Finite Group Databasebeta

← #27538736

Subgroups — #27538736

Order 13568.

546 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285227C211yesmaximal, characteristic, complemented
2#20283486C211yesmaximal, characteristic
3#20276882C211yesmaximal, characteristic, complemented
4#20276878C211yesmaximal, characteristic, complemented
5#20276875C211yesmaximal, characteristic, complemented
6#20276871C211yesmaximal, characteristic, complemented
7#20276866C211yesmaximal, characteristic, complemented
8#20276861C211yesmaximal, characteristic, complemented
9#20276856C211yesmaximal, characteristic, complemented
10#20276851C211yesmaximal, characteristic, complemented
11#20276497C211yesmaximal, characteristic, complemented
12C2×#16810441C211yesmaximal, characteristic, complemented
13#20276423C211yesmaximal, characteristic
14#20276420C211yesmaximal, characteristic
15C53×#2158C211yesmaximal, characteristic, complemented
16C2×#13133239C2211yescharacteristic
17#16811176—21no—
18#16811091—21no—
19C22×#11719908C2211yescharacteristic
20#16811027C2211yescharacteristic
21#16811027C2211yescharacteristic
22#16811024C2211yescharacteristic
23#16811024C2211yescharacteristic
24#16811023—21no—
25#16811020—21no—
26#16810717C2211yescharacteristic
27#16810717C2211yescharacteristic
28#16810710—21no—
29#16810710—21no—
30#16810705C2211yescharacteristic
31#16810705C2211yescharacteristic
32#16810661C2211yescharacteristic
33#16810659—21no—
34#16810617—21no—
35C2×C8×Dic53C2211yescharacteristic
36#16810450C2211yescharacteristic
37#16810450C2211yescharacteristic
38#16810448C2211yescharacteristic
39#16810448C2211yescharacteristic
40#16810441C2211yescharacteristic
41#16810441C2211yescharacteristic
42#16810441C2211yescharacteristic
43#16810441C2211yescharacteristic
44#16810428C2211yescharacteristic
45#16810428C2211yescharacteristic
46#16810426C2211yescharacteristic
47#16810426C2211yescharacteristic
48#16810425C2211yescharacteristic
49#16810425C2211yescharacteristic
50#16810423C2211yescharacteristic
51#16810423C2211yescharacteristic
52C22×C16×C53C2211yescharacteristic
53C53×#427C2211yescharacteristic
54C53×#427C2211yescharacteristic
55C53×#367C2211yescharacteristic
56C53×#367C2211yescharacteristic
57C53×#366C2211yescharacteristic
58C53×#366C2211yescharacteristic
59C2×#11719945—21no—
60#13133239—22no—
61#13133239Q814yescomplemented
62C2×Dic212C2311yescharacteristic
63C2×#11719913—21no—
64C2×D424C2311yescharacteristic
65C22×#88322D411yescharacteristic
66#13133216—21no—
67#13133216—21no—
68#13133208—21no—
69#13133208—21no—
70#13133194—41no—
71#13133194—41no—
72C4×Dic106—41no—
73C4×Dic106—41no—
74#13133182—21no—
75#13133182—21no—
76#13133179—21no—
77#13133179—21no—
78#13133177—41no—
79#13133177—41no—
80#13133176—21no—
81#13133174—41no—
82C2×C4×Dic53D411yescharacteristic
83C42×D53—41no—
84C2×#11719908—22no—
85C2×#11719908C2311yescharacteristic
