Finite Group Databasebeta

← #27540989

Subgroups — #27540989

Order 13568.

546 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287084C211yesmaximal, characteristic, complemented
2#20276946C211yesmaximal, characteristic, complemented
3#20276920C211yesmaximal, characteristic, complemented
4#20276893C211yesmaximal, characteristic, complemented
5#20276879C211yesmaximal, characteristic, complemented
6#20276809C211yesmaximal, characteristic, complemented
7#20276804C211yesmaximal, characteristic, complemented
8#20276774C211yesmaximal, characteristic, complemented
9#20276655C211yesmaximal, characteristic, complemented
10#20276649C211yesmaximal, characteristic, complemented
11C4×#13133163C211yesmaximal, characteristic, complemented
12D4×#11719908C211yesmaximal, characteristic, complemented
13#20276422C211yesmaximal, characteristic, complemented
14C2×#16810427C211yesmaximal, characteristic, complemented
15C53×#3285C211yesmaximal, characteristic
16#16811313—21no—
17C2×#13133163C2211yescharacteristic
18C2×#13133163C2211yescharacteristic
19C2×#13133161C2211yescharacteristic
20C22×#11719908C2211yescharacteristic
21C22×#11719908C2211yescharacteristic
22#16811025C2211yescharacteristic
23#16810965—21no—
24#16810954—21no—
25#16810931—21no—
26#16810864C2211yescharacteristic
27#16810864C2211yescharacteristic
28C4×Dic212C2211yescharacteristic
29#16810661C2211yescharacteristic
30C2×#13133151C2211yescharacteristic
31#16810450C2211yescharacteristic
32#16810448C2211yescharacteristic
33#16810448C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35#16810427C2211yescharacteristic
36#16810427C2211yescharacteristic
37#16810427C2211yescharacteristic
38#16810427C2211yescharacteristic
39#16810427C2211yescharacteristic
40#16810427C2211yescharacteristic
41#16810426C2211yescharacteristic
42#16810424C2211yescharacteristic
43#16810424C2211yescharacteristic
44#16810424C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46C22⋊#11719908C2211yescharacteristic
47C4×#11719908C2211yescharacteristic
48C2×#135×C53C2211yescharacteristic
49C22×Q16×C53C2211yescharacteristic
50C53×#548—21no—
51C53×#495C2211yescharacteristic
52C53×#467C2211yescharacteristic
53C53×#466C2211yescharacteristic
54C53×#464—21no—
55C53×#452—21no—
56C53×#448—21no—
57C4×Q16×C53C2211yescharacteristic
58C8×D4×C53C2211yescharacteristic
59C2×Dic212C2311yescharacteristic
60#13133210—41no—
61#13133209—21no—
62#13133207—41no—
63#13133198—41no—
64#13133196—21no—
65#13133196—21no—
66C4×Dic106—21no—
67#13133176—21no—
68#13133173—41no—
69C2×C4⋊Dic53—21no—
70#13133163—22no—
71#13133163—41no—
72#13133161—41no—
73C2×#11719908—41no—
74C2×#11719908—41no—
75C2×#11719908C2311yescharacteristic
76C2×#11719908C2311yescharacteristic
77C2×#11719908C2311yescharacteristic
78C2×#11719908C2311yescharacteristic
79#13133151—21no—
80#13133151C2311yescharacteristic
81#13133151C2311yescharacteristic
82#13133151C2311yescharacteristic
83#13133149—21no—
84#13133149—21no—
85#13133142—21no—
86#13133142—21no—
87#13133142—21no—
88C2×#41×C53—21no—
