Finite Group Databasebeta

← #27425550

Subgroups — #27425550

Order 13568.

706 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
1C2×#16810900C211yesmaximal, characteristic, complemented
2C4×#13133242C211yesmaximal, characteristic
3#20283381C211yesmaximal, characteristic, complemented
4#20276158C212yesmaximal, complemented
5#20276005C212yesmaximal
6#20275705C211yesmaximal, characteristic, complemented
7#20275497C212yesmaximal, complemented
8#20275488C211yesmaximal, characteristic, complemented
9#20275376C212yesmaximal, complemented
10#20274704C211yesmaximal, characteristic, complemented
11C53×#3079C211yesmaximal, characteristic, complemented
12C2×#13133242C2211yescharacteristic
13C2×C4×D212C2211yescharacteristic
14#16810900—21no—
15#16810900—21no—
16#16810900—22no—
17#16810733—21no—
18#16810721—21no—
19C4×#11719924—21no—
20C4×#11719922—21no—
21#16810684—22no—
22#16810680—21no—
23#16810676—21no—
24C2×#13133152C2212yes—
25#16810666C2211yescharacteristic
26C2×#13133150C2212yes—
27#16810647C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29#16810627C2211yescharacteristic
30#16810624C2211yescharacteristic
31C2×#13133137C2212yes—
32C2×#13133133C2212yes—
33#16810590C2212yes—
34#16810579C2212yes—
35C4×#11719917C2211yescharacteristic
36#16810392C2212yes—
37#16810392C2212yes—
38#16810391C2212yes—
39#16810387C2212yes—
40C2×C4⋊Q8×C53C2211yescharacteristic
41C4×#41×C53C2211yescharacteristic
42C53×#477—21no—
43C53×#476—21no—
44C53×#475—21no—
45C53×#474—21no—
46C53×#428C2212yes—
47C53×#423C2211yescharacteristic
48C53×#417C2212yes—
49C22×D212C2311yescharacteristic
50C22×C4×D53—21no—
51#13133242—22no—
52#13133242—22no—
53C2×#11719924—21no—
54C2×#11719922—21no—
55C2×#11719917C2311yescharacteristic
56C2×#11719917C2312yes—
57C2×#11719917C2311yescharacteristic
58C22×#88322C2312yes—
59C4×D212—21no—
60C4×D212—21no—
61C4×D212—21no—
62C4×D212—21no—
63C2×C4⋊Dic53C2311yescharacteristic
64C2×#11719921—21no—
65#13133152—24no—
66#13133150—21no—
67#13133150—22no—
68#13133150—22no—
69#13133148—21no—
70C4⋊#88322—21no—
71C4⋊#88322—21no—
72C4⋊#88322D411yescharacteristic
73C4⋊#88322D411yescharacteristic
74C4⋊#88322D412yes—
75#13133141—21no—
76#13133138—21no—
77#13133137—21no—
78#13133137—21no—
79#13133137—22no—
80#13133137—22no—
81#13133133—24no—
82C22⋊#88322—22no—
83C22⋊#88322—22no—
84#13133128—21no—
85#13133128—22no—
86C4×#88322—21no—
87C4×#88322—22no—
88C2×#41×C53C2311yescharacteristic
89C22×Q8×C53—21no—
90C2×M16×C53C2312yes—
91C4⋊Q8×C53—21no—
92C4⋊Q8×C53—21no—
93C4⋊Q8×C53—21no—
94C4⋊Q8×C53—21no—
95C4×Q8×C53D411yescharacteristic
96C4×Q8×C53D411yescharacteristic
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53D411yescharacteristic
100C4×D4×C53D411yescharacteristic
101#117×C53—21no—
102#117×C53C2311yescharacteristic
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53C2312yes—
105C2×C42×C53—21no—
106C2×C42×C53C2311yescharacteristic
107#107×C53—22no—
108#106×C53—22no—
109C4⋊C8×C53—21no—
