Finite Group Databasebeta

← #27280401

Subgroups — #27280401

Order 13568.

686 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
1#20283164C211yesmaximal, characteristic
2#20274659C211yesmaximal, characteristic
3#20273235C211yesmaximal, characteristic, complemented
4#20273079C211yesmaximal, characteristic, complemented
5#20272592C211yesmaximal, characteristic, complemented
6#20272562C211yesmaximal, characteristic, complemented
7C2×C53×#452C211yesmaximal, characteristic
8C2×#13133224—21no—
9C2×#13133144—21no—
10C2×#13133144—21no—
11#16810590—22no—
12#16810590—21no—
13#16810587—21no—
14#16810587C2211yescharacteristic
15#16810587C2211yescharacteristic
16C2×C22⋊#88322—21no—
17C2×C22⋊#88322C2211yescharacteristic
18C4×C22⋊Dic53C2211yescharacteristic
19#16810360—22no—
20#16810349—21no—
21#16810349—21no—
22C22×#41×C53C2211yescharacteristic
23C22×Q16×C53C411yescharacteristic, complemented
24C22×SD16×C53C411yescharacteristic, complemented
25C2×C22⋊Q8×C53C2211yescharacteristic
26C53×#452—21no—
27C53×#452—21no—
28C53×#452—21no—
29C53×#452—21no—
30C2×#103×C53C411yescharacteristic, complemented
31C2×#102×C53C411yescharacteristic, complemented
32C2×C22⋊C8×C53C2211yescharacteristic
33#13133224—42no—
34#13133224—42no—
35C2×#11719917—22no—
36C2×#11719917—21no—
37C2×#11719917—21no—
38C22×#88322—22no—
39C22×#88322—21no—
40C2×C22⋊Dic53—21no—
41C2×C4×Dic53—21no—
42#13133144—42no—
43#13133144—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×C53D411yescharacteristic
59C2×#41×C53D411yescharacteristic
60C2×#41×C53D411yescharacteristic
61C2×#41×C53D411yescharacteristic
62C22×Q8×C53C2×C411yescharacteristic
63C22×Q8×C53C2×C411yescharacteristic
64C22×D4×C53—21no—
65C22×D4×C53C2×C411yescharacteristic
66C23×C4×C53—21no—
67C23×C4×C53C2311yescharacteristic
68C2×Q16×C53—21no—
69C2×Q16×C53—21no—
70C2×Q16×C53—42no—
71C2×SD16×C53—21no—
72C2×SD16×C53—21no—
73C2×SD16×C53—42no—
74C22×C8×C53C2×C411yescharacteristic
75C22×C8×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—
141C2×C8×C53—21no—
142C2×C8×C53—21no—
143C2×C8×C53—41no—
144C2×C8×C53—41no—
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—22no—
152C22⋊Dic53—41no—
153C4×Dic53—41no—
154C4×Dic53—41no—
155#11719917—44no—
156#11719917—41no—
157#11719917—41no—
158#11719917—41no—
159#11719917—41no—
160C2×#88322—21no—
161C2×#88322—22no—
162C2×#88322—21no—
163C2×#88322—44no—
164C2×#88322—41no—
165C2×#88322—41no—
166C2×#88322—41no—
167C23×C53—21no—
168C23×C53—21no—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53—41no—
172C23×C53—41no—
173C23×C53—42no—
174C23×C53—41no—
175C23×C53—41no—
176C23×C53—81no—
177C23×C53C4×D411yescharacteristic
178C23×C53#12111yescharacteristic
179C23×C53#12311yescharacteristic
180Q8×C53—21no—
181Q8×C53—21no—
182Q8×C53—21no—
183Q8×C53—42no—
184Q8×C53—41no—
185Q8×C53—41no—
186Q8×C53—81no—
187Q8×C53#9912yes—
188D4×C53—24no—
189D4×C53—21no—
190D4×C53—21no—
191D4×C53—21no—
192D4×C53—42no—
193D4×C53—42no—
194D4×C53—42no—
195D4×C53—42no—
196D4×C53#9912yes—
197C2×C4×C53—24no—
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—21no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
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—41no—
216C2×C4×C53—41no—
217C2×C4×C53—81no—
