Finite Group Databasebeta

← #27286404

Subgroups — #27286404

Order 13568.

728 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810900C211yesmaximal, characteristic, complemented
2#20279628C211yesmaximal, characteristic
3#20275677C211yesmaximal, characteristic, complemented
4C2×D4⋊#88322C211yesmaximal, characteristic, complemented
5C2×#16810350C211yesmaximal, characteristic, complemented
6#20272599C211yesmaximal, characteristic, complemented
7C2×C53×#461C211yesmaximal, characteristic, complemented
8C2×#13133191—21no—
9C2×#13133188—21no—
10C2×C4×D212C2211yescharacteristic
11#16810900—22no—
12#16810900—22no—
13C2×#13133150—21no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133137—21no—
16C2×C4×#88322C2211yescharacteristic
17#16810539—22no—
18#16810493C2211yescharacteristic
19#16810483—21no—
20#16810480—21no—
21#16810385—22no—
22D4⋊#88322—22no—
23#16810350—22no—
24C22×SD16×C53—21no—
25C2×C4⋊Q8×C53C2211yescharacteristic
26C2×C4×D4×C53C2211yescharacteristic
27C53×#461—22no—
28C53×#461—22no—
29C2×C4⋊C8×C53C2211yescharacteristic
30C2×#103×C53—21no—
31C22×D212D411yescharacteristic
32C22×C22⋊D53—22no—
33C22×C4×D53—21no—
34C22×#88322—21no—
35C22×#88322—21no—
36#13133191—42no—
37#13133191—42no—
38#13133188—42no—
39#13133188—42no—
40C4×D212—21no—
41C4×D212—21no—
42C4×D212—21no—
43C4×D212—21no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53D411yescharacteristic
47C2×C4⋊Dic53D412yes—
48C2×C22⋊Dic53—22no—
49C2×#11719921—22no—
50C2×#11719921—21no—
51#13133150—24no—
52C4⋊#88322—21no—
53C4⋊#88322D412yescomplemented
54#13133137—24no—
55C4×#88322—21no—
56C4×#88322—21no—
57#13133124—22no—
58C22×Q8×C53—21no—
59C22×D4×C53D411yescharacteristic
60C23×C4×C53—21no—
61C2×SD16×C53—24no—
62C2×SD16×C53—42no—
63C2×SD16×C53—42no—
64C22×C8×C53—21no—
65C4⋊Q8×C53—21no—
66C4⋊Q8×C53—21no—
67C4⋊Q8×C53—21no—
68C4⋊Q8×C53D412yescomplemented
69C4×D4×C53—21no—
70C4×D4×C53—21no—
71C4×D4×C53—21no—
72C4×D4×C53—21no—
73C2×C4⋊C4×C53—21no—
74C2×C4⋊C4×C53—21no—
75C2×C4⋊C4×C53D411yescharacteristic
76C2×C22⋊C4×C53—21no—
77C2×C42×C53C2311yescharacteristic
78C4⋊C8×C53—21no—
79C4⋊C8×C53D412yescomplemented
80#103×C53—24no—
81C24×C53—21no—
82C23×D53—21no—
83C2×Q8×C53—22no—
84C2×Q8×C53—41no—
85C2×Q8×C53—41no—
86C2×Q8×C53—41no—
87C2×Q8×C53—41no—
88C2×Q8×C53—41no—
89C2×D4×C53—21no—
90C2×D4×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—21no—
93C2×D4×C53—21no—
94C2×D4×C53D812yes—
95C22×C4×C53—21no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C2×C22⋊D53—44no—
107C2×C22⋊D53—44no—
108C2×C22⋊D53—82no—
109C22×Dic53—21no—
110C22×Dic53—22no—
111C2×D212—21no—
112C2×D212—21no—
113C2×D212—21no—
114C2×D212—21no—
115C2×D212—21no—
116C2×D212SD1612yescomplemented
117C2×C4×D53—41no—
118C2×C4×D53—41no—
119C2×C4×D53—41no—
120C2×C4×D53—41no—
121C2×C4×D53—41no—
122C2×C4×D53—41no—
123SD16×C53—48no—
124C2×C8×C53—22no—
125C2×C8×C53—41no—
126C2×C8×C53—41no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—22no—