110C4⋊C8×C53—21no—
111#103×C53—22no—
112#102×C53—22no—
113C23×D53—21no—
114#41×C53—21no—
115#41×C53—21no—
116#41×C53—21no—
117#41×C53—21no—
118C2×Q8×C53—41no—
119C2×Q8×C53—41no—
120C2×Q8×C53—82no—
121C2×Q8×C53C2×D411yescharacteristic
122C2×D4×C53—21no—
123C2×D4×C53C2×D411yescharacteristic
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—22no—
128C22×C4×C53#4111yescharacteristic
129C22×C4×C53#4111yescharacteristic
130C22×C4×C53C2411yescharacteristic
131C22×Dic53—21no—
132C2×D212—21no—
133C2×D212—21no—
134C2×D212—21no—
135C2×D212—21no—
136C2×D212—21no—
137C2×D212#4112yes—
138C2×C4×D53—41no—
139C2×C4×D53—41no—
140C2×C4×D53—82no—
141M16×C53—22no—
142M16×C53—22no—
143C2×C8×C53—22no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53—41no—
149C4⋊C4×C53—41no—
150C4⋊C4×C53C2×D411yescharacteristic
151C4⋊C4×C53C2×D411yescharacteristic
152C22⋊C4×C53—21no—
153C22⋊C4×C53—21no—
154C22⋊C4×C53—21no—
155C42×C53—21no—
156C42×C53—21no—
157C42×C53—21no—
158C42×C53C2×D411yescharacteristic
159C42×C53C2×D411yescharacteristic
160C42×C53C2×D411yescharacteristic
161C42×C53C2×D411yescharacteristic
162#11719924—24no—
163#11719922—24no—
164#11719921—41no—
165#11719921—41no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168#11719917—21no—
169#11719917—22no—
170#11719917—21no—
171#11719917—21no—
172#11719917—22no—
173#11719917#4112yes—
174C2×#88322—21no—
175C2×#88322—22no—
176C2×#88322—22no—
177C2×#88322—22no—
178C2×#88322—21no—
179C2×#88322C2×D412yes—
180C2×#88322C2×D412yes—
181C23×C53—21no—
182C23×C53C2×#4111yescharacteristic
183C22×D53—22no—
184C22×D53—41no—
185C22×D53—41no—
186C22×D53—81no—
187C22×D53—81no—
188Q8×C53—21no—
189Q8×C53—21no—
190Q8×C53—82no—
191Q8×C53—82no—
192D4×C53—21no—
193D4×C53—21no—
194D4×C53—41no—
195D4×C53—41no—
196C2×C4×C53—22no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53—81no—
206C2×C4×C53—82no—
207C2×C4×C53#12111yescharacteristic
208C2×C4×C53#12111yescharacteristic
209C2×C4×C53#12111yescharacteristic
210C2×C4×C53#12111yescharacteristic
211C2×C4×C53C22×D411yescharacteristic
212C2×C4×C53C22×D411yescharacteristic
213C2×C4×C53C2×#4111yescharacteristic
214C2×C4×C53C2×#4111yescharacteristic
215C2×C4×C532−1+411yescharacteristic
216C2×C4×C532−1+411yescharacteristic
217C2×Dic53—41no—
218C2×Dic53—81no—
219D212—22no—
220D212—22no—
221D212—41no—
222D212—41no—
223C4×D53—82no—
224C4×D53—82no—
225C8×C53—42no—
226#88322—22no—
227#88322—22no—
228#88322—22no—
229#88322—41no—
230#88322—42no—
231#24720—531nomaximal, Sylow
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—41no—
235C22×C53—41no—
236C22×C53—41no—
237C22×C53C2×#12111yescharacteristic
238C22×C53#55211yescharacteristic
239C22×C53#55611yescharacteristic
240C2×D53—42no—
241C2×D53—82no—
242C2×D53—81no—
243C2×D53—81no—
244C2×D53—82no—
245C4×C53—21no—
246C4×C53—21no—
247C4×C53—21no—
248C4×C53—41no—
249C4×C53—41no—
250C4×C53—41no—
251C4×C53—82no—
252C4×C53#47212yes—
253C4×C53#52511yescharacteristic
254C4×C53#53611yescharacteristic
