Finite Group Databasebeta

← #27301688

Subgroups — #27301688

Order 13568.

630 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283179C211yesmaximal, characteristic
2#20276390C211yesmaximal, characteristic, complemented
3#20276369C211yesmaximal, characteristic, complemented
4#20274690C211yesmaximal, characteristic
5#20272844C211yesmaximal, characteristic, complemented
6#20272836C211yesmaximal, characteristic, complemented
7C53×#3011C211yesmaximal, characteristic
8#16811101—21no—
9#16810640—21no—
10#16810632—21no—
11#16810607C2211yescharacteristic
12#16810597—22no—
13#16810597—22no—
14#16810597—21no—
15#16810597C2211yescharacteristic
16#16810597C2211yescharacteristic
17#16810597C2211yescharacteristic
18#16810418—22no—
19#16810413—22no—
20C2×2−1+4×C53C2211yescharacteristic
21C2×#137×C53C411yescharacteristic, complemented
22C22×Q16×C53C411yescharacteristic, complemented
23C53×#536C2211yescharacteristic
24C53×#452—21no—
25C53×#452—21no—
26C53×#451—21no—
27C53×#451—21no—
28C53×#419C411yescharacteristic, complemented
29C53×#416C411yescharacteristic, complemented
30C53×#408C2211yescharacteristic
31C2×#11719917—21no—
32Q8×Dic53—42no—
33Q8×Dic53—42no—
34#13133176—24no—
35#13133176—21no—
36#13133176—21no—
37#13133176—21no—
38#13133144—42no—
39#13133141—42no—
40#13133136—21no—
41#13133134—21no—
42#13133132—24no—
43#13133132—21no—
44#13133132—21no—
45#13133132—21no—
46#13133132—21no—
47#13133132D412yes—
48#13133132D412yes—
492−1+4×C53—42no—
502−1+4×C53—42no—
51C2×#41×C53—21no—
52C2×#41×C53—21no—
53C2×#41×C53—21no—
54C2×#41×C53C2×C411yescharacteristic
55C2×#41×C53D411yescharacteristic
56C2×#41×C53D411yescharacteristic
57C2×#41×C53C2311yescharacteristic
58C22×Q8×C53—21no—
59C22×Q8×C53C2×C411yescharacteristic
60C22×Q8×C53C2×C411yescharacteristic
61C22×Q8×C53D411yescharacteristic
62C22×Q8×C53D411yescharacteristic
63#137×C53—42no—
64C2×Q16×C53—21no—
65C2×Q16×C53—21no—
66C2×Q16×C53—21no—
67C2×Q16×C53—42no—
68C2×SD16×C53—21no—
69C2×M16×C53C2×C411yescharacteristic
70C22×C8×C53C2×C411yescharacteristic
71C4⋊Q8×C53—21no—
72#124×C53—21no—
73#123×C53—21no—
74#123×C53—21no—
75C22⋊Q8×C53—21no—
76C22⋊Q8×C53—21no—
77#117×C53C2×C411yescharacteristic
78#103×C53—21no—
79#103×C53—21no—
80#103×C53—21no—
81#102×C53—21no—
82C22⋊C8×C53—21no—
83C22⋊C8×C53—21no—
84#41×C53—22no—
85#41×C53—22no—
86#41×C53—41no—
87#41×C53—41no—
88#41×C53—41no—
89#41×C53—41no—
90#41×C53—41no—
91#41×C53—41no—
92#41×C53—44no—
93#41×C53—44no—
94#41×C53—44no—
95C2×Q8×C53—22no—
96C2×Q8×C53—22no—
97C2×Q8×C53—21no—
98C2×Q8×C53—21no—
99C2×Q8×C53—21no—
100C2×Q8×C53—22no—
101C2×Q8×C53—22no—
102C2×Q8×C53—21no—
103C2×Q8×C53—21no—
104C2×Q8×C53—41no—
105C2×Q8×C53—41no—
106C2×Q8×C53—41no—
107C2×Q8×C53—41no—
108C2×Q8×C53—44no—
109C2×Q8×C53C22⋊C411yescharacteristic
110C2×Q8×C53C22⋊C411yescharacteristic
111C2×Q8×C53C22⋊C411yescharacteristic
112C2×Q8×C53#4111yescharacteristic
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53C22⋊C411yescharacteristic
117C2×D4×C53C2×D411yescharacteristic
118C2×D4×C53C2×D411yescharacteristic
119C2×D4×C53#4111yescharacteristic
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×C53C22×C411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130Q16×C53—42no—
131Q16×C53—42no—
132Q16×C53—42no—
133SD16×C53—42no—
134M16×C53—41no—
135C2×C8×C53—21no—
136C2×C8×C53—21no—
137C2×C8×C53—41no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—41no—
