Finite Group Databasebeta

← #27537456

Subgroups — #27537456

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD32×D53C211yesmaximal, characteristic
2#20283578C211yesmaximal, characteristic, complemented
3#20283562C211yesmaximal, characteristic
4#20276582C211yesmaximal, characteristic, complemented
5#20276579C211yesmaximal, characteristic, complemented
6#20276577C211yesmaximal, characteristic, complemented
7#20276576C211yesmaximal, characteristic, complemented
8C4×#13133162C211yesmaximal, characteristic, complemented
9SD32×Dic53C211yesmaximal, characteristic, complemented
10#20276566C211yesmaximal, characteristic, complemented
11#20276498C211yesmaximal, characteristic, complemented
12#20276433C211yesmaximal, characteristic, complemented
13#20276429C211yesmaximal, characteristic
14#20276310C211yesmaximal, characteristic, complemented
15C4×SD32×C53C211yesmaximal, characteristic, complemented
16C2×Q16×D53C2211yescharacteristic
17C2×D8×D53C2211yescharacteristic
18#16811099—21no—
19#16811095—21no—
20C2×#13133162C2211yescharacteristic
21C2×#13133161C2211yescharacteristic
22C2×#13133160C2211yescharacteristic
23SD32×D53C418yescomplemented
24C2×#13133156C2211yescharacteristic
25C2×C16×D53C2211yescharacteristic
26#16810733—21no—
27#16810732—21no—
28#16810728—21no—
29#16810727—21no—
30C4×Dic212C2211yescharacteristic, complemented
31Q16×Dic53C2211yescharacteristic, complemented
32#16810717C2211yescharacteristic, complemented
33C4×D424C2211yescharacteristic
34D8×Dic53C2211yescharacteristic
35#16810705C2211yescharacteristic
36#16810660C2211yescharacteristic
37#16810660C2211yescharacteristic
38#16810630—21no—
39#16810626—21no—
40#16810577C2211yescharacteristic
41#16810451C2211yescharacteristic, complemented
42#16810448C2211yescharacteristic, complemented
43#16810448C2211yescharacteristic, complemented
44#16810428C2211yescharacteristic, complemented
45#16810427C2211yescharacteristic, complemented
46#16810426C2211yescharacteristic, complemented
47#16810425C2211yescharacteristic
48#16810424C2211yescharacteristic
49#16810423C2211yescharacteristic
50#16810399C2211yescharacteristic, complemented
51#16810398C2211yescharacteristic, complemented
52C16×Dic53C2211yescharacteristic, complemented
53C2×SD32×C53C2211yescharacteristic
54C4×Q16×C53C2211yescharacteristic, complemented
55C4×D8×C53C2211yescharacteristic
56C53×#367C2211yescharacteristic, complemented
57C53×#358C2211yescharacteristic, complemented
58C53×#357C2211yescharacteristic
59C4×C16×C53C2211yescharacteristic, complemented
60C2×Q8×D53—21no—
61C2×D4×D53—21no—
62C2×Dic212C2311yescharacteristic
63C2×#11719925—21no—
64Q16×D53—22no—
65Q16×D53C2×C414yescomplemented
66C2×D424C2311yescharacteristic
67C2×#11719922—21no—
68D8×D53—22no—
69D8×D53C2×C414yes—
70C2×#11719912D411yescharacteristic
71C2×C8×D53C2311yescharacteristic
72Q8×Dic53—21no—
73C4×Dic106—21no—
74C4×Dic106—41no—
75#13133182—21no—
76#13133182—41no—
77C4×D212—21no—
78D4×Dic53—21no—
79#13133179—21no—
80C4×C22⋊D53—41no—
81#13133177—41no—
82#13133175—21no—
83#13133175—21no—
84#13133175—41no—
85#13133174—41no—
86#13133174D411yescharacteristic
87C2×#11719921—41no—
88C22⋊C4×D53—41no—
89#13133162C2×C414yescomplemented
90#13133161C2×C414yescomplemented
91#13133160C2×C414yescomplemented
