Finite Group Databasebeta

← #27362663

Subgroups — #27362663

Order 13568.

1224 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285836C211yesmaximal, characteristic, complemented
2#20284715C211yesmaximal, characteristic, complemented
3#20284179C211yesmaximal, characteristic, complemented
4#20273950C214yesmaximal, complemented
5#20273804C214yesmaximal, complemented
6#20273387C211yesmaximal, characteristic, complemented
7#20273140C211yesmaximal, characteristic, complemented
8#20273106C211yesmaximal, characteristic, complemented
9C2×C53×#547C211yesmaximal, characteristic, complemented
10C2×#13133243—21no—
11#16811173C2211yescharacteristic
12#16811026C2212yescomplemented
13#16810971C2212yescomplemented
14#16810936—21no—
15#16810936—22no—
16#16810903C2212yescomplemented
17#16810900—22no—
18#16810838—22no—
19#16810828C2212yescomplemented
20#16810818C2212yescomplemented
21#16810818C2212yescomplemented
22#16810757—21no—
23C4×#11719922—22no—
24#16810676C2211yescharacteristic
25#16810676C2211yescharacteristic
26#16810647C2211yescharacteristic
27#16810622—21no—
28#16810369C2214yescomplemented
29D4⋊#88322C2214yescomplemented
30D4⋊#88322C2214yescomplemented
31C22×#41×C53—21no—
32C53×#547—22no—
33C53×#547—22no—
34C53×#547—22no—
35C53×#547C2214yes—
36C2×C4⋊Q8×C53C2211yescharacteristic
37C2×#123×C53—22no—
38C2×C22⋊Q8×C53—21no—
39C2×#121×C53—21no—
40C2×C4×D4×C53C2211yescharacteristic
41C2×C4×D4×C53C2211yescharacteristic
42C22×C4⋊C4×C53—21no—
43C2×#11719945—21no—
44#13133243—44no—
45C2×#11719923—22no—
46C2×#11719922—22no—
47C2×#11719922—22no—
48C2×#11719917—21no—
49#13133217C2311yescharacteristic
50#13133214C2311yescharacteristic
51#13133205—21no—
52#13133191—21no—
53#13133191—21no—
54C4×D212—21no—
55C4×D212—21no—
56#13133150—24no—
57#13133150—22no—
58C4⋊#88322—21no—
59C4⋊#88322—21no—
60C4⋊#88322C2312yes—
61C4⋊#88322C2312yes—
62#13133140—22no—
63#13133138—22no—
64#13133137—22no—
65#13133137—22no—
66#13133137—24no—
67C22⋊#88322—21no—
68C22⋊#88322—21no—
69C4×#88322—21no—
70C4×#88322C2312yes—
71C2×#41×C53—24no—
72C2×#41×C53—24no—
73C2×#41×C53—44no—
74C2×#41×C53—44no—
75C22×Q8×C53—21no—
76C22×D4×C53—22no—
77C22×D4×C53D411yescharacteristic
78C22×D4×C53D411yescharacteristic
79C23×C4×C53—22no—
80C23×C4×C53—21no—
81C23×C4×C53—21no—
82C4⋊Q8×C53—21no—
83C4⋊Q8×C53—21no—
84C4⋊Q8×C53D412yes—
85C4⋊Q8×C53C2312yes—
86#123×C53—28no—
87#123×C53—28no—
88C22⋊Q8×C53—24no—
89C22⋊Q8×C53—24no—
90#121×C53—24no—
91#121×C53—24no—
92C4×D4×C53—21no—
93C4×D4×C53—21no—
94C4×D4×C53—21no—
95C4×D4×C53—21no—
96C4×D4×C53D412yes—
97C4×D4×C53D412yes—
98C4×D4×C53C2312yes—
99C4×D4×C53C2312yes—
100C2×C4⋊C4×C53—22no—
101C2×C4⋊C4×C53—24no—
102C2×C4⋊C4×C53—24no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C4⋊C4×C53D411yescharacteristic
106C2×C4⋊C4×C53D412yes—
107C2×C22⋊C4×C53—22no—
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C42×C53C2311yescharacteristic
