Finite Group Databasebeta

← #27490741

Subgroups — #27490741

Order 13568.

966 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810862C211yesmaximal, characteristic, complemented
2C2×#16810590C211yesmaximal, characteristic
3#20280444C211yesmaximal, characteristic
4#20276146C212yesmaximal, complemented
5#20275911C212yesmaximal, complemented
6#20275908C212yesmaximal, complemented
7#20275723C212yesmaximal, complemented
8#20275149C211yesmaximal, characteristic, complemented
9#20275146C211yesmaximal, characteristic, complemented
10#20274738C211yesmaximal, characteristic, complemented
11C53×#2616C211yesmaximal, characteristic
12C22×#11719925C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14C23×#88322C2211yescharacteristic
15C2×#13133196C2211yescharacteristic
16C2×#13133196C2212yes—
17#16810862—24no—
18C2×#13133150C2211yescharacteristic
19C2×#13133148C2212yes—
20C2×#13133141C2211yescharacteristic
21C2×#13133141C2212yes—
22C2×#13133140C2211yescharacteristic
23C2×#13133140C2212yes—
24#16810590—24no—
25C2×C22⋊#88322C2212yes—
26C2×C22⋊#88322C2212yes—
27#16810542—21no—
28#16810537—22no—
29#16810537—21no—
30#16810483C2211yescharacteristic
31#16810483C2212yes—
32#16810385C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35C22×#41×C53C2211yescharacteristic
36C2×C22⋊Q8×C53C2211yescharacteristic
37C2×C22⋊Q8×C53C2212yes—
38C53×#395—21no—
39C53×#394—22no—
40C53×#394—21no—
41C53×#380C2211yescharacteristic
42C53×#380C2212yes—
43C22×Dic106C2311yescharacteristic
44C2×#11719925—22no—
45C2×#11719925—42no—
46C2×#11719917—22no—
47C2×#11719917—42no—
48C22×#88322—42no—
49C22×#88322D414yes—
50C22×#88322C2312yes—
51C22×#88322C2311yescharacteristic
52C22×#88322C2311yescharacteristic
53#13133196—22no—
54#13133196—22no—
55#13133196—42no—
56#13133196—42no—
57C2×C4⋊Dic53C2312yes—
58C2×C4⋊Dic53C2311yescharacteristic
59C2×#11719919—22no—
60C2×#11719919—21no—
61C2×C22⋊Dic53—22no—
62C2×C22⋊Dic53—21no—
63C2×C22⋊Dic53—22no—
64C2×C22⋊Dic53—21no—
65#13133150—22no—
66#13133148—42no—
67#13133141—22no—
68#13133141—42no—
69#13133140—22no—
70#13133140—42no—
71C22⋊#88322—24no—
72C22⋊#88322—24no—
73#13133124—22no—
74#13133124—21no—
75C2×#41×C53—22no—
76C2×#41×C53—22no—
77C2×#41×C53—42no—
78C2×#41×C53—42no—
79C22×Q8×C53D411yescharacteristic
80C22×Q8×C53C2311yescharacteristic
81C22×D4×C53D412yes—
82C22×D4×C53D411yescharacteristic
83C23×C4×C53C2312yes—
84C23×C4×C53C2311yescharacteristic
85C22⋊Q8×C53—22no—
86C22⋊Q8×C53—22no—
87C22⋊Q8×C53—42no—
88C22⋊Q8×C53—42no—
89C2×C4⋊C4×C53—22no—
90C2×C4⋊C4×C53—21no—
91C2×C4⋊C4×C53C2312yes—
92C2×C4⋊C4×C53C2311yescharacteristic
93C2×C22⋊C4×C53—22no—
94C2×C22⋊C4×C53—21no—
95C2×C22⋊C4×C53—22no—
96C2×C22⋊C4×C53—21no—
97#95×C53—22no—
98#95×C53—21no—
99C24×C53C2×D412yes—
100C24×C53C2×D411yescharacteristic
101#41×C53—44no—
102#41×C53—44no—
103#41×C53—48no—
104C2×Q8×C53—22no—
105C2×Q8×C53—21no—
106C2×Q8×C53—21no—
107C2×Q8×C53—22no—
108C2×Q8×C53—22no—
109C2×Q8×C53—42no—
110C2×D4×C53—24no—
111C2×D4×C53—22no—
112C2×D4×C53—42no—
113C2×D4×C53—42no—
114C2×D4×C53—42no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53—22no—
125C22×C4×C53—21no—
126C22×C4×C53C2×D412yes—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D412yes—
129C22×C4×C53#4112yes—
130C22×C4×C53C2411yescharacteristic
131C22×Dic53—21no—
132C22×Dic53—22no—
133C22×Dic53—21no—
134C2×Dic106—21no—
135C2×Dic106—21no—
136C2×Dic106—22no—
137C2×Dic106—22no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—44no—
143C4⋊C4×C53—42no—
144C22⋊C4×C53—44no—
145C22⋊C4×C53—42no—
146C22⋊C4×C53—81no—
