Finite Group Databasebeta

← #27366865

Subgroups — #27366865

Order 13568.

760 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284711C211yesmaximal, characteristic
2C2×#16810900C211yesmaximal, characteristic, complemented
3M16×D212C211yesmaximal, characteristic, complemented
4#20274201C212yesmaximal, complemented
5#20274177C212yesmaximal, complemented
6#20274077C212yesmaximal, complemented
7#20274053C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273392C211yesmaximal, characteristic, complemented
10#20273387C211yesmaximal, characteristic, complemented
11C53×#3152C211yesmaximal, characteristic, complemented
12C2×M16×D53—21no—
13C2×C4×D212C2211yescharacteristic
14#16810934—21no—
15#16810900—21no—
16#16810900—21no—
17#16810900C2212yes—
18#16810900C2211yescharacteristic
19#16810900C2211yescharacteristic
20#16810900C2212yes—
21#16810815C2211yescharacteristic
22#16810815C2211yescharacteristic
23#16810756—21no—
24C4×#11719922—21no—
25C4×#11719922—21no—
26#16810702C2212yes—
27#16810696—22no—
28C8×D212C2212yes—
29#16810676C2211yescharacteristic
30C2×#13133150—21no—
31#16810647C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33C2×#13133137—21no—
34#16810586—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810369C2212yescomplemented
38#16810369C2212yescomplemented
39#16810369C2212yescomplemented
40#16810367C2212yescomplemented
41C2×#137×C53—21no—
42C2×#137×C53—21no—
43C53×#532C2211yescharacteristic
44C2×C4⋊Q8×C53C2211yescharacteristic
45C53×#497—22no—
46C53×#494C2212yescomplemented
47C53×#492C2212yescomplemented
48C4×M16×C53C2211yescharacteristic
49C22×D212D411yescharacteristic
50C22×C4×D53—21no—
51C2×#11719922—21no—
52C2×#11719922—21no—
53C2×#11719917—21no—
54C2×#11719912—22no—
55M16×D53—44no—
56C22×#88322—21no—
57C2×C8×D53—22no—
58C4×D212—21no—
59C4×D212—21no—
60C4×D212D411yescharacteristic
61C4×D212D411yescharacteristic
62C4×D212C2311yescharacteristic
63C4×D212C2311yescharacteristic
64C2×C4⋊Dic53D411yescharacteristic
65C2×#11719921—21no—
66#13133150—22no—
67#13133150—22no—
68#13133150—21no—
69#13133150—21no—
70C4⋊#88322—21no—
71C4⋊#88322—21no—
72C4⋊#88322C2311yescharacteristic
73C4⋊#88322C2311yescharacteristic
74C4⋊#88322C2312yes—
75C4⋊#88322C2312yes—
76#13133138—21no—
77#13133138—21no—
78#13133137—21no—
79#13133137—21no—
80#13133137—22no—
81#13133137—22no—
82C22⋊#88322—21no—
83#13133129—22no—
84#13133129—22no—
85C4×#88322—21no—
86C2×#41×C53—21no—
87C22×Q8×C53—21no—
88#137×C53—44no—
89#137×C53—44no—
90C2×Q16×C53—22no—
91C2×Q16×C53—22no—
92C2×SD16×C53—22no—
93C2×SD16×C53—22no—
94C2×M16×C53D411yescharacteristic
95C2×M16×C53D411yescharacteristic
96C4⋊Q8×C53—21no—
97C4⋊Q8×C53—21no—
98C4⋊Q8×C53—21no—
99C4⋊Q8×C53C2312yes—
100C4⋊Q8×C53C2311yescharacteristic
101#127×C53C2311yescharacteristic
102#124×C53—21no—
103#121×C53—21no—
104#121×C53—21no—
105C4×D4×C53—21no—
106C4×D4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C42×C53C2311yescharacteristic
110#97×C53D412yes—
111C4×C8×C53C2312yes—
112C23×D53—21no—
113#41×C53—42no—
