Finite Group Databasebeta

← #27401696

Subgroups — #27401696

Order 13568.

692 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283369C211yesmaximal, characteristic
2C2×#16810659C211yesmaximal, characteristic
3#20275465C212yesmaximal, complemented
4#20275314C212yesmaximal, complemented
5#20275311C212yesmaximal, complemented
6#20274814C211yesmaximal, characteristic
7#20274812C211yesmaximal, characteristic
8#20274751C211yesmaximal, characteristic
9#20274600C212yesmaximal, complemented
10#20274494C211yesmaximal, characteristic
11C53×#2924C211yesmaximal, characteristic
12C23×#88322C2211yescharacteristic
13C2×#13133176C2211yescharacteristic
14#16810681—22no—
15#16810677—21no—
16#16810673—21no—
17C2×#13133152C2212yes—
18#16810659C411yescharacteristic, complemented
19#16810659C411yescharacteristic, complemented
20#16810659C411yescharacteristic, complemented
21#16810659C411yescharacteristic, complemented
22#16810659C2211yescharacteristic
23#16810659C2211yescharacteristic
24#16810659C2211yescharacteristic
25#16810659C2211yescharacteristic
26C2×#13133150C411yescharacteristic, complemented
27C2×#13133150C411yescharacteristic, complemented
28C2×#13133148C411yescharacteristic, complemented
29C2×#13133148C411yescharacteristic, complemented
30#16810649C2212yes—
31#16810587C2211yescharacteristic
32#16810578C2212yes—
33C2×#13133127C2212yes—
34C4×C22⋊Dic53C2211yescharacteristic
35#16810387C2212yes—
36#16810384C2212yes—
37#16810384C2212yes—
38#16810382C2212yes—
39#16810382C2212yes—
40#16810377C2212yes—
41C2×#117×C53C2211yescharacteristic
42C53×#433—22no—
43C53×#432—21no—
44C53×#431—21no—
45C53×#424C2212yes—
46C53×#407C2211yescharacteristic
47C2×C22⋊C8×C53C2211yescharacteristic
48C53×#405C2212yes—
49C2×#11719917C2312yes—
50C22×#88322—24no—
51C22×#88322C2×C411yescharacteristic
52C22×#88322C2×C411yescharacteristic
53C22×#88322C2×C411yescharacteristic
54C22×#88322C2×C411yescharacteristic
55C22×#88322C2311yescharacteristic
56C22×#88322C2311yescharacteristic
57#13133176—21no—
58#13133176—21no—
59#13133176C2×C411yescharacteristic
60#13133176C2×C411yescharacteristic
61#13133176C2311yescharacteristic
62#13133176C2311yescharacteristic
63C2×C4⋊Dic53C2×C411yescharacteristic
64C2×C4⋊Dic53C2×C411yescharacteristic
65C2×C22⋊Dic53—21no—
66C2×C22⋊Dic53—21no—
67C2×C4×Dic53—21no—
68C2×C4×Dic53C2312yes—
69#13133152—24no—
70#13133150C2×C411yescharacteristic
71#13133150C2×C411yescharacteristic
72#13133150C2×C411yescharacteristic
73#13133150C2×C411yescharacteristic
74#13133150C2×C411yescharacteristic
75#13133150C2×C411yescharacteristic
76#13133150C2×C411yescharacteristic
77#13133150C2×C411yescharacteristic
78#13133148C2×C411yescharacteristic
79#13133148C2×C411yescharacteristic
80#13133148C2×C411yescharacteristic
81#13133148C2×C411yescharacteristic
82#13133148C2×C411yescharacteristic
83#13133148C2×C411yescharacteristic
84#13133148C2×C411yescharacteristic
85#13133148C2×C411yescharacteristic
86#13133146—22no—
87#13133146—22no—
88C22⋊#88322—21no—
89C22⋊#88322—21no—
90#13133127—22no—
91#13133127—24no—
92C8×Dic53—22no—
