Finite Group Databasebeta

← #27541183

Subgroups — #27541183

Order 13568.

570 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#20287108C211yesmaximal, characteristic, complemented
2#20276951C211yesmaximal, characteristic, complemented
3#20276915C211yesmaximal, characteristic, complemented
4#20276893C211yesmaximal, characteristic, complemented
5#20276852C211yesmaximal, characteristic, complemented
6#20276826C211yesmaximal, characteristic, complemented
7#20276804C211yesmaximal, characteristic, complemented
8#20276774C211yesmaximal, characteristic, complemented
9#20276655C211yesmaximal, characteristic, complemented
10#20276644C211yesmaximal, characteristic, complemented
11#20276579C211yesmaximal, characteristic, complemented
12#20276541C211yesmaximal, characteristic, complemented
13#20276427C211yesmaximal, characteristic, complemented
14C2×#16810424C211yesmaximal, characteristic, complemented
15C53×#3280C211yesmaximal, characteristic
16#16811330—21no—
17C2×#13133163C2211yescharacteristic, complemented
18C2×#13133161C2211yescharacteristic
19C2×#13133161C2211yescharacteristic
20C2×#13133157C2211yescharacteristic, complemented
21C22×#11719908C2211yescharacteristic
22#16811028C2211yescharacteristic, complemented
23#16810959—21no—
24#16810959—21no—
25#16810919—21no—
26#16810864C2211yescharacteristic
27#16810864C2211yescharacteristic, complemented
28C4×Dic212C2211yescharacteristic, complemented
29C2×#13133151C2211yescharacteristic
30C2×#13133151C2211yescharacteristic, complemented
31#16810448C2211yescharacteristic, complemented
32#16810448C2211yescharacteristic, complemented
33#16810448C2211yescharacteristic, complemented
34C4⋊#11719908C2211yescharacteristic, complemented
35#16810427C2211yescharacteristic, complemented
36#16810427C2211yescharacteristic, complemented
37#16810427C2211yescharacteristic, complemented
38#16810426C2211yescharacteristic, complemented
39#16810424C2211yescharacteristic
40#16810424C2211yescharacteristic
41#16810424C2211yescharacteristic
42#16810424C2211yescharacteristic
43#16810424C2211yescharacteristic
44#16810424C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46C22⋊#11719908C2211yescharacteristic, complemented
47#16810399C2211yescharacteristic, complemented
48C2×#135×C53C2211yescharacteristic
49C22×D8×C53C2211yescharacteristic
50C53×#546—21no—
51C53×#495C2211yescharacteristic
52C53×#467C2211yescharacteristic
53C53×#466C2211yescharacteristic
54C53×#463—21no—
55C53×#450—21no—
56C53×#449—21no—
57C4×D8×C53C2211yescharacteristic
58C8×D4×C53C2211yescharacteristic
59C2×Dic212C2311yescharacteristic
60#13133217—21no—
61#13133208—41no—
62#13133196—21no—
63#13133196—21no—
64#13133194—41no—
65#13133194—41no—
66C4×Dic106—21no—
67D4×Dic53—41no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70#13133163—41no—
71#13133161—22no—
72#13133161—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#13133142—21no—
86#13133142—21no—
87#13133142—21no—
88C2×#41×C53—21no—
89C22×D4×C53—21no—
90#135×C53—22no—
91#135×C53—41no—
92C2×Q16×C53C2311yescharacteristic
93C2×SD16×C53—21no—
94C2×SD16×C53—21no—
95C2×D8×C53—21no—
96C2×D8×C53—22no—
97C2×D8×C53—41no—
98C2×D8×C53C2311yescharacteristic
99C2×D8×C53C2311yescharacteristic
100C22×C8×C53C2311yescharacteristic
101C22×C8×C53C2311yescharacteristic
102#124×C53—21no—
103#123×C53—41no—
104C22⋊Q8×C53—21no—
105#121×C53—21no—
106#121×C53—41no—
107#120×C53—41no—
108C4×D4×C53—21no—
109C4×D4×C53D411yescharacteristic
110C2×C22⋊C4×C53—41no—
111#107×C53C2311yescharacteristic
112C4⋊C8×C53D411yescharacteristic
113#103×C53—21no—
114#102×C53—21no—
115#102×C53—21no—
116C22⋊C8×C53D411yescharacteristic
117C22⋊C8×C53D411yescharacteristic
