Finite Group Databasebeta

← #27350005

Subgroups — #27350005

Order 13568.

1210 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811102C211yesmaximal, characteristic
2#20283741C212yesmaximal, complemented
3#20273057C212yesmaximal, complemented
4#20273050C212yesmaximal, complemented
5#20273046C212yesmaximal, complemented
6#20273043C212yesmaximal, complemented
7#20273033C211yesmaximal, characteristic, complemented
8C2×#16810348C212yesmaximal, complemented
9C53×#1459C211yesmaximal, characteristic, complemented
10C22×#11719947C411yescharacteristic
11#16811102—24no—
12#16811102—24no—
13C2×C4×Dic106C411yescharacteristic
14C2×#13133182C2211yescharacteristic
15C2×D4×Dic53C411yescharacteristic
16C2×C4×C22⋊D53C411yescharacteristic
17C2×C4×C22⋊D53C2211yescharacteristic
18C2×#13133177C411yescharacteristic
19C2×#13133177C2211yescharacteristic
20C2×#13133174C411yescharacteristic
21C2×#13133173C411yescharacteristic
22C2×#13133173C2211yescharacteristic
23C22×#11719919C411yescharacteristic
24C22×#11719919C2211yescharacteristic
25C2×C4⋊C4×D53C2211yescharacteristic
26#16810694—24no—
27#16810691—24no—
28#16810677C2214yes—
29#16810585C2214yes—
30C2×C22⋊#88322C2212yes—
31#16810349C2212yes—
32#16810348C2214yes—
33#16810348C2214yes—
34#16810347C2212yes—
35C2×C4×D4×C53C2211yescharacteristic
36C2×C22⋊C8×C53C2212yes—
37C53×#324C2212yes—
38C53×#323C2212yes—
39C2×#11719947—24no—
40C2×#11719947—24no—
41C2×#11719947—24no—
42C22×Dic106C2×C411yescharacteristic
43C22×C22⋊D53C2×C411yescharacteristic
44C22×C22⋊D53C2×C411yescharacteristic
45C23×Dic53C2×C411yescharacteristic
46C23×Dic53C2×C411yescharacteristic
47C23×Dic53C2×C411yescharacteristic
48C23×Dic53C2×C411yescharacteristic
49C22×C4×D53C2×C411yescharacteristic
50C22×C4×D53C2×C411yescharacteristic
51C22×C4×D53C2311yescharacteristic
52C2×#11719912—24no—
53C22×#88322—22no—
54C4×Dic106—22no—
55C4×Dic106—22no—
56#13133182—24no—
57D4×Dic53—24no—
58C4×C22⋊D53—22no—
59C4×C22⋊D53—22no—
60C4×C22⋊D53D414yes—
61C4×C22⋊D53D414yes—
62#13133177—24no—
63#13133177—24no—
64#13133174—22no—
65#13133174—22no—
66#13133173—24no—
67#13133173—24no—
68C2×C4⋊Dic53C2×C411yescharacteristic
69C2×#11719919—22no—
70C2×#11719919—22no—
71C2×#11719919—22no—
72C2×#11719919—22no—
73C2×#11719919C2×C411yescharacteristic
74C2×#11719919C2×C411yescharacteristic
75C2×#11719919C2×C411yescharacteristic
76C2×#11719919C2×C411yescharacteristic
77C2×#11719919C2×C411yescharacteristic
78C2×#11719919C2×C411yescharacteristic
79C2×#11719919C2311yescharacteristic
80C2×#11719919C2311yescharacteristic
81C4⋊C4×D53—24no—
82C2×C22⋊Dic53C2×C411yescharacteristic
83C2×C22⋊Dic53C2×C411yescharacteristic
84C2×#11719921C2×C411yescharacteristic
85C2×#11719921C2×C411yescharacteristic
86C2×C4×Dic53C2×C411yescharacteristic
87C2×C4×Dic53C2×C411yescharacteristic
88C2×C4×Dic53C2×C411yescharacteristic
89C2×C4×Dic53C2×C411yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133146—24no—
92C22⋊#88322—22no—
93C22⋊#88322C2314yes—
94#13133129—24no—
95#13133127—24no—
96C22×D4×C53C2×C411yescharacteristic
97C23×C4×C53C2×C411yescharacteristic
98C23×C4×C53C2311yescharacteristic
99C22×C8×C53—22no—
100C4×D4×C53—22no—