147C22⋊C4×C53—82no—
148C22⋊Dic53—44no—
149C22⋊Dic53—42no—
150C22⋊Dic53—81no—
151C22⋊Dic53—82no—
152#11719925—44no—
153C4⋊Dic53—21no—
154C4⋊Dic53—21no—
155C4⋊Dic53—22no—
156C4⋊Dic53—22no—
157#11719919—44no—
158#11719919—42no—
159#11719917—44no—
160C2×#88322—22no—
161C2×#88322—22no—
162C2×#88322—22no—
163C2×#88322—22no—
164C2×#88322—22no—
165C2×#88322—44no—
166C2×#88322C2×D412yes—
167C2×#88322C2×D412yes—
168C23×C53—22no—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53—22no—
172C23×C53—22no—
173C23×C53—22no—
174C23×C53—22no—
175C23×C53—22no—
176C23×C53—22no—
177C23×C53—22no—
178C23×C53—22no—
179C23×C53#12112yes—
180C23×C53C22×D411yescharacteristic
181Q8×C53—44no—
182Q8×C53—41no—
183Q8×C53—41no—
184Q8×C53—42no—
185D4×C53—44no—
186D4×C53—48no—
187C2×C4×C53—22no—
188C2×C4×C53—22no—
189C2×C4×C53—22no—
190C2×C4×C53—24no—
191C2×C4×C53—22no—
192C2×C4×C53—22no—
193C2×C4×C53—22no—
194C2×C4×C53—42no—
195C2×C4×C53—42no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×C4×C53—41no—
203C2×C4×C53—42no—
204C2×C4×C53—41no—
205C2×C4×C53—42no—
206C2×C4×C53#12112yes—
207C2×C4×C53C22×D411yescharacteristic
208C2×C4×C53C2×#4111yescharacteristic
209C2×C4×C53H(5,𝔽2)12yes—
210C2×C4×C53H(5,𝔽2)12yes—
211C2×Dic53—41no—
212C2×Dic53—41no—
213C2×Dic53—42no—
214C2×Dic53—42no—
215C2×Dic53—41no—
216C2×Dic53—42no—
217C2×Dic53—41no—
218C2×Dic53—42no—
219Dic106—41no—
220Dic106—41no—
221Dic106—42no—
222#88322—22no—
223#88322—22no—
224#88322—42no—
225#29737—531nomaximal, Sylow
226C22×C53—22no—
227C22×C53—22no—
228C22×C53—22no—
229C22×C53—42no—
230C22×C53—42no—
231C22×C53—42no—
232C22×C53—42no—
233C22×C53—42no—
234C22×C53—42no—
235C22×C53—42no—
236C22×C53—42no—
237C22×C53—42no—
238C22×C53C2×#12111yescharacteristic
239C22×C53#53412yes—
240C22×C53#57711yescharacteristic
241C22×C53#57711yescharacteristic
242C22×C53#57812yes—
243C4×C53—22no—
244C4×C53—42no—
245C4×C53—81no—
246C4×C53—82no—
247C4×C53—81no—
248C4×C53#53411yescharacteristic
249C4×C53#53511yescharacteristic
250C4×C53#53812yes—
251Dic53—81no—
252Dic53—82no—
253Dic53—81no—
254C2×#467—531no—
255C2×#408—531no—
256#2616—531no—
257#2616D5311yescharacteristic
258#2076—532no—
259#2035—532no—
260#2035—532no—
261#2001—532no—
262#1894—531no—
263#1894—531no—
264#1807—531no—
265C2×C53—42no—
266C2×C53—82no—
267C2×C53#261711yescharacteristic
268C2×C53#305911yescharacteristic
269C2×C53#306011yescharacteristic
270C2×C53#320112yes—
271C2×C53#320112yes—
272C22×#41C2×D5311yescharacteristic
273C22×Q16—531no—
274C22×M16—531no—
275C23×C8—531no—
276C2×C22⋊Q8—531no—
277C2×C22⋊Q8—532no—
278C2×C22⋊Q8C2×D5311yescharacteristic
279C2×C22⋊Q8C2×D5312yes—
280#467—1064no—
281C2×#107—531no—
282C2×#106—532no—
283C2×#103—531no—
284C2×#103—531no—
285C2×#103—532no—
286C2×#103—532no—
287#408—1064no—
288C2×C22⋊C8—532no—
289C2×C22⋊C8—532no—
290#395—21no—
291#395—1061no—
292#394—22no—
293#394—21no—
294#394—1062no—
295#394—1061no—
296#380—531no—
297#380—532no—
298#380C2×D5311yescharacteristic
299#380C2×D5312yes—
300#340—532no—
301#340—532no—
302#340—532no—
303C53#2973711yescharacteristic, complemented, Sylow
304C2×#41—22no—
305C2×#41—22no—
306C2×#41—42no—
307C2×#41—42no—
308C22×Q8—531no—
309C22×Q8C22⋊D5311yescharacteristic
310C22×Q8C22×D5311yescharacteristic
311C22×D4C22⋊D5312yes—
312C22×D4C22⋊D5311yescharacteristic
313C23×C4C22×D5312yes—
314C23×C4C22×D5311yescharacteristic
315C2×Q16—1062no—
316C2×Q16—2122no—
317C2×M16—1062no—
318C2×M16—2122no—
319C22×C8—531no—
320C22×C8—531no—
321C22×C8—534no—
322C22×C8—532no—
323C22×C8—2122no—
324C22⋊Q8—22no—
325C22⋊Q8—22no—
