Finite Group Databasebeta

← #27366331

Subgroups — #27366331

Order 13568.

802 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285141C211yesmaximal, characteristic, complemented
2#20284715C211yesmaximal, characteristic, complemented
3C2×D4×#88322C211yesmaximal, characteristic, complemented
4#20274078C212yesmaximal, complemented
5#20274068C212yesmaximal, complemented
6#20274058C212yesmaximal, complemented
7#20274054C212yesmaximal, complemented
8#20273461C211yesmaximal, characteristic, complemented
9#20273364C211yesmaximal, characteristic, complemented
10#20273362C211yesmaximal, characteristic, complemented
11C53×#3100C211yesmaximal, characteristic, complemented
12C2×#13133237—21no—
13C2×#13133237—21no—
14C23×#88322—21no—
15#16811173C2211yescharacteristic
16#16811005—21no—
17#16811005—21no—
18#16811002C2211yescharacteristic, complemented
19#16810999C2211yescharacteristic, complemented
20#16810997C2211yescharacteristic, complemented
21#16810995C2211yescharacteristic, complemented
22#16810936—21no—
23#16810903C2212yes—
24#16810818C2212yes—
25#16810757—21no—
26C4×#11719922—22no—
27D4×#88322—21no—
28D4×#88322—21no—
29D4×#88322C2212yes—
30D4×#88322C2212yes—
31#16810676C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33C2×C22⋊#88322—21no—
34C2×C4×#88322C2211yescharacteristic
35#16810375C2212yescomplemented
36#16810373C2212yescomplemented
37#16810369C2212yescomplemented
38#16810368C2212yescomplemented
39#16810367C2212yescomplemented
40#16810366C2212yescomplemented
41C53×#532C2211yescharacteristic
42C2×C4×D4×C53C2211yescharacteristic
43C53×#480—21no—
44C53×#474C2212yes—
45C53×#459C2212yes—
46C53×#447—21no—
47C4×D8×C53—22no—
48C53×#431C2211yescharacteristic
49C2×#11719945—21no—
50#13133237—44no—
51#13133237—44no—
52C2×#11719925—21no—
53C2×#11719925—21no—
54C2×#11719923—21no—
55C2×#11719923—21no—
56C2×#11719924—21no—
57C2×#11719924—21no—
58C2×#11719922—21no—
59C2×#11719922—21no—
60C2×#11719922—22no—
61C22×#88322—22no—
62C22×#88322—22no—
63C22×#88322—21no—
64C22×#88322—44no—
65C22×#88322D411yescharacteristic
66C22×#88322D411yescharacteristic
67#13133217C2311yescharacteristic
68#13133214C2311yescharacteristic
69#13133205—21no—
70#13133191—21no—
71#13133191—21no—
72C4×D212—21no—
73C4×D212—21no—
74#13133150—22no—
75C4⋊#88322—21no—
76C4⋊#88322C2312yes—
77C4⋊#88322C2312yes—
78#13133138—22no—
79#13133137—22no—
80C22⋊#88322—22no—
81C22⋊#88322—22no—
82C22⋊#88322—21no—
83C4×#88322—21no—
84C4×#88322D411yescharacteristic
85C4×#88322D411yescharacteristic
86C4×#88322C2311yescharacteristic
87C4×#88322C2311yescharacteristic
88C2×#41×C53—21no—
89C22×D4×C53D411yescharacteristic
90C23×C4×C53—21no—
91C2×D8×C53—22no—
92C4⋊Q8×C53C2311yescharacteristic
93#127×C53C2311yescharacteristic
94#124×C53—21no—
95#121×C53—21no—
96#121×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53—21no—
100C4×D4×C53—21no—
101C4×D4×C53D412yes—
102C4×D4×C53C2312yes—
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C22⋊C4×C53—21no—
