Finite Group Databasebeta

← #27438790

Subgroups — #27438790

Order 13568.

1102 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811033C211yesmaximal, characteristic, complemented
2C2×#16810903C211yesmaximal, characteristic, complemented
3#20279888C211yesmaximal, characteristic, complemented
4#20275901C212yesmaximal, complemented
5#20275894C212yesmaximal, complemented
6#20275673C211yesmaximal, characteristic, complemented
7#20275514C211yesmaximal, characteristic, complemented
8#20274914C212yesmaximal, complemented
9#20274903C212yesmaximal, complemented
10#20274540C211yesmaximal, characteristic, complemented
11C2×C53×#553C211yesmaximal, characteristic
12C2×#13133217C2211yescharacteristic
13#16811033—24no—
14#16810903—24no—
15C2×#13133150C2212yes—
16C2×#13133150C2212yes—
17C2×#13133148C2212yes—
18C2×C4⋊#88322C2211yescharacteristic
19C2×#13133138C2212yes—
20C2×#13133138C2212yes—
21C2×C22⋊#88322C2212yes—
22C2×C22⋊#88322C2212yes—
23C2×#13133128C2211yescharacteristic
24#16810535—21no—
25#16810535—21no—
26#16810526—21no—
27#16810488—21no—
28#16810483C2212yes—
29#16810483C2212yes—
30C4×C4⋊Dic53C2211yescharacteristic
31#16810385C2212yes—
32#16810385C2212yes—
33#16810385C2212yes—
34C53×#553—24no—
35C53×#553—24no—
36C2×#126×C53—21no—
37C2×#125×C53C2211yescharacteristic
38C2×#123×C53—21no—
39C2×#121×C53—21no—
40C2×#121×C53C2211yescharacteristic
41C2×#121×C53C2211yescharacteristic
42C2×C4×D4×C53—21no—
43C2×C4×D4×C53C2211yescharacteristic
44C2×C4×D4×C53C2211yescharacteristic
45C2×#117×C53C2211yescharacteristic
46C22×C4⋊C4×C53C2211yescharacteristic
47C22×Dic106—21no—
48C22×#88322C2312yes—
49C22×#88322C2312yes—
50#13133217—22no—
51#13133217—22no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53C2312yes—
54C2×C4⋊Dic53C2312yes—
55C2×C22⋊Dic53—22no—
56C2×C22⋊Dic53—22no—
57C2×C4×Dic53—21no—
58#13133150—24no—
59#13133150—24no—
60#13133148—24no—
61C4⋊#88322—22no—
62#13133138—24no—
63#13133138—24no—
64C22⋊#88322—24no—
65C22⋊#88322—24no—
66#13133128—22no—
67#13133124—21no—
68#13133124—22no—
69#13133124—22no—
70C22×D4×C53—21no—
71C22×D4×C53—21no—
72C22×D4×C53D411yescharacteristic
73C22×D4×C53C2311yescharacteristic
74C23×C4×C53—21no—
75C23×C4×C53—21no—
76C23×C4×C53C2311yescharacteristic
77C23×C4×C53C2311yescharacteristic
78#126×C53—28no—
79#125×C53—22no—
80#125×C53D414yes—
81#123×C53—28no—
82#121×C53—28no—
83#121×C53—22no—
84#121×C53—22no—
85#121×C53—22no—
86#121×C53—22no—
87C4×D4×C53—28no—
88C4×D4×C53—22no—
89C4×D4×C53—22no—
90C4×D4×C53—22no—
91C4×D4×C53D414yes—
92#117×C53—22no—
93#117×C53—22no—
94C2×C4⋊C4×C53—22no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53D411yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—21no—
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×C53—21no—
112C24×C53C2×D411yescharacteristic
113C24×C53C2×D411yescharacteristic
114C2×D4×C53—24no—
115C2×D4×C53—22no—
116C2×D4×C53—24no—
117C2×D4×C53—22no—
118C2×D4×C53—21no—
119C2×D4×C53—22no—
120C2×D4×C53—22no—
121C2×D4×C53—41no—
122C2×D4×C53—42no—
123C2×D4×C53—42no—
124C2×D4×C53—82no—
125C2×D4×C53C2×D412yes—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—24no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—24no—
139C22×C4×C53—82no—
140C22×C4×C53—82no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2411yescharacteristic
146C22×Dic53—22no—
147C22×Dic53—22no—
148C2×Dic106—42no—
149C2×Dic106—82no—
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—24no—
163C22⋊C4×C53—24no—
164C22⋊C4×C53—22no—
165C22⋊C4×C53—22no—
166C42×C53—24no—
167C42×C53C2×D412yes—
168C42×C53C2×D412yes—
169C22⋊Dic53—82no—
170C22⋊Dic53—82no—
