Finite Group Databasebeta

← #27350002

Subgroups — #27350002

Order 13568.

1210 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811103C211yesmaximal, characteristic
2#20283743C212yesmaximal, complemented
3#20273058C212yesmaximal, complemented
4#20273050C212yesmaximal, complemented
5#20273049C211yesmaximal, characteristic, complemented
6#20273043C212yesmaximal, complemented
7#20273030C212yesmaximal, complemented
8#324×D53C212yesmaximal, complemented
9C53×#1464C211yesmaximal, characteristic, complemented
10C22×#11719945C411yescharacteristic
11#16811103—24no—
12#16811103—24no—
13C2×C4×Dic106C411yescharacteristic
14C2×C4×Dic106C2211yescharacteristic
15C2×C4×D212C411yescharacteristic
16C2×C4×D212C2211yescharacteristic
17C2×#13133177C411yescharacteristic
18C2×#13133177C411yescharacteristic
19C2×#13133177C2211yescharacteristic
20C2×#13133177C2211yescharacteristic
21C2×#13133175C411yescharacteristic
22C2×#13133175C411yescharacteristic
23C22×C4⋊Dic53C411yescharacteristic
24C2×C4⋊C4×D53C2211yescharacteristic
25C2×C4⋊C4×D53C2211yescharacteristic
26#16810696—24no—
27#16810693—24no—
28#16810680C2212yes—
29#16810585C2214yes—
30C22⋊C8×D53C2214yes—
31#16810349C2212yes—
32#16810348C2214yes—
33#16810348C2214yes—
34#16810347C2212yes—
35C2×#117×C53C2211yescharacteristic
36C53×#432C2212yes—
37C53×#324C2212yes—
38C53×#323C2212yes—
39C2×#11719945—24no—
40C2×#11719945—22no—
41C2×#11719945—22no—
42C2×#11719945—24no—
43C22×Dic106C2×C411yescharacteristic
44C22×D212C2×C411yescharacteristic
45C22×C22⋊D53C2×C411yescharacteristic
46C22×C22⋊D53C2×C411yescharacteristic
47C23×Dic53C2×C411yescharacteristic
48C23×Dic53C2×C411yescharacteristic
49C22×C4×D53C2×C411yescharacteristic
50C22×C4×D53C2×C411yescharacteristic
51C22×C4×D53C2×C411yescharacteristic
52C22×C4×D53C2×C411yescharacteristic
53C22×C4×D53C2311yescharacteristic
54C22×C4×D53C2311yescharacteristic
55C2×#11719912—24no—
56C2×C8×D53—24no—
57C4×Dic106—22no—
58C4×Dic106—22no—
59C4×Dic106—22no—
60C4×Dic106—22no—
61C4×D212—22no—
62C4×D212—22no—
63C4×D212D414yes—
64C4×D212D414yes—
65#13133177—24no—
66#13133177—24no—
67#13133177—24no—
68#13133177—24no—
69#13133175—24no—
70#13133175—24no—
71C2×C4⋊Dic53—22no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53C2×C411yescharacteristic
74C2×C4⋊Dic53C2×C411yescharacteristic
75C2×C4⋊Dic53C2×C411yescharacteristic
76C2×C4⋊Dic53C2×C411yescharacteristic
77C2×#11719919C2×C411yescharacteristic
78C2×#11719919C2×C411yescharacteristic
79C2×#11719919C2311yescharacteristic
80C2×#11719919C2311yescharacteristic
81C4⋊C4×D53—24no—
82C4⋊C4×D53—24no—
83C2×#11719921C2×C411yescharacteristic
84C2×#11719921C2×C411yescharacteristic
85C2×#11719921C2×C411yescharacteristic
86C2×#11719921C2×C411yescharacteristic
87C2×C4×Dic53C2×C411yescharacteristic
88C2×C4×Dic53C2×C411yescharacteristic
89C2×C4×Dic53C2×C411yescharacteristic
90C2×C4×Dic53C2×C411yescharacteristic
91C4⋊#88322—22no—
92C22⋊#88322C2314yes—
93#13133129—24no—
94#13133129—24no—
95#13133128—22no—
96C23×C4×C53C2×C411yescharacteristic
97#117×C53—22no—
98#117×C53—22no—
99C2×C4⋊C4×C53C2×C411yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C22⋊C4×C53C2×C411yescharacteristic
102C2×C22⋊C4×C53C2311yescharacteristic