129C4⋊C4×C53—22no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—41no—
132C4⋊C4×C53—41no—
133C22⋊C4×C53—41no—
134C22⋊C4×C53—41no—
135C42×C53C2×D411yescharacteristic
136C42×C53C2×D411yescharacteristic
137C42×C53C2×D411yescharacteristic
138C42×C53#4111yescharacteristic
139C22⋊Dic53—44no—
140#11719921—44no—
141#11719921—41no—
142#11719921—41no—
143C4⋊Dic53—21no—
144C4⋊Dic53—21no—
145C4⋊Dic53—41no—
146C4⋊Dic53—41no—
147C4⋊Dic53—41no—
148C4⋊Dic53—41no—
149C4⋊Dic53—42no—
150C2×#88322—22no—
151C2×#88322—22no—
152C2×#88322—41no—
153C2×#88322—41no—
154C2×#88322—41no—
155C2×#88322—41no—
156C23×C53—22no—
157C23×C53—41no—
158C23×C53—41no—
159C23×C53—41no—
160C23×C53—41no—
161C23×C53—41no—
162C23×C53—41no—
163C23×C53#12011yescharacteristic
164C22×D53—22no—
165C22×D53—41no—
166C22×D53—41no—
167C22×D53—41no—
168C22×D53—41no—
169C22×D53—41no—
170C22×D53—41no—
171Q8×C53—42no—
172Q8×C53—42no—
173Q8×C53—81no—
174Q8×C53—81no—
175D4×C53—22no—
176D4×C53—22no—
177D4×C53—41no—
178D4×C53—41no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—41no—
182C2×C4×C53—42no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—81no—
186C2×C4×C53—81no—
187C2×C4×C53—81no—
188C2×C4×C53—81no—
189C2×C4×C53—81no—
190C2×C4×C53—81no—
191C2×C4×C53#12011yescharacteristic
192C2×C4×C53#12011yescharacteristic
193C2×C4×C53#12111yescharacteristic
194C2×C4×C53#12111yescharacteristic
195C2×C4×C53#12111yescharacteristic
196C2×C4×C53#12711yescharacteristic
197C2×C4×C53C2×D811yescharacteristic
198C2×C4×C53C2×SD1611yescharacteristic
199C2×C4×C53#13511yescharacteristic
200C2×C4×C53#13511yescharacteristic
201C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
202C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
203C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
204C2×C4×C53#13711yescharacteristic
205C22⋊D53—88no—
206C2×Dic53—41no—
207C2×Dic53—41no—
208C2×Dic53—42no—
209C2×Dic53—41no—
210C2×Dic53—42no—
211C2×Dic53—42no—
212D212—22no—
213D212—22no—
214D212—41no—
215D212—41no—
216C4×D53—81no—
217C4×D53—81no—
218C4×D53—81no—
219C4×D53—81no—
220C8×C53—42no—
221#88322—42no—
222#88322—42no—
223#10231—531nomaximal, Sylow
224C22×C53—42no—
225C22×C53—42no—
226C22×C53—42no—
227C22×C53—81no—
228C22×C53—81no—
229C22×C53—81no—
230C22×C53—81no—
231C22×C53#39211yescharacteristic
232C22×C53#44711yescharacteristic
233C22×C53#44911yescharacteristic
234C22×C53#45011yescharacteristic
235C22×C53#45211yescharacteristic
236C22×C53#45511yescharacteristic
237C22×C53#45611yescharacteristic
238C2×D53—42no—
239C2×D53—42no—
240C2×D53—42no—
241C2×D53—81no—
242C2×D53—81no—
243C2×D53—81no—
244C2×D53—81no—
245C4×C53—21no—
246C4×C53—21no—
247C4×C53—82no—
248C4×C53—81no—
249C4×C53#44711yescharacteristic
250C4×C53#44811yescharacteristic
251C4×C53#44911yescharacteristic
252C4×C53#44911yescharacteristic
253C4×C53#46511yescharacteristic
254C4×C53#46611yescharacteristic
255C4×C53#46811yescharacteristic
256C4×C53#46911yescharacteristic
257Dic53—82no—