171C4⋊Dic53—22no—
172C4⋊Dic53—22no—
173C4⋊Dic53—22no—
174C4⋊Dic53—22no—
175C4⋊Dic53—42no—
176C4×Dic53—42no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322#4114yes—
183C23×C53—21no—
184C23×C53—22no—
185C23×C53—21no—
186C23×C53—22no—
187C23×C53—21no—
188C23×C53—22no—
189C23×C53—22no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—22no—
193C23×C53—42no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—42no—
197C23×C53#13712yes—
198C23×C53C22×D411yescharacteristic
199D4×C53—22no—
200D4×C53—22no—
201D4×C53—44no—
202D4×C53—48no—
203D4×C53—42no—
204D4×C53—84no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—84no—
226C2×C4×C53—84no—
227C2×C4×C53#12012yes—
228C2×C4×C53#12012yes—
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C2×#4112yes—
232C2×C4×C532−1+412yes—
233C2×Dic53—42no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239Dic106—84no—
240#88322—24no—
241#88322—42no—
242#25713—531nomaximal, Sylow
243C22×C53—24no—
244C22×C53—22no—
245C22×C53—42no—
246C22×C53—41no—
247C22×C53—41no—
248C22×C53—42no—
249C22×C53—42no—
250C22×C53—41no—
251C22×C53—42no—
252C22×C53—41no—
253C22×C53—42no—
254C22×C53—42no—
255C22×C53—41no—
256C22×C53—82no—
257C22×C53—82no—
258C22×C53C2×#12011yescharacteristic
259C22×C53#54712yes—
260C22×C53C2×#13711yescharacteristic
261C22×C53#57511yescharacteristic
262C22×C53#57612yes—
263C4×C53—22no—
264C4×C53—22no—
265C4×C53—41no—
266C4×C53—41no—
267C4×C53—42no—
268C4×C53—42no—
269C4×C53—42no—
270C4×C53#53211yescharacteristic
271C4×C53#53611yescharacteristic
272C4×C53#54812yes—
273Dic53—82no—
274Dic53—82no—
275C2×#553D5311yescharacteristic
276C2×#501—531no—
277C2×#474—531no—
278#2563—531no—
279#2034—532no—
280#2033—532no—
281#1989—531no—
282#1955—531no—
283#1854—532no—
284#1852—532no—
285#1762—531no—
286C2×C53—42no—
287C2×C53—81no—
288C2×C53—82no—
289C2×C53#243211yescharacteristic
290C2×C53#300711yescharacteristic
291C2×C53#300811yescharacteristic
292C2×C53#334612yes—
293C2×C53#334712yes—
294#553—24no—
295#553—24no—
296C2×C4⋊Q8—531no—
297C2×#126—21no—
298C2×#125C2×D5311yescharacteristic
299C2×#123—21no—
300C2×#121—21no—
301C2×#121C2×D5311yescharacteristic
302C2×#121C2×D5311yescharacteristic
303C2×C4×D4—21no—
304C2×C4×D4C2×D5311yescharacteristic
305C2×C4×D4C2×D5311yescharacteristic
306C2×#117C2×D5311yescharacteristic
307C22×C4⋊C4C2×D5311yescharacteristic
308#501—1064no—
309#474—1064no—
310C2×#107—532no—
311C2×#107—532no—
312C2×#106—532no—
313C2×C4⋊C8—531no—
314C2×#102—532no—
315C2×#102—532no—
316C2×C22⋊C8—532no—
317C2×C22⋊C8—532no—
318C2×#97—531no—
319#393—1061no—
320#393—1061no—
321#390—1061no—
322#382—1061no—
323#380—532no—
324#380—532no—
325C4×C4⋊C4—531no—
326#340—532no—
327#340—532no—
328#340—532no—
329C53#2571311yescharacteristic, complemented, Sylow
330C22×Q8—1061no—
331C22×D4—21no—
332C22×D4—21no—
333C22×D4C22⋊D5311yescharacteristic
334C22×D4C22×D5311yescharacteristic
335C23×C4—21no—
336C23×C4—21no—
337C23×C4C22×D5311yescharacteristic
338C23×C4C22×D5311yescharacteristic
339C22×C8—532no—
340C22×C8—532no—
341C4⋊Q8—1062no—
342C4⋊Q8—1062no—
343#126—28no—
344#125—22no—
345#125C22⋊D5314yes—
346#123—28no—
347#121—28no—
348#121—22no—
349#121—22no—
350#121—22no—
351#121—22no—
352C4×D4—28no—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4C22⋊D5314yes—
357#117—22no—
358#117—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—532no—
365C2×C4⋊C4—532no—
366C2×C4⋊C4—1061no—
367C2×C4⋊C4C22⋊D5311yescharacteristic
368C2×C4⋊C4C22×D5311yescharacteristic
369C2×C4⋊C4C22×D5311yescharacteristic
370C2×C4⋊C4C22×D5311yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—1062no—
