Finite Group Databasebeta

← #27538776

Subgroups — #27538776

Order 13568.

556 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285248C211yesmaximal, characteristic, complemented
2#20283562C211yesmaximal, characteristic
3#20276874C211yesmaximal, characteristic, complemented
4#20276873C211yesmaximal, characteristic, complemented
5#20276872C211yesmaximal, characteristic, complemented
6#20276871C211yesmaximal, characteristic, complemented
7#20276861C211yesmaximal, characteristic, complemented
8#20276850C211yesmaximal, characteristic, complemented
9#20276849C211yesmaximal, characteristic, complemented
10#20276502C211yesmaximal, characteristic, complemented
11#20276498C211yesmaximal, characteristic, complemented
12#20276472C211yesmaximal, characteristic, complemented
13#20276429C211yesmaximal, characteristic
14#357×D53C211yesmaximal, characteristic
15C53×#2226C211yesmaximal, characteristic, complemented
16C2×D8×D53C2211yescharacteristic
17#16811250—21no—
18#16811095—21no—
19C2×#13133156C2211yescharacteristic
20C2×C16×D53C2211yescharacteristic
21#16811031—21no—
22#16811029—21no—
23#16811027C2211yescharacteristic, complemented
24#16811025C2211yescharacteristic, complemented
25#16811024C2211yescharacteristic, complemented
26#16810733—21no—
27#16810732—21no—
28C4×D424C2211yescharacteristic
29D8×Dic53C2211yescharacteristic
30#16810705C2211yescharacteristic
31#16810664—21no—
32#16810660C2211yescharacteristic
33#16810660C2211yescharacteristic
34#107×D53C2211yescharacteristic
35#16810626—21no—
36#16810577C2211yescharacteristic
37#16810451C2211yescharacteristic, complemented
38#16810450C2211yescharacteristic, complemented
39#16810449C2211yescharacteristic, complemented
40#16810448C2211yescharacteristic, complemented
41#16810448C2211yescharacteristic, complemented
42#16810448C2211yescharacteristic, complemented
43C4⋊#11719908C2211yescharacteristic, complemented
44#16810441C2211yescharacteristic, complemented
45#16810441C2211yescharacteristic, complemented
46#16810425C2211yescharacteristic
47#16810425C2211yescharacteristic
48#16810424C2211yescharacteristic
49#16810424C2211yescharacteristic
50#16810423C2211yescharacteristic
51#16810423C2211yescharacteristic
52C53×#500C2211yescharacteristic, complemented
53C4×D8×C53C2211yescharacteristic
54C53×#367C2211yescharacteristic, complemented
55C53×#367C2211yescharacteristic, complemented
56C53×C4⋊C16C2211yescharacteristic, complemented
57C53×#357C2211yescharacteristic
58C53×#357C2211yescharacteristic
59C2×D4×D53—21no—
60C2×D424C2311yescharacteristic
61C2×#11719922—21no—
62D8×D53—22no—
63D8×D53Q814yescomplemented
64C2×#11719912D411yescharacteristic
65C2×C8×D53C2311yescharacteristic
66#13133216—21no—
67#13133215—41no—
68#13133209—21no—
69#13133208—21no—
70#13133207—41no—
71#13133195—41no—
72#13133193—41no—
73C4×D212—21no—
74D4×Dic53—21no—
75#13133179—21no—
76C4×C22⋊D53—41no—
77#13133177—41no—
78#13133175—21no—
79#13133175—21no—
80#13133174D411yescharacteristic
81C4⋊C4×D53—21no—
82C2×#11719921—41no—
83C22⋊C4×D53—41no—
84#13133156—22no—
85C2×#11719908C2311yescharacteristic
86C2×#11719908C2311yescharacteristic
87C16×D53—22no—
88#13133151C2311yescharacteristic
89#13133151C2311yescharacteristic
90#13133151C2311yescharacteristic
91#13133150—21no—
92#13133150—21no—
93#13133149—21no—
94#13133148—21no—
95#13133148—21no—
96#13133148—21no—
97#13133139—21no—
98#13133138—21no—
99#13133137—21no—
100#13133128D411yescharacteristic
101#13133127D411yescharacteristic
102C8×Dic53C2311yescharacteristic
103C2×D8×C53C2311yescharacteristic
104C4⋊Q8×C53—21no—
105C4×D4×C53—21no—
106C2×C16×C53C2311yescharacteristic
107C2×C16×C53C2311yescharacteristic
108#107×C53—21no—
109#107×C53C2311yescharacteristic