140C4⋊C4×C53—41no—
141C22⋊C4×C53—21no—
142C22⋊C4×C53—41no—
143C22⋊C4×C53—41no—
144C42×C53—21no—
145C22⋊Dic53—22no—
146C22⋊Dic53—22no—
147C22⋊Dic53—41no—
148C4⋊Dic53—44no—
149C4⋊Dic53—41no—
150C4⋊Dic53—41no—
151C4⋊Dic53—41no—
152C4×Dic53—44no—
153C4×Dic53—41no—
154C4×Dic53—41no—
155C4×Dic53—41no—
156#11719917—41no—
157#11719917—41no—
158C2×#88322—41no—
159C23×C53—21no—
160C23×C53C4×D411yescharacteristic
161C23×C53#12011yescharacteristic
162C23×C53#12111yescharacteristic
163Q8×C53—24no—
164Q8×C53—21no—
165Q8×C53—21no—
166Q8×C53—21no—
167Q8×C53—21no—
168Q8×C53—41no—
169Q8×C53—42no—
170Q8×C53—42no—
171Q8×C53—41no—
172Q8×C53—41no—
173Q8×C53—81no—
174Q8×C53#9912yes—
175Q8×C53#9912yes—
176D4×C53—21no—
177D4×C53—21no—
178D4×C53—41no—
179D4×C53—41no—
180D4×C53—41no—
181D4×C53—42no—
182D4×C53—42no—
183D4×C53—42no—
184C2×C4×C53—24no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—42no—
195C2×C4×C53—42no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—81no—
202C2×C4×C53C2×C22⋊C411yescharacteristic
203C2×C4×C53C4×D411yescharacteristic
204C2×C4×C53#12111yescharacteristic
205C2×C4×C53#12311yescharacteristic
206C2×Dic53—42no—
207C2×Dic53—42no—
208C2×Dic53—81no—
209C8×C53—41no—
210C8×C53—41no—
211#88322—81no—
212#13101—531nomaximal, Sylow
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—21no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53—81no—
221C22×C53#38611yescharacteristic
222C4×C53—21no—
223C4×C53—22no—
224C4×C53—22no—
225C4×C53—21no—
226C4×C53—21no—
227C4×C53—41no—
228C4×C53—81no—
229C4×C53—81no—
230C4×C53C2×#9911yescharacteristic
231C4×C53C2×#9911yescharacteristic
232Dic53—82no—
233Dic53—82no—
234Dic53—161no—
235#3011D5311yescharacteristic
236#2884—531no—
237#2129—531no—
238#2123—531no—
239#1796—531no—
240#1395—531no—
241#1391—531no—
242C2×C53—21no—
243C2×C53—41no—
244C2×C53—41no—
245C2×C53—41no—
246C2×C53—81no—
247C2×C53#189811yescharacteristic
248C2×2−1+4C2×D5311yescharacteristic
249C2×#137Dic5311yescharacteristic, complemented
250C22×Q16Dic5311yescharacteristic, complemented
251#536C2×D5311yescharacteristic
252#519—1061no—
253#452—21no—
254#452—21no—
255#451—21no—
256#451—21no—
257#421—1061no—
258#419—1061no—
259#419Dic5311yescharacteristic, complemented
260#416Dic5311yescharacteristic, complemented
261#413—531no—
262#410—531no—
263#410—531no—
264#410—531no—
265#410—1062no—
266#410—1062no—
267#410—1061no—
268#408C2×D5311yescharacteristic
269#354—1062no—
270#352—1062no—
271C53#1310111yescharacteristic, complemented, Sylow
2722−1+4—42no—
2732−1+4—42no—
274C2×#41—21no—
275C2×#41—21no—
276C2×#41—21no—
277C2×#41C2×Dic5311yescharacteristic
278C2×#41C22⋊D5311yescharacteristic
279C2×#41C22⋊D5311yescharacteristic
280C2×#41C22×D5311yescharacteristic
281C22×Q8—21no—
282C22×Q8C2×Dic5311yescharacteristic
283C22×Q8C2×Dic5311yescharacteristic
284C22×Q8C22⋊D5311yescharacteristic
285C22×Q8C22⋊D5311yescharacteristic
286#137—42no—
287C2×Q16—21no—
288C2×Q16—21no—
289C2×Q16—21no—
290C2×Q16—42no—
291C2×SD16—21no—
292C2×M16—1061no—
293C2×M16C2×Dic5311yescharacteristic
294C22×C8C2×Dic5311yescharacteristic
295C4⋊Q8—21no—
296#124—21no—
297#123—21no—
298#123—21no—
299C22⋊Q8—21no—
300C22⋊Q8—21no—
301C4×Q8—2122no—
302C4×Q8—2122no—
303#117—1064no—
304#117—1061no—
305#117—1061no—
306#117—1061no—
307#117C2×Dic5311yescharacteristic
308#104—2122no—
309#103—21no—
310#103—21no—
311#103—21no—
