Finite Group Databasebeta

← #27539649

Subgroups — #27539649

Order 13568.

558 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285296C211yesmaximal, characteristic, complemented
2#20285289C211yesmaximal, characteristic
3#20276996C211yesmaximal, characteristic, complemented
4#20276969C211yesmaximal, characteristic, complemented
5#20276941C211yesmaximal, characteristic, complemented
6#20276937C211yesmaximal, characteristic, complemented
7#20276813C211yesmaximal, characteristic, complemented
8#20276811C211yesmaximal, characteristic, complemented
9#20276809C211yesmaximal, characteristic, complemented
10#20276570C211yesmaximal, characteristic, complemented
11SD32×Dic53C211yesmaximal, characteristic, complemented
12#20276566C211yesmaximal, characteristic, complemented
13#20276564C211yesmaximal, characteristic, complemented
14C2×C16×Dic53C211yesmaximal, characteristic, complemented
15C53×#2217C211yesmaximal, characteristic, complemented
16#16811175—21no—
17#16811168—21no—
18C2×#13133162C2211yescharacteristic
19C2×#13133161C2211yescharacteristic
20C2×#13133160C2211yescharacteristic
21C22×#11719908C2211yescharacteristic
22#16811027C2211yescharacteristic
23#16811027C2211yescharacteristic
24#16811001C2211yescharacteristic
25#16810998C2211yescharacteristic
26#16810972—21no—
27#16810968—21no—
28#16810864C2211yescharacteristic
29#16810862—21no—
30#16810859C2211yescharacteristic
31#16810857—21no—
32Q16×Dic53C2211yescharacteristic
33#16810717C2211yescharacteristic
34#16810715—21no—
35D8×Dic53C2211yescharacteristic
36#16810705C2211yescharacteristic
37#16810703—21no—
38C2×C8×Dic53C2211yescharacteristic
39#16810451C2211yescharacteristic
40#16810450C2211yescharacteristic
41#16810450C2211yescharacteristic
42#16810428C2211yescharacteristic
43#16810427C2211yescharacteristic
44#16810426C2211yescharacteristic
45#16810425C2211yescharacteristic
46#16810424C2211yescharacteristic
47#16810423C2211yescharacteristic
48C16×Dic53C2211yescharacteristic
49C16×Dic53C2211yescharacteristic
50C16×Dic53C2211yescharacteristic
51C16×Dic53C2211yescharacteristic
52C2×SD32×C53C2211yescharacteristic
53C22×C16×C53C2211yescharacteristic
54C53×#467C2211yescharacteristic
55C53×#466C2211yescharacteristic
56C53×#367C2211yescharacteristic
57C53×#358C2211yescharacteristic
58C53×#357C2211yescharacteristic
59C2×#11719947—41no—
60C2×Dic212C2311yescharacteristic
61C2×#11719925—21no—
62C2×D424C2311yescharacteristic
63C2×#11719922—21no—
64C22×#88322D411yescharacteristic
65#13133216—21no—
66#13133216—21no—
67#13133215—21no—
68#13133213—21no—
69#13133207—41no—
70#13133203—41no—
71#13133196—21no—
72#13133194—41no—
73#13133191—21no—
74#13133189—41no—
75Q8×Dic53—21no—
76#13133182—21no—
77#13133182—41no—
78D4×Dic53—21no—
79#13133179—21no—
80#13133177—41no—
81#13133173—41no—
82C2×C4×Dic53D411yescharacteristic
83#13133162—22no—
84#13133161—22no—
85#13133160—22no—
86C2×#11719908—22no—
87C2×#11719908C2311yescharacteristic
88C2×#11719908C2311yescharacteristic
89#13133151C2311yescharacteristic
