Finite Group Databasebeta

← #27281143

Subgroups — #27281143

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283164C211yesmaximal, characteristic
2#20274655C211yesmaximal, characteristic
3#20273287C211yesmaximal, characteristic, complemented
4#20273079C211yesmaximal, characteristic, complemented
5#20272620C211yesmaximal, characteristic, complemented
6#20272602C211yesmaximal, characteristic, complemented
7C53×#3007C211yesmaximal, characteristic
8C2×#13133224—21no—
9C2×#13133141—21no—
10C2×#13133138—21no—
11#16810590—22no—
12#16810590—21no—
13#16810587—21no—
14#16810587C2211yescharacteristic
15C2×C22⋊#88322—21no—
16C2×C22⋊#88322C2211yescharacteristic
17C2×C22⋊#88322C2211yescharacteristic
18#16810483C2211yescharacteristic
19#16810360—21no—
20#16810360—21no—
21#16810349—22no—
22C22×#41×C53C2211yescharacteristic
23C2×#137×C53C411yescharacteristic, complemented
24C2×#137×C53C411yescharacteristic, complemented
25C2×C22⋊Q8×C53C2211yescharacteristic
26C53×#452—21no—
27C53×#452—21no—
28C53×#451—21no—
29C53×#450—21no—
30C53×#417C411yescharacteristic, complemented
31C53×#417C411yescharacteristic, complemented
32C53×#407C2211yescharacteristic
33#13133224—42no—
34#13133224—42no—
35C2×#11719917—22no—
36C2×#11719917—21no—
37C22×#88322—22no—
38C22×#88322—21no—
39C22×#88322—21no—
40C2×C4⋊Dic53—21no—
41C2×C22⋊Dic53—21no—
42#13133141—42no—
43#13133138—42no—
44C22⋊#88322—22no—
45C22⋊#88322—22no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C22⋊#88322D411yescharacteristic
51C22⋊#88322D411yescharacteristic
52C22⋊#88322D412yes—
53#13133124—21no—
54C2×#41×C53—21no—
55C2×#41×C53—21no—
56C2×#41×C53—42no—
57C2×#41×C53—42no—
58C2×#41×C53C2×C411yescharacteristic
59C2×#41×C53C2×C411yescharacteristic
60C2×#41×C53D411yescharacteristic
61C2×#41×C53D411yescharacteristic
62C22×Q8×C53C2×C411yescharacteristic
63C22×Q8×C53D411yescharacteristic
64C22×D4×C53—21no—
65C22×D4×C53D411yescharacteristic
66C23×C4×C53—21no—
67C23×C4×C53C2311yescharacteristic
68#137×C53—42no—
69#137×C53—42no—
70C2×Q16×C53—21no—
71C2×Q16×C53—21no—
72C2×SD16×C53—21no—
73C2×SD16×C53—21no—
74C2×M16×C53C2×C411yescharacteristic
75C2×M16×C53C2×C411yescharacteristic
76C22⋊Q8×C53—21no—
77C22⋊Q8×C53—21no—
78C22⋊Q8×C53—21no—
79C22⋊Q8×C53—21no—
80C2×C4⋊C4×C53—21no—
81C2×C4⋊C4×C53C2×C411yescharacteristic
82C2×C22⋊C4×C53—21no—
83#103×C53—21no—
84#103×C53—21no—
85#102×C53—21no—
86#102×C53—21no—
87C22⋊C8×C53—21no—
88C22⋊C8×C53—21no—
89C24×C53—21no—
90C24×C53#4111yescharacteristic
91#41×C53—22no—
92#41×C53—22no—
93#41×C53—22no—
94#41×C53—22no—
95#41×C53—41no—
96#41×C53—41no—
97#41×C53—41no—
98#41×C53—41no—
99#41×C53—44no—
100#41×C53—44no—
101C2×Q8×C53—21no—
102C2×Q8×C53—22no—
103C2×Q8×C53—21no—
104C2×Q8×C53—21no—
105C2×Q8×C53—41no—
106C2×Q8×C53—41no—
107C2×Q8×C53—41no—
108C2×Q8×C53C22⋊C411yescharacteristic
109C2×Q8×C53C22⋊C411yescharacteristic
110C2×D4×C53—21no—
111C2×D4×C53—22no—
112C2×D4×C53—21no—
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53—41no—
117C2×D4×C53—44no—
118C2×D4×C53C22⋊C411yescharacteristic
119C2×D4×C53C22⋊C411yescharacteristic
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—41no—
128C22×C4×C53—41no—
129C22×C4×C53—44no—
130C22×C4×C53C22×C411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53#4111yescharacteristic
136C22×Dic53—41no—
137Q16×C53—42no—
138Q16×C53—42no—
139SD16×C53—42no—
140SD16×C53—42no—
141M16×C53—41no—
142M16×C53—41no—
143C2×C8×C53—21no—
144C2×C8×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—41no—
148C4⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C22⋊C4×C53—41no—
151C22⋊Dic53—81no—
152C4⋊Dic53—41no—
153C4⋊Dic53—41no—
154#11719917—44no—
155#11719917—41no—