93#13133124—21no—
94C23×C4×C53C2311yescharacteristic
95C2×M16×C53C2312yes—
96C22×C8×C53C2312yes—
97#117×C53—21no—
98#117×C53—21no—
99#117×C53C2×C411yescharacteristic
100#117×C53C2×C411yescharacteristic
101#117×C53C2311yescharacteristic
102#117×C53C2311yescharacteristic
103C2×C4⋊C4×C53C2×C411yescharacteristic
104C2×C4⋊C4×C53C2×C411yescharacteristic
105C2×C22⋊C4×C53—21no—
106C2×C42×C53—21no—
107C4⋊C8×C53—22no—
108C4⋊C8×C53—22no—
109C22⋊C8×C53—21no—
110C22⋊C8×C53—21no—
111C22⋊C8×C53D411yescharacteristic
112C22⋊C8×C53D411yescharacteristic
113C22⋊C8×C53D412yes—
114#97×C53—22no—
115C4×C8×C53—22no—
116C24×C53#4111yescharacteristic
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53C22×C411yescharacteristic
122C22×C4×C53C22×C411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53#4111yescharacteristic
126C22×C4×C53C2411yescharacteristic
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—22no—
130M16×C53—22no—
131M16×C53—22no—
132C2×C8×C53—22no—
133C2×C8×C53—22no—
134C2×C8×C53—22no—
135C2×C8×C53C2×D412yes—
136C2×C8×C53C2×D412yes—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53C22×C411yescharacteristic
140C4⋊C4×C53C22×C411yescharacteristic
141C4⋊C4×C53C22×C411yescharacteristic
142C4⋊C4×C53C22×C411yescharacteristic
143C22⋊C4×C53—21no—
144C22⋊C4×C53—21no—
145C22⋊C4×C53—21no—
146C22⋊C4×C53—21no—
147C42×C53—21no—
148C42×C53—21no—
149C42×C53—21no—
150C42×C53—21no—
151C22⋊Dic53—21no—
152C22⋊Dic53—21no—
153C22⋊Dic53—21no—
154C22⋊Dic53—21no—
155C22⋊Dic53—42no—
156C4⋊Dic53—21no—
157C4⋊Dic53—21no—
158C4⋊Dic53C22×C411yescharacteristic
159C4⋊Dic53C22×C411yescharacteristic
160C4⋊Dic53C22×C411yescharacteristic
161C4⋊Dic53C22×C411yescharacteristic
162C4×Dic53—21no—
163C4×Dic53—21no—
164C4×Dic53—21no—
165C4×Dic53—21no—
166C4×Dic53—22no—
167C4×Dic53—22no—
168#11719917—22no—
169#11719917—22no—
170C2×#88322—22no—
171C2×#88322—22no—
172C2×#88322—22no—
173C2×#88322—22no—
174C2×#88322—22no—
175C2×#88322—22no—
176C2×#88322C22×C411yescharacteristic
177C2×#88322C22×C411yescharacteristic
178C2×#88322C22×C411yescharacteristic
179C2×#88322C22×C411yescharacteristic
180C2×#88322#4114yes—
181C23×C53—22no—
182C23×C53—22no—
183C23×C53—41no—
184C23×C53C4×D411yescharacteristic
185C23×C53C4×D411yescharacteristic
186C23×C53C2×#4111yescharacteristic
187C2×C4×C53—22no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—22no—
193C2×C4×C53—22no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—42no—
199C2×C4×C53—81no—
200C2×C4×C53C4×D411yescharacteristic
201C2×C4×C53C4×D411yescharacteristic
202C2×C4×C53C23×C411yescharacteristic
203C2×C4×C53C22×D411yescharacteristic
204C2×C4×C53C2×#4112yes—
205C2×C4×C532−1+412yes—
206C2×Dic53—21no—
207C2×Dic53—21no—
208C2×Dic53—21no—
209C2×Dic53—21no—
210C2×Dic53—42no—
211C2×Dic53—42no—
212C2×Dic53—42no—
213C2×Dic53—42no—
214C2×Dic53—81no—
215C8×C53—22no—
216C8×C53—22no—
217C8×C53—42no—
218#88322—21no—
219#88322—21no—
220#88322—42no—
221#88322#11712yes—
222#88322#11712yes—
223#22607—531nomaximal, Sylow
224C22×C53—21no—
225C22×C53—21no—
226C22×C53—21no—
227C22×C53—21no—
228C22×C53—21no—
