Finite Group Databasebeta

← #27370158

Subgroups — #27370158

Order 13568.

672 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#20285656C211yesmaximal, characteristic
2#20284845C211yesmaximal, characteristic, complemented
3#20283804C211yesmaximal, characteristic, complemented
4#20274281C212yesmaximal, complemented
5#20274185C212yesmaximal, complemented
6#20274157C212yesmaximal, complemented
7#20274061C212yesmaximal, complemented
8#20273498C211yesmaximal, characteristic, complemented
9#20273429C211yesmaximal, characteristic
10#20273413C211yesmaximal, characteristic, complemented
11C2×C53×#460C211yesmaximal, characteristic
12#16811180C2211yescharacteristic
13#16810995C2212yes—
14#16810969C2212yes—
15#16810953—21no—
16#16810908C2211yescharacteristic, complemented
17#16810908C2211yescharacteristic, complemented
18#16810833C2211yescharacteristic, complemented
19#16810833C2211yescharacteristic, complemented
20#16810820—22no—
21#16810779—21no—
22#16810774—21no—
23#16810774—21no—
24#16810754—22no—
25#16810754C2212yescomplemented
26Q8×#88322C2212yescomplemented
27#16810728—21no—
28#16810728—21no—
29#16810680—21no—
30#16810680C2211yescharacteristic
31#16810680C2211yescharacteristic
32#16810676—21no—
33#16810676C2211yescharacteristic
34#16810375C2212yescomplemented
35#16810373C2212yescomplemented
36#16810370C2212yescomplemented
37#16810368C2212yescomplemented
38#16810367C2212yes—
39#16810367C2212yes—
40C22×SD16×C53—21no—
41C2×#127×C53C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C53×#460—21no—
44C53×#460—21no—
45C53×#460C2212yes—
46C53×#460C2212yes—
47C2×C4⋊C8×C53C2211yescharacteristic
48C2×#102×C53—21no—
49C2×#11719925—21no—
50C2×#11719925—21no—
51C2×#11719923—22no—
52C2×#11719923—22no—
53#13133217C2311yescharacteristic
54#13133217C2311yescharacteristic
55#13133209—21no—
56#13133196—21no—
57#13133196—21no—
58C4×Dic106—21no—
59C4×Dic106—21no—
60#13133176—21no—
61#13133148—21no—
62#13133148—21no—
63C4⋊#88322—22no—
64C4⋊#88322—22no—
65C4⋊#88322D412yes—
66C4⋊#88322C2312yes—
67C4⋊#88322C2311yescharacteristic
68C4⋊#88322C2311yescharacteristic
69#13133141—21no—
70#13133141—21no—
71#13133140—21no—
72#13133140—21no—
73#13133138—22no—
74#13133138—22no—
75C22⋊#88322—21no—
76C22⋊#88322—21no—
77C22⋊#88322D411yescharacteristic
78C22⋊#88322D411yescharacteristic
79#13133128—21no—
80#13133128—21no—
81#13133128—22no—
82C4×#88322—22no—
83C4×#88322C2312yes—
84C22×Q8×C53D411yescharacteristic
85C22×D4×C53—21no—
86C22×D4×C53—21no—
87C22×D4×C53—21no—
88C2×SD16×C53—22no—
89C2×SD16×C53—22no—
90C2×SD16×C53—44no—
91C22×C8×C53—21no—
92#127×C53—21no—
93#127×C53—21no—
94#127×C53—21no—
95#127×C53C2312yes—
96C4×Q8×C53—21no—
97C4×Q8×C53—21no—
98C4×Q8×C53D411yescharacteristic
99C4×Q8×C53D411yescharacteristic
100C4×Q8×C53C2311yescharacteristic
101C4×Q8×C53C2311yescharacteristic
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C42×C53—21no—
105C2×C42×C53C2311yescharacteristic
106C4⋊C8×C53—21no—
107C4⋊C8×C53C2312yes—
108#102×C53—22no—
109#102×C53—22no—
110C24×C53—41no—
111C2×Q8×C53—21no—
112C2×Q8×C53—22no—
113C2×Q8×C53—41no—
114C2×Q8×C53C2×D411yescharacteristic
115C2×Q8×C53C2×D411yescharacteristic
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53—22no—
119C2×D4×C53—42no—
120C2×D4×C53—42no—
121C2×D4×C53—41no—
122C2×D4×C53—41no—
123C2×D4×C53—41no—
124C2×D4×C53—82no—
125C2×D4×C53—82no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53#4111yescharacteristic
131C2×Dic106—21no—
132C2×Dic106—21no—
133SD16×C53—44no—
134SD16×C53—44no—
135C2×C8×C53—22no—
136C2×C8×C53—42no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53C2×D411yescharacteristic
143C4⋊C4×C53C2×D411yescharacteristic
