Finite Group Databasebeta

← #27538737

Subgroups — #27538737

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#20285228C211yesmaximal, characteristic, complemented
2C4×#13133240C211yesmaximal, characteristic
3#20276879C211yesmaximal, characteristic, complemented
4#20276873C211yesmaximal, characteristic, complemented
5#20276869C211yesmaximal, characteristic, complemented
6#20276865C211yesmaximal, characteristic, complemented
7#20276861C211yesmaximal, characteristic, complemented
8#20276855C211yesmaximal, characteristic, complemented
9#20276851C211yesmaximal, characteristic, complemented
10#20276498C211yesmaximal, characteristic, complemented
11#20276496C211yesmaximal, characteristic, complemented
12C4⋊C16×D53C211yesmaximal, characteristic, complemented
13#20276421C211yesmaximal, characteristic
14#20276421C211yesmaximal, characteristic
15C53×#2229C211yesmaximal, characteristic, complemented
16C2×#13133240C2211yescharacteristic
17#16811179—21no—
18C4×#11719949—21no—
19C2×C16×D53C2211yescharacteristic
20C2×C16×D53C2211yescharacteristic
21#16811027C2211yescharacteristic
22#16811027C2211yescharacteristic
23#16811025C2211yescharacteristic
24#16811022—21no—
25#16811020—21no—
26Q16×Dic53C2211yescharacteristic
27C4×#11719924—21no—
28#16810708—21no—
29C4×D424C2211yescharacteristic
30#16810705C2211yescharacteristic
31#16810660C2211yescharacteristic
32#16810660C2211yescharacteristic
33#16810660C2211yescharacteristic
34#16810658—21no—
35#16810618—21no—
36C4×C8×D53C2211yescharacteristic
37#16810451C2211yescharacteristic
38#16810450C2211yescharacteristic
39#16810450C2211yescharacteristic
40#16810449C2211yescharacteristic
41#16810448C2211yescharacteristic
42#16810448C2211yescharacteristic
43C4⋊#11719908C2211yescharacteristic
44#16810441C2211yescharacteristic
45#16810441C2211yescharacteristic
46#16810427C2211yescharacteristic
47#16810427C2211yescharacteristic
48#16810425C2211yescharacteristic
49#16810425C2211yescharacteristic
50#16810423C2211yescharacteristic
51#16810423C2211yescharacteristic
52C53×#499C2211yescharacteristic
53C4×Q16×C53C2211yescharacteristic
54C53×#367C2211yescharacteristic
55C53×#366C2211yescharacteristic
56C53×C4⋊C16C2211yescharacteristic
57C53×#358C2211yescharacteristic
58C53×#358C2211yescharacteristic
59C2×#11719949—21no—
60#13133240—22no—
61#13133240Q814yescomplemented
62C2×#11719924—21no—
63C2×D424C2311yescharacteristic
64C2×C8×D53D411yescharacteristic
65C2×C8×D53C2311yescharacteristic
66#13133216—21no—
67#13133216—21no—
68#13133209—21no—
69#13133195—41no—
70#13133195—41no—
71Q8×Dic53—21no—
72#13133182—41no—
73#13133182—41no—
74C4×D212—21no—
75C4×D212—41no—
76#13133179—21no—
77#13133179—41no—
78#13133175—21no—
79#13133175—21no—
80#13133175—21no—
81#13133175—41no—
82C42×D53—41no—
83C42×D53D411yescharacteristic
84C2×#11719908C2311yescharacteristic
85C2×#11719908C2311yescharacteristic
86C16×D53—22no—
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#13133141—21no—
98#13133139—21no—
99#13133137—21no—
100C4×#88322D411yescharacteristic
101C8×Dic53D411yescharacteristic
102C8×Dic53C2311yescharacteristic
103C2×Q16×C53C2311yescharacteristic
104#125×C53—21no—
105C4×Q8×C53—21no—
106C2×C16×C53C2311yescharacteristic
107C2×C16×C53C2311yescharacteristic
108#107×C53C2311yescharacteristic
109#107×C53C2311yescharacteristic
110#107×C53C2311yescharacteristic
111#106×C53—21no—
112#103×C53—21no—
113C4×C8×C53C2311yescharacteristic
114C2×Q8×C53—21no—
115#11719949—24no—
116#11719949—42no—
117C2×D212—21no—
118C2×D212—41no—
119C2×C4×D53—41no—
120C2×C4×D53—41no—
121C2×C4×D53—41no—
122C2×C4×D53D811yescharacteristic
123C2×C4×D53D811yescharacteristic
