Finite Group Databasebeta

← #27538745

Subgroups — #27538745

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285228C211yesmaximal, characteristic, complemented
2C4×D8×D53C211yesmaximal, characteristic
3#20276873C211yesmaximal, characteristic, complemented
4#20276872C211yesmaximal, characteristic, complemented
5#20276869C211yesmaximal, characteristic, complemented
6#20276861C211yesmaximal, characteristic, complemented
7#20276860C211yesmaximal, characteristic, complemented
8#20276851C211yesmaximal, characteristic, complemented
9#20276850C211yesmaximal, characteristic, complemented
10#20276502C211yesmaximal, characteristic, complemented
11#20276500C211yesmaximal, characteristic, complemented
12#20276469C211yesmaximal, characteristic, complemented
13#20276429C211yesmaximal, characteristic
14#20276429C211yesmaximal, characteristic
15C53×#2228C211yesmaximal, characteristic, complemented
16C2×D8×D53C2211yescharacteristic
17#16811179—21no—
18C4×D4×D53—21no—
19C2×#13133156C2211yescharacteristic
20C2×#13133156C2211yescharacteristic
21#16811027C2211yescharacteristic, complemented
22#16811027C2211yescharacteristic, complemented
23#16811025C2211yescharacteristic, complemented
24#16811022—21no—
25#16811020—21no—
26C4×D424C2211yescharacteristic
27D8×Dic53C2211yescharacteristic
28#16810705C2211yescharacteristic
29C4×#11719922—21no—
30#16810703—21no—
31#16810660C2211yescharacteristic
32#16810660C2211yescharacteristic
33#16810658—21no—
34#107×D53C2211yescharacteristic
35#102×D53—21no—
36C4×C8×D53C2211yescharacteristic
37#16810451C2211yescharacteristic, complemented
38#16810450C2211yescharacteristic, complemented
39#16810450C2211yescharacteristic, complemented
40#16810449C2211yescharacteristic, 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×#499C2211yescharacteristic, complemented
53C4×D8×C53C2211yescharacteristic
54C53×#367C2211yescharacteristic, complemented
55C53×#366C2211yescharacteristic, complemented
56C53×C4⋊C16C2211yescharacteristic, complemented
57C53×#357C2211yescharacteristic
58C53×#357C2211yescharacteristic
59C2×D4×D53—21no—
60C22×C4×D53—41no—
61C2×D424C2311yescharacteristic
62C2×#11719922—21no—
63D8×D53—22no—
64D8×D53Q814yescomplemented
65C2×C8×D53D411yescharacteristic
66C2×C8×D53C2311yescharacteristic
67#13133216—21no—
68#13133216—21no—
69#13133209—21no—
70#13133195—41no—
71#13133195—41no—
72#13133182—41no—
73#13133182—41no—
74C4×D212—21no—
75D4×Dic53—21no—
76#13133179—21no—
77C4×C22⋊D53—41no—
78#13133177—41no—
79#13133175—21no—
80#13133175—21no—
81C4⋊C4×D53—21no—
82C22⋊C4×D53—41no—
83C42×D53D411yescharacteristic
84#13133156—22no—
85#13133156—22no—
86C2×#11719908C2311yescharacteristic
87C2×#11719908C2311yescharacteristic
88#13133151C2311yescharacteristic
89#13133151C2311yescharacteristic
90#13133151C2311yescharacteristic
91#13133150—21no—
92#13133150—21no—
93#13133150—21no—
94#13133149—21no—
95#13133148—21no—
96#13133148—21no—
97#13133139—21no—
98#13133138—21no—
99#13133137—21no—
100C4×#88322D411yescharacteristic
101C8×Dic53D411yescharacteristic
102C8×Dic53C2311yescharacteristic
103C2×D8×C53C2311yescharacteristic
104#125×C53—21no—
105C4×D4×C53—21no—
106C2×C16×C53C2311yescharacteristic
107C2×C16×C53C2311yescharacteristic
108#107×C53C2311yescharacteristic
109#107×C53C2311yescharacteristic
110#107×C53C2311yescharacteristic
111#106×C53—21no—
112#102×C53—21no—
113C4×C8×C53C2311yescharacteristic
114C23×D53—41no—
115C2×D4×C53—21no—
116C22×C4×C53—41no—
117C2×C22⋊D53—41no—
118C22×Dic53—41no—
119D4×D53—24no—
120D4×D53—42no—
121C2×D212—21no—
