Finite Group Databasebeta

← #27366876

Subgroups — #27366876

Order 13568.

706 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284719C211yesmaximal, characteristic
2C2×#16810825C211yesmaximal, characteristic
3M16×D212C211yesmaximal, characteristic
4#20274225C212yesmaximal, complemented
5#20274211C212yesmaximal, complemented
6#20274101C212yesmaximal, complemented
7#20274087C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273405C211yesmaximal, characteristic, complemented
10#20273400C211yesmaximal, characteristic, complemented
11C53×#3165C211yesmaximal, characteristic, complemented
12C22×#11719924—21no—
13C2×M16×D53—21no—
14C2×C4×D212C2211yescharacteristic
15#16810951—21no—
16#16810910C2211yescharacteristic
17#16810910C2211yescharacteristic
18#16810825—21no—
19#16810825—21no—
20#16810825C2212yes—
21#16810825C2211yescharacteristic
22#16810825C2211yescharacteristic
23#16810825C2212yes—
24#16810773—21no—
25C4×#11719924—21no—
26C4×#11719924—21no—
27#16810702C2212yes—
28#16810696—22no—
29C8×D212C2212yes—
30#16810676C2211yescharacteristic
31#16810647C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33C2×#13133141—21no—
34#16810586—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810372C2212yescomplemented
38#16810369C2212yescomplemented
39#16810369C2212yescomplemented
40#16810369C2212yescomplemented
41C53×#533C2211yescharacteristic
42C2×C4⋊Q8×C53C2211yescharacteristic
43C53×#501—22no—
44C53×#500C2212yescomplemented
45C53×#498C2212yescomplemented
46C53×#428—21no—
47C53×#428—21no—
48C4×M16×C53C2211yescharacteristic
49C22×D212D411yescharacteristic
50C22×C4×D53—21no—
51C2×#11719924—21no—
52C2×#11719924—21no—
53C2×#11719924—22no—
54C2×#11719924—22no—
55C2×#11719924—44no—
56C2×#11719917—21no—
57C2×#11719912—22no—
58M16×D53—44no—
59C22×#88322—21no—
60C2×C8×D53—22no—
61C4×D212—21no—
62C4×D212—21no—
63C4×D212D411yescharacteristic
64C4×D212D411yescharacteristic
65C4×D212C2311yescharacteristic
66C4×D212C2311yescharacteristic
67C2×C4⋊Dic53D411yescharacteristic
68C2×#11719921—21no—
69#13133148—21no—
70#13133148—21no—
71C4⋊#88322—21no—
72C4⋊#88322—21no—
73C4⋊#88322C2312yes—
74C4⋊#88322C2311yescharacteristic
75C4⋊#88322C2311yescharacteristic
76C4⋊#88322C2312yes—
77#13133141—22no—
78#13133141—22no—
79#13133141—21no—
80#13133141—21no—
81#13133137—21no—
82#13133137—21no—
83C22⋊#88322—21no—
84#13133129—22no—
85#13133129—22no—
86C4×#88322—21no—
87C22×Q8×C53—21no—
88C2×M16×C53D411yescharacteristic
89C2×M16×C53D411yescharacteristic
90C4⋊Q8×C53—21no—
91C4⋊Q8×C53—21no—
92C4⋊Q8×C53—21no—
93C4⋊Q8×C53C2311yescharacteristic
94C4⋊Q8×C53C2312yes—
95C4⋊Q8×C53C2311yescharacteristic
96#125×C53—21no—
97C22⋊Q8×C53—21no—
98C22⋊Q8×C53—21no—
99C4×Q8×C53—21no—
100C4×Q8×C53—21no—
101#117×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C42×C53C2311yescharacteristic
105#107×C53—22no—
106#107×C53—22no—
107#106×C53—22no—