101C4×D4×C53—22no—
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C22⋊C4×C53C2×C411yescharacteristic
104C2×C22⋊C4×C53C2311yescharacteristic
105C2×C42×C53C2×C411yescharacteristic
106C22⋊C8×C53—22no—
107C22⋊C8×C53C2314yes—
108C24×C53C22×C411yescharacteristic
109C24×C53C22×C411yescharacteristic
110C23×D53C22×C411yescharacteristic
111C2×D4×C53—22no—
112C2×D4×C53—22no—
113C2×D4×C53M1614yescomplemented
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—81no—
117C22×C4×C53—81no—
118C22×C4×C53C22×C411yescharacteristic
119C22×C4×C53C22×C411yescharacteristic
120C22×C4×C53C22×C411yescharacteristic
121C22×C4×C53C22×C411yescharacteristic
122C22×C4×C53C2×D412yes—
123C22×C4×C53C2×D412yes—
124C22×C4×C53C2411yescharacteristic
125C2×C22⋊D53—22no—
126C2×C22⋊D53—22no—
127C2×C22⋊D53—22no—
128C2×C22⋊D53—22no—
129C2×C22⋊D53—22no—
130C2×C22⋊D53C22⋊C414yes—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—81no—
139C22×Dic53—81no—
140C22×Dic53M1614yescomplemented
141C22×Dic53C22×C411yescharacteristic
142C22×Dic53C22×C411yescharacteristic
143C22×Dic53C22×C411yescharacteristic
144C22×Dic53C22×C411yescharacteristic
145C22×Dic53C22×C411yescharacteristic
146C22×Dic53C22×C411yescharacteristic
147C22×Dic53C22×C411yescharacteristic
148#11719947—232no—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—22no—
153C2×C4×D53—81no—
154C2×C4×D53—81no—
155C2×C4×D53C2×D414yes—
156C2×Dic106—22no—
157C2×Dic106—22no—
158C2×Dic106—22no—
159C2×C8×C53—24no—
160C2×C8×C53—82no—
161C4⋊C4×C53—22no—
162C22⋊C4×C53—22no—
163C22⋊C4×C53—22no—
164C42×C53—21no—
165C42×C53—21no—
166C22⋊Dic53—22no—
167C22⋊Dic53C22⋊C414yes—
168#11719921—22no—
169#11719921C22⋊C412yes—
170#11719921C22⋊C412yes—
171C4⋊Dic53—22no—
172#11719919—21no—
173#11719919—21no—
174#11719919—21no—
175#11719919—21no—
176#11719919—22no—
177#11719919—22no—
178#11719919—22no—
179#11719919—22no—
180#11719919—22no—
181#11719919C22⋊C412yes—
182#11719919C22⋊C412yes—
183C4×Dic53—22no—
184C4×Dic53—22no—
185C4×Dic53—22no—
186C4×Dic53—22no—
187C4×Dic53C2×D414yes—
188C2×#88322—24no—
189C2×#88322—82no—
190#11719912—84no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53C2×C22⋊C412yes—
198C23×C53C2×M1612yes—
199C23×C53C2×M1612yes—
200C23×C53C2×M1612yes—
201C23×C53C23×C411yescharacteristic
202C22×D53—21no—
203C22×D53—21no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53C2×C22⋊C412yes—
207D4×C53—28no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—81no—
219C2×C4×C53—81no—
220C2×C4×C53C2×C22⋊C411yescharacteristic
221C2×C4×C53C2×C22⋊C411yescharacteristic
222C2×C4×C53C2×M1612yes—
223C2×C4×C53C22×D412yes—
224C22⋊D53—28no—
225C22⋊D53—28no—
226C2×Dic53—28no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—28no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—82no—
241C2×Dic53—82no—
242C2×Dic53C2×C22⋊C412yes—
243C2×Dic53C2×C22⋊C412yes—
244C2×Dic53C2×C22⋊C412yes—
245C2×Dic53C2×M1612yes—
246C2×Dic53C2×M1612yes—
247C4×D53—28no—
248C4×D53—44no—
249C4×D53—82no—
250Dic106—28no—
251C8×C53—44no—
252#88322—44no—
253#17368—531nomaximal, Sylow
254C22×C53—21no—
255C22×C53—22no—
256C22×C53—21no—
257C22×C53—22no—
258C22×C53—22no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53#40714yes—