90#13133151C2311yescharacteristic
91#13133150—21no—
92#13133150—21no—
93#13133142—21no—
94#13133141—21no—
95#13133140—21no—
96#13133139—21no—
97#13133138—21no—
98#13133137—21no—
99C8×Dic53D411yescharacteristic
100C8×Dic53D411yescharacteristic
101C8×Dic53C2311yescharacteristic
102C8×Dic53C2311yescharacteristic
103C2×Q16×C53C2311yescharacteristic
104C2×D8×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C22⋊Q8×C53—21no—
107#121×C53—21no—
108SD32×C53—22no—
109C2×C16×C53—22no—
110C2×C16×C53C2311yescharacteristic
111C2×C16×C53C2311yescharacteristic
112#107×C53C2311yescharacteristic
113#107×C53C2311yescharacteristic
114#103×C53—21no—
115#102×C53—21no—
116C2×Q8×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—41no—
119C22×C4×C53C2×D411yescharacteristic
120C2×C22⋊D53—41no—
121C2×C22⋊D53—41no—
122C2×C22⋊D53—41no—
123C22×Dic53—41no—
124C22×Dic53D811yescharacteristic
125C22×Dic53D811yescharacteristic
126#11719947—84no—
127C2×D212—21no—
128C2×C4×D53—41no—
129C2×Dic106—21no—
130C2×Dic106—41no—
131C2×Dic106—41no—
132Q16×C53—21no—
133Q16×C53—21no—
134D8×C53—21no—
135D8×C53—21no—
136C2×C8×C53—21no—
137C2×C8×C53—21no—
138C2×C8×C53C2×D411yescharacteristic
139C2×C8×C53C2411yescharacteristic
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—41no—
143C22⋊C4×C53—41no—
144C22⋊C4×C53—41no—
145C22⋊Dic53—41no—
146C22⋊Dic53—41no—
147#11719925—42no—
148#11719922—42no—
149#11719921—41no—
150#11719921—41no—
151C4⋊Dic53—21no—
152C4⋊Dic53—21no—
153C4⋊Dic53—41no—
154#11719919—41no—
155#11719919—41no—
156#11719919—41no—
157#11719919—41no—
158#11719919—41no—
159#11719919—41no—
160C4×Dic53—41no—
161C4×Dic53—41no—
162C4×Dic53D811yescharacteristic
163C4×Dic53D811yescharacteristic
164C4×Dic53C2×D411yescharacteristic
165C4×Dic53C2×D411yescharacteristic
166C2×#88322—22no—
167C2×#88322C2×D411yescharacteristic
168C2×#88322C2×D411yescharacteristic
169Dic212—21no—
170Dic212—21no—
171D424—21no—
172D424—21no—
173C16×C53—21no—
174C16×C53#4112yescomplemented
175#11719908—21no—
176#11719908#4112yescomplemented
177C23×C53—41no—
178C23×C53C2×D811yescharacteristic
179C22×D53—41no—
180Q8×C53—41no—
181Q8×C53—41no—
182D4×C53—41no—
183D4×C53—41no—
184D4×C53—82no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53C2×D811yescharacteristic
191C2×C4×C53C22×D411yescharacteristic
192C22⋊D53—82no—
193C22⋊D53—82no—
194C22⋊D53—82no—
195C2×Dic53—22no—
196C2×Dic53—41no—
197C2×Dic53—41no—
198C2×Dic53—41no—
199C2×Dic53—82no—
200C2×Dic53SD3214yescomplemented
201C2×Dic53C2×D811yescharacteristic
202C2×Dic53C2×D811yescharacteristic
203C2×Dic53C2×D811yescharacteristic
204C2×Dic53C2×D811yescharacteristic
205D212—41no—
206D212—41no—
207C4×D53—82no—
208Dic106—41no—
209Dic106—41no—
210Dic106—82no—
211Dic106—82no—
212C8×C53—21no—
213C8×C53C2×#4111yescharacteristic
214C8×C53C2×#4111yescharacteristic
215#88322—21no—
216#88322#13512yes—
217#33736—531nomaximal, Sylow
218C22×C53—21no—
219C22×C53—21no—
220C22×C53—81no—
221C22×C53—81no—
222C22×C53—81no—
223C22×C53C2×SD3212yes—
224C22×C53C22×D811yescharacteristic
225C2×D53—81no—