89C22×Q8×C53—21no—
90#135×C53—22no—
91#135×C53—41no—
92C2×Q16×C53—21no—
93C2×Q16×C53—22no—
94C2×Q16×C53—41no—
95C2×Q16×C53C2311yescharacteristic
96C2×Q16×C53C2311yescharacteristic
97C2×SD16×C53—21no—
98C2×SD16×C53—21no—
99C2×D8×C53C2311yescharacteristic
100C22×C8×C53C2311yescharacteristic
101C22×C8×C53C2311yescharacteristic
102#124×C53—21no—
103#124×C53—41no—
104C22⋊Q8×C53—21no—
105C22⋊Q8×C53—41no—
106#121×C53—21no—
107#121×C53—41no—
108C4×Q8×C53—21no—
109C4×D4×C53—41no—
110C4×D4×C53D411yescharacteristic
111#107×C53C2311yescharacteristic
112C4⋊C8×C53D411yescharacteristic
113#103×C53—21no—
114#103×C53—21no—
115#102×C53—21no—
116C22⋊C8×C53D411yescharacteristic
117C22⋊C8×C53D411yescharacteristic
118C4×C8×C53C2311yescharacteristic
119#41×C53—42no—
120#41×C53—81no—
121C2×Q8×C53—21no—
122C2×Q8×C53—21no—
123C2×Q8×C53—42no—
124C2×Q8×C53—41no—
125C2×Q8×C53—81no—
126C2×D4×C53—21no—
127C2×D4×C53—41no—
128C2×D4×C53—41no—
129C2×D4×C53D811yescharacteristic
130C22×C4×C53—41no—
131C22×C4×C53—41no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×Dic53—41no—
135C2×Dic106—21no—
136Q16×C53—21no—
137Q16×C53—21no—
138Q16×C53—21no—
139Q16×C53—42no—
140Q16×C53#4112yes—
141SD16×C53—42no—
142SD16×C53—81no—
143D8×C53—21no—
144D8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53C2×D411yescharacteristic
150C2×C8×C53C2×D411yescharacteristic
151C2×C8×C53C2×D411yescharacteristic
152C2×C8×C53C2411yescharacteristic
153C4⋊C4×C53—21no—
154C4⋊C4×C53—41no—
155C4⋊C4×C53—41no—
156C4⋊C4×C53D811yescharacteristic
157C22⋊C4×C53—41no—
158C22⋊C4×C53—41no—
159C22⋊C4×C53—41no—
160C22⋊C4×C53—41no—
161C22⋊C4×C53D811yescharacteristic
162C22⋊C4×C53D811yescharacteristic
163C42×C53—41no—
164C42×C53C2×D411yescharacteristic
165C22⋊Dic53—41no—
166C22⋊Dic53—41no—
167C22⋊Dic53—41no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—41no—
172#11719919—41no—
173#11719919—41no—
174#11719919—41no—
175#11719919—41no—
176#11719919—41no—
177C4×Dic53—41no—
178C4×Dic53—41no—
179Dic212—21no—
180Dic212—21no—
181#11719908—21no—
182#11719908—21no—
183#11719908—21no—
184#11719908—21no—
185C23×C53—41no—
186C23×C53C2×D811yescharacteristic
187C23×C53C2×D811yescharacteristic
188Q8×C53—22no—
189Q8×C53—41no—
190Q8×C53—41no—
191Q8×C53—41no—
192Q8×C53—82no—
193D4×C53—22no—
194D4×C53—41no—
195D4×C53—41no—
196D4×C53—82no—
197D4×C53—82no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—82no—
206C2×C4×C53—82no—
207C2×C4×C53C2×D811yescharacteristic
208C2×C4×C53C2×D811yescharacteristic
209C2×C4×C53C2×D811yescharacteristic
210C2×C4×C53C2×D811yescharacteristic
211C2×C4×C53C22×D411yescharacteristic
212C2×Dic53—41no—
213C2×Dic53—41no—
214C2×Dic53—41no—
215C2×Dic53—41no—
216C2×Dic53—82no—
217Dic106—41no—
218Dic106—41no—
219C8×C53—21no—
220C8×C53—21no—
221C8×C53—21no—
222C8×C53C2×#4111yescharacteristic
223C8×C53H(5,𝔽2)11yescharacteristic
224#33867—531nomaximal, Sylow
225C22×C53—21no—
226C22×C53—21no—
227C22×C53—21no—
228C22×C53—21no—
229C22×C53—21no—
230C22×C53—81no—
231C22×C53—81no—
232C22×C53—81no—
233C22×C53#50812yes—
