Finite Group Databasebeta

← #27541004

Subgroups — #27541004

Order 13568.

558 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287118C211yesmaximal, characteristic, complemented
2#20276946C211yesmaximal, characteristic, complemented
3#20276920C211yesmaximal, characteristic, complemented
4#20276898C211yesmaximal, characteristic, complemented
5#20276879C211yesmaximal, characteristic, complemented
6#20276813C211yesmaximal, characteristic, complemented
7#20276799C211yesmaximal, characteristic, complemented
8#20276767C211yesmaximal, characteristic, complemented
9#20276638C211yesmaximal, characteristic, complemented
10#20276632C211yesmaximal, characteristic, complemented
11C4×#13133160C211yesmaximal, characteristic, complemented
12D4×#11719908C211yesmaximal, characteristic, complemented
13#20276422C211yesmaximal, characteristic, complemented
14C2×#16810427C211yesmaximal, characteristic, complemented
15C53×#3285C211yesmaximal, characteristic, complemented
16#16811319—21no—
17C2×#13133160C2211yescharacteristic
18C2×#13133160C2211yescharacteristic
19C2×#13133158C2211yescharacteristic
20C22×#11719908C2211yescharacteristic
21C22×#11719908C2211yescharacteristic
22#16811025C2211yescharacteristic
23#16810948—21no—
24#16810942—21no—
25#16810924—21no—
26#16810859C2211yescharacteristic
27#16810859C2211yescharacteristic
28C4×D424C2211yescharacteristic
29#16810661C2211yescharacteristic
30C2×#13133151C2211yescharacteristic
31#16810450C2211yescharacteristic
32#16810450C2211yescharacteristic
33#16810448C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35#16810427C2211yescharacteristic
36#16810427C2211yescharacteristic
37#16810427C2211yescharacteristic
38#16810427C2211yescharacteristic
39#16810427C2211yescharacteristic
40#16810427C2211yescharacteristic
41#16810424C2211yescharacteristic
42#16810424C2211yescharacteristic
43#16810424C2211yescharacteristic
44#16810423C2211yescharacteristic
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×D424C2311yescharacteristic
60#13133212—41no—
61#13133209—21no—
62#13133199—41no—
63#13133191—21no—
64#13133191—21no—
65#13133189—41no—
66C4×D212—21no—
67#13133177—41no—
68#13133176—21no—
69C2×C4⋊Dic53—21no—
70#13133160—22no—
71#13133160—41no—
72#13133158—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#13133139—21no—
86#13133139—21no—
87#13133139—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
134C2×C22⋊D53—41no—
135C2×C22⋊D53—41no—
136C22×Dic53—41no—
137C2×D212—21no—
138C2×C4×D53—41no—
139Q16×C53—21no—
140Q16×C53—21no—
141Q16×C53—21no—
142Q16×C53—42no—
143Q16×C53#4112yes—
144SD16×C53—42no—
145SD16×C53—81no—
146D8×C53—21no—
147D8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53C2×D411yescharacteristic
153C2×C8×C53C2×D411yescharacteristic
154C2×C8×C53C2×D411yescharacteristic
155C2×C8×C53C2411yescharacteristic
156C4⋊C4×C53—21no—
157C4⋊C4×C53—41no—
158C4⋊C4×C53—41no—
159C4⋊C4×C53D811yescharacteristic
160C22⋊C4×C53—41no—
161C22⋊C4×C53—41no—
162C22⋊C4×C53—41no—
163C22⋊C4×C53—41no—
164C22⋊C4×C53D811yescharacteristic
165C22⋊C4×C53D811yescharacteristic
166C42×C53—41no—
167C42×C53C2×D411yescharacteristic
168C22⋊Dic53—41no—
169#11719921—41no—
170#11719921—41no—
171#11719921—41no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—41no—
176#11719919—41no—
177#11719919—41no—
178C4×Dic53—41no—
179D424—21no—
180D424—21no—
181#11719908—21no—
182#11719908—21no—
183#11719908—21no—
184#11719908—21no—
185C23×C53—41no—
186C23×C53C2×D811yescharacteristic
187C23×C53C2×D811yescharacteristic
188C22×D53—41no—