108#106×C53—22no—
109#97×C53D412yes—
110C4×C8×C53C2312yes—
111C23×D53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—22no—
114C2×Q8×C53—21no—
115C2×Q8×C53—42no—
116C2×Q8×C53—42no—
117C2×Q8×C53—41no—
118C22×C4×C53—41no—
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53#4111yescharacteristic
122C22×Dic53—21no—
123C2×D212—22no—
124C2×D212—21no—
125C2×D212—41no—
126C2×D212C2×D411yescharacteristic
127C2×D212C2×D411yescharacteristic
128C2×C4×D53—21no—
129C2×C4×D53—21no—
130C2×C4×D53—21no—
131C2×C4×D53—21no—
132C2×C4×D53—42no—
133C2×C4×D53—42no—
134M16×C53—22no—
135M16×C53SD1614yescomplemented
136C2×C8×C53C2×D412yes—
137C2×C8×C53C2×D412yes—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—41no—
145C4⋊C4×C53—41no—
146C4⋊C4×C53—41no—
147C4⋊C4×C53—41no—
148C22⋊C4×C53—41no—
149C42×C53—41no—
150C42×C53C2×D411yescharacteristic
151C42×C53C2×D411yescharacteristic
152C42×C53#4111yescharacteristic
153C42×C53C2411yescharacteristic
154#11719924—42no—
155#11719924—44no—
156#11719924—42no—
157#11719924—44no—
158#11719921—21no—
159#11719921—21no—
160#11719921—21no—
161#11719921—21no—
162C4⋊Dic53—21no—
163C4⋊Dic53C2×D411yescharacteristic
164C4⋊Dic53C2×D411yescharacteristic
165#11719917—42no—
166C2×#88322—21no—
167C2×#88322—21no—
168C2×#88322—22no—
169C2×#88322—22no—
170C2×#88322—42no—
171#11719912—44no—
172C8×D53—44no—
173C23×C53#12111yescharacteristic
174C22×D53—21no—
175C22×D53—21no—
176C22×D53—42no—
177C22×D53—42no—
178C22×D53—81no—
179Q8×C53—42no—
180Q8×C53—41no—
181Q8×C53—42no—
182Q8×C53—41no—
183Q8×C53—82no—
184Q8×C53—81no—
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—82no—
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53#12111yescharacteristic
194C2×C4×C53#12111yescharacteristic
195C2×C4×C53C2×SD1612yes—
196C2×C4×C53#13712yes—
197C2×C4×C53C22×D411yescharacteristic
198C2×C4×C53C22×D411yescharacteristic
199C2×C4×C53C2×#4111yescharacteristic
200C2×Dic53—21no—
201C2×Dic53—21no—
202C2×Dic53—42no—
203D212—21no—
204D212—21no—
205D212—21no—
206D212—42no—
207D212AGL(1,ℤ/8)12yes—
208C4×D53—41no—
209C4×D53—41no—
210C4×D53—41no—
211C4×D53—41no—
212C4×D53—42no—
213C4×D53—42no—
214C8×C53—22no—
215C8×C53C2×SD1614yescomplemented
216#88322—42no—
217#88322—42no—
218#88322—41no—
219#88322—41no—
220#19164—531nomaximal, Sylow
221C22×C53—21no—
222C22×C53—21no—
223C22×C53#46112yes—
224C22×C53C2×#12111yescharacteristic
225C2×D53—22no—
226C2×D53—41no—
227C2×D53—41no—
228C2×D53—41no—
229C2×D53—41no—
230C2×D53—41no—
231C2×D53—84no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—82no—
236C4×C53—81no—
237C4×C53—81no—
238C4×C53#46112yes—
239C4×C53C22×SD1611yescharacteristic
240C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
241C4×C53C2×#13711yescharacteristic
242C4×C53#57511yescharacteristic
243Dic53—41no—
244Dic53—41no—
245#3165D5311yescharacteristic, complemented
246#3101—531no—
247C2×#461—531no—
248D4×M16—531no—