218C2×C4×C53C2×C22⋊C411yescharacteristic
219C2×C4×C53C4×D411yescharacteristic
220C2×C4×C53#12011yescharacteristic
221C2×C4×C53#12111yescharacteristic
222C2×Dic53—42no—
223C2×Dic53—81no—
224C2×Dic53—81no—
225C8×C53—41no—
226C8×C53—41no—
227#88322—41no—
228#88322—42no—
229#88322—41no—
230#88322—81no—
231#9239—531nomaximal, Sylow
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—22no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—41no—
242C22×C53—41no—
243C22×C53—41no—
244C22×C53—41no—
245C22×C53—41no—
246C22×C53—81no—
247C22×C53—81no—
248C22×C53—82no—
249C22×C53—82no—
250C22×C53#38611yescharacteristic
251C22×C53#44311yescharacteristic
252C22×C53#44411yescharacteristic
253C4×C53—21no—
254C4×C53—21no—
255C4×C53—22no—
256C4×C53—21no—
257C4×C53—41no—
258C4×C53—81no—
259C4×C53—81no—
260C4×C53C2×#9911yescharacteristic
261C4×C53C2×#9911yescharacteristic
262Dic53—82no—
263C2×#452D5311yescharacteristic
264#2880—531no—
265#1789—531no—
266#1508—531no—
267#1472—531no—
268#1293—531no—
269#1282—531no—
270C2×C53—21no—
271C2×C53—21no—
272C2×C53—41no—
273C2×C53—41no—
274C2×C53—42no—
275C2×C53—81no—
276C2×C53#187611yescharacteristic
277C2×C53#189811yescharacteristic
278C2×C53#190211yescharacteristic
279C22×#41C2×D5311yescharacteristic
280C22×Q16Dic5311yescharacteristic, complemented
281C22×SD16Dic5311yescharacteristic, complemented
282C2×#131—1061no—
283C2×C22⋊Q8C2×D5311yescharacteristic
284#452—21no—
285#452—21no—
286#452—21no—
287#452—21no—
288C2×#104—1061no—
289C2×#104—1061no—
290C2×#103Dic5311yescharacteristic, complemented
291C2×#102Dic5311yescharacteristic, complemented
292#408—1062no—
293#408—1061no—
294#407—531no—
295#407—531no—
296#407—1061no—
297C2×C22⋊C8—531no—
298C2×C22⋊C8—1061no—
299C2×C22⋊C8C2×D5311yescharacteristic
300C4×C22⋊C4—531no—
301#328—1062no—
302#324—1061no—
303#324—1061no—
304C53#923911yescharacteristic, complemented, Sylow
305C2×#41—21no—
306C2×#41—21no—
307C2×#41—42no—
308C2×#41—42no—
309C2×#41C22⋊D5311yescharacteristic
310C2×#41C22⋊D5311yescharacteristic
311C2×#41C22⋊D5311yescharacteristic
312C2×#41C22⋊D5311yescharacteristic
313C22×Q8C2×Dic5311yescharacteristic
314C22×Q8C2×Dic5311yescharacteristic
315C22×D4—21no—
316C22×D4C2×Dic5311yescharacteristic
317C23×C4—21no—
318C23×C4C22×D5311yescharacteristic
319C2×Q16—21no—
320C2×Q16—21no—
321C2×Q16—42no—
322C2×SD16—21no—
323C2×SD16—21no—
324C2×SD16—42no—
325#131—2122no—
326#131—2122no—
327C2×M16—1062no—
328C2×M16—1061no—
329C2×M16—1061no—
330C22×C8—1062no—
331C22×C8—1061no—
332C22×C8C2×Dic5311yescharacteristic
333C22×C8C2×Dic5311yescharacteristic
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8—21no—
337C22⋊Q8—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4C2×Dic5311yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—1061no—
342C2×C42—1061no—
343#104—2122no—
344#104—2122no—
345#103—21no—
346#103—21no—
347#102—21no—
348#102—21no—
349C22⋊C8—21no—
350C22⋊C8—21no—
351C22⋊C8—531no—
352C22⋊C8—531no—
353C22⋊C8—532no—
354C22⋊C8—1062no—
355C22⋊C8—1062no—
356C22⋊C8—1061no—
357C22⋊C8—1061no—
358C22⋊C8—1061no—
359C22⋊C8—1061no—
360#95—1061no—
361C24—21no—
362C24#1171994711yescharacteristic
363#41—22no—
364#41—22no—
365#41—22no—
366#41—22no—
367#41—41no—
368#41—41no—
369#41—41no—
370#41—41no—
371#41—44no—