122C2×C4×D53—41no—
123C2×C4×D53—41no—
124C2×C4×D53—41no—
125C2×C4×D53—84no—
126C2×C4×D53D811yescharacteristic
127C2×C4×D53D811yescharacteristic
128C2×C4×D53C2×D411yescharacteristic
129C2×Dic106—41no—
130C2×Dic106—41no—
131D8×C53—21no—
132D8×C53C2×Q812yes—
133C2×C8×C53C2×D411yescharacteristic
134C2×C8×C53C2411yescharacteristic
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—41no—
139C4⋊C4×C53—41no—
140C22⋊C4×C53—41no—
141C42×C53C2×D411yescharacteristic
142C22⋊Dic53—41no—
143#11719922—24no—
144#11719921—41no—
145#11719921—41no—
146#11719921—41no—
147#11719921—41no—
148C4⋊Dic53—21no—
149C4⋊Dic53—21no—
150C4⋊Dic53—21no—
151#11719919—41no—
152#11719919—41no—
153#11719919—41no—
154#11719919—41no—
155#11719919—41no—
156C4×Dic53—41no—
157C4×Dic53—41no—
158C4×Dic53D811yescharacteristic
159C4×Dic53D811yescharacteristic
160C4×Dic53C2×D411yescharacteristic
161C2×#88322C2×D411yescharacteristic
162C2×#88322C2×D411yescharacteristic
163D424—21no—
164D424C2×Q812yes—
165C8×D53—22no—
166C8×D53C2×Q812yes—
167C8×D53#4112yes—
168C16×C53—21no—
169C16×C53—21no—
170#11719908—21no—
171#11719908—21no—
172C23×C53—41no—
173C22×D53—41no—
174C22×D53—44no—
175C22×D53—82no—
176C22×D53—82no—
177C22×D53C2×D811yescharacteristic
178D4×C53—22no—
179D4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—82no—
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—82no—
192C2×Dic53C2×D811yescharacteristic
193C2×Dic53C2×D811yescharacteristic
194C2×Dic53C2×D811yescharacteristic
195D212—22no—
196D212—41no—
197C4×D53—22no—
198C4×D53—41no—
199C4×D53—82no—
200C4×D53—82no—
201C4×D53—82no—
202C4×D53C22⋊Q812yes—
203C4×D53#13712yes—
204Dic106—82no—
205Dic106—82no—
206C8×C53—21no—
207C8×C53C22×Q811yescharacteristic
208C8×C53C2×#4111yescharacteristic
209#88322—21no—
210#88322C22⋊Q812yes—
211#33627—531nomaximal, Sylow
212C22×C53—42no—
213C22×C53—81no—
214C22×C53—81no—
215C22×C53C22×D811yescharacteristic
216C2×D53—21no—
217C2×D53—21no—
218C2×D53—42no—
219C2×D53—88no—
220C2×D53—81no—
221C2×D53—81no—
222C2×D53#47412yes—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—81no—
226C4×C53—81no—
227C4×C53—81no—
228C4×C53C2×C22⋊Q811yescharacteristic
229C4×C53C2×#13711yescharacteristic
230Dic53—21no—
231Dic53—81no—
232Dic53—81no—
233Dic53—81no—
234Dic53#47412yes—
235Dic53#50912yes—
236#3160—531no—
237C2×C4×D8—531no—
238#2228—531no—
239#2228—531no—
240#2228—531no—
241#2228D5311yescharacteristic, complemented
242#2226—531no—
243#2226—531no—
244#2224—531no—
245#2224—531no—
246#2159—531no—
247#2159—531no—
248#2150—531no—
249#2140—531no—
250#2140—531no—
251C2×C53—82no—
252C2×C53C2×#47411yescharacteristic
253C2×C53C2×#50911yescharacteristic
254C2×C53#341411yescharacteristic
255D53—22no—
256D53—82no—
257C22×D8—531no—
258#533—1061no—
259C2×C4×D4—1061no—
260C2×M32—531no—
261C2×M32—531no—
262#500—531no—
263#500—531no—
264#499—531no—
265#499—1061no—
266#499C2×D5311yescharacteristic, complemented
267#498—1061no—
268C4×D8—531no—
269C4×D8—531no—
270C4×D8—531no—
271C4×D8—1061no—
272C4×D8—1061no—
273C4×D8C2×D5311yescharacteristic
274#427—531no—
275#427—531no—
276#427—1061no—
277C2×#107—531no—
278C2×#102—1061no—
279C2×C4×C8—531no—
280#367—531no—
281#367—531no—
282#367—531no—
283#367C2×D5311yescharacteristic, complemented
284#366—531no—
285#366—531no—
286#366—531no—
287#366C2×D5311yescharacteristic, complemented
288C4⋊C16—531no—
289C4⋊C16—531no—
290C4⋊C16—531no—
291C4⋊C16C2×D5311yescharacteristic, complemented
292#357—531no—
293#357—531no—
294#357—531no—
295#357—531no—
296#357—531no—
297#357—531no—