156#11719917—41no—
157C2×#88322—21no—
158C2×#88322—22no—
159C2×#88322—21no—
160C2×#88322—44no—
161C2×#88322—41no—
162C2×#88322—41no—
163C2×#88322—41no—
164C2×#88322—41no—
165C2×#88322—41no—
166C23×C53—21no—
167C23×C53—21no—
168C23×C53—21no—
169C23×C53—21no—
170C23×C53—41no—
171C23×C53—41no—
172C23×C53—42no—
173C23×C53—41no—
174C23×C53—41no—
175C23×C53—81no—
176C23×C53C4×D411yescharacteristic
177C23×C53#12111yescharacteristic
178C23×C53#12311yescharacteristic
179Q8×C53—21no—
180Q8×C53—21no—
181Q8×C53—21no—
182Q8×C53—42no—
183Q8×C53—41no—
184Q8×C53—41no—
185Q8×C53—81no—
186Q8×C53#9912yes—
187D4×C53—24no—
188D4×C53—21no—
189D4×C53—21no—
190D4×C53—21no—
191D4×C53—42no—
192D4×C53—42no—
193D4×C53—42no—
194D4×C53—42no—
195D4×C53#9912yes—
196C2×C4×C53—24no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—81no—
216C2×C4×C53C2×C22⋊C411yescharacteristic
217C2×C4×C53C4×D411yescharacteristic
218C2×C4×C53#12011yescharacteristic
219C2×C4×C53#12111yescharacteristic
220C2×Dic53—81no—
221C2×Dic53—81no—
222C2×Dic53—81no—
223C8×C53—41no—
224C8×C53—41no—
225#88322—41no—
226#88322—42no—
227#88322—41no—
228#88322—81no—
229#9345—531nomaximal, Sylow
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—22no—
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—41no—
240C22×C53—41no—
241C22×C53—41no—
242C22×C53—41no—
243C22×C53—41no—
244C22×C53—81no—
245C22×C53—81no—
246C22×C53—82no—
247C22×C53—82no—
248C22×C53#38611yescharacteristic
249C22×C53#47111yescharacteristic
250C22×C53#47311yescharacteristic
251C4×C53—21no—
252C4×C53—21no—
253C4×C53—22no—
254C4×C53—21no—
255C4×C53—41no—
256C4×C53—81no—
257C4×C53—81no—
258C4×C53C2×#9911yescharacteristic
259C4×C53C2×#9911yescharacteristic
260Dic53—161no—
261#3007D5311yescharacteristic
262#2880—531no—
263#1788—531no—
264#1520—531no—
265#1472—531no—
266#1305—531no—
267#1298—531no—
268C2×C53—21no—
269C2×C53—21no—
270C2×C53—41no—
271C2×C53—41no—
272C2×C53—42no—
273C2×C53—81no—
274C2×C53#188111yescharacteristic
275C2×C53#189811yescharacteristic
276C2×C53#190811yescharacteristic
277C22×#41C2×D5311yescharacteristic
278C2×#137Dic5311yescharacteristic, complemented
279C2×#137Dic5311yescharacteristic, complemented
280C2×#131—1061no—
281C2×C22⋊Q8C2×D5311yescharacteristic
282#452—21no—
283#452—21no—
284#451—21no—
285#450—21no—
286#417Dic5311yescharacteristic, complemented
287#417Dic5311yescharacteristic, complemented
288C2×#103—1061no—
289C2×#102—1061no—
290#408—1062no—
291#408—1061no—
292#407—531no—
293#407—1061no—
294#407C2×D5311yescharacteristic
295C2×C22⋊C8—531no—
296C2×C22⋊C8—531no—
297C2×C22⋊C8—1061no—
298#380—531no—
299#328—1061no—
300#328—1061no—
301#324—1062no—
302C53#934511yescharacteristic, complemented, Sylow
303C2×#41—21no—
304C2×#41—21no—
305C2×#41—42no—
306C2×#41—42no—
307C2×#41C2×Dic5311yescharacteristic
308C2×#41C2×Dic5311yescharacteristic
309C2×#41C22⋊D5311yescharacteristic
310C2×#41C22⋊D5311yescharacteristic
311C22×Q8C2×Dic5311yescharacteristic
312C22×Q8C22⋊D5311yescharacteristic
313C22×D4—21no—
314C22×D4C22⋊D5311yescharacteristic
315C23×C4—21no—
316C23×C4C22×D5311yescharacteristic
317#137—42no—
318#137—42no—
319C2×Q16—21no—
320C2×Q16—21no—
321C2×SD16—21no—
322C2×SD16—21no—
323#131—2122no—
324#131—2122no—
325C2×M16—1062no—
326C2×M16—1061no—
327C2×M16C2×Dic5311yescharacteristic
328C2×M16C2×Dic5311yescharacteristic
329C22×C8—1062no—
330C22×C8—1061no—
331C22×C8—1061no—
332C22⋊Q8—21no—
333C22⋊Q8—21no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—1061no—
338C2×C4⋊C4C2×Dic5311yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—1061no—
341#103—21no—
342#103—21no—
343#103—2122no—
344#102—21no—
345#102—21no—
