Finite Group Databasebeta

← #27538889

Subgroups — #27538889

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284292C211yesmaximal, characteristic
2#427×D53C211yesmaximal, characteristic, complemented
3#20276922C211yesmaximal, characteristic, complemented
4#20276921C211yesmaximal, characteristic, complemented
5#20276916C211yesmaximal, characteristic, complemented
6#20276911C211yesmaximal, characteristic, complemented
7#20276905C211yesmaximal, characteristic, complemented
8#20276904C211yesmaximal, characteristic, complemented
9#20276899C211yesmaximal, characteristic, complemented
10#20276894C211yesmaximal, characteristic, complemented
11#20276499C211yesmaximal, characteristic, complemented
12#20276497C211yesmaximal, characteristic, complemented
13#20276422C211yesmaximal, characteristic, complemented
14C2×#16810402C211yesmaximal, characteristic, complemented
15C53×#2138C211yesmaximal, characteristic, complemented
16C22×C8×D53C2211yescharacteristic
17#16811130—21no—
18#117×D53—21no—
19C22×#11719908C2211yescharacteristic
20#16810861C2211yescharacteristic
21#16810861C2211yescharacteristic
22#16810856C2211yescharacteristic
23#16810856C2211yescharacteristic
24#16810849—21no—
25#16810849—21no—
26#16810661C2211yescharacteristic
27#16810661C2211yescharacteristic
28#16810660C2211yescharacteristic
29#16810660C2211yescharacteristic
30#16810659—21no—
31#16810658—21no—
32#107×D53C2211yescharacteristic
33#107×D53C2211yescharacteristic
34#106×D53—21no—
35#16810619—21no—
36#16810451C2211yescharacteristic, complemented
37#16810451C2211yescharacteristic, complemented
38#16810450C2211yescharacteristic, complemented
39#16810450C2211yescharacteristic, complemented
40#16810449C2211yescharacteristic, complemented
41#16810449C2211yescharacteristic, complemented
42#16810448C2211yescharacteristic, complemented
43#16810448C2211yescharacteristic, complemented
44#16810427C2211yescharacteristic, complemented
45#16810427C2211yescharacteristic, complemented
46#16810424C2211yescharacteristic, complemented
47#16810424C2211yescharacteristic, complemented
48#16810402C2211yescharacteristic
49#16810402C2211yescharacteristic
50#16810402C2211yescharacteristic
51#16810402C2211yescharacteristic
52C2×#135×C53C2211yescharacteristic
53C22×C16×C53C2211yescharacteristic
54C53×#427C2211yescharacteristic
55C53×#358C2211yescharacteristic, complemented
56C53×#358C2211yescharacteristic, complemented
57C53×#357C2211yescharacteristic, complemented
58C53×#357C2211yescharacteristic, complemented
59C22×C4×D53D411yescharacteristic
60C22×#88322D411yescharacteristic
61C2×C8×D53—22no—
62C2×C8×D53—22no—
63C2×C8×D53D411yescharacteristic
64C2×C8×D53D411yescharacteristic
65C2×C8×D53C2311yescharacteristic
66C2×C8×D53C2311yescharacteristic
67#13133198—41no—
68#13133193—21no—
69#13133193—21no—
70#13133188—21no—
71#13133188—21no—
72#13133185—41no—
73C4×C22⋊D53—41no—
74C4×C22⋊D53—41no—
75#13133176—21no—
76#13133176—21no—
77#13133175—21no—
78#13133175—21no—
79#13133174—41no—
80#13133173—41no—
81C4⋊C4×D53—21no—
82C4⋊C4×D53—21no—
83C22⋊C4×D53—41no—
84C42×D53—41no—
85C2×#11719908—22no—
86C2×#11719908C2311yescharacteristic
87C2×#11719908C2311yescharacteristic
88#13133151C2311yescharacteristic
89#13133151C2311yescharacteristic
90#13133151C2311yescharacteristic
