Finite Group Databasebeta

← #27537427

Subgroups — #27537427

Order 13568.

696 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811057C211yesmaximal, characteristic
2#20283585C211yesmaximal, characteristic, complemented
3#20283562C211yesmaximal, characteristic, complemented
4#20276591C211yesmaximal, characteristic, complemented
5#20276589C211yesmaximal, characteristic, complemented
6#20276579C211yesmaximal, characteristic, complemented
7#20276578C211yesmaximal, characteristic, complemented
8#20276570C211yesmaximal, characteristic, complemented
9C4×#13133162C211yesmaximal, characteristic, complemented
10#20276559C211yesmaximal, characteristic, complemented
11#20276500C211yesmaximal, characteristic, complemented
12#20276419C211yesmaximal, characteristic, complemented
13#357×D53C211yesmaximal, characteristic, complemented
14#20276310C211yesmaximal, characteristic, complemented
15C4×D16×C53C211yesmaximal, characteristic, complemented
16C2×#13133236C2211yescharacteristic
17C2×D8×D53C2211yescharacteristic
18#16811102—21no—
19#16811095—21no—
20#16811057C418yescomplemented
21C2×#13133162C2211yescharacteristic
22C2×#13133161C2211yescharacteristic
23C2×#13133158C2211yescharacteristic
24C2×#13133156C2211yescharacteristic
25C2×C16×D53C2211yescharacteristic
26#16810733—21no—
27#16810732—21no—
28#16810723—21no—
29#16810720—21no—
30C4×Dic212C2211yescharacteristic, complemented
31#16810717C2211yescharacteristic, complemented
32C4×D424C2211yescharacteristic
33D8×Dic53C2211yescharacteristic
34D8×Dic53C2211yescharacteristic, complemented
35#16810705C2211yescharacteristic
36#16810660C2211yescharacteristic
37#107×D53C2211yescharacteristic
38#16810626—21no—
39#16810621—21no—
40#16810577C2211yescharacteristic
41#16810451C2211yescharacteristic, complemented
42#16810450C2211yescharacteristic, complemented
43#16810448C2211yescharacteristic, complemented
44#16810428C2211yescharacteristic, complemented
45#16810426C2211yescharacteristic, complemented
46#16810425C2211yescharacteristic
47#16810424C2211yescharacteristic
48#16810424C2211yescharacteristic, complemented
49#16810423C2211yescharacteristic
50#16810399C2211yescharacteristic, complemented
51#16810398C2211yescharacteristic, complemented
52C16×Dic53C2211yescharacteristic, complemented
53C2×D16×C53C2211yescharacteristic
54C4×D8×C53C2211yescharacteristic
55C4×D8×C53C2211yescharacteristic, complemented
56C53×#366C2211yescharacteristic, complemented
57C53×#357C2211yescharacteristic
58C53×#357C2211yescharacteristic, complemented
59C4×C16×C53C2211yescharacteristic, complemented
60C2×#11719947—21no—
61C2×D4×D53—21no—
62#13133236—22no—
63#13133236C2×C414yescomplemented
64C2×Dic212C2311yescharacteristic
65C2×#11719923—21no—
66C2×D424C2311yescharacteristic
67C2×#11719922—21no—
68D8×D53—22no—
69D8×D53C2×C414yescomplemented
70C2×#11719912D411yescharacteristic
71C2×C8×D53C2311yescharacteristic
72C4×Dic106—21no—
73#13133182—21no—
74C4×D212—21no—
75D4×Dic53—21no—
76D4×Dic53—21no—
77#13133179—21no—
78C4×C22⋊D53—41no—
79C4×C22⋊D53—41no—
80#13133177—41no—
81#13133177—41no—
82#13133175—21no—
83#13133174D411yescharacteristic
84#13133173—41no—
85C2×#11719919—41no—
86C4⋊C4×D53—21no—
87C2×#11719921—41no—
88C22⋊C4×D53—41no—
89#13133162C2×C414yescomplemented
90#13133161C2×C414yescomplemented
91#13133158C2×C414yescomplemented
92#13133156C2×C414yescomplemented
93C2×#11719908C2311yescharacteristic
94C2×#11719908C2311yescharacteristic
95C16×D53—22no—
96#13133151C2311yescharacteristic
97#13133151C2311yescharacteristic
98#13133150—21no—
99#13133148—21no—
100#13133142—21no—
101#13133140—21no—
102#13133139—21no—
103#13133138—21no—
104#13133138—21no—
105#13133137—21no—
106#13133128D411yescharacteristic
107#13133127D411yescharacteristic
108C8×Dic53C2311yescharacteristic
109C2×D8×C53C2311yescharacteristic
110C2×D8×C53C2311yescharacteristic
111C4×D4×C53—21no—
112C4×D4×C53—21no—
113D16×C53C2×C414yescomplemented
114C2×C16×C53C2311yescharacteristic
115C2×C16×C53C2311yescharacteristic
116#107×C53C2311yescharacteristic
117#107×C53C2311yescharacteristic
118#102×C53—21no—
119#102×C53—21no—
