Finite Group Databasebeta

← #27355484

Subgroups — #27355484

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284755C211yesmaximal, characteristic
2#20283816C211yesmaximal, characteristic
3#20283805C211yesmaximal, characteristic
4#20273623C211yesmaximal, characteristic, complemented
5#20273611C211yesmaximal, characteristic, complemented
6#20273585C211yesmaximal, characteristic, complemented
7#20273585C211yesmaximal, characteristic, complemented
8#20273564C211yesmaximal, characteristic, complemented
9#20273564C211yesmaximal, characteristic, complemented
10#20273533C211yesmaximal, characteristic, complemented
11#20273521C211yesmaximal, characteristic, complemented
12#20273187C211yesmaximal, characteristic
13#20273109C211yesmaximal, characteristic
14#20273015C211yesmaximal, characteristic
15C53×#1572C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18#16810959C411yescharacteristic
19#16810959C411yescharacteristic
20#16810919C411yescharacteristic
21#16810919C411yescharacteristic
22#16810834C411yescharacteristic
23#16810834C411yescharacteristic
24#16810781C411yescharacteristic
25#16810781C411yescharacteristic
26#16810753—21no—
27#16810751—21no—
28#16810751—21no—
29#16810750C2211yescharacteristic
30#16810750C2211yescharacteristic
31#16810750C2211yescharacteristic
32#16810750C2211yescharacteristic
33C8×Dic106—21no—
34#16810747C2211yescharacteristic
35#16810747C2211yescharacteristic
36#16810747C2211yescharacteristic
37#16810747C2211yescharacteristic
38C4×Dic212C2211yescharacteristic
39C4×Dic212C2211yescharacteristic
40C4×Dic212C2211yescharacteristic
41C4×Dic212C2211yescharacteristic
42#16810681—21no—
43#16810681—21no—
44#16810677—21no—
45#16810676C2211yescharacteristic
46#16810676C2211yescharacteristic
47#16810673—21no—
48#16810353C2211yescharacteristic
49#16810353C2211yescharacteristic
50#16810353C2211yescharacteristic
51#16810353C2211yescharacteristic
52#16810353C2211yescharacteristic
53#16810353C2211yescharacteristic
54#16810353C2211yescharacteristic
55#16810353C2211yescharacteristic
56#16810345C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810344C2211yescharacteristic
59C8×#88322C2211yescharacteristic
60C53×#432C2211yescharacteristic
61C53×#431C2211yescharacteristic
62C53×#431C2211yescharacteristic
63C53×#335C2211yescharacteristic
64C53×#335C2211yescharacteristic
65C53×#334C2211yescharacteristic
66C53×#334C2211yescharacteristic
67C22×Dic106C2×C411yescharacteristic
68C2×Dic212C2×C411yescharacteristic
69C2×Dic212C2×C411yescharacteristic
70C2×Dic212C2×C411yescharacteristic
71C2×Dic212C2×C411yescharacteristic
72#13133217C2×C411yescharacteristic
73#13133217C2×C411yescharacteristic
74#13133209—21no—
75#13133196—21no—
76#13133196C2×C411yescharacteristic
77#13133196C2×C411yescharacteristic
78C4×Dic106—21no—
79C4×Dic106—21no—
80C4×Dic106—21no—
81C4×Dic106C2×C411yescharacteristic
82C4×Dic106C2×C411yescharacteristic
83C4×Dic106C2311yescharacteristic
84C4×Dic106C2311yescharacteristic
85C4×Dic106C2311yescharacteristic
86C4×Dic106C2311yescharacteristic
87#13133176—21no—
88C2×C4⋊Dic53C2×C411yescharacteristic
89C2×#11719919—21no—
90C2×C4×Dic53—21no—
91#13133151C2×C411yescharacteristic
92#13133151C2×C411yescharacteristic
93#13133151C2×C411yescharacteristic
94#13133151C2×C411yescharacteristic
95C4⋊#88322—21no—
96C4⋊#88322D411yescharacteristic
