Finite Group Databasebeta

← #27442156

Subgroups — #27442156

Order 13568.

696 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277598C211yesmaximal, characteristic
2#20276282C211yesmaximal, characteristic, complemented
3#20276217C211yesmaximal, characteristic, complemented
4#20276217C211yesmaximal, characteristic, complemented
5#20276050C211yesmaximal, characteristic, complemented
6#20275673C211yesmaximal, characteristic, complemented
7#20275481C211yesmaximal, characteristic, complemented
8#20275290C211yesmaximal, characteristic, complemented
9#20275290C211yesmaximal, characteristic, complemented
10#20274908C211yesmaximal, characteristic
11#20274900C211yesmaximal, characteristic
12#20274807C211yesmaximal, characteristic
13#20274746C211yesmaximal, characteristic
14#20274434C211yesmaximal, characteristic
15C53×#2442C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25#16810521C2211yescharacteristic
26#16810521C2211yescharacteristic
27#16810511—21no—
28#16810511—21no—
29#16810487—21no—
30#16810483C2211yescharacteristic
31C4×C4⋊Dic53C2211yescharacteristic
32C4×C4⋊Dic53C2211yescharacteristic
33#16810468—21no—
34#16810385C2211yescharacteristic
35#16810385C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#125×C53C2211yescharacteristic
49C22×C4⋊C4×C53C2211yescharacteristic
50C53×#400—21no—
51C53×#398—21no—
52C53×#389C2211yescharacteristic
53C53×#389C2211yescharacteristic
54C53×#385—21no—
55C53×#382C2211yescharacteristic
56C53×#380—21no—
57C53×#380C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C2×#11719917—22no—
60C2×#11719917—22no—
61C2×#11719917—41no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133176—22no—
67#13133176Q814yes—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×C22⋊Dic53—21no—
75C2×C22⋊Dic53—21no—
76C2×C4×Dic53—21no—
77C2×C4×Dic53—21no—
78C2×C4×Dic53—21no—
79#13133150—41no—
80#13133148—22no—
81#13133148—22no—
82#13133148—41no—
83C4⋊#88322—22no—
84C4⋊#88322—41no—
85C22⋊#88322—22no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C23×C4×C53—21no—
92C23×C4×C53C2311yescharacteristic
93#125×C53—22no—
94#125×C53—41no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—22no—
101C2×C4⋊C4×C53—41no—
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53D411yescharacteristic
108C2×C22⋊C4×C53D411yescharacteristic
109C2×C42×C53C2311yescharacteristic
110C2×C42×C53C2311yescharacteristic
111#95×C53—21no—
112#95×C53—21no—
113#95×C53—21no—
114#95×C53—21no—
115#95×C53—21no—
116#95×C53D411yescharacteristic
117#95×C53D411yescharacteristic
118C24×C53C2×D411yescharacteristic
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53—81no—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×Q812yes—
136C22×C4×C53#4112yes—
137C22×C4×C53C2411yescharacteristic
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—42no—
150C4⋊C4×C53—42no—
151C4⋊C4×C53—42no—
152C4⋊C4×C53—81no—
153C4⋊C4×C53—81no—
154C4⋊C4×C53#4112yes—
155C22⋊C4×C53—41no—
156C22⋊C4×C53—41no—
157C22⋊C4×C53—42no—
158C42×C53—21no—
159C42×C53—21no—
160C42×C53—21no—
161C42×C53—21no—
162C22⋊Dic53—24no—
163C22⋊Dic53—81no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53C2×Q812yes—
171C4⋊Dic53C2×Q812yes—
172#11719919—81no—
173#11719919—81no—
174C4×Dic53—24no—
175C4×Dic53—42no—
176C4×Dic53—42no—
177#11719917—44no—
178C2×#88322—21no—
179C2×#88322—21no—
180C2×#88322—21no—
181C2×#88322—21no—
182C2×#88322—21no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322#4112yes—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53C22⋊Q812yes—
197C23×C53C22×D411yescharacteristic
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
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—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
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×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—84no—
229C2×C4×C53C22⋊Q812yes—
230C2×C4×C53C22×Q811yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53H(5,𝔽2)11yescharacteristic
235C2×C4×C532−1+411yescharacteristic
236C2×Dic53—22no—
237C2×Dic53—22no—
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—41no—
247C2×Dic53—41no—
248#88322—22no—
249#88322—41no—
250#88322—41no—
251#88322—41no—
252#26019—531nomaximal, Sylow
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53C2×C22⋊Q811yescharacteristic
261C22×C53#54611yescharacteristic
262C22×C53#54711yescharacteristic
263C22×C53#57511yescharacteristic
264C22×C53#57511yescharacteristic
265C22×C53#57711yescharacteristic
266C22×C53#57711yescharacteristic
267C4×C53—22no—
268C4×C53—41no—
269C4×C53—41no—
270C4×C53—41no—
271C4×C53—41no—
272C4×C53—42no—
273C4×C53—81no—
274C4×C53—81no—
275C4×C53—81no—
276C4×C53#53311yescharacteristic
277C4×C53#54411yescharacteristic
278C4×C53#55111yescharacteristic
279C4×C53#55211yescharacteristic
280Dic53—42no—
281Dic53—42no—
282Dic53—81no—
283Dic53—81no—
284#2442D5311yescharacteristic
285#2346—531no—
286#2101—531no—
287#2088—531no—
288#2088—531no—
289#2059—531no—
290#1989—531no—
291#1949—531no—
292#1919—531no—
293#1919—531no—
294#1853—531no—
295#1851—531no—
296#1825—531no—
297#1809—531no—
298#1736—531no—
299C2×C53—42no—
300C2×C53#243811yescharacteristic
