Finite Group Databasebeta

← #27276590

Subgroups — #27276590

Order 13568.

1040 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274643C211yesmaximal, characteristic, complemented
2#20274633C211yesmaximal, characteristic, complemented
3#20273248C212yesmaximal, complemented
4#20273244C212yesmaximal, complemented
5C2×C53×#537C211yesmaximal, characteristic
6C2×#13133136—21no—
7C2×#13133134—21no—
8C2×#13133132—21no—
9C2×#13133132—21no—
10#16810587C2212yes—
11#16810486C2212yes—
12#16810360—24no—
13#16810357—24no—
14C22×#41×C53C411yescharacteristic, complemented
15C22×#41×C53C2211yescharacteristic
16C53×#537—24no—
17C53×#537—24no—
18C2×C22⋊Q8×C53—21no—
19C2×C22⋊Q8×C53C411yescharacteristic, complemented
20C2×C22⋊Q8×C53C2211yescharacteristic
21C2×#121×C53—21no—
22C2×#121×C53—21no—
23C2×#121×C53—21no—
24C2×#121×C53C411yescharacteristic, complemented
25C2×#121×C53C2211yescharacteristic
26C22×C4⋊C4×C53C411yescharacteristic, complemented
27C2×#11719917—22no—
28C2×C22⋊Dic53—22no—
29#13133136—42no—
30#13133134—42no—
31#13133132—42no—
32#13133132—42no—
33C22⋊#88322—22no—
34C22⋊#88322—22no—
35#13133124—22no—
36#13133124—22no—
37C2×#41×C53—22no—
38C2×#41×C53—22no—
39C2×#41×C53—22no—
40C2×#41×C53—22no—
41C2×#41×C53—42no—
42C2×#41×C53—42no—
43C2×#41×C53—42no—
44C2×#41×C53—42no—
45C22×Q8×C53C2×C411yescharacteristic
46C22×Q8×C53D411yescharacteristic
47C22×D4×C53—21no—
48C22×D4×C53—21no—
49C22×D4×C53—21no—
50C22×D4×C53C2×C411yescharacteristic
51C22×D4×C53D411yescharacteristic
52C22×D4×C53D411yescharacteristic
53C22×D4×C53D411yescharacteristic
54C23×C4×C53—21no—
55C23×C4×C53—21no—
56C23×C4×C53C2×C411yescharacteristic
57C23×C4×C53C2×C411yescharacteristic
58C23×C4×C53C2311yescharacteristic
59C22⋊Q8×C53—28no—
60C22⋊Q8×C53—22no—
61C22⋊Q8×C53—22no—
62C22⋊Q8×C53—22no—
63C22⋊Q8×C53D414yes—
64#121×C53—28no—
65#121×C53—28no—
66#121×C53—28no—
67#121×C53—22no—
68#121×C53—22no—
69#121×C53—22no—
70#121×C53D414yes—
71C2×C4⋊C4×C53—22no—
72C2×C4⋊C4×C53—22no—
73C2×C4⋊C4×C53—21no—
74C2×C4⋊C4×C53—21no—
75C2×C4⋊C4×C53—21no—
76C2×C4⋊C4×C53C2×C411yescharacteristic
77C2×C4⋊C4×C53C2×C411yescharacteristic
78C2×C22⋊C4×C53—21no—
79C2×C22⋊C4×C53—21no—
80C2×C22⋊C4×C53—21no—
81C2×C22⋊C4×C53—21no—
82C24×C53—21no—
83C24×C53C22⋊C411yescharacteristic
84C24×C53C22⋊C411yescharacteristic
85C24×C53C2×D411yescharacteristic
86#41×C53—48no—
87#41×C53—48no—
88#41×C53—48no—
89#41×C53—48no—
90C2×Q8×C53—22no—
91C2×Q8×C53—21no—
92C2×Q8×C53—22no—
93C2×Q8×C53—41no—
94C2×Q8×C53—41no—
95C2×Q8×C53—41no—
96C2×Q8×C53C22⋊C412yes—
97C2×D4×C53—24no—
98C2×D4×C53—22no—
99C2×D4×C53—22no—
100C2×D4×C53—22no—
101C2×D4×C53—22no—
102C2×D4×C53—24no—
103C2×D4×C53—24no—
104C2×D4×C53—22no—
105C2×D4×C53—21no—
106C2×D4×C53—22no—
107C2×D4×C53—41no—
108C2×D4×C53—41no—
109C2×D4×C53—41no—
110C2×D4×C53—41no—
111C2×D4×C53—41no—
112C2×D4×C53—42no—
113C2×D4×C53—42no—
114C2×D4×C53—82no—
115C2×D4×C53—82no—
116C2×D4×C53C22⋊C412yes—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—24no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—24no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—41no—
129C22×C4×C53—41no—
130C22×C4×C53—41no—
131C22×C4×C53—41no—
132C22×C4×C53—42no—
133C22×C4×C53—42no—
134C22×C4×C53—82no—
135C22×C4×C53C22⋊C411yescharacteristic
136C22×C4×C53C22⋊C411yescharacteristic
137C22×C4×C53C22×C411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D412yes—
140C22×C4×C53C2×D412yes—
141C22×Dic53—42no—
142C4⋊C4×C53—24no—
