Finite Group Databasebeta

← #27400241

Subgroups — #27400241

Order 13568.

1132 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811313C211yesmaximal, characteristic
2#20277586C212yesmaximal, complemented
3#20275699C212yesmaximal, complemented
4#20275170C214yesmaximal
5#20274791C212yesmaximal, complemented
6C4×#13133132C211yesmaximal, characteristic
7#20274446C212yesmaximal, complemented
8C53×#1799C211yesmaximal, characteristic
9#16811313—24no—
10#16811313—22no—
11#16811313—22no—
12C2×#13133210C412yescomplemented
13C2×#13133209C411yescharacteristic, complemented
14C2×#13133207C412yescomplemented
15C2×#13133198C2212yes—
16C2×#13133196C411yescharacteristic, complemented
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C411yescharacteristic, complemented
19C2×#13133176C411yescharacteristic, complemented
20C2×#13133173C2214yes—
21C2×#13133173C2212yes—
22C22×C4⋊Dic53C2211yescharacteristic
23C2×#13133132C2212yes—
24#16810519—22no—
25#16810506—24no—
26#16810484C2214yes—
27C4⋊C4×Dic53—22no—
28C4×C22⋊Dic53C2212yes—
29#16810390C2211yescharacteristic
30#16810390C2211yescharacteristic
31#16810389C2214yes—
32#16810389C2214yes—
33C2×C4×D4×C53C2211yescharacteristic
34C2×#99×C53C2212yes—
35C53×#380C2212yes—
36C53×#342C2212yes—
37C22×Dic106C2×C411yescharacteristic
38C23×Dic53C2312yes—
39#13133210—24no—
40#13133210—24no—
41#13133209—22no—
42#13133209—22no—
43#13133207—24no—
44#13133207—24no—
45#13133198—24no—
46#13133198—24no—
47#13133196—22no—
48#13133196—22no—
49#13133196—22no—
50#13133196—22no—
51C4×Dic106—22no—
52C4×Dic106—21no—
53C4×Dic106—21no—
54#13133176—22no—
55#13133176—21no—
56#13133176—21no—
57#13133173—28no—
58#13133173—24no—
59#13133173—24no—
60#13133173—44no—
61C2×C4⋊Dic53C2×C411yescharacteristic
62C2×C4⋊Dic53C2×C411yescharacteristic
63C2×C4⋊Dic53C2×C411yescharacteristic
64C2×C4⋊Dic53C2×C412yes—
65C2×C4⋊Dic53D414yes—
66C2×C4⋊Dic53D412yes—
67C2×C4⋊Dic53D412yes—
68C2×#11719919—24no—
69C2×#11719919C2×C412yes—
70C2×#11719919C2×C412yes—
71C2×#11719919C2×C412yes—
72C2×#11719919C2×C412yes—
73C2×#11719919C2×C412yes—
74C2×C22⋊Dic53C2×C412yes—
75C2×C22⋊Dic53C2×C412yes—
76C2×C22⋊Dic53C2312yes—
77C2×C22⋊Dic53C2312yes—
78C2×C22⋊Dic53C2312yes—
79C2×C4×Dic53—22no—
80C2×C4×Dic53—24no—
81C2×C4×Dic53C2×C412yes—
82C2×C4×Dic53C2×C412yes—
83#13133132—24no—
84#13133124—22no—
85#13133124—24no—
86#13133124—24no—
87C22×D4×C53C2311yescharacteristic
88C23×C4×C53C2×C411yescharacteristic
89C23×C4×C53C2311yescharacteristic
90C4×D4×C53—22no—
91C4×D4×C53—22no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53C2×C411yescharacteristic
94C2×C22⋊C4×C53C2×C411yescharacteristic
95C2×C22⋊C4×C53C2314yes—
96C2×C22⋊C4×C53C2311yescharacteristic
97C2×C42×C53C2×C411yescharacteristic
98#99×C53—42no—
99#95×C53—22no—
100C24×C53C22×C411yescharacteristic
101C24×C53C2411yescharacteristic
102C2×D4×C53—22no—
103C2×D4×C53—22no—
104C2×D4×C53—21no—
