Finite Group Databasebeta

← #27538734

Subgroups — #27538734

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#135×Dic53C211yesmaximal, characteristic
3#20276881C211yesmaximal, characteristic, complemented
4#20276877C211yesmaximal, characteristic, complemented
5#20276874C211yesmaximal, characteristic, complemented
6#20276870C211yesmaximal, characteristic, complemented
7#20276865C211yesmaximal, characteristic, complemented
8#20276860C211yesmaximal, characteristic, complemented
9#20276855C211yesmaximal, characteristic, complemented
10#20276850C211yesmaximal, characteristic, complemented
11#20276499C211yesmaximal, characteristic, complemented
12#20276497C211yesmaximal, characteristic, complemented
13C2×#16810441C211yesmaximal, characteristic, complemented
14#20276422C211yesmaximal, characteristic
15C53×#2138C211yesmaximal, characteristic
16#16811176—21no—
17#41×Dic53—21no—
18C22×#11719908C2211yescharacteristic
19#16811027C2211yescharacteristic
20#16811027C2211yescharacteristic
21#16811024C2211yescharacteristic
22#16811024C2211yescharacteristic
23#16811023—21no—
24#16811020—21no—
25Q16×Dic53C2211yescharacteristic
26Q16×Dic53C2211yescharacteristic
27SD16×Dic53—21no—
28SD16×Dic53—21no—
29D8×Dic53C2211yescharacteristic
30D8×Dic53C2211yescharacteristic
31#16810661C2211yescharacteristic
32#16810661C2211yescharacteristic
33#16810659—21no—
34#16810619—21no—
35C2×C8×Dic53C2211yescharacteristic
36#16810451C2211yescharacteristic
37#16810451C2211yescharacteristic
38#16810450C2211yescharacteristic
39#16810450C2211yescharacteristic
40#16810449C2211yescharacteristic
41#16810449C2211yescharacteristic
42#16810448C2211yescharacteristic
43#16810448C2211yescharacteristic
44#16810441C2211yescharacteristic
45#16810441C2211yescharacteristic
46#16810441C2211yescharacteristic
47#16810441C2211yescharacteristic
48#16810427C2211yescharacteristic
49#16810427C2211yescharacteristic
50#16810424C2211yescharacteristic
51#16810424C2211yescharacteristic
52C2×#135×C53C2211yescharacteristic
53C22×C16×C53C2211yescharacteristic
54C53×#427C2211yescharacteristic
55C53×#358C2211yescharacteristic
56C53×#358C2211yescharacteristic
57C53×#357C2211yescharacteristic
58C53×#357C2211yescharacteristic
59C22×#88322D411yescharacteristic
60#13133216—21no—
61#13133216—21no—
62#13133208—21no—
63#13133208—21no—
64#13133194—41no—
65#13133194—41no—
66Q8×Dic53—21no—
67Q8×Dic53—21no—
68C4×Dic106—41no—
69C4×Dic106—41no—
70D4×Dic53—21no—
71D4×Dic53—21no—
72D4×Dic53—41no—
73D4×Dic53—41no—
74#13133176—21no—
75#13133176—21no—
76#13133176—41no—
77C2×C4×Dic53—41no—
78C2×C4×Dic53D411yescharacteristic
79C2×#11719908—22no—
80C2×#11719908C2311yescharacteristic
81C2×#11719908C2311yescharacteristic
82#13133151C2311yescharacteristic
83#13133151C2311yescharacteristic
84#13133151C2311yescharacteristic
85#13133151C2311yescharacteristic
86#13133150—21no—
87#13133150—21no—
88#13133149—21no—
89#13133149—21no—
90#13133148—21no—
91#13133148—21no—
92#13133141—21no—
93#13133141—21no—
94#13133138—21no—
95#13133138—21no—
96C8×Dic53D411yescharacteristic
97C8×Dic53D411yescharacteristic
98C8×Dic53C2311yescharacteristic
99C8×Dic53C2311yescharacteristic
100C2×#41×C53—21no—
101#135×C53—22no—
102#135×C53Q814yescomplemented
103C2×Q16×C53C2311yescharacteristic
104C2×SD16×C53—21no—