124C2×C4×D53C2×D411yescharacteristic
125C2×Dic106—41no—
126C2×Dic106—41no—
127Q16×C53—21no—
128Q16×C53C2×Q812yes—
129C2×C8×C53C2×D411yescharacteristic
130C2×C8×C53C2411yescharacteristic
131C4⋊C4×C53—21no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—41no—
135C4⋊C4×C53—41no—
136C4⋊C4×C53—41no—
137C42×C53—41no—
138C42×C53C2×D411yescharacteristic
139#11719924—24no—
140#11719921—41no—
141#11719921—41no—
142#11719921—41no—
143#11719921—41no—
144#11719921—41no—
145C4⋊Dic53—21no—
146C4⋊Dic53—21no—
147C4⋊Dic53—21no—
148C4⋊Dic53—41no—
149#11719919—41no—
150#11719919—41no—
151#11719919—41no—
152#11719919—41no—
153C4×Dic53—41no—
154C4×Dic53—41no—
155C4×Dic53—41no—
156C4×Dic53—41no—
157C4×Dic53D811yescharacteristic
158C4×Dic53D811yescharacteristic
159C4×Dic53C2×D411yescharacteristic
160C2×#88322C2×D411yescharacteristic
161C2×#88322C2×D411yescharacteristic
162D424—21no—
163D424C2×Q812yes—
164C8×D53—22no—
165C8×D53C2×Q812yes—
166C8×D53#4112yes—
167C16×C53—21no—
168C16×C53—21no—
169#11719908—21no—
170#11719908—21no—
171C22×D53—41no—
172C22×D53C2×D811yescharacteristic
173Q8×C53—22no—
174Q8×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53C2×D811yescharacteristic
180C2×C4×C53C2×D811yescharacteristic
181C2×C4×C53C22×D411yescharacteristic
182C2×Dic53—41no—
183C2×Dic53—41no—
184C2×Dic53—41no—
185C2×Dic53C2×D811yescharacteristic
186C2×Dic53C2×D811yescharacteristic
187C2×Dic53C2×D811yescharacteristic
188D212—22no—
189D212—44no—
190D212—41no—
191C4×D53—22no—
192C4×D53—41no—
193C4×D53—44no—
194C4×D53—82no—
195C4×D53—82no—
196C4×D53C22⋊Q812yes—
197C4×D53#13712yes—
198Dic106—82no—
199Dic106—82no—
200C8×C53—21no—
201C8×C53C22×Q811yescharacteristic
202C8×C53C2×#4111yescharacteristic
203#88322—21no—
204#88322C22⋊Q812yes—
205#33626—531nomaximal, Sylow
206C22×C53C22×D811yescharacteristic
207C2×D53—21no—
208C2×D53—21no—
209C2×D53—42no—
210C2×D53—81no—
211C2×D53—81no—
212C2×D53#47412yes—
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—42no—
216C4×C53—81no—
217C4×C53—81no—
218C4×C53—81no—
219C4×C53C2×C22⋊Q811yescharacteristic
220C4×C53C2×#13711yescharacteristic
221Dic53—21no—
222Dic53—81no—
223Dic53—81no—
224Dic53—81no—
225Dic53#47412yes—
226Dic53#50812yes—
227#3160—531no—
228C4×#135—531no—
229#2229—531no—
230#2229D5311yescharacteristic, complemented
231#2228—531no—
232#2228—531no—
233#2227—531no—
234#2226—531no—
235#2225—531no—
236#2224—531no—
237#2158—531no—
238#2158—531no—
239C2×C4⋊C16—531no—
240#2138—531no—
241#2138—531no—
242C2×C53C2×#47411yescharacteristic
243C2×C53C2×#50811yescharacteristic
244C2×C53#341411yescharacteristic
245D53—22no—
246D53—82no—
247C2×#135—531no—
248#533—1061no—
249C4×#41—1061no—
250C22×C16—531no—
251C22×C16—531no—
252#500—531no—
253#500—531no—
254#499—531no—
255#499—1061no—
256#499C2×D5311yescharacteristic
257#498—1061no—
258C4×Q16—531no—
259C4×Q16C2×D5311yescharacteristic
260C4×SD16—1061no—
261C4×SD16—1061no—
262C4×D8—531no—
263C4×D8—531no—
264#427—531no—
265#427—531no—
266#427—531no—
267#427—1061no—
268#416—1061no—
269C2×C4×C8—531no—
270#367—531no—
271#367—531no—
272#367—531no—
273#367C2×D5311yescharacteristic
274#366—531no—
275#366—531no—
276#366—531no—
277#366C2×D5311yescharacteristic
278C4⋊C16—531no—
279C4⋊C16—531no—
280C4⋊C16—531no—
281C4⋊C16C2×D5311yescharacteristic
282#358—531no—
283#358—531no—
284#358C2×D5311yescharacteristic
285#358C2×D5311yescharacteristic
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×Q16C22×D5311yescharacteristic
295C2×SD16—1061no—
296C2×D8—531no—
297C22×C8—531no—
298C22×C8—531no—
299C4⋊Q8—1061no—
300C4⋊Q8—1061no—
301#125—21no—
302#125—1061no—
303C22⋊Q8—2121no—
304C22⋊Q8—2121no—
