Finite Group Databasebeta

← #27541093

Subgroups — #27541093

Order 13568.

580 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287063C211yesmaximal, characteristic, complemented
2#20276951C211yesmaximal, characteristic, complemented
3#20276915C211yesmaximal, characteristic, complemented
4#20276893C211yesmaximal, characteristic, complemented
5#20276867C211yesmaximal, characteristic, complemented
6#20276830C211yesmaximal, characteristic, complemented
7#20276799C211yesmaximal, characteristic, complemented
8#20276767C211yesmaximal, characteristic, complemented
9#20276638C211yesmaximal, characteristic, complemented
10#20276632C211yesmaximal, characteristic, complemented
11#20276577C211yesmaximal, characteristic, complemented
12#20276541C211yesmaximal, characteristic, complemented
13#20276427C211yesmaximal, characteristic, complemented
14C2×#16810427C211yesmaximal, characteristic, complemented
15C53×#3285C211yesmaximal, characteristic, complemented
16#16811277—21no—
17C2×#13133160C2211yescharacteristic
18C2×#13133160C2211yescharacteristic
19C2×#13133158C2211yescharacteristic
20C2×#13133157C2211yescharacteristic
21C22×#11719908C2211yescharacteristic
22#16811028C2211yescharacteristic
23#16810937—21no—
24#16810937—21no—
25#16810904—21no—
26#16810859C2211yescharacteristic
27#16810859C2211yescharacteristic
28C4×D424C2211yescharacteristic
29C2×#13133151C2211yescharacteristic
30C2×#13133151C2211yescharacteristic
31#16810448C2211yescharacteristic
32#16810448C2211yescharacteristic
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
47#16810399C2211yescharacteristic
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×#11719947—41no—
60C2×D424C2311yescharacteristic
61#13133217—21no—
62#13133201—41no—
63#13133201—41no—
64#13133197—41no—
65#13133191—21no—
66#13133191—21no—
67C4×D212—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70#13133160—22no—
71#13133160—41no—
72#13133158—41no—
73#13133157—41no—
74C2×#11719908—41no—
75C2×#11719908C2311yescharacteristic
76C2×#11719908C2311yescharacteristic
77C2×#11719908C2311yescharacteristic
78C2×#11719908C2311yescharacteristic
79#13133151—21no—
80#13133151—21no—
81#13133151—21no—
82#13133151C2311yescharacteristic
83#13133151C2311yescharacteristic
84#13133151C2311yescharacteristic
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—
137C22×Dic53—41no—
138#11719947—84no—
139C2×D212—21no—
140C2×C4×D53—41no—
141C2×Dic106—41no—
142Q16×C53—21no—
143Q16×C53—21no—
144Q16×C53—21no—
145Q16×C53—42no—
146Q16×C53#4112yescomplemented
147SD16×C53—42no—
148SD16×C53—81no—
149D8×C53—21no—
150D8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53C2×D411yescharacteristic
156C2×C8×C53C2×D411yescharacteristic
157C2×C8×C53C2×D411yescharacteristic
158C2×C8×C53C2411yescharacteristic
159C4⋊C4×C53—21no—
160C4⋊C4×C53—41no—
161C4⋊C4×C53—41no—
162C4⋊C4×C53D811yescharacteristic
163C22⋊C4×C53—41no—
164C22⋊C4×C53—41no—
165C22⋊C4×C53—41no—
166C22⋊C4×C53—41no—
167C22⋊C4×C53D811yescharacteristic
168C22⋊C4×C53D811yescharacteristic
169C42×C53—41no—
170C42×C53C2×D411yescharacteristic
171#11719921—41no—
172#11719921—41no—
173#11719921—41no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—41no—
178C4⋊Dic53—41no—
179C4⋊Dic53—41no—
180D424—21no—
181D424—21no—
182#11719908—21no—
183#11719908—21no—
184#11719908—21no—
185#11719908—21no—
186C23×C53—41no—
187C23×C53C2×D811yescharacteristic
188C23×C53C2×D811yescharacteristic
189C22×D53—41no—
190Q8×C53—22no—
191Q8×C53—41no—
192Q8×C53—41no—
193Q8×C53—41no—
194Q8×C53—82no—
195D4×C53—22no—
