Finite Group Databasebeta

← #27447896

Subgroups — #27447896

Order 13568.

840 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278417C211yesmaximal, characteristic, complemented
2#20277751C211yesmaximal, characteristic
3#20276279C211yesmaximal, characteristic, complemented
4#20276208C211yesmaximal, characteristic, complemented
5#20276102C211yesmaximal, characteristic, complemented
6#20275852C211yesmaximal, characteristic, complemented
7#20275666C211yesmaximal, characteristic, complemented
8#20275294C211yesmaximal, characteristic, complemented
9#20275153C211yesmaximal, characteristic
10#20275030C211yesmaximal, characteristic
11#20274979C211yesmaximal, characteristic, complemented
12#20274916C211yesmaximal, characteristic, complemented
13#20274716C211yesmaximal, characteristic, complemented
14#20274435C211yesmaximal, characteristic, complemented
15C53×#1851C211yesmaximal, characteristic, complemented
16C2×#13133217C2211yescharacteristic
17C2×#13133194—21no—
18C2×#13133191C2211yescharacteristic
19C2×C4×D212C2211yescharacteristic
20C2×#13133176C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28#16810555—21no—
29#16810544—21no—
30#16810525C2211yescharacteristic
31#16810523—21no—
32#16810521C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810512—21no—
36#16810499—21no—
37#16810498—21no—
38#16810493C2211yescharacteristic
39#16810483C2211yescharacteristic
40C4×C4⋊Dic53C2211yescharacteristic
41#16810470—21no—
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C22×M16×C53C2211yescharacteristic
53C2×C4⋊C8×C53C2211yescharacteristic
54C2×C4⋊C8×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C4×C22⋊C4×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106—21no—
60C22×D212C2311yescharacteristic
61C22×C22⋊D53—21no—
62C23×Dic53—21no—
63C22×C4×D53—21no—
64#13133217—22no—
65#13133217—41no—
66#13133194—44no—
67#13133191—22no—
68#13133191Q814yescomplemented
69C4×D212—22no—
70C4×D212—41no—
71#13133176—22no—
72#13133176—41no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—22no—
78C2×C4⋊Dic53—41no—
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—21no—
89C2×C22⋊Dic53—21no—
90C2×C22⋊Dic53—21no—
91C2×#11719921—21no—
92C2×#11719921—21no—
93C2×#11719921—21no—
94C2×#11719921—21no—
95C2×C4×Dic53—21no—
96C2×C4×Dic53—21no—
97#13133151—41no—
98#13133149—22no—
99#13133139—22no—
100#13133139—41no—
101#13133139—41no—
102#13133139—41no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108C23×C4×C53C2311yescharacteristic
109C2×M16×C53—22no—
110C2×M16×C53—41no—
111C2×M16×C53—41no—
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C4⋊C8×C53—41no—
121C4⋊C8×C53—41no—
122C22⋊C8×C53—22no—
123#95×C53D411yescharacteristic
124#95×C53D411yescharacteristic
125C24×C53C2×D411yescharacteristic
126C23×D53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×Q812yes—
137C22×C4×C53#4112yes—
138C22×C4×C53C2411yescharacteristic
139C2×C22⋊D53—24no—
140C2×C22⋊D53—81no—
141C2×C22⋊D53—81no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—42no—
150C22×Dic53—42no—
151C22×Dic53—81no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212C2×Q812yes—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C2×C4×D53—81no—
161C2×Dic106—42no—
162C2×Dic106—42no—
163C2×Dic106—81no—
164M16×C53—44no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C22⋊C4×C53—22no—
178C22⋊C4×C53—22no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—21no—
183C22⋊Dic53—42no—
184C22⋊Dic53—42no—
185#11719921—24no—
186#11719921—42no—
187#11719921—42no—
188#11719921—81no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—42no—
201C4⋊Dic53—42no—
202C4⋊Dic53—42no—
203C4⋊Dic53—81no—
204C4⋊Dic53C2×Q812yes—
205C4⋊Dic53#4112yes—
206C4⋊Dic53#4112yes—
207#11719919—42no—
208#11719919—42no—
209#11719919—81no—
210C4×Dic53—42no—