105C2×C42×C53C2311yescharacteristic
106#107×C53—22no—
107C4⋊C8×C53C2312yes—
108#102×C53—22no—
109#102×C53—22no—
110C22⋊C8×C53—21no—
111C4×C8×C53—21no—
112C24×C53—21no—
113#41×C53—42no—
114#41×C53—42no—
115C2×Q8×C53—21no—
116C2×D4×C53—24no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—41no—
122C2×D4×C53C2×D412yes—
123C22×C4×C53—21no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—42no—
127C22×C4×C53—42no—
128C22×C4×C53—41no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53#4111yescharacteristic
132C2×C22⋊D53—41no—
133#11719945—42no—
134#11719945—42no—
135C2×D212—21no—
136C2×D212—21no—
137C2×D212—21no—
138C2×C4×D53—41no—
139C2×Dic106—21no—
140D8×C53—44no—
141C2×C8×C53—22no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53C2×D412yes—
146C22⋊C4×C53—22no—
147C22⋊C4×C53—22no—
148C22⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C42×C53C2×D411yescharacteristic
151C42×C53C2×D411yescharacteristic
152C42×C53#4111yescharacteristic
153C42×C53C2411yescharacteristic
154#11719925—42no—
155#11719925—42no—
156#11719924—42no—
157#11719924—42no—
158#11719923—42no—
159#11719923—42no—
160#11719922—42no—
161#11719922—44no—
162#11719922—42no—
163#11719921—41no—
164#11719921—41no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C2×#88322—22no—
168C2×#88322—22no—
169C2×#88322—22no—
170C2×#88322—42no—
171C2×#88322—44no—
172C2×#88322—44no—
173C2×#88322D814yescomplemented
174C2×#88322C2×D411yescharacteristic
175C2×#88322C2×D411yescharacteristic
176C2×#88322C2×D411yescharacteristic
177C2×#88322C2×D411yescharacteristic
178C23×C53—24no—
179C23×C53—22no—
180C23×C53—42no—
181C23×C53—42no—
182C23×C53—41no—
183C23×C53#12111yescharacteristic
184C22×D53—41no—
185Q8×C53—41no—
186Q8×C53—41no—
187D4×C53—24no—
188D4×C53—22no—
189D4×C53—41no—
190D4×C53—42no—
191D4×C53—41no—
192D4×C53—81no—
193D4×C53—82no—
194D4×C53#13514yescomplemented
195C2×C4×C53—22no—
196C2×C4×C53—22no—
197C2×C4×C53—41no—
198C2×C4×C53—42no—
199C2×C4×C53—42no—
200C2×C4×C53—42no—
201C2×C4×C53—44no—
202C2×C4×C53—41no—
203C2×C4×C53—82no—
204C2×C4×C53#12111yescharacteristic
205C2×C4×C53#12111yescharacteristic
206C2×C4×C53#12111yescharacteristic
207C2×C4×C53C2×D812yes—
208C2×C4×C53AGL(1,ℤ/8)12yes—
209C2×C4×C53C22×D411yescharacteristic
210C2×C4×C53C22×D411yescharacteristic
211C2×C4×C53C2×#4111yescharacteristic
212C22⋊D53—82no—
213C2×Dic53—41no—
214D212—41no—
215D212—42no—
216D212—41no—
217D212—81no—
218C4×D53—82no—
219Dic106—41no—
220Dic106—41no—
221C8×C53—42no—
222#88322—21no—
223#88322—21no—
224#88322—42no—
225#88322—42no—
226#88322C2×D812yes—
227#88322C2×D812yes—
228#19101—531nomaximal, Sylow
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—24no—
232C22×C53—48no—
233C22×C53—42no—
234C22×C53—42no—
235C22×C53—81no—
236C22×C53—81no—
237C22×C53—81no—
238C22×C53#45912yes—
239C22×C53C2×#12111yescharacteristic
240C2×D53—81no—
241C2×D53—81no—
242C2×D53—81no—
243C4×C53—21no—
244C4×C53—21no—
245C4×C53—21no—
246C4×C53—42no—
247C4×C53—81no—
248C4×C53#45912yes—