105C2×D4×C53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—24no—
109C22×C4×C53—42no—
110C22×C4×C53—42no—
111C22×C4×C53C22×C411yescharacteristic
112C22×C4×C53C22×C411yescharacteristic
113C22×C4×C53C22×C411yescharacteristic
114C22×C4×C53C22×C411yescharacteristic
115C22×C4×C53C22×C411yescharacteristic
116C22×C4×C53C2×D412yes—
117C22×C4×C53C2×D412yes—
118C22×Dic53—24no—
119C22×Dic53—22no—
120C22×Dic53—22no—
121C22×Dic53—22no—
122C22×Dic53—22no—
123C22×Dic53C22×C412yes—
124C22×Dic53C22×C412yes—
125C22×Dic53C22×C412yes—
126C22×Dic53C22×C412yes—
127C22×Dic53C2×D414yes—
128C22×Dic53C2×D414yes—
129C2×Dic106—21no—
130C2×Dic106—21no—
131C2×Dic106—42no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53—44no—
134C22⋊C4×C53—22no—
135C22⋊C4×C53—22no—
136C22⋊C4×C53—24no—
137C22⋊C4×C53—24no—
138C42×C53—21no—
139C42×C53—21no—
140C22⋊Dic53—24no—
141C22⋊Dic53—22no—
142C22⋊Dic53—22no—
143C22⋊Dic53—22no—
144C22⋊Dic53—24no—
145C22⋊Dic53—22no—
146C22⋊Dic53—22no—
147C22⋊Dic53#4114yes—
148C4⋊Dic53—21no—
149C4⋊Dic53—21no—
150C4⋊Dic53C22⋊C412yes—
151C4⋊Dic53C22⋊C411yescharacteristic
152C4⋊Dic53C22⋊C411yescharacteristic
153C4⋊Dic53C22⋊C412yes—
154C4⋊Dic53C22⋊C412yes—
155C4⋊Dic53C22⋊C414yes—
156C4⋊Dic53C22⋊C414yes—
157#11719919—24no—
158#11719919—24no—
159#11719919—24no—
160#11719919—22no—
161#11719919—22no—
162#11719919—24no—
163#11719919—48no—
164C4×Dic53—22no—
165C4×Dic53—22no—
166C4×Dic53—22no—
167C4×Dic53—22no—
168C4×Dic53—48no—
169C4×Dic53—44no—
170C23×C53—21no—
171C23×C53—21no—
172C23×C53—21no—
173C23×C53—41no—
174C23×C53—41no—
175C23×C53—41no—
176C23×C53—41no—
177C23×C53C23×C411yescharacteristic
178C23×C53C22×D412yes—
179C23×C53C2×#4112yes—
180C23×C53H(5,𝔽2)11yescharacteristic
181C23×C532−1+411yescharacteristic
182D4×C53—42no—
183D4×C53—42no—
184C2×C4×C53—22no—
185C2×C4×C53—24no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—41no—
192C2×C4×C53—42no—
193C2×C4×C53—42no—
194C2×C4×C53—42no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—44no—
198C2×C4×C53—44no—
199C2×C4×C53—44no—
200C2×C4×C53C2×C22⋊C411yescharacteristic
201C2×C4×C53C2×C22⋊C412yes—
202C2×C4×C53C2×C22⋊C411yescharacteristic
203C2×Dic53—22no—
204C2×Dic53—22no—
205C2×Dic53—22no—
206C2×Dic53—22no—
207C2×Dic53—24no—
208C2×Dic53—24no—
209C2×Dic53—24no—
210C2×Dic53—24no—
211C2×Dic53—22no—
212C2×Dic53—22no—
213C2×Dic53—24no—
214C2×Dic53—42no—
215C2×Dic53—42no—
216C2×Dic53—42no—
217C2×Dic53—42no—
218C2×Dic53—42no—
219C2×Dic53—42no—
220C2×Dic53—42no—
221C2×Dic53C2×C22⋊C412yes—
222C2×Dic53C2×C22⋊C412yes—
223C2×Dic53C2×C22⋊C414yes—
224Dic106—42no—
225Dic106—42no—
226#22478—531nomaximal, Sylow
227C22×C53—22no—
228C22×C53—21no—
229C22×C53—22no—
230C22×C53—21no—
231C22×C53—42no—
232C22×C53—41no—
233C22×C53—41no—
234C22×C53—41no—
235C22×C53—41no—
236C22×C53—41no—
237C22×C53—81no—
238C22×C53—81no—
239C22×C53C22×C22⋊C411yescharacteristic
240C22×C53C4×#4111yescharacteristic
241C22×C53#52011yescharacteristic
