Finite Group Databasebeta

← #27350001

Subgroups — #27350001

Order 13568.

1210 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811104C211yesmaximal, characteristic
2#20283741C212yesmaximal, complemented
3#20273057C212yesmaximal, complemented
4#20273052C212yesmaximal, complemented
5#20273048C212yesmaximal, complemented
6#20273043C212yesmaximal, complemented
7#20273031C212yesmaximal, complemented
8C2×#16810349C211yesmaximal, characteristic, complemented
9C2×C53×#324C211yesmaximal, characteristic, complemented
10C22×#11719945C411yescharacteristic
11#16811104—24no—
12#16811104—24no—
13C2×#13133182C411yescharacteristic
14C2×#13133182C2211yescharacteristic
15C2×#13133179C411yescharacteristic
16C2×#13133179C2211yescharacteristic
17C2×C4×C22⋊D53C411yescharacteristic
18C2×C4×C22⋊D53C411yescharacteristic
19C2×C4×C22⋊D53C2211yescharacteristic
20C2×C4×C22⋊D53C2211yescharacteristic
21C2×#13133176C411yescharacteristic
22C2×#13133175C411yescharacteristic
23C2×#13133175C2211yescharacteristic
24C2×C4⋊C4×D53C411yescharacteristic
25C2×C4⋊C4×D53C2211yescharacteristic
26#16810694—24no—
27#16810691—24no—
28#16810677C2214yes—
29#16810585C2214yes—
30C2×C22⋊#88322C2212yes—
31#16810349C2212yes—
32#16810349C2212yes—
33#16810348C2214yes—
34#16810346C2214yes—
35C22×C4⋊C4×C53C2211yescharacteristic
36C2×C22⋊C8×C53C2212yes—
37C53×#324C2212yes—
38C53×#324C2212yes—
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
47C22×C4×D53C2×C411yescharacteristic
48C22×C4×D53C2×C411yescharacteristic
49C22×C4×D53C2×C411yescharacteristic
50C22×C4×D53C2×C411yescharacteristic
51C22×C4×D53C2×C411yescharacteristic
52C22×C4×D53C2311yescharacteristic
53C2×#11719912—24no—
54C22×#88322—22no—
55#13133182—24no—
56#13133182—24no—
57#13133179—24no—
58#13133179—24no—
59C4×C22⋊D53—22no—
60C4×C22⋊D53—22no—
61C4×C22⋊D53—22no—
62C4×C22⋊D53—22no—
63C4×C22⋊D53—22no—
64C4×C22⋊D53—22no—
65C4×C22⋊D53D414yes—
66C4×C22⋊D53D414yes—
67#13133176—22no—
68#13133176—22no—
69#13133175—24no—
70#13133175—24no—
71C2×C4⋊Dic53C2×C411yescharacteristic
72C2×C4⋊Dic53C2×C411yescharacteristic
73C2×#11719919C2×C411yescharacteristic
74C2×#11719919C2×C411yescharacteristic
75C2×#11719919C2×C411yescharacteristic
76C2×#11719919C2311yescharacteristic
77C4⋊C4×D53—24no—
78C4⋊C4×D53—24no—
79C2×C22⋊Dic53C2×C411yescharacteristic
80C2×C22⋊Dic53C2×C411yescharacteristic
81C2×#11719921C2×C411yescharacteristic
82C2×#11719921C2×C411yescharacteristic
83C2×#11719921C2×C411yescharacteristic
84C2×#11719921C2311yescharacteristic
85C2×C4×Dic53C2×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—
96C23×C4×C53C2×C411yescharacteristic
97C23×C4×C53C2×C411yescharacteristic
98C23×C4×C53C2311yescharacteristic
99C22×C8×C53—22no—
100C2×C4⋊C4×C53—22no—
101C2×C4⋊C4×C53—22no—
102C2×C4⋊C4×C53C2×C411yescharacteristic
103C2×C4⋊C4×C53C2×C411yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C22⋊C8×C53—22no—
107C22⋊C8×C53C2314yes—
108C24×C53C22×C411yescharacteristic
109C23×D53C22×C411yescharacteristic
110C23×D53C22×C411yescharacteristic
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—22no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—81no—