103C2×C42×C53C2×C411yescharacteristic
104C2×C42×C53C2311yescharacteristic
105C4⋊C8×C53—22no—
106C22⋊C8×C53C2314yes—
107#97×C53—22no—
108C24×C53C22×C411yescharacteristic
109C23×D53C22×C411yescharacteristic
110C23×D53C22×C411yescharacteristic
111C22×C4×C53—22no—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53C22×C411yescharacteristic
116C22×C4×C53C22×C411yescharacteristic
117C22×C4×C53C22×C411yescharacteristic
118C22×C4×C53C22×C411yescharacteristic
119C22×C4×C53C22×C411yescharacteristic
120C22×C4×C53C2411yescharacteristic
121C2×C22⋊D53—22no—
122C2×C22⋊D53—22no—
123C2×C22⋊D53—22no—
124C2×C22⋊D53—22no—
125C2×C22⋊D53—22no—
126C2×C22⋊D53M1614yescomplemented
127C22×Dic53—22no—
128C22×Dic53—22no—
129C22×Dic53—81no—
130C22×Dic53—81no—
131C22×Dic53C22×C411yescharacteristic
132C22×Dic53C22×C411yescharacteristic
133C22×Dic53C22×C411yescharacteristic
134C22×Dic53C22×C411yescharacteristic
135C22×Dic53C22×C411yescharacteristic
136C22×Dic53C22×C411yescharacteristic
137#11719945—216no—
138#11719945—216no—
139C2×D212—22no—
140C2×D212—22no—
141C2×D212C22⋊C412yes—
142C2×D212C22⋊C412yes—
143C2×C4×D53—22no—
144C2×C4×D53—22no—
145C2×C4×D53—22no—
146C2×C4×D53—22no—
147C2×C4×D53—22no—
148C2×C4×D53—22no—
149C2×C4×D53—22no—
150C2×C4×D53—81no—
151C2×C4×D53—81no—
152C2×C4×D53—81no—
153C2×C4×D53—81no—
154C2×C4×D53M1614yescomplemented
155C2×C4×D53C2×D414yes—
156C2×C4×D53C2×D414yes—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—22no—
160C2×Dic106—22no—
161C2×C8×C53—24no—
162C4⋊C4×C53—22no—
163C4⋊C4×C53—22no—
164C22⋊C4×C53—22no—
165C22⋊C4×C53—22no—
166C42×C53—21no—
167C42×C53—21no—
168C42×C53C2×D412yes—
169C42×C53C2×D412yes—
170#11719921—22no—
171#11719921—22no—
172#11719921C22⋊C414yes—
173#11719921C22⋊C414yes—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—22no—
177C4⋊Dic53—22no—
178C4⋊Dic53C22⋊C412yes—
179C4⋊Dic53C22⋊C412yes—
180#11719919—22no—
181#11719919—22no—
182#11719919—22no—
183#11719919—22no—
184C4×Dic53—22no—
185C4×Dic53—22no—
186C4×Dic53—22no—
187C4×Dic53—22no—
188C2×#88322—24no—
189#11719912—84no—
190C8×D53—84no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53C2×M1612yes—
196C23×C53C23×C411yescharacteristic
197C22×D53—21no—
198C22×D53—21no—
199C22×D53—41no—
200C22×D53—41no—
201C22×D53—41no—
202C22×D53—41no—
203C22×D53C2×C22⋊C412yes—
204C22×D53C2×C22⋊C412yes—
205C22×D53C2×M1612yes—
206C22×D53C2×M1612yes—
207C2×C4×C53—24no—
208C2×C4×C53—24no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53C2×C22⋊C411yescharacteristic
220C2×C4×C53C2×C22⋊C411yescharacteristic
221C2×C4×C53C2×C22⋊C412yes—
222C2×C4×C53C2×M1612yes—
223C2×C4×C53C22×D412yes—
224C22⋊D53—28no—
225C22⋊D53—28no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—82no—
237C2×Dic53—82no—
238C2×Dic53C2×C22⋊C412yes—
239C2×Dic53C2×C22⋊C412yes—
240C2×Dic53C2×M1612yes—
241C2×Dic53C2×M1612yes—
242D212—24no—
243D212—24no—
244C4×D53—28no—
245C4×D53—28no—
246C4×D53—44no—
247C4×D53—44no—
248C4×D53—82no—
249C4×D53—82no—
250Dic106—24no—
251Dic106—24no—
252C8×C53—44no—
253#88322—44no—
254#17368—531nomaximal, Sylow
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—22no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53C22×C22⋊C411yescharacteristic
