Finite Group Databasebeta

← #27366924

Subgroups — #27366924

Order 13568.

696 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285109C211yesmaximal, characteristic, complemented
2#20284715C211yesmaximal, characteristic
3C2×#16810913C211yesmaximal, characteristic, complemented
4#20274290C212yesmaximal, complemented
5#20274248C212yesmaximal, complemented
6#20274166C212yesmaximal, complemented
7#20274124C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273397C211yesmaximal, characteristic, complemented
10#20273393C211yesmaximal, characteristic, complemented
11D4×M16×C53C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13C2×#13133217C2211yescharacteristic
14#16810983—22no—
15#16810974C2212yescomplemented
16#16810970C2212yescomplemented
17#16810936—21no—
18#16810913—21no—
19#16810913—21no—
20#16810913C2212yes—
21#16810913C2212yes—
22#16810903C2211yescharacteristic
23#16810903C2211yescharacteristic
24#16810818C2211yescharacteristic
25#16810818C2211yescharacteristic
26#16810757—21no—
27C4×#11719922—21no—
28C4×#11719922—21no—
29#16810676C2211yescharacteristic
30C2×#13133148—21no—
31C2×C4⋊#88322C2211yescharacteristic
32#16810625—21no—
33#16810625—21no—
34C2×#13133138—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810371C2212yescomplemented
38#16810371C2212yescomplemented
39#16810370C2212yescomplemented
40#16810366C2212yescomplemented
41C22×M16×C53—21no—
42C2×C4×D4×C53C2211yescharacteristic
43C53×#436C2212yes—
44C53×#435—22no—
45C8×D4×C53C2212yes—
46C53×#423C2211yescharacteristic
47C53×#407—21no—
48C4×M16×C53C2211yescharacteristic
49C2×#11719945—21no—
50C22×Dic106—21no—
51C2×#11719922—21no—
52C2×#11719922—21no—
53C22×#88322—21no—
54#13133217—21no—
55#13133217—21no—
56#13133217—21no—
57#13133217C2311yescharacteristic
58#13133217C2312yes—
59#13133214C2311yescharacteristic
60#13133205—21no—
61#13133191—21no—
62#13133191—21no—
63C4×D212—21no—
64C4×D212—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67#13133150—21no—
68#13133150—21no—
69#13133148—22no—
70#13133148—22no—
71C4⋊#88322—21no—
72C4⋊#88322C2311yescharacteristic
73C4⋊#88322C2311yescharacteristic
74C4⋊#88322C2312yes—
75#13133142—22no—
76#13133142—22no—
77#13133139—22no—
78#13133139—22no—
79#13133138—21no—
80#13133138—21no—
81#13133138—22no—
82#13133138—22no—
83#13133137—21no—
84#13133137—21no—
85C22⋊#88322—21no—
86C4×#88322—21no—
87C22×D4×C53D411yescharacteristic
88C23×C4×C53—21no—
89C2×M16×C53—22no—
90C2×M16×C53—21no—
91C2×M16×C53—44no—
92C2×M16×C53D411yescharacteristic
93C2×M16×C53D411yescharacteristic
94C22×C8×C53—22no—
95C4×D4×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53D411yescharacteristic
98C4×D4×C53D411yescharacteristic
99C4×D4×C53C2311yescharacteristic
100C4×D4×C53C2311yescharacteristic
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C22⋊C4×C53—21no—
103C2×C42×C53C2311yescharacteristic
104C4⋊C8×C53—21no—