110#107×C53C2311yescharacteristic
111#107×C53C2311yescharacteristic
112#102×C53—21no—
113C4×C8×C53C2311yescharacteristic
114C23×D53—41no—
115C2×Q8×C53—41no—
116C2×D4×C53—21no—
117C22×C4×C53—41no—
118C2×C22⋊D53—41no—
119C22×Dic53—41no—
120D4×D53—24no—
121D4×D53—42no—
122C2×D212—21no—
123C2×C4×D53—41no—
124C2×C4×D53—41no—
125C2×C4×D53—41no—
126C2×C4×D53C2×D411yescharacteristic
127C2×Dic106—41no—
128D8×C53—21no—
129D8×C53C2×Q812yes—
130C2×C8×C53C2×D411yescharacteristic
131C2×C8×C53C2411yescharacteristic
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—41no—
136C22⋊C4×C53—41no—
137C42×C53C2×D411yescharacteristic
138C22⋊Dic53—41no—
139#11719922—24no—
140#11719921—41no—
141#11719921—41no—
142#11719921—41no—
143#11719921—41no—
144#11719921D811yescharacteristic
145#11719921D811yescharacteristic
146C4⋊Dic53—21no—
147C4⋊Dic53—21no—
148C4⋊Dic53—21no—
149C4⋊Dic53—41no—
150#11719919—41no—
151#11719919—41no—
152#11719919—41no—
153#11719919—41no—
154#11719919—41no—
155#11719919D811yescharacteristic
156#11719919D811yescharacteristic
157C4×Dic53—41no—
158C4×Dic53—41no—
159C4×Dic53C2×D411yescharacteristic
160C2×#88322C2×D411yescharacteristic
161C2×#88322C2×D411yescharacteristic
162D424—21no—
163D424C2×Q812yes—
164#11719912—22no—
165C8×D53C2×Q812yes—
166C8×D53#4112yes—
167C16×C53—21no—
168C16×C53—21no—
169#11719908—21no—
170#11719908—21no—
171C23×C53—41no—
172C22×D53—41no—
173C22×D53—44no—
174C22×D53—161no—
175C22×D53—161no—
176C22×D53C2×D811yescharacteristic
177Q8×C53—161no—
178D4×C53—22no—
179D4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—161no—
184C2×C4×C53C2×D811yescharacteristic
185C2×C4×C53C2×D811yescharacteristic
186C2×C4×C53C22×D411yescharacteristic
187C22⋊D53—44no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—161no—
192C2×Dic53C2×D811yescharacteristic
193C2×Dic53C2×D811yescharacteristic
194C2×Dic53C2×D811yescharacteristic
195D212—22no—
196D212—41no—
197C4×D53—161no—
198C4×D53—161no—
199C4×D53—161no—
200C4×D53C22⋊Q812yes—
201C4×D53#13712yes—
202Dic106—161no—
203C8×C53—21no—
204C8×C53C22×Q811yescharacteristic
205C8×C53C2×#4111yescharacteristic
206#88322—21no—
207#88322C22⋊Q812yes—
208#33631—531nomaximal, Sylow
209C22×C53—42no—
210C22×C53—81no—
211C22×C53—81no—
212C22×C53C22×D811yescharacteristic
213C2×D53—21no—
214C2×D53—21no—
215C2×D53—42no—
216C2×D53—81no—
217C2×D53—81no—
218C2×D53—164no—
219C2×D53#47412yes—
220C4×C53—21no—
221C4×C53—21no—
222C4×C53—81no—
223C4×C53—81no—
224C4×C53—81no—
225C4×C53C2×C22⋊Q811yescharacteristic
226C4×C53C2×#13711yescharacteristic
227Dic53—21no—
228Dic53—21no—
229Dic53—81no—
230Dic53—81no—
231Dic53—81no—
232Dic53#47412yes—
233#3164—531no—
234#2948—531no—
235#2228—531no—
236#2228—531no—
237#2228—531no—
238#2228—531no—
239#2226—531no—
240#2226D5311yescharacteristic, complemented
241#2224—531no—
242#2224—531no—
243#2159—531no—
244#2158—531no—
245#2151—531no—
246#2140—531no—
247C2×#357—531no—
248C2×C53—82no—
249C2×C53C2×#47411yescharacteristic
250C2×C53#341511yescharacteristic
251C2×C53#341611yescharacteristic
252D53—22no—
253D53—82no—
254C22×D8—531no—
255#544—1061no—
256#518—1061no—
257C2×M32—531no—
258C22×C16—531no—
259#501—1061no—
260#501—1061no—
261#500—531no—
262#500C2×D5311yescharacteristic, complemented
263#499—531no—
264#499—531no—
265#442—1061no—
266#442—1061no—
267C4×D8—531no—
268C4×D8—531no—
269C4×D8—531no—
270C4×D8C2×D5311yescharacteristic
271#428—1061no—
272#427—531no—
273#427—531no—
274C2×#107—531no—
275#418—1061no—
276#405—531no—
277#367—531no—
278#367—531no—
279#367C2×D5311yescharacteristic, complemented
280#367C2×D5311yescharacteristic, complemented
281#366—531no—
282#366—531no—
283#366—531no—
284#366—531no—
285C4⋊C16—531no—