211C4×Dic53—81no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53C22⋊Q812yes—
219C23×C53C22×D411yescharacteristic
220C22×D53—22no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53C22⋊Q812yes—
244C2×C4×C53AGL(1,ℤ/8)12yes—
245C2×C4×C53#13712yes—
246C2×C4×C53C22×Q811yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53H(5,𝔽2)11yescharacteristic
251C2×C4×C53H(5,𝔽2)11yescharacteristic
252C22⋊D53—84no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—84no—
275D212—22no—
276D212—22no—
277D212—41no—
278D212—41no—
279C4×D53—84no—
280Dic106—84no—
281C8×C53—41no—
282C8×C53—41no—
283C8×C53—41no—
284C8×C53—41no—
285#26425—531nomaximal, Sylow
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53C2×C22⋊Q811yescharacteristic
294C22×C53#54611yescharacteristic
295C22×C53#54611yescharacteristic
296C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C53C2×#13711yescharacteristic
298C22×C53#57611yescharacteristic
299C22×C53#57711yescharacteristic
300C2×D53—42no—
301C2×D53—42no—
302C2×D53—42no—
303C2×D53—81no—
304C2×D53—81no—
305C2×D53—81no—
306C2×D53—81no—
307C4×C53—22no—
308C4×C53—22no—
309C4×C53—22no—
310C4×C53—41no—
311C4×C53—41no—
312C4×C53#52911yescharacteristic
313C4×C53#53811yescharacteristic
314C4×C53#55411yescharacteristic
315C4×C53#55411yescharacteristic
316Dic53—42no—
317Dic53—42no—
318Dic53—42no—
319Dic53—81no—
320Dic53—81no—
321Dic53—81no—
322Dic53—81no—
323#2440—531no—
324#2363—531no—
325#2100—531no—
326#2087—531no—
327#2067—531no—
328#2023—531no—
329#1988—531no—
330#1919—531no—
331#1895—531no—
332#1874—531no—
333#1864—531no—
334#1854—531no—
335#1851D5311yescharacteristic, complemented
336#1802—531no—
337#1736—531no—
338C2×C53—42no—
339C2×C53#245011yescharacteristic
340C2×C53#307911yescharacteristic
341C2×C53#308111yescharacteristic
342C2×C53#331411yescharacteristic
343C2×C53#331811yescharacteristic
344C2×C53#332911yescharacteristic
345C2×C53#333211yescharacteristic
346D53—82no—
347C22×M16C2×D5311yescharacteristic
348C2×C4⋊Q8—531no—
349C2×C22⋊Q8—1061no—
350C2×#121—531no—
351C2×C4×D4—531no—
352C2×#117—531no—
353C22×C4⋊C4—531no—
354C2×#107—531no—
355C2×#106—531no—
356C2×C4⋊C8C2×D5311yescharacteristic
357C2×C4⋊C8C2×D5311yescharacteristic
358C2×#102—531no—
359C2×#102—531no—
360C2×#102—531no—
361C2×#102—531no—
362C2×C22⋊C8C2×D5311yescharacteristic
363#398—1061no—
364#395—1061no—
365#390—531no—
366#390—1061no—
367#389—531no—
368#389—531no—
369#389—531no—
370#388—1061no—
371#386—1061no—
372#385—1061no—
373#384—531no—
374#380—531no—
375C4×C4⋊C4—531no—
376C4×C22⋊C4—1061no—
377C4×C22⋊C4C2×D5311yescharacteristic
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#336C2×D5311yescharacteristic
389#336C2×D5311yescharacteristic
390C53#2642511yescharacteristic, complemented, Sylow
391C22×Q8—1061no—
392C22×D4—531no—
393C22×D4—1061no—
394C23×C4—1061no—
395C23×C4—1061no—
396C23×C4C22×D5311yescharacteristic
397C2×M16—22no—
398C2×M16—41no—
399C2×M16—41no—
400C22×C8C22×D5311yescharacteristic
401C22×C8C22×D5311yescharacteristic
402C22×C8C22×D5311yescharacteristic
403C22×C8C22×D5311yescharacteristic
404C4⋊Q8—1062no—
405C4⋊Q8—2121no—
406C22⋊Q8—2124no—
407#121—534no—
408#121—1062no—
409C4×D4—1062no—
410C4×D4—2121no—
411#117—1062no—
412#117—2121no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—1062no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—2121no—
429C2×C22⋊C4—1061no—
430C2×C22⋊C4—1061no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4—1061no—
435C2×C22⋊C4D21211yescharacteristic
436C2×C22⋊C4D21211yescharacteristic
437C2×C42—1061no—
438C2×C42—1061no—
439C2×C42C22×D5311yescharacteristic
440C2×C42C22×D5311yescharacteristic
441#107—2121no—
442#106—1062no—
443C4⋊C8—41no—
444C4⋊C8—41no—
445#102—1062no—
446#102—2121no—
447#102—2121no—
448#102—2121no—
449C22⋊C8—22no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95D21211yescharacteristic