117C22×C4×C53—81no—
118C22×C4×C53M1614yescomplemented
119C22×C4×C53C22×C411yescharacteristic
120C22×C4×C53C22×C411yescharacteristic
121C22×C4×C53C22×C411yescharacteristic
122C22×C4×C53C22×C411yescharacteristic
123C22×C4×C53C22×C411yescharacteristic
124C22×C4×C53C2×D412yes—
125C22×C4×C53C2×D412yes—
126C22×C4×C53C2411yescharacteristic
127C2×C22⋊D53—21no—
128C2×C22⋊D53—21no—
129C2×C22⋊D53—22no—
130C2×C22⋊D53—22no—
131C2×C22⋊D53—22no—
132C2×C22⋊D53C22⋊C412yes—
133C2×C22⋊D53C22⋊C412yes—
134C2×C22⋊D53M1614yescomplemented
135C22×Dic53C22×C411yescharacteristic
136C22×Dic53C22×C411yescharacteristic
137C22×Dic53C22×C411yescharacteristic
138C22×Dic53C22×C411yescharacteristic
139C22×Dic53C22×C411yescharacteristic
140C22×Dic53C22×C411yescharacteristic
141#11719945—216no—
142#11719945—216no—
143C2×D212—22no—
144C2×D212—22no—
145C2×D212—22no—
146C2×C4×D53—22no—
147C2×C4×D53—22no—
148C2×C4×D53—22no—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—22no—
153C2×C4×D53—22no—
154C2×C4×D53—22no—
155C2×C4×D53—81no—
156C2×C4×D53—81no—
157C2×C4×D53—81no—
158C2×C4×D53—81no—
159C2×C4×D53C2×D414yes—
160C2×Dic106—22no—
161C2×Dic106—22no—
162C2×Dic106—22no—
163C2×C8×C53—24no—
164C2×C8×C53—82no—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53—22no—
169C22⋊Dic53—21no—
170C22⋊Dic53—21no—
171C22⋊Dic53C22⋊C412yes—
172C22⋊Dic53C22⋊C412yes—
173#11719921—22no—
174#11719921—22no—
175#11719921—22no—
176#11719921C22⋊C414yes—
177C4⋊Dic53—22no—
178C4⋊Dic53—22no—
179#11719919—22no—
180#11719919—22no—
181#11719919—22no—
182#11719919C22⋊C414yes—
183C4×Dic53—22no—
184C4×Dic53—22no—
185C4×Dic53—22no—
186C4×Dic53—22no—
187C4×Dic53—22no—
188C4×Dic53C2×D414yes—
189C2×#88322—24no—
190C2×#88322—82no—
191#11719912—84no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53C2×C22⋊C411yescharacteristic
195C23×C53C2×C22⋊C411yescharacteristic
196C23×C53C2×M1612yes—
197C23×C53C2×M1612yes—
198C23×C53C23×C411yescharacteristic
199C22×D53—21no—
200C22×D53—21no—
201C22×D53—21no—
202C22×D53—21no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53—41no—
207C22×D53C2×C22⋊C412yes—
208C22×D53C2×M1612yes—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—24no—
214C2×C4×C53—24no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—81no—
224C2×C4×C53—81no—
225C2×C4×C53C2×C22⋊C412yes—
226C2×C4×C53C2×M1612yes—
227C2×C4×C53C2×M1612yes—
228C2×C4×C53C22×D412yes—
229C22⋊D53—24no—
230C22⋊D53—24no—
231C22⋊D53—24no—
232C22⋊D53—24no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53C2×C22⋊C412yes—
244C2×Dic53C2×C22⋊C412yes—
245C2×Dic53C2×C22⋊C412yes—
246C2×Dic53C2×M1612yes—
247D212—28no—
248C4×D53—28no—
249C4×D53—28no—
250C4×D53—44no—
251C4×D53—82no—
252C4×D53—82no—
253C4×D53—82no—
254Dic106—28no—
255C8×C53—44no—
256#88322—44no—
257#17368—531nomaximal, Sylow
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—21no—
263C22×C53—21no—
264C22×C53—42no—
265C22×C53—42no—
266C22×C53#40712yes—
267C22×C53#40712yes—
268C22×C53C22×C22⋊C411yescharacteristic
269C22×C53C22×M1611yescharacteristic
270C22×C53C22×M1611yescharacteristic
271C2×D53—22no—
272C2×D53—22no—
273C2×D53—22no—