226C2×D53—81no—
227C2×D53—81no—
228C4×C53—21no—
229C4×C53—81no—
230C4×C53—81no—
231C4×C53—81no—
232C4×C53#53211yescharacteristic
233C4×C53C2×#13511yescharacteristic
234Dic53—21no—
235Dic53—81no—
236Dic53—81no—
237Dic53—81no—
238Dic53C2×SD3212yes—
239Dic53C2×SD3212yes—
240Dic53#50812yes—
241#3170—531no—
242#3169—531no—
243#2254—531no—
244#2248—531no—
245#2241—531no—
246#2240—531no—
247#2217—531no—
248#2217—531no—
249#2217—531no—
250#2217D5311yescharacteristic, complemented
251C4×SD32—531no—
252C4×SD32—531no—
253C4×SD32—531no—
254C4×SD32—531no—
255C2×C4×C16—531no—
256C2×C53—22no—
257C2×C53—161no—
258C2×C53#316911yescharacteristic
259C2×C53C22×SD3211yescharacteristic
260C2×C53C2×#50811yescharacteristic
261D53—161no—
262#533—1061no—
263#532—1061no—
264C2×SD32—531no—
265C2×SD32—531no—
266C2×SD32—531no—
267C2×SD32C2×D5311yescharacteristic
268C22×C16—531no—
269C22×C16C2×D5311yescharacteristic
270#500—531no—
271#500—531no—
272#494—531no—
273#493—531no—
274#487—1061no—
275#486—1061no—
276#467—531no—
277#467—1061no—
278#467C2×D5311yescharacteristic
279#466—531no—
280#466—1061no—
281#466C2×D5311yescharacteristic
282C4×Q16—531no—
283C4×Q16—531no—
284C4×Q16—1061no—
285C4×D8—531no—
286C4×D8—531no—
287C4×D8—1061no—
288C2×C4×C8—531no—
289#367—531no—
290#367—531no—
291#367—531no—
292#367C2×D5311yescharacteristic
293#358—531no—
294#358—531no—
295#358—531no—
296#358C2×D5311yescharacteristic
297#357—531no—
298#357—531no—
299#357—531no—
300#357C2×D5311yescharacteristic
301C4×C16—531no—
302C4×C16—531no—
303C4×C16—531no—
304C4×C16—531no—
305C53#3373611yescharacteristic, complemented, Sylow
306C2×#41—2121no—
307C2×Q16—531no—
308C2×Q16—1061no—
309C2×Q16C22×D5311yescharacteristic
310C2×D8—531no—
311C2×D8—1061no—
312C2×D8C22×D5311yescharacteristic
313C22×C8—531no—
314C22×C8C22×D5311yescharacteristic
315C4⋊Q8—1061no—
316C4⋊Q8—1061no—
317C4⋊Q8—1061no—
318#127—1061no—
319#125—2121no—
320#124—2121no—
321C22⋊Q8—21no—
322C22⋊Q8—1061no—
323C22⋊Q8—2121no—
324#121—21no—
325#121—1061no—
326#121—2121no—
327C4×Q8—1061no—
328C4×Q8—1061no—
329C4×Q8—2121no—
330C4×D4—1061no—
331C4×D4—1061no—
332C4×D4—2121no—
333#117—2121no—
334C2×C42—531no—
335SD32—22no—
336SD32—1062no—
337SD32—1062no—
338SD32—1062no—
339C2×C16—22no—
340C2×C16—531no—
341C2×C16—531no—
342C2×C16—1062no—
343C2×C16C22×D5311yescharacteristic
344C2×C16C22×D5311yescharacteristic
345#107—531no—
346#107—531no—
347#107—1061no—
348#107—1061no—
349#107C22×D5311yescharacteristic
350#107C22×D5311yescharacteristic
351#103—21no—
352#103—1061no—
353#103—1061no—
354#103—1061no—
355#102—21no—
356#102—1061no—
357#102—1061no—
358#102—1061no—
359C4×C8—531no—
360C4×C8—531no—
361C4×C8—531no—
362C4×C8—531no—
363#41—4244no—
364C2×Q8—21no—
365C2×Q8—1061no—
366C2×Q8—2121no—
367C2×Q8—2121no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—1061no—
371C2×D4—2121no—
372C2×D4—2121no—
373C2×D4—2121no—
374C22×C4—531no—
375C22×C4—531no—
376C22×C4—2121no—
377C22×C4—2121no—
378C22×C4D4×D5311yescharacteristic
379Q16—21no—
380Q16—21no—
381Q16—1061no—