118C4×C8×C53C2311yescharacteristic
119C24×C53—41no—
120#41×C53—42no—
121#41×C53—81no—
122C2×Q8×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—42no—
126C2×D4×C53—41no—
127C2×D4×C53—41no—
128C2×D4×C53—41no—
129C2×D4×C53—81no—
130C2×D4×C53D811yescharacteristic
131C22×C4×C53—41no—
132C22×C4×C53—41no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×Dic53—41no—
136C22×Dic53—41no—
137C2×Dic106—21no—
138C2×Dic106—41no—
139Q16×C53—21no—
140Q16×C53—21no—
141SD16×C53—42no—
142SD16×C53—81no—
143D8×C53—21no—
144D8×C53—21no—
145D8×C53—21no—
146D8×C53—42no—
147D8×C53#4112yes—
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×C53D811yescharacteristic
159C22⋊C4×C53—41no—
160C22⋊C4×C53—41no—
161C22⋊C4×C53—41no—
162C22⋊C4×C53—41no—
163C22⋊C4×C53—41no—
164C22⋊C4×C53D811yescharacteristic
165C22⋊C4×C53D811yescharacteristic
166C42×C53C2×D411yescharacteristic
167C22⋊Dic53—41no—
168C22⋊Dic53—41no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—41no—
173C4⋊Dic53—41no—
174C4⋊Dic53—41no—
175#11719919—41no—
176#11719919—41no—
177#11719919—41no—
178C4×Dic53—41no—
179Dic212—21no—
180Dic212—21no—
181#11719908—21no—
182#11719908—21no—
183#11719908—21no—
184#11719908—21no—
185C23×C53—41no—
186C23×C53—41no—
187C23×C53—82no—
188C23×C53—82no—
189C23×C53—161no—
190C23×C53C2×D811yescharacteristic
191C23×C53C2×D811yescharacteristic
192Q8×C53—41no—
193Q8×C53—41no—
194D4×C53—22no—
195D4×C53—22no—
196D4×C53—41no—
197D4×C53—41no—
198D4×C53—41no—
199D4×C53—82no—
200D4×C53—82no—
201D4×C53—161no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—82no—
209C2×C4×C53—161no—
210C2×C4×C53C2×D811yescharacteristic
211C2×C4×C53C2×D811yescharacteristic
212C2×C4×C53C2×D811yescharacteristic
213C2×C4×C53C2×D811yescharacteristic
214C2×C4×C53C22×D411yescharacteristic
215C2×Dic53—41no—
216C2×Dic53—41no—
217C2×Dic53—41no—
218C2×Dic53—41no—
219C2×Dic53—161no—
220C2×Dic53—161no—
221Dic106—41no—
222Dic106—41no—
223Dic106—161no—
224C8×C53—21no—
225C8×C53—21no—
226C8×C53—21no—
227C8×C53C2×#4111yescharacteristic
228C8×C53H(5,𝔽2)11yescharacteristic
229#33881—531nomaximal, Sylow
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—42no—
236C22×C53—81no—
237C22×C53—81no—
238C22×C53—81no—
239C22×C53—81no—
240C22×C53—81no—
241C22×C53—164no—
242C22×C53#51012yes—
243C22×C53C22×D811yescharacteristic
244C4×C53—21no—
245C4×C53—21no—
246C4×C53—21no—
247C4×C53—21no—
248C4×C53—81no—
249C4×C53—81no—
250C4×C53#54611yescharacteristic
251C4×C53#57711yescharacteristic
252Dic53—81no—
253Dic53—81no—
254Dic53—81no—
255Dic53—81no—
256#3338—531no—
257#3280D5311yescharacteristic
258#2243—531no—
259#2236—531no—
260#2232—531no—
261#2224—531no—
262#2220—531no—
263#2216—531no—
264#2211—531no—
265#2189—531no—
266#2187—531no—
267#2175—531no—
268#2168—531no—
269#2139—531no—
270C2×#357—531no—
271C2×C53—22no—
272C2×C53—41no—
273C2×C53—82no—
274C2×C53—161no—
275C2×C53#329411yescharacteristic
276C2×C53C2×#51011yescharacteristic
277C2×C53#341511yescharacteristic
278C2×#135C2×D5311yescharacteristic
279C22×D8C2×D5311yescharacteristic
280#554—1061no—
281#546—21no—
282C2×Q32—531no—
283C2×SD32—531no—
284C2×SD32—531no—
285C2×M32—531no—
286C22×C16—531no—
287#500—531no—
288#495C2×D5311yescharacteristic
289#484—1061no—
290#484—1061no—
291#477—1061no—
292#467—531no—
293#467—531no—
294#467C2×D5311yescharacteristic
295#466C2×D5311yescharacteristic
296#463—21no—
297#450—21no—
298#449—21no—
299C4×Q16—531no—
300C4×D8C2×D5311yescharacteristic
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#357—531no—
313#357—531no—
314#357—531no—
315#357—531no—
316#357—531no—
317#357—531no—
318C22⋊C16—531no—
319C22⋊C16—531no—
320#346—531no—
321C53#3388111yescharacteristic, complemented, Sylow
322C2×#41—21no—
