Finite Group Databasebeta

← #27400645

Subgroups — #27400645

Order 13568.

1114 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811330C211yesmaximal, characteristic
2#20277590C212yesmaximal, complemented
3#20275700C212yesmaximal, complemented
4#20275698C214yesmaximal, complemented
5#20274703C211yesmaximal, characteristic
6#20274701C212yesmaximal
7#99×Dic53C212yesmaximal
8C4×#99×C53C211yesmaximal, characteristic
9#16811330—24no—
10#16811330—22no—
11#16811330—22no—
12C2×#13133217C411yescharacteristic, complemented
13C2×#13133208C412yescomplemented
14C2×#13133196C411yescharacteristic, complemented
15C2×#13133196C2211yescharacteristic
16C2×#13133194C412yescomplemented
17C2×#13133194C2212yes—
18C2×C4×Dic106C411yescharacteristic, complemented
19C2×D4×Dic53C2212yes—
20C22×C4⋊Dic53C411yescharacteristic, complemented
21C22×C4⋊Dic53C2211yescharacteristic
22C2×#13133132C2212yes—
23#16810521—22no—
24#16810506—24no—
25#16810483C2212yes—
26#16810481C2214yes—
27C4×C4⋊Dic53—22no—
28C22⋊C4×Dic53C2214yes—
29#16810390C2212yes—
30#16810389C2214yes—
31#16810389C2214yes—
32C2×C4×D4×C53C2211yescharacteristic
33C2×#99×C53C2211yescharacteristic
34C2×#99×C53C2211yescharacteristic
35C4×C22⋊C4×C53C2212yes—
36C53×#342C2212yes—
37C22×Dic106C2×C411yescharacteristic
38C22×Dic106C2×C412yes—
39C23×Dic53C2×C412yes—
40C23×Dic53C2312yes—
41#13133217—22no—
42#13133217—22no—
43#13133208—24no—
44#13133208—24no—
45#13133196—22no—
46#13133196—22no—
47#13133196—22no—
48#13133196—22no—
49#13133194—24no—
50#13133194—24no—
51#13133194—24no—
52#13133194—24no—
53C4×Dic106—22no—
54C4×Dic106—21no—
55C4×Dic106—21no—
56D4×Dic53—24no—
57D4×Dic53—24no—
58C2×C4⋊Dic53—22no—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53—22no—
62C2×C4⋊Dic53C2×C412yes—
63C2×C4⋊Dic53C2×C411yescharacteristic
64C2×C4⋊Dic53C2×C412yes—
65C2×C4⋊Dic53C2×C411yescharacteristic
66C2×C4⋊Dic53C2×C412yes—
67C2×C4⋊Dic53C2×C411yescharacteristic
68C2×C4⋊Dic53D414yes—
69C2×C4⋊Dic53D412yes—
70C2×C4⋊Dic53D412yes—
71C2×#11719919—24no—
72C2×#11719919C2×C412yes—
73C2×#11719919C2×C412yes—
74C2×#11719919C2×C412yes—
75C2×C22⋊Dic53C2×C412yes—
76C2×C22⋊Dic53C2314yes—
77C2×C22⋊Dic53C2312yes—
78C2×C4×Dic53—24no—
79C2×C4×Dic53C2×C412yes—
80#13133132—24no—
81#13133124—22no—
82#13133124—24no—
83#13133124—24no—
84C22×D4×C53C2311yescharacteristic
85C23×C4×C53C2×C411yescharacteristic
86C23×C4×C53C2311yescharacteristic
87C4×D4×C53—22no—
88C4×D4×C53—22no—
89C2×C4⋊C4×C53C2×C411yescharacteristic
90C2×C22⋊C4×C53C2×C411yescharacteristic
91C2×C22⋊C4×C53C2312yes—
92C2×C22⋊C4×C53C2312yes—
93C2×C22⋊C4×C53C2311yescharacteristic
94C2×C42×C53—22no—
95C2×C42×C53C2×C411yescharacteristic
96#99×C53—22no—
97#99×C53Q814yescomplemented
98#95×C53—22no—
99C24×C53C22×C411yescharacteristic