144C42×C53—21no—
145C42×C53—21no—
146C42×C53—21no—
147C42×C53—21no—
148C42×C53C2×D411yescharacteristic
149C42×C53C2×D411yescharacteristic
150C42×C53#4111yescharacteristic
151C42×C53C2411yescharacteristic
152C22⋊Dic53—41no—
153#11719925—42no—
154#11719925—42no—
155#11719923—82no—
156#11719923—82no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161#11719919—41no—
162#11719919—41no—
163C4×Dic53—41no—
164C2×#88322—22no—
165C2×#88322—21no—
166C2×#88322—21no—
167C2×#88322C2×D412yes—
168C2×#88322C2×D412yes—
169C23×C53—42no—
170C23×C53—82no—
171C23×C53—82no—
172C23×C53—82no—
173C23×C53#12111yescharacteristic
174Q8×C53—21no—
175Q8×C53—21no—
176Q8×C53—21no—
177Q8×C53—42no—
178Q8×C53AGL(1,ℤ/8)12yes—
179D4×C53—42no—
180D4×C53—42no—
181D4×C53—42no—
182D4×C53—42no—
183D4×C53—82no—
184D4×C53—82no—
185D4×C53—82no—
186D4×C53—82no—
187C2×C4×C53—21no—
188C2×C4×C53—22no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—22no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—42no—
196C2×C4×C53—42no—
197C2×C4×C53#12111yescharacteristic
198C2×C4×C53#12111yescharacteristic
199C2×C4×C53#12111yescharacteristic
200C2×C4×C53C22×D411yescharacteristic
201C2×C4×C53C22×D411yescharacteristic
202C2×C4×C53C2×#4111yescharacteristic
203C2×Dic53—41no—
204C2×Dic53—41no—
205Dic106—41no—
206Dic106—41no—
207Dic106—42no—
208C8×C53—42no—
209#88322—22no—
210#88322—22no—
211#88322—41no—
212#88322—41no—
213#88322—42no—
214#19468—531nomaximal, Sylow
215C22×C53—21no—
216C22×C53—21no—
217C22×C53—82no—
218C22×C53—82no—
219C22×C53—82no—
220C22×C53—164no—
221C22×C53#47112yes—
222C22×C53C2×#12111yescharacteristic
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—22no—
226C4×C53—21no—
227C4×C53—21no—
228C4×C53—41no—
229C4×C53—41no—
230C4×C53—41no—
231C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
232C4×C53#57511yescharacteristic
233C4×C53#57712yes—
234Dic53—81no—
235Dic53—81no—
236#3208—531no—
237#3113—531no—
238C2×#460D5311yescharacteristic
239#2984—531no—
240#1702—532no—
241#1686—532no—
242#1682—532no—
243#1666—532no—
244#1573—531no—
245#1555—531no—
246#1552—531no—
247C2×C53—22no—
248C2×C53—162no—
249C2×C53#304311yescharacteristic
250C2×C53#305511yescharacteristic
251C2×C53C2×#47111yescharacteristic
252C22×SD16—21no—
253#533—531no—
254C2×#127C2×D5311yescharacteristic
255C2×C4×Q8C2×D5311yescharacteristic
256#492—532no—
257#486—532no—
258#483—1061no—
259#475—531no—
260#475—531no—
261#462—531no—
262#462—531no—
263#460—21no—
264#460—21no—
265#460—1062no—
266#460C2×D5312yes—
267#460C2×D5312yes—
268#451—1061no—
269#450—1061no—
270#450—1061no—
271#446—532no—
272#446—1062no—
273C8×Q8—532no—
274#441—1061no—
275#441—1061no—
276#432—531no—
277#432—531no—
278#432—1061no—
279#431—531no—
280#431—1061no—
281C2×C4⋊C8C2×D5311yescharacteristic
282C2×#102—21no—
283#335—532no—
284#334—532no—
285#332—532no—
286#332—532no—
287#331—532no—
288#331—532no—
289C53#1946811yescharacteristic, complemented, Sylow
290C22×Q8C22⋊D5311yescharacteristic
291C22×D4—21no—
292C22×D4—21no—
293C22×D4—21no—
294C2×Q16—1061no—
295C2×Q16—1061no—
296C2×SD16—22no—
297C2×SD16—22no—
298C2×SD16—44no—
299C2×SD16—1062no—
300C2×SD16—1062no—
301C22×C8—21no—
302C4⋊Q8—531no—
303C4⋊Q8—531no—
304#127—21no—
305#127—21no—
306#127—21no—
307#127C22×D5312yes—
308#125—1061no—
309C22⋊Q8—1061no—
310C22⋊Q8—1061no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—1061no—
314C4×Q8—1061no—
315C4×Q8C22⋊D5311yescharacteristic
316C4×Q8C22⋊D5311yescharacteristic
317C4×Q8C22×D5311yescharacteristic
318C4×Q8C22×D5311yescharacteristic
319#117—1061no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4C22⋊D5311yescharacteristic
322C2×C42—21no—