196D4×C53—41no—
197D4×C53—41no—
198D4×C53—82no—
199D4×C53—82no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—82no—
208C2×C4×C53—82no—
209C2×C4×C53C2×D811yescharacteristic
210C2×C4×C53C2×D811yescharacteristic
211C2×C4×C53C2×D811yescharacteristic
212C2×C4×C53C2×D811yescharacteristic
213C2×C4×C53C22×D411yescharacteristic
214C22⋊D53—82no—
215C22⋊D53—82no—
216C2×Dic53—41no—
217C2×Dic53—41no—
218C2×Dic53—41no—
219C2×Dic53—82no—
220C2×Dic53—82no—
221D212—41no—
222D212—41no—
223C4×D53—82no—
224Dic106—82no—
225C8×C53—21no—
226C8×C53—21no—
227C8×C53—21no—
228C8×C53C2×#4111yescharacteristic
229C8×C53H(5,𝔽2)11yescharacteristic
230#33875—531nomaximal, Sylow
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—81no—
237C22×C53—81no—
238C22×C53—81no—
239C22×C53#50912yes—
240C22×C53C22×D811yescharacteristic
241C2×D53—81no—
242C2×D53—81no—
243C2×D53—81no—
244C4×C53—21no—
245C4×C53—21no—
246C4×C53—21no—
247C4×C53—21no—
248C4×C53—42no—
249C4×C53—81no—
250C4×C53—81no—
251C4×C53#54611yescharacteristic
252C4×C53#57711yescharacteristic
253Dic53—81no—
254Dic53—81no—
255Dic53—81no—
256#3334—531no—
257#3285D5311yescharacteristic, complemented
258#2243—531no—
259#2236—531no—
260#2232—531no—
261#2227—531no—
262#2220—531no—
263#2215—531no—
264#2210—531no—
265#2186—531no—
266#2185—531no—
267#2175—531no—
268#2168—531no—
269#2139—531no—
270C2×#358—531no—
271C2×C53—22no—
272C2×C53—41no—
273C2×C53—161no—
274C2×C53#329411yescharacteristic
275C2×C53C2×#50911yescharacteristic
276C2×C53#341711yescharacteristic
277D53—161no—
278C2×#135C2×D5311yescharacteristic
279C22×Q16C2×D5311yescharacteristic
280#548—21no—
281#547—1061no—
282C2×SD32—531no—
283C2×SD32—531no—
284C2×D16—531no—
285C2×M32—531no—
286C22×C16—531no—
287#500—531no—
288#495C2×D5311yescharacteristic
289#480—1061no—
290#480—1061no—
291#474—1061no—
292#467C2×D5311yescharacteristic
293#466—531no—
294#466—531no—
295#466C2×D5311yescharacteristic
296#464—21no—
297#452—21no—
298#448—21no—
299C4×Q16C2×D5311yescharacteristic
300C4×D8—531no—
301C8×D4C2×D5311yescharacteristic
302C2×#107—531no—
303C2×#107—531no—
304#366—531no—
305#366—531no—
306#366—531no—
307C4⋊C16—531no—
308#358—531no—
309#358—531no—
310#358—531no—
311#358—531no—
312#358—531no—
313#358—531no—
314#357—531no—
315#357—531no—
316#357—531no—
317#357—531no—
318C22⋊C16—531no—
319C22⋊C16—531no—
320#346—531no—
321C53#3387511yescharacteristic, complemented, Sylow
322C2×#41—21no—
323C2×#41—2121no—
324C22×Q8—21no—
325#135—22no—
326#135—41no—
327C2×Q16—22no—
328C2×Q16—21no—
329C2×Q16—41no—
330C2×Q16C22×D5311yescharacteristic
331C2×Q16C22×D5311yescharacteristic
332C2×SD16—21no—
333C2×SD16—21no—
334C2×D8—531no—
335C2×D8C22×D5311yescharacteristic
336C22×C8C22×D5311yescharacteristic
337C22×C8C22×D5311yescharacteristic
338C4⋊Q8—1061no—
339#124—21no—
340#124—41no—
341#123—2121no—
342#123—2121no—
343C22⋊Q8—21no—
344C22⋊Q8—41no—
345C22⋊Q8—2121no—
346#121—21no—
347#121—41no—
348#121—1061no—
349#121—1061no—
350C4×Q8—21no—
351C4×D4—41no—
352C4×D4—1061no—
353C4×D4C22⋊D5311yescharacteristic
354C2×C4⋊C4—1061no—
355C2×C4⋊C4—1061no—
356SD32—1062no—
357SD32—2121no—
358D16—2121no—
359M32—2121no—
360C2×C16—531no—
361C2×C16—531no—
362C2×C16—531no—
363C2×C16—531no—
364C2×C16—2121no—
365#107—531no—
366#107—531no—
367#107—531no—
368#107—1061no—
369#107—1061no—
370#107—1061no—
371#107C22×D5311yescharacteristic
372C4⋊C8C22⋊D5311yescharacteristic
373#103—21no—
374#103—21no—
375#102—21no—
376#102—1061no—
377#102—1061no—
378#102—1061no—
379C22⋊C8C22⋊D5311yescharacteristic