105C4⋊C8×C53C2312yes—
106C22⋊C8×C53—22no—
107C22⋊C8×C53—22no—
108#97×C53D412yes—
109C4×C8×C53C2312yes—
110C24×C53—21no—
111C2×D4×C53—22no—
112C2×D4×C53—21no—
113C2×D4×C53—41no—
114C2×D4×C53C2×D411yescharacteristic
115C2×D4×C53C2×D411yescharacteristic
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—42no—
122C22×C4×C53—42no—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53#4111yescharacteristic
126C2×C22⋊D53—41no—
127C22×Dic53—41no—
128#11719945—42no—
129#11719945—42no—
130C2×D212—21no—
131C2×D212—21no—
132C2×D212—21no—
133C2×C4×D53—41no—
134C2×Dic106—21no—
135C2×Dic106—22no—
136C2×Dic106—41no—
137C2×Dic106—82no—
138M16×C53—22no—
139M16×C53—22no—
140M16×C53—44no—
141M16×C53—42no—
142C2×C8×C53—22no—
143C2×C8×C53—44no—
144C2×C8×C53C2×D412yes—
145C2×C8×C53C2×D412yes—
146C4⋊C4×C53—21no—
147C4⋊C4×C53C2×D411yescharacteristic
148C4⋊C4×C53C2×D411yescharacteristic
149C22⋊C4×C53—21no—
150C22⋊C4×C53—21no—
151C22⋊C4×C53—21no—
152C22⋊C4×C53—21no—
153C42×C53C2×D411yescharacteristic
154C42×C53C2×D411yescharacteristic
155C42×C53#4111yescharacteristic
156C42×C53C2411yescharacteristic
157#11719922—42no—
158#11719922—42no—
159#11719921—41no—
160#11719921—41no—
161C4⋊Dic53—22no—
162C4⋊Dic53—21no—
163C4⋊Dic53—22no—
164C4⋊Dic53—21no—
165C4⋊Dic53—41no—
166C4⋊Dic53—41no—
167C2×#88322—21no—
168C2×#88322—21no—
169C2×#88322—22no—
170C2×#88322—42no—
171C23×C53—21no—
172C23×C53—21no—
173C23×C53—42no—
174C23×C53—42no—
175C23×C53—81no—
176C23×C53#12111yescharacteristic
177C22×D53—41no—
178D4×C53—21no—
179D4×C53—21no—
180D4×C53—21no—
181D4×C53—42no—
182D4×C53#13712yes—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—41no—
188C2×C4×C53—42no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—42no—
194C2×C4×C53—42no—
195C2×C4×C53#12111yescharacteristic
196C2×C4×C53#12111yescharacteristic
197C2×C4×C53#12111yescharacteristic
198C2×C4×C53#13512yes—
199C2×C4×C53AGL(1,ℤ/8)12yes—
200C2×C4×C53C22×D411yescharacteristic
201C2×C4×C53C22×D411yescharacteristic
202C2×C4×C53C2×#4111yescharacteristic
203C22⋊D53—82no—
204C2×Dic53—42no—
205C2×Dic53—41no—
206C2×Dic53—82no—
207D212—41no—
208D212—41no—
209D212—41no—
210D212—41no—
211D212—81no—
212C4×D53—82no—
213Dic106—41no—
214Dic106—41no—
215Dic106—82no—
216Dic106—82no—
217C8×C53—22no—
218C8×C53—22no—
219C8×C53—42no—
220#88322—41no—
221#88322—42no—
222#88322—41no—
223#19168—531nomaximal, Sylow
224C22×C53—21no—
225C22×C53—21no—
226C22×C53—22no—
227C22×C53—41no—
228C22×C53—41no—
229C22×C53—41no—
230C22×C53—41no—
231C22×C53—41no—
232C22×C53—84no—
233C22×C53#46312yes—
234C22×C53C2×#12111yescharacteristic
235C2×D53—81no—
236C2×D53—81no—
237C2×D53—81no—
238C4×C53—21no—
239C4×C53—21no—
240C4×C53—21no—
241C4×C53—41no—
242C4×C53—41no—
243C4×C53#46312yes—
244C4×C53C2×#13511yescharacteristic
245C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
246C4×C53C2×#13711yescharacteristic
247C4×C53#57511yescharacteristic