298#357C2×D5311yescharacteristic
299#357C2×D5311yescharacteristic
300C53#3362711yescharacteristic, complemented, Sylow
301C22×D4—1061no—
302C23×C4—2121no—
303C2×D8—534no—
304C2×D8—531no—
305C2×D8—1062no—
306C2×D8—1061no—
307C2×D8C22×D5311yescharacteristic
308C22×C8—531no—
309C22×C8—531no—
310C4⋊Q8—1061no—
311C4⋊Q8—1061no—
312#125—21no—
313#125—1061no—
314C22⋊Q8—2121no—
315C22⋊Q8—2121no—
316C4×Q8—2121no—
317C4×Q8—2121no—
318C4×D4—21no—
319C4×D4—1061no—
320C4×D4—1061no—
321C4×D4—1061no—
322C4×D4—2121no—
323C4×D4—2121no—
324#117—1061no—
325#117—1061no—
326C2×C4⋊C4—1061no—
327C2×C22⋊C4—2121no—
328C2×C42—531no—
329M32—1062no—
330M32—1062no—
331C2×C16—531no—
332C2×C16—531no—
333C2×C16C22×D5311yescharacteristic
334C2×C16C22×D5311yescharacteristic
335#107—531no—
336#107—531no—
337#107—531no—
338#107—1061no—
339#107—1061no—
340#107—1061no—
341#107C22×D5311yescharacteristic
342#107C22×D5311yescharacteristic
343#107C22×D5311yescharacteristic
344#106—21no—
345#106—1061no—
346#106—1061no—
347#106—1061no—
348#102—21no—
349#102—1061no—
350#102—1061no—
351#102—1061no—
352C4×C8—531no—
353C4×C8—531no—
354C4×C8—531no—
355C4×C8C22×D5311yescharacteristic
356C24—2121no—
357C2×Q8—2121no—
358C2×Q8—2121no—
359C2×D4—21no—
360C2×D4—1064no—
361C2×D4—1061no—
362C2×D4—2122no—
363C2×D4—2121no—
364C22×C4—41no—
365C22×C4—531no—
366C22×C4—531no—
367C22×C4—531no—
368C22×C4—4244no—
369C22×C4—2121no—
370C22×C4—2121no—
371C22×C4—2121no—
372C22×C4—2121no—
373D8—21no—
374D8—532no—
375D8—1064no—
376D8—1061no—
377D8Q8×D5312yes—
378C2×C8—531no—
379C2×C8—532no—
380C2×C8—532no—
381C2×C8—531no—
382C2×C8—1062no—
383C2×C8D4×D5311yescharacteristic
384C2×C8C23×D5311yescharacteristic
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—1061no—
391C4⋊C4—1061no—
392C4⋊C4—1061no—
393C4⋊C4—2121no—
394C4⋊C4—2121no—
395C4⋊C4—2121no—
396C4⋊C4—2121no—
397C4⋊C4—2121no—
398C22⋊C4—41no—
399C22⋊C4—2121no—
400C22⋊C4—2121no—
401C22⋊C4—2121no—
402C22⋊C4—2121no—
403C22⋊C4—2121no—
404C42—531no—
405C42—531no—
406C42—531no—
407C42—2121no—
408C42—2121no—
409C42D4×D5311yescharacteristic
410C16—21no—
411C16—21no—
412C16—1061no—
413C16—1061no—
414C23—41no—
415C23—531no—
416C23—4242no—
417C23—4242no—
418C23—2124no—
419C23—2121no—
420Q8—4242no—
421Q8—4242no—
422D4—22no—
423D4—41no—
424D4—1062no—
425D4—2124no—
426D4—2121no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—82no—
431C2×C4—531no—
432C2×C4—531no—
433C2×C4—532no—
434C2×C4—532no—
435C2×C4—531no—
436C2×C4—1062no—
437C2×C4—4242no—
438C2×C4—4242no—
439C2×C4—4242no—
440C2×C4—4242no—
441C2×C4—2121no—
442C2×C4—2121no—
443C2×C4—2121no—
444C2×C4—2121no—
445C2×C4D8×D5311yescharacteristic
446C2×C4D8×D5311yescharacteristic
447C2×C4C2×D4×D5311yescharacteristic
448C8—21no—
449C8—532no—
450C8—1061no—
451C8C2×Q8×D5311yescharacteristic
452C8#41×D5311yescharacteristic
453C22—42no—
454C22—81no—
455C22—81no—
456C22—532no—
457C22—1061no—
458C22—1061no—
459C22—4248no—
460C22—4241no—
461C22—4241no—
462C22—2122no—
463C22C2×D8×D5311yescharacteristic
464C4—21no—
465C4—21no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—532no—
470C4—532no—
471C4—1061no—
472C4—4241no—
473C4—4241no—
474C4—4241no—
475C4C22⋊Q8×D5311yescharacteristic
476C4#137×D5311yescharacteristic
477C2—82no—
478C2—1062no—
479C2—4242no—
480C2#474×D5311yescharacteristic
481C2#509×D5311yescharacteristic
482C2#2028784811yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2228C2
socleC22×C53C22×D8
prime radicalC2×D8×D53C22
hypercenter#2228D53
hypocenterC53#33627
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2228