258Dic53—81no—
259C2×#474—531no—
260C2×#461D5311yescharacteristic, complemented
261#2537—531no—
262#1990—531no—
263C2×D4⋊C8—531no—
264C2×D4⋊C8—531no—
265#1296—531no—
266C2×C53—82no—
267C2×C53#163911yescharacteristic
268C2×C53#164611yescharacteristic
269C2×C53#164711yescharacteristic
270C2×C53#165411yescharacteristic
271C2×C53#199911yescharacteristic
272C2×C53#200011yescharacteristic
273C2×C53#200711yescharacteristic
274D53—82no—
275C22×SD16—21no—
276C2×C4⋊Q8C2×D5311yescharacteristic
277C2×#121—1061no—
278C2×#121—1061no—
279C2×C4×D4—531no—
280C2×C4×D4C2×D5311yescharacteristic
281#474—1062no—
282#474—1062no—
283#461—22no—
284#461—22no—
285C2×#107—1061no—
286C2×C4⋊C8—531no—
287C2×C4⋊C8C2×D5311yescharacteristic
288C2×#103—21no—
289C2×#102—1061no—
290C2×C4×C8—531no—
291#394—1062no—
292#384—531no—
293#380—1061no—
294#380—1061no—
295#340—1062no—
296D4⋊C8—1062no—
297D4⋊C8—1062no—
298C53#1023111yescharacteristic, complemented, Sylow
299C22×Q8—21no—
300C22×D4—531no—
301C22×D4—1062no—
302C22×D4C22⋊D5311yescharacteristic
303C23×C4—21no—
304C23×C4—1061no—
305C2×SD16—24no—
306C2×SD16—42no—
307C2×SD16—42no—
308C22×C8—21no—
309C22×C8—1061no—
310C22×C8—1061no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8C22⋊D5312yescomplemented
315#121—2122no—
316#121—2122no—
317#121—2122no—
318#121—2122no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—21no—
323C4×D4—1061no—
324C4×D4—1061no—
325C4×D4—1061no—
326C4×D4—1061no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—531no—
330C2×C4⋊C4—532no—
331C2×C4⋊C4—1061no—
332C2×C4⋊C4—1061no—
333C2×C4⋊C4C22⋊D5311yescharacteristic
334C2×C22⋊C4—21no—
335C2×C22⋊C4—1062no—
336C2×C22⋊C4—1062no—
337C2×C22⋊C4—1061no—
338C2×C42C22×D5311yescharacteristic
339#107—1064no—
340C4⋊C8—21no—
341C4⋊C8—532no—
342C4⋊C8—1061no—
343C4⋊C8C22⋊D5312yescomplemented
344#103—24no—
345#102—1064no—
346C4×C8—1061no—
347C4×C8—1061no—
348#95—1062no—
349C24—21no—
350C24—1061no—
351C2×Q8—22no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—532no—
363C2×D4—1061no—
364C2×D4—1061no—
365C2×D4—1061no—
366C2×D4—1061no—
367C2×D4—1061no—
368C2×D4—4242no—
369C2×D4—2124no—
370C2×D4—2124no—
371C2×D4#1171992212yes—
372C22×C4—21no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—1061no—
381C22×C4—1062no—
382C22×C4—2121no—
383C22×C4—2121no—
384C22×C4—2121no—
385C22×C4—2121no—
386C22×C4—2121no—
387C22×C4—2121no—
388C22×C4D4×D5311yescharacteristic
389C22×C4D4×D5311yescharacteristic
390C22×C4C2×C22⋊D5311yescharacteristic
391SD16—48no—
392C2×C8—22no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—1062no—
396C2×C8—1062no—
397C2×C8—2121no—
398C2×C8—2121no—
399C2×C8—2121no—
400C2×C8—2121no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—1061no—
408C4⋊C4—1061no—
409C4⋊C4—2122no—
410C4⋊C4—2121no—
411C4⋊C4—2121no—
412C4⋊C4—2121no—
413C4⋊C4—2121no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—2124no—
417C22⋊C4—2124no—
418C22⋊C4—2121no—
419C22⋊C4—2121no—
420C42D4×D5311yescharacteristic