323C22×D4—21no—
324#135—22no—
325#135—41no—
326C2×Q16—531no—
327C2×Q16C22×D5311yescharacteristic
328C2×SD16—21no—
329C2×SD16—21no—
330C2×D8—22no—
331C2×D8—21no—
332C2×D8—41no—
333C2×D8C22×D5311yescharacteristic
334C2×D8C22×D5311yescharacteristic
335C22×C8C22×D5311yescharacteristic
336C22×C8C22×D5311yescharacteristic
337C4⋊Q8—1061no—
338#125—2121no—
339#124—21no—
340#123—41no—
341C22⋊Q8—21no—
342C22⋊Q8—1061no—
343C22⋊Q8—1061no—
344C22⋊Q8—2121no—
345C22⋊Q8—2121no—
346#121—21no—
347#121—41no—
348#120—41no—
349C4×Q8—1061no—
350C4×D4—21no—
351C4×D4—2121no—
352C4×D4C22⋊D5311yescharacteristic
353C2×C4⋊C4—1061no—
354C2×C4⋊C4—1061no—
355C2×C22⋊C4—41no—
356Q32—2121no—
357SD32—1062no—
358SD32—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—1061no—
375#103—1061no—
376#103—1061no—
377#102—21no—
378#102—21no—
379C22⋊C8C22⋊D5311yescharacteristic
380C22⋊C8C22⋊D5311yescharacteristic
381C4×C8C22×D5311yescharacteristic
382C24—41no—
383#41—42no—
384#41—81no—
385C2×Q8—21no—
386C2×Q8—1061no—
387C2×Q8—2121no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—42no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—81no—
395C2×D4#1171992211yescharacteristic
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—2121no—
399C22×C4—2121no—
400C22×C4C2×C22⋊D5311yescharacteristic
401C22×C4C2×C22⋊D5311yescharacteristic
402Q16—21no—
403Q16—21no—
404Q16—1061no—
405Q16—1061no—
406SD16—42no—
407SD16—81no—
408D8—21no—
409D8—21no—
410D8—21no—
411D8—42no—
412D8#1171994712yes—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8C2×C22⋊D5311yescharacteristic
418C2×C8C2×C22⋊D5311yescharacteristic
419C2×C8C2×C22⋊D5311yescharacteristic
420C2×C8C23×D5311yescharacteristic
421C4⋊C4—21no—
422C4⋊C4—41no—
423C4⋊C4—1061no—
424C4⋊C4—1061no—
425C4⋊C4—1061no—
426C4⋊C4—2121no—
427C4⋊C4—2121no—
428C4⋊C4—2121no—
429C4⋊C4—2121no—
430C4⋊C4—2121no—
431C4⋊C4—2121no—
432C4⋊C4#1171992211yescharacteristic
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—2121no—
439C22⋊C4—2121no—
440C22⋊C4#1171992211yescharacteristic
441C22⋊C4#1171992211yescharacteristic
442C42—2121no—
443C42C2×C22⋊D5311yescharacteristic
444C16—1061no—
445C16—1061no—
446C16—1061no—
447C16—1061no—
448C23—41no—
449C23—41no—
450C23—82no—
451C23—82no—
452C23—161no—
453C23C2×#1171992211yescharacteristic
454C23C2×#1171992211yescharacteristic
455Q8—41no—
456Q8—41no—
457Q8—8481no—
458Q8—2121no—
459Q8—2121no—
460D4—22no—
461D4—22no—
462D4—41no—
463D4—41no—
464D4—41no—
465D4—82no—
466D4—82no—
467D4—161no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—82no—
475C2×C4—161no—
476C2×C4—8481no—
477C2×C4—8481no—
478C2×C4—2121no—
479C2×C4—2121no—
480C2×C4—2121no—
481C2×C4—2121no—
482C2×C4C2×#1171992211yescharacteristic
483C2×C4C2×#1171992211yescharacteristic
484C2×C4C2×#1171992211yescharacteristic
485C2×C4C2×#1171992211yescharacteristic
486C2×C4C22×C22⋊D5311yescharacteristic
487C8—21no—
488C8—21no—
489C8—21no—
490C8C2×#1171994711yescharacteristic
491C8#1313326211yescharacteristic
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—42no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—164no—
504C22#1681106312yes—
505C22C22×#1171992211yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—81no—
511C4—81no—
512C4—4241no—
513C4—4241no—
514C4—4241no—
515C4—4241no—
516C4#1681126711yescharacteristic
517C4#1681143411yescharacteristic
518C2—22no—
519C2—41no—
520C2—82no—
521C2—161no—
522C2#2028652811yescharacteristic
523C2C2×#1681106311yescharacteristic
524C2#2028785411yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
Fitting, prime radicalC53×#3280C2
socleC22×C53C22×D8
hypercenter#3280D53
hypocenterC53#33881
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3280