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—44no—
232C22×C53#44412yes—
233C22×C53C2×C4×D411yescharacteristic
234C22×C53#55212yes—
235C4×C53—21no—
236C4×C53—21no—
237C4×C53—41no—
238C4×C53—41no—
239C4×C53—41no—
240C4×C53—41no—
241C4×C53C2×#11711yescharacteristic
242C4×C53#51911yescharacteristic
243C4×C53#54712yes—
244Dic53—41no—
245Dic53—41no—
246Dic53—41no—
247Dic53—41no—
248Dic53—82no—
249#2924—531no—
250#2924D5311yescharacteristic
251C2×#427—531no—
252#1946—532no—
253#1922—532no—
254#1922—532no—
255#1826—531no—
256#1826—531no—
257#1810—531no—
258#1773—532no—
259#1752—531no—
260C2×C53—22no—
261C2×C53—22no—
262C2×C53—41no—
263C2×C53#234411yescharacteristic
264C2×C53C2×#44411yescharacteristic
265C2×C53#295711yescharacteristic
266C23×C8—531no—
267C2×#117—531no—
268C2×#117C2×D5311yescharacteristic
269#433—22no—
270#433—1062no—
271#432—21no—
272#432—1061no—
273#431—21no—
274#431—1061no—
275C2×#108—532no—
276#427—531no—
277#427—531no—
278#427—531no—
279#427—531no—
280#427—531no—
281#427—531no—
282#427—531no—
283#427—531no—
284C2×#107—531no—
285C2×#107—531no—
286C2×#106—531no—
287C2×#106—531no—
288#424—532no—
289#424C2×D5312yes—
290#407—531no—
291#407C2×D5311yescharacteristic
292C2×C22⋊C8C2×D5311yescharacteristic
293#405—532no—
294#405C2×D5312yes—
295C2×#97—532no—
296C4×C22⋊C4—531no—
297#341—532no—
298#339—532no—
299#339—532no—
300#339—532no—
301#339—532no—
302#336—532no—
303C53#2260711yescharacteristic, complemented, Sylow
304C23×C4C22×D5311yescharacteristic
305C2×M16—532no—
306C2×M16C22×D5312yes—
307C22×C8—531no—
308C22×C8—531no—
309C22×C8—531no—
310C22×C8—531no—
311C22×C8—531no—
312C22×C8—531no—
313C22×C8—1064no—
314C22×C8C22×D5312yes—
315#117—21no—
316#117—21no—
317#117—531no—
318#117—531no—
319#117—531no—
320#117—531no—
321#117—1061no—
322#117—1061no—
323#117C4×D5311yescharacteristic
324#117C4×D5311yescharacteristic
325#117C22×D5311yescharacteristic
326#117C22×D5311yescharacteristic
327C2×C4⋊C4—531no—
328C2×C4⋊C4—531no—
329C2×C4⋊C4C4×D5311yescharacteristic
330C2×C4⋊C4C4×D5311yescharacteristic
331C2×C22⋊C4—21no—
332C2×C22⋊C4—1061no—
333C2×C22⋊C4—1061no—
334C2×C42—21no—
335C2×C42—532no—
336C2×C42—1061no—
337#108—1064no—
338#107—531no—
339#107—531no—
340#107—531no—
341#107—531no—
342#107—531no—
343#107—531no—
344#107—531no—
345#107—531no—
346#106—531no—
347#106—531no—
348#106—531no—
349#106—531no—
350#106—531no—
351#106—531no—
352#106—531no—
353#106—531no—
354C4⋊C8—22no—
355C4⋊C8—22no—
356C4⋊C8—1062no—
357C4⋊C8—1062no—
358C22⋊C8—21no—
359C22⋊C8—21no—
360C22⋊C8—1061no—
361C22⋊C8—1061no—
362C22⋊C8C22⋊D5311yescharacteristic
363C22⋊C8C22⋊D5311yescharacteristic
364C22⋊C8C22⋊D5312yes—
365#97—22no—
366#97—1064no—
367#97—1062no—
368C4×C8—22no—
369C4×C8—1062no—
370#95—1061no—
371C24#1171994511yescharacteristic
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—1061no—
377C22×C4—1061no—
378C22×C4—1062no—
379C22×C4C2×C4×D5311yescharacteristic
380C22×C4C2×C4×D5311yescharacteristic
381C22×C4#1171994511yescharacteristic
382C22×C4C2×C22⋊D5311yescharacteristic
383C22×C4C2×C22⋊D5311yescharacteristic