266C22×C53C22×C22⋊C411yescharacteristic
267C22×C53C22×M1611yescharacteristic
268C22×C53C22×M1611yescharacteristic
269C2×D53—22no—
270C2×D53—22no—
271C2×D53—22no—
272C2×D53—42no—
273C2×D53—42no—
274C4×C53—22no—
275C4×C53—22no—
276C4×C53—21no—
277C4×C53—21no—
278C4×C53—41no—
279C4×C53—41no—
280Dic53—22no—
281Dic53—22no—
282Dic53—22no—
283Dic53—22no—
284Dic53—22no—
285Dic53—41no—
286Dic53—41no—
287C2×#520—531no—
288#2973—532no—
289#1467—532no—
290#1465—532no—
291#1464—532no—
292#1463—532no—
293#1459—531no—
294#1459D5311yescharacteristic, complemented
295C2×#324—532no—
296C2×C53—21no—
297C2×C53—21no—
298C2×C53—24no—
299C2×C53—42no—
300C2×C53C2×#40711yescharacteristic
301C2×C53#288311yescharacteristic
302C2×C53#288911yescharacteristic
303D53—42no—
304C22×#41—531no—
305#520—1064no—
306#520—1064no—
307C2×C4×Q8—531no—
308C2×C4×Q8—531no—
309C2×C4×D4—531no—
310C2×C4×D4—531no—
311C2×C4×D4—531no—
312C2×C4×D4—531no—
313C2×C4×D4—531no—
314C2×C4×D4C2×D5311yescharacteristic
315C2×#117—531no—
316C2×#117—531no—
317C2×#117—531no—
318C22×C4⋊C4—531no—
319C22×C4⋊C4—531no—
320C22×C4⋊C4—531no—
321#435—1064no—
322#435—1064no—
323#432—534no—
324#407—534no—
325C2×C22⋊C8—532no—
326C2×C22⋊C8C2×D5312yes—
327#324—532no—
328#324—534no—
329#324—534no—
330#324C2×D5312yes—
331#323—532no—
332#323C2×D5312yes—
333C53#1736811yescharacteristic, complemented, Sylow
334C2×#41—1064no—
335C2×#41—1064no—
336C2×#41—1064no—
337C22×Q8—531no—
338C22×D4—531no—
339C22×D4—531no—
340C22×D4C4×D5311yescharacteristic
341C23×C4—531no—
342C23×C4—531no—
343C23×C4—531no—
344C23×C4—531no—
345C23×C4—531no—
346C23×C4—531no—
347C23×C4—531no—
348C23×C4C4×D5311yescharacteristic
349C23×C4C22×D5311yescharacteristic
350C2×M16—1064no—
351C22×C8—22no—
352C22×C8—1062no—
353C4×Q8—1062no—
354C4×Q8—1062no—
355C4×Q8—1064no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—534no—
359C4×D4—534no—
360C4×D4—1064no—
361C4×D4—1064no—
362C4×D4—1062no—
363C4×D4—1062no—
364C4×D4—1064no—
365#117—1062no—
366#117—1062no—
367#117—1064no—
368#117—1064no—
369C2×C4⋊C4—531no—
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—1062no—
380C2×C4⋊C4—1062no—
381C2×C4⋊C4—1062no—
382C2×C4⋊C4—1062no—
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×C42—531no—
395C2×C42C4×D5311yescharacteristic
396C4⋊C8—1064no—
397C22⋊C8—22no—
398C22⋊C8—534no—
399C22⋊C8—1064no—
400C22⋊C8—1062no—
401C22⋊C8C22×D5314yes—
402#97—1064no—
403C24—531no—
404C24C2×C4×D5311yescharacteristic
405C24C2×C4×D5311yescharacteristic
406#41—10632no—
407C2×Q8—1062no—
408C2×Q8—1062no—
409C2×Q8—1062no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—534no—
413C2×D4—1062no—
414C2×D4—1062no—
415C2×D4—1062no—
416C2×D4—1062no—
417C2×D4—1062no—
418C2×D4#1171991214yescomplemented
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—534no—
424C22×C4—531no—
425C22×C4—531no—
426C22×C4—531no—
427C22×C4—531no—
428C22×C4—531no—
429C22×C4—534no—
430C22×C4—531no—
431C22×C4—531no—
432C22×C4—1062no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—1062no—
439C22×C4—1062no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—4241no—
444C22×C4—4241no—
445C22×C4—4241no—