249#1693—532no—
250#1691—532no—
251#1673—532no—
252#1671—532no—
253#1565—531no—
254#1550—531no—
255#1549—531no—
256C2×C53—22no—
257C2×C53C2×#46111yescharacteristic
258C2×C53#305511yescharacteristic
259C2×C53#306911yescharacteristic
260D53—42no—
261D53—81no—
262C22×SD16—1061no—
263C22×M16—1061no—
264#533C2×D5311yescharacteristic
265C2×C4⋊Q8C2×D5311yescharacteristic
266C2×C4×D4—531no—
267#501—22no—
268#500C2×D5312yescomplemented
269#498C2×D5312yescomplemented
270#483—1061no—
271#476—531no—
272#476—531no—
273#461—531no—
274#461—531no—
275#461—532no—
276#461—532no—
277#461—1061no—
278#461—1061no—
279#450—1061no—
280C4×SD16—1061no—
281C4×SD16—1061no—
282#436—532no—
283#435—1062no—
284C8×D4—532no—
285#431—531no—
286#428—21no—
287#428—21no—
288#423—531no—
289C2×C4⋊C8—531no—
290C2×#103—1061no—
291#407—1061no—
292C4×M16C2×D5311yescharacteristic
293#335—532no—
294#334—532no—
295#333—532no—
296#332—532no—
297#332—532no—
298#332—532no—
299C53#1916411yescharacteristic, complemented, Sylow
300C22×Q8—21no—
301C22×D4—531no—
302C23×C4—1061no—
303C2×SD16—1062no—
304C2×SD16—1062no—
305C2×SD16—1061no—
306C2×SD16—1061no—
307C2×SD16—2124no—
308C2×M16—1061no—
309C2×M16—1062no—
310C2×M16—2124no—
311C2×M16C22⋊D5311yescharacteristic
312C2×M16C22⋊D5311yescharacteristic
313C22×C8—1061no—
314C22×C8—1062no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318C4⋊Q8C22×D5312yes—
319C4⋊Q8C22×D5311yescharacteristic
320C4⋊Q8C22×D5311yescharacteristic
321#125—21no—
322C22⋊Q8—21no—
323C22⋊Q8—21no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×D4—531no—
327C4×D4—531no—
328C4×D4—531no—
329C4×D4—531no—
330C4×D4—1061no—
331C4×D4—1061no—
332#117—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—531no—
336C2×C22⋊C4—1061no—
337C2×C42C22×D5311yescharacteristic
338#107—22no—
339#107—22no—
340#106—22no—
341#106—22no—
342#106—1061no—
343#106—1061no—
344C4⋊C8—532no—
345C4⋊C8—531no—
346C4⋊C8—531no—
347C4⋊C8—532no—
348C4⋊C8—1061no—
349C4⋊C8—1061no—
350#103—1062no—
351#103—1062no—
352#103—1061no—
353#103—1061no—
354#102—1061no—
355#102—1061no—
356C22⋊C8—1061no—
357C22⋊C8—1062no—
358C22⋊C8—1062no—
359#97C22⋊D5312yes—
360C4×C8—1061no—
361C4×C8C22×D5312yes—
362C24—1061no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—21no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—41no—
369C2×D4—531no—
370C2×D4—531no—
371C2×D4—1062no—
372C2×D4—1061no—
373C2×D4—2121no—
374C22×C4—41no—
375C22×C4—1061no—
376C22×C4—1061no—
377C22×C4—1061no—
378C22×C4—1061no—
379C22×C4—1061no—
380C22×C4—2122no—
381C22×C4—2122no—
382C22×C4D4×D5311yescharacteristic
383C22×C4#1171994711yescharacteristic
384C22×C4C2×C22⋊D5311yescharacteristic
385SD16—2124no—
386SD16—2122no—
387SD16—2124no—
388SD16—2122no—
389M16—22no—
390M16—2122no—
391M16—2124no—
392M16#1171992414yescomplemented
393C2×C8—1061no—
394C2×C8—1061no—
395C2×C8—1062no—
396C2×C8—1062no—
397C2×C8—2122no—