91#13133151C2311yescharacteristic
92#13133150—21no—
93#13133150—21no—
94#13133149—21no—
95#13133149—21no—
96#13133148—21no—
97#13133148—21no—
98#13133141—21no—
99#13133141—21no—
100#13133138—21no—
101#13133138—21no—
102C2×#41×C53—21no—
103#135×C53—22no—
104#135×C53—22no—
105C2×Q16×C53C2311yescharacteristic
106C2×SD16×C53—21no—
107C2×D8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109#117×C53—21no—
110C2×C16×C53—22no—
111C2×C16×C53C2311yescharacteristic
112C2×C16×C53C2311yescharacteristic
113#107×C53C2311yescharacteristic
114#107×C53C2311yescharacteristic
115#106×C53—21no—
116C23×D53D811yescharacteristic
117#41×C53—42no—
118#41×C53—42no—
119C2×Q8×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—41no—
122C22×C4×C53—41no—
123C22×C4×C53C2×D411yescharacteristic
124C2×C22⋊D53—41no—
125C2×C22⋊D53—41no—
126C22×Dic53D811yescharacteristic
127C2×C4×D53—22no—
128C2×C4×D53—22no—
129C2×C4×D53—41no—
130C2×C4×D53—41no—
131C2×C4×D53D811yescharacteristic
132C2×C4×D53D811yescharacteristic
133C2×C4×D53C2×D411yescharacteristic
134C2×C4×D53C2×D411yescharacteristic
135Q16×C53—21no—
136Q16×C53—21no—
137SD16×C53—42no—
138D8×C53—21no—
139D8×C53—21no—
140C2×C8×C53C2×D411yescharacteristic
141C2×C8×C53#4112yes—
142C2×C8×C53#4112yes—
143C2×C8×C53C2411yescharacteristic
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C22⋊C4×C53—41no—
147C42×C53—41no—
148C22⋊Dic53—41no—
149C22⋊Dic53—41no—
150C22⋊Dic53—41no—
151C22⋊Dic53—41no—
152#11719921—41no—
153#11719921—41no—
154#11719921—41no—
155#11719921—41no—
156C4⋊Dic53—21no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160#11719919—41no—
161#11719919—41no—
162#11719919—41no—
163#11719919—41no—
164C4×Dic53—41no—
165C4×Dic53—41no—
166C4×Dic53—41no—
167C4×Dic53—41no—
168C2×#88322—22no—
169C2×#88322C2×D411yescharacteristic
170C2×#88322C2×D411yescharacteristic
171C8×D53—22no—
172C8×D53—22no—
173C8×D53—21no—
174C8×D53—21no—
175C8×D53—21no—
176C8×D53—21no—
177C16×C53—21no—
178C16×C53—21no—
179#11719908—21no—
180#11719908—21no—
181C23×C53—41no—
182C23×C53C2×D811yescharacteristic
183C22×D53—22no—
184C22×D53—22no—
185C22×D53—41no—
186C22×D53C2×D811yescharacteristic
187C22×D53C2×D811yescharacteristic
188Q8×C53—41no—
189Q8×C53—41no—
190D4×C53—41no—
191D4×C53—41no—
192D4×C53—82no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—82no—
197C2×C4×C53#12312yes—
198C2×C4×C53C2×D811yescharacteristic
199C2×C4×C53#13712yes—
200C2×C4×C53C22×D411yescharacteristic
201C22⋊D53—82no—
202C22⋊D53—82no—
203C2×Dic53—22no—
204C2×Dic53—41no—
205C2×Dic53—41no—
206C2×Dic53—41no—
207C2×Dic53—41no—
208C2×Dic53C2×D811yescharacteristic
209C2×Dic53C2×D811yescharacteristic
210C4×D53—21no—
211C4×D53—21no—
212C4×D53—21no—
213C4×D53—21no—
214C4×D53—22no—
215C4×D53—22no—
216C4×D53—82no—
217C4×D53—82no—
218C8×C53#12312yes—
219C8×C53C2×#4111yescharacteristic
220C8×C53C2×#4111yescharacteristic
221#88322—21no—
222#88322—21no—
223#33645—531nomaximal, Sylow
224C22×C53—21no—
225C22×C53—21no—
226C22×C53—81no—
227C22×C53—81no—
228C22×C53—81no—
229C22×C53#48012yes—
230C22×C53C22×D811yescharacteristic
231C2×D53—21no—
232C2×D53—21no—
233C2×D53—21no—
234C2×D53—21no—
235C2×D53—21no—
236C2×D53—44no—
237C2×D53#50812yes—
238C4×C53—81no—
239C4×C53—81no—
240C4×C53—81no—
241C4×C53#48012yes—
242C4×C53C2×#12311yescharacteristic