421C42#1171994711yescharacteristic
422C42C2×C22⋊D5311yescharacteristic
423C42C2×C22⋊D5311yescharacteristic
424C23—22no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—1062no—
432C23—2121no—
433C23—2121no—
434C23—2121no—
435C23—2121no—
436C23—2121no—
437C23—2121no—
438C23#1313318511yescharacteristic
439Q8—42no—
440Q8—42no—
441Q8—81no—
442Q8—81no—
443D4—22no—
444D4—22no—
445D4—41no—
446D4—41no—
447D4—1062no—
448D4—1062no—
449D4—4248no—
450D4—2121no—
451D4—2121no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—4241no—
465C2×C4—4241no—
466C2×C4—4241no—
467C2×C4—4241no—
468C2×C4—2121no—
469C2×C4—2122no—
470C2×C4—2121no—
471C2×C4—2122no—
472C2×C4—2121no—
473C2×C4—2122no—
474C2×C4#1313318511yescharacteristic
475C2×C4#1313318611yescharacteristic
476C2×C4#1313318811yescharacteristic
477C2×C4#1313318911yescharacteristic
478C2×C4#1313318911yescharacteristic
479C2×C4#1313321311yescharacteristic
480C2×C4C2×#1171992211yescharacteristic
481C2×C4SD16×D5311yescharacteristic
482C2×C4#1313323711yescharacteristic
483C2×C4#1313323811yescharacteristic
484C2×C4#1313324211yescharacteristic
485C2×C4#1313324311yescharacteristic
486C2×C4#1313324411yescharacteristic
487C2×C4#1313324611yescharacteristic
488C8—42no—
489C8—2122no—
490C8—2122no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—4241no—
499C22—4241no—
500C22—4241no—
501C22—4241no—
502C22—2122no—
503C22—2122no—
504C22—2122no—
505C22#1681053111yescharacteristic
506C22#1681075611yescharacteristic
507C22#1681076611yescharacteristic
508C22#1681077211yescharacteristic
509C22#1681078211yescharacteristic
510C22#1681079911yescharacteristic
511C22#1681080411yescharacteristic
512C4—21no—
513C4—21no—
514C4—81no—
515C4—82no—
516C4—4242no—
517C4—4241no—
518C4#1681075711yescharacteristic
519C4#1681076011yescharacteristic
520C4#1681076711yescharacteristic
521C4#1681076811yescharacteristic
522C4#1681084911yescharacteristic
523C4#1681085711yescharacteristic
524C4#1681086611yescharacteristic
525C4#1681087411yescharacteristic
526C2—82no—
527C2—4242no—
528C2#2027389211yescharacteristic
529C2#2027394111yescharacteristic
530C2#2027394811yescharacteristic
531C2#2027399711yescharacteristic
532C2#2027571111yescharacteristic
533C2#2027571511yescharacteristic
534C2#2027576011yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#461C2
socleC23×C53#120
prime radical#20279628C2
hypercenterC2×#461D53
hypocenterC53#10231
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C2×#16810900C2semidirect
3. #20275677C2semidirect
4. C2×D4⋊#88322C2semidirect
5. C2×#16810350C2semidirect
6. #20272599C2semidirect
7. C2×C53×#461C2semidirect
53. C4⋊#88322D4semidirect
68. C4⋊Q8×C53D4semidirect
79. C4⋊C8×C53D4semidirect
116. C2×D212SD16semidirect
260. C2×#461D53semidirect
298. C53#10231semidirect
314. C4⋊Q8C22⋊D53semidirect
343. C4⋊C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109002
#202796282
#202756772
C2×D4⋊#883222
C2×#168103502
#202725992
C2×C53×#4612
#1023153