86C2×#11719908C2311yescharacteristic
87#13133151C2311yescharacteristic
88#13133151C2311yescharacteristic
89#13133150—21no—
90#13133150—21no—
91#13133149—21no—
92#13133148—21no—
93#13133148—21no—
94#13133140—21no—
95#13133140—21no—
96#13133137—21no—
97#13133137—21no—
98C8×Dic53D411yescharacteristic
99C8×Dic53D411yescharacteristic
100C8×Dic53C2311yescharacteristic
101C8×Dic53C2311yescharacteristic
102C22×C8×C53C2311yescharacteristic
103#117×C53—21no—
104#117×C53—21no—
105C2×C16×C53—22no—
106C2×C16×C53C2311yescharacteristic
107C2×C16×C53C2311yescharacteristic
108#107×C53C2311yescharacteristic
109#107×C53C2311yescharacteristic
110#107×C53C2311yescharacteristic
111#107×C53C2311yescharacteristic
112#106×C53—21no—
113#106×C53—21no—
114C22×C4×C53C2×D411yescharacteristic
115C2×C22⋊D53—41no—
116C22×Dic53D811yescharacteristic
117C22×Dic53D811yescharacteristic
118#11719945—24no—
119#11719945—42no—
120C2×D212—21no—
121C2×C4×D53—41no—
122C2×C4×D53—41no—
123C2×C4×D53—41no—
124C2×Dic106—21no—
125C2×Dic106—41no—
126C2×Dic106—41no—
127C2×C8×C53C2×D411yescharacteristic
128C2×C8×C53C2×Q812yes—
129C2×C8×C53#4112yes—
130C2×C8×C53C2411yescharacteristic
131C4⋊C4×C53—21no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C22⋊C4×C53—41no—
136C22⋊C4×C53—41no—
137C42×C53—41no—
138C42×C53—41no—
139C22⋊Dic53—41no—
140#11719921—41no—
141#11719921—41no—
142C4⋊Dic53—21no—
143C4⋊Dic53—21no—
144C4⋊Dic53—41no—
145C4⋊Dic53—41no—
146#11719919—41no—
147#11719919—41no—
148#11719919—41no—
149#11719919—41no—
150#11719919—41no—
151#11719919—41no—
152C4×Dic53—41no—
153C4×Dic53—41no—
154C4×Dic53—41no—
155C4×Dic53D811yescharacteristic
156C4×Dic53D811yescharacteristic
157C4×Dic53C2×D411yescharacteristic
158C4×Dic53C2×D411yescharacteristic
159C2×#88322—22no—
160C2×#88322C2×D411yescharacteristic
161C2×#88322C2×D411yescharacteristic
162Dic212—21no—
163Dic212C2×Q812yes—
164D424—21no—
165D424C2×Q812yes—
166#11719913—24no—
167C16×C53—21no—
168C16×C53—21no—
169#11719908—21no—
170#11719908—21no—
171C23×C53C2×D811yescharacteristic
172C22×D53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53C22⋊Q812yes—
178C2×C4×C53C2×D811yescharacteristic
179C2×C4×C53#13712yes—
180C2×C4×C53C22×D411yescharacteristic
181C22⋊D53—44no—
182C2×Dic53—22no—
183C2×Dic53—41no—
184C2×Dic53—41no—
185C2×Dic53—41no—
186C2×Dic53—41no—
187C2×Dic53C2×D811yescharacteristic
188C2×Dic53C2×D811yescharacteristic
189C2×Dic53C2×D811yescharacteristic
190C2×Dic53C2×D811yescharacteristic
191D212—22no—
192D212—41no—
193C4×D53—44no—
194C4×D53—82no—
195C4×D53—82no—
196Dic106—22no—
197Dic106—41no—
198Dic106—82no—
199Dic106—82no—
200C8×C53C22⋊Q812yes—
201C8×C53C22×Q811yescharacteristic
202C8×C53C2×#4111yescharacteristic
203#88322—21no—
204#88322—21no—
205#33626—531nomaximal, Sylow
206C22×C53—21no—
207C22×C53—21no—
208C22×C53#47412yes—
209C22×C53C22×D811yescharacteristic