248Dic53—81no—
249Dic53—82no—
250#3141—531no—
251#3100—531no—
252C2×#476—531no—
253D4×M16D5311yescharacteristic, complemented
254#1703—532no—
255#1697—532no—
256#1683—532no—
257#1677—532no—
258#1565—531no—
259#1548—531no—
260#1547—531no—
261C2×C53—22no—
262C2×C53—42no—
263C2×C53—81no—
264C2×C53C2×#46311yescharacteristic
265C2×C53#305611yescharacteristic
266C2×C53#306811yescharacteristic
267D53—161no—
268C22×M16—21no—
269#532—531no—
270C2×C4⋊Q8—531no—
271C2×C4×D4C2×D5311yescharacteristic
272#489—1062no—
273#487—532no—
274#486—532no—
275#480—1061no—
276#476—532no—
277#476—532no—
278#476—1061no—
279#476—1061no—
280#474—531no—
281#474—531no—
282#459—531no—
283#459—531no—
284#447—1061no—
285C4×D8—1061no—
286C4×D8—1061no—
287#436C2×D5312yes—
288#435—22no—
289C8×D4C2×D5312yes—
290#431—531no—
291C2×#106—1061no—
292#423C2×D5311yescharacteristic
293C2×C4⋊C8—531no—
294#417—1061no—
295#417—1061no—
296C2×#102—1061no—
297#407—21no—
298C4×M16C2×D5311yescharacteristic
299#335—532no—
300#334—532no—
301#333—532no—
302#333—532no—
303#332—532no—
304#331—532no—
305C53#1916811yescharacteristic, complemented, Sylow
306C2×#41—1061no—
307C22×Q8—1061no—
308C22×D4D21211yescharacteristic
309C23×C4—21no—
310C2×D8—1061no—
311C2×D8—1061no—
312C2×M16—21no—
313C2×M16—22no—
314C2×M16—44no—
315C2×M16C22⋊D5311yescharacteristic
316C2×M16C22⋊D5311yescharacteristic
317C22×C8—22no—
318C22×C8—1061no—
319C4⋊Q8—532no—
320C4⋊Q8—531no—
321C4⋊Q8—1061no—
322C4⋊Q8—1061no—
323C4⋊Q8—1061no—
324#127—531no—
325#124—1061no—
326#121—1061no—
327#121—1061no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—1061no—
331C4×D4—1061no—
332C4×D4D21211yescharacteristic
333C4×D4D21211yescharacteristic
334C4×D4C22×D5311yescharacteristic
335C4×D4C22×D5311yescharacteristic
336C2×C4⋊C4—1061no—
337C2×C4⋊C4—1061no—
338C2×C4⋊C4D21211yescharacteristic
339C2×C22⋊C4—21no—
340C2×C42C22×D5311yescharacteristic
341#107—1061no—
342#107—1061no—
343#106—1062no—
344#106—1062no—
345C4⋊C8—21no—
346C4⋊C8—532no—
347C4⋊C8—531no—
348C4⋊C8—531no—
349C4⋊C8—1061no—
350C4⋊C8C22×D5312yes—
351#103—1062no—
352#103—1062no—
353#102—1062no—
354#102—1062no—
355#102—1061no—
356#102—1061no—
357#102—1062no—
358#102—1062no—
359#102—1061no—
360#102—1061no—
361C22⋊C8—22no—
362C22⋊C8—22no—
363C22⋊C8—1061no—
364#97C22⋊D5312yes—
365C4×C8—1061no—
366C4×C8C22×D5312yes—
367C24—21no—
368#41—2122no—
369#41—2122no—
370C2×Q8—1061no—
371C2×Q8—1062no—
372C2×Q8—4242no—
373C2×Q8—2121no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—41no—
377C2×D4—1061no—
378C2×D4—1061no—
379C2×D4—1061no—
380C2×D4—2121no—
381C2×D4C2×D21211yescharacteristic
382C2×D4C2×D21211yescharacteristic
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—2121no—
391C22×C4—2121no—
392C22×C4C2×D21211yescharacteristic
393C22×C4#1171994511yescharacteristic
394C22×C4C2×C22⋊D5311yescharacteristic
395D8—2122no—
396D8—2122no—
397M16—22no—
398M16—22no—