143C4⋊C4×C53—24no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—24no—
147C4⋊C4×C53C22⋊C412yes—
148C4⋊C4×C53C22⋊C412yes—
149C22⋊C4×C53—24no—
150C22⋊C4×C53—24no—
151C22⋊C4×C53—24no—
152C22⋊C4×C53—24no—
153C22⋊Dic53—42no—
154C22⋊Dic53—42no—
155#11719917—42no—
156#11719917—42no—
157C2×#88322—42no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—22no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—41no—
167C23×C53—41no—
168C23×C53—41no—
169C23×C53—41no—
170C23×C53—42no—
171C23×C53—42no—
172C23×C53—41no—
173C23×C53—41no—
174C23×C53—41no—
175C23×C53—41no—
176C23×C53#9912yes—
177C23×C53C2×C22⋊C411yescharacteristic
178C23×C53#12012yes—
179Q8×C53—44no—
180Q8×C53—42no—
181Q8×C53—42no—
182D4×C53—44no—
183D4×C53—44no—
184D4×C53—42no—
185D4×C53—42no—
186D4×C53—44no—
187D4×C53—48no—
188D4×C53—84no—
189D4×C53—84no—
190C2×C4×C53—24no—
191C2×C4×C53—24no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—22no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—42no—
205C2×C4×C53—42no—
206C2×C4×C53—48no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—84no—
214C2×C4×C53#9912yes—
215C2×C4×C53C2×C22⋊C411yescharacteristic
216C2×C4×C53C2×C22⋊C411yescharacteristic
217C2×C4×C53#12012yes—
218C2×Dic53—82no—
219C2×Dic53—82no—
220C2×Dic53—82no—
221#88322—82no—
222#8517—531nomaximal, Sylow
223C22×C53—21no—
224C22×C53—21no—
225C22×C53—22no—
226C22×C53—22no—
227C22×C53—22no—
228C22×C53—22no—
229C22×C53—42no—
230C22×C53—42no—
231C22×C53—41no—
232C22×C53—41no—
233C22×C53—41no—
234C22×C53—42no—
235C22×C53—41no—
236C22×C53—42no—
237C22×C53—41no—
238C22×C53—42no—
239C22×C53—81no—
240C22×C53—81no—
241C22×C53—81no—
242C22×C53—81no—
243C22×C53—82no—
244C22×C53—82no—
245C22×C53#37911yescharacteristic
246C22×C53C2×#9911yescharacteristic
247C22×C53C2×#9911yescharacteristic
248C4×C53—22no—
249C4×C53—22no—
250C4×C53—21no—
251C4×C53—21no—
252C4×C53—42no—
253C4×C53—42no—
254C4×C53—41no—
255C4×C53—42no—
256Dic53—162no—
257C2×#537D5311yescharacteristic
258#1785—531no—
259#1782—531no—
260#1511—532no—
261#1510—532no—
262C2×C53—21no—
263C2×C53—21no—
264C2×C53—42no—
265C2×C53—42no—
266C2×C53—81no—
267C2×C53—82no—
268C2×C53#178111yescharacteristic
269C2×C53#179511yescharacteristic
270C2×C53#179611yescharacteristic
271C22×#41Dic5311yescharacteristic, complemented
272C22×#41C2×D5311yescharacteristic
273#537—24no—
274#537—24no—
275C2×C22⋊Q8—21no—
276C2×C22⋊Q8Dic5311yescharacteristic, complemented
277C2×C22⋊Q8C2×D5311yescharacteristic
278C2×#121—21no—
279C2×#121—21no—
280C2×#121—21no—
281C2×#121Dic5311yescharacteristic, complemented
282C2×#121C2×D5311yescharacteristic
283C22×C4⋊C4Dic5311yescharacteristic, complemented
284C2×#101—1061no—
285C2×#100—1061no—
286C2×#99—1061no—
287C2×#99—1061no—
288#407—532no—
289#381—532no—
290#328—1064no—
291#327—1064no—
292C53#851711yescharacteristic, complemented, Sylow
293C2×#41—22no—
294C2×#41—22no—
295C2×#41—22no—
296C2×#41—22no—
297C2×#41—42no—
298C2×#41—42no—
299C2×#41—42no—
300C2×#41—42no—
301C22×Q8C2×Dic5311yescharacteristic
302C22×Q8C22⋊D5311yescharacteristic
303C22×D4—21no—
304C22×D4—21no—
305C22×D4—21no—
306C22×D4C2×Dic5311yescharacteristic
307C22×D4C22⋊D5311yescharacteristic
308C22×D4C22⋊D5311yescharacteristic
309C22×D4C22⋊D5311yescharacteristic
310C23×C4—21no—
311C23×C4—21no—
312C23×C4C2×Dic5311yescharacteristic
313C23×C4C2×Dic5311yescharacteristic
314C23×C4C22×D5311yescharacteristic
315C2×M16—1062no—
316C22⋊Q8—28no—
317C22⋊Q8—22no—