382Q16—1061no—
383Q16—2122no—
384D8—21no—
385D8—21no—
386D8—1061no—
387D8—1061no—
388D8—2122no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—531no—
392C2×C8—531no—
393C2×C8—1062no—
394C2×C8D4×D5311yescharacteristic
395C2×C8C23×D5311yescharacteristic
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—41no—
399C4⋊C4—1061no—
400C4⋊C4—1061no—
401C4⋊C4—2121no—
402C4⋊C4—2121no—
403C4⋊C4—2121no—
404C4⋊C4—2121no—
405C4⋊C4—2121no—
406C4⋊C4—2121no—
407C4⋊C4—2121no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—2121no—
411C22⋊C4—2121no—
412C22⋊C4—2121no—
413C22⋊C4—2121no—
414C42—531no—
415C42—531no—
416C42—531no—
417C42—531no—
418C42—2121no—
419C42—2121no—
420C16—21no—
421C16—532no—
422C16—1061no—
423C16#1171994712yescomplemented
424C23—41no—
425C23—2121no—
426C23D8×D5311yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—4242no—
430Q8—4242no—
431Q8—2121no—
432Q8—2121no—
433D4—41no—
434D4—41no—
435D4—82no—
436D4—4242no—
437D4—4242no—
438D4—4242no—
439D4—2121no—
440D4—2121no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—534no—
447C2×C4—531no—
448C2×C4—531no—
449C2×C4—531no—
450C2×C4—531no—
451C2×C4—1062no—
452C2×C4—4242no—
453C2×C4—4242no—
454C2×C4—2121no—
455C2×C4—2121no—
456C2×C4—2121no—
457C2×C4D8×D5311yescharacteristic
458C2×C4C2×D4×D5311yescharacteristic
459C8—21no—
460C8—532no—
461C8—1061no—
462C8#41×D5311yescharacteristic
463C8C2×#1171994711yescharacteristic
464C22—21no—
465C22—21no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—4241no—
470C22—4241no—
471C22—4241no—
472C22SD32×D5312yes—
473C22C2×D8×D5311yescharacteristic
474C4—21no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—532no—
479C4—532no—
480C4—532no—
481C4—1061no—
482C4—4241no—
483C4—4241no—
484C4—4241no—
485C4#1681116811yescharacteristic
486C4#135×D5311yescharacteristic
487C2—22no—
488C2—161no—
489C2—8481no—
490C2#2028528911yescharacteristic
491C2C2×SD32×D5311yescharacteristic
492C2#508×D5311yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2217C2
socleC22×C53C22×D8
prime radical#20285289C2
hypercenter#2217D53
hypocenterC53#33736
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2217

Complements

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

NH ≅ G/Ntype
1. #20285296C2semidirect
3. #20276996C2semidirect
4. #20276969C2semidirect
5. #20276941C2semidirect
6. #20276937C2semidirect
7. #20276813C2semidirect
8. #20276811C2semidirect
9. #20276809C2semidirect
10. #20276570C2semidirect
11. SD32×Dic53C2semidirect
12. #20276566C2semidirect
13. #20276564C2semidirect
14. C2×C16×Dic53C2semidirect
15. C53×#2217C2semidirect
174. C16×C53#41semidirect
176. #11719908#41semidirect
200. C2×Dic53SD32semidirect
250. #2217D53semidirect
305. C53#33736semidirect
423. C16#11719947semidirect

Maximal subgroups

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

subgroupindex
#202852962
#202852892
#202769962
#202769692
#202769412
#202769372
#202768132
#202768112
#202768092
#202765702
SD32×Dic532
#202765662
#202765642
C2×C16×Dic532
C53×#22172
#3373653