255C4×C53#54711yescharacteristic
256C4×C53#54811yescharacteristic
257Dic53—81no—
258#3079D5311yescharacteristic, complemented
259C2×#474—531no—
260C4×AGL(1,ℤ/8)—531no—
261#2925—531no—
262#2078—532no—
263#2051—532no—
264#1997—531no—
265#1952—532no—
266#1950—531no—
267#1931—532no—
268#1800—531no—
269C2×C53—21no—
270C2×C53—21no—
271C2×C53—81no—
272C2×C53#241511yescharacteristic
273C2×C53C2×#47211yescharacteristic
274C2×C53#306911yescharacteristic
275D53—82no—
276C2×AGL(1,ℤ/8)—531no—
277C2×C4⋊Q8C2×D5311yescharacteristic
278C4×#41C2×D5311yescharacteristic
279C2×C4×D4—531no—
280#477—21no—
281#476—21no—
282#475—21no—
283#474—21no—
284#474—1061no—
285#474—1061no—
286#474—1062no—
287#442—1061no—
288#440—1061no—
289C4×SD16—1061no—
290C4×D8—1061no—
291#433—1062no—
292#432—1061no—
293#431—1061no—
294C2×#108—532no—
295#428—531no—
296#428C2×D5312yes—
297C2×#107—532no—
298#423—531no—
299#423C2×D5311yescharacteristic
300C2×C4⋊C8—531no—
301#418—531no—
302#417—531no—
303#417C2×D5312yes—
304C2×#102—532no—
305C2×#100—532no—
306#408—532no—
307#405—532no—
308C4×M16—531no—
309#343—532no—
310#343—532no—
311#343—532no—
312#341—532no—
313C53#2472011yescharacteristic, complemented, Sylow
314C2×#41C22×D5311yescharacteristic
315C22×Q8—21no—
316C22×D4—531no—
317C23×C4—1061no—
318AGL(1,ℤ/8)—1062no—
319AGL(1,ℤ/8)—1062no—
320C2×SD16—1061no—
321C2×D8—1061no—
322C2×M16—531no—
323C2×M16—531no—
324C2×M16—532no—
325C2×M16C22×D5312yes—
326C22×C8—532no—
327C4⋊Q8—21no—
328C4⋊Q8—21no—
329C4⋊Q8—21no—
330C4⋊Q8—21no—
331C4×Q8C22⋊D5311yescharacteristic
332C4×Q8C22⋊D5311yescharacteristic
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—1061no—
336C4×D4—1061no—
337C4×D4—1061no—
338C4×D4—1061no—
339C4×D4C22⋊D5311yescharacteristic
340C4×D4C22⋊D5311yescharacteristic
341#117—21no—
342#117C22×D5311yescharacteristic
343C2×C4⋊C4—21no—
344C2×C4⋊C4—531no—
345C2×C4⋊C4C22×D5312yes—
346C2×C22⋊C4—1061no—
347C2×C42—21no—
348C2×C42C22×D5311yescharacteristic
349#108—1064no—
350#107—22no—
351#107—1062no—
352#107—1062no—
353#107—1061no—
354#106—22no—
355#106—1061no—
356C4⋊C8—21no—
357C4⋊C8—21no—
358C4⋊C8—531no—
359C4⋊C8—531no—
360C4⋊C8—532no—
361C4⋊C8—1061no—
362C4⋊C8—1061no—
363#103—22no—
364#103—1061no—
365#102—22no—
366#102—1062no—
367#102—1062no—
368#102—1061no—
369#102—1061no—
370#102—1061no—
371#100—1064no—
372C22⋊C8—1062no—
373C22⋊C8—1062no—
374#97—1062no—
375#97—1061no—
376C4×C8—1062no—
377C4×C8—1061no—
378C24—1061no—
379#41—21no—
380#41—21no—
381#41—21no—
382#41—21no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—82no—
386C2×Q8C2×C22⋊D5311yescharacteristic
387C2×D4—21no—
388C2×D4—532no—
389C2×D4—1061no—
390C2×D4—1061no—
391C2×D4—1061no—
392C2×D4—1061no—
393C2×D4—1061no—
394C2×D4C2×C22⋊D5311yescharacteristic
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—1061no—
400C22×C4—4242no—
401C22×C4—2121no—
402C22×C4—2121no—
403C22×C4#1171994711yescharacteristic
404C22×C4#1171994711yescharacteristic
405C22×C4C23×D5311yescharacteristic
