Finite Group Databasebeta

← #27417773

Subgroups — #27417773

Order 13568.

1118 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810956C211yesmaximal, characteristic, complemented
2C22×#13133150C211yesmaximal, characteristic, complemented
3#20279032C211yesmaximal, characteristic
4#20276040C212yesmaximal, complemented
5#20275915C212yesmaximal, complemented
6#20275139C211yesmaximal, characteristic, complemented
7#20275008C211yesmaximal, characteristic
8#20274969C212yesmaximal, complemented
9#20274903C212yesmaximal, complemented
10#20274805C211yesmaximal, characteristic, complemented
11C2×C53×#554C211yesmaximal, characteristic
12C23×#88322C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14#16810956—24no—
15C2×#13133150—24no—
16C2×#13133150C2212yes—
17C2×#13133150C2212yes—
18C2×#13133150C2212yes—
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×C22⋊#88322C2211yescharacteristic
22C2×C22⋊#88322C2212yes—
23#16810562—21no—
24#16810559—21no—
25#16810521C2212yes—
26#16810503—21no—
27#16810495—21no—
28#16810483C2211yescharacteristic
29#16810483C2212yes—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
34C53×#554—24no—
35C53×#554—24no—
36C2×C4⋊Q8×C53C2211yescharacteristic
37C2×#125×C53—21no—
38C2×C22⋊Q8×C53—21no—
39C2×C22⋊Q8×C53—21no—
40C2×C22⋊Q8×C53C2211yescharacteristic
41C2×C22⋊Q8×C53C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C2×C4×D4×C53—21no—
44C2×C4×D4×C53C2211yescharacteristic
45C22×C4⋊C4×C53C2211yescharacteristic
46C22×C4⋊C4×C53C2211yescharacteristic
47C23×Dic53—21no—
48C22×#88322—24no—
49C22×#88322—22no—
50C22×#88322C2312yes—
51C22×#88322C2312yes—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53C2312yes—
56C2×C4⋊Dic53C2312yes—
57C2×#11719919—22no—
58C2×C22⋊Dic53—21no—
59C2×C22⋊Dic53—22no—
60#13133150—24no—
61#13133150—24no—
62#13133150—24no—
63#13133141—24no—
64#13133141—24no—
65C22⋊#88322—24no—
66C22⋊#88322—22no—
67#13133124—22no—
68#13133124—21no—
69#13133124—22no—
70C22×Q8×C53—21no—
71C22×Q8×C53C2311yescharacteristic
72C22×D4×C53D411yescharacteristic
73C23×C4×C53—21no—
74C23×C4×C53—21no—
75C23×C4×C53C2311yescharacteristic
76C23×C4×C53C2311yescharacteristic
77C4⋊Q8×C53—22no—
78C4⋊Q8×C53—22no—
79#125×C53—28no—
80C22⋊Q8×C53—28no—
81C22⋊Q8×C53—28no—
82C22⋊Q8×C53—22no—
83C22⋊Q8×C53—22no—
84C22⋊Q8×C53—22no—
85C22⋊Q8×C53D414yes—
86C4×Q8×C53—22no—
87C4×Q8×C53—22no—
88C4×D4×C53—28no—
89C4×D4×C53—22no—
90C4×D4×C53—22no—
91C2×C4⋊C4×C53—22no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53D414yes—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53D411yescharacteristic
108C2×C22⋊C4×C53D411yescharacteristic
109C2×C42×C53—21no—
110C2×C42×C53C2311yescharacteristic
111C24×C53C2×D411yescharacteristic
112C24×C53C2×D411yescharacteristic
113C2×Q8×C53—24no—
114C2×Q8×C53—21no—
115C2×Q8×C53—22no—
116C2×Q8×C53—22no—
117C2×Q8×C53—82no—
118C2×Q8×C53C2×D412yes—
119C2×D4×C53—22no—
120C2×D4×C53—22no—
121C2×D4×C53Q1614yes—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—24no—
129C22×C4×C53—24no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—82no—
135C22×C4×C53—82no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D412yes—
140C22×C4×C53C2×D412yes—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—42no—
146C22×Dic53—82no—
147C4⋊C4×C53—24no—
148C4⋊C4×C53—24no—
149C4⋊C4×C53—24no—
150C4⋊C4×C53—24no—
151C4⋊C4×C53—24no—
152C4⋊C4×C53—24no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53C2×D412yes—
158C4⋊C4×C53C2×D412yes—
159C4⋊C4×C53C2×D412yes—
160C22⋊C4×C53—24no—
161C22⋊C4×C53—24no—
162C22⋊C4×C53—22no—
163C22⋊C4×C53—22no—
164C42×C53—24no—
165C42×C53—21no—
166C42×C53—21no—
167C22⋊Dic53—42no—
168C22⋊Dic53—82no—
169C4⋊Dic53—22no—
