Finite Group Databasebeta

← #27366862

Subgroups — #27366862

Order 13568.

844 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285168C211yesmaximal, characteristic, complemented
2C2×#16810818C211yesmaximal, characteristic, complemented
3D4×#11719917C211yesmaximal, characteristic, complemented
4#20274192C212yesmaximal, complemented
5#20274182C212yesmaximal, complemented
6#20274068C212yesmaximal, complemented
7#20274058C212yesmaximal, complemented
8#20273468C211yesmaximal, characteristic, complemented
9#20273392C211yesmaximal, characteristic, complemented
10#20273382C211yesmaximal, characteristic, complemented
11C53×#3100C211yesmaximal, characteristic, complemented
12C2×#13133242—21no—
13C2×#13133242—21no—
14C22×#11719922—21no—
15C22×#11719917—21no—
16C2×#13133214C2211yescharacteristic
17#16811013—22no—
18#16810999C2212yescomplemented
19#16810997C2212yescomplemented
20#16810818—21no—
21#16810818—21no—
22#16810818C2212yes—
23#16810818C2212yes—
24#16810701C2212yescomplemented
25#16810695—22no—
26D4×#88322C2212yescomplemented
27#16810647C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133137—21no—
30#16810587—21no—
31C4×#11719917C2211yescharacteristic
32#16810375C2212yescomplemented
33#16810373C2212yescomplemented
34#16810369C2212yescomplemented
35#16810367C2212yescomplemented
36#16810366C2212yescomplemented
37#16810366C2212yescomplemented
38C53×#532C2211yescharacteristic
39C2×C4×D4×C53C2211yescharacteristic
40C53×#480—21no—
41C53×#474C2211yescharacteristic, complemented
42C53×#474C2211yescharacteristic, complemented
43C53×#459C2211yescharacteristic, complemented
44C53×#459C2211yescharacteristic, complemented
45C53×#447—21no—
46C4×D8×C53—21no—
47C4×D8×C53—21no—
48C53×#431C2211yescharacteristic
49C22×D212—21no—
50C22×D212—21no—
51C22×D212—21no—
52#13133242—44no—
53#13133242—44no—
54C2×#11719924—22no—
55C2×#11719924—22no—
56C2×#11719922—22no—
57C2×#11719922—22no—
58C2×#11719922—22no—
59C2×#11719922—22no—
60C2×#11719922—44no—
61C2×#11719917—22no—
62C2×#11719917—21no—
63C2×#11719917—44no—
64C2×#11719917D411yescharacteristic
65C2×#11719917D411yescharacteristic
66C22×#88322—22no—
67C22×#88322—21no—
68#13133214—21no—
69#13133214—21no—
70#13133214—21no—
71#13133214C2312yes—
72C4⋊#88322—21no—
73C4⋊#88322—21no—
74C4⋊#88322C2312yes—
75C4⋊#88322C2312yes—
76#13133137—22no—
77#13133137—22no—
78C22⋊#88322—22no—
79C22⋊#88322—22no—
80#13133128D412yes—
81C4×#88322C2312yes—
82C2×#41×C53—21no—
83C22×D4×C53D411yescharacteristic
84C23×C4×C53—21no—
85C2×D8×C53—21no—
86C2×D8×C53—21no—
87C4⋊Q8×C53C2311yescharacteristic
88#127×C53C2311yescharacteristic
89#124×C53—21no—
90#121×C53—21no—
91#121×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53—21no—
94C4×D4×C53—21no—
95C4×D4×C53—21no—
96C4×D4×C53D411yescharacteristic
97C4×D4×C53D411yescharacteristic
98C4×D4×C53C2311yescharacteristic
99C4×D4×C53C2311yescharacteristic
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C22⋊C4×C53—21no—
102C2×C42×C53C2311yescharacteristic
103#107×C53—21no—
104#107×C53—21no—
105C4⋊C8×C53C2311yescharacteristic
106C4⋊C8×C53C2311yescharacteristic
107#102×C53—21no—
108#102×C53—21no—
109#102×C53—21no—
110#102×C53—21no—
111C22⋊C8×C53—21no—
112C4×C8×C53—21no—