120C4×C8×C53C2311yescharacteristic
121C23×D53—41no—
122C2×D4×C53—21no—
123C2×D4×C53—21no—
124C22×C4×C53—41no—
125C22×C4×C53—41no—
126C2×C22⋊D53—41no—
127C2×C22⋊D53—41no—
128C22×Dic53—41no—
129C22×Dic53—41no—
130C22×Dic53—41no—
131#11719947—24no—
132#11719947—42no—
133D4×D53—24no—
134D4×D53—42no—
135C2×D212—21no—
136C2×C4×D53—41no—
137C2×C4×D53—41no—
138C2×C4×D53—41no—
139C2×C4×D53C2×D411yescharacteristic
140C2×Dic106—21no—
141D8×C53—21no—
142D8×C53—21no—
143D8×C53C22×C412yes—
144D8×C53C22×C412yes—
145C2×C8×C53C2×D411yescharacteristic
146C2×C8×C53C2411yescharacteristic
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C22⋊C4×C53—41no—
150C22⋊C4×C53—41no—
151C42×C53C2×D411yescharacteristic
152C22⋊Dic53—41no—
153C22⋊Dic53—41no—
154#11719923—24no—
155#11719922—24no—
156#11719921—41no—
157#11719921—41no—
158#11719921—41no—
159#11719921D811yescharacteristic
160#11719921D811yescharacteristic
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163#11719919—41no—
164#11719919—41no—
165#11719919—41no—
166#11719919D811yescharacteristic
167#11719919D811yescharacteristic
168C4×Dic53—41no—
169C4×Dic53—41no—
170C4×Dic53—41no—
171C4×Dic53C2×D411yescharacteristic
172C2×#88322C2×D411yescharacteristic
173C2×#88322C2×D411yescharacteristic
174Dic212—21no—
175Dic212C22×C412yes—
176D424—21no—
177D424C22×C412yes—
178#11719912—22no—
179C8×D53C22×C412yes—
180C8×D53#4112yes—
181C16×C53—21no—
182C16×C53C22×C412yes—
183#11719908—21no—
184#11719908C22×C412yes—
185C23×C53—41no—
186C23×C53—41no—
187C22×D53—41no—
188C22×D53—44no—
189C22×D53—161no—
190C22×D53—161no—
191C22×D53C2×D811yescharacteristic
192D4×C53—22no—
193D4×C53—22no—
194D4×C53—41no—
195D4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—161no—
199C2×C4×C53—161no—
200C2×C4×C53C2×D811yescharacteristic
201C2×C4×C53C2×D811yescharacteristic
202C2×C4×C53C22×D411yescharacteristic
203C22⋊D53—44no—
204C22⋊D53—44no—
205C2×Dic53—41no—
206C2×Dic53—44no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53—161no—
210C2×Dic53—161no—
211C2×Dic53C2×D811yescharacteristic
212C2×Dic53C2×D811yescharacteristic
213C2×Dic53C2×D811yescharacteristic
214D212—22no—
215D212—41no—
216C4×D53—161no—
217C4×D53—161no—
218C4×D53—161no—
219C4×D53C4×D412yes—
220C4×D53#13512yes—
221Dic106—22no—
222Dic106—41no—
223C8×C53—21no—
224C8×C53C23×C411yescharacteristic
225C8×C53C2×#4111yescharacteristic
226#88322—21no—
227#88322C4×D412yes—
228#33454—531nomaximal, Sylow
229C22×C53—42no—
230C22×C53—42no—
231C22×C53—81no—
232C22×C53—81no—
233C22×C53—81no—
234C22×C53—81no—
235C22×C53C22×D811yescharacteristic
236C2×D53—21no—
237C2×D53—21no—
238C2×D53—42no—
239C2×D53—81no—
240C2×D53—81no—
241C2×D53—164no—
242C2×D53C4×D812yes—
243C4×C53—21no—
244C4×C53—21no—
245C4×C53—81no—
246C4×C53—81no—
247C4×C53C2×C4×D411yescharacteristic
248C4×C53C2×#13511yescharacteristic
249Dic53—21no—
250Dic53—21no—
251Dic53—42no—
252Dic53—81no—
253Dic53—81no—
254Dic53C4×D812yes—
255C2×#509—531no—
256#2951—531no—
257#2948—531no—
258#2177—531no—
259#2177—531no—
260#2175—531no—
261#2175—531no—
262C4×SD32—531no—
263C4×SD32—531no—
264C4×D16—531no—
265C4×D16D5311yescharacteristic, complemented
266#2159—531no—
267#2138—531no—
268C2×#357—531no—
269#2108—531no—
270C2×C53—82no—
271C2×C53—82no—
272C2×C53C2×C4×D811yescharacteristic
273C2×C53#341511yescharacteristic
274C2×C53#341611yescharacteristic
275D53—22no—
276D53—82no—
277C2×#135—531no—
278C22×D8—531no—
279#520—1061no—
280#518—1061no—
281#509—538no—
282C2×SD32—531no—
283C2×SD32—531no—
284C2×D16—531no—
285C2×D16C2×D5311yescharacteristic
286C2×M32—531no—
287C22×C16—531no—
288#442—1061no—
289#442—1061no—
290#440—1061no—
291#440—1061no—
292C4×Q16—531no—
293C4×Q16—531no—
294C4×D8—531no—
295C4×D8—531no—
296C4×D8—531no—
297C4×D8—531no—
298C4×D8C2×D5311yescharacteristic
299C4×D8C2×D5311yescharacteristic, complemented
300#427—531no—
301C2×#107—531no—
302#418—1061no—
303#417—1061no—
304#405—531no—
305#367—531no—
306#367—531no—
307#366—531no—
308#366C2×D5311yescharacteristic, complemented
309#358—531no—
310#358—531no—
311#357—531no—
312#357—531no—
313#357—531no—
314#357—531no—
