Finite Group Databasebeta

← #27489524

Subgroups — #27489524

Order 13568.

1268 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810964C211yesmaximal, characteristic, complemented
2C2×#16810779C211yesmaximal, characteristic, complemented
3#20280444C211yesmaximal, characteristic
4#20275915C212yesmaximal, complemented
5#20275908C212yesmaximal, complemented
6#20275723C212yesmaximal, complemented
7#20275723C212yesmaximal, complemented
8#20274810C211yesmaximal, characteristic, complemented
9#20274650C211yesmaximal, characteristic, complemented
10#20274627C211yesmaximal, characteristic
11C2×C53×#536C211yesmaximal, characteristic
12C22×#11719925C2212yes—
13C2×#13133196C2211yescharacteristic
14C2×#13133196C2212yes—
15#16810964—24no—
16#16810779—24no—
17C2×#13133150C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×#13133141C2212yes—
22C2×#13133140C2211yescharacteristic
23C2×#13133140C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×C22⋊#88322C2212yes—
27C2×C22⋊#88322C2212yes—
28#16810542—21no—
29#16810537—22no—
30#16810537—21no—
31#16810483C2211yescharacteristic
32#16810483C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35C22×#41×C53C2211yescharacteristic
36C23×Q8×C53C2211yescharacteristic
37C53×#536—24no—
38C53×#536—24no—
39C2×C4⋊Q8×C53—21no—
40C2×#124×C53—21no—
41C2×#123×C53—22no—
42C2×C22⋊Q8×C53C2212yes—
43C2×C22⋊Q8×C53C2212yes—
44C2×#117×C53C2211yescharacteristic
45C22×Dic106C2311yescharacteristic
46C2×#11719925—24no—
47C2×#11719925—42no—
48C2×#11719925—42no—
49C22×#88322C2311yescharacteristic
50C22×#88322C2311yescharacteristic
51C22×#88322C2312yes—
52#13133196—22no—
53#13133196—22no—
54#13133196—42no—
55#13133196—42no—
56C2×C4⋊Dic53C2312yes—
57C2×C4⋊Dic53C2311yescharacteristic
58C2×#11719919—22no—
59C2×#11719919—21no—
60C2×C22⋊Dic53—22no—
61C2×C22⋊Dic53—21no—
62C2×C22⋊Dic53—22no—
63C2×C22⋊Dic53—21no—
64#13133150—22no—
65#13133148—22no—
66#13133141—24no—
67#13133141—42no—
68#13133141—42no—
69#13133140—22no—
70#13133140—22no—
71C22⋊#88322—22no—
72C22⋊#88322—22no—
73C22⋊#88322—42no—
74C22⋊#88322—42no—
75#13133124—22no—
76#13133124—21no—
77C2×#41×C53—22no—
78C2×#41×C53—22no—
79C2×#41×C53—42no—
80C2×#41×C53—42no—
81C22×Q8×C53—21no—
82C22×Q8×C53—22no—
83C22×Q8×C53—42no—
84C22×Q8×C53—42no—
85C22×Q8×C53D414yes—
86C22×Q8×C53D411yescharacteristic
87C22×Q8×C53C2312yes—
88C22×D4×C53D412yes—
89C22×D4×C53D411yescharacteristic
90C23×C4×C53—21no—
91C23×C4×C53C2312yes—
92C23×C4×C53C2311yescharacteristic
93C4⋊Q8×C53—28no—
94#124×C53—28no—
95#123×C53—216no—
96C22⋊Q8×C53—24no—
97C22⋊Q8×C53—24no—
98C22⋊Q8×C53—24no—
99C22⋊Q8×C53—24no—
100#117×C53—22no—
101#117×C53D414yes—
102C2×C4⋊C4×C53—22no—
103C2×C4⋊C4×C53—22no—
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—22no—
108C2×C22⋊C4×C53—22no—
109C2×C42×C53—21no—
110C24×C53C2×D412yes—
111C24×C53C2×D411yescharacteristic
112#41×C53—48no—
113#41×C53—44no—
114#41×C53—44no—
115C2×Q8×C53—24no—
116C2×Q8×C53—22no—
117C2×Q8×C53—22no—
118C2×Q8×C53—22no—
119C2×Q8×C53—22no—
120C2×Q8×C53—22no—
121C2×Q8×C53—22no—
122C2×Q8×C53—42no—
123C2×Q8×C53—48no—
124C2×Q8×C53—44no—
125C2×Q8×C53—44no—
126C2×Q8×C53—44no—
127C2×Q8×C53C2×D414yes—
128C2×D4×C53—24no—
129C2×D4×C53—22no—
130C2×D4×C53—42no—
131C2×D4×C53—42no—
132C2×D4×C53—42no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—24no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—22no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—44no—
146C22×C4×C53C2×D412yes—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D412yes—
149C22×C4×C53C2×D412yes—
150C22×C4×C53C2411yescharacteristic
151C22×Dic53—21no—
152C22×Dic53—22no—
153C22×Dic53—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—22no—