113C24×C53—21no—
114C23×D53—41no—
115#41×C53—42no—
116#41×C53—42no—
117C2×Q8×C53—21no—
118C2×D4×C53—22no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—41no—
124C2×D4×C53—41no—
125C2×D4×C53C2×D411yescharacteristic
126C2×D4×C53C2×D411yescharacteristic
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—42no—
133C22×C4×C53—42no—
134C22×C4×C53—41no—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53#4111yescharacteristic
138C2×D212—22no—
139C2×D212—22no—
140C2×D212—22no—
141C2×D212—42no—
142C2×D212—42no—
143C2×D212—42no—
144C2×D212—42no—
145C2×D212—41no—
146C2×D212—41no—
147C2×D212—41no—
148C2×D212—82no—
149D8×C53—42no—
150D8×C53—42no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53C2×D411yescharacteristic
157C4⋊C4×C53C2×D411yescharacteristic
158C22⋊C4×C53—21no—
159C22⋊C4×C53—21no—
160C22⋊C4×C53—21no—
161C22⋊C4×C53—21no—
162C22⋊C4×C53—41no—
163C22⋊C4×C53—41no—
164C42×C53C2×D411yescharacteristic
165C42×C53C2×D411yescharacteristic
166C42×C53#4111yescharacteristic
167C42×C53C2411yescharacteristic
168#11719924—44no—
169#11719924—44no—
170#11719922—44no—
171#11719922—44no—
172#11719922—44no—
173#11719922—44no—
174#11719917—22no—
175#11719917—42no—
176#11719917—44no—
177#11719917D814yescomplemented
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—42no—
181C2×#88322—44no—
182C2×#88322C2×D412yes—
183C2×#88322C2×D412yes—
184C23×C53—21no—
185C23×C53—21no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53—41no—
189C23×C53—81no—
190C23×C53#12111yescharacteristic
191C22×D53—42no—
192C22×D53—82no—
193C22×D53—82no—
194C22×D53—82no—
195Q8×C53—41no—
196Q8×C53—41no—
197D4×C53—21no—
198D4×C53—21no—
199D4×C53—21no—
200D4×C53—42no—
201D4×C53—41no—
202D4×C53—41no—
203D4×C53—41no—
204D4×C53—41no—
205D4×C53—81no—
206D4×C53—82no—
207D4×C53AGL(1,ℤ/8)12yescomplemented
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—41no—
213C2×C4×C53—42no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—41no—
221C2×C4×C53—82no—
222C2×C4×C53#12111yescharacteristic
223C2×C4×C53#12111yescharacteristic
224C2×C4×C53#12111yescharacteristic
225C2×C4×C53C2×D812yes—
226C2×C4×C53AGL(1,ℤ/8)12yes—
227C2×C4×C53C22×D411yescharacteristic
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230D212—42no—
231D212—42no—
232D212—42no—
233D212—42no—
234D212—82no—
235D212—82no—
236D212—82no—
237D212—82no—
238C8×C53—41no—
239C8×C53—41no—
240#88322—22no—
241#88322—42no—
242#88322—42no—
243#88322C2×D814yescomplemented
244#19162—531nomaximal, Sylow
245C22×C53—21no—
246C22×C53—21no—
247C22×C53—22no—
248C22×C53—41no—
249C22×C53—41no—
250C22×C53—41no—
251C22×C53—41no—
252C22×C53—41no—
253C22×C53—84no—
254C22×C53—81no—
255C22×C53—81no—
256C22×C53—81no—
257C22×C53#45912yes—
258C22×C53C2×#12111yescharacteristic
259C2×D53—82no—
260C2×D53—82no—
261C2×D53—82no—
262C2×D53—164no—
263C4×C53—21no—
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—41no—
267C4×C53—41no—
268C4×C53—81no—
269C4×C53#45912yes—
270C4×C53C22×D811yescharacteristic
271C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
272C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
273C4×C53#57511yescharacteristic
274#3151—531no—
275#3100D5311yescharacteristic, complemented
276C2×#459—531no—
277D4×M16—531no—