111C24×C53—21no—
112C24×C53—21no—
113#41×C53—48no—
114#41×C53—416no—
115#41×C53—48no—
116C2×Q8×C53—22no—
117C2×Q8×C53—22no—
118C2×Q8×C53—42no—
119C2×Q8×C53—42no—
120C2×D4×C53—28no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—22no—
124C2×D4×C53—42no—
125C2×D4×C53—48no—
126C2×D4×C53D814yescomplemented
127C2×D4×C53C2×D412yes—
128C2×D4×C53C2×D412yes—
129C22×C4×C53—28no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—44no—
138C22×C4×C53—48no—
139C22×C4×C53—42no—
140C22×C4×C53—42no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C2×C22⋊D53—41no—
145#11719945—42no—
146#11719945—42no—
147C2×D212—21no—
148C2×D212—21no—
149C2×D212—21no—
150C2×C4×D53—41no—
151C2×Dic106—21no—
152C4⋊C4×C53—28no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53C2×D412yes—
159C4⋊C4×C53C2×D412yes—
160C4⋊C4×C53C2×D414yes—
161C22⋊C4×C53—28no—
162C22⋊C4×C53—22no—
163C22⋊C4×C53—22no—
164C22⋊C4×C53—22no—
165C22⋊C4×C53—22no—
166C42×C53C2×D411yescharacteristic
167C42×C53C2×D411yescharacteristic
168C42×C53C2×D411yescharacteristic
169C42×C53C2411yescharacteristic
170#11719923—44no—
171#11719922—44no—
172#11719922—44no—
173#11719921—41no—
174#11719921—41no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177#11719917—42no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C23×C53—22no—
182C23×C53—24no—
183C23×C53—22no—
184C23×C53—42no—
185C23×C53—42no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53—42no—
189C23×C53#12011yescharacteristic
190C22×D53—41no—
191Q8×C53—42no—
192Q8×C53—42no—
193Q8×C53—44no—
194D4×C53—22no—
195D4×C53—22no—
196D4×C53—22no—
197D4×C53—24no—
198D4×C53—42no—
199D4×C53—416no—
200D4×C53C2×D814yes—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—24no—
204C2×C4×C53—22no—
205C2×C4×C53—41no—
206C2×C4×C53—44no—
207C2×C4×C53—42no—
208C2×C4×C53—416no—
209C2×C4×C53—42no—
210C2×C4×C53—44no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—44no—
214C2×C4×C53—44no—
215C2×C4×C53#12011yescharacteristic
216C2×C4×C53#12011yescharacteristic
217C2×C4×C53#12011yescharacteristic
218C2×C4×C53C2×D812yes—
219C2×C4×C53AGL(1,ℤ/8)12yes—
220C2×C4×C53C22×D411yescharacteristic
221C2×C4×C53C22×D411yescharacteristic
222C2×C4×C53C22×D411yescharacteristic
223C22⋊D53—82no—
224C2×Dic53—41no—
225D212—42no—
226D212—41no—
227D212—41no—
228D212—42no—
229C4×D53—82no—
230Dic106—41no—
231Dic106—41no—
232#88322—42no—
233#88322—42no—
234#88322—42no—
235#18756—531nomaximal, Sylow
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—24no—
239C22×C53—48no—
240C22×C53—42no—
241C22×C53—42no—
242C22×C53—42no—
243C22×C53—42no—
244C22×C53—42no—
245C22×C53—84no—
246C22×C53#44712yes—
247C22×C53C2×#12011yescharacteristic
248C2×D53—81no—
249C2×D53—81no—
250C2×D53—81no—
251C4×C53—21no—
252C4×C53—21no—
253C4×C53—21no—
254C4×C53—42no—
255C4×C53—44no—
256C4×C53—42no—
257C4×C53#44712yes—
258C4×C53C22×D811yescharacteristic
259C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