274C2×D53—22no—
275C2×D53—42no—
276C2×D53—42no—
277C2×D53—42no—
278C2×D53—41no—
279C2×D53—41no—
280C2×D53—42no—
281C4×C53—22no—
282C4×C53—22no—
283C4×C53—22no—
284C4×C53—22no—
285C4×C53—41no—
286C4×C53—41no—
287Dic53—22no—
288Dic53—22no—
289Dic53—22no—
290Dic53—22no—
291Dic53—41no—
292Dic53—41no—
293C2×#520—531no—
294#2973—532no—
295#1467—532no—
296#1465—532no—
297#1464—532no—
298#1463—532no—
299#1459—532no—
300C2×#324—531no—
301C2×#324D5311yescharacteristic, complemented
302C2×C53—21no—
303C2×C53—21no—
304C2×C53—22no—
305C2×C53—22no—
306C2×C53C2×#40711yescharacteristic
307C2×C53#288311yescharacteristic
308C2×C53#288911yescharacteristic
309D53—42no—
310D53—42no—
311C22×#41—531no—
312#520—1064no—
313#520—1064no—
314C2×C4×Q8—531no—
315C2×C4×Q8—531no—
316C2×C4×D4—531no—
317C2×C4×D4—531no—
318C2×C4×D4—531no—
319C2×C4×D4—531no—
320C2×C4×D4—531no—
321C2×C4×D4—531no—
322C2×#117—531no—
323C2×#117—531no—
324C2×#117—531no—
325C22×C4⋊C4—531no—
326C22×C4⋊C4—531no—
327C22×C4⋊C4C2×D5311yescharacteristic
328#435—1064no—
329#435—1064no—
330#432—534no—
331#407—534no—
332C2×C22⋊C8—532no—
333C2×C22⋊C8C2×D5312yes—
334#324—532no—
335#324—532no—
336#324—534no—
337#324C2×D5312yes—
338#324C2×D5312yes—
339#323—534no—
340C53#1736811yescharacteristic, complemented, Sylow
341C2×#41—1062no—
342C2×#41—1062no—
343C2×#41—1064no—
344C2×#41—1064no—
345C22×Q8—531no—
346C22×D4—531no—
347C22×D4—531no—
348C22×D4—531no—
349C23×C4—531no—
350C23×C4—531no—
351C23×C4—531no—
352C23×C4—531no—
353C23×C4—531no—
354C23×C4—531no—
355C23×C4C4×D5311yescharacteristic
356C23×C4C4×D5311yescharacteristic
357C23×C4C22×D5311yescharacteristic
358C2×M16—1064no—
359C22×C8—22no—
360C22×C8—1062no—
361C4×Q8—1064no—
362C4×Q8—1064no—
363C4×D4—534no—
364C4×D4—534no—
365C4×D4—1064no—
366C4×D4—1062no—
367C4×D4—1062no—
368C4×D4—1064no—
369C4×D4—1062no—
370C4×D4—1062no—
371C4×D4—1062no—
372C4×D4—1062no—
373#117—1064no—
374#117—1064no—
375#117—1062no—
376#117—1062no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—1064no—
386C2×C4⋊C4—1064no—
387C2×C4⋊C4C4×D5311yescharacteristic
388C2×C4⋊C4C4×D5311yescharacteristic
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C22⋊C4—531no—
392C2×C22⋊C4—531no—
393C2×C22⋊C4—531no—
394C2×C22⋊C4—531no—
395C2×C22⋊C4—531no—
396C2×C22⋊C4—531no—
397C2×C42—531no—
398C2×C42—531no—
399C2×C42—531no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—531no—
403C4⋊C8—1064no—
404C22⋊C8—22no—
405C22⋊C8—534no—
406C22⋊C8—1064no—
407C22⋊C8—1062no—
408C22⋊C8C22×D5314yes—
409#97—1064no—
410C24—531no—
411C24—531no—
412C24C2×C4×D5311yescharacteristic
413#41—10616no—
414#41—10616no—
415C2×Q8—1062no—
416C2×Q8—1062no—
417C2×Q8—1062no—
418C2×D4—532no—
419C2×D4—532no—
420C2×D4—534no—
421C2×D4—1062no—
422C2×D4—1062no—
423C2×D4—1061no—
424C2×D4—1061no—
425C2×D4—1062no—
426C2×D4—1062no—
427C2×D4—1062no—
428C2×D4—1062no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4—531no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—534no—
441C22×C4—531no—
442C22×C4—531no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—4241no—
453C22×C4—4241no—
454C22×C4—4241no—
