Finite Group Databasebeta

← #27422836

Subgroups — #27422836

Order 13568.

1034 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810762C211yesmaximal, characteristic
2C2×#16810680C211yesmaximal, characteristic, complemented
3#20283046C211yesmaximal, characteristic
4#20276110C212yesmaximal, complemented
5#20276030C212yesmaximal, complemented
6#20275599C212yesmaximal, complemented
7#20275408C212yesmaximal, complemented
8#20275032C211yesmaximal, characteristic
9#20274903C211yesmaximal, characteristic, complemented
10#20274650C211yesmaximal, characteristic, complemented
11C2×C53×#391C211yesmaximal, characteristic, complemented
12C22×#11719924C2212yes—
13C2×#13133200—21no—
14C2×#13133191C2211yescharacteristic
15#16810762—24no—
16#16810680—24no—
17C2×#13133150C2212yes—
18C2×C4⋊#88322C2212yes—
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×#13133141C2212yes—
22C2×#13133137C2212yes—
23C2×C22⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25C2×#13133128C2212yes—
26#16810525C2211yescharacteristic
27#16810525C2211yescharacteristic
28#16810521C2211yescharacteristic
29#16810521C2211yescharacteristic
30#16810514—21no—
31#16810514—21no—
32#16810502—21no—
33#16810483C2211yescharacteristic
34#16810385C2212yes—
35#16810385C2212yes—
36C23×Q8×C53C2211yescharacteristic
37C22×C4⋊C4×C53C2211yescharacteristic
38C22×C42×C53C2211yescharacteristic
39C53×#391—21no—
40C53×#391—21no—
41C53×#391C2212yes—
42C53×#391C2212yes—
43C2×#95×C53—21no—
44C2×#95×C53—21no—
45C22×D212C2311yescharacteristic
46C22×C22⋊D53—21no—
47C2×#11719924—24no—
48C2×#11719924—24no—
49C2×#11719924—42no—
50C22×#88322C2312yes—
51C22×#88322C2312yes—
52#13133200—44no—
53#13133191—22no—
54#13133191—22no—
55C2×C4⋊Dic53C2311yescharacteristic
56C2×C4⋊Dic53C2311yescharacteristic
57C2×C4⋊Dic53C2311yescharacteristic
58C2×#11719919—21no—
59C2×#11719919—21no—
60C2×C22⋊Dic53—21no—
61C2×#11719921—21no—
62C2×#11719921—21no—
63C2×#11719921—21no—
64C2×#11719921—21no—
65C2×#11719921—21no—
66#13133150—42no—
67C4⋊#88322—24no—
68#13133141—24no—
69#13133141—24no—
70#13133141—42no—
71#13133137—42no—
72C22⋊#88322—22no—
73C22⋊#88322D414yes—
74#13133128—24no—
75#13133124—21no—
76#13133124—21no—
77#13133124—21no—
78C22×Q8×C53—24no—
79C22×Q8×C53—21no—
80C22×Q8×C53—22no—
81C22×Q8×C53—22no—
82C22×Q8×C53—42no—
83C22×Q8×C53C2312yes—
84C23×C4×C53—21no—
85C23×C4×C53—21no—
86C23×C4×C53D411yescharacteristic
87C23×C4×C53D411yescharacteristic
88C23×C4×C53C2311yescharacteristic
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53—42no—
91C2×C4⋊C4×C53C2312yes—
92C2×C42×C53—22no—
93C2×C42×C53—22no—
94C2×C42×C53D411yescharacteristic
95C2×C42×C53D411yescharacteristic
96C2×C42×C53C2311yescharacteristic
97C2×C42×C53C2311yescharacteristic
98#95×C53—22no—
99#95×C53—22no—
100#95×C53—22no—
101#95×C53—22no—
102C24×C53C2×D411yescharacteristic
103C23×D53—21no—
104C2×Q8×C53—24no—
105C2×Q8×C53—22no—
106C2×Q8×C53—22no—
107C2×Q8×C53—22no—
108C2×Q8×C53—22no—
109C2×Q8×C53—22no—
110C2×Q8×C53—48no—
111C2×Q8×C53—48no—
112C2×Q8×C53—42no—
113C2×Q8×C53—42no—
114C2×Q8×C53#4114yes—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—24no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—41no—
126C22×C4×C53—42no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53—42no—
130C22×C4×C53—42no—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D412yes—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53#4112yes—
138C22×C4×C53C2411yescharacteristic
139C2×C22⋊D53—42no—
140C2×C22⋊D53—42no—
141C2×C22⋊D53—42no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212—22no—
148C2×D212—22no—
149C4⋊C4×C53—22no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—42no—
152C42×C53—22no—
153C42×C53—22no—
154C42×C53—21no—
155C42×C53—21no—
156C42×C53—21no—
157C42×C53—21no—
158C22⋊Dic53—42no—
159#11719924—48no—
160#11719921—42no—
161#11719921—42no—
162#11719921—42no—
163#11719921—42no—