100C24×C53C2411yescharacteristic
101C2×D4×C53—22no—
102C2×D4×C53—22no—
103C2×D4×C53C2×Q812yes—
104C2×D4×C53#4112yes—
105C22×C4×C53—22no—
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—22no—
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—22no—
119C22×Dic53—22no—
120C22×Dic53—24no—
121C22×Dic53—82no—
122C22×Dic53—82no—
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—24no—
132C2×Dic106—42no—
133C2×Dic106—82no—
134C4⋊C4×C53—22no—
135C22⋊C4×C53—22no—
136C22⋊C4×C53—22no—
137C22⋊C4×C53—22no—
138C22⋊C4×C53—22no—
139C22⋊C4×C53—22no—
140C22⋊C4×C53C2×Q814yes—
141C42×C53—21no—
142C42×C53—21no—
143C42×C53—44no—
144C22⋊Dic53—24no—
145C22⋊Dic53—24no—
146C22⋊Dic53—24no—
147C22⋊Dic53—24no—
148C4⋊Dic53—21no—
149C4⋊Dic53—21no—
150C4⋊Dic53—22no—
151C4⋊Dic53—22no—
152C4⋊Dic53—24no—
153C4⋊Dic53—44no—
154C4⋊Dic53C22⋊C412yes—
155C4⋊Dic53C22⋊C411yescharacteristic
156C4⋊Dic53C22⋊C411yescharacteristic
157C4⋊Dic53C22⋊C414yes—
158C4⋊Dic53C22⋊C414yes—
159C4⋊Dic53C22⋊C412yes—
160C4⋊Dic53C22⋊C412yes—
161#11719919—24no—
162#11719919—22no—
163#11719919—22no—
164#11719919—24no—
165#11719919—84no—
166C4×Dic53—22no—
167C4×Dic53—22no—
168C4×Dic53—84no—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53—41no—
172C23×C53—41no—
173C23×C53—41no—
174C23×C53—41no—
175C23×C53C4×Q812yes—
176C23×C53C23×C411yescharacteristic
177C23×C53C22×D412yes—
178C23×C53C22×Q811yescharacteristic
179C23×C53C2×#4111yescharacteristic
180C23×C532−1+412yes—
181D4×C53—24no—
182D4×C53—24no—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—22no—
186C2×C4×C53—24no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—24no—
190C2×C4×C53—41no—
191C2×C4×C53—42no—
192C2×C4×C53—42no—
193C2×C4×C53—42no—
194C2×C4×C53—42no—
195C2×C4×C53—42no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—42no—
199C2×C4×C53—42no—
200C2×C4×C53—42no—
201C2×C4×C53C2×C22⋊C411yescharacteristic
202C2×C4×C53C2×C22⋊C412yes—
203C2×C4×C53C2×C22⋊C411yescharacteristic
204C2×C4×C53C4×Q812yes—
205C2×Dic53—22no—
206C2×Dic53—22no—
207C2×Dic53—42no—
208C2×Dic53—42no—
209C2×Dic53—44no—
210C2×Dic53—44no—
211C2×Dic53—42no—
212C2×Dic53—42no—
213C2×Dic53—44no—
214C2×Dic53—44no—
215C2×Dic53—44no—
216C2×Dic53—82no—
217C2×Dic53—82no—
218C2×Dic53C2×C22⋊C414yes—
219C2×Dic53C2×C22⋊C412yes—
220C2×Dic53C2×C22⋊C412yes—
221Dic106—42no—
222Dic106—42no—
223Dic106—84no—
224#22513—531nomaximal, Sylow
225C22×C53—22no—
226C22×C53—21no—
227C22×C53—22no—
228C22×C53—21no—
229C22×C53—22no—
230C22×C53—41no—
231C22×C53—41no—
232C22×C53—42no—
233C22×C53—42no—
234C22×C53—42no—
235C22×C53—41no—
236C22×C53—41no—
237C22×C53C22×C22⋊C411yescharacteristic
238C22×C53C2×C4×Q811yescharacteristic
239C22×C53#51911yescharacteristic
240C22×C53#54812yes—
241C22×C53D4×Q812yes—