263C22×C53C22×M1611yescharacteristic
264C22×C53C22×M1611yescharacteristic
265C2×D53—22no—
266C2×D53—22no—
267C2×D53—22no—
268C2×D53—22no—
269C2×D53—22no—
270C2×D53—42no—
271C2×D53—42no—
272C2×D53—42no—
273C2×D53—42no—
274C2×D53#40714yes—
275C4×C53—21no—
276C4×C53—21no—
277C4×C53—22no—
278C4×C53—22no—
279C4×C53—22no—
280C4×C53—41no—
281C4×C53—41no—
282Dic53—22no—
283Dic53—22no—
284Dic53—22no—
285Dic53—22no—
286Dic53—41no—
287Dic53—41no—
288C2×#520—531no—
289#2973—532no—
290#1467—532no—
291#1465—532no—
292#1464—531no—
293#1464D5311yescharacteristic, complemented
294#1463—532no—
295#1459—532no—
296C2×#324—532no—
297C2×C53—21no—
298C2×C53—21no—
299C2×C53—42no—
300C2×C53C2×#40711yescharacteristic
301C2×C53#288311yescharacteristic
302C2×C53#288911yescharacteristic
303D53—24no—
304D53—42no—
305C22×#41—531no—
306#520—1064no—
307#520—1064no—
308C2×C4×Q8—531no—
309C2×C4×Q8—531no—
310C2×C4×D4—531no—
311C2×C4×D4—531no—
312C2×C4×D4—531no—
313C2×C4×D4—531no—
314C2×C4×D4—531no—
315C2×C4×D4—531no—
316C2×#117—531no—
317C2×#117—531no—
318C2×#117C2×D5311yescharacteristic
319C22×C4⋊C4—531no—
320C22×C4⋊C4—531no—
321C22×C4⋊C4—531no—
322#435—1064no—
323#435—1064no—
324#432—532no—
325#432C2×D5312yes—
326#407—534no—
327C2×C22⋊C8—534no—
328#324—532no—
329#324—534no—
330#324—534no—
331#324C2×D5312yes—
332#323—532no—
333#323C2×D5312yes—
334C53#1736811yescharacteristic, complemented, Sylow
335C2×#41—1064no—
336C2×#41—1062no—
337C2×#41—1062no—
338C2×#41—1064no—
339C22×Q8—531no—
340C22×D4—531no—
341C22×D4—531no—
342C22×D4—531no—
343C23×C4—531no—
344C23×C4—531no—
345C23×C4—531no—
346C23×C4—531no—
347C23×C4—531no—
348C23×C4—531no—
349C23×C4—531no—
350C23×C4—531no—
351C23×C4C4×D5311yescharacteristic
352C2×M16—1064no—
353C22×C8—1064no—
354C4×Q8—1062no—
355C4×Q8—1062no—
356C4×Q8—1062no—
357C4×Q8—1062no—
358C4×D4—534no—
359C4×D4—534no—
360C4×D4—1062no—
361C4×D4—1062no—
362C4×D4—1064no—
363C4×D4—1064no—
364C4×D4—1064no—
365C4×D4—1064no—
366#117—22no—
367#117—22no—
368#117—1064no—
369#117—1064no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—1064no—
379C2×C4⋊C4—1064no—
380C2×C4⋊C4—1062no—
381C2×C4⋊C4—1062no—
382C2×C4⋊C4C4×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C22⋊C4—531no—
385C2×C22⋊C4—531no—
386C2×C22⋊C4—531no—
387C2×C22⋊C4—531no—
388C2×C22⋊C4C4×D5311yescharacteristic
389C2×C22⋊C4C22×D5311yescharacteristic
390C2×C42—531no—
391C2×C42—531no—
392C2×C42—531no—
393C2×C42—531no—
394C2×C42C4×D5311yescharacteristic
395C2×C42C22×D5311yescharacteristic
396C4⋊C8—22no—
397C4⋊C8—1062no—
398C22⋊C8—534no—
399C22⋊C8—1064no—
400C22⋊C8—1064no—
401C22⋊C8C22×D5314yes—
402#97—22no—
403#97—1062no—
404C24—531no—
405C24—531no—
406C24C2×C4×D5311yescharacteristic
407#41—10616no—
408#41—10616no—
409C2×Q8—1062no—
410C2×Q8—1062no—
411C2×Q8—1061no—
412C2×Q8—1061no—
413C2×D4—534no—
414C2×D4—532no—
415C2×D4—532no—
416C2×D4—1062no—
417C2×D4—1062no—
418C2×D4—1062no—
419C2×D4—1062no—
420C2×D4—1062no—
421C2×D4—1062no—
422C2×D4—1062no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—531no—
428C22×C4—531no—
429C22×C4—531no—
430C22×C4—531no—
431C22×C4—534no—
432C22×C4—534no—
433C22×C4—534no—
434C22×C4—531no—
435C22×C4—531no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—1062no—