278#1687—532no—
279#1685—532no—
280#1667—532no—
281#1665—532no—
282#1565—531no—
283#1547—531no—
284#1545—531no—
285C2×C53—22no—
286C2×C53—42no—
287C2×C53—81no—
288C2×C53—161no—
289C2×C53C2×#45911yescharacteristic
290C2×C53#305511yescharacteristic
291C2×C53#306811yescharacteristic
292D53—162no—
293C2×AGL(1,ℤ/8)—1061no—
294C2×AGL(1,ℤ/8)—1061no—
295C22×D8—1061no—
296C22×M16—1061no—
297#532C2×D5311yescharacteristic
298C2×#127—531no—
299C2×C4×D4C2×D5311yescharacteristic
300#496—1062no—
301#493—532no—
302#492—532no—
303#480—21no—
304#474C2×D5311yescharacteristic, complemented
305#474C2×D5311yescharacteristic, complemented
306#459—532no—
307#459—532no—
308#459—1061no—
309#459—1061no—
310#459C2×D5311yescharacteristic, complemented
311#459C2×D5311yescharacteristic, complemented
312#447—21no—
313C4×D8—21no—
314C4×D8—21no—
315#436—532no—
316#435—1062no—
317C8×D4—532no—
318#431C2×D5311yescharacteristic
319#423—531no—
320C2×C4⋊C8—531no—
321C2×#102—1061no—
322#407—1061no—
323C4×M16—531no—
324#335—532no—
325#334—532no—
326#332—532no—
327#331—532no—
328#331—532no—
329#331—532no—
330C53#1916211yescharacteristic, complemented, Sylow
331C2×#41—21no—
332C22×D4—1061no—
333C22×D4—1061no—
334C22×D4—1061no—
335C22×D4C22⋊D5311yescharacteristic
336C23×C4—21no—
337AGL(1,ℤ/8)—2124no—
338AGL(1,ℤ/8)—2124no—
339C2×SD16—1062no—
340C2×SD16—1062no—
341C2×D8—21no—
342C2×D8—21no—
343C2×D8—1062no—
344C2×D8—1062no—
345C2×D8—1062no—
346C2×D8—1062no—
347C2×D8—2124no—
348C2×M16—531no—
349C2×M16—531no—
350C2×M16—1062no—
351C2×M16—1061no—
352C2×M16—2124no—
353C22×C8—1062no—
354C22×C8—1061no—
355C4⋊Q8C22×D5311yescharacteristic
356#127—532no—
357#127—1061no—
358#127—1061no—
359#127—1061no—
360#127C22×D5311yescharacteristic
361#124—21no—
362#121—21no—
363#121—21no—
364C4×D4—21no—
365C4×D4—21no—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4C22⋊D5311yescharacteristic
369C4×D4C22⋊D5311yescharacteristic
370C4×D4C22×D5311yescharacteristic
371C4×D4C22×D5311yescharacteristic
372C2×C4⋊C4C22⋊D5311yescharacteristic
373C2×C22⋊C4—21no—
374C2×C42C22×D5311yescharacteristic
375#107—21no—
376#107—21no—
377C4⋊C8—532no—
378C4⋊C8—532no—
379C4⋊C8—1061no—
380C4⋊C8—1061no—
381C4⋊C8C22×D5311yescharacteristic
382C4⋊C8C22×D5311yescharacteristic
383#102—21no—
384#102—21no—
385#102—21no—
386#102—21no—
387#102—1062no—
388#102—1062no—
389C22⋊C8—21no—
390C22⋊C8—1062no—
391C22⋊C8—1062no—
392#97—532no—
393C4×C8—21no—
394C4×C8—532no—
395C24—21no—
396C24—2121no—
397#41—42no—
398#41—42no—
399C2×Q8—21no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—1062no—
408C2×D4—1062no—
409C2×D4—1062no—
410C2×D4—4242no—
411C2×D4—2122no—
412C2×D4—2122no—
413C2×D4—2121no—
414C2×D4—2121no—
415C2×D4—2121no—
416C2×D4—2122no—
417C2×D4—2122no—
418C2×D4C2×C22⋊D5311yescharacteristic
419C2×D4C2×C22⋊D5311yescharacteristic
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4D4×D5311yescharacteristic
429C22×C4#1171994711yescharacteristic
430C22×C4C2×C22⋊D5311yescharacteristic
431SD16—2124no—
432SD16—2124no—
433D8—42no—
434D8—42no—
435D8—2124no—
436D8—2124no—
437D8—2124no—
438D8—2124no—
439M16—534no—
440M16—1062no—
441M16—2124no—
442M16—2122no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—532no—
446C2×C8—532no—