189Q8×C53—22no—
190Q8×C53—41no—
191Q8×C53—41no—
192Q8×C53—41no—
193Q8×C53—82no—
194D4×C53—22no—
195D4×C53—41no—
196D4×C53—41no—
197D4×C53—82no—
198D4×C53—82no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—82no—
207C2×C4×C53—82no—
208C2×C4×C53C2×D811yescharacteristic
209C2×C4×C53C2×D811yescharacteristic
210C2×C4×C53C2×D811yescharacteristic
211C2×C4×C53C2×D811yescharacteristic
212C2×C4×C53C22×D411yescharacteristic
213C22⋊D53—82no—
214C22⋊D53—161no—
215C2×Dic53—41no—
216C2×Dic53—41no—
217C2×Dic53—41no—
218C2×Dic53—161no—
219D212—41no—
220D212—41no—
221C4×D53—161no—
222C8×C53—21no—
223C8×C53—21no—
224C8×C53—21no—
225C8×C53C2×#4111yescharacteristic
226C8×C53H(5,𝔽2)11yescharacteristic
227#33868—531nomaximal, Sylow
228C22×C53—21no—
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—81no—
234C22×C53—81no—
235C22×C53—81no—
236C22×C53#50812yes—
237C22×C53C22×D811yescharacteristic
238C2×D53—81no—
239C2×D53—81no—
240C2×D53—81no—
241C4×C53—21no—
242C4×C53—21no—
243C4×C53—21no—
244C4×C53—21no—
245C4×C53—42no—
246C4×C53—81no—
247C4×C53—81no—
248C4×C53#54611yescharacteristic
249C4×C53#57711yescharacteristic
250Dic53—81no—
251Dic53—81no—
252Dic53—81no—
253#3339—531no—
254#3285D5311yescharacteristic, complemented
255#2242—531no—
256#2237—531no—
257#2233—531no—
258#2229—531no—
259#2217—531no—
260#2215—531no—
261#2210—531no—
262#2186—531no—
263#2185—531no—
264C4×SD32—531no—
265D4×C16—531no—
266#2138—531no—
267C2×#358—531no—
268C2×C53—22no—
269C2×C53—41no—
270C2×C53—161no—
271C2×C53#329411yescharacteristic
272C2×C53C2×#50811yescharacteristic
273C2×C53#341611yescharacteristic
274D53—161no—
275C2×#135C2×D5311yescharacteristic
276C22×Q16C2×D5311yescharacteristic
277#553—1061no—
278#548—21no—
279C2×SD32—531no—
280C2×SD32—531no—
281C2×D16—531no—
282C22×C16—531no—
283C22×C16—531no—
284#499—531no—
285#495C2×D5311yescharacteristic
286#482—1061no—
287#481—1061no—
288#478—1061no—
289#467C2×D5311yescharacteristic
290#466—531no—
291#466—531no—
292#466C2×D5311yescharacteristic
293#464—21no—
294#452—21no—
295#448—21no—
296C4×Q16C2×D5311yescharacteristic
297C4×D8—531no—
298C8×D4C2×D5311yescharacteristic
299#427—531no—
300C2×#107—531no—
301#367—531no—
302#367—531no—
303#366—531no—
304C4⋊C16—531no—
305#358—531no—
306#358—531no—
307#358—531no—
308#358—531no—
309#358—531no—
310#358—531no—
311#357—531no—
312#357—531no—
313#357—531no—
314#357—531no—
315C22⋊C16—531no—
316C22⋊C16—531no—
317C4×C16—531no—
318C53#3386811yescharacteristic, complemented, Sylow
319C2×#41—21no—
320C22×Q8—21no—
321#135—22no—
322#135—41no—
323C2×Q16—22no—
324C2×Q16—21no—
325C2×Q16—41no—
326C2×Q16C22×D5311yescharacteristic
327C2×Q16C22×D5311yescharacteristic
328C2×SD16—21no—
329C2×SD16—21no—
330C2×D8—531no—
331C2×D8C22×D5311yescharacteristic
332C22×C8C22×D5311yescharacteristic
333C22×C8C22×D5311yescharacteristic
334#126—2121no—
335#125—1061no—
336#124—21no—
337#124—41no—
338#123—2121no—
339C22⋊Q8—21no—
340C22⋊Q8—41no—
341#121—21no—
342#121—41no—
343#121—1061no—
344#121—1061no—
345#121—2121no—
346C4×Q8—21no—
347C4×D4—41no—
348C4×D4—1061no—
349C4×D4—2121no—
350C4×D4C22⋊D5311yescharacteristic
351#117—1061no—
352C2×C4⋊C4—1061no—
353SD32—1062no—
354SD32—2121no—
355D16—2121no—
356C2×C16—531no—
357C2×C16—531no—
358C2×C16—531no—
359C2×C16—531no—
360C2×C16—2121no—
361C2×C16—2121no—
362#107—531no—
363#107—531no—
364#107—531no—
365#107—1061no—
366#107C22×D5311yescharacteristic
367#106—1061no—
368#106—1061no—
369C4⋊C8C22⋊D5311yescharacteristic
370#103—21no—
371#103—21no—
372#102—21no—
373#102—1061no—