249C4×C53C22×D811yescharacteristic
250C4×C53C2×#13512yes—
251C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
252Dic53—81no—
253#3147—531no—
254#3100—531no—
255#3100D5311yescharacteristic, complemented
256C2×C8×D4—531no—
257#1669—532no—
258#1667—532no—
259#1665—532no—
260#1665—532no—
261#1564—531no—
262#1540—531no—
263#1540—531no—
264C2×C53—22no—
265C2×C53—44no—
266C2×C53—161no—
267C2×C53C2×#45911yescharacteristic
268C2×C53#305011yescharacteristic
269C2×C53#306711yescharacteristic
270D53—161no—
271C2×#135—1061no—
272C2×#135—1061no—
273C23×C8—1061no—
274#532—531no—
275#532C2×D5311yescharacteristic
276C2×C4×D4C2×D5311yescharacteristic
277#495—1061no—
278#495—1061no—
279#494—531no—
280#493—531no—
281#492—531no—
282#492—531no—
283#480—21no—
284#480—1061no—
285#474—532no—
286#474C2×D5312yes—
287#459—532no—
288#459C2×D5312yes—
289#447—21no—
290#447—1061no—
291C4×D8—22no—
292C4×D8—1062no—
293C8×D4—532no—
294C8×D4—532no—
295C8×D4—1061no—
296C8×D4—1061no—
297#431—531no—
298#431C2×D5311yescharacteristic
299C2×C4⋊C8—531no—
300C2×C22⋊C8—1061no—
301C2×C4×C8—531no—
302#335—532no—
303#334—532no—
304#332—532no—
305#332—532no—
306#331—532no—
307#331—532no—
308C53#1910111yescharacteristic, complemented, Sylow
309C2×#41—21no—
310C2×#41—1061no—
311C22×D4C22⋊D5311yescharacteristic
312C23×C4—21no—
313#135—2124no—
314#135—2124no—
315C2×Q16—1061no—
316C2×Q16—1061no—
317C2×SD16—1061no—
318C2×SD16—1061no—
319C2×SD16—1061no—
320C2×SD16—1061no—
321C2×D8—22no—
322C2×D8—1061no—
323C2×D8—1061no—
324C2×D8—1062no—
325C22×C8—531no—
326C22×C8—531no—
327C22×C8—1062no—
328C22×C8—1062no—
329C22×C8—1061no—
330C22×C8—2124no—
331C4⋊Q8—531no—
332C4⋊Q8C22×D5311yescharacteristic
333#127—531no—
334#127C22×D5311yescharacteristic
335#124—21no—
336#124—1061no—
337#121—21no—
338#121—21no—
339#121—1061no—
340#121—1061no—
341C4×D4—21no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—21no—
345C4×D4—1061no—
346C4×D4—1061no—
347C4×D4C22⋊D5312yes—
348C4×D4C22×D5312yes—
349C2×C4⋊C4C22⋊D5311yescharacteristic
350C2×C22⋊C4—21no—
351C2×C42C22×D5311yescharacteristic
352#107—22no—
353#107—1062no—
354C4⋊C8—532no—
355C4⋊C8—532no—
356C4⋊C8—1061no—
357C4⋊C8C22×D5312yes—
358#102—22no—
359#102—22no—
360#102—1062no—
361#102—1062no—
362C22⋊C8—21no—
363C22⋊C8—1062no—
364C22⋊C8—1062no—
365C22⋊C8—1061no—
366C4×C8—21no—
367C4×C8—531no—
368C4×C8—531no—
369C4×C8—531no—
370C4×C8—531no—
371C4×C8—1061no—
372C24—21no—
373#41—42no—
374#41—42no—
375#41—2122no—
376#41—2122no—
377C2×Q8—21no—
378C2×Q8—1061no—
379C2×D4—24no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—41no—
385C2×D4—1061no—
386C2×D4—1061no—
387C2×D4—1061no—
388C2×D4—2121no—
389C2×D4C2×C22⋊D5312yes—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—2121no—
397C22×C4D4×D5311yescharacteristic
398C22×C4#1171994711yescharacteristic
399C22×C4C2×C22⋊D5311yescharacteristic
400Q16—2122no—
401Q16—2122no—
402SD16—2122no—
403SD16—2122no—
404SD16—2122no—
405SD16—2122no—
406D8—44no—
407D8—2122no—
408D8—2122no—
409D8—2124no—