446C22×C4—4241no—
447C22×C4C2×C4×D5311yescharacteristic
448C22×C4C2×C4×D5311yescharacteristic
449C22×C4C2×C4×D5311yescharacteristic
450C22×C4C2×C4×D5311yescharacteristic
451C22×C4D4×D5312yes—
452C22×C4D4×D5312yes—
453C22×C4C23×D5311yescharacteristic
454M16—4244no—
455C2×C8—24no—
456C2×C8—82no—
457C2×C8—1064no—
458C2×C8—4242no—
459C4⋊C4—22no—
460C4⋊C4—532no—
461C4⋊C4—532no—
462C4⋊C4—1062no—
463C4⋊C4—1062no—
464C4⋊C4—1062no—
465C4⋊C4—1062no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1062no—
471C4⋊C4—1062no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—534no—
475C22⋊C4—532no—
476C22⋊C4—532no—
477C22⋊C4—1062no—
478C22⋊C4—1062no—
479C42—21no—
480C42—21no—
481C42—534no—
482C42—1062no—
483C42—1062no—
484C42—1062no—
485C42—1062no—
486C23—21no—
487C23—21no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—532no—
493C23—1061no—
494C23—1061no—
495C23—2121no—
496C23—2121no—
497C23C22⋊C4×D5312yes—
498C23M16×D5312yes—
499C23C2×#1171991212yes—
500C23C2×#1171991212yes—
501C23C22×C4×D5311yescharacteristic
502Q8—1068no—
503D4—28no—
504D4—1068no—
505D4—1068no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—532no—
519C2×C4—532no—
520C2×C4—532no—
521C2×C4—532no—
522C2×C4—532no—
523C2×C4—1068no—
524C2×C4—1068no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—1068no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—4242no—
535C2×C4—4242no—
536C2×C4—4242no—
537C2×C4—2124no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4C22⋊C4×D5311yescharacteristic
543C2×C4C22⋊C4×D5311yescharacteristic
544C2×C4C2×#1171991212yes—
545C2×C4C2×D4×D5312yes—
546C8—44no—
547C8—2124no—
548C22—21no—
549C22—22no—
550C22—21no—
551C22—22no—
552C22—22no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—41no—
558C22—41no—
559C22—1062no—
560C22—1062no—
561C22—1062no—
562C22—2122no—
563C22—2122no—
564C22#1681058514yes—
565C22C2×C22⋊C4×D5311yescharacteristic
566C22C2×M16×D5311yescharacteristic
567C22C22×#1171991211yescharacteristic
568C4—22no—
569C4—22no—
570C4—21no—
571C4—21no—
572C4—41no—
573C4—41no—
574C4—1062no—
575C4—1062no—
576C4—1062no—
577C4—1062no—
578C4—1062no—
579C4—2121no—
580C4—2121no—
581C2—21no—
582C2—21no—
583C2—24no—
584C2—42no—
585C2—2122no—
586C2C2×#1681058511yescharacteristic
587C2#2028317311yescharacteristic
588C2#2028319611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1459C2
socleC22×C53C22×C22⋊C4
prime radicalC22×#11719947C4
cosocleC22×C4×C53C24
hypercenter#1459D53
hypocenterC53#17368
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1459

Complements

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

NH ≅ G/Ntype
2. #20283741C2semidirect
3. #20273057C2semidirect
4. #20273050C2semidirect
5. #20273046C2semidirect
6. #20273043C2semidirect
7. #20273033C2semidirect
8. C2×#16810348C2semidirect
9. C53×#1459C2semidirect
113. C2×D4×C53M16semidirect
140. C22×Dic53M16semidirect
294. #1459D53semidirect
333. C53#17368semidirect
418. C2×D4#11719912semidirect

Maximal subgroups

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

subgroupindex
C2×#168111022
#202837412
#202730572
#202730502
#202730462
#202730432
#202730332
C2×#168103482
C53×#14592
#1736853