398C2×C8—2124no—
399C2×C8C2×C22⋊D5312yes—
400C2×C8C2×C22⋊D5312yes—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—531no—
412C4⋊C4—531no—
413C4⋊C4—1061no—
414C22⋊C4—41no—
415C22⋊C4—1061no—
416C22⋊C4—1061no—
417C22⋊C4—1061no—
418C22⋊C4—1061no—
419C42—41no—
420C42D4×D5311yescharacteristic
421C42#1171994711yescharacteristic
422C42C2×C22⋊D5311yescharacteristic
423C42C23×D5311yescharacteristic
424C23—1061no—
425C23—1061no—
426C23—4241no—
427C23—2122no—
428C23—2122no—
429C23#1313318811yescharacteristic
430Q8—42no—
431Q8—41no—
432Q8—41no—
433Q8—42no—
434Q8—81no—
435Q8—82no—
436D4—532no—
437D4—1061no—
438D4—1061no—
439D4—1061no—
440D4—2122no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—82no—
448C2×C4—1061no—
449C2×C4—1061no—
450C2×C4—2122no—
451C2×C4—2122no—
452C2×C4—2122no—
453C2×C4—2121no—
454C2×C4—2121no—
455C2×C4—2121no—
456C2×C4—2121no—
457C2×C4#1313318811yescharacteristic
458C2×C4#1313318811yescharacteristic
459C2×C4#1313318811yescharacteristic
460C2×C4C2×#1171992412yes—
461C2×C4#1313324812yes—
462C2×C4C2×D4×D5311yescharacteristic
463C2×C4C22×C22⋊D5311yescharacteristic
464C2×C4C2×#1171994711yescharacteristic
465C8—22no—
466C8—2122no—
467C8—2122no—
468C8—2121no—
469C8—2121no—
470C8C2×#1171992414yescomplemented
471C22—21no—
472C22—21no—
473C22—1062no—
474C22—4244no—
475C22—2121no—
476C22—2121no—
477C22—2121no—
478C22—2121no—
479C22—2121no—
480C22#1681082512yes—
481C22C2×#1313318811yescharacteristic
482C4—21no—
483C4—21no—
484C4—21no—
485C4—81no—
486C4—82no—
487C4—81no—
488C4—2121no—
489C4—2121no—
490C4#1681082512yes—
491C4C22×#1171992411yescharacteristic
492C4AGL(1,ℤ/8)×D5311yescharacteristic
493C4C2×#1313324811yescharacteristic
494C4#1681141911yescharacteristic
495C2—22no—
496C2—4241no—
497C2—2122no—
498C2C2×#1681082511yescharacteristic
499C2#2028432811yescharacteristic
500C2#2028444811yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3165C2
socleC22×C53C2×#121
prime radicalC22×D212D4
hypercenter#3165D53
hypocenterC53#19164
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
4. #20274225C2semidirect
5. #20274211C2semidirect
6. #20274101C2semidirect
7. #20274087C2semidirect
8. #20273467C2semidirect
9. #20273405C2semidirect
10. #20273400C2semidirect
11. C53×#3165C2semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810372C22semidirect
38. #16810369C22semidirect
39. #16810369C22semidirect
40. #16810369C22semidirect
44. C53×#500C22semidirect
45. C53×#498C22semidirect
135. M16×C53SD16semidirect
215. C8×C53C2×SD16semidirect
245. #3165D53semidirect
268. #500C2×D53semidirect
269. #498C2×D53semidirect
299. C53#19164semidirect
392. M16#11719924semidirect
470. C8C2×#11719924semidirect

Maximal subgroups

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

subgroupindex
#202847192
C2×#168108252
M16×D2122
#202742252
#202742112
#202741012
#202740872
#202734672
#202734052
#202734002
C53×#31652
#1916453