312#103—2122no—
313#102—21no—
314#101—1061no—
315#100—1061no—
316#99—532no—
317#99—532no—
318#99—1064no—
319#99—1061no—
320#99—1061no—
321#99—1061no—
322#99—1061no—
323C22⋊C8—21no—
324C22⋊C8—21no—
325#41—22no—
326#41—22no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—41no—
331#41—41no—
332#41—41no—
333#41—44no—
334#41—44no—
335#41—44no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—44no—
350C2×Q8C22⋊Dic5311yescharacteristic
351C2×Q8C22⋊Dic5311yescharacteristic
352C2×Q8C22⋊Dic5311yescharacteristic
353C2×Q8#1171994711yescharacteristic
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4C22⋊Dic5311yescharacteristic
358C2×D4D4×D5311yescharacteristic
359C2×D4D4×D5311yescharacteristic
360C2×D4#1171994711yescharacteristic
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4C22×Dic5311yescharacteristic
369C22×C4C2×C22⋊D5311yescharacteristic
370C22×C4C2×C22⋊D5311yescharacteristic
371Q16—42no—
372Q16—42no—
373Q16—42no—
374SD16—42no—
375M16—41no—
376M16—2121no—
377M16—2121no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—41no—
381C2×C8—2121no—
382C4⋊C4—21no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—2121no—
386C4⋊C4—2124no—
387C4⋊C4—2121no—
388C4⋊C4—2121no—
389C22⋊C4—21no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—1062no—
393C22⋊C4—1062no—
394C22⋊C4—2121no—
395C42—21no—
396C42—2121no—
397C42—2124no—
398C42—2121no—
399C42—2121no—
400C23—21no—
401C23D4×Dic5311yescharacteristic
402C23#1313318511yescharacteristic
403C23#1313318911yescharacteristic
404Q8—24no—
405Q8—21no—
406Q8—21no—
407Q8—21no—
408Q8—21no—
409Q8—41no—
410Q8—42no—
411Q8—42no—
412Q8—41no—
413Q8—41no—
414Q8—81no—
415Q8#1313313212yes—
416Q8#1313313212yes—
417D4—21no—
418D4—21no—
419D4—41no—
420D4—41no—
421D4—41no—
422D4—42no—
423D4—42no—
424D4—42no—
425C2×C4—24no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—81no—
443C2×C4—4241no—
444C2×C4—2122no—
445C2×C4—2122no—
446C2×C4C2×C22⋊Dic5311yescharacteristic
447C2×C4D4×Dic5311yescharacteristic
448C2×C4#1313318911yescharacteristic
449C2×C4#1313319811yescharacteristic
450C8—41no—
451C8—41no—
452C8—4241no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—81no—
461C22#1681050311yescharacteristic
462C4—21no—
463C4—22no—
464C4—22no—
465C4—21no—
466C4—21no—
467C4—41no—
468C4—81no—
469C4—81no—
470C4—8481no—
471C4—4242no—
472C4—4242no—
473C4C2×#1313313211yescharacteristic
474C4C2×#1313313211yescharacteristic
475C2—21no—
476C2—41no—
477C2—41no—
478C2—41no—
479C2—81no—
480C2#2027516911yescharacteristic
subgroupG/H
centerC2#20275169
commutatorC22×C4×C53C22×C4
FrattiniC2×#41C22×D53
FittingC53×#3011C2
socleC2×C53#1898
prime radicalC2×2−1+4×C53C22
cosocleC2×#41×C53C23
hypercenter#3011D53
hypocenterC53#13101
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ #3011

Complements

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

NH ≅ G/Ntype
2. #20276390C2semidirect
3. #20276369C2semidirect
5. #20272844C2semidirect
6. #20272836C2semidirect
21. C2×#137×C53C4semidirect
22. C22×Q16×C53C4semidirect
28. C53×#419C4semidirect
29. C53×#416C4semidirect
249. C2×#137Dic53semidirect
250. C22×Q16Dic53semidirect
259. #419Dic53semidirect
260. #416Dic53semidirect
271. C53#13101semidirect

Maximal subgroups

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

subgroupindex
#202831792
#202763902
#202763692
#202746902
#202728442
#202728362
C53×#30112
#1310153