Finite Group Databasebeta

← #27279647

Subgroups — #27279647

Order 13568.

984 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283002C211yesmaximal, characteristic
2C2×#16810361C212yesmaximal, complemented
3#20272938C211yesmaximal, characteristic
4#20272595C212yesmaximal, complemented
5C22×C22⋊C8×C53C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic, complemented
7C2×#13133196C411yescharacteristic, complemented
8C2×#13133196C2211yescharacteristic
9C22×C22⋊Dic53—21no—
10#16810496—21no—
11#16810496—21no—
12#16810496—21no—
13#16810483C411yescharacteristic, complemented
14#16810483C411yescharacteristic, complemented
15#16810483C2211yescharacteristic
16#16810479—21no—
17C2×#13133124—21no—
18C2×#13133124C2212yes—
19#16810361—24no—
20C24×C4×C53C2211yescharacteristic
21C23×C8×C53—21no—
22C2×C22⋊C8×C53—21no—
23C2×C22⋊C8×C53—24no—
24C2×C22⋊C8×C53C2212yes—
25C22×Dic106C2×C411yescharacteristic
26C23×Dic53—22no—
27#13133196—22no—
28#13133196—22no—
29#13133196—42no—
30#13133196—42no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×C22⋊Dic53—24no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53—21no—
44C2×C22⋊Dic53—42no—
45C2×C22⋊Dic53—42no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—22no—
50#13133124—22no—
51#13133124—42no—
52C25×C53D411yescharacteristic
53C23×C4×C53—22no—
54C23×C4×C53—22no—
55C23×C4×C53—42no—
56C23×C4×C53—42no—
57C23×C4×C53C2×C411yescharacteristic
58C23×C4×C53C2×C411yescharacteristic
59C23×C4×C53D411yescharacteristic
60C23×C4×C53D411yescharacteristic
61C23×C4×C53D411yescharacteristic
62C23×C4×C53C2311yescharacteristic
63C22×C8×C53—24no—
64C22×C8×C53—22no—
65C22×C8×C53—42no—
66C22×C8×C53—42no—
67C22⋊C8×C53—24no—
68C22⋊C8×C53—22no—
69C22⋊C8×C53D414yes—
70C24×C53—22no—
71C24×C53—22no—
72C24×C53—22no—
73C24×C53—22no—
74C24×C53—41no—
75C24×C53—41no—
76C24×C53C22⋊C411yescharacteristic
77C24×C53C22⋊C411yescharacteristic
78C24×C53SD1612yes—
79C24×C53Q1612yes—
80C24×C53C2×D411yescharacteristic
81C22×C4×C53—22no—
82C22×C4×C53—22no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—22no—
86C22×C4×C53—22no—
87C22×C4×C53—22no—
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53—22no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—48no—
101C22×C4×C53—48no—
102C22×C4×C53C22⋊C411yescharacteristic
103C22×C4×C53C22⋊C411yescharacteristic
104C22×C4×C53C22×C411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D412yes—
107C22×C4×C53#4112yes—
108C22×Dic53—24no—
109C22×Dic53—21no—
110C22×Dic53—21no—
111C22×Dic53—21no—
112C22×Dic53—21no—
113C22×Dic53—42no—
114C22×Dic53—42no—
115C22×Dic53—42no—
116C22×Dic53—42no—
117C22×Dic53—42no—
118C22×Dic53—42no—
119C2×Dic106—22no—
120C2×Dic106—41no—
121C2×Dic106—41no—
122C2×C8×C53—24no—
123C2×C8×C53—48no—
124C2×C8×C53—42no—
125C2×C8×C53—42no—
126C22⋊Dic53—42no—
127C22⋊Dic53—48no—
128C22⋊Dic53—42no—
129C22⋊Dic53—42no—
130C22⋊Dic53—42no—
131C22⋊Dic53—42no—
132C22⋊Dic53—42no—
133C4⋊Dic53—22no—
134C4⋊Dic53—41no—
135C4⋊Dic53—41no—
136#11719919—42no—
137#11719919—42no—
138#11719919—41no—
139#11719919—41no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—21no—
143C23×C53—21no—
144C23×C53—21no—
145C23×C53—21no—
146C23×C53—21no—
147C23×C53—21no—
148C23×C53—24no—
149C23×C53—24no—
150C23×C53—21no—
151C23×C53—21no—
152C23×C53—21no—
153C23×C53—21no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—24no—
157C23×C53—24no—
158C23×C53—42no—
159C23×C53—42no—
160C23×C53—44no—
161C23×C53—44no—
162C23×C53—44no—
163C23×C53—44no—
164C23×C53#9912yes—
165C23×C53#10312yes—
166C23×C53#10312yes—