243C4×C53C2×#13711yescharacteristic
244Dic53—21no—
245Dic53—21no—
246Dic53—81no—
247Dic53—81no—
248Dic53—81no—
249Dic53—81no—
250#3050—531no—
251C2×#427—531no—
252#2237—531no—
253#2237—531no—
254#2236—531no—
255#2235—531no—
256#2234—531no—
257#2234—531no—
258#2233—531no—
259#2232—531no—
260#2158—531no—
261#2158—531no—
262#2138—531no—
263#2138D5311yescharacteristic, complemented
264C2×C22⋊C16—531no—
265C2×C53—22no—
266C2×C53—161no—
267C2×C53C2×#48011yescharacteristic
268C2×C53C2×#50811yescharacteristic
269C2×C53#341411yescharacteristic
270D53—22no—
271D53—41no—
272C2×#135C2×D5311yescharacteristic
273C23×C8—531no—
274#525—1061no—
275C2×#117—1061no—
276C22×C16—531no—
277C22×C16C2×D5311yescharacteristic
278#467—531no—
279#467—531no—
280#466—531no—
281#466—531no—
282#465—1061no—
283#465—1061no—
284#427—531no—
285#427—531no—
286#427—531no—
287#427—531no—
288#427—1061no—
289#427—1061no—
290#427C2×D5311yescharacteristic
291C2×#107—531no—
292C2×#107—531no—
293C2×#106—1061no—
294#416—1061no—
295#367—531no—
296#367—531no—
297#367—531no—
298#367—531no—
299#366—531no—
300#366—531no—
301#366—531no—
302#366—531no—
303#358—531no—
304#358—531no—
305#358C2×D5311yescharacteristic, complemented
306#358C2×D5311yescharacteristic, complemented
307#357—531no—
308#357—531no—
309#357C2×D5311yescharacteristic, complemented
310#357C2×D5311yescharacteristic, complemented
311C22⋊C16—531no—
312C22⋊C16—531no—
313C22⋊C16—531no—
314C22⋊C16—531no—
315C53#3364511yescharacteristic, complemented, Sylow
316C2×#41—21no—
317C23×C4—531no—
318#135—22no—
319#135—22no—
320C2×Q16C22×D5311yescharacteristic
321C2×SD16—21no—
322C2×D8C22×D5311yescharacteristic
323C22×C8—531no—
324C22×C8—531no—
325C22×C8—531no—
326C22×C8—531no—
327C22×C8—531no—
328C22×C8—1062no—
329C22×C8—1062no—
330C22×C8C22×D5311yescharacteristic
331#123—2121no—
332C22⋊Q8—1061no—
333C22⋊Q8—1061no—
334#121—1061no—
335#121—1061no—
336#120—2121no—
337C4×D4—2121no—
338C4×D4—2121no—
339#117—21no—
340#117—1061no—
341#117—1061no—
342#117—1061no—
343#117—1061no—
344#117—2121no—
345#117—2121no—
346C2×C4⋊C4—1061no—
347C2×C4⋊C4—1061no—
348C2×C22⋊C4—2121no—
349C2×C42—2121no—
350C2×C16—22no—
351C2×C16—531no—
352C2×C16—531no—
353C2×C16—1062no—
354C2×C16C22×D5311yescharacteristic
355C2×C16C22×D5311yescharacteristic
356#107—531no—
357#107—531no—
358#107—531no—
359#107—531no—
360#107—1061no—
361#107—1061no—
362#107C22×D5311yescharacteristic
363#107C22×D5311yescharacteristic
364#106—21no—
365#106—1061no—
366#106—1061no—
367#106—1061no—
368#106—1061no—
369#103—1061no—
370#103—1061no—
371#102—1061no—
372#102—1061no—
373C24—531no—
374#41—42no—
375#41—42no—
376C2×Q8—21no—
377C2×D4—21no—
378C2×D4—41no—
379C2×D4—2121no—
380C2×D4—2121no—
381C22×C4—41no—
382C22×C4—531no—
383C22×C4—531no—
384C22×C4—531no—
385C22×C4—531no—
386C22×C4—531no—
387C22×C4—1062no—
388C22×C4—1062no—
389C22×C4—2121no—
390C22×C4—2121no—
391C22×C4D4×D5311yescharacteristic
392Q16—21no—
393Q16—21no—
394SD16—42no—
395D8—21no—
396D8—21no—
397C2×C8—531no—
398C2×C8—531no—
399C2×C8—1062no—
400C2×C8—1062no—
401C2×C8—1062no—
402C2×C8—1061no—
403C2×C8—1061no—
404C2×C8—1061no—
405C2×C8—1061no—
406C2×C8#1171994512yes—
407C2×C8D4×D5311yescharacteristic
408C2×C8#1171994712yes—
409C2×C8C23×D5311yescharacteristic