97C4⋊#88322D411yescharacteristic
98#13133146—21no—
99#13133146—21no—
100#13133146—21no—
101#13133146—21no—
102#13133146—21no—
103#13133146—21no—
104#13133146—21no—
105#13133146—21no—
106#13133142C2×C411yescharacteristic
107#13133142C2×C411yescharacteristic
108#13133142C2×C411yescharacteristic
109#13133142C2×C411yescharacteristic
110#13133142C2×C411yescharacteristic
111#13133142C2×C411yescharacteristic
112#13133142C2×C411yescharacteristic
113#13133142C2×C411yescharacteristic
114C22⋊#88322—21no—
115C22⋊#88322D411yescharacteristic
116C22⋊#88322D411yescharacteristic
117#13133127—21no—
118#13133127—21no—
119#13133127—21no—
120#13133127—21no—
121C4×#88322C2311yescharacteristic
122C4×#88322C2311yescharacteristic
123C4×#88322C2311yescharacteristic
124C4×#88322C2311yescharacteristic
125C8×Dic53—21no—
126C8×Dic53—21no—
127C8×Dic53—21no—
128C8×Dic53—21no—
129C2×C42×C53C2311yescharacteristic
130C4⋊C8×C53C2×C411yescharacteristic
131C4⋊C8×C53C2×C411yescharacteristic
132C4⋊C8×C53C2311yescharacteristic
133C4⋊C8×C53C2311yescharacteristic
134C4⋊C8×C53C2311yescharacteristic
135C4⋊C8×C53C2311yescharacteristic
136C22⋊C8×C53—21no—
137C22⋊C8×C53—21no—
138C22⋊C8×C53C2×C411yescharacteristic
139C22⋊C8×C53C2×C411yescharacteristic
140#97×C53—21no—
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C22×C4×C53C22×C411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53#4111yescharacteristic
147C22×Dic53—21no—
148C22×Dic53—21no—
149C2×Dic106—22no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—41no—
153C2×Dic106C22×C411yescharacteristic
154C2×Dic106C22×C411yescharacteristic
155C2×Dic106C22×C411yescharacteristic
156C2×Dic106C22×C411yescharacteristic
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53C22×C411yescharacteristic
162C2×C8×C53C22×C411yescharacteristic
163C2×C8×C53C22×C411yescharacteristic
164C2×C8×C53C22×C411yescharacteristic
165C42×C53C22×C411yescharacteristic
166C42×C53C2×D411yescharacteristic
167C42×C53#4111yescharacteristic
168C42×C53C2411yescharacteristic
169C22⋊Dic53—21no—
170C22⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53C22×C411yescharacteristic
176C4⋊Dic53C22×C411yescharacteristic
177C4⋊Dic53C22×C411yescharacteristic
178C4⋊Dic53C22×C411yescharacteristic
179#11719919—21no—
180#11719919—21no—
181#11719919—21no—
182#11719919—21no—
183#11719919—21no—
184#11719919—21no—
185#11719919—21no—
186#11719919—21no—
187C4×Dic53—21no—
188C4×Dic53—21no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322C2×D411yescharacteristic
196C2×#88322C2×D411yescharacteristic
197C2×#88322C2×D411yescharacteristic
198C2×#88322C2×D411yescharacteristic
199Dic212—22no—
200Dic212—22no—
201Dic212—41no—
202Dic212—41no—
203C23×C53C4×D411yescharacteristic
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—41no—
209C2×C4×C53C4×D411yescharacteristic
210C2×C4×C53C4×D411yescharacteristic
211C2×C4×C53C4×D411yescharacteristic
212C2×C4×C53C23×C411yescharacteristic
213C2×C4×C53C22×D411yescharacteristic
214C2×C4×C53C2×#4111yescharacteristic
215C2×Dic53—21no—
216C2×Dic53—21no—
217C2×Dic53—21no—
218C2×Dic53—21no—
219C2×Dic53—21no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—42no—
224C2×Dic53—42no—
225Dic106—21no—
226Dic106—21no—
227Dic106—21no—
228Dic106—21no—
229Dic106—21no—
230Dic106—21no—
231Dic106—21no—
232Dic106—42no—
233Dic106—42no—
234Dic106#13112yes—