114#41×C53—42no—
115C2×Q8×C53—21no—
116C2×Q8×C53—22no—
117C2×Q8×C53—41no—
118C2×Q8×C53—42no—
119C2×Q8×C53—42no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—41no—
124C22×C4×C53—41no—
125C22×C4×C53—41no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53#4111yescharacteristic
129C22×Dic53—21no—
130C2×D212—22no—
131C2×D212—21no—
132C2×D212—41no—
133C2×D212C2×D411yescharacteristic
134C2×D212C2×D411yescharacteristic
135C2×C4×D53—21no—
136C2×C4×D53—21no—
137C2×C4×D53—21no—
138C2×C4×D53—21no—
139C2×C4×D53—42no—
140C2×C4×D53—42no—
141Q16×C53—44no—
142Q16×C53—44no—
143SD16×C53—44no—
144SD16×C53—44no—
145M16×C53—22no—
146M16×C53D814yescomplemented
147C2×C8×C53C2×D412yes—
148C2×C8×C53C2×D412yes—
149C4⋊C4×C53—22no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—41no—
154C4⋊C4×C53—41no—
155C22⋊C4×C53—41no—
156C22⋊C4×C53—41no—
157C42×C53C2×D411yescharacteristic
158C42×C53C2×D411yescharacteristic
159C42×C53#4111yescharacteristic
160C42×C53C2411yescharacteristic
161#11719922—42no—
162#11719922—42no—
163#11719921—21no—
164#11719921—21no—
165#11719921—21no—
166#11719921—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53C2×D411yescharacteristic
169C4⋊Dic53C2×D411yescharacteristic
170#11719917—42no—
171C2×#88322—21no—
172C2×#88322—21no—
173C2×#88322—22no—
174C2×#88322—22no—
175C2×#88322—42no—
176#11719912—44no—
177C8×D53—44no—
178C23×C53—41no—
179C23×C53#12111yescharacteristic
180C22×D53—21no—
181C22×D53—21no—
182C22×D53—42no—
183C22×D53—42no—
184C22×D53—81no—
185Q8×C53—42no—
186Q8×C53—41no—
187Q8×C53—42no—
188Q8×C53—41no—
189Q8×C53—82no—
190D4×C53—41no—
191D4×C53—41no—
192D4×C53—41no—
193D4×C53—41no—
194D4×C53—81no—
195D4×C53—82no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—42no—
201C2×C4×C53—82no—
202C2×C4×C53—82no—
203C2×C4×C53#12111yescharacteristic
204C2×C4×C53#12111yescharacteristic
205C2×C4×C53#12111yescharacteristic
206C2×C4×C53C2×D812yes—
207C2×C4×C53AGL(1,ℤ/8)12yes—
208C2×C4×C53C22×D411yescharacteristic
209C2×C4×C53C22×D411yescharacteristic
210C2×C4×C53C2×#4111yescharacteristic
211C2×Dic53—21no—
212C2×Dic53—21no—
213C2×Dic53—42no—
214D212—21no—
215D212—21no—
216D212—21no—
217D212—42no—
218D212#13712yescomplemented
219C4×D53—41no—
220C4×D53—41no—
221C4×D53—41no—
222C4×D53—41no—
223C4×D53—42no—
224C4×D53—42no—
225C8×C53—22no—
226C8×C53C2×D814yes—
227#88322—42no—
228#88322—41no—
229#88322—42no—
230#88322—41no—
231#19163—531nomaximal, Sylow
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—81no—
235C22×C53—81no—
236C22×C53—81no—
237C22×C53#45912yes—
238C22×C53C2×#12111yescharacteristic
239C2×D53—22no—
240C2×D53—41no—
241C2×D53—41no—
242C2×D53—41no—
243C2×D53—41no—
244C2×D53—41no—
245C2×D53—84no—
246C4×C53—21no—
247C4×C53—21no—
248C4×C53—21no—
249C4×C53—81no—
250C4×C53—82no—
251C4×C53#45912yes—
252C4×C53C22×D811yescharacteristic
253C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C53C2×#13711yescharacteristic
255C4×C53#57511yescharacteristic
256Dic53—41no—
257Dic53—41no—
258#3152D5311yescharacteristic, complemented