164#11719921—42no—
165C4⋊Dic53—21no—
166C4⋊Dic53—22no—
167C4⋊Dic53—21no—
168C4⋊Dic53—22no—
169#11719919—42no—
170#11719919—42no—
171C2×#88322—22no—
172C2×#88322—22no—
173C2×#88322—22no—
174C2×#88322—22no—
175C2×#88322—22no—
176C2×#88322C2×D414yes—
177C23×C53—21no—
178C23×C53—21no—
179C23×C53—21no—
180C23×C53—21no—
181C23×C53#12112yes—
182C23×C53AGL(1,ℤ/8)12yes—
183C23×C53#13712yes—
184C23×C53C22×D411yescharacteristic
185C22×D53—41no—
186C22×D53—41no—
187C22×D53—41no—
188C22×D53—41no—
189C22×D53—41no—
190C22×D53—41no—
191C22×D53—41no—
192Q8×C53—24no—
193Q8×C53—24no—
194Q8×C53—48no—
195Q8×C53—42no—
196Q8×C53—42no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—24no—
202C2×C4×C53—24no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—24no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—44no—
214C2×C4×C53—48no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—44no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—84no—
222C2×C4×C53#12112yes—
223C2×C4×C53#13712yes—
224C2×C4×C53C22×D411yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C53C2×#4112yes—
227C2×C4×C53H(5,𝔽2)12yes—
228C22⋊D53—84no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238D212—42no—
239D212—42no—
240#88322—24no—
241#88322—42no—
242#24509—531nomaximal, Sylow
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—22no—
247C22×C53—22no—
248C22×C53—22no—
249C22×C53#46814yes—
250C22×C53C2×#12111yescharacteristic
251C22×C53#52511yescharacteristic
252C22×C53#53511yescharacteristic
253C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
254C22×C53C2×#13711yescharacteristic
255C22×C53C2×#13711yescharacteristic
256C22×C53#57511yescharacteristic
257C2×D53—81no—
258C2×D53—81no—
259C2×D53—81no—
260C2×D53—81no—
261C2×D53—81no—
262C2×D53—81no—
263C2×D53—81no—
264C4×C53—22no—
265C4×C53—22no—
266C4×C53—44no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53—41no—
270C4×C53—82no—
271C4×C53#54612yes—
272C4×C53#55112yes—
273Dic53—81no—
274Dic53—81no—
275Dic53—81no—
276C2×#448—531no—
277C2×#432—531no—
278#2854—531no—
279C2×#391D5311yescharacteristic, complemented
280#2068—532no—
281#2055—532no—
282#1973—532no—
283#1935—532no—
284#1874—531no—
285#1852—531no—
286#1787—531no—
287C2×C53—24no—
288C2×C53#241111yescharacteristic
289C2×C53C2×#46811yescharacteristic
290C2×C53#305611yescharacteristic
291C2×C53#311211yescharacteristic
292C2×C53#311711yescharacteristic
293C2×C53#321311yescharacteristic
294C2×C53#321811yescharacteristic
295D53—161no—
296C23×Q8C2×D5311yescharacteristic
297C22×SD16—532no—
298C2×#123—1061no—
299C2×#121—531no—
300C22×C4⋊C4C2×D5311yescharacteristic
301C22×C42C2×D5311yescharacteristic
302#448—1064no—
303#432—1064no—
304C2×#107—532no—
305C2×C4⋊C8—532no—
306C2×#103—532no—
307C2×#103—532no—
308C2×#103—532no—
309C2×#102—532no—
310C2×C22⋊C8—531no—
311C2×C22⋊C8—531no—
312C2×#97—532no—
313#391—21no—
314#391—21no—
315#391C2×D5312yes—
316#391C2×D5312yes—
317#390—531no—
318#390—531no—
319#389—531no—
320#389—531no—
321#388—1061no—
322#388—1061no—
323#386—1061no—
324#380—531no—
325C2×#95—21no—
326C2×#95—21no—
327#340—532no—
328#340—532no—
329C53#2450911yescharacteristic, complemented, Sylow
330C22×Q8—24no—
331C22×Q8—21no—
332C22×Q8—22no—
333C22×Q8—22no—
334C22×Q8—42no—
335C22×Q8C22×D5312yes—
336C22×D4—531no—
337C22×D4—1061no—
338C23×C4—21no—
339C23×C4—21no—
340C23×C4C22⋊D5311yescharacteristic
341C23×C4C22⋊D5311yescharacteristic
342C23×C4C22×D5311yescharacteristic
343C2×SD16—1064no—
344C2×SD16—1064no—
345C2×SD16—2122no—
346C22×C8—532no—
347C22×C8—532no—
348#123—2124no—
349#121—1062no—
350#121—1062no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—531no—
354C2×C4⋊C4—531no—
355C2×C4⋊C4—531no—
356C2×C4⋊C4—1061no—
357C2×C4⋊C4—1061no—
358C2×C4⋊C4C22×D5312yes—
359C2×C22⋊C4—1061no—
360C2×C22⋊C4—1061no—
361C2×C22⋊C4—1061no—
362C2×C22⋊C4—1061no—
363C2×C22⋊C4—1061no—