92#13133156—22no—
93C2×#11719908C2311yescharacteristic
94C2×#11719908C2311yescharacteristic
95C16×D53C2×C414yescomplemented
96#13133151C2311yescharacteristic
97#13133151C2311yescharacteristic
98#13133148—21no—
99#13133148—21no—
100#13133142—21no—
101#13133141—21no—
102#13133140—21no—
103#13133139—21no—
104#13133138—21no—
105#13133137—21no—
106#13133128D411yescharacteristic
107#13133127D411yescharacteristic
108C8×Dic53C2311yescharacteristic
109C2×Q16×C53C2311yescharacteristic
110C2×D8×C53C2311yescharacteristic
111C4×Q8×C53—21no—
112C4×D4×C53—21no—
113SD32×C53C2×C414yescomplemented
114C2×C16×C53C2311yescharacteristic
115C2×C16×C53C2311yescharacteristic
116#107×C53C2311yescharacteristic
117#107×C53C2311yescharacteristic
118#103×C53—21no—
119#102×C53—21no—
120C4×C8×C53C2311yescharacteristic
121C23×D53—41no—
122C2×Q8×C53—21no—
123C2×D4×C53—21no—
124C22×C4×C53—41no—
125C2×C22⋊D53—41no—
126C22×Dic53—41no—
127Q8×D53—24no—
128Q8×D53—42no—
129D4×D53—24no—
130D4×D53—42no—
131C2×D212—21no—
132C2×C4×D53—41no—
133C2×C4×D53—41no—
134C2×C4×D53—41no—
135C2×C4×D53C2×D411yescharacteristic
136C2×Dic106—21no—
137C2×Dic106—41no—
138Q16×C53—21no—
139Q16×C53C22×C412yes—
140D8×C53—21no—
141D8×C53C22×C412yes—
142C2×C8×C53C2×D411yescharacteristic
143C2×C8×C53C2411yescharacteristic
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—41no—
147C22⋊C4×C53—41no—
148C42×C53—41no—
149C42×C53C2×D411yescharacteristic
150C22⋊Dic53—41no—
151#11719925—24no—
152#11719922—24no—
153#11719921—41no—
154#11719921—41no—
155#11719921—41no—
156#11719921—41no—
157#11719921D811yescharacteristic
158#11719921D811yescharacteristic
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—41no—
162#11719919—41no—
163#11719919—41no—
164#11719919D811yescharacteristic
165#11719919D811yescharacteristic
166C4×Dic53—41no—
167C4×Dic53—41no—
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—
186C22×D53—41no—
187C22×D53—44no—
188C22×D53—161no—
189C22×D53—161no—
190C22×D53C2×D811yescharacteristic
191Q8×C53—22no—
192Q8×C53—41no—
193D4×C53—22no—
194D4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—161no—
199C2×C4×C53C2×D811yescharacteristic
200C2×C4×C53C2×D811yescharacteristic
201C2×C4×C53C22×D411yescharacteristic
202C22⋊D53—44no—
203C2×Dic53—41no—
204C2×Dic53—41no—
205C2×Dic53—41no—
206C2×Dic53—161no—
207C2×Dic53C2×D811yescharacteristic
208C2×Dic53C2×D811yescharacteristic
209C2×Dic53C2×D811yescharacteristic
210D212—22no—
211D212—41no—
212C4×D53—44no—
213C4×D53—161no—
214C4×D53—161no—
215C4×D53C4×D412yes—
216C4×D53#13512yes—
217Dic106—22no—
218Dic106—41no—
219Dic106—44no—
220C8×C53—21no—
221C8×C53C23×C411yescharacteristic
222C8×C53C2×#4111yescharacteristic
223#88322—21no—
224#88322C4×D412yes—
225#33457—531nomaximal, Sylow
226C22×C53—42no—
227C22×C53—81no—
228C22×C53—81no—
229C22×C53C22×D811yescharacteristic
230C2×D53—21no—
231C2×D53—21no—
232C2×D53—42no—
233C2×D53—81no—
234C2×D53—81no—
235C2×D53—164no—
236C2×D53C4×D812yes—
237C4×C53—21no—
238C4×C53—21no—
239C4×C53—42no—
240C4×C53—81no—
241C4×C53—81no—
242C4×C53C2×C4×D411yescharacteristic
243C4×C53C2×#13511yescharacteristic
244Dic53—21no—
245Dic53—21no—
246Dic53—42no—
247Dic53—81no—
248Dic53—81no—
249Dic53C4×D812yes—
250C22×SD32—531no—
251#2950—531no—