399M16—44no—
400M16—42no—
401C2×C8—22no—
402C2×C8—44no—
403C2×C8—1061no—
404C2×C8—1061no—
405C2×C8—1062no—
406C2×C8—2122no—
407C2×C8C2×C22⋊D5312yes—
408C2×C8C2×C22⋊D5312yes—
409C4⋊C4—21no—
410C4⋊C4—1062no—
411C4⋊C4—1062no—
412C4⋊C4—1061no—
413C4⋊C4—1061no—
414C4⋊C4—2121no—
415C4⋊C4—2121no—
416C4⋊C4C2×D21211yescharacteristic
417C4⋊C4C2×D21211yescharacteristic
418C22⋊C4—21no—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—2121no—
423C22⋊C4—2121no—
424C42C2×D21211yescharacteristic
425C42#1171994511yescharacteristic
426C42C2×C22⋊D5311yescharacteristic
427C42C23×D5311yescharacteristic
428C23—21no—
429C23—21no—
430C23—42no—
431C23—42no—
432C23—81no—
433C23—2121no—
434C23#1313319111yescharacteristic
435Q8—4242no—
436Q8—4242no—
437Q8—2121no—
438Q8—2121no—
439D4—21no—
440D4—21no—
441D4—21no—
442D4—42no—
443D4—4241no—
444D4—4242no—
445D4—2121no—
446D4—2121no—
447D4—2121no—
448D4—2121no—
449D4#1313325012yes—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—4242no—
463C2×C4—4242no—
464C2×C4—2121no—
465C2×C4—2122no—
466C2×C4#1313319111yescharacteristic
467C2×C4#1313319111yescharacteristic
468C2×C4#1313319111yescharacteristic
469C2×C4#1313323712yes—
470C2×C4#1313324312yes—
471C2×C4C22×C22⋊D5311yescharacteristic
472C2×C4C22×D21211yescharacteristic
473C2×C4C2×#1171994511yescharacteristic
474C8—22no—
475C8—22no—
476C8—42no—
477C8—2121no—
478C8—2122no—
479C8—2121no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—84no—
489C22—4241no—
490C22—4241no—
491C22—4241no—
492C22#1681083812yes—
493C22C2×#1313319111yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—41no—
498C4—41no—
499C4—4242no—
500C4—4241no—
501C4#1681083812yes—
502C4C2×#1313323711yescharacteristic
503C4C2×#1313324311yescharacteristic
504C4C2×#1313325011yescharacteristic
505C4#1681142311yescharacteristic
506C2—22no—
507C2—42no—
508C2—81no—
509C2—8481no—
510C2C2×#1681083811yescharacteristic
511C2#2028434511yescharacteristic
512C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingD4×M16×C53C2
socleC22×C53C2×#121
prime radical#20284715C2
hypercenterD4×M16D53
hypocenterC53#19168
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
1. #20285109C2semidirect
3. C2×#16810913C2semidirect
4. #20274290C2semidirect
5. #20274248C2semidirect
6. #20274166C2semidirect
7. #20274124C2semidirect
8. #20273467C2semidirect
9. #20273397C2semidirect
10. #20273393C2semidirect
11. D4×M16×C53C2semidirect
15. #16810974C22semidirect
16. #16810970C22semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810371C22semidirect
38. #16810371C22semidirect
39. #16810370C22semidirect
40. #16810366C22semidirect
253. D4×M16D53semidirect
305. C53#19168semidirect

Maximal subgroups

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

subgroupindex
#202851092
#202847152
C2×#168109132
#202742902
#202742482
#202741662
#202741242
#202734672
#202733972
#202733932
D4×M16×C532
#1916853