235C8×C53—21no—
236C8×C53—21no—
237C8×C53—21no—
238C8×C53—21no—
239C8×C53—41no—
240C8×C53—41no—
241C8×C53—41no—
242C8×C53—41no—
243#88322—21no—
244#88322—21no—
245#88322—21no—
246#88322—21no—
247#88322—41no—
248#88322—41no—
249#18032—531nomaximal, Sylow
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53C2×C4×D411yescharacteristic
254C4×C53—21no—
255C4×C53—21no—
256C4×C53—21no—
257C4×C53—21no—
258C4×C53C2×#13111yescharacteristic
259C4×C53#56811yescharacteristic
260C4×C53#57811yescharacteristic
261C4×C53#57811yescharacteristic
262Dic53—22no—
263Dic53—41no—
264Dic53—41no—
265Dic53—41no—
266Dic53—41no—
267Dic53—41no—
268Dic53—41no—
269Dic53—41no—
270#3104—531no—
271#2986—531no—
272#2984—531no—
273#1593—531no—
274#1591—531no—
275#1587—531no—
276#1587—531no—
277#1584—531no—
278#1584—531no—
279#1579—531no—
280C8×Q16—531no—
281#1572D5311yescharacteristic
282#1497—531no—
283#1479—531no—
284#1454—531no—
285C2×C53—41no—
286C2×C53#293311yescharacteristic
287C2×C53#295011yescharacteristic
288C2×C53#295511yescharacteristic
289#533—531no—
290C2×C4×Q8—531no—
291#484—531no—
292#484—531no—
293#477—531no—
294#477—531no—
295#462—531no—
296#462—531no—
297#451—531no—
298#451—531no—
299#446—531no—
300#446—531no—
301#446—531no—
302#446—531no—
303#446—1061no—
304#446—1061no—
305#446—1061no—
306C8×Q8—531no—
307C8×Q8—531no—
308C8×Q8—531no—
309C8×Q8—531no—
310C8×Q8—1061no—
311C4×Q16—531no—
312C4×Q16—531no—
313C4×Q16—531no—
314C4×Q16—531no—
315#433—1061no—
316#433—1061no—
317#432—1061no—
318#432C2×D5311yescharacteristic
319#431—531no—
320#431—531no—
321#431—1061no—
322#431C2×D5311yescharacteristic
323#431C2×D5311yescharacteristic
324#335C2×D5311yescharacteristic
325#335C2×D5311yescharacteristic
326#334C2×D5311yescharacteristic
327#334C2×D5311yescharacteristic
328Q8⋊C8—531no—
329Q8⋊C8—531no—
330Q8⋊C8—531no—
331Q8⋊C8—531no—
332Q8⋊C8—531no—
333Q8⋊C8—531no—
334Q8⋊C8—531no—
335Q8⋊C8—531no—
336#322—531no—
337#322—531no—
338#322—531no—
339C82—531no—
340C53#1803211yescharacteristic, complemented, Sylow
341C22×Q8—531no—
342C2×Q16—531no—
343C2×Q16—531no—
344C2×Q16—531no—
345C2×Q16—531no—
346C4⋊Q8—531no—
347C4⋊Q8—531no—
348#125—1061no—
349C22⋊Q8—531no—
350C22⋊Q8—531no—
351C22⋊Q8—1061no—
352C4×Q8—531no—
353C4×Q8—531no—
354C4×Q8—531no—
355C4×Q8—531no—
356C4×Q8—531no—
357C4×Q8—531no—
358C4×Q8—1061no—
359C4×Q8—1061no—
360C4×Q8—1061no—
361#117—1061no—
362C2×C4⋊C4—531no—
363C2×C4⋊C4—1061no—
364C2×C42—1061no—
365C2×C42C22×D5311yescharacteristic
366#107—531no—
367#107—531no—
368#107—531no—
369#107—531no—
370C4⋊C8—531no—
371C4⋊C8—531no—
372C4⋊C8—1061no—
373C4⋊C8—1061no—
374C4⋊C8—1061no—
375C4⋊C8—1061no—
376C4⋊C8—1061no—
377C4⋊C8—1061no—
378C4⋊C8—1061no—
379C4⋊C8—1061no—
380C4⋊C8—1061no—
381C4⋊C8C4×D5311yescharacteristic
382C4⋊C8C4×D5311yescharacteristic
383C4⋊C8C22×D5311yescharacteristic
384C4⋊C8C22×D5311yescharacteristic
385C4⋊C8C22×D5311yescharacteristic
386C4⋊C8C22×D5311yescharacteristic
387#103—531no—
388#103—531no—
389#103—531no—
390#103—531no—
391#103—531no—
392#103—531no—
393#103—531no—
394#103—531no—
395C22⋊C8—21no—
396C22⋊C8—21no—
397C22⋊C8—531no—
398C22⋊C8—531no—
399C22⋊C8—1061no—
400C22⋊C8C4×D5311yescharacteristic
401C22⋊C8C4×D5311yescharacteristic
402#97—21no—
403#97—1061no—
404#97—1061no—
405#97—1061no—
406#97—1061no—
407C4×C8—21no—
408C4×C8—531no—
409C4×C8—531no—