157C2×Dic106—22no—
158C4⋊C4×C53—28no—
159C4⋊C4×C53—28no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53C2×D412yes—
163C4⋊C4×C53C2×D412yes—
164C22⋊C4×C53—28no—
165C22⋊C4×C53—28no—
166C22⋊C4×C53—24no—
167C42×C53—24no—
168C22⋊Dic53—44no—
169C22⋊Dic53—42no—
170C22⋊Dic53—81no—
171C22⋊Dic53—82no—
172#11719925—48no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—22no—
176C4⋊Dic53—22no—
177#11719919—44no—
178#11719919—42no—
179C2×#88322—21no—
180C2×#88322—21no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322—22no—
186C23×C53—22no—
187C23×C53—21no—
188C23×C53—22no—
189C23×C53—21no—
190C23×C53—22no—
191C23×C53—22no—
192C23×C53—22no—
193C23×C53—22no—
194C23×C53—22no—
195C23×C53—22no—
196C23×C53—22no—
197C23×C53#12012yes—
198C23×C53C22×D411yescharacteristic
199Q8×C53—24no—
200Q8×C53—24no—
201Q8×C53—48no—
202Q8×C53—44no—
203Q8×C53—42no—
204Q8×C53—42no—
205D4×C53—44no—
206D4×C53—48no—
207C2×C4×C53—22no—
208C2×C4×C53—24no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—24no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—48no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—41no—
221C2×C4×C53—42no—
222C2×C4×C53—41no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53#12012yes—
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53H(5,𝔽2)12yes—
231C2×C4×C53H(5,𝔽2)12yes—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—41no—
238C2×Dic53—42no—
239C2×Dic53—41no—
240Dic106—42no—
241Dic106—41no—
242Dic106—41no—
243#88322—42no—
244#88322—41no—
245#88322—41no—
246#29605—531nomaximal, Sylow
247C22×C53—22no—
248C22×C53—22no—
249C22×C53—22no—
250C22×C53—42no—
251C22×C53—42no—
252C22×C53—42no—
253C22×C53—42no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53C2×#12011yescharacteristic
260C22×C53#53412yes—
261C22×C53#57711yescharacteristic
262C22×C53#57812yes—
263C22×C53#57811yescharacteristic
264C4×C53—22no—
265C4×C53—42no—
266C4×C53—44no—
267C4×C53—42no—
268C4×C53—42no—
269C4×C53#53412yes—
270C4×C53#53511yescharacteristic
271C4×C53#53511yescharacteristic
272Dic53—81no—
273Dic53—82no—
274Dic53—81no—
275C2×#536D5311yescharacteristic
276C2×#485—531no—
277C2×#451—531no—
278#2616—531no—
279#2036—532no—
280#2035—532no—
281#2001—532no—
282#2001—532no—
283#1825—531no—
284#1787—531no—
285#1780—531no—
286C2×C53—42no—
287C2×C53—82no—
288C2×C53#261311yescharacteristic
289C2×C53#301111yescharacteristic
290C2×C53#301311yescharacteristic
291C2×C53#320112yes—
292C2×C53#320212yes—
293C22×#41C2×D5311yescharacteristic
294C23×Q8C2×D5311yescharacteristic
295C22×Q16—532no—
296#536—24no—
297#536—24no—
298C2×C4⋊Q8—21no—
299C2×#124—21no—
300C2×#123—22no—
301C2×C22⋊Q8—531no—
302C2×C22⋊Q8—532no—
303C2×C22⋊Q8C2×D5312yes—
304C2×C22⋊Q8C2×D5312yes—
305C2×#117C2×D5311yescharacteristic
306#485—1064no—
307#451—1064no—
308C2×#107—531no—
309C2×#106—531no—
310C2×#103—532no—
311C2×#103—531no—
312C2×#103—531no—
313C2×#103—532no—
314C2×#103—532no—
315C2×C22⋊C8—531no—
316C2×C22⋊C8—531no—
317C2×C22⋊C8—532no—
318C2×C22⋊C8—532no—
319#395—1061no—
320#394—1062no—
321#394—1061no—
322#380—531no—
323#380—532no—
324#340—532no—
325#340—532no—
326C53#2960511yescharacteristic, complemented, Sylow
327C2×#41—22no—
328C2×#41—22no—
329C2×#41—42no—
330C2×#41—42no—
331C22×Q8—21no—
332C22×Q8—22no—
333C22×Q8—42no—
334C22×Q8—42no—
335C22×Q8—531no—
336C22×Q8C22⋊D5314yes—
337C22×Q8C22⋊D5311yescharacteristic
338C22×Q8C22×D5312yes—
339C22×D4C22⋊D5312yes—
340C22×D4C22⋊D5311yescharacteristic
341C23×C4—21no—
342C23×C4C22×D5312yes—
343C23×C4C22×D5311yescharacteristic
344C2×Q16—1064no—
345C2×Q16—2122no—
346C2×Q16—2122no—
347C22×C8—532no—
348C22×C8—531no—
349C22×C8—531no—
350C4⋊Q8—28no—
351#124—28no—
352#123—216no—
353C22⋊Q8—24no—
354C22⋊Q8—24no—