323C2×C42C22×D5311yescharacteristic
324#106—1061no—
325#106—1061no—
326C4⋊C8—21no—
327C4⋊C8—532no—
328C4⋊C8—532no—
329C4⋊C8—531no—
330C4⋊C8—531no—
331C4⋊C8—1062no—
332C4⋊C8—1062no—
333C4⋊C8C22×D5312yes—
334#103—1061no—
335#103—1061no—
336#103—1061no—
337#103—1061no—
338#102—22no—
339#102—22no—
340#102—1062no—
341#102—1062no—
342C22⋊C8—531no—
343C22⋊C8—531no—
344C22⋊C8—1061no—
345C22⋊C8—1061no—
346#97—1062no—
347#97—1061no—
348#97—1061no—
349C4×C8—532no—
350C4×C8—1062no—
351C24—41no—
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—41no—
355C2×Q8—1061no—
356C2×Q8—1061no—
357C2×Q8C2×C22⋊D5311yescharacteristic
358C2×Q8C2×C22⋊D5311yescharacteristic
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—82no—
368C2×D4—82no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4D4×D5311yescharacteristic
372C22×C4#1171994711yescharacteristic
373C22×C4C2×C22⋊D5311yescharacteristic
374Q16—2122no—
375Q16—2122no—
376SD16—44no—
377SD16—44no—
378SD16—4242no—
379SD16—4242no—
380C2×C8—22no—
381C2×C8—42no—
382C2×C8—532no—
383C2×C8—532no—
384C2×C8—1062no—
385C2×C8—1061no—
386C2×C8—1061no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—1061no—
393C4⋊C4—1061no—
394C4⋊C4—1061no—
395C4⋊C4—1061no—
396C4⋊C4—2121no—
397C4⋊C4—2121no—
398C4⋊C4C2×C22⋊D5311yescharacteristic
399C4⋊C4C2×C22⋊D5311yescharacteristic
400C22⋊C4—2121no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—2121no—
406C42D4×D5311yescharacteristic
407C42#1171994711yescharacteristic
408C42C2×C22⋊D5311yescharacteristic
409C42C23×D5311yescharacteristic
410C23—42no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23#1313318911yescharacteristic
415Q8—21no—
416Q8—21no—
417Q8—21no—
418Q8—42no—
419Q8—2121no—
420Q8—2121no—
421Q8—2122no—
422Q8#1313324312yes—
423D4—42no—
424D4—42no—
425D4—42no—
426D4—42no—
427D4—82no—
428D4—82no—
429D4—82no—
430D4—82no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—2121no—
442C2×C4—2121no—
443C2×C4#1313318911yescharacteristic
444C2×C4#1313318911yescharacteristic
445C2×C4#1313318911yescharacteristic
446C2×C4C2×D4×D5311yescharacteristic
447C2×C4C22×C22⋊D5311yescharacteristic
448C2×C4C2×#1171994711yescharacteristic
449C8—42no—
450C8—1062no—
451C8—1062no—
452C8—2121no—
453C8—2121no—
454C8—2122no—
455C22—21no—
456C22—21no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—164no—
461C22#1681088412yes—
462C22C2×#1313318911yescharacteristic
463C4—21no—
464C4—21no—
465C4—22no—
466C4—21no—
467C4—21no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—4241no—
472C4—4241no—
473C4C2×#1313324311yescharacteristic
474C4#1681142111yescharacteristic
475C4#1681142912yes—
476C2—22no—
477C2—162no—
478C2#2028423311yescharacteristic
479C2#2028433011yescharacteristic
480C2C2×#1681088411yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#460C2
socleC22×C53C2×#121
prime radicalC2×#127×C53C22
hypercenterC2×#460D53
hypocenterC53#19468
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. #20284845C2semidirect
3. #20283804C2semidirect
4. #20274281C2semidirect
5. #20274185C2semidirect
6. #20274157C2semidirect
7. #20274061C2semidirect
8. #20273498C2semidirect
10. #20273413C2semidirect
16. #16810908C22semidirect
17. #16810908C22semidirect
18. #16810833C22semidirect
19. #16810833C22semidirect
25. #16810754C22semidirect
26. Q8×#88322C22semidirect
34. #16810375C22semidirect
35. #16810373C22semidirect
36. #16810370C22semidirect
37. #16810368C22semidirect
289. C53#19468semidirect

Maximal subgroups

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

subgroupindex
#202856562
#202848452
#202838042
#202742812
#202741852
#202741572
#202740612
#202734982
#202734292
#202734132
C2×C53×#4602
#1946853