Complements

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

NH ≅ G/Ntype
1. #20285228C2semidirect
3. #20276873C2semidirect
4. #20276872C2semidirect
5. #20276869C2semidirect
6. #20276861C2semidirect
7. #20276860C2semidirect
8. #20276851C2semidirect
9. #20276850C2semidirect
10. #20276502C2semidirect
11. #20276500C2semidirect
12. #20276469C2semidirect
15. C53×#2228C2semidirect
21. #16811027C22semidirect
22. #16811027C22semidirect
23. #16811025C22semidirect
37. #16810451C22semidirect
38. #16810450C22semidirect
39. #16810450C22semidirect
40. #16810449C22semidirect
41. #16810448C22semidirect
42. #16810448C22semidirect
43. C4⋊#11719908C22semidirect
44. #16810441C22semidirect
45. #16810441C22semidirect
52. C53×#499C22semidirect
54. C53×#367C22semidirect
55. C53×#366C22semidirect
56. C53×C4⋊C16C22semidirect
64. D8×D53Q8semidirect
241. #2228D53semidirect
266. #499C2×D53semidirect
283. #367C2×D53semidirect
287. #366C2×D53semidirect
291. C4⋊C16C2×D53semidirect
300. C53#33627semidirect

Maximal subgroups

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

subgroupindex
#202852282
C4×D8×D532
#202768732
#202768722
#202768692
#202768612
#202768602
#202768512
#202768502
#202765022
#202765002
#202764692
#202764292
C53×#22282
#3362753