364C2×C22⋊C4—1061no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42C22⋊D5311yescharacteristic
368C2×C42C22⋊D5311yescharacteristic
369C2×C42C22×D5311yescharacteristic
370C2×C42C22×D5311yescharacteristic
371#107—2122no—
372C4⋊C8—1064no—
373#103—1064no—
374#103—1064no—
375#103—2122no—
376#102—2122no—
377C22⋊C8—534no—
378C22⋊C8—1062no—
379#97—1064no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—1061no—
385#95—1061no—
386#95—1061no—
387C24—1061no—
388C24C2×C22⋊D5311yescharacteristic
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—48no—
396C2×Q8—48no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8#1171994514yes—
400C2×D4—1061no—
401C2×D4—1061no—
402C2×D4—1062no—
403C2×D4—1062no—
404C2×D4—2122no—
405C2×D4—2122no—
406C2×D4—2122no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—1061no—
424C22×C4—1061no—
425C22×C4—1061no—
426C22×C4D4×D5312yes—
427C22×C4#1171994712yes—
428C22×C4C2×C22⋊D5311yescharacteristic
429C22×C4C2×C22⋊D5311yescharacteristic
430C22×C4C2×C22⋊D5311yescharacteristic
431C22×C4C2×C22⋊D5311yescharacteristic
432C22×C4C2×C22⋊D5311yescharacteristic
433C22×C4C23×D5311yescharacteristic
434SD16—2128no—
435C2×C8—534no—
436C2×C8—1062no—
437C2×C8—1062no—
438C2×C8—1062no—
439C2×C8—1062no—
440C2×C8—1062no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—1061no—
445C4⋊C4—1061no—
446C4⋊C4—1062no—
447C4⋊C4—1062no—
448C4⋊C4—2122no—
449C4⋊C4—2122no—
450C22⋊C4—2122no—
451C22⋊C4—2122no—
452C22⋊C4—2122no—
453C22⋊C4—2122no—
454C22⋊C4—2122no—
455C22⋊C4—2122no—
456C42—22no—
457C42—22no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—2121no—
467C23—2121no—
468C23—2121no—
469C23—2121no—
470C23—2121no—
471C23—2121no—
472C23—2121no—
473C23#1313318912yes—
474C23#1313324212yes—
475C23#1313324812yes—
476C23C22×C22⋊D5311yescharacteristic
477Q8—24no—
478Q8—24no—
479Q8—48no—
480Q8—42no—
481Q8—42no—
482D4—4244no—
483D4—2122no—
484D4—2122no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—44no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—84no—
510C2×C4—2121no—
511C2×C4—2121no—
512C2×C4—2121no—
513C2×C4—2121no—
514C2×C4—2121no—
515C2×C4—2121no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4#1313318912yes—
520C2×C4#1313324812yes—
521C2×C4C2×D4×D5311yescharacteristic
522C2×C4C2×#1171994711yescharacteristic
523C2×C4C2×#1171994512yes—
524C2×C4#1313326212yes—
525C8—1064no—
526C8—2122no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—4241no—
534C22—4241no—
535C22—4241no—
536C22—4241no—
537C22—4241no—
538C22—4241no—
539C22—4241no—
540C22#1681087114yes—
541C22C2×#1313318911yescharacteristic
542C22#1681113411yescharacteristic
543C22#1681119011yescharacteristic
544C22C2×#1313324211yescharacteristic
545C22C2×#1313324811yescharacteristic
546C22C2×#1313324811yescharacteristic
547C22#1681142111yescharacteristic
548C4—22no—
549C4—22no—
550C4—44no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—82no—
555C4—4241no—
556C4—4241no—
557C4—4241no—
558C4#1681126312yes—
559C4#1681129912yes—
560C2—24no—
561C2—8481no—
562C2#2027810011yescharacteristic
563C2C2×#1681087111yescharacteristic
564C2#2028434111yescharacteristic
565C2#2028483711yescharacteristic
566C2#2028488611yescharacteristic
567C2#2028570211yescharacteristic
568C2#2028574211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#391C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenterC2×#391D53
hypocenterC53#24509
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. C2×#16810680C2semidirect
4. #20276110C2semidirect
5. #20276030C2semidirect
6. #20275599C2semidirect
7. #20275408C2semidirect
9. #20274903C2semidirect
10. #20274650C2semidirect
11. C2×C53×#391C2semidirect
279. C2×#391D53semidirect
329. C53#24509semidirect

Maximal subgroups

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

subgroupindex
C2×#168107622
C2×#168106802
#202830462
#202761102
#202760302
#202755992
#202754082
#202750322
#202749032
#202746502
C2×C53×#3912
#2450953