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—1061no—
413C4⋊C4—1061no—
414C4⋊C4—1061no—
415C4⋊C4—1061no—
416C4⋊C4—2121no—
417C4⋊C4—2121no—
418C4⋊C4—2121no—
419C4⋊C4—2121no—
420C22⋊C4—41no—
421C22⋊C4—2121no—
422C22⋊C4—2121no—
423C22⋊C4—2121no—
424C22⋊C4—2121no—
425C22⋊C4—2121no—
426C22⋊C4—2121no—
427C22⋊C4—2121no—
428C22⋊C4—2121no—
429C42—41no—
430C42—2121no—
431C42—2121no—
432C42—2121no—
433C42—2121no—
434C16—21no—
435C16—21no—
436C16—1061no—
437C16—1061no—
438C23—41no—
439C23—531no—
440C23—531no—
441C23—1062no—
442C23—1062no—
443C23—2121no—
444C23D8×D5311yescharacteristic
445Q8—41no—
446Q8—41no—
447D4—41no—
448D4—41no—
449D4—82no—
450D4—4242no—
451D4—4242no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—82no—
456C2×C4—531no—
457C2×C4—531no—
458C2×C4—1062no—
459C2×C4—1061no—
460C2×C4—1061no—
461C2×C4—1061no—
462C2×C4—1061no—
463C2×C4—1062no—
464C2×C4—1062no—
465C2×C4—4242no—
466C2×C4—4242no—
467C2×C4—2121no—
468C2×C4—2121no—
469C2×C4—2121no—
470C2×C4—2121no—
471C2×C4#1313319912yes—
472C2×C4D8×D5311yescharacteristic
473C2×C4#1313324612yes—
474C2×C4C2×D4×D5311yescharacteristic
475C8—1061no—
476C8—1061no—
477C8#1313319912yes—
478C8C2×#1171994711yescharacteristic
479C8C2×#1171994511yescharacteristic
480C22—21no—
481C22—21no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—532no—
486C22—1061no—
487C22—1061no—
488C22—1061no—
489C22—1061no—
490C22—1061no—
491C22—2124no—
492C22#1681093512yes—
493C22C2×D8×D5311yescharacteristic
494C4—81no—
495C4—81no—
496C4—81no—
497C4—1061no—
498C4—1061no—
499C4—4241no—
500C4—4241no—
501C4—4241no—
502C4—4241no—
503C4#1681093512yes—
504C4C2×#1313319911yescharacteristic
505C4C2×#1313324611yescharacteristic
506C2—22no—
507C2—161no—
508C2—1062no—
509C2—2121no—
510C2C2×#1681093511yescharacteristic
511C2#508×D5311yescharacteristic
512C2#2028784811yescharacteristic
subgroupG/H
centerC2×C4D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2138C2
socleC22×C53C22×D8
prime radical#20284292C2
hypercenter#2138D53
hypocenterC53#33645
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
2. #427×D53C2semidirect
3. #20276922C2semidirect
4. #20276921C2semidirect
5. #20276916C2semidirect
6. #20276911C2semidirect
7. #20276905C2semidirect
8. #20276904C2semidirect
9. #20276899C2semidirect
10. #20276894C2semidirect
11. #20276499C2semidirect
12. #20276497C2semidirect
13. #20276422C2semidirect
14. C2×#16810402C2semidirect
15. C53×#2138C2semidirect
36. #16810451C22semidirect
37. #16810451C22semidirect
38. #16810450C22semidirect
39. #16810450C22semidirect
40. #16810449C22semidirect
41. #16810449C22semidirect
42. #16810448C22semidirect
43. #16810448C22semidirect
44. #16810427C22semidirect
45. #16810427C22semidirect
46. #16810424C22semidirect
47. #16810424C22semidirect
55. C53×#358C22semidirect
56. C53×#358C22semidirect
57. C53×#357C22semidirect
58. C53×#357C22semidirect
263. #2138D53semidirect
305. #358C2×D53semidirect
306. #358C2×D53semidirect
309. #357C2×D53semidirect
310. #357C2×D53semidirect
315. C53#33645semidirect

Maximal subgroups

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

subgroupindex
#202842922
#427×D532
#202769222
#202769212
#202769162
#202769112
#202769052
#202769042
#202768992
#202768942
#202764992
#202764972
#202764222
C2×#168104022
C53×#21382
#3364553