234C22×C53C22×D811yescharacteristic
235C4×C53—21no—
236C4×C53—21no—
237C4×C53—21no—
238C4×C53—21no—
239C4×C53—42no—
240C4×C53—81no—
241C4×C53—81no—
242C4×C53#54611yescharacteristic
243C4×C53#57711yescharacteristic
244Dic53—81no—
245Dic53—81no—
246Dic53—81no—
247Dic53—81no—
248#3336—531no—
249#3285D5311yescharacteristic
250#2242—531no—
251#2237—531no—
252#2232—531no—
253#2229—531no—
254#2217—531no—
255#2216—531no—
256#2211—531no—
257#2189—531no—
258#2188—531no—
259C4×Q32—531no—
260D4×C16—531no—
261#2138—531no—
262C2×#358—531no—
263C2×C53—22no—
264C2×C53—41no—
265C2×C53—161no—
266C2×C53#329411yescharacteristic
267C2×C53C2×#50811yescharacteristic
268C2×C53#341711yescharacteristic
269C2×#135C2×D5311yescharacteristic
270C22×Q16C2×D5311yescharacteristic
271#552—1061no—
272#548—21no—
273C2×Q32—531no—
274C2×Q32—531no—
275C2×SD32—531no—
276C22×C16—531no—
277C22×C16—531no—
278#499—531no—
279#495C2×D5311yescharacteristic
280#485—1061no—
281#483—1061no—
282#479—1061no—
283#467—531no—
284#467—531no—
285#467C2×D5311yescharacteristic
286#466C2×D5311yescharacteristic
287#464—21no—
288#452—21no—
289#448—21no—
290C4×Q16—531no—
291C4×Q16C2×D5311yescharacteristic
292C8×D4C2×D5311yescharacteristic
293#427—531no—
294C2×#107—531no—
295#367—531no—
296#366—531no—
297#366—531no—
298C4⋊C16—531no—
299#358—531no—
300#358—531no—
301#358—531no—
302#358—531no—
303#358—531no—
304#358—531no—
305#358—531no—
306#357—531no—
307#357—531no—
308#357—531no—
309C22⋊C16—531no—
310C22⋊C16—531no—
311C4×C16—531no—
312C53#3386711yescharacteristic, complemented, Sylow
313C2×#41—21no—
314C22×Q8—21no—
315#135—22no—
316#135—41no—
317C2×Q16—22no—
318C2×Q16—21no—
319C2×Q16—41no—
320C2×Q16—531no—
321C2×Q16C22×D5311yescharacteristic
322C2×Q16C22×D5311yescharacteristic
323C2×SD16—21no—
324C2×SD16—21no—
325C2×D8C22×D5311yescharacteristic
326C22×C8C22×D5311yescharacteristic
327C22×C8C22×D5311yescharacteristic
328#126—2121no—
329#125—1061no—
330#125—2121no—
331#124—21no—
332#124—41no—
333#123—2121no—
334C22⋊Q8—21no—
335C22⋊Q8—41no—
336C22⋊Q8—1061no—
337C22⋊Q8—1061no—
338#121—21no—
339#121—41no—
340C4×Q8—21no—
341C4×Q8—1061no—
342C4×D4—41no—
343C4×D4C22⋊D5311yescharacteristic
344#117—1061no—
345#117—2121no—
346C2×C4⋊C4—1061no—
347Q32—1062no—
348Q32—2121no—
349SD32—2121no—
350C2×C16—531no—
351C2×C16—531no—
352C2×C16—531no—
353C2×C16—531no—
354C2×C16—2121no—
355C2×C16—2121no—
356#107—531no—
357#107—531no—
358#107—531no—
359#107—1061no—
360#107C22×D5311yescharacteristic
361#106—1061no—
362#106—1061no—
363C4⋊C8C22⋊D5311yescharacteristic
364#103—21no—
365#103—21no—
366#103—1061no—
367#103—1061no—
368#103—1061no—
369#102—21no—
370C22⋊C8C22⋊D5311yescharacteristic
371C22⋊C8C22⋊D5311yescharacteristic
372C4×C8C22×D5311yescharacteristic
373#41—42no—
374#41—81no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—42no—
378C2×Q8—41no—
379C2×Q8—81no—
380C2×Q8—1061no—
381C2×D4—21no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4#1171992211yescharacteristic
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—2121no—
388C22×C4C2×C22⋊D5311yescharacteristic