260C4×C53#57512yes—
261Dic53—81no—
262C2×#547D5311yescharacteristic, complemented
263#3226—531no—
264#3100—531no—
265#3036—531no—
266#1647—534no—
267#1625—534no—
268#1546—531no—
269#1486—531no—
270#1478—531no—
271C2×C53—22no—
272C2×C53—44no—
273C2×C53—82no—
274C2×C53C2×#44711yescharacteristic
275C2×C53#300511yescharacteristic
276C2×C53#302011yescharacteristic
277D53—161no—
278C22×#41—21no—
279C2×AGL(1,ℤ/8)—1061no—
280#547—22no—
281#547—22no—
282#547—22no—
283#547C2×D5314yes—
284#532—531no—
285C2×C4⋊Q8C2×D5311yescharacteristic
286C2×#123—22no—
287C2×C22⋊Q8—21no—
288C2×#121—21no—
289C2×C4×D4C2×D5311yescharacteristic
290C2×C4×D4C2×D5311yescharacteristic
291C22×C4⋊C4—21no—
292#500—532no—
293#486—532no—
294#480—1061no—
295#480—1062no—
296#474—532no—
297#474—1062no—
298#463—1062no—
299#461—532no—
300#459—532no—
301#459—532no—
302#447—1061no—
303C4×D8—1062no—
304#431—531no—
305#431—531no—
306#423—531no—
307#417—1061no—
308#332—534no—
309D4⋊C8—534no—
310D4⋊C8—534no—
311C53#1875611yescharacteristic, complemented, Sylow
312C2×#41—24no—
313C2×#41—24no—
314C2×#41—44no—
315C2×#41—44no—
316C2×#41—1061no—
317C22×Q8—21no—
318C22×D4—22no—
319C22×D4C22⋊D5311yescharacteristic
320C22×D4C22⋊D5311yescharacteristic
321C23×C4—22no—
322C23×C4—21no—
323C23×C4—21no—
324AGL(1,ℤ/8)—2124no—
325C2×SD16—1062no—
326C2×D8—1062no—
327C2×D8—1062no—
328C2×M16—1061no—
329C4⋊Q8—21no—
330C4⋊Q8—21no—
331C4⋊Q8—531no—
332C4⋊Q8C22⋊D5312yes—
333C4⋊Q8C22×D5312yes—
334#127—531no—
335#124—1061no—
336#123—28no—
337#123—28no—
338C22⋊Q8—24no—
339C22⋊Q8—24no—
340#121—24no—
341#121—24no—
342#121—1061no—
343#121—1061no—
344C4×D4—21no—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—1061no—
349C4×D4—1061no—
350C4×D4C22⋊D5312yes—
351C4×D4C22⋊D5312yes—
352C4×D4C22×D5312yes—
353C4×D4C22×D5312yes—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4C22⋊D5311yescharacteristic
359C2×C4⋊C4C22⋊D5311yescharacteristic
360C2×C4⋊C4C22⋊D5312yes—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C42C22×D5311yescharacteristic
365#107—1062no—
366#107—1064no—
367C4⋊C8—532no—
368C4⋊C8—532no—
369C4⋊C8—1061no—
370C4⋊C8—1061no—
371#103—1062no—
372#102—1062no—
373#102—1062no—
374#102—1062no—
375#102—1064no—
376C22⋊C8—1061no—
377C22⋊C8—1061no—
378C4×C8—532no—
379C4×C8—1061no—
380C24—21no—
381C24—21no—
382#41—416no—
383#41—48no—
384#41—48no—
385#41—2122no—
386#41—2122no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—1061no—
392C2×D4—28no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—42no—
397C2×D4—48no—
398C2×D4—1061no—
399C2×D4—1061no—
400C2×D4—1061no—
401C2×D4—2121no—
402C2×D4#1171992214yescomplemented
403C2×D4C2×C22⋊D5312yes—
404C2×D4C2×C22⋊D5312yes—
405C22×C4—28no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—44no—
414C22×C4—48no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—2121no—
418C22×C4C2×C22⋊D5311yescharacteristic
419C22×C4C2×C22⋊D5311yescharacteristic