259#3100—531no—
260C2×#474—531no—
261D4×M16—531no—
262#1689—532no—
263#1685—532no—
264#1669—532no—
265#1665—532no—
266#1565—531no—
267#1547—531no—
268#1546—531no—
269C2×C53—22no—
270C2×C53—161no—
271C2×C53C2×#45911yescharacteristic
272C2×C53#305611yescharacteristic
273C2×C53#306911yescharacteristic
274D53—42no—
275D53—81no—
276C2×#137—21no—
277C2×#137—21no—
278C22×M16—1061no—
279#532C2×D5311yescharacteristic
280C2×C4⋊Q8C2×D5311yescharacteristic
281C2×C4×D4—531no—
282#497—22no—
283#494C2×D5312yescomplemented
284#492C2×D5312yescomplemented
285#480—1061no—
286#474—531no—
287#474—531no—
288#474—532no—
289#474—532no—
290#474—1061no—
291#474—1061no—
292#459—531no—
293#459—531no—
294#447—1061no—
295C4×D8—1061no—
296C4×D8—1061no—
297#436—532no—
298#435—1062no—
299C8×D4—532no—
300#431—531no—
301C2×#107—1061no—
302#423—531no—
303C2×C4⋊C8—531no—
304C2×#102—1061no—
305#407—1061no—
306C4×M16C2×D5311yescharacteristic
307#335—532no—
308#334—532no—
309#332—532no—
310#332—532no—
311#332—532no—
312#331—532no—
313C53#1916311yescharacteristic, complemented, Sylow
314C2×#41—21no—
315C22×Q8—21no—
316C22×D4—531no—
317C23×C4—1061no—
318#137—44no—
319#137—44no—
320C2×Q16—22no—
321C2×Q16—22no—
322C2×SD16—22no—
323C2×SD16—22no—
324C2×D8—1061no—
325C2×D8—1061no—
326C2×M16—1061no—
327C2×M16—1062no—
328C2×M16—2124no—
329C2×M16C22⋊D5311yescharacteristic
330C2×M16C22⋊D5311yescharacteristic
331C22×C8—1061no—
332C22×C8—1062no—
333C4⋊Q8—21no—
334C4⋊Q8—21no—
335C4⋊Q8—21no—
336C4⋊Q8C22×D5312yes—
337C4⋊Q8C22×D5311yescharacteristic
338#127C22×D5311yescharacteristic
339#124—21no—
340#121—21no—
341#121—21no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—531no—
345C4×D4—531no—
346C4×D4—531no—
347C4×D4—531no—
348C4×D4—1061no—
349C4×D4—1061no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—531no—
353C2×C22⋊C4—1061no—
354C2×C42C22×D5311yescharacteristic
355#107—1062no—
356#107—1062no—
357#107—1061no—
358#107—1061no—
359C4⋊C8—532no—
360C4⋊C8—532no—
361C4⋊C8—531no—
362C4⋊C8—531no—
363C4⋊C8—1061no—
364C4⋊C8—1061no—
365#102—1062no—
366#102—1062no—
367#102—1061no—
368#102—1061no—
369#102—1061no—
370#102—1061no—
371C22⋊C8—1061no—
372C22⋊C8—1062no—
373C22⋊C8—1062no—
374#97C22⋊D5312yes—
375C4×C8—1061no—
376C4×C8C22×D5312yes—
377C24—1061no—
378#41—42no—
379#41—42no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—41no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—41no—
389C2×D4—531no—
390C2×D4—531no—
391C2×D4—1062no—
392C2×D4—1061no—
393C2×D4—2121no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—1061no—
397C22×C4—1061no—
398C22×C4—1061no—
399C22×C4—1061no—
400C22×C4—1061no—
401C22×C4—2122no—
402C22×C4—2122no—
403C22×C4D4×D5311yescharacteristic
404C22×C4#1171994711yescharacteristic
405C22×C4C2×C22⋊D5311yescharacteristic
406Q16—44no—
407Q16—44no—
408SD16—44no—
409SD16—44no—
410D8—2122no—
411D8—2122no—
412M16—22no—
413M16—2122no—
414M16—2124no—
415M16#1171992214yescomplemented
416C2×C8—1061no—
417C2×C8—1061no—
418C2×C8—1062no—
419C2×C8—1062no—
420C2×C8—2122no—