242C22×C53#54612yes—
243C22×C53#54712yes—
244C4×C53—21no—
245C4×C53—22no—
246C4×C53—41no—
247C4×C53—41no—
248C4×C53—41no—
249C4×C53—84no—
250C4×C53#41012yes—
251Dic53—22no—
252Dic53—24no—
253Dic53—22no—
254Dic53—44no—
255Dic53—42no—
256Dic53—42no—
257Dic53—82no—
258C2×#552—531no—
259#2344—532no—
260#1996—532no—
261#1899—534no—
262#1820—532no—
263#1799D5311yescharacteristic
264C4×#99—531no—
265#1740—532no—
266C2×C53—22no—
267C2×C53—22no—
268C2×C53—21no—
269C2×C53—21no—
270C2×C53—81no—
271C2×C53#233911yescharacteristic
272C2×C53C2×#41011yescharacteristic
273C2×C53#288411yescharacteristic
274#552—1062no—
275#552—1064no—
276#552—1062no—
277C2×#126—532no—
278C2×#125—532no—
279C2×#125—531no—
280C2×#123—532no—
281C2×C22⋊Q8—531no—
282C2×C22⋊Q8—531no—
283C2×C4×Q8—531no—
284C2×C4×D4C2×D5311yescharacteristic
285C2×#117—531no—
286C2×#117—532no—
287C2×#117—534no—
288C22×C4⋊C4—531no—
289C2×#99—532no—
290C2×#99C2×D5312yes—
291#389—1062no—
292#387—1064no—
293#381—534no—
294#380C2×D5312yes—
295C4×C4⋊C4—1062no—
296C4×C22⋊C4—532no—
297#342—531no—
298#342—534no—
299#342—534no—
300#342—531no—
301#342C2×D5312yes—
302C53#2247811yescharacteristic, complemented, Sylow
303C22×Q8—531no—
304C22×D4C22×D5311yescharacteristic
305C23×C4—532no—
306C23×C4C4×D5311yescharacteristic
307C23×C4C22×D5311yescharacteristic
308#126—1064no—
309#126—1064no—
310#125—1062no—
311#125—1062no—
312#125—1064no—
313#125—1064no—
314#123—1064no—
315#123—1064no—
316C22⋊Q8—1062no—
317C22⋊Q8—1062no—
318C22⋊Q8—1062no—
319C22⋊Q8—1062no—
320C4×Q8—1062no—
321C4×Q8—1061no—
322C4×Q8—1061no—
323C4×D4—22no—
324C4×D4—22no—
325#117—1064no—
326#117—1064no—
327#117—1062no—
328#117—1061no—
329#117—1061no—
330#117—1068no—
331#117—2124no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—531no—
334C2×C4⋊C4—531no—
335C2×C4⋊C4—532no—
336C2×C4⋊C4—532no—
337C2×C4⋊C4—534no—
338C2×C4⋊C4—532no—
339C2×C4⋊C4—532no—
340C2×C4⋊C4—531no—
341C2×C4⋊C4—532no—
342C2×C4⋊C4—532no—
343C2×C4⋊C4—532no—
344C2×C4⋊C4—532no—
345C2×C4⋊C4—1064no—
346C2×C4⋊C4C4×D5311yescharacteristic
347C2×C22⋊C4—532no—
348C2×C22⋊C4—532no—
349C2×C22⋊C4—532no—
350C2×C22⋊C4—532no—
351C2×C22⋊C4—532no—
352C2×C22⋊C4C4×D5311yescharacteristic
353C2×C22⋊C4C22×D5314yes—
354C2×C22⋊C4C22×D5311yescharacteristic
355C2×C42—532no—
356C2×C42—532no—
357C2×C42—1064no—
358C2×C42—1062no—
359C2×C42C4×D5311yescharacteristic
360#99—42no—
361#99—1064no—
362#95—22no—
363#95—1064no—
364#95—1064no—
365#95—1062no—
366C24C2×C4×D5311yescharacteristic
367C24C23×D5311yescharacteristic
368C2×Q8—1061no—
369C2×Q8—1061no—
370C2×Q8—2122no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—21no—
374C2×D4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—532no—
381C22×C4—532no—
382C22×C4—534no—
383C22×C4—534no—
384C22×C4—532no—
385C22×C4—532no—
386C22×C4—1064no—
387C22×C4—1062no—
388C22×C4—1062no—
389C22×C4—1062no—
390C22×C4—1062no—
391C22×C4C2×C4×D5311yescharacteristic