318C22⋊Q8—22no—
319C22⋊Q8—22no—
320C22⋊Q8C22⋊D5314yes—
321#121—28no—
322#121—28no—
323#121—28no—
324#121—22no—
325#121—22no—
326#121—22no—
327#121C22⋊D5314yes—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4C2×Dic5311yescharacteristic
334C2×C4⋊C4C2×Dic5311yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—1062no—
340#101—2122no—
341#100—2122no—
342#99—2122no—
343#99—2122no—
344C22⋊C8—1062no—
345C22⋊C8—1062no—
346#95—1062no—
347#95—1062no—
348C24—21no—
349C24C22⋊Dic5311yescharacteristic
350C24C22⋊Dic5311yescharacteristic
351C24C2×C22⋊D5311yescharacteristic
352#41—48no—
353#41—48no—
354#41—48no—
355#41—48no—
356C2×Q8—22no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8C22⋊Dic5312yes—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—82no—
381C2×D4—82no—
382C2×D4C22⋊Dic5312yes—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—82no—
401C22×C4—2122no—
402C22×C4C22⋊Dic5311yescharacteristic
403C22×C4C22⋊Dic5311yescharacteristic
404C22×C4C22×Dic5311yescharacteristic
405C22×C4C2×C22⋊D5311yescharacteristic
406C22×C4C2×C22⋊D5312yes—
407C22×C4C2×C22⋊D5312yes—
408M16—2122no—
409M16—2122no—
410C2×C8—2122no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—24no—
416C4⋊C4C22⋊Dic5312yes—
417C4⋊C4C22⋊Dic5312yes—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—2122no—
423C22⋊C4—2122no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—22no—
430C23—21no—
431C23—21no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23—42no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23#1313313212yes—
443C23C2×C22⋊Dic5311yescharacteristic
444C23#1313318612yes—
445Q8—44no—
446Q8—42no—
447Q8—42no—
448D4—44no—
449D4—44no—
450D4—42no—
451D4—42no—
452D4—44no—
453D4—48no—
454D4—84no—
455D4—84no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—48no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—84no—
480C2×C4—4242no—
481C2×C4—4242no—
482C2×C4—4242no—
483C2×C4#1313313212yes—
484C2×C4C2×C22⋊Dic5311yescharacteristic
485C2×C4C2×C22⋊Dic5311yescharacteristic
486C2×C4#1313318612yes—
487C8—4242no—
488C22—21no—
489C22—21no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—42no—
495C22—42no—
496C22—41no—
497C22—41no—
498C22—42no—
499C22—41no—
500C22—41no—
501C22—42no—
502C22—42no—
503C22—41no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—82no—
509C22—82no—
510C22#1681047911yescharacteristic
511C22C2×#1313313211yescharacteristic
512C22C2×#1313313211yescharacteristic
513C4—22no—
514C4—22no—
515C4—21no—
516C4—21no—
517C4—41no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—8482no—
522C2—21no—
523C2—21no—
524C2—42no—
525C2—42no—
526C2—82no—
527C2—81no—
528C2#2027463011yescharacteristic
529C2#2027468511yescharacteristic
530C2#2027469011yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC2×C53×#537C2
socleC22×C53#379
cosocleC23×C4×C53C23
hypercenterC2×#537D53
hypocenterC53#8517
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #20274643C2semidirect
2. #20274633C2semidirect
3. #20273248C2semidirect
4. #20273244C2semidirect
14. C22×#41×C53C4semidirect
19. C2×C22⋊Q8×C53C4semidirect
24. C2×#121×C53C4semidirect
26. C22×C4⋊C4×C53C4semidirect
271. C22×#41Dic53semidirect
276. C2×C22⋊Q8Dic53semidirect
281. C2×#121Dic53semidirect
283. C22×C4⋊C4Dic53semidirect
292. C53#8517semidirect

Maximal subgroups

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

subgroupindex
#202746432
#202746332
#202732482
#202732442
C2×C53×#5372
#851753