456#95D21211yescharacteristic
457C24—1061no—
458C24C2×D21211yescharacteristic
459C2×Q8—4241no—
460C2×Q8—2122no—
461C2×Q8—2122no—
462C2×D4—532no—
463C2×D4—1064no—
464C2×D4—1061no—
465C2×D4—1061no—
466C2×D4—1061no—
467C2×D4—1061no—
468C2×D4—1061no—
469C2×D4—4241no—
470C2×D4—4241no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—4241no—
483C22×C4—4241no—
484C22×C4—2122no—
485C22×C4—2122no—
486C22×C4—2122no—
487C22×C4—2122no—
488C22×C4C2×D21211yescharacteristic
489C22×C4C2×D21211yescharacteristic
490C22×C4C2×D21211yescharacteristic
491C22×C4C2×D21211yescharacteristic
492C22×C4C2×D21211yescharacteristic
493C22×C4#1171994712yes—
494C22×C4Q8×D5312yes—
495C22×C4C23×D5311yescharacteristic
496M16—44no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C4⋊C4—532no—
510C4⋊C4—532no—
511C4⋊C4—532no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—4241no—
524C4⋊C4—4241no—
525C4⋊C4—2122no—
526C4⋊C4—2122no—
527C4⋊C4—2122no—
528C4⋊C4—2122no—
529C4⋊C4—2122no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—1064no—
533C22⋊C4—4241no—
534C22⋊C4—2122no—
535C22⋊C4—2122no—
536C22⋊C4—2122no—
537C22⋊C4—2122no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—4241no—
543C42—2122no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—1062no—
551C23—2121no—
552C23—2121no—
553C23—2121no—
554C23—2121no—
555C23—2121no—
556C23—2121no—
557C23#1313319712yes—
558C23C22×D21211yescharacteristic
559Q8—4244no—
560D4—1062no—
561D4—1062no—
562D4—4244no—
563D4—2121no—
564D4—2121no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—1062no—
582C2×C4—1062no—
583C2×C4—1062no—
584C2×C4—4244no—
585C2×C4—4244no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4#1313319712yes—
605C2×C4#1313324512yes—
606C2×C4#1313325012yes—
607C2×C4C2×Q8×D5311yescharacteristic
608C2×C4#41×D5311yescharacteristic
609C2×C4#41×D5311yescharacteristic
610C2×C4C2×#1171994711yescharacteristic
611C2×C4#1313326211yescharacteristic
612C2×C4#1313326211yescharacteristic
613C8—41no—
614C8—41no—
615C8—41no—
616C8—41no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—4241no—
625C22—4241no—
626C22—4241no—
627C22—4241no—
628C22—2122no—
629C22—2122no—
630C22—2122no—
631C22C2×#1313319711yescharacteristic
632C22#1681126811yescharacteristic
633C22#1681126811yescharacteristic
634C22C2×#1313324511yescharacteristic
635C22C2×#1313325011yescharacteristic
636C22#1681142811yescharacteristic
637C22#1681143511yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—41no—
642C4—41no—
643C4—4241no—
644C4—4241no—
645C4—4241no—
646C4—4241no—
647C4—2122no—
648C4—2122no—
649C4—2122no—
650C4#1681115011yescharacteristic
651C4#1681120811yescharacteristic
652C4#1681132711yescharacteristic
653C4#1681132711yescharacteristic
654C2—42no—
655C2—4242no—
656C2#2027853711yescharacteristic
657C2#2028453111yescharacteristic
658C2#2028454711yescharacteristic
659C2#2028679811yescharacteristic
660C2#2028685811yescharacteristic
661C2#2028701011yescharacteristic
662C2#2028704311yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1851C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1851D53
hypocenterC53#26425
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
1. #20278417C2semidirect
3. #20276279C2semidirect
4. #20276208C2semidirect
5. #20276102C2semidirect
6. #20275852C2semidirect
7. #20275666C2semidirect
8. #20275294C2semidirect
11. #20274979C2semidirect
12. #20274916C2semidirect
13. #20274716C2semidirect
14. #20274435C2semidirect
15. C53×#1851C2semidirect
68. #13133191Q8semidirect
335. #1851D53semidirect
390. C53#26425semidirect

Maximal subgroups

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

subgroupindex
#202784172
#202777512
#202762792
#202762082
#202761022
#202758522
#202756662
#202752942
#202751532
#202750302
#202749792
#202749162
#202747162
#202744352
C53×#18512
#2642553