414C22×C4—1061no—
415C22×C4—1061no—
416C22×C4—1061no—
417C22×C4—1061no—
418C22×C4—4241no—
419C22×C4—4241no—
420C22×C4—2122no—
421C22×C4C2×D21211yescharacteristic
422C22×C4C2×D21211yescharacteristic
423C22×C4C2×D21211yescharacteristic
424C22×C4C2×D21211yescharacteristic
425C22×C4C2×D21211yescharacteristic
426C22×C4#1171994512yes—
427C22×C4#1171994512yes—
428C22×C4C23×D5311yescharacteristic
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—1061no—
444C4⋊C4—1061no—
445C4⋊C4—1061no—
446C4⋊C4—1061no—
447C4⋊C4—1061no—
448C4⋊C4—1061no—
449C4⋊C4—1061no—
450C4⋊C4—1061no—
451C4⋊C4—1061no—
452C4⋊C4—1061no—
453C4⋊C4—1061no—
454C4⋊C4—1061no—
455C4⋊C4—1061no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—4241no—
460C4⋊C4—4241no—
461C4⋊C4—4241no—
462C4⋊C4—4241no—
463C4⋊C4—4241no—
464C4⋊C4—4241no—
465C4⋊C4—4241no—
466C4⋊C4—4241no—
467C4⋊C4—2122no—
468C4⋊C4—2122no—
469C4⋊C4—2122no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—4241no—
473C22⋊C4—4241no—
474C22⋊C4—2122no—
475C42—21no—
476C42—21no—
477C42—2122no—
478C42#1171994712yes—
479C42#1171994712yes—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23#1313320512yes—
487C23C22×D21211yescharacteristic
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—4244no—
505C2×C4—2121no—
506C2×C4—2121no—
507C2×C4—2121no—
508C2×C4—2121no—
509C2×C4—2121no—
510C2×C4—2121no—
511C2×C4—2121no—
512C2×C4—2121no—
513C2×C4—2121no—
514C2×C4—2121no—
515C2×C4—2121no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4—2121no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4#1313320112yes—
530C2×C4#1313320112yes—
531C2×C4#1313320512yes—
532C2×C4C2×#1171994711yescharacteristic
533C2×C4C2×#1171994711yescharacteristic
534C2×C4C2×#1171994511yescharacteristic
535C2×C4C2×#1171994511yescharacteristic
536C2×C4#1313326411yescharacteristic
537C2×C4#1313326411yescharacteristic
538C8—41no—
539C8—41no—
540C8—41no—
541C8—41no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22C2×#1313320111yescharacteristic
550C22C2×#1313320511yescharacteristic
551C22#1681119911yescharacteristic
552C22#1681142811yescharacteristic
553C22#1681143511yescharacteristic
554C22#1681143511yescharacteristic
555C22#1681144011yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—41no—
560C4—41no—
561C4—4241no—
562C4—4241no—
563C4—4241no—
564C4—4241no—
565C4—4241no—
566C4—4241no—
567C4—4241no—
568C4—4241no—
569C4#1681131311yescharacteristic
570C4#1681131311yescharacteristic
571C4#1681131311yescharacteristic
572C4#1681131311yescharacteristic
573C2—42no—
574C2#2027920011yescharacteristic
575C2#2028469511yescharacteristic
576C2#2028470611yescharacteristic
577C2#2028507111yescharacteristic
578C2#2028508211yescharacteristic
579C2#2028597011yescharacteristic
580C2#2028598511yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1750C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1750D53
hypocenterC53#27970
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1750

Complements

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

NH ≅ G/Ntype
3. #20276284C2semidirect
4. #20276284C2semidirect
5. #20276279C2semidirect
6. #20276274C2semidirect
7. #20275684C2semidirect
8. #20275680C2semidirect
9. #20274849C2semidirect
10. #20274843C2semidirect
349. C53#27970semidirect

Maximal subgroups

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

subgroupindex
#202805452
#202773382
#202762842
#202762792
#202762742
#202756842
#202756802
#202748492
#202748432
#202748112
#202748062
#202748022
C53×#17502
#2797053