389C22×C4C2×C22⋊D5311yescharacteristic
390Q16—21no—
391Q16—21no—
392Q16—21no—
393Q16—42no—
394Q16—1061no—
395Q16—1061no—
396Q16#1171994712yes—
397SD16—42no—
398SD16—81no—
399D8—21no—
400D8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8C2×C22⋊D5311yescharacteristic
406C2×C8C2×C22⋊D5311yescharacteristic
407C2×C8C2×C22⋊D5311yescharacteristic
408C2×C8C23×D5311yescharacteristic
409C4⋊C4—21no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—1061no—
413C4⋊C4—1061no—
414C4⋊C4—1061no—
415C4⋊C4—2121no—
416C4⋊C4—2121no—
417C4⋊C4—2121no—
418C4⋊C4—2121no—
419C4⋊C4—2121no—
420C4⋊C4—2121no—
421C4⋊C4#1171992211yescharacteristic
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—2121no—
427C22⋊C4—2121no—
428C22⋊C4—2121no—
429C22⋊C4#1171992211yescharacteristic
430C22⋊C4#1171992211yescharacteristic
431C42—41no—
432C42—2121no—
433C42—2121no—
434C42C2×C22⋊D5311yescharacteristic
435C16—1061no—
436C16—1061no—
437C16—1061no—
438C16—1061no—
439C23—41no—
440C23C2×#1171992211yescharacteristic
441C23C2×#1171992211yescharacteristic
442Q8—22no—
443Q8—41no—
444Q8—41no—
445Q8—41no—
446Q8—82no—
447Q8—2121no—
448Q8—2121no—
449D4—22no—
450D4—41no—
451D4—41no—
452D4—82no—
453D4—82no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—4242no—
464C2×C4—2121no—
465C2×C4—2121no—
466C2×C4—2121no—
467C2×C4—2121no—
468C2×C4C2×#1171992211yescharacteristic
469C2×C4C2×#1171992211yescharacteristic
470C2×C4C2×#1171992211yescharacteristic
471C2×C4C2×#1171992211yescharacteristic
472C2×C4C22×C22⋊D5311yescharacteristic
473C8—21no—
474C8—21no—
475C8—21no—
476C8C2×#1171994711yescharacteristic
477C8#1313326211yescharacteristic
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22#1681105312yes—
487C22C22×#1171992211yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—42no—
493C4—81no—
494C4—81no—
495C4—4241no—
496C4—4241no—
497C4—4241no—
498C4—4241no—
499C4#1681126711yescharacteristic
500C4#1681143411yescharacteristic
501C2—22no—
502C2—41no—
503C2—161no—
504C2#2028652811yescharacteristic
505C2C2×#1681105311yescharacteristic
506C2#2028786611yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
Fitting, prime radicalC53×#3285C2
socleC22×C53C22×D8
hypercenter#3285D53
hypocenterC53#33867
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3285

Complements

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

NH ≅ G/Ntype
1. #20287084C2semidirect
2. #20276946C2semidirect
3. #20276920C2semidirect
4. #20276893C2semidirect
5. #20276879C2semidirect
6. #20276809C2semidirect
7. #20276804C2semidirect
8. #20276774C2semidirect
9. #20276655C2semidirect
10. #20276649C2semidirect
11. C4×#13133163C2semidirect
12. D4×#11719908C2semidirect
13. #20276422C2semidirect
14. C2×#16810427C2semidirect
312. C53#33867semidirect

Maximal subgroups

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

subgroupindex
#202870842
#202769462
#202769202
#202768932
#202768792
#202768092
#202768042
#202767742
#202766552
#202766492
C4×#131331632
D4×#117199082
#202764222
C2×#168104272
C53×#32852
#3386753