242C4×C53—22no—
243C4×C53—21no—
244C4×C53—22no—
245C4×C53—44no—
246C4×C53—41no—
247C4×C53—41no—
248C4×C53—82no—
249C4×C53#41012yes—
250Dic53—24no—
251Dic53—22no—
252Dic53—22no—
253Dic53—42no—
254Dic53—42no—
255Dic53—84no—
256C2×#554—531no—
257#2345—532no—
258#1996—532no—
259#1996—534no—
260#1799—532no—
261#1799—531no—
262C4×#99—532no—
263C4×#99D5311yescharacteristic
264C2×C53—22no—
265C2×C53—22no—
266C2×C53—21no—
267C2×C53—21no—
268C2×C53—42no—
269C2×C53Q8×C22⋊C411yescharacteristic
270C2×C53C2×#41011yescharacteristic
271C2×C53#288311yescharacteristic
272#554—1062no—
273#554—1064no—
274#554—1062no—
275C2×C4⋊Q8—531no—
276C2×#125—532no—
277C2×C22⋊Q8—531no—
278C2×C22⋊Q8—531no—
279C2×C22⋊Q8—532no—
280C2×C22⋊Q8—532no—
281C2×C4×Q8—531no—
282C2×C4×D4—532no—
283C2×C4×D4C2×D5311yescharacteristic
284C22×C4⋊C4—531no—
285C22×C4⋊C4—531no—
286C2×#99—532no—
287C2×#99C2×D5311yescharacteristic
288C2×#99C2×D5311yescharacteristic
289#389—1062no—
290#387—1064no—
291#380—532no—
292#380—534no—
293C4×C4⋊C4—1062no—
294C4×C22⋊C4—534no—
295C4×C22⋊C4C2×D5312yes—
296#342—534no—
297#342—532no—
298#342—534no—
299#342C2×D5312yes—
300C53#2251311yescharacteristic, complemented, Sylow
301C22×Q8—532no—
302C22×Q8—531no—
303C22×D4C22×D5311yescharacteristic
304C23×C4—532no—
305C23×C4—532no—
306C23×C4C4×D5311yescharacteristic
307C23×C4C22×D5311yescharacteristic
308C4⋊Q8—1062no—
309C4⋊Q8—1062no—
310#125—1064no—
311#125—1064no—
312C22⋊Q8—1064no—
313C22⋊Q8—1064no—
314C22⋊Q8—1064no—
315C22⋊Q8—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—
325C4×D4—1064no—
326C4×D4—1064no—
327C2×C4⋊C4—531no—
328C2×C4⋊C4—532no—
329C2×C4⋊C4—531no—
330C2×C4⋊C4—532no—
331C2×C4⋊C4—534no—
332C2×C4⋊C4—532no—
333C2×C4⋊C4—532no—
334C2×C4⋊C4—531no—
335C2×C4⋊C4—532no—
336C2×C4⋊C4—532no—
337C2×C4⋊C4—532no—
338C2×C4⋊C4—532no—
339C2×C4⋊C4—1062no—
340C2×C4⋊C4—1061no—
341C2×C4⋊C4—1061no—
342C2×C4⋊C4—1064no—
343C2×C4⋊C4—1062no—
344C2×C4⋊C4C4×D5311yescharacteristic
345C2×C22⋊C4—532no—
346C2×C22⋊C4—532no—
347C2×C22⋊C4—534no—
348C2×C22⋊C4C4×D5311yescharacteristic
349C2×C22⋊C4C22×D5312yes—
350C2×C22⋊C4C22×D5311yescharacteristic
351C2×C22⋊C4C22×D5312yes—
352C2×C42—22no—
353C2×C42—532no—
354C2×C42—1064no—
355C2×C42C4×D5311yescharacteristic
356#99—22no—
357#99—1064no—
358#99Dic10614yescomplemented
359#95—22no—
360#95—1064no—
361#95—1064no—
362#95—1062no—
363C24C2×C4×D5311yescharacteristic
364C24C23×D5311yescharacteristic
365C2×Q8—1064no—
366C2×Q8—1061no—
367C2×Q8—1061no—
368C2×Q8—4242no—
369C2×Q8—2122no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4C2×Dic10612yes—
373C2×D4#1171994512yes—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
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—1062no—
387C22×C4—1062no—
388C22×C4—1064no—
