Finite Group Databasebeta

← #27354173

Subgroups — #27354173

Order 13568.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284791C211yesmaximal, characteristic
2#20283864C211yesmaximal, characteristic
3#20283795C211yesmaximal, characteristic, complemented
4#20273596C211yesmaximal, characteristic, complemented
5#20273589C211yesmaximal, characteristic, complemented
6#20273568C211yesmaximal, characteristic, complemented
7#20273561C211yesmaximal, characteristic, complemented
8#20273531C211yesmaximal, characteristic, complemented
9#20273524C211yesmaximal, characteristic, complemented
10#20273519C211yesmaximal, characteristic, complemented
11#20273518C211yesmaximal, characteristic, complemented
12#20273503C211yesmaximal, characteristic, complemented
13#20273145C211yesmaximal, characteristic, complemented
14#20273015C211yesmaximal, characteristic, complemented
15C53×#1482C211yesmaximal, characteristic
16C2×C4×Dic106C2211yescharacteristic
17#16810951C411yescharacteristic
18#16810934C411yescharacteristic
19#16810927C411yescharacteristic
20#16810927C411yescharacteristic
21#16810842C411yescharacteristic
22#16810842C411yescharacteristic
23#16810778C411yescharacteristic
24#16810761C411yescharacteristic
25#16810754C2211yescharacteristic
26Q8×#88322C2211yescharacteristic
27C8×Dic106—21no—
28C8×Dic106—21no—
29#16810747C2211yescharacteristic
30#16810747C2211yescharacteristic
31#16810747C2211yescharacteristic
32#16810747C2211yescharacteristic
33C4×#11719925C2211yescharacteristic
34C4×#11719925C2211yescharacteristic
35C4×#11719923C2211yescharacteristic
36C4×#11719923C2211yescharacteristic
37#16810701—21no—
38#16810701C2211yescharacteristic
39#16810695—21no—
40D4×#88322C2211yescharacteristic
41#16810684—21no—
42#16810676C2211yescharacteristic
43#16810676C2211yescharacteristic
44#16810673—21no—
45#16810673—21no—
46#16810650—21no—
47#16810647C2211yescharacteristic
48#16810375C2211yescharacteristic
49#16810375C2211yescharacteristic
50#16810373C2211yescharacteristic
51#16810373C2211yescharacteristic
52#16810355C2211yescharacteristic
53#16810355C2211yescharacteristic
54#16810355C2211yescharacteristic
55#16810355C2211yescharacteristic
56#16810345C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810344C2211yescharacteristic
59C8×#88322C2211yescharacteristic
60C53×#529C2211yescharacteristic
61C53×#431C2211yescharacteristic
62C53×#431C2211yescharacteristic
63C53×Q8⋊C8C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×D4⋊C8C2211yescharacteristic
66C53×D4⋊C8C2211yescharacteristic
67C22×Dic106C2×C411yescharacteristic
68C2×#11719925C2×C411yescharacteristic
69C2×#11719925C2×C411yescharacteristic
70C2×#11719923C2×C411yescharacteristic
71C2×#11719923C2×C411yescharacteristic
72C2×#11719917—21no—
73C2×#11719917—21no—
74C22×#88322—21no—
75C4×Dic106—21no—
76C4×Dic106—21no—
77C4×Dic106C2×C411yescharacteristic
78C4×Dic106C2×C411yescharacteristic
79C4×Dic106C2311yescharacteristic
80C4×Dic106C2311yescharacteristic
81C2×C4⋊Dic53C2×C411yescharacteristic
82C2×#11719919—21no—
83C2×C4×Dic53—21no—
84#13133150C2×C411yescharacteristic
85#13133150C2×C411yescharacteristic
86#13133148C2×C411yescharacteristic
87#13133148C2×C411yescharacteristic
88C4⋊#88322—21no—
89C4⋊#88322—21no—
90C4⋊#88322—21no—
91C4⋊#88322C2×C411yescharacteristic
92C4⋊#88322C2×C411yescharacteristic
93C4⋊#88322D411yescharacteristic
94C4⋊#88322D411yescharacteristic