315#357C2×D5311yescharacteristic
316#357C2×D5311yescharacteristic, complemented
317#346—531no—
318#346—531no—
319C4×C16—531no—
320C4×C16C2×D5311yescharacteristic, complemented
321C53#3345411yescharacteristic, complemented, Sylow
322C2×#41—1061no—
323C22×D4—1061no—
324#135—534no—
325#135—1062no—
326C2×Q16—531no—
327C2×SD16—1061no—
328C2×D8—534no—
329C2×D8—531no—
330C2×D8—1062no—
331C2×D8—1061no—
332C2×D8C22×D5311yescharacteristic
333C2×D8C22×D5311yescharacteristic
334C2×M16—531no—
335C22×C8—531no—
336C4×Q8—1061no—
337C4×Q8—1061no—
338C4×D4—21no—
339C4×D4—21no—
340C4×D4—1061no—
341C4×D4—1061no—
342C4×D4—1061no—
343C4×D4—1061no—
344C4×D4—2121no—
345C4×D4—2121no—
346C4×D4—2121no—
347C4×D4—2121no—
348#117—531no—
349#117—1061no—
350#117—2121no—
351C2×C4⋊C4—1061no—
352C2×C4⋊C4—2121no—
353C2×C22⋊C4—2121no—
354C2×C22⋊C4—2121no—
355SD32—534no—
356SD32—534no—
357D16—534no—
358D16C4×D5314yescomplemented
359M32—534no—
360C2×C16—531no—
361C2×C16—531no—
362C2×C16—1062no—
363C2×C16C22×D5311yescharacteristic
364C2×C16C22×D5311yescharacteristic
365#107—531no—
366#107—531no—
367#107—1061no—
368#107C22×D5311yescharacteristic
369#107C22×D5311yescharacteristic
370#106—1061no—
371#103—1061no—
372#103—1061no—
373#102—21no—
374#102—21no—
375#102—1061no—
376#102—1061no—
377#102—1061no—
378#102—1061no—
379#97—531no—
380#97—531no—
381C4×C8—531no—
382C4×C8C22×D5311yescharacteristic
383C24—2121no—
384#41—1064no—
385#41—2122no—
386C2×Q8—1061no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—1064no—
390C2×D4—1061no—
391C2×D4—2122no—
392C2×D4—2121no—
393C2×D4—2121no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—531no—
397C22×C4—2121no—
398C22×C4—2121no—
399C22×C4—2121no—
400C22×C4—2121no—
401C22×C4—2121no—
402C22×C4—2121no—
403Q16—532no—
404Q16—1061no—
405SD16—1064no—
406D8—21no—
407D8—21no—
408D8—532no—
409D8—1064no—
410D8—1061no—
411D8C2×C4×D5312yes—
412D8C2×C4×D5312yes—
413M16—1062no—
414C2×C8—531no—
415C2×C8—532no—
416C2×C8—532no—
417C2×C8—531no—
418C2×C8D4×D5311yescharacteristic
419C2×C8C23×D5311yescharacteristic
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—531no—
423C4⋊C4—531no—
424C4⋊C4—1061no—
425C4⋊C4—1061no—
426C4⋊C4—2121no—
427C4⋊C4—2121no—
428C4⋊C4—2121no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—531no—
432C22⋊C4—531no—
433C22⋊C4—2121no—
434C22⋊C4—2121no—
435C22⋊C4—2121no—
436C22⋊C4—2121no—
437C22⋊C4—2121no—
438C42—531no—
439C42—2121no—
440C42—2121no—
441C42—2121no—
442C42D4×D5311yescharacteristic
443C16—21no—
444C16—532no—
445C16—1061no—
446C16C2×C4×D5312yes—
447C23—41no—
448C23—41no—
449C23—531no—
450C23—8481no—
451C23—8481no—
452C23—2124no—
453C23—2121no—
454Q8—1062no—
455Q8—2121no—
456D4—22no—
457D4—22no—
458D4—41no—
459D4—41no—
460D4—1062no—
461D4—2124no—
462D4—2124no—
463D4—2121no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—161no—
467C2×C4—161no—
468C2×C4—531no—
469C2×C4—532no—
470C2×C4—532no—
471C2×C4—531no—
472C2×C4—531no—
473C2×C4—8481no—
474C2×C4—8481no—
475C2×C4—8481no—
476C2×C4—8481no—
477C2×C4—8481no—
478C2×C4—2124no—
479C2×C4—2121no—
480C2×C4—2121no—
481C2×C4—2121no—
482C2×C4D8×D5311yescharacteristic
483C2×C4D8×D5311yescharacteristic
484C2×C4C2×D4×D5311yescharacteristic
485C8—21no—
486C8—532no—
487C8—1061no—
488C8C22×C4×D5311yescharacteristic
489C8#41×D5311yescharacteristic
490C22—42no—
491C22—42no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—532no—
497C22—8484no—
498C22—1061no—
499C22—1061no—
500C22—4241no—
501C22—4241no—
502C22—2122no—
503C22C2×D8×D5311yescharacteristic
504C4—21no—
505C4—21no—
506C4—81no—
507C4—81no—
508C4—532no—
509C4—1061no—
510C4—1061no—
511C4—4241no—
512C4—4241no—
513C4—2122no—
514C4C4×D4×D5311yescharacteristic
515C4#135×D5311yescharacteristic
516C2—82no—
517C2—82no—
518C2—1062no—
519C2—4242no—
520C2C4×D8×D5311yescharacteristic
521C2#2028785211yescharacteristic
522C2#2028785811yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC4×D16×C53C2
socleC22×C53C22×D8
prime radicalC2×#16811057C2
cosocleC2×C8×C53C24
hypercenterC4×D16D53
hypocenterC53#33454
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×D16