105C2×D8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107#117×C53—21no—
108C2×C16×C53—22no—
109C2×C16×C53C2311yescharacteristic
110C2×C16×C53C2311yescharacteristic
111#107×C53C2311yescharacteristic
112#107×C53C2311yescharacteristic
113#106×C53—21no—
114#41×C53—24no—
115#41×C53—42no—
116C2×Q8×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53C2×D411yescharacteristic
121C22×Dic53—41no—
122C22×Dic53—41no—
123C22×Dic53D811yescharacteristic
124C22×Dic53D811yescharacteristic
125C2×Dic106—41no—
126C2×Dic106—41no—
127Q16×C53—21no—
128Q16×C53C2×Q812yes—
129SD16×C53—24no—
130D8×C53—21no—
131D8×C53C2×Q812yes—
132C2×C8×C53C2×D411yescharacteristic
133C2×C8×C53C2×Q812yes—
134C2×C8×C53#4112yes—
135C2×C8×C53C2411yescharacteristic
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C22⋊C4×C53—41no—
139C42×C53—41no—
140C22⋊Dic53—41no—
141C22⋊Dic53—41no—
142C22⋊Dic53—41no—
143C22⋊Dic53—41no—
144C4⋊Dic53—21no—
145C4⋊Dic53—21no—
146C4⋊Dic53—21no—
147C4⋊Dic53—21no—
148C4⋊Dic53—41no—
149C4⋊Dic53—41no—
150C4⋊Dic53—41no—
151C4⋊Dic53—41no—
152#11719919—41no—
153#11719919—41no—
154#11719919—41no—
155#11719919—41no—
156C4×Dic53—41no—
157C4×Dic53—41no—
158C4×Dic53—41no—
159C4×Dic53—41no—
160C4×Dic53D811yescharacteristic
161C4×Dic53D811yescharacteristic
162C4×Dic53C2×D411yescharacteristic
163C4×Dic53C2×D411yescharacteristic
164C2×#88322—22no—
165C2×#88322C2×D411yescharacteristic
166C2×#88322C2×D411yescharacteristic
167C16×C53—21no—
168C16×C53—21no—
169#11719908—21no—
170#11719908—21no—
171C23×C53—41no—
172C23×C53C2×D811yescharacteristic
173Q8×C53—22no—
174Q8×C53—41no—
175D4×C53—22no—
176D4×C53—44no—
177D4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—44no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53C22⋊Q812yes—
183C2×C4×C53C2×D811yescharacteristic
184C2×C4×C53#13712yes—
185C2×C4×C53C22×D411yescharacteristic
186C2×Dic53—22no—
187C2×Dic53—41no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—82no—
193C2×Dic53—82no—
194C2×Dic53C2×D811yescharacteristic
195C2×Dic53C2×D811yescharacteristic
196C2×Dic53C2×D811yescharacteristic
197C2×Dic53C2×D811yescharacteristic
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—42no—
209C22×C53—81no—
210C22×C53—81no—
211C22×C53#47412yes—
212C22×C53C22×D811yescharacteristic
213C4×C53—42no—
214C4×C53—81no—
215C4×C53—81no—
216C4×C53#47412yes—
217C4×C53C2×C22⋊Q811yescharacteristic
218C4×C53C2×#13711yescharacteristic
219Dic53—21no—
220Dic53—21no—
221Dic53—21no—
222Dic53—81no—
223Dic53—81no—
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#2158—531no—
239C2×C4⋊C16—531no—
240#2138—531no—
241#2138D5311yescharacteristic
242C2×C53—22no—
243C2×C53—82no—
244C2×C53C2×#47411yescharacteristic
245C2×C53C2×#50811yescharacteristic
246C2×C53#341411yescharacteristic
247C2×#135C2×D5311yescharacteristic
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—531no—
266#427—1061no—
267#427C2×D5311yescharacteristic
268#416—1061no—
269C2×C4×C8—531no—
270#367—531no—
271#367—531no—
272#367—531no—
273#367—531no—
274#366—531no—
275#366—531no—
276#366—531no—
277#366—531no—
278C4⋊C16—531no—
279C4⋊C16—531no—
280C4⋊C16—531no—
281C4⋊C16—531no—
282#358—531no—
283#358—531no—
284#358C2×D5311yescharacteristic
285#358C2×D5311yescharacteristic
286#357—531no—
287#357—531no—
288#357C2×D5311yescharacteristic