286C4⋊C16—531no—
287C4⋊C16—531no—
288C4⋊C16C2×D5311yescharacteristic, complemented
289#357—531no—
290#357—531no—
291#357—531no—
292#357—531no—
293#357—531no—
294#357—531no—
295#357C2×D5311yescharacteristic
296#357C2×D5311yescharacteristic
297C53#3363111yescharacteristic, complemented, Sylow
298C22×D4—1061no—
299C2×D8—534no—
300C2×D8—531no—
301C2×D8—1062no—
302C2×D8—1061no—
303C2×D8C22×D5311yescharacteristic
304C2×M16—531no—
305C22×C8—531no—
306C4⋊Q8—21no—
307C4⋊Q8—1061no—
308C4⋊Q8—2121no—
309#125—1061no—
310#125—1061no—
311#125—2121no—
312C22⋊Q8—2121no—
313C22⋊Q8—2121no—
314C4×D4—21no—
315C4×D4—1061no—
316C4×D4—1061no—
317C4×D4—1061no—
318C4×D4—2121no—
319C4×D4—2121no—
320#117—531no—
321#117—1061no—
322#117—1061no—
323C2×C4⋊C4—1061no—
324C2×C22⋊C4—2121no—
325C2×C22⋊C4—2121no—
326M32—1062no—
327C2×C16—531no—
328C2×C16—531no—
329C2×C16—1062no—
330C2×C16C22×D5311yescharacteristic
331C2×C16C22×D5311yescharacteristic
332#107—21no—
333#107—531no—
334#107—531no—
335#107—531no—
336#107—1061no—
337#107—1061no—
338#107C22×D5311yescharacteristic
339#107C22×D5311yescharacteristic
340#107C22×D5311yescharacteristic
341#106—1061no—
342#106—1061no—
343#106—1061no—
344#106—1061no—
345#102—21no—
346#102—1061no—
347#102—1061no—
348#102—1061no—
349#97—531no—
350#97—531no—
351C4×C8—531no—
352C4×C8C22×D5311yescharacteristic
353C24—2121no—
354C2×Q8—41no—
355C2×Q8—2121no—
356C2×D4—21no—
357C2×D4—1064no—
358C2×D4—1061no—
359C2×D4—2122no—
360C2×D4—2121no—
361C22×C4—41no—
362C22×C4—531no—
363C22×C4—2121no—
364C22×C4—2121no—
365C22×C4—2121no—
366C22×C4—2121no—
367D8—21no—
368D8—532no—
369D8—1064no—
370D8—1061no—
371D8Q8×D5312yes—
372M16—1062no—
373C2×C8—531no—
374C2×C8—532no—
375C2×C8—532no—
376C2×C8—531no—
377C2×C8D4×D5311yescharacteristic
378C2×C8C23×D5311yescharacteristic
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—41no—
383C4⋊C4—531no—
384C4⋊C4—531no—
385C4⋊C4—1061no—
386C4⋊C4—1061no—
387C4⋊C4—1061no—
388C4⋊C4—2121no—
389C4⋊C4—2121no—
390C4⋊C4—2121no—
391C4⋊C4—2121no—
392C4⋊C4—2121no—
393C4⋊C4—2121no—
394C22⋊C4—41no—
395C22⋊C4—531no—
396C22⋊C4—531no—
397C22⋊C4—2121no—
398C22⋊C4—2121no—
399C22⋊C4—2121no—
400C22⋊C4—2121no—
401C22⋊C4—2121no—
402C42—531no—
403C42—2121no—
404C42—2121no—
405C42D4×D5311yescharacteristic
406C16—21no—
407C16—21no—
408C16—1061no—
409C16—1061no—
410C23—41no—
411C23—531no—
412C23—8481no—
413C23—8481no—
414C23—2124no—
415C23—2121no—
416Q8—161no—
417Q8—8481no—
418D4—22no—
419D4—41no—
420D4—1062no—
421D4—2124no—
422D4—2121no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—161no—
427C2×C4—531no—
428C2×C4—532no—
429C2×C4—532no—
430C2×C4—531no—
431C2×C4—531no—
432C2×C4—8481no—
433C2×C4—8481no—
434C2×C4—8481no—
435C2×C4—8481no—
436C2×C4—2121no—
437C2×C4—2121no—
438C2×C4—2121no—
439C2×C4D8×D5311yescharacteristic
440C2×C4D8×D5311yescharacteristic
441C2×C4C2×D4×D5311yescharacteristic
442C8—21no—
443C8—532no—
444C8—1061no—
445C8C2×Q8×D5311yescharacteristic
446C8#41×D5311yescharacteristic
447C22—42no—
448C22—81no—
449C22—81no—
450C22—532no—
451C22—8484no—
452C22—1061no—
453C22—1061no—
454C22—4241no—
455C22—4241no—
456C22—2122no—
457C22C2×D8×D5311yescharacteristic
458C4—21no—
459C4—21no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—532no—
464C4—1061no—
465C4—1061no—
466C4—4241no—
467C4—4241no—
468C4—4241no—
469C4C22⋊Q8×D5311yescharacteristic
470C4#137×D5311yescharacteristic
471C2—82no—
472C2—1062no—
473C2—4242no—
474C2#474×D5311yescharacteristic
475C2#2028785511yescharacteristic
476C2#2028785711yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2226C2
socleC22×C53C22×D8
prime radicalC2×D8×D53C22
hypercenter#2226D53
hypocenterC53#33631
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2226