455C22×C4—4241no—
456C22×C4#1171991214yescomplemented
457C22×C4C2×C4×D5311yescharacteristic
458C22×C4C2×C4×D5311yescharacteristic
459C22×C4C2×C4×D5311yescharacteristic
460C22×C4C2×C4×D5311yescharacteristic
461C22×C4C2×C4×D5311yescharacteristic
462C22×C4D4×D5312yes—
463C22×C4D4×D5312yes—
464C22×C4C23×D5311yescharacteristic
465M16—4244no—
466C2×C8—24no—
467C2×C8—82no—
468C2×C8—1064no—
469C2×C8—4242no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—534no—
475C4⋊C4—1062no—
476C4⋊C4—1062no—
477C4⋊C4—1062no—
478C4⋊C4—1062no—
479C4⋊C4—1062no—
480C22⋊C4—532no—
481C22⋊C4—532no—
482C22⋊C4—534no—
483C22⋊C4—1062no—
484C22⋊C4—1062no—
485C22⋊C4—1061no—
486C22⋊C4—1061no—
487C22⋊C4—1062no—
488C42—534no—
489C42—1062no—
490C42—1062no—
491C42—1062no—
492C42—1062no—
493C42—1062no—
494C23—41no—
495C23—41no—
496C23—532no—
497C23—532no—
498C23—1061no—
499C23—1061no—
500C23—1061no—
501C23—1061no—
502C23—2121no—
503C23—2121no—
504C23—2121no—
505C23—2121no—
506C23C22⋊C4×D5311yescharacteristic
507C23C22⋊C4×D5311yescharacteristic
508C23M16×D5312yes—
509C23C2×#1171991212yes—
510C23C22×C4×D5311yescharacteristic
511Q8—1068no—
512D4—1068no—
513D4—1064no—
514D4—1064no—
515D4—1064no—
516D4—1064no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—532no—
534C2×C4—532no—
535C2×C4—532no—
536C2×C4—532no—
537C2×C4—1068no—
538C2×C4—1068no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—4242no—
546C2×C4—4242no—
547C2×C4—4242no—
548C2×C4—2124no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4C22⋊C4×D5312yes—
554C2×C4C2×#1171991212yes—
555C2×C4C2×#1171991212yes—
556C2×C4C2×D4×D5312yes—
557C8—44no—
558C8—2124no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—42no—
566C22—42no—
567C22—1062no—
568C22—1062no—
569C22—1062no—
570C22—1062no—
571C22—2122no—
572C22—2122no—
573C22—2122no—
574C22—2122no—
575C22—2121no—
576C22—2121no—
577C22#1681058512yes—
578C22#1681058512yes—
579C22C2×C22⋊C4×D5311yescharacteristic
580C22C2×M16×D5311yescharacteristic
581C22C22×#1171991211yescharacteristic
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—41no—
587C4—41no—
588C4—1062no—
589C4—1062no—
590C4—1062no—
591C4—1062no—
592C4—2121no—
593C4—2121no—
594C2—21no—
595C2—21no—
596C2—22no—
597C2—22no—
598C2—2122no—
599C2—2122no—
600C2C2×#1681058511yescharacteristic
601C2#2028317311yescharacteristic
602C2#2028319511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#324C2
socleC22×C53C22×C22⋊C4
prime radicalC22×#11719945C4
cosocleC22×C4×C53C24
hypercenterC2×#324D53
hypocenterC53#17368
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #20283741C2semidirect
3. #20273057C2semidirect
4. #20273052C2semidirect
5. #20273048C2semidirect
6. #20273043C2semidirect
7. #20273031C2semidirect
8. C2×#16810349C2semidirect
9. C2×C53×#324C2semidirect
118. C22×C4×C53M16semidirect
134. C2×C22⋊D53M16semidirect
301. C2×#324D53semidirect
340. C53#17368semidirect
456. C22×C4#11719912semidirect

Maximal subgroups

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

subgroupindex
C2×#168111042
#202837412
#202730572
#202730522
#202730482
#202730432
#202730312
C2×#168103492
C2×C53×#3242
#1736853