420C22×C4C2×C22⋊D5311yescharacteristic
421SD16—2124no—
422D8—2124no—
423D8—2124no—
424M16—2122no—
425C2×C8—1062no—
426C2×C8—1062no—
427C2×C8—1062no—
428C4⋊C4—28no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—1061no—
435C4⋊C4—1061no—
436C4⋊C4C2×C22⋊D5312yes—
437C4⋊C4C2×C22⋊D5312yes—
438C4⋊C4C2×C22⋊D5314yes—
439C22⋊C4—28no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—2121no—
445C22⋊C4—2121no—
446C42C2×C22⋊D5311yescharacteristic
447C42C2×C22⋊D5311yescharacteristic
448C42C2×C22⋊D5311yescharacteristic
449C42C23×D5311yescharacteristic
450C23—24no—
451C23—22no—
452C23—22no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—2121no—
459C23#1313318611yescharacteristic
460Q8—44no—
461Q8—42no—
462Q8—42no—
463Q8—2121no—
464Q8—2121no—
465D4—24no—
466D4—22no—
467D4—22no—
468D4—22no—
469D4—42no—
470D4—416no—
471D4—4242no—
472D4—2122no—
473D4—2121no—
474D4—2121no—
475D4—2122no—
476D4C2×#1171992214yes—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—416no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—4242no—
492C2×C4—2121no—
493C2×C4#1313318611yescharacteristic
494C2×C4#1313318611yescharacteristic
495C2×C4#1313318611yescharacteristic
496C2×C4C2×#1171992212yes—
497C2×C4#1313324312yes—
498C2×C4C22×C22⋊D5311yescharacteristic
499C2×C4C22×C22⋊D5311yescharacteristic
500C2×C4C22×C22⋊D5311yescharacteristic
501C8—2122no—
502C8—2122no—
503C8—2122no—
504C22—21no—
505C22—21no—
506C22—24no—
507C22—48no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—84no—
514C22—4241no—
515C22—4241no—
516C22—4241no—
517C22#1681075712yes—
518C22C2×#1313318611yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—42no—
523C4—42no—
524C4—44no—
525C4—4241no—
526C4#1681075712yes—
527C4C22×#1171992211yescharacteristic
528C4C2×#1313324311yescharacteristic
529C4#1681142112yes—
530C2—22no—
531C2—44no—
532C2—82no—
533C2—8481no—
534C2C2×#1681075711yescharacteristic
535C2#2028395711yescharacteristic
536C2#2028407911yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#547C2
socleC22×C53C2×#120
hypercenterC2×#547D53
hypocenterC53#18756
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #20285836C2semidirect
2. #20284715C2semidirect
3. #20284179C2semidirect
4. #20273950C2semidirect
5. #20273804C2semidirect
6. #20273387C2semidirect
7. #20273140C2semidirect
8. #20273106C2semidirect
9. C2×C53×#547C2semidirect
12. #16811026C22semidirect
13. #16810971C22semidirect
16. #16810903C22semidirect
19. #16810828C22semidirect
20. #16810818C22semidirect
21. #16810818C22semidirect
28. #16810369C22semidirect
29. D4⋊#88322C22semidirect
30. D4⋊#88322C22semidirect
126. C2×D4×C53D8semidirect
262. C2×#547D53semidirect
311. C53#18756semidirect
402. C2×D4#11719922semidirect

Maximal subgroups

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

subgroupindex
#202858362
#202847152
#202841792
#202739502
#202738042
#202733872
#202731402
#202731062
C2×C53×#5472
#1875653