Complements

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

NH ≅ G/Ntype
1. #20287108C2semidirect
2. #20276951C2semidirect
3. #20276915C2semidirect
4. #20276893C2semidirect
5. #20276852C2semidirect
6. #20276826C2semidirect
7. #20276804C2semidirect
8. #20276774C2semidirect
9. #20276655C2semidirect
10. #20276644C2semidirect
11. #20276579C2semidirect
12. #20276541C2semidirect
13. #20276427C2semidirect
14. C2×#16810424C2semidirect
17. C2×#13133163C22semidirect
20. C2×#13133157C22semidirect
22. #16811028C22semidirect
27. #16810864C22semidirect
28. C4×Dic212C22semidirect
30. C2×#13133151C22semidirect
31. #16810448C22semidirect
32. #16810448C22semidirect
33. #16810448C22semidirect
34. C4⋊#11719908C22semidirect
35. #16810427C22semidirect
36. #16810427C22semidirect
37. #16810427C22semidirect
38. #16810426C22semidirect
46. C22⋊#11719908C22semidirect
47. #16810399C22semidirect
321. C53#33881semidirect

Maximal subgroups

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

subgroupindex
#202871082
#202769512
#202769152
#202768932
#202768522
#202768262
#202768042
#202767742
#202766552
#202766442
#202765792
#202765412
#202764272
C2×#168104242
C53×#32802
#3388153