305C4×Q8—21no—
306C4×Q8—1061no—
307C4×Q8—2121no—
308C4×Q8—2121no—
309C4×D4—1061no—
310C4×D4—1061no—
311C4×D4—2121no—
312C4×D4—2121no—
313#117—1061no—
314#117—1061no—
315#117—1061no—
316#117—2121no—
317C2×C42—531no—
318C2×C42—2121no—
319C2×C16—531no—
320C2×C16—531no—
321C2×C16—1062no—
322C2×C16—1062no—
323C2×C16C22×D5311yescharacteristic
324C2×C16C22×D5311yescharacteristic
325#107—531no—
326#107—531no—
327#107—531no—
328#107—1061no—
329#107—1061no—
330#107C22×D5311yescharacteristic
331#107C22×D5311yescharacteristic
332#107C22×D5311yescharacteristic
333#106—21no—
334#106—1061no—
335#106—1061no—
336#106—1061no—
337#106—1061no—
338#103—21no—
339#103—1061no—
340#102—1061no—
341#102—1061no—
342C4×C8—531no—
343C4×C8—531no—
344C4×C8—531no—
345C4×C8C22×D5311yescharacteristic
346#41—1064no—
347#41—2122no—
348C2×Q8—21no—
349C2×Q8—2121no—
350C2×Q8—2121no—
351C2×D4—1061no—
352C2×D4—2121no—
353C22×C4—531no—
354C22×C4—531no—
355C22×C4—531no—
356C22×C4—2121no—
357C22×C4—2121no—
358C22×C4—2121no—
359Q16—21no—
360Q16Q8×D5312yes—
361SD16—1064no—
362D8—532no—
363D8—1061no—
364C2×C8—531no—
365C2×C8—532no—
366C2×C8—532no—
367C2×C8—531no—
368C2×C8—1062no—
369C2×C8D4×D5311yescharacteristic
370C2×C8C23×D5311yescharacteristic
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—1061no—
378C4⋊C4—1061no—
379C4⋊C4—1061no—
380C4⋊C4—2121no—
381C4⋊C4—2121no—
382C4⋊C4—2121no—
383C4⋊C4—2121no—
384C4⋊C4—2121no—
385C22⋊C4—2121no—
386C22⋊C4—2121no—
387C22⋊C4—2121no—
388C22⋊C4—2121no—
389C22⋊C4—2121no—
390C42—41no—
391C42—531no—
392C42—531no—
393C42—531no—
394C42—2121no—
395C42—2121no—
396C42—2121no—
397C42—2121no—
398C42D4×D5311yescharacteristic
399C16—21no—
400C16—21no—
401C16—1061no—
402C16—1061no—
403C23—531no—
404C23—2121no—
405Q8—22no—
406Q8—41no—
407Q8—4242no—
408Q8—4242no—
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—532no—
419C2×C4—532no—
420C2×C4—531no—
421C2×C4—1062no—
422C2×C4—4242no—
423C2×C4—4242no—
424C2×C4—2121no—
425C2×C4—2124no—
426C2×C4—2121no—
427C2×C4—2121no—
428C2×C4—2121no—
429C2×C4D8×D5311yescharacteristic
430C2×C4D8×D5311yescharacteristic
431C2×C4C2×D4×D5311yescharacteristic
432C8—21no—
433C8—532no—
434C8—1061no—
435C8C2×Q8×D5311yescharacteristic
436C8#41×D5311yescharacteristic
437C22—532no—
438C22—1061no—
439C22—1061no—
440C22—4241no—
441C22—4241no—
442C22—2122no—
443C22C2×D8×D5311yescharacteristic
444C4—21no—
445C4—21no—
446C4—42no—
447C4—81no—
448C4—81no—
449C4—81no—
450C4—532no—
451C4—532no—
452C4—1061no—
453C4—4241no—
454C4—4241no—
455C4—4241no—
456C4C22⋊Q8×D5311yescharacteristic
457C4#137×D5311yescharacteristic
458C2—1062no—
459C2—4242no—
460C2#474×D5311yescharacteristic
461C2#508×D5311yescharacteristic
462C2#2028785011yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2229C2
socleC22×C53C22×D8
prime radicalC2×#13133240C22
hypercenter#2229D53
hypocenterC53#33626
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2229

Complements

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

NH ≅ G/Ntype
1. #20285228C2semidirect
3. #20276879C2semidirect
4. #20276873C2semidirect
5. #20276869C2semidirect
6. #20276865C2semidirect
7. #20276861C2semidirect
8. #20276855C2semidirect
9. #20276851C2semidirect
10. #20276498C2semidirect
11. #20276496C2semidirect
12. C4⋊C16×D53C2semidirect
15. C53×#2229C2semidirect
61. #13133240Q8semidirect
230. #2229D53semidirect
290. C53#33626semidirect

Maximal subgroups

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

subgroupindex
#202852282
C4×#131332402
#202768792
#202768732
#202768692
#202768652
#202768612
#202768552
#202768512
#202764982
#202764962
C4⋊C16×D532
#202764212
C53×#22292
#3362653