374#102—1061no—
375#102—1061no—
376C22⋊C8C22⋊D5311yescharacteristic
377C22⋊C8C22⋊D5311yescharacteristic
378C4×C8C22×D5311yescharacteristic
379#41—42no—
380#41—81no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—42no—
384C2×Q8—41no—
385C2×Q8—81no—
386C2×D4—21no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—1061no—
390C2×D4—2121no—
391C2×D4—2121no—
392C2×D4#1171992211yescharacteristic
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—2121no—
396C22×C4—2121no—
397C22×C4C2×C22⋊D5311yescharacteristic
398C22×C4C2×C22⋊D5311yescharacteristic
399Q16—21no—
400Q16—21no—
401Q16—21no—
402Q16—42no—
403Q16#1171994712yes—
404SD16—42no—
405SD16—81no—
406D8—21no—
407D8—21no—
408D8—1061no—
409D8—1061no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8C2×C22⋊D5311yescharacteristic
415C2×C8C2×C22⋊D5311yescharacteristic
416C2×C8C2×C22⋊D5311yescharacteristic
417C2×C8C23×D5311yescharacteristic
418C4⋊C4—21no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—1061no—
422C4⋊C4—1061no—
423C4⋊C4—1061no—
424C4⋊C4—2121no—
425C4⋊C4—2121no—
426C4⋊C4—2121no—
427C4⋊C4#1171992211yescharacteristic
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—2121no—
433C22⋊C4—2121no—
434C22⋊C4—2121no—
435C22⋊C4—2121no—
436C22⋊C4#1171992211yescharacteristic
437C22⋊C4#1171992211yescharacteristic
438C42—41no—
439C42—2121no—
440C42C2×C22⋊D5311yescharacteristic
441C16—1061no—
442C16—1061no—
443C16—1061no—
444C16—1061no—
445C23—41no—
446C23—2121no—
447C23C2×#1171992211yescharacteristic
448C23C2×#1171992211yescharacteristic
449Q8—22no—
450Q8—41no—
451Q8—41no—
452Q8—41no—
453Q8—82no—
454D4—22no—
455D4—41no—
456D4—41no—
457D4—82no—
458D4—82no—
459D4—8481no—
460D4—4242no—
461D4—2121no—
462D4—2121no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—8481no—
473C2×C4—8481no—
474C2×C4—2121no—
475C2×C4—2121no—
476C2×C4—2121no—
477C2×C4C2×#1171992211yescharacteristic
478C2×C4C2×#1171992211yescharacteristic
479C2×C4C2×#1171992211yescharacteristic
480C2×C4C2×#1171992211yescharacteristic
481C2×C4C22×C22⋊D5311yescharacteristic
482C8—21no—
483C8—21no—
484C8—21no—
485C8C2×#1171994711yescharacteristic
486C8#1313326211yescharacteristic
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—4241no—
496C22—4241no—
497C22—4241no—
498C22#1681105312yes—
499C22C22×#1171992211yescharacteristic
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—42no—
505C4—81no—
506C4—81no—
507C4—4241no—
508C4—4241no—
509C4—4241no—
510C4#1681126711yescharacteristic
511C4#1681143411yescharacteristic
512C2—22no—
513C2—41no—
514C2—161no—
515C2—8481no—
516C2#2028652811yescharacteristic
517C2C2×#1681105311yescharacteristic
518C2#2028785911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3285C2
socleC22×C53C22×D8
hypercenter#3285D53
hypocenterC53#33868
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. #20287118C2semidirect
2. #20276946C2semidirect
3. #20276920C2semidirect
4. #20276898C2semidirect
5. #20276879C2semidirect
6. #20276813C2semidirect
7. #20276799C2semidirect
8. #20276767C2semidirect
9. #20276638C2semidirect
10. #20276632C2semidirect
11. C4×#13133160C2semidirect
12. D4×#11719908C2semidirect
13. #20276422C2semidirect
14. C2×#16810427C2semidirect
15. C53×#3285C2semidirect
254. #3285D53semidirect
318. C53#33868semidirect

Maximal subgroups

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

subgroupindex
#202871182
#202769462
#202769202
#202768982
#202768792
#202768132
#202767992
#202767672
#202766382
#202766322
C4×#131331602
D4×#117199082
#202764222
C2×#168104272
C53×#32852
#3386853