447C2×C8—1062no—
448C2×C8—1062no—
449C2×C8—2124no—
450C2×C8—2122no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4C2×C22⋊D5311yescharacteristic
455C4⋊C4C2×C22⋊D5311yescharacteristic
456C22⋊C4—21no—
457C22⋊C4—21no—
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C42D4×D5311yescharacteristic
463C42#1171994711yescharacteristic
464C42C2×C22⋊D5311yescharacteristic
465C42C23×D5311yescharacteristic
466C23—21no—
467C23—21no—
468C23—42no—
469C23—42no—
470C23—41no—
471C23—81no—
472C23—4242no—
473C23—4242no—
474C23—4242no—
475C23—2122no—
476C23#1313318911yescharacteristic
477Q8—41no—
478Q8—41no—
479D4—21no—
480D4—21no—
481D4—21no—
482D4—42no—
483D4—41no—
484D4—41no—
485D4—41no—
486D4—41no—
487D4—81no—
488D4—82no—
489D4—4242no—
490D4—4242no—
491D4—4242no—
492D4—4242no—
493D4—2122no—
494D4—2122no—
495D4—2122no—
496D4—2122no—
497D4#1313324212yescomplemented
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—82no—
512C2×C4#1313318911yescharacteristic
513C2×C4#1313318911yescharacteristic
514C2×C4#1313318911yescharacteristic
515C2×C4D8×D5312yes—
516C2×C4#1313324112yes—
517C2×C4C2×D4×D5311yescharacteristic
518C2×C4C22×C22⋊D5311yescharacteristic
519C2×C4C2×#1171994711yescharacteristic
520C8—41no—
521C8—41no—
522C8—534no—
523C8—1062no—
524C8—2122no—
525C8—2122no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—84no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—8484no—
539C22—4242no—
540C22—4242no—
541C22—4242no—
542C22#1681081612yes—
543C22C2×#1313318911yescharacteristic
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—81no—
550C4#1681081612yes—
551C4C2×D8×D5311yescharacteristic
552C4C2×#1313324111yescharacteristic
553C4C2×#1313324211yescharacteristic
554C4#1681142111yescharacteristic
555C2—22no—
556C2—42no—
557C2—81no—
558C2—161no—
559C2—8482no—
560C2C2×#1681081611yescharacteristic
561C2#2028433611yescharacteristic
562C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3100C2
socleC22×C53C2×#121
hypercenter#3100D53
hypocenterC53#19162
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #20285168C2semidirect
2. C2×#16810818C2semidirect
3. D4×#11719917C2semidirect
4. #20274192C2semidirect
5. #20274182C2semidirect
6. #20274068C2semidirect
7. #20274058C2semidirect
8. #20273468C2semidirect
9. #20273392C2semidirect
10. #20273382C2semidirect
11. C53×#3100C2semidirect
18. #16810999C22semidirect
19. #16810997C22semidirect
24. #16810701C22semidirect
26. D4×#88322C22semidirect
32. #16810375C22semidirect
33. #16810373C22semidirect
34. #16810369C22semidirect
35. #16810367C22semidirect
36. #16810366C22semidirect
37. #16810366C22semidirect
41. C53×#474C22semidirect
42. C53×#474C22semidirect
43. C53×#459C22semidirect
44. C53×#459C22semidirect
177. #11719917D8semidirect
207. D4×C53AGL(1,ℤ/8)semidirect
243. #88322C2×D8semidirect
275. #3100D53semidirect
304. #474C2×D53semidirect
305. #474C2×D53semidirect
310. #459C2×D53semidirect
311. #459C2×D53semidirect
330. C53#19162semidirect
497. D4#13133242semidirect

Maximal subgroups

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

subgroupindex
#202851682
C2×#168108182
D4×#117199172
#202741922
#202741822
#202740682
#202740582
#202734682
#202733922
#202733822
C53×#31002
#1916253