384C22×C4C23×D5311yescharacteristic
385M16—22no—
386M16—22no—
387M16—1062no—
388M16—1062no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—531no—
393C2×C8—531no—
394C2×C8—534no—
395C2×C8—531no—
396C2×C8—531no—
397C2×C8—1062no—
398C2×C8—1062no—
399C2×C8—1062no—
400C2×C8—1062no—
401C2×C8—1062no—
402C2×C8—1062no—
403C2×C8C2×C22⋊D5312yes—
404C2×C8C2×C22⋊D5312yes—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—531no—
408C4⋊C4—531no—
409C4⋊C4—531no—
410C4⋊C4—531no—
411C4⋊C4—1061no—
412C4⋊C4—1061no—
413C4⋊C4C2×C4×D5311yescharacteristic
414C4⋊C4C2×C4×D5311yescharacteristic
415C4⋊C4C2×C4×D5311yescharacteristic
416C4⋊C4C2×C4×D5311yescharacteristic
417C22⋊C4—21no—
418C22⋊C4—21no—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—1061no—
422C22⋊C4—1061no—
423C22⋊C4—1061no—
424C22⋊C4—1061no—
425C22⋊C4—2122no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—1061no—
431C42—1061no—
432C42—1061no—
433C42—1061no—
434C42—1062no—
435C42—1062no—
436C23—22no—
437C23—22no—
438C23—41no—
439C23C4×C22⋊D5311yescharacteristic
440C23C4×C22⋊D5311yescharacteristic
441C23C2×#1171994511yescharacteristic
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—42no—
454C2×C4—81no—
455C2×C4—1061no—
456C2×C4—1061no—
457C2×C4—1061no—
458C2×C4—1061no—
459C2×C4—4241no—
460C2×C4—2122no—
461C2×C4—2122no—
462C2×C4—2122no—
463C2×C4—2122no—
464C2×C4C4×C22⋊D5311yescharacteristic
465C2×C4C4×C22⋊D5311yescharacteristic
466C2×C4C22×C4×D5311yescharacteristic
467C2×C4C22×C22⋊D5311yescharacteristic
468C2×C4#41×D5312yes—
469C2×C4#1313326412yes—
470C8—22no—
471C8—22no—
472C8—42no—
473C8—532no—
474C8—532no—
475C8—1061no—
476C8—1061no—
477C8—2122no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—44no—
486C22#1681074312yes—
487C22C2×C4×C22⋊D5311yescharacteristic
488C22#1681130612yes—
489C4—21no—
490C4—21no—
491C4—41no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—4242no—
496C4—2121no—
497C4—2121no—
498C4—2121no—
499C4—2121no—
500C4#117×D5311yescharacteristic
501C4#1681110011yescharacteristic
502C4#1681127212yes—
503C2—22no—
504C2—22no—
505C2—41no—
506C2#2027758511yescharacteristic
507C2C2×#1681074311yescharacteristic
508C2#2028363011yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2924C2
socleC22×C53C2×C4×D4
prime radicalC24×C53#41
cosocleC22×C4×C53C24
hypercenter#2924D53
hypocenterC53#22607
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ #2924

Complements

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

NH ≅ G/Ntype
3. #20275465C2semidirect
4. #20275314C2semidirect
5. #20275311C2semidirect
9. #20274600C2semidirect
18. #16810659C4semidirect
19. #16810659C4semidirect
20. #16810659C4semidirect
21. #16810659C4semidirect
26. C2×#13133150C4semidirect
27. C2×#13133150C4semidirect
28. C2×#13133148C4semidirect
29. C2×#13133148C4semidirect
303. C53#22607semidirect

Maximal subgroups

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

subgroupindex
#202833692
C2×#168106592
#202754652
#202753142
#202753112
#202748142
#202748122
#202747512
#202746002
#202744942
C53×#29242
#2260753