95C4⋊#88322C2311yescharacteristic
96C4⋊#88322C2311yescharacteristic
97#13133146—21no—
98#13133146—21no—
99#13133146—21no—
100#13133146—21no—
101#13133141C2×C411yescharacteristic
102#13133141C2×C411yescharacteristic
103#13133140C2×C411yescharacteristic
104#13133140C2×C411yescharacteristic
105#13133140C2×C411yescharacteristic
106#13133140C2×C411yescharacteristic
107#13133138C2×C411yescharacteristic
108#13133138C2×C411yescharacteristic
109C22⋊#88322—21no—
110C22⋊#88322—21no—
111C22⋊#88322C2×C411yescharacteristic
112C22⋊#88322C2×C411yescharacteristic
113C22⋊#88322D411yescharacteristic
114C22⋊#88322D411yescharacteristic
115#13133128—21no—
116C4×#88322—21no—
117C4×#88322C2311yescharacteristic
118C4×#88322C2311yescharacteristic
119C4×#88322C2311yescharacteristic
120C4×#88322C2311yescharacteristic
121C8×Dic53—21no—
122C8×Dic53—21no—
123C8×Dic53—21no—
124C8×Dic53—21no—
125#126×C53—21no—
126#125×C53C2×C411yescharacteristic
127#124×C53C2×C411yescharacteristic
128#123×C53—21no—
129C22⋊Q8×C53C2×C411yescharacteristic
130#121×C53C2×C411yescharacteristic
131C4×Q8×C53C2311yescharacteristic
132C4×D4×C53—21no—
133C4×D4×C53C2311yescharacteristic
134#117×C53—21no—
135C2×C42×C53C2311yescharacteristic
136C4⋊C8×C53—21no—
137C4⋊C8×C53C2311yescharacteristic
138C4⋊C8×C53C2311yescharacteristic
139C22⋊C8×C53—21no—
140C22⋊C8×C53—21no—
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×D4×C53—21no—
146C2×D4×C53C22×C411yescharacteristic
147C22×C4×C53—21no—
148C22×C4×C53C22×C411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53#4111yescharacteristic
151C22×Dic53—21no—
152C22×Dic53—21no—
153C2×Dic106—21no—
154C2×Dic106—22no—
155C2×Dic106—22no—
156C2×Dic106C22×C411yescharacteristic
157C2×Dic106C22×C411yescharacteristic
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53C22×C411yescharacteristic
167C4⋊C4×C53C22×C411yescharacteristic
168C22⋊C4×C53—21no—
169C22⋊C4×C53—21no—
170C22⋊C4×C53—21no—
171C22⋊C4×C53—21no—
172C22⋊C4×C53—21no—
173C42×C53—21no—
174C42×C53C22×C411yescharacteristic
175C42×C53C2×D411yescharacteristic
176C42×C53#4111yescharacteristic
177C42×C53C2411yescharacteristic
178#11719925—22no—
179#11719925—22no—
180#11719923—22no—
181#11719923—22no—
182C4⋊Dic53—21no—
183C4⋊Dic53C22×C411yescharacteristic
184C4⋊Dic53C22×C411yescharacteristic
185#11719919—21no—
186#11719919—21no—
187#11719919—21no—
188#11719919—21no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193#11719917—81no—
194#11719917—81no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—81no—
198C2×#88322C22×C411yescharacteristic
199C2×#88322C22×C411yescharacteristic
200C2×#88322C22×C411yescharacteristic
201C2×#88322C22×C411yescharacteristic
202C2×#88322C2×D411yescharacteristic
203C2×#88322C2×D411yescharacteristic
204C2×#88322C2×D411yescharacteristic
205C2×#88322C2×D411yescharacteristic
206C23×C53—21no—
207C23×C53C4×D411yescharacteristic
208Q8×C53—21no—
209Q8×C53—21no—
210D4×C53—21no—
211D4×C53—21no—
212D4×C53—81no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53C4×D411yescharacteristic
224C2×C4×C53C4×D411yescharacteristic
225C2×C4×C53C4×D411yescharacteristic
226C2×C4×C53C23×C411yescharacteristic
227C2×C4×C53C22×D411yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×Dic53—24no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—81no—
235Dic106—24no—
236Dic106—21no—
237Dic106—21no—
238Dic106#13112yes—
239Dic106#13112yes—