167C23×C53#10412yes—
168C23×C53C2×C22⋊C411yescharacteristic
169C23×C53#12112yes—
170C23×C53C2×SD1611yescharacteristic
171C23×C53C2×Q1611yescharacteristic
172C23×C53#13512yes—
173C2×C4×C53—24no—
174C2×C4×C53—24no—
175C2×C4×C53—21no—
176C2×C4×C53—21no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53—44no—
187C2×C4×C53—44no—
188C2×C4×C53—44no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53#9912yes—
192C2×C4×C53C4×D412yes—
193C2×C4×C53#12011yescharacteristic
194C2×C4×C53#12311yescharacteristic
195C2×Dic53—44no—
196C2×Dic53—44no—
197C2×Dic53—44no—
198C2×Dic53—41no—
199C2×Dic53—41no—
200C2×Dic53—41no—
201C2×Dic53—41no—
202C2×Dic53—41no—
203C2×Dic53—41no—
204C2×Dic53—41no—
205C2×Dic53—41no—
206C2×Dic53—41no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53—41no—
210C2×Dic53—82no—
211C2×Dic53—82no—
212C2×Dic53—82no—
213C2×Dic53—82no—
214Dic106—82no—
215C8×C53—44no—
216#9111—531nomaximal, Sylow
217C22×C53—22no—
218C22×C53—22no—
219C22×C53—22no—
220C22×C53—22no—
221C22×C53—22no—
222C22×C53—21no—
223C22×C53—21no—
224C22×C53—22no—
225C22×C53—22no—
226C22×C53—22no—
227C22×C53—22no—
228C22×C53—21no—
229C22×C53—22no—
230C22×C53—21no—
231C22×C53—22no—
232C22×C53—22no—
233C22×C53—22no—
234C22×C53—48no—
235C22×C53—48no—
236C22×C53—41no—
237C22×C53—41no—
238C22×C53—41no—
239C22×C53—41no—
240C22×C53#32814yescomplemented
241C22×C53#38611yescharacteristic
242C22×C53C2×#9911yescharacteristic
243C22×C53C2×#9911yescharacteristic
244C22×C53C2×#10311yescharacteristic
245C22×C53#41611yescharacteristic
246C22×C53C2×#10411yescharacteristic
247C22×C53#42111yescharacteristic
248C22×C53#46512yes—
249C22×C53#46712yes—
250C4×C53—21no—
251C4×C53—22no—
252C4×C53—22no—
253C4×C53—21no—
254C4×C53—41no—
255Dic53—84no—
256Dic53—81no—
257Dic53—81no—
258Dic53—81no—
259Dic53—81no—
260C22×C22⋊C8D5311yescharacteristic
261#2848—531no—
262C2×#328—532no—
263C22⋊#95—531no—
264#1294—532no—
265C2×C53—22no—
266C2×C53—24no—
267C2×C53—22no—
268C2×C53—42no—
269C2×C53C2×#32811yescharacteristic
270C2×C53#150111yescharacteristic
271C2×C53#151611yescharacteristic
272C2×C53#152011yescharacteristic
273C2×C53#187111yescharacteristic
274C2×C53#189211yescharacteristic
275C2×C53#189811yescharacteristic
276C24×C4C2×D5311yescharacteristic
277C23×C8—21no—
278C2×C22⋊Q8—531no—
279C2×C22⋊Q8—531no—
280C2×C22⋊Q8—531no—
281C22×C22⋊C4—1061no—
282C2×C22⋊C8—21no—
283C2×C22⋊C8—24no—
284C2×C22⋊C8C2×D5312yes—
285#385—1061no—
286#385—1061no—
287#385—1061no—
288#380—531no—
289#380—531no—
290#380—531no—
291#379—1061no—
292C2×#95—532no—
293C2×#95—1061no—
294#328—1064no—
295C53#911111yescharacteristic, complemented, Sylow
296C25D21211yescharacteristic
297C22×Q8—531no—
298C23×C4—22no—
299C23×C4—22no—
300C23×C4—42no—
301C23×C4—42no—
302C23×C4—1062no—
303C23×C4C4×D5311yescharacteristic
304C23×C4C4×D5311yescharacteristic
305C23×C4D21211yescharacteristic
306C23×C4C22⋊D5311yescharacteristic
307C23×C4C22⋊D5311yescharacteristic
308C23×C4C22×D5311yescharacteristic
309C22×C8—24no—
310C22×C8—22no—
311C22×C8—42no—
312C22×C8—42no—
313C22⋊Q8—1062no—
314C22⋊Q8—1062no—
315C22⋊Q8—2122no—
316C22⋊Q8—2122no—
317C2×C4⋊C4—531no—
318C2×C4⋊C4—531no—
319C2×C4⋊C4—531no—
320C2×C4⋊C4—1061no—
321C2×C4⋊C4—1061no—
322C2×C4⋊C4—1061no—
323C2×C22⋊C4—1064no—
324C2×C22⋊C4—1061no—
325C2×C22⋊C4—1061no—
326C2×C22⋊C4—1061no—
327C2×C22⋊C4—1061no—
328C2×C22⋊C4—1061no—
329C2×C22⋊C4—1061no—
330C2×C22⋊C4—2122no—
331C2×C22⋊C4—2122no—
332C22⋊C8—24no—
333C22⋊C8—22no—
334C22⋊C8C22⋊D5314yes—
335#95—1061no—
336#95—1061no—
337#95—1061no—