252#2948—531no—
253#2175—531no—
254#2175—531no—
255#2175—531no—
256#2175—531no—
257C4×SD32—531no—
258C4×SD32—531no—
259C4×SD32—531no—
260C4×SD32D5311yescharacteristic, complemented
261#2158—531no—
262#2141—531no—
263#2140—531no—
264#2108—531no—
265C2×C53—82no—
266C2×C53C2×C4×D811yescharacteristic
267C2×C53#341611yescharacteristic
268C2×C53#341611yescharacteristic
269D53—22no—
270D53—82no—
271C22×Q16—531no—
272C22×D8—531no—
273#519—1061no—
274#518—1061no—
275C2×SD32—531no—
276C2×SD32—531no—
277C2×SD32—538no—
278C2×SD32—531no—
279C2×SD32C2×D5311yescharacteristic
280C2×M32—531no—
281C22×C16—531no—
282#442—1061no—
283#442—1061no—
284#441—1061no—
285#441—1061no—
286C4×Q16—531no—
287C4×Q16—531no—
288C4×Q16—531no—
289C4×Q16C2×D5311yescharacteristic, complemented
290C4×D8—531no—
291C4×D8—531no—
292C4×D8—531no—
293C4×D8C2×D5311yescharacteristic
294#427—531no—
295#427—531no—
296#419—1061no—
297#418—1061no—
298#405—531no—
299#367—531no—
300#367C2×D5311yescharacteristic, complemented
301#366—531no—
302#366—531no—
303#358—531no—
304#358—531no—
305#358—531no—
306#358C2×D5311yescharacteristic, complemented
307#357—531no—
308#357—531no—
309#357—531no—
310#357C2×D5311yescharacteristic
311#346—531no—
312#346—531no—
313C4×C16—531no—
314C4×C16C2×D5311yescharacteristic, complemented
315C53#3345711yescharacteristic, complemented, Sylow
316C22×Q8—1061no—
317C22×D4—1061no—
318C2×Q16—534no—
319C2×Q16—531no—
320C2×Q16—1062no—
321C2×Q16—1061no—
322C2×Q16C22×D5311yescharacteristic
323C2×D8—534no—
324C2×D8—531no—
325C2×D8—1062no—
326C2×D8—1061no—
327C2×D8C22×D5311yescharacteristic
328C2×M16—531no—
329C22×C8—531no—
330C4×Q8—21no—
331C4×Q8—1061no—
332C4×Q8—1061no—
333C4×Q8—1061no—
334C4×Q8—2121no—
335C4×Q8—2121no—
336C4×D4—21no—
337C4×D4—1061no—
338C4×D4—1061no—
339C4×D4—1061no—
340C4×D4—2121no—
341C4×D4—2121no—
342#117—531no—
343#117—1061no—
344#117—1061no—
345#117—2121no—
346#117—2121no—
347C2×C22⋊C4—2121no—
348C2×C22⋊C4—2121no—
349SD32—534no—
350SD32—534no—
351SD32—534no—
352SD32C4×D5314yescomplemented
353M32—1062no—
354C2×C16—531no—
355C2×C16—534no—
356C2×C16—531no—
357C2×C16C22×D5311yescharacteristic
358C2×C16C22×D5311yescharacteristic
359#107—531no—
360#107—531no—
361#107C22×D5311yescharacteristic
362#107C22×D5311yescharacteristic
363#106—1061no—
364#106—1061no—
365#103—21no—
366#103—1061no—
367#103—1061no—
368#103—1061no—
369#102—21no—
370#102—1061no—
371#102—1061no—
372#102—1061no—
373#97—531no—
374#97—531no—
375C4×C8—531no—
376C4×C8C22×D5311yescharacteristic
377C24—2121no—
378C2×Q8—21no—
379C2×Q8—1064no—
380C2×Q8—1061no—
381C2×Q8—2121no—
382C2×Q8—2122no—
383C2×D4—21no—
384C2×D4—1064no—
385C2×D4—1061no—
386C2×D4—2122no—
387C2×D4—2121no—
388C22×C4—41no—
389C22×C4—531no—
390C22×C4—2121no—
391C22×C4—2121no—
392C22×C4—2121no—
393C22×C4—2121no—
394Q16—21no—
395Q16—532no—
396Q16—1064no—
397Q16—1061no—
398Q16C2×C4×D5312yes—
399D8—21no—
400D8—532no—
401D8—1064no—
402D8—1061no—
403D8C2×C4×D5312yes—
404M16—1062no—
405C2×C8—531no—
406C2×C8—532no—
407C2×C8—532no—
408C2×C8—531no—
409C2×C8D4×D5311yescharacteristic
410C2×C8C23×D5311yescharacteristic
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—41no—
414C4⋊C4—531no—
415C4⋊C4—531no—
416C4⋊C4—1061no—
417C4⋊C4—1061no—
418C4⋊C4—2121no—
419C4⋊C4—2121no—