240C8×C53—41no—
241C8×C53—41no—
242C8×C53—41no—
243C8×C53—41no—
244#88322—21no—
245#88322—21no—
246#88322—21no—
247#88322—21no—
248#88322—21no—
249#88322—21no—
250#88322—21no—
251#88322—21no—
252#88322—41no—
253#88322—41no—
254#17919—531nomaximal, Sylow
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—21no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53C2×C4×D411yescharacteristic
262C4×C53—21no—
263C4×C53—21no—
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—41no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53C2×#13111yescharacteristic
270C4×C53C2×#13111yescharacteristic
271C4×C53#57711yescharacteristic
272C4×C53#57811yescharacteristic
273Dic53—22no—
274Dic53—22no—
275Dic53—41no—
276Dic53—41no—
277#3108—531no—
278#2993—531no—
279#2982—531no—
280#1588—531no—
281#1587—531no—
282#1584—531no—
283#1583—531no—
284#1579—531no—
285SD16⋊C8—531no—
286C8×Q16—531no—
287C8×SD16—531no—
288#1574—531no—
289#1487—531no—
290#1482D5311yescharacteristic
291#1454—531no—
292C2×C53—41no—
293C2×C53—81no—
294C2×C53#292911yescharacteristic
295C2×C53#294711yescharacteristic
296C2×C53#295711yescharacteristic
297#529C2×D5311yescharacteristic
298C2×C4×Q8—531no—
299#483—531no—
300#480—531no—
301#479—531no—
302#479—531no—
303#464—531no—
304#464—531no—
305#451—531no—
306#448—531no—
307#446—531no—
308C8×Q8—531no—
309C8×Q8—531no—
310C8×Q8—531no—
311C8×Q8—531no—
312C8×Q8—531no—
313C8×Q8—1061no—
314C8×Q8—1061no—
315C4×Q16—531no—
316C4×Q16—531no—
317C4×SD16—531no—
318C4×SD16—531no—
319#436—531no—
320#436—1061no—
321#435—1061no—
322C8×D4—531no—
323#433—1061no—
324#431—531no—
325#431—531no—
326#431—1061no—
327#431—1061no—
328#431C2×D5311yescharacteristic
329#431C2×D5311yescharacteristic
330#424—1061no—
331#423—531no—
332#335—531no—
333#335—531no—
334#334—531no—
335#334—531no—
336Q8⋊C8—531no—
337Q8⋊C8—531no—
338Q8⋊C8—531no—
339Q8⋊C8—531no—
340Q8⋊C8C2×D5311yescharacteristic
341Q8⋊C8C2×D5311yescharacteristic
342D4⋊C8C2×D5311yescharacteristic
343D4⋊C8C2×D5311yescharacteristic
344#322—531no—
345#322—531no—
346#322—531no—
347C82—531no—
348C53#1791911yescharacteristic, complemented, Sylow
349C22×Q8—531no—
350C2×Q16—531no—
351C2×Q16—531no—
352C2×SD16—531no—
353C2×SD16—531no—
354C2×M16—1061no—
355C2×M16—1061no—
356C22×C8—1061no—
357#126—21no—
358#125C4×D5311yescharacteristic
359#124C4×D5311yescharacteristic
360#123—21no—
361C22⋊Q8C4×D5311yescharacteristic
362#121C4×D5311yescharacteristic
363C4×Q8—531no—
364C4×Q8—531no—
365C4×Q8—531no—
366C4×Q8—531no—
367C4×Q8—1061no—
368C4×Q8—1061no—
369C4×Q8C22×D5311yescharacteristic
370C4×D4—21no—
371C4×D4C22×D5311yescharacteristic
372#117—21no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—1061no—
375C2×C42—1061no—
376C2×C42C22×D5311yescharacteristic
377#107—531no—
378#107—531no—
379#106—531no—
380#106—531no—
381C4⋊C8—21no—
382C4⋊C8—531no—
383C4⋊C8—531no—
384C4⋊C8—531no—
385C4⋊C8—531no—
386C4⋊C8—531no—
387C4⋊C8—531no—
388C4⋊C8—1061no—
389C4⋊C8—1061no—
390C4⋊C8—1061no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8C22×D5311yescharacteristic
396C4⋊C8C22×D5311yescharacteristic
397#103—531no—
398#103—531no—
399#103—531no—
400#103—531no—
401#103—531no—
402#103—531no—
403#102—531no—
404#102—531no—
405C22⋊C8—21no—
406C22⋊C8—21no—
407C22⋊C8—531no—
408C22⋊C8—531no—
409C22⋊C8—531no—
410C22⋊C8—531no—
411C22⋊C8—1061no—
412C22⋊C8—1061no—
413#97—1061no—
414C4×C8—21no—
415C4×C8—531no—