338#95—1062no—
339#95—1062no—
340#95—2122no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—41no—
346C24—41no—
347C24#1171991312yes—
348C24Dic21212yes—
349C24#1171992111yescharacteristic
350C24#1171992111yescharacteristic
351C24C2×D21211yescharacteristic
352C2×Q8—1062no—
353C2×Q8—2121no—
354C2×Q8—2121no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—1064no—
377C22×C4—1061no—
378C22×C4—1061no—
379C22×C4—1061no—
380C22×C4—1061no—
381C22×C4—2122no—
382C22×C4—2122no—
383C22×C4—2122no—
384C22×C4—2122no—
385C22×C4—2122no—
386C22×C4—2122no—
387C22×C4#1171992111yescharacteristic
388C22×C4#1171992111yescharacteristic
389C22×C4C2×C4×D5311yescharacteristic
390C22×C4#1171994512yes—
391C22×C4C2×C22⋊D5311yescharacteristic
392C22×C4C2×C22⋊D5312yes—
393C2×C8—24no—
394C2×C8—48no—
395C2×C8—42no—
396C2×C8—42no—
397C4⋊C4—1062no—
398C4⋊C4—2121no—
399C4⋊C4—2121no—
400C4⋊C4—2122no—
401C4⋊C4—2122no—
402C4⋊C4—2121no—
403C4⋊C4—2121no—
404C22⋊C4—2128no—
405C22⋊C4—2122no—
406C22⋊C4—2122no—
407C22⋊C4—2122no—
408C22⋊C4—2122no—
409C22⋊C4—2122no—
410C22⋊C4—2122no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—42no—
430C23—42no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23#1313313112yes—
436C23#1313314212yes—
437C23#1313314212yes—
438C23#1313314312yes—
439C23C2×#1171992111yescharacteristic
440C23#1313319112yes—
441C23C2×#1171991311yescharacteristic
442C23C2×Dic21211yescharacteristic
443C23#1313323912yes—
444Q8—4242no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—4242no—
464C2×C4—4242no—
465C2×C4—4242no—
466C2×C4—4242no—
467C2×C4—2124no—
468C2×C4—2124no—
469C2×C4—2124no—
470C2×C4—2121no—
471C2×C4—2121no—
472C2×C4—2121no—
473C2×C4—2121no—
474C2×C4—2121no—
475C2×C4—2121no—
476C2×C4—2121no—
477C2×C4—2121no—
478C2×C4—2121no—
479C2×C4—2121no—
480C2×C4—2121no—
481C2×C4—2121no—
482C2×C4#1313313112yes—
483C2×C4C4×C22⋊D5312yes—
484C2×C4#1313318611yescharacteristic
485C2×C4#1313320011yescharacteristic
486C8—44no—
487C22—21no—
488C22—21no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—21no—
496C22—22no—
497C22—21no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—48no—
505C22—48no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22#1681036114yescomplemented
511C22#1681050211yescharacteristic
512C22C2×#1313313111yescharacteristic
513C22C2×#1313313111yescharacteristic
514C22C2×#1313314211yescharacteristic
515C22#1681062011yescharacteristic
516C22C2×#1313314311yescharacteristic
517C22#1681063911yescharacteristic
518C22#1681085212yes—
519C22#1681086412yes—
520C4—21no—
521C4—22no—
522C4—21no—
523C4—22no—
524C4—41no—
525C4—4244no—
526C4—4241no—
527C4—4241no—
528C4—4241no—
529C4—4241no—
530C2—22no—
531C2—22no—
532C2—24no—
533C2—42no—
534C2C2×#1681036111yescharacteristic
535C2#2027320711yescharacteristic
536C2#2027327211yescharacteristic
537C2#2027328811yescharacteristic
538C2#2027501711yescharacteristic
539C2#2027513811yescharacteristic
540C2#2027516811yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC22×C22⋊C8×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenterC22×C22⋊C8D53
hypocenterC53#9111
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#16810361C2semidirect
4. #20272595C2semidirect
6. C2×#13133196C4semidirect
7. C2×#13133196C4semidirect
13. #16810483C4semidirect
14. #16810483C4semidirect
240. C22×C53#328semidirect
295. C53#9111semidirect
510. C22#16810361semidirect

Maximal subgroups

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

subgroupindex
#202830022
C2×#168103612
#202729382
#202725952
C22×C22⋊C8×C532
#911153