392C22×C4C2×C4×D5311yescharacteristic
393C22×C4C2×C4×D5311yescharacteristic
394C22×C4C2×C4×D5311yescharacteristic
395C22×C4C2×C4×D5311yescharacteristic
396C22×C4D4×D5312yes—
397C22×C4D4×D5312yes—
398C4⋊C4—22no—
399C4⋊C4—44no—
400C4⋊C4—532no—
401C4⋊C4—531no—
402C4⋊C4—531no—
403C4⋊C4—532no—
404C4⋊C4—532no—
405C4⋊C4—534no—
406C4⋊C4—534no—
407C4⋊C4—1064no—
408C4⋊C4—1064no—
409C4⋊C4—1062no—
410C4⋊C4—1062no—
411C4⋊C4—1064no—
412C4⋊C4—1061no—
413C4⋊C4—1061no—
414C4⋊C4—1064no—
415C4⋊C4—2128no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—534no—
421C22⋊C4—1062no—
422C22⋊C4—1062no—
423C22⋊C4—1064no—
424C22⋊C4—1064no—
425C22⋊C4—1062no—
426C22⋊C4—1062no—
427C22⋊C4—1062no—
428C42—21no—
429C42—21no—
430C42—1062no—
431C42—1062no—
432C42—1062no—
433C42—1062no—
434C42—2124no—
435C42—2128no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23C22×C4×D5311yescharacteristic
444C23C2×D4×D5312yes—
445C23#41×D5312yes—
446C23#1313326211yescharacteristic
447C23#1313326511yescharacteristic
448Q8—2122no—
449Q8—2122no—
450D4—42no—
451D4—42no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—532no—
469C2×C4—532no—
470C2×C4—534no—
471C2×C4—1064no—
472C2×C4—1064no—
473C2×C4—1064no—
474C2×C4—1064no—
475C2×C4—1062no—
476C2×C4—1062no—
477C2×C4—1062no—
478C2×C4—1062no—
479C2×C4—1064no—
480C2×C4—1062no—
481C2×C4—1062no—
482C2×C4—2122no—
483C2×C4—2122no—
484C2×C4—2122no—
485C2×C4—2122no—
486C2×C4—2122no—
487C2×C4—2122no—
488C2×C4—2122no—
489C2×C4C22⋊C4×D5311yescharacteristic
490C2×C4C22⋊C4×D5312yes—
491C2×C4C22⋊C4×D5311yescharacteristic
492C22—22no—
493C22—21no—
494C22—22no—
495C22—21no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—81no—
503C22—81no—
504C22C2×C22⋊C4×D5311yescharacteristic
505C22#1681109111yescharacteristic
506C22#1681110411yescharacteristic
507C22#1681127012yes—
508C22#1681127512yes—
509C4—21no—
510C4—22no—
511C4—41no—
512C4—41no—
513C4—41no—
514C4—84no—
515C4—1062no—
516C4—1062no—
517C4—1064no—
518C4—4242no—
519C4—2124no—
520C4—2122no—
521C4—2122no—
522C4#1681059512yes—
523C2—22no—
524C2—22no—
525C2—21no—
526C2—21no—
527C2—81no—
528C2#2027754311yescharacteristic
529C2C2×#1681059511yescharacteristic
530C2#2028317711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC53×#1799C2
socleC22×C53C22×C22⋊C4
prime radicalC22×D4×C53C23
cosocleC24×C53C24
hypercenter#1799D53
hypocenterC53#22478
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #20277586C2semidirect
3. #20275699C2semidirect
5. #20274791C2semidirect
7. #20274446C2semidirect
12. C2×#13133210C4semidirect
13. C2×#13133209C4semidirect
14. C2×#13133207C4semidirect
16. C2×#13133196C4semidirect
18. C2×C4×Dic106C4semidirect
19. C2×#13133176C4semidirect
302. C53#22478semidirect

Maximal subgroups

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

subgroupindex
C2×#168113132
#202775862
#202756992
#202751702
#202747912
C4×#131331322
#202744462
C53×#17992
#2247853