170C4⋊Dic53—22no—
171C4⋊Dic53—22no—
172C4⋊Dic53—22no—
173C4⋊Dic53—42no—
174#11719919—82no—
175C2×#88322—28no—
176C2×#88322—22no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322#4114yes—
182C23×C53—22no—
183C23×C53—22no—
184C23×C53—21no—
185C23×C53—22no—
186C23×C53—21no—
187C23×C53—22no—
188C23×C53#12012yes—
189C23×C53C2×Q1612yes—
190C23×C53C2×Q1612yes—
191C23×C53#13712yes—
192C23×C53C22×D411yescharacteristic
193Q8×C53—24no—
194Q8×C53—41no—
195Q8×C53—41no—
196Q8×C53—84no—
197D4×C53—28no—
198C2×C4×C53—21no—
199C2×C4×C53—22no—
200C2×C4×C53—21no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—24no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—21no—
208C2×C4×C53—22no—
209C2×C4×C53—21no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—42no—
219C2×C4×C53—84no—
220C2×C4×C53—84no—
221C2×C4×C53#12012yes—
222C2×C4×C53C2×Q1612yes—
223C2×C4×C53C22×D411yescharacteristic
224C2×C4×C53C22×D411yescharacteristic
225C2×C4×C53C2×#4112yes—
226C2×C4×C532−1+412yes—
227C2×Dic53—42no—
228C2×Dic53—42no—
229C2×Dic53—42no—
230C2×Dic53—42no—
231C2×Dic53—42no—
232C2×Dic53—42no—
233C2×Dic53—84no—
234#88322—24no—
235#88322—42no—
236#24098—531nomaximal, Sylow
237C22×C53—24no—
238C22×C53—24no—
239C22×C53—22no—
240C22×C53—22no—
241C22×C53—24no—
242C22×C53—22no—
243C22×C53—42no—
244C22×C53—41no—
245C22×C53—41no—
246C22×C53#45114yes—
247C22×C53C2×#12011yescharacteristic
248C22×C53#54712yes—
249C22×C53C22×Q1611yescharacteristic
250C22×C53C2×#13711yescharacteristic
251C22×C53#57612yes—
252C4×C53—22no—
253C4×C53—22no—
254C4×C53—41no—
255C4×C53—42no—
256C4×C53—41no—
257C4×C53—42no—
258C4×C53—42no—
259C4×C53—42no—
260C4×C53#52511yescharacteristic
261C4×C53#53611yescharacteristic
262C4×C53#54712yes—
263Dic53—82no—
264Dic53—82no—
265C2×#554D5311yescharacteristic
266C2×#484—531no—
267C22×#107—531no—
268#2487—531no—
269#2057—532no—
270#2036—532no—
271#1892—531no—
272#1869—531no—
273#1863—532no—
274#1852—532no—
275#1824—531no—
276C2×C53—24no—
277C2×C53—42no—
278C2×C53#240611yescharacteristic
279C2×C53C2×#45111yescharacteristic
280C2×C53#300811yescharacteristic
281C2×C53#333812yes—
282C2×C53#334612yes—
283C23×C8—531no—
284#554—24no—
285#554—24no—
286C2×C4⋊Q8C2×D5311yescharacteristic
287C2×#125—21no—
288C2×C22⋊Q8—21no—
289C2×C22⋊Q8—21no—
290C2×C22⋊Q8C2×D5311yescharacteristic
291C2×C22⋊Q8C2×D5311yescharacteristic
292C2×C4×Q8C2×D5311yescharacteristic
293C2×C4×D4—21no—
294C2×C4×D4C2×D5311yescharacteristic
295C22×C4⋊C4—531no—
296C22×C4⋊C4C2×D5311yescharacteristic
297C22×C4⋊C4C2×D5311yescharacteristic
298#484—1064no—
299C2×#107—532no—
300C2×#107—532no—
301C2×#107—532no—
302C2×#107—1064no—
303C2×#103—532no—
304C2×#103—532no—
305C2×C22⋊C8—531no—
306C2×C22⋊C8—532no—
307#400—1061no—
308#399—1061no—
309#389—532no—
310#386—1061no—
311#385—1061no—
312#380—531no—
313#380—532no—
314#340—532no—
315#340—532no—
316#340—532no—
317#336—532no—
318C53#2409811yescharacteristic, complemented, Sylow
319C22×Q8—21no—
320C22×Q8C22×D5311yescharacteristic
321C22×D4C22⋊D5311yescharacteristic
322C23×C4—21no—
323C23×C4—21no—
324C23×C4—1061no—
325C23×C4C22×D5311yescharacteristic
326C23×C4C22×D5311yescharacteristic
327C22×C8—532no—
328C22×C8—532no—
329C22×C8—1064no—
330C22×C8—1062no—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333#125—28no—
334C22⋊Q8—28no—
335C22⋊Q8—28no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8C22⋊D5314yes—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×D4—28no—
343C4×D4—22no—
344C4×D4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—532no—
355C2×C4⋊C4—532no—
356C2×C4⋊C4—1062no—
357C2×C4⋊C4—1062no—
358C2×C4⋊C4—1062no—
359C2×C4⋊C4—1061no—
360C2×C4⋊C4C22⋊D5311yescharacteristic