380C22⋊C8C22⋊D5311yescharacteristic
381C4×C8C22×D5311yescharacteristic
382#41—42no—
383#41—81no—
384#41—4244no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—42no—
388C2×Q8—41no—
389C2×Q8—81no—
390C2×Q8—2121no—
391C2×D4—21no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—1061no—
395C2×D4—2121no—
396C2×D4—2121no—
397C2×D4#1171992211yescharacteristic
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—2121no—
401C22×C4—2121no—
402C22×C4—2121no—
403C22×C4C2×C22⋊D5311yescharacteristic
404C22×C4C2×C22⋊D5311yescharacteristic
405Q16—21no—
406Q16—21no—
407Q16—21no—
408Q16—42no—
409Q16#1171994712yescomplemented
410SD16—42no—
411SD16—81no—
412D8—21no—
413D8—21no—
414D8—1061no—
415D8—1061no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8C2×C22⋊D5311yescharacteristic
421C2×C8C2×C22⋊D5311yescharacteristic
422C2×C8C2×C22⋊D5311yescharacteristic
423C2×C8C23×D5311yescharacteristic
424C4⋊C4—21no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—1061no—
428C4⋊C4—1061no—
429C4⋊C4—1061no—
430C4⋊C4—2121no—
431C4⋊C4—2121no—
432C4⋊C4—2121no—
433C4⋊C4#1171992211yescharacteristic
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—2121no—
439C22⋊C4—2121no—
440C22⋊C4—2121no—
441C22⋊C4#1171992211yescharacteristic
442C22⋊C4#1171992211yescharacteristic
443C42—41no—
444C42C2×C22⋊D5311yescharacteristic
445C16—1061no—
446C16—1061no—
447C16—1061no—
448C16—1061no—
449C23—41no—
450C23—2121no—
451C23C2×#1171992211yescharacteristic
452C23C2×#1171992211yescharacteristic
453Q8—22no—
454Q8—41no—
455Q8—41no—
456Q8—41no—
457Q8—82no—
458Q8—4242no—
459D4—22no—
460D4—41no—
461D4—41no—
462D4—82no—
463D4—82no—
464D4—4242no—
465D4—4242no—
466D4—2121no—
467D4—2121no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—4242no—
478C2×C4—4242no—
479C2×C4—4242no—
480C2×C4—2121no—
481C2×C4—2121no—
482C2×C4—2121no—
483C2×C4C2×#1171992211yescharacteristic
484C2×C4C2×#1171992211yescharacteristic
485C2×C4C2×#1171992211yescharacteristic
486C2×C4C2×#1171992211yescharacteristic
487C2×C4C22×C22⋊D5311yescharacteristic
488C8—21no—
489C8—21no—
490C8—21no—
491C8C2×#1171994711yescharacteristic
492C8#1313326211yescharacteristic
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—4241no—
502C22—4241no—
503C22—4241no—
504C22#1681105812yes—
505C22C22×#1171992211yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—42no—
511C4—81no—
512C4—81no—
513C4—4241no—
514C4—4241no—
515C4—4241no—
516C4#1681126711yescharacteristic
517C4#1681143411yescharacteristic
518C2—22no—
519C2—41no—
520C2—161no—
521C2—8481no—
522C2#2028652811yescharacteristic
523C2C2×#1681105811yescharacteristic
524C2#2028786611yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3285C2
socleC22×C53C22×D8
hypercenter#3285D53
hypocenterC53#33875
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. #20287063C2semidirect
2. #20276951C2semidirect
3. #20276915C2semidirect
4. #20276893C2semidirect
5. #20276867C2semidirect
6. #20276830C2semidirect
7. #20276799C2semidirect
8. #20276767C2semidirect
9. #20276638C2semidirect
10. #20276632C2semidirect
11. #20276577C2semidirect
12. #20276541C2semidirect
13. #20276427C2semidirect
14. C2×#16810427C2semidirect
15. C53×#3285C2semidirect
146. Q16×C53#41semidirect
257. #3285D53semidirect
321. C53#33875semidirect
409. Q16#11719947semidirect

Maximal subgroups

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

subgroupindex
#202870632
#202769512
#202769152
#202768932
#202768672
#202768302
#202767992
#202767672
#202766382
#202766322
#202765772
#202765412
#202764272
C2×#168104272
C53×#32852
#3387553