301C2×C53#308011yescharacteristic
302C2×C53#308411yescharacteristic
303C2×C53#332111yescharacteristic
304C2×C53#332311yescharacteristic
305C2×C53#334811yescharacteristic
306C2×C53#335111yescharacteristic
307C22×M16—531no—
308C2×#125C2×D5311yescharacteristic
309C2×#117—531no—
310C22×C4⋊C4C2×D5311yescharacteristic
311C2×#107—531no—
312C2×#106—531no—
313C2×#106—531no—
314C2×#106—531no—
315C2×C4⋊C8—531no—
316C2×C4⋊C8—531no—
317C2×C22⋊C8—531no—
318#400—21no—
319#398—21no—
320#389—531no—
321#389—531no—
322#389C2×D5311yescharacteristic
323#389C2×D5311yescharacteristic
324#388—1061no—
325#388—1061no—
326#385—21no—
327#382—1061no—
328#382C2×D5311yescharacteristic
329#380—21no—
330#380—531no—
331#380C2×D5311yescharacteristic
332C4×C4⋊C4—531no—
333C4×C4⋊C4—531no—
334C4×C22⋊C4C2×D5311yescharacteristic
335#376—1061no—
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#340—531no—
348#336—531no—
349#336—531no—
350C53#2601911yescharacteristic, complemented, Sylow
351C23×C4—21no—
352C23×C4C22×D5311yescharacteristic
353C2×M16—1062no—
354C2×M16—1062no—
355C2×M16—2121no—
356C22×C8—531no—
357C22×C8—531no—
358C22×C8—531no—
359C22×C8—531no—
360#125—22no—
361#125—41no—
362#117—534no—
363#117—1062no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—41no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4C22×D5311yescharacteristic
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C4⋊C4C22×D5311yescharacteristic
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1061no—
383C2×C22⋊C4—1061no—
384C2×C22⋊C4C22⋊D5311yescharacteristic
385C2×C22⋊C4C22⋊D5311yescharacteristic
386C2×C42—1061no—
387C2×C42—1061no—
388C2×C42—1061no—
389C2×C42C22×D5311yescharacteristic
390C2×C42C22×D5311yescharacteristic
391#107—2121no—
392#106—1062no—
393#106—1062no—
394#106—2121no—
395C4⋊C8—1062no—
396C4⋊C8—2121no—
397C22⋊C8—1062no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—1061no—
404#95—1061no—
405#95—1061no—
406#95—1061no—
407#95—1061no—
408#95C22⋊D5311yescharacteristic
409#95C22⋊D5311yescharacteristic
410C24C2×C22⋊D5311yescharacteristic
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—81no—
422C22×C4—1061no—
423C22×C4—1061no—
424C22×C4—1061no—
425C22×C4—1061no—
426C22×C4Q8×D5312yes—
427C22×C4#1171994912yes—
428C22×C4C2×C22⋊D5311yescharacteristic
429C22×C4C2×C22⋊D5311yescharacteristic
430C22×C4C2×C22⋊D5311yescharacteristic
431C22×C4C2×C22⋊D5311yescharacteristic
432C22×C4C2×C22⋊D5311yescharacteristic
433C22×C4C23×D5311yescharacteristic
434M16—2124no—
435C2×C8—532no—
436C2×C8—1061no—
437C2×C8—1061no—
438C2×C8—1061no—
439C2×C8—1061no—
440C2×C8—1061no—
441C2×C8—1061no—
442C2×C8—1061no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4—532no—
460C4⋊C4—532no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—4241no—
468C4⋊C4—4241no—
469C4⋊C4#1171994712yes—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—42no—
473C22⋊C4—1064no—
474C22⋊C4—4241no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—1064no—
480C42—2122no—
481C42—2122no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23#1313319312yes—
489C23C22×C22⋊D5311yescharacteristic
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—84no—
521C2×C4—1062no—
522C2×C4—1062no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4—2121no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4#1313319312yes—
534C2×C4C2×Q8×D5311yescharacteristic
535C2×C4#41×D5311yescharacteristic
536C2×C4C2×#1171994711yescharacteristic
537C2×C4C2×#1171994911yescharacteristic
538C2×C4#1313326211yescharacteristic
539C2×C4#1313326411yescharacteristic
540C8—1062no—
541C8—2121no—
542C8—2121no—
543C8—2121no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22C2×#1313319311yescharacteristic
552C22#1681126711yescharacteristic
553C22#1681127211yescharacteristic
554C22#1681142111yescharacteristic
555C22#1681142111yescharacteristic
556C22#1681143111yescharacteristic
557C22#1681143411yescharacteristic
558C4—22no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—42no—
563C4—41no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—4241no—
568C4—4241no—
569C4—2122no—
570C4—2122no—
571C4#1681117511yescharacteristic
572C4#1681125211yescharacteristic
573C4#1681130011yescharacteristic
574C4#1681131511yescharacteristic
575C2—42no—
576C2#2027838711yescharacteristic
577C2#2028453311yescharacteristic
578C2#2028457311yescharacteristic
579C2#2028690111yescharacteristic
580C2#2028693111yescharacteristic
581C2#2028722411yescharacteristic
582C2#2028726911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2442C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2442D53
hypocenterC53#26019
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
2. #20276282C2semidirect
3. #20276217C2semidirect
4. #20276217C2semidirect
5. #20276050C2semidirect
6. #20275673C2semidirect
7. #20275481C2semidirect
8. #20275290C2semidirect
9. #20275290C2semidirect
350. C53#26019semidirect

Maximal subgroups

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

subgroupindex
#202775982
#202762822
#202762172
#202760502
#202756732
#202754812
#202752902
#202749082
#202749002
#202748072
#202747462
#202744342
C53×#24422
#2601953