289#357C2×D5311yescharacteristic
290C53#3362611yescharacteristic, complemented, Sylow
291C2×#41—21no—
292#135—22no—
293#135Dic10614yescomplemented
294C2×Q16C22×D5311yescharacteristic
295C2×SD16—21no—
296C2×D8C22×D5311yescharacteristic
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—1061no—
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—531no—
328#107—531no—
329#107—1061no—
330#107—1061no—
331#107C22×D5311yescharacteristic
332#107C22×D5311yescharacteristic
333#106—21no—
334#106—1061no—
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—24no—
347#41—42no—
348C2×Q8—21no—
349C2×Q8—2121no—
350C2×Q8—2121no—
351C2×D4—21no—
352C2×D4—41no—
353C22×C4—41no—
354C22×C4—531no—
355C22×C4—531no—
356C22×C4—2121no—
357C22×C4—2121no—
358C22×C4D4×D5311yescharacteristic
359Q16—21no—
360Q16C2×Dic10612yes—
361SD16—24no—
362D8—21no—
363D8C2×Dic10612yes—
364C2×C8—531no—
365C2×C8—531no—
366C2×C8—1062no—
367C2×C8C2×Dic10612yes—
368C2×C8D4×D5311yescharacteristic
369C2×C8#1171994712yes—
370C2×C8C23×D5311yescharacteristic
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—1061no—
374C4⋊C4—1061no—
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—2121no—
387C22⋊C4—2121no—
388C22⋊C4—2121no—
389C22⋊C4—2121no—
390C42—41no—
391C42—531no—
392C42—531no—
393C42—531no—
394C42—531no—
395C42—2121no—
396C42—2121no—
397C42—2121no—
398C42—2121no—
399C16—21no—
400C16—21no—
401C16—1061no—
402C16—1061no—
403C23—41no—
404C23D8×D5311yescharacteristic
405Q8—22no—
406Q8—41no—
407Q8—4242no—
408Q8—4242no—
409D4—22no—
410D4—44no—
411D4—41no—
412C2×C4—41no—
413C2×C4—44no—
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—2121no—
428C2×C4#1313319412yes—
429C2×C4D8×D5311yescharacteristic
430C2×C4#1313324612yes—
431C2×C4C2×D4×D5311yescharacteristic
432C8—1061no—
433C8—1061no—
434C8#1313319412yes—
435C8C22×Dic10611yescharacteristic
436C8C2×#1171994711yescharacteristic
437C22—21no—
438C22—21no—
439C22—42no—
440C22—81no—
441C22—81no—
442C22#1681090112yes—
443C22C2×D8×D5311yescharacteristic
444C4—42no—
445C4—81no—
446C4—81no—
447C4—532no—
448C4—1061no—
449C4—1061no—
450C4—1061no—
451C4—4241no—
452C4—4241no—
453C4—4241no—
454C4—4241no—
455C4#1681090112yes—
456C4C2×#1313319411yescharacteristic
457C4C2×#1313324611yescharacteristic
458C2—22no—
459C2—82no—
460C2C2×#1681090111yescharacteristic
461C2#508×D5311yescharacteristic
462C2#2028784811yescharacteristic
subgroupG/H
centerC2×C4D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2138C2
socleC22×C53C22×D8
prime radicalC2×#135×C53C22
hypercenter#2138D53
hypocenterC53#33626
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. #20285227C2semidirect
3. #20276881C2semidirect
4. #20276877C2semidirect
5. #20276874C2semidirect
6. #20276870C2semidirect
7. #20276865C2semidirect
8. #20276860C2semidirect
9. #20276855C2semidirect
10. #20276850C2semidirect
11. #20276499C2semidirect
12. #20276497C2semidirect
13. C2×#16810441C2semidirect
102. #135×C53Q8semidirect
290. C53#33626semidirect
293. #135Dic106semidirect

Maximal subgroups

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

subgroupindex
#202852272
#135×Dic532
#202768812
#202768772
#202768742
#202768702
#202768652
#202768602
#202768552
#202768502
#202764992
#202764972
C2×#168104412
#202764222
C53×#21382
#3362653