Complements

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

NH ≅ G/Ntype
2. #20283585C2semidirect
3. #20283562C2semidirect
4. #20276591C2semidirect
5. #20276589C2semidirect
6. #20276579C2semidirect
7. #20276578C2semidirect
8. #20276570C2semidirect
9. C4×#13133162C2semidirect
10. #20276559C2semidirect
11. #20276500C2semidirect
12. #20276419C2semidirect
13. #357×D53C2semidirect
14. #20276310C2semidirect
15. C4×D16×C53C2semidirect
20. #16811057C4semidirect
30. C4×Dic212C22semidirect
31. #16810717C22semidirect
34. D8×Dic53C22semidirect
41. #16810451C22semidirect
42. #16810450C22semidirect
43. #16810448C22semidirect
44. #16810428C22semidirect
45. #16810426C22semidirect
48. #16810424C22semidirect
50. #16810399C22semidirect
51. #16810398C22semidirect
52. C16×Dic53C22semidirect
55. C4×D8×C53C22semidirect
56. C53×#366C22semidirect
58. C53×#357C22semidirect
59. C4×C16×C53C22semidirect
63. #13133236C2×C4semidirect
69. D8×D53C2×C4semidirect
89. #13133162C2×C4semidirect
90. #13133161C2×C4semidirect
91. #13133158C2×C4semidirect
92. #13133156C2×C4semidirect
113. D16×C53C2×C4semidirect
265. C4×D16D53semidirect
299. C4×D8C2×D53semidirect
308. #366C2×D53semidirect
316. #357C2×D53semidirect
320. C4×C16C2×D53semidirect
321. C53#33454semidirect
358. D16C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168110572
#202835852
#202835622
#202765912
#202765892
#202765792
#202765782
#202765702
C4×#131331622
#202765592
#202765002
#202764192
#357×D532
#202763102
C4×D16×C532
#3345453