326C22⋊Q8—42no—
327C22⋊Q8—42no—
328C22⋊Q8—1062no—
329C22⋊Q8—1062no—
330C22⋊Q8—2122no—
331C22⋊Q8—2122no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—532no—
335C2×C4⋊C4—531no—
336C2×C4⋊C4—1062no—
337C2×C4⋊C4—1061no—
338C2×C4⋊C4C22×D5312yes—
339C2×C4⋊C4C22×D5311yescharacteristic
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—1062no—
345C2×C22⋊C4—1061no—
346C2×C22⋊C4—1062no—
347C2×C22⋊C4—1061no—
348#107—1062no—
349#106—2122no—
350#103—1062no—
351#103—1062no—
352#103—2122no—
353#103—2122no—
354C22⋊C8—1064no—
355C22⋊C8—1064no—
356#95—22no—
357#95—21no—
358#95—1062no—
359#95—1061no—
360C24C2×C22⋊D5312yes—
361C24C2×C22⋊D5311yescharacteristic
362#41—44no—
363#41—44no—
364#41—48no—
365C2×Q8—22no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—42no—
371C2×Q8—1061no—
372C2×Q8—1061no—
373C2×Q8—1062no—
374C2×Q8—1062no—
375C2×D4—24no—
376C2×D4—22no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—1061no—
392C22×C4—1062no—
393C22×C4—1061no—
394C22×C4D4×D5312yes—
395C22×C4#1171994712yes—
396C22×C4C2×C22⋊D5312yes—
397C22×C4C2×C22⋊D5311yescharacteristic
398C22×C4C23×D5311yescharacteristic
399Q16—2124no—
400M16—2124no—
401C2×C8—532no—
402C2×C8—532no—
403C2×C8—1062no—
404C2×C8—1062no—
405C2×C8—1062no—
406C2×C8—1062no—
407C2×C8—1062no—
408C2×C8—2124no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—44no—
414C4⋊C4—42no—
415C4⋊C4—1061no—
416C4⋊C4—1061no—
417C4⋊C4—1062no—
418C4⋊C4—1062no—
419C4⋊C4—2124no—
420C4⋊C4—2122no—
421C22⋊C4—44no—
422C22⋊C4—42no—
423C22⋊C4—81no—
424C22⋊C4—82no—
425C22⋊C4—4241no—
426C22⋊C4—4242no—
427C22⋊C4—2124no—
428C22⋊C4—2122no—
429C23—21no—
430C23—21no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23#1313318912yes—
441C23C22×C22⋊D5311yescharacteristic
442Q8—44no—
443Q8—41no—
444Q8—41no—
445Q8—42no—
446Q8—2121no—
447Q8—2121no—
448Q8—2122no—
449D4—44no—
450D4—48no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—2121no—
471C2×C4—2121no—
472C2×C4—2122no—
473C2×C4—2121no—
474C2×C4—2121no—
475C2×C4—2122no—
476C2×C4—2122no—
477C2×C4—2122no—
478C2×C4#1313318912yes—
479C2×C4C2×D4×D5311yescharacteristic
480C2×C4C2×#1171994711yescharacteristic
481C2×C4#1313326212yes—
482C2×C4#1313326212yes—
483C8—1062no—
484C8—1062no—
485C8—2122no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22C2×#1313318911yescharacteristic
499C22#1681118212yes—
500C22#1681143111yescharacteristic
501C22#1681143411yescharacteristic
502C22#1681143812yes—
503C4—22no—
504C4—42no—
505C4—81no—
506C4—82no—
507C4—81no—
508C4—4241no—
509C4—4242no—
510C4—4241no—
511C4#1681118311yescharacteristic
512C4#1681118811yescharacteristic
513C4#1681120812yes—
514C2—42no—
515C2—82no—
516C2#2028045311yescharacteristic
517C2#2028436711yescharacteristic
518C2#2028437411yescharacteristic
519C2#2028559112yes—
520C2#2028559712yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2616C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
hypercenter#2616D53
hypocenterC53#29737
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2616

Complements

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

NH ≅ G/Ntype
1. C2×#16810862C2semidirect
4. #20276146C2semidirect
5. #20275911C2semidirect
6. #20275908C2semidirect
7. #20275723C2semidirect
8. #20275149C2semidirect
9. #20275146C2semidirect
10. #20274738C2semidirect
303. C53#29737semidirect

Maximal subgroups

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

subgroupindex
C2×#168108622
C2×#168105902
#202804442
#202761462
#202759112
#202759082
#202757232
#202751492
#202751462
#202747382
C53×#26162
#2973753