Complements

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

NH ≅ G/Ntype
1. #20285248C2semidirect
3. #20276874C2semidirect
4. #20276873C2semidirect
5. #20276872C2semidirect
6. #20276871C2semidirect
7. #20276861C2semidirect
8. #20276850C2semidirect
9. #20276849C2semidirect
10. #20276502C2semidirect
11. #20276498C2semidirect
12. #20276472C2semidirect
15. C53×#2226C2semidirect
23. #16811027C22semidirect
24. #16811025C22semidirect
25. #16811024C22semidirect
37. #16810451C22semidirect
38. #16810450C22semidirect
39. #16810449C22semidirect
40. #16810448C22semidirect
41. #16810448C22semidirect
42. #16810448C22semidirect
43. C4⋊#11719908C22semidirect
44. #16810441C22semidirect
45. #16810441C22semidirect
52. C53×#500C22semidirect
54. C53×#367C22semidirect
55. C53×#367C22semidirect
56. C53×C4⋊C16C22semidirect
63. D8×D53Q8semidirect
240. #2226D53semidirect
262. #500C2×D53semidirect
279. #367C2×D53semidirect
280. #367C2×D53semidirect
288. C4⋊C16C2×D53semidirect
297. C53#33631semidirect

Maximal subgroups

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

subgroupindex
#202852482
#202835622
#202768742
#202768732
#202768722
#202768712
#202768612
#202768502
#202768492
#202765022
#202764982
#202764722
#202764292
#357×D532
C53×#22262
#3363153