355C22⋊Q8—24no—
356C22⋊Q8—24no—
357C22⋊Q8—1062no—
358C22⋊Q8—1062no—
359C22⋊Q8—2122no—
360C22⋊Q8—2122no—
361#117—22no—
362#117C22⋊D5314yes—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—532no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—1062no—
368C2×C4⋊C4—1061no—
369C2×C4⋊C4C22×D5311yescharacteristic
370C2×C4⋊C4C22×D5311yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—1062no—
375C2×C22⋊C4—1061no—
376C2×C22⋊C4—1062no—
377C2×C22⋊C4—1061no—
378C2×C42—21no—
379#107—1062no—
380#106—1062no—
381#103—1064no—
382#103—1062no—
383#103—1062no—
384#103—2122no—
385#103—2122no—
386C22⋊C8—1062no—
387C22⋊C8—1062no—
388C22⋊C8—2122no—
389C22⋊C8—2122no—
390#95—1062no—
391#95—1061no—
392C24C2×C22⋊D5312yes—
393C24C2×C22⋊D5311yescharacteristic
394#41—48no—
395#41—44no—
396#41—44no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—42no—
405C2×Q8—48no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—1061no—
410C2×Q8—1061no—
411C2×Q8—1062no—
412C2×Q8—1062no—
413C2×Q8C2×C22⋊D5314yes—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—44no—
432C22×C4—1061no—
433C22×C4—1062no—
434C22×C4—1061no—
435C22×C4C2×C22⋊D5312yes—
436C22×C4C2×C22⋊D5311yescharacteristic
437C22×C4C2×C22⋊D5312yes—
438C22×C4C2×C22⋊D5312yes—
439C22×C4C23×D5311yescharacteristic
440Q16—2128no—
441C2×C8—1061no—
442C2×C8—1061no—
443C2×C8—1062no—
444C2×C8—1062no—
445C2×C8—1062no—
446C2×C8—1062no—
447C2×C8—1062no—
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—1061no—
453C4⋊C4—1061no—
454C4⋊C4—1062no—
455C4⋊C4—1062no—
456C4⋊C4—2124no—
457C4⋊C4—2122no—
458C4⋊C4C2×C22⋊D5312yes—
459C4⋊C4C2×C22⋊D5312yes—
460C22⋊C4—28no—
461C22⋊C4—28no—
462C22⋊C4—24no—
463C22⋊C4—4241no—
464C22⋊C4—4242no—
465C22⋊C4—2124no—
466C22⋊C4—2122no—
467C42—24no—
468C23—21no—
469C23—22no—
470C23—22no—
471C23—21no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23#1313318612yes—
480C23C22×C22⋊D5311yescharacteristic
481Q8—24no—
482Q8—24no—
483Q8—48no—
484Q8—44no—
485Q8—42no—
486Q8—42no—
487Q8—2122no—
488Q8—2121no—
489Q8—2121no—
490D4—44no—
491D4—48no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—48no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—2121no—
513C2×C4—2121no—
514C2×C4—2121no—
515C2×C4—2122no—
516C2×C4—2122no—
517C2×C4—2122no—
518C2×C4—2121no—
519C2×C4—2122no—
520C2×C4#1313318612yes—
521C2×C4C22×C22⋊D5311yescharacteristic
522C2×C4C22×C22⋊D5311yescharacteristic
523C2×C4#1313326212yes—
524C2×C4#1313326212yes—
525C8—2122no—
526C8—2121no—
527C8—2121no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22C2×#1313318611yescharacteristic
541C22#1681118212yes—
542C22#1681143111yescharacteristic
543C22#1681143812yes—
544C22#1681143811yescharacteristic
545C4—22no—
546C4—42no—
547C4—42no—
548C4—44no—
549C4—42no—
550C4—4241no—
551C4—4241no—
552C4—4242no—
553C4#1681118212yes—
554C4#1681119011yescharacteristic
555C4#1681119011yescharacteristic
556C2—42no—
557C2—82no—
558C2#2028042711yescharacteristic
559C2#2028401011yescharacteristic
560C2#2028403011yescharacteristic
561C2#2028559712yes—
562C2#2028560612yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#536C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
hypercenterC2×#536D53
hypocenterC53#29605
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×#16810964C2semidirect
2. C2×#16810779C2semidirect
4. #20275915C2semidirect
5. #20275908C2semidirect
6. #20275723C2semidirect
7. #20275723C2semidirect
8. #20274810C2semidirect
9. #20274650C2semidirect
326. C53#29605semidirect

Maximal subgroups

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

subgroupindex
C2×#168109642
C2×#168107792
#202804442
#202759152
#202759082
#202757232
#202748102
#202746502
#202746272
C2×C53×#5362
#2960553