210C2×D53—42no—
211C2×D53—81no—
212C2×D53—81no—
213C4×C53—81no—
214C4×C53—81no—
215C4×C53—81no—
216C4×C53—81no—
217C4×C53#47412yes—
218C4×C53C2×C22⋊Q811yescharacteristic
219C4×C53C2×#13711yescharacteristic
220Dic53—21no—
221Dic53—21no—
222Dic53—21no—
223Dic53—42no—
224Dic53—81no—
225Dic53—81no—
226Dic53#50812yes—
227#3160—531no—
228C4×#135—531no—
229#2229—531no—
230#2229—531no—
231#2228—531no—
232#2228—531no—
233#2227—531no—
234#2226—531no—
235#2225—531no—
236#2224—531no—
237#2158—531no—
238#2158D5311yescharacteristic, complemented
239C2×C4⋊C16—531no—
240#2138—531no—
241#2138—531no—
242C2×C53—22no—
243C2×C53C2×#47411yescharacteristic
244C2×C53C2×#50811yescharacteristic
245C2×C53#341411yescharacteristic
246D53—82no—
247C2×#135—531no—
248#533—1061no—
249C4×#41—1061no—
250C22×C16—531no—
251C22×C16C2×D5311yescharacteristic
252#500—531no—
253#500—531no—
254#499—531no—
255#499—531no—
256#499—1061no—
257#498—1061no—
258C4×Q16—531no—
259C4×Q16—531no—
260C4×SD16—1061no—
261C4×SD16—1061no—
262C4×D8—531no—
263C4×D8—531no—
264#427—531no—
265#427—1061no—
266#427C2×D5311yescharacteristic
267#427C2×D5311yescharacteristic
268#416—1061no—
269C2×C4×C8—531no—
270#367—531no—
271#367—531no—
272#367C2×D5311yescharacteristic
273#367C2×D5311yescharacteristic
274#366—531no—
275#366—531no—
276#366C2×D5311yescharacteristic
277#366C2×D5311yescharacteristic
278C4⋊C16—531no—
279C4⋊C16—531no—
280C4⋊C16—531no—
281C4⋊C16—531no—
282#358—531no—
283#358—531no—
284#358—531no—
285#358—531no—
286#357—531no—
287#357—531no—
288#357—531no—
289#357—531no—
290C53#3362611yescharacteristic, complemented, Sylow
291C2×#41—1061no—
292#135—534no—
293#135—1062no—
294C2×Q16—531no—
295C2×SD16—1061no—
296C2×D8—531no—
297C22×C8—531no—
298C22×C8C22×D5311yescharacteristic
299C4⋊Q8—1061no—
300C4⋊Q8—1061no—
301#125—1061no—
302#125—1061no—
303C22⋊Q8—2121no—
304C22⋊Q8—2121no—
305C4×Q8—1061no—
306C4×Q8—1061no—
307C4×Q8—2121no—
308C4×Q8—2121no—
309C4×D4—1061no—
310C4×D4—1061no—
311C4×D4—2121no—
312C4×D4—2121no—
313#117—21no—
314#117—21no—
315#117—1061no—
316#117—2121no—
317C2×C42—531no—
318C2×C42—2121no—
319C2×C16—22no—
320C2×C16—531no—
321C2×C16—531no—
322C2×C16—1062no—
323C2×C16C22×D5311yescharacteristic
324C2×C16C22×D5311yescharacteristic
325#107—531no—
326#107—531no—
327#107—1061no—
328#107—1061no—
329#107C22×D5311yescharacteristic
330#107C22×D5311yescharacteristic
331#107C22×D5311yescharacteristic
332#107C22×D5311yescharacteristic
333#106—21no—
334#106—21no—
335#106—1061no—
336#106—1061no—
337#106—1061no—
338#103—1061no—
339#103—1061no—
340#102—1061no—
341#102—1061no—
342C4×C8—531no—
343C4×C8—531no—
344C4×C8—531no—
345C4×C8—531no—
346#41—1064no—
347#41—2122no—
348C2×Q8—1061no—
349C2×Q8—2121no—
350C2×Q8—2121no—
351C2×D4—1061no—
352C2×D4—2121no—
353C22×C4—531no—
354C22×C4—531no—
355C22×C4—2121no—
356C22×C4—2121no—
357C22×C4—2121no—
358C22×C4D4×D5311yescharacteristic