376C2×C22⋊C4—1062no—
377C2×C22⋊C4C22⋊D5311yescharacteristic
378C2×C22⋊C4C22⋊D5311yescharacteristic
379C2×C42—21no—
380C2×C42—1061no—
381C2×C42C22×D5311yescharacteristic
382#107—1064no—
383#107—1064no—
384#106—1064no—
385C4⋊C8—1062no—
386#102—1064no—
387#102—1064no—
388C22⋊C8—1064no—
389C22⋊C8—1064no—
390#97—1062no—
391#95—1061no—
392#95—1062no—
393#95—1062no—
394C24—21no—
395C24C2×C22⋊D5311yescharacteristic
396C24C2×C22⋊D5311yescharacteristic
397C2×Q8—4242no—
398C2×Q8—2122no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—41no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—82no—
410C2×D4C2×C22⋊D5312yes—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4—1062no—
427C22×C4—1062no—
428C22×C4C2×C22⋊D5311yescharacteristic
429C22×C4C2×C22⋊D5311yescharacteristic
430C22×C4C2×C22⋊D5311yescharacteristic
431C22×C4C2×C22⋊D5311yescharacteristic
432C22×C4C23×D5311yescharacteristic
433C2×C8—534no—
434C2×C8—1062no—
435C2×C8—1062no—
436C2×C8—1062no—
437C2×C8—1062no—
438C2×C8—1062no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—1062no—
447C4⋊C4—1062no—
448C4⋊C4—1062no—
449C4⋊C4—1062no—
450C4⋊C4—2122no—
451C4⋊C4C2×C22⋊D5312yes—
452C4⋊C4C2×C22⋊D5312yes—
453C4⋊C4C2×C22⋊D5312yes—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—4242no—
461C22⋊C4—4242no—
462C42—24no—
463C42—2122no—
464C42C2×C22⋊D5312yes—
465C42C2×C22⋊D5312yes—
466C23—22no—
467C23—21no—
468C23—21no—
469C23—22no—
470C23—22no—
471C23—21no—
472C23—22no—
473C23—21no—
474C23—21no—
475C23—22no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23#1313324712yes—
481C23C22×C22⋊D5311yescharacteristic
482Q8—4244no—
483D4—22no—
484D4—22no—
485D4—44no—
486D4—48no—
487D4—42no—
488D4—84no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—84no—
510C2×C4—84no—
511C2×C4—2122no—
512C2×C4—2122no—
513C2×C4—2122no—
514C2×C4—2122no—
515C2×C4—2122no—
516C2×C4—2122no—
517C2×C4#1313318612yes—
518C2×C4#1313318612yes—
519C2×C4C22×C22⋊D5311yescharacteristic
520C2×C4C22×C22⋊D5311yescharacteristic
521C2×C4#41×D5312yes—
522C2×C4#1313326412yes—
523C8—1064no—
524C8—2122no—
525C22—22no—
526C22—24no—
527C22—41no—
528C22—41no—
529C22—42no—
530C22—41no—
531C22—42no—
532C22—41no—
533C22—42no—
534C22—42no—
535C22—41no—
536C22—42no—
537C22—42no—
538C22—82no—
539C22—82no—
540C22C2×#1313318611yescharacteristic
541C22#1681127212yes—
542C22C2×#1313324711yescharacteristic
543C22#1681142111yescharacteristic
544C22#1681142612yes—
545C4—22no—
546C4—22no—
547C4—41no—
548C4—41no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—4242no—
553C4—4242no—
554C4#1681116911yescharacteristic
555C4#1681119411yescharacteristic
556C4#1681128012yes—
557C2—42no—
558C2—81no—
559C2—82no—
560C2#2027833911yescharacteristic
561C2#2028397511yescharacteristic
562C2#2028398311yescharacteristic
563C2#2028720012yes—
564C2#2028720912yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#553C2
socleC23×C53C22×D4
hypercenterC2×#553D53
hypocenterC53#25713
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#553

Complements

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

NH ≅ G/Ntype
1. C2×#16811033C2semidirect
2. C2×#16810903C2semidirect
3. #20279888C2semidirect
4. #20275901C2semidirect
5. #20275894C2semidirect
6. #20275673C2semidirect
7. #20275514C2semidirect
8. #20274914C2semidirect
9. #20274903C2semidirect
10. #20274540C2semidirect
329. C53#25713semidirect

Maximal subgroups

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

subgroupindex
C2×#168110332
C2×#168109032
#202798882
#202759012
#202758942
#202756732
#202755142
#202749142
#202749032
#202745402
C2×C53×#5532
#2571353