410C4×C8—531no—
411C4×C8—531no—
412C4×C8—1061no—
413C4×C8—1061no—
414C4×C8—1061no—
415C4×C8—1061no—
416C4×C8C22×D5311yescharacteristic
417C4×C8C22×D5311yescharacteristic
418C2×Q8—531no—
419C2×Q8—531no—
420C2×Q8—531no—
421C2×Q8—531no—
422C2×Q8—1062no—
423C2×Q8—1061no—
424C2×Q8—1061no—
425C2×Q8—2121no—
426C22×C4—1061no—
427C22×C4—1061no—
428C22×C4C2×C4×D5311yescharacteristic
429C22×C4D4×D5311yescharacteristic
430C22×C4#1171994911yescharacteristic
431Q16—1062no—
432Q16—1062no—
433Q16—2121no—
434Q16—2121no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—531no—
440C2×C8—531no—
441C2×C8—531no—
442C2×C8—531no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8C2×C4×D5311yescharacteristic
446C2×C8C2×C4×D5311yescharacteristic
447C2×C8C2×C4×D5311yescharacteristic
448C2×C8C2×C4×D5311yescharacteristic
449C4⋊C4—531no—
450C4⋊C4—531no—
451C4⋊C4—531no—
452C4⋊C4—531no—
453C4⋊C4—1061no—
454C4⋊C4—1061no—
455C4⋊C4—1061no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C22⋊C4—1061no—
466C22⋊C4—1061no—
467C42—1061no—
468C42—1061no—
469C42—1061no—
470C42—1061no—
471C42—1061no—
472C42—1061no—
473C42C2×C4×D5311yescharacteristic
474C42D4×D5311yescharacteristic
475C42#1171994911yescharacteristic
476C42C23×D5311yescharacteristic
477C23#1313317911yescharacteristic
478Q8—532no—
479Q8—1061no—
480Q8—1061no—
481Q8—1061no—
482Q8—1061no—
483Q8—1061no—
484Q8—1061no—
485Q8—1061no—
486Q8—2122no—
487Q8—2122no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—41no—
493C2×C4—1061no—
494C2×C4—1061no—
495C2×C4—1061no—
496C2×C4—1061no—
497C2×C4—1061no—
498C2×C4—1061no—
499C2×C4—1061no—
500C2×C4—1061no—
501C2×C4—2122no—
502C2×C4—2122no—
503C2×C4#1313317911yescharacteristic
504C2×C4#1313317911yescharacteristic
505C2×C4#1313317911yescharacteristic
506C2×C4C22×C4×D5311yescharacteristic
507C2×C4C2×D4×D5311yescharacteristic
508C2×C4C2×#1171994911yescharacteristic
509C8—21no—
510C8—21no—
511C8—21no—
512C8—21no—
513C8—41no—
514C8—41no—
515C8—41no—
516C8—41no—
517C8—1061no—
518C8—1061no—
519C8—1061no—
520C8—1061no—
521C8—2121no—
522C8—2121no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22C2×#1313317911yescharacteristic
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—1062no—
532C4—2121no—
533C4—2121no—
534C4—2121no—
535C4—2121no—
536C4—2121no—
537C4—2121no—
538C4—2121no—
539C4#131×D5311yescharacteristic
540C4#1681138811yescharacteristic
541C4#1681143711yescharacteristic
542C4#1681143711yescharacteristic
543C2—41no—
544C2#2028343911yescharacteristic
545C2#2028358211yescharacteristic
546C2#2028361711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1572C2
socleC22×C53C2×C4×D4
prime radicalC23×C53C4×D4
cosocleC42×C53C24
hypercenter#1572D53
hypocenterC53#18032
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1572

Complements

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

NH ≅ G/Ntype
4. #20273623C2semidirect
5. #20273611C2semidirect
6. #20273585C2semidirect
7. #20273585C2semidirect
8. #20273564C2semidirect
9. #20273564C2semidirect
10. #20273533C2semidirect
11. #20273521C2semidirect
340. C53#18032semidirect

Maximal subgroups

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

subgroupindex
#202847552
#202838162
#202838052
#202736232
#202736112
#202735852
#202735642
#202735332
#202735212
#202731872
#202731092
#202730152
C53×#15722
#1803253