389C22×C4—4242no—
390C22×C4—4242no—
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—532no—
400C4⋊C4—531no—
401C4⋊C4—531no—
402C4⋊C4—534no—
403C4⋊C4—534no—
404C4⋊C4—532no—
405C4⋊C4—532no—
406C4⋊C4—1062no—
407C4⋊C4—1062no—
408C4⋊C4—1064no—
409C4⋊C4—1062no—
410C4⋊C4—1062no—
411C4⋊C4—1064no—
412C4⋊C4—1061no—
413C4⋊C4—1061no—
414C4⋊C4—1064no—
415C4⋊C4—4244no—
416C4⋊C4—2124no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—1064no—
423C22⋊C4—1064no—
424C22⋊C4—1064no—
425C22⋊C4—1064no—
426C22⋊C4C2×Dic10614yes—
427C42—21no—
428C42—21no—
429C42—44no—
430C42—1062no—
431C42—1062no—
432C42—4244no—
433C23—21no—
434C23—21no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23C4×Dic10612yes—
440C23C22×C4×D5311yescharacteristic
441C23C2×D4×D5312yes—
442C23C22×Dic10611yescharacteristic
443C23C2×#1171994511yescharacteristic
444C23#1313326412yes—
445Q8—4244no—
446Q8—2122no—
447Q8—2122no—
448D4—24no—
449D4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—532no—
469C2×C4—532no—
470C2×C4—534no—
471C2×C4—1062no—
472C2×C4—1062no—
473C2×C4—4242no—
474C2×C4—4242no—
475C2×C4—2124no—
476C2×C4—2124no—
477C2×C4—2122no—
478C2×C4—2122no—
479C2×C4—2124no—
480C2×C4—2124no—
481C2×C4—2124no—
482C2×C4—2122no—
483C2×C4—2122no—
484C2×C4C22⋊C4×D5311yescharacteristic
485C2×C4C22⋊C4×D5312yes—
486C2×C4C22⋊C4×D5311yescharacteristic
487C2×C4C4×Dic10612yes—
488C22—22no—
489C22—21no—
490C22—22no—
491C22—21no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—41no—
499C22—41no—
500C22C2×C22⋊C4×D5311yescharacteristic
501C22C2×C4×Dic10611yescharacteristic
502C22#1681110011yescharacteristic
503C22#1681128212yes—
504C22D4×Dic10612yes—
505C4—21no—
506C4—22no—
507C4—22no—
508C4—44no—
509C4—41no—
510C4—41no—
511C4—82no—
512C4—1062no—
513C4—1062no—
514C4—1064no—
515C4—4244no—
516C4—2122no—
517C4—2122no—
518C4#1681059512yes—
519C2—22no—
520C2—22no—
521C2—21no—
522C2—21no—
523C2—42no—
524C2C22⋊C4×Dic10611yescharacteristic
525C2C2×#1681059511yescharacteristic
526C2#2028317411yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC4×#99×C53C2
socleC22×C53C22×C22⋊C4
prime radicalC22×D4×C53C23
cosocleC24×C53C24
hypercenterC4×#99D53
hypocenterC53#22513
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. #20277590C2semidirect
3. #20275700C2semidirect
4. #20275698C2semidirect
12. C2×#13133217C4semidirect
13. C2×#13133208C4semidirect
14. C2×#13133196C4semidirect
16. C2×#13133194C4semidirect
18. C2×C4×Dic106C4semidirect
20. C22×C4⋊Dic53C4semidirect
97. #99×C53Q8semidirect
300. C53#22513semidirect
358. #99Dic106semidirect

Maximal subgroups

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

subgroupindex
C2×#168113302
#202775902
#202757002
#202756982
#202747032
#202747012
#99×Dic532
C4×#99×C532
#2251353