439C22×C4—1062no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—4241no—
446C22×C4—4241no—
447C22×C4—4241no—
448C22×C4—4241no—
449C22×C4—4241no—
450C22×C4—4241no—
451C22×C4C2×C4×D5311yescharacteristic
452C22×C4C2×C4×D5311yescharacteristic
453C22×C4C2×C4×D5311yescharacteristic
454C22×C4C2×C4×D5311yescharacteristic
455C22×C4C2×C4×D5311yescharacteristic
456C22×C4C23×D5311yescharacteristic
457M16—4244no—
458C2×C8—24no—
459C2×C8—1064no—
460C2×C8—4244no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—532no—
464C4⋊C4—532no—
465C4⋊C4—1062no—
466C4⋊C4—1062no—
467C4⋊C4—1062no—
468C4⋊C4—1062no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1062no—
472C4⋊C4—1062no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—534no—
476C22⋊C4—534no—
477C22⋊C4—1062no—
478C22⋊C4—1062no—
479C42—21no—
480C42—21no—
481C42—1062no—
482C42—1062no—
483C42—1062no—
484C42—1062no—
485C42D4×D5312yes—
486C42D4×D5312yes—
487C23—21no—
488C23—21no—
489C23—41no—
490C23—41no—
491C23—532no—
492C23—532no—
493C23—532no—
494C23—532no—
495C23—1061no—
496C23—1061no—
497C23—2121no—
498C23—2121no—
499C23—2121no—
500C23—2121no—
501C23M16×D5312yes—
502C23C22×C4×D5311yescharacteristic
503Q8—1064no—
504Q8—1064no—
505D4—1064no—
506D4—1064no—
507D4—1068no—
508D4—1068no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—532no—
522C2×C4—532no—
523C2×C4—532no—
524C2×C4—532no—
525C2×C4—1068no—
526C2×C4—1068no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—4242no—
534C2×C4—4242no—
535C2×C4—4242no—
536C2×C4—4242no—
537C2×C4—2124no—
538C2×C4—2124no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4C22⋊C4×D5311yescharacteristic
544C2×C4C22⋊C4×D5311yescharacteristic
545C2×C4C22⋊C4×D5312yes—
546C2×C4M16×D5312yes—
547C2×C4C2×D4×D5312yes—
548C8—44no—
549C8—2124no—
550C22—21no—
551C22—21no—
552C22—22no—
553C22—42no—
554C22—42no—
555C22—41no—
556C22—41no—
557C22—534no—
558C22—1062no—
559C22—1062no—
560C22—1062no—
561C22—1062no—
562C22—1062no—
563C22—2122no—
564C22—2122no—
565C22—2122no—
566C22—2122no—
567C22C2×C22⋊C4×D5311yescharacteristic
568C22C2×M16×D5311yescharacteristic
569C22C2×M16×D5311yescharacteristic
570C4—22no—
571C4—21no—
572C4—21no—
573C4—22no—
574C4—22no—
575C4—41no—
576C4—41no—
577C4—1062no—
578C4—1062no—
579C4—1062no—
580C4—1062no—
581C4—2121no—
582C4—2121no—
583C2—21no—
584C2—21no—
585C2—42no—
586C2—1064no—
587C2—2122no—
588C2#407×D5311yescharacteristic
589C2#2028317411yescharacteristic
590C2#2028319611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1464C2
socleC22×C53C22×C22⋊C4
prime radicalC22×#11719945C4
cosocleC22×C4×C53C24
hypercenter#1464D53
hypocenterC53#17368
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #20283743C2semidirect
3. #20273058C2semidirect
4. #20273050C2semidirect
5. #20273049C2semidirect
6. #20273043C2semidirect
7. #20273030C2semidirect
8. #324×D53C2semidirect
9. C53×#1464C2semidirect
126. C2×C22⋊D53M16semidirect
154. C2×C4×D53M16semidirect
293. #1464D53semidirect
334. C53#17368semidirect

Maximal subgroups

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

subgroupindex
C2×#168111032
#202837432
#202730582
#202730502
#202730492
#202730432
#202730302
#324×D532
C53×#14642
#1736853