420C4⋊C4—2121no—
421C22⋊C4—41no—
422C22⋊C4—531no—
423C22⋊C4—531no—
424C22⋊C4—2121no—
425C22⋊C4—2121no—
426C22⋊C4—2121no—
427C22⋊C4—2121no—
428C22⋊C4—2121no—
429C42—41no—
430C42—531no—
431C42—2121no—
432C42—2121no—
433C42—2121no—
434C42—2121no—
435C42—2121no—
436C42D4×D5311yescharacteristic
437C16—21no—
438C16—532no—
439C16—1061no—
440C16C2×C4×D5312yes—
441C23—41no—
442C23—531no—
443C23—8481no—
444C23—8481no—
445C23—2124no—
446C23—2121no—
447Q8—22no—
448Q8—41no—
449Q8—1062no—
450Q8—2124no—
451Q8—2121no—
452D4—22no—
453D4—41no—
454D4—1062no—
455D4—2124no—
456D4—2121no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—161no—
461C2×C4—531no—
462C2×C4—532no—
463C2×C4—532no—
464C2×C4—531no—
465C2×C4—531no—
466C2×C4—8481no—
467C2×C4—8481no—
468C2×C4—8481no—
469C2×C4—2124no—
470C2×C4—2121no—
471C2×C4—2121no—
472C2×C4—2121no—
473C2×C4D8×D5311yescharacteristic
474C2×C4D8×D5311yescharacteristic
475C2×C4C2×D4×D5311yescharacteristic
476C8—21no—
477C8—532no—
478C8—1061no—
479C8C22×C4×D5311yescharacteristic
480C8#41×D5311yescharacteristic
481C22—42no—
482C22—81no—
483C22—81no—
484C22—532no—
485C22—8484no—
486C22—1061no—
487C22—1061no—
488C22—4241no—
489C22—4241no—
490C22—2122no—
491C22C2×D8×D5311yescharacteristic
492C4—21no—
493C4—21no—
494C4—42no—
495C4—81no—
496C4—81no—
497C4—532no—
498C4—1061no—
499C4—1061no—
500C4—4241no—
501C4—4241no—
502C4—2122no—
503C4C4×D4×D5311yescharacteristic
504C4#135×D5311yescharacteristic
505C2—82no—
506C2—1062no—
507C2—4242no—
508C2C4×D8×D5311yescharacteristic
509C2#2028785711yescharacteristic
510C2#2028785811yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC4×SD32×C53C2
socleC22×C53C22×D8
prime radicalC2×D8×D53C22
cosocleC2×C8×C53C24
hypercenterC4×SD32D53
hypocenterC53#33457
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×SD32

Complements

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

NH ≅ G/Ntype
2. #20283578C2semidirect
4. #20276582C2semidirect
5. #20276579C2semidirect
6. #20276577C2semidirect
7. #20276576C2semidirect
8. C4×#13133162C2semidirect
9. SD32×Dic53C2semidirect
10. #20276566C2semidirect
11. #20276498C2semidirect
12. #20276433C2semidirect
14. #20276310C2semidirect
15. C4×SD32×C53C2semidirect
23. SD32×D53C4semidirect
30. C4×Dic212C22semidirect
31. Q16×Dic53C22semidirect
32. #16810717C22semidirect
41. #16810451C22semidirect
42. #16810448C22semidirect
43. #16810448C22semidirect
44. #16810428C22semidirect
45. #16810427C22semidirect
46. #16810426C22semidirect
50. #16810399C22semidirect
51. #16810398C22semidirect
52. C16×Dic53C22semidirect
54. C4×Q16×C53C22semidirect
56. C53×#367C22semidirect
57. C53×#358C22semidirect
59. C4×C16×C53C22semidirect
65. Q16×D53C2×C4semidirect
89. #13133162C2×C4semidirect
90. #13133161C2×C4semidirect
91. #13133160C2×C4semidirect
95. C16×D53C2×C4semidirect
113. SD32×C53C2×C4semidirect
260. C4×SD32D53semidirect
289. C4×Q16C2×D53semidirect
300. #367C2×D53semidirect
306. #358C2×D53semidirect
314. C4×C16C2×D53semidirect
315. C53#33457semidirect
352. SD32C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×SD32×D532
#202835782
#202835622
#202765822
#202765792
#202765772
#202765762
C4×#131331622
SD32×Dic532
#202765662
#202764982
#202764332
#202764292
#202763102
C4×SD32×C532
#3345753