416C4×C8—531no—
417C4×C8—531no—
418C4×C8—531no—
419C4×C8—1061no—
420C4×C8—1061no—
421C4×C8—1061no—
422C4×C8—1061no—
423C4×C8—1061no—
424C4×C8C22×D5311yescharacteristic
425C4×C8C22×D5311yescharacteristic
426C2×Q8—531no—
427C2×Q8—531no—
428C2×Q8—1062no—
429C2×Q8—1062no—
430C2×Q8—1061no—
431C2×Q8C2×C4×D5311yescharacteristic
432C2×D4—21no—
433C2×D4C2×C4×D5311yescharacteristic
434C22×C4—21no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4C2×C4×D5311yescharacteristic
438C22×C4D4×D5311yescharacteristic
439C22×C4#1171994711yescharacteristic
440Q16—1062no—
441Q16—1062no—
442SD16—1062no—
443SD16—1062no—
444M16—4241no—
445M16—4241no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—531no—
451C2×C8—531no—
452C2×C8—531no—
453C2×C8—531no—
454C2×C8—531no—
455C2×C8—531no—
456C2×C8—531no—
457C2×C8—531no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—4241no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—531no—
466C4⋊C4—531no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4C2×C4×D5311yescharacteristic
473C4⋊C4C2×C4×D5311yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C42—21no—
480C42—1061no—
481C42—1061no—
482C42—1061no—
483C42—1061no—
484C42C2×C4×D5311yescharacteristic
485C42D4×D5311yescharacteristic
486C42#1171994711yescharacteristic
487C42C23×D5311yescharacteristic
488C23—21no—
489C23#1313317711yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—532no—
493Q8—532no—
494Q8—1064no—
495Q8—1061no—
496Q8—1061no—
497D4—21no—
498D4—21no—
499D4—81no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—1064no—
511C2×C4—1061no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—4241no—
516C2×C4#1313317711yescharacteristic
517C2×C4#1313317711yescharacteristic
518C2×C4#1313317711yescharacteristic
519C2×C4C22×C4×D5311yescharacteristic
520C2×C4C2×D4×D5311yescharacteristic
521C2×C4C2×#1171994711yescharacteristic
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—1061no—
527C8—1061no—
528C8—1061no—
529C8—1061no—
530C8—1061no—
531C8—1061no—
532C8—1061no—
533C8—1061no—
534C8—2121no—
535C8—2121no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22C2×#1313317711yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—1062no—
551C4—1062no—
552C4—2121no—
553C4—2121no—
554C4#131×D5311yescharacteristic
555C4#131×D5311yescharacteristic
556C4#1681143011yescharacteristic
557C4#1681143711yescharacteristic
558C2—41no—
559C2—81no—
560C2#2028341211yescharacteristic
561C2#2028356111yescharacteristic
562C2#2028362811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1482C2
socleC22×C53C2×C4×D4
prime radical#121×C53C2×C4
cosocleC42×C53C24
hypercenter#1482D53
hypocenterC53#17919
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1482

Complements

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

NH ≅ G/Ntype
3. #20283795C2semidirect
4. #20273596C2semidirect
5. #20273589C2semidirect
6. #20273568C2semidirect
7. #20273561C2semidirect
8. #20273531C2semidirect
9. #20273524C2semidirect
10. #20273519C2semidirect
11. #20273518C2semidirect
12. #20273503C2semidirect
13. #20273145C2semidirect
14. #20273015C2semidirect
348. C53#17919semidirect

Maximal subgroups

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

subgroupindex
#202847912
#202838642
#202837952
#202735962
#202735892
#202735682
#202735612
#202735312
#202735242
#202735192
#202735182
#202735032
#202731452
#202730152
C53×#14822
#1791953