359Q16—532no—
360Q16—1061no—
361SD16—1064no—
362D8—532no—
363D8—1061no—
364C2×C8—531no—
365C2×C8—531no—
366C2×C8—1062no—
367C2×C8#1171994512yes—
368C2×C8D4×D5311yescharacteristic
369C2×C8Q8×D5312yes—
370C2×C8C23×D5311yescharacteristic
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—1061no—
376C4⋊C4—1061no—
377C4⋊C4—2121no—
378C4⋊C4—2121no—
379C4⋊C4—2121no—
380C4⋊C4—2121no—
381C4⋊C4—2121no—
382C4⋊C4—2121no—
383C4⋊C4—2121no—
384C4⋊C4—2121no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—2121no—
388C22⋊C4—2121no—
389C22⋊C4—2121no—
390C42—41no—
391C42—41no—
392C42—531no—
393C42—531no—
394C42—531no—
395C42—531no—
396C42—2121no—
397C42—2121no—
398C42—2121no—
399C16—21no—
400C16—21no—
401C16—1061no—
402C16—1061no—
403C23—2121no—
404C23D8×D5311yescharacteristic
405Q8—1062no—
406Q8—4242no—
407Q8—4242no—
408Q8—2121no—
409D4—1062no—
410D4—2124no—
411D4—2121no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—531no—
417C2×C4—531no—
418C2×C4—531no—
419C2×C4—531no—
420C2×C4—1062no—
421C2×C4—4242no—
422C2×C4—4242no—
423C2×C4—2121no—
424C2×C4—2121no—
425C2×C4—2121no—
426C2×C4—2121no—
427C2×C4—2124no—
428C2×C4#1313319512yes—
429C2×C4D8×D5311yescharacteristic
430C2×C4#1313324912yes—
431C2×C4C2×D4×D5311yescharacteristic
432C8—1061no—
433C8—1061no—
434C8#1313319512yes—
435C8C2×Q8×D5311yescharacteristic
436C8C2×#1171994511yescharacteristic
437C22—21no—
438C22—21no—
439C22—4241no—
440C22—4241no—
441C22—2122no—
442C22#1681090212yes—
443C22C2×D8×D5311yescharacteristic
444C4—81no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—532no—
449C4—1061no—
450C4—1061no—
451C4—1061no—
452C4—4241no—
453C4—4241no—
454C4—2122no—
455C4#1681090212yes—
456C4C2×#1313319511yescharacteristic
457C4C2×#1313324911yescharacteristic
458C2—22no—
459C2—4242no—
460C2C2×#1681090211yescharacteristic
461C2#508×D5311yescharacteristic
462C2#2028784811yescharacteristic
subgroupG/H
centerC2×C4D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2158C2
socleC22×C53C22×D8
prime radicalC2×#13133239C22
hypercenter#2158D53
hypocenterC53#33626
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2158

Complements

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

NH ≅ G/Ntype
1. #20285227C2semidirect
3. #20276882C2semidirect
4. #20276878C2semidirect
5. #20276875C2semidirect
6. #20276871C2semidirect
7. #20276866C2semidirect
8. #20276861C2semidirect
9. #20276856C2semidirect
10. #20276851C2semidirect
11. #20276497C2semidirect
12. C2×#16810441C2semidirect
15. C53×#2158C2semidirect
61. #13133239Q8semidirect
238. #2158D53semidirect
290. C53#33626semidirect

Maximal subgroups

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

subgroupindex
#202852272
#202834862
#202768822
#202768782
#202768752
#202768712
#202768662
#202768612
#202768562
#202768512
#202764972
C2×#168104412
#202764232
#202764202
C53×#21582
#3362653