406SD16—1064no—
407D8—1064no—
408M16—22no—
409M16—22no—
410M16—532no—
411M16—1061no—
412M16—1062no—
413M16—1062no—
414M16—1061no—
415M16—1061no—
416C2×C8—22no—
417C2×C8—532no—
418C2×C8—532no—
419C2×C8—1061no—
420C2×C8—1062no—
421C2×C8—1062no—
422C2×C8—1062no—
423C2×C8—1061no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—1061no—
431C4⋊C4—1061no—
432C4⋊C4C2×C22⋊D5311yescharacteristic
433C4⋊C4C2×C22⋊D5311yescharacteristic
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—2121no—
438C22⋊C4—2121no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42D4×D5311yescharacteristic
443C42D4×D5311yescharacteristic
444C42C2×C22⋊D5311yescharacteristic
445C42C2×C22⋊D5311yescharacteristic
446C23—21no—
447C23—1062no—
448C23—4241no—
449C23—4241no—
450C23—2121no—
451C23—2121no—
452C23C2×#1171994711yescharacteristic
453Q8—21no—
454Q8—21no—
455Q8—82no—
456Q8—82no—
457D4—21no—
458D4—21no—
459D4—41no—
460D4—41no—
461D4—1062no—
462D4—1062no—
463D4—2121no—
464D4—2121no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—4242no—
477C2×C4—4242no—
478C2×C4—4241no—
479C2×C4—2121no—
480C2×C4#1313318911yescharacteristic
481C2×C4#1313318911yescharacteristic
482C2×C4#1313318911yescharacteristic
483C2×C4#1313318911yescharacteristic
484C2×C4C2×D4×D5311yescharacteristic
485C2×C4C22×C22⋊D5311yescharacteristic
486C2×C4#41×D5311yescharacteristic
487C2×C4#41×D5311yescharacteristic
488C2×C4#1313326511yescharacteristic
489C2×C4#1313326511yescharacteristic
490C8—42no—
491C8—1062no—
492C8—1062no—
493C8—1062no—
494C8—2122no—
495C8—2121no—
496C22—21no—
497C22—21no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—4242no—
502C22—4242no—
503C22—4241no—
504C22—4241no—
505C22—2122no—
506C22C2×#1313318911yescharacteristic
507C22#1681130711yescharacteristic
508C22#1681134411yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—82no—
516C4—4241no—
517C4#1681089112yes—
518C4#1681113011yescharacteristic
519C4#1681119711yescharacteristic
520C4#1681127511yescharacteristic
521C4#1681128311yescharacteristic
522C2—21no—
523C2—21no—
524C2—81no—
525C2—4242no—
526C2#2027814511yescharacteristic
527C2C2×#1681089111yescharacteristic
528C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#3079C2
socleC22×C53C2×#121
prime radicalC2×#13133242C22
hypercenter#3079D53
hypocenterC53#24720
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3079

Complements

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

NH ≅ G/Ntype
1. C2×#16810900C2semidirect
3. #20283381C2semidirect
4. #20276158C2semidirect
6. #20275705C2semidirect
7. #20275497C2semidirect
8. #20275488C2semidirect
9. #20275376C2semidirect
10. #20274704C2semidirect
11. C53×#3079C2semidirect
258. #3079D53semidirect
313. C53#24720semidirect

Maximal subgroups

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

subgroupindex
C2×#168109002
C4×#131332422
#202833812
#202761582
#202760052
#202757052
#202754972
#202754882
#202753762
#202747042
C53×#30792
#2472053