410C2×C8—22no—
411C2×C8—534no—
412C2×C8—531no—
413C2×C8—531no—
414C2×C8—531no—
415C2×C8—531no—
416C2×C8—1062no—
417C2×C8—1062no—
418C2×C8—1062no—
419C2×C8—2124no—
420C2×C8—2124no—
421C2×C8—2122no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—1061no—
426C4⋊C4—1061no—
427C4⋊C4C2×C22⋊D5312yes—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—2121no—
433C22⋊C4—2121no—
434C42D4×D5311yescharacteristic
435C42#1171994711yescharacteristic
436C42C2×C22⋊D5311yescharacteristic
437C42C23×D5311yescharacteristic
438C23—24no—
439C23—22no—
440C23—42no—
441C23—42no—
442C23—41no—
443C23—2121no—
444C23#1313318911yescharacteristic
445Q8—41no—
446Q8—41no—
447Q8—2121no—
448Q8—2121no—
449D4—24no—
450D4—22no—
451D4—41no—
452D4—42no—
453D4—41no—
454D4—81no—
455D4—82no—
456D4—4241no—
457D4—4242no—
458D4—2121no—
459D4—2122no—
460D4—2121no—
461D4#1313323714yescomplemented
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4—4242no—
472C2×C4—2121no—
473C2×C4#1313318911yescharacteristic
474C2×C4#1313318911yescharacteristic
475C2×C4#1313318911yescharacteristic
476C2×C4D8×D5312yes—
477C2×C4#1313324112yes—
478C2×C4C2×D4×D5311yescharacteristic
479C2×C4C22×C22⋊D5311yescharacteristic
480C2×C4C2×#1171994711yescharacteristic
481C8—42no—
482C8—532no—
483C8—532no—
484C8—1061no—
485C8—1061no—
486C8—2122no—
487C8—2122no—
488C22—21no—
489C22—21no—
490C22—24no—
491C22—48no—
492C22—42no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—4241no—
498C22—4241no—
499C22—4241no—
500C22#1681081612yes—
501C22C2×#1313318911yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—42no—
506C4—81no—
507C4—4241no—
508C4#1681081612yes—
509C4C2×D8×D5311yescharacteristic
510C4C2×#1313323712yes—
511C4C2×#1313324111yescharacteristic
512C2—22no—
513C2—44no—
514C2—161no—
515C2—8481no—
516C2C2×#1681081611yescharacteristic
517C2#2028429311yescharacteristic
518C2#2028443211yescharacteristic
subgroupG/H
centerC2×C4#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3100C2
socleC22×C53C2×#121
hypercenter#3100D53
hypocenterC53#19101
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #20285141C2semidirect
2. #20284715C2semidirect
3. C2×D4×#88322C2semidirect
4. #20274078C2semidirect
5. #20274068C2semidirect
6. #20274058C2semidirect
7. #20274054C2semidirect
8. #20273461C2semidirect
9. #20273364C2semidirect
10. #20273362C2semidirect
11. C53×#3100C2semidirect
18. #16811002C22semidirect
19. #16810999C22semidirect
20. #16810997C22semidirect
21. #16810995C22semidirect
35. #16810375C22semidirect
36. #16810373C22semidirect
37. #16810369C22semidirect
38. #16810368C22semidirect
39. #16810367C22semidirect
40. #16810366C22semidirect
173. C2×#88322D8semidirect
194. D4×C53#135semidirect
255. #3100D53semidirect
308. C53#19101semidirect
461. D4#13133237semidirect

Maximal subgroups

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

subgroupindex
#202851412
#202847152
C2×D4×#883222
#202740782
#202740682
#202740582
#202740542
#202734612
#202733642
#202733622
C53×#31002
#1910153