421C2×C8—2124no—
422C2×C8C2×C22⋊D5312yes—
423C2×C8C2×C22⋊D5312yes—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—531no—
431C4⋊C4—531no—
432C4⋊C4—1061no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—1061no—
436C22⋊C4—1061no—
437C22⋊C4—1061no—
438C22⋊C4—1061no—
439C42D4×D5311yescharacteristic
440C42#1171994711yescharacteristic
441C42C2×C22⋊D5311yescharacteristic
442C42C23×D5311yescharacteristic
443C23—41no—
444C23—1061no—
445C23—1061no—
446C23—4241no—
447C23—2122no—
448C23—2122no—
449C23#1313318811yescharacteristic
450Q8—42no—
451Q8—41no—
452Q8—42no—
453Q8—41no—
454Q8—82no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—41no—
459D4—81no—
460D4—82no—
461D4—532no—
462D4—1061no—
463D4—1061no—
464D4—1061no—
465D4—2122no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—1061no—
474C2×C4—1061no—
475C2×C4—2122no—
476C2×C4—2122no—
477C2×C4—2122no—
478C2×C4—2121no—
479C2×C4—2121no—
480C2×C4—2121no—
481C2×C4—2121no—
482C2×C4#1313318811yescharacteristic
483C2×C4#1313318811yescharacteristic
484C2×C4#1313318811yescharacteristic
485C2×C4C2×#1171992212yes—
486C2×C4#1313324312yes—
487C2×C4C2×D4×D5311yescharacteristic
488C2×C4C22×C22⋊D5311yescharacteristic
489C2×C4C2×#1171994711yescharacteristic
490C8—22no—
491C8—2122no—
492C8—2121no—
493C8—2122no—
494C8—2121no—
495C8C2×#1171992214yes—
496C22—21no—
497C22—21no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—1062no—
502C22—4244no—
503C22—2121no—
504C22—2121no—
505C22—2121no—
506C22—2121no—
507C22—2121no—
508C22#1681081512yes—
509C22C2×#1313318811yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—81no—
514C4—82no—
515C4—2121no—
516C4—2121no—
517C4#1681081512yes—
518C4C22×#1171992211yescharacteristic
519C4C2×#1313324311yescharacteristic
520C4#137×D5311yescharacteristic
521C4#1681142211yescharacteristic
522C2—22no—
523C2—161no—
524C2—4241no—
525C2—2122no—
526C2C2×#1681081511yescharacteristic
527C2#2028433911yescharacteristic
528C2#2028444811yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3152C2
socleC22×C53C2×#121
prime radical#20284711C2
hypercenter#3152D53
hypocenterC53#19163
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
2. C2×#16810900C2semidirect
3. M16×D212C2semidirect
4. #20274201C2semidirect
5. #20274177C2semidirect
6. #20274077C2semidirect
7. #20274053C2semidirect
8. #20273467C2semidirect
9. #20273392C2semidirect
10. #20273387C2semidirect
11. C53×#3152C2semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810369C22semidirect
38. #16810369C22semidirect
39. #16810369C22semidirect
40. #16810367C22semidirect
46. C53×#494C22semidirect
47. C53×#492C22semidirect
146. M16×C53D8semidirect
218. D212#137semidirect
258. #3152D53semidirect
283. #494C2×D53semidirect
284. #492C2×D53semidirect
313. C53#19163semidirect
415. M16#11719922semidirect

Maximal subgroups

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

subgroupindex
#202847112
C2×#168109002
M16×D2122
#202742012
#202741772
#202740772
#202740532
#202734672
#202733922
#202733872
C53×#31522
#1916353