346#102—2122no—
347C22⋊C8—21no—
348C22⋊C8—21no—
349C22⋊C8—532no—
350C22⋊C8—531no—
351C22⋊C8—531no—
352C22⋊C8—1062no—
353C22⋊C8—1062no—
354C22⋊C8—1061no—
355C22⋊C8—1061no—
356C22⋊C8—1061no—
357C22⋊C8—1061no—
358#95—1061no—
359C24—21no—
360C24#1171994711yescharacteristic
361#41—22no—
362#41—22no—
363#41—22no—
364#41—22no—
365#41—41no—
366#41—41no—
367#41—41no—
368#41—41no—
369#41—44no—
370#41—44no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8C22⋊Dic5311yescharacteristic
379C2×Q8C22⋊Dic5311yescharacteristic
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—44no—
388C2×D4C22⋊Dic5311yescharacteristic
389C2×D4C22⋊Dic5311yescharacteristic
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—44no—
400C22×C4—2121no—
401C22×C4D4×D5311yescharacteristic
402C22×C4D4×D5311yescharacteristic
403C22×C4#1171994711yescharacteristic
404C22×C4C22×Dic5311yescharacteristic
405C22×C4C2×C22⋊D5311yescharacteristic
406C22×C4C2×C22⋊D5311yescharacteristic
407Q16—42no—
408Q16—42no—
409SD16—42no—
410SD16—42no—
411M16—41no—
412M16—41no—
413M16—2124no—
414M16—2121no—
415M16—2121no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—1061no—
419C2×C8—1062no—
420C2×C8—1061no—
421C2×C8—2121no—
422C2×C8—2121no—
423C2×C8—2121no—
424C2×C8—2124no—
425C2×C8—2121no—
426C2×C8—2121no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—2121no—
432C4⋊C4—2121no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—4241no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—41no—
441C23—41no—
442C23—42no—
443C23—41no—
444C23—41no—
445C23—81no—
446C23D4×Dic5311yescharacteristic
447C23#1313318911yescharacteristic
448C23#1313319811yescharacteristic
449Q8—21no—
450Q8—21no—
451Q8—21no—
452Q8—42no—
453Q8—41no—
454Q8—41no—
455Q8—81no—
456Q8#1313313212yes—
457D4—24no—
458D4—21no—
459D4—21no—
460D4—21no—
461D4—42no—
462D4—42no—
463D4—42no—
464D4—42no—
465D4#1313313212yes—
466C2×C4—24no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—81no—
486C2×C4—4241no—
487C2×C4—4241no—
488C2×C4—4241no—
489C2×C4C2×C22⋊Dic5311yescharacteristic
490C2×C4D4×Dic5311yescharacteristic
491C2×C4#1313318511yescharacteristic
492C2×C4#1313318911yescharacteristic
493C8—41no—
494C8—41no—
495C8—4241no—
496C8—2121no—
497C8—2122no—
498C8—2121no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—22no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—81no—
514C22—81no—
515C22—82no—
516C22—82no—
517C22#1681050311yescharacteristic
518C22#1681088811yescharacteristic
519C22#1681089611yescharacteristic
520C4—21no—
521C4—21no—
522C4—22no—
523C4—21no—
524C4—41no—
525C4—81no—
526C4—81no—
527C4—8481no—
528C4C2×#1313313211yescharacteristic
529C4C2×#1313313211yescharacteristic
530C2—21no—
531C2—21no—
532C2—41no—
533C2—41no—
534C2—42no—
535C2—81no—
536C2#2027507911yescharacteristic
537C2#2027516911yescharacteristic
538C2#2027523211yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#3007C2
socleC22×C53#386
prime radicalC22×#41×C53C22
cosocleC23×C4×C53C23
hypercenter#3007D53
hypocenterC53#9345
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3007

Complements

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

NH ≅ G/Ntype
3. #20273287C2semidirect
4. #20273079C2semidirect
5. #20272620C2semidirect
6. #20272602C2semidirect
23. C2×#137×C53C4semidirect
24. C2×#137×C53C4semidirect
30. C53×#417C4semidirect
31. C53×#417C4semidirect
278. C2×#137Dic53semidirect
279. C2×#137Dic53semidirect
286. #417Dic53semidirect
287. #417Dic53semidirect
302. C53#9345semidirect

Maximal subgroups

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

subgroupindex
#202831642
#202746552
#202732872
#202730792
#202726202
#202726022
C53×#30072
#934553