361C2×C4⋊C4C22⋊D5314yes—
362C2×C4⋊C4C22×D5311yescharacteristic
363C2×C4⋊C4C22×D5311yescharacteristic
364C2×C4⋊C4C22×D5311yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—1062no—
368C2×C22⋊C4—1061no—
369C2×C22⋊C4C22⋊D5311yescharacteristic
370C2×C22⋊C4C22⋊D5311yescharacteristic
371C2×C42—21no—
372C2×C42C22×D5311yescharacteristic
373#107—1064no—
374#107—1064no—
375#107—1064no—
376#103—1064no—
377#103—1064no—
378C22⋊C8—1064no—
379C22⋊C8—1062no—
380#95—1062no—
381#95—1061no—
382#95—1062no—
383C24C2×C22⋊D5311yescharacteristic
384C24C2×C22⋊D5311yescharacteristic
385C2×Q8—24no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—82no—
390C2×Q8C2×C22⋊D5312yes—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4#1171992514yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4—1062no—
409C22×C4—1062no—
410C22×C4—4242no—
411C22×C4—2122no—
412C22×C4C2×C22⋊D5311yescharacteristic
413C22×C4C2×C22⋊D5311yescharacteristic
414C22×C4C2×C22⋊D5311yescharacteristic
415C22×C4C2×C22⋊D5312yes—
416C22×C4C2×C22⋊D5312yes—
417C22×C4C2×C22⋊D5311yescharacteristic
418C22×C4C23×D5311yescharacteristic
419C2×C8—534no—
420C2×C8—1068no—
421C2×C8—1062no—
422C2×C8—1062no—
423C2×C8—1062no—
424C2×C8—1062no—
425C2×C8—1062no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—1062no—
437C4⋊C4—1062no—
438C4⋊C4—1062no—
439C4⋊C4—1062no—
440C4⋊C4—4242no—
441C4⋊C4—2122no—
442C4⋊C4C2×C22⋊D5312yes—
443C4⋊C4C2×C22⋊D5312yes—
444C4⋊C4C2×C22⋊D5312yes—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—4242no—
450C22⋊C4—2122no—
451C42—24no—
452C42—21no—
453C42—21no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—21no—
458C23—21no—
459C23—22no—
460C23#1313318612yes—
461C23C2×#1171992512yes—
462C23C2×#1171992512yes—
463C23#1313324812yes—
464C23C22×C22⋊D5311yescharacteristic
465Q8—24no—
466Q8—41no—
467Q8—41no—
468Q8—84no—
469D4—28no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—84no—
492C2×C4—84no—
493C2×C4—4244no—
494C2×C4—2122no—
495C2×C4—2122no—
496C2×C4—2122no—
497C2×C4—2122no—
498C2×C4—2122no—
499C2×C4—2122no—
500C2×C4#1313318612yes—
501C2×C4C2×#1171992512yes—
502C2×C4C22×C22⋊D5311yescharacteristic
503C2×C4C22×C22⋊D5311yescharacteristic
504C2×C4#41×D5312yes—
505C2×C4#1313326412yes—
506C8—1064no—
507C8—2122no—
508C22—22no—
509C22—22no—
510C22—24no—
511C22—24no—
512C22—22no—
513C22—24no—
514C22—41no—
515C22—41no—
516C22—42no—
517C22#1681077914yes—
518C22C2×#1313318611yescharacteristic
519C22#1681127212yes—
520C22C22×#1171992511yescharacteristic
521C22C2×#1313324811yescharacteristic
522C22#1681142612yes—
523C4—22no—
524C4—22no—
525C4—41no—
526C4—42no—
527C4—41no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—4242no—
532C4—4242no—
533C4#1681113011yescharacteristic
534C4#1681119411yescharacteristic
535C4#1681127212yes—
536C2—24no—
537C2—42no—
538C2#2027804611yescharacteristic
539C2C2×#1681077911yescharacteristic
540C2#2028398311yescharacteristic
541C2#2028710312yes—
542C2#2028719412yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#554C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenterC2×#554D53
hypocenterC53#24098
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#16810956C2semidirect
2. C22×#13133150C2semidirect
4. #20276040C2semidirect
5. #20275915C2semidirect
6. #20275139C2semidirect
8. #20274969C2semidirect
9. #20274903C2semidirect
10. #20274805C2semidirect
318. C53#24098semidirect

Maximal subgroups

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

subgroupindex
C2×#168109562
C22×#131331502
#202790322
#202760402
#202759152
#202751392
#202750082
#202749692
#202749032
#202748052
C2×C53×#5542
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