372#41—44no—
373C2×Q8—21no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8C22⋊Dic5311yescharacteristic
381C2×Q8C22⋊Dic5311yescharacteristic
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—41no—
389C2×D4—44no—
390C2×D4C22⋊Dic5311yescharacteristic
391C2×D4C22⋊Dic5311yescharacteristic
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—44no—
402C22×C4—2121no—
403C22×C4D4×D5311yescharacteristic
404C22×C4D4×D5311yescharacteristic
405C22×C4#1171994711yescharacteristic
406C22×C4C22×Dic5311yescharacteristic
407C22×C4C2×C22⋊D5311yescharacteristic
408C22×C4C2×C22⋊D5311yescharacteristic
409Q16—42no—
410Q16—42no—
411SD16—42no—
412SD16—42no—
413M16—2121no—
414M16—2121no—
415M16—2124no—
416M16—2121no—
417M16—2121no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—1061no—
423C2×C8—1061no—
424C2×C8—1062no—
425C2×C8—2121no—
426C2×C8—2124no—
427C2×C8—2121no—
428C2×C8—2121no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—1062no—
436C22⋊C4—2121no—
437C42—2121no—
438C42—2121no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—41no—
447C23—41no—
448C23—81no—
449C23D4×Dic5311yescharacteristic
450C23#1313318911yescharacteristic
451C23#1313319811yescharacteristic
452Q8—21no—
453Q8—21no—
454Q8—21no—
455Q8—42no—
456Q8—41no—
457Q8—41no—
458Q8—81no—
459Q8#1313313212yes—
460D4—24no—
461D4—21no—
462D4—21no—
463D4—21no—
464D4—42no—
465D4—42no—
466D4—42no—
467D4—42no—
468D4#1313313212yes—
469C2×C4—24no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—81no—
490C2×C4—4241no—
491C2×C4—4241no—
492C2×C4—2122no—
493C2×C4C2×C22⋊Dic5311yescharacteristic
494C2×C4D4×Dic5311yescharacteristic
495C2×C4#1313318511yescharacteristic
496C2×C4#1313318911yescharacteristic
497C8—41no—
498C8—41no—
499C8—4241no—
500C8—2121no—
501C8—2121no—
502C8—2122no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—22no—
510C22—21no—
511C22—21no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—81no—
518C22—81no—
519C22—82no—
520C22—82no—
521C22#1681050311yescharacteristic
522C22#1681074011yescharacteristic
523C22#1681074511yescharacteristic
524C4—21no—
525C4—21no—
526C4—22no—
527C4—21no—
528C4—41no—
529C4—81no—
530C4—81no—
531C4—4242no—
532C4C2×#1313313211yescharacteristic
533C4C2×#1313313211yescharacteristic
534C2—21no—
535C2—21no—
536C2—41no—
537C2—41no—
538C2—42no—
539C2—81no—
540C2#2027504411yescharacteristic
541C2#2027516911yescharacteristic
542C2#2027519011yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#452C2
socleC22×C53#386
prime radicalC22×#41×C53C22
cosocleC23×C4×C53C23
hypercenterC2×#452D53
hypocenterC53#9239
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#452

Complements

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

NH ≅ G/Ntype
3. #20273235C2semidirect
4. #20273079C2semidirect
5. #20272592C2semidirect
6. #20272562C2semidirect
23. C22×Q16×C53C4semidirect
24. C22×SD16×C53C4semidirect
30. C2×#103×C53C4semidirect
31. C2×#102×C53C4semidirect
280. C22×Q16Dic53semidirect
281. C22×SD16Dic53semidirect
290. C2×#103Dic53semidirect
291. C2×#102Dic53semidirect
304. C53#9239semidirect

Maximal subgroups

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

subgroupindex
#202831642
#202746592
#202732352
#202730792
#202725922
#202725622
C2×C53×#4522
#923953