Finite Group Databasebeta

← #27441206

Subgroups — #27441206

Order 13568.

994 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810900C211yesmaximal, characteristic, complemented
2C2×#16810617C211yesmaximal, characteristic
3#20277546C211yesmaximal, characteristic
4#20276266C211yesmaximal, characteristic, complemented
5#20276107C211yesmaximal, characteristic, complemented
6#20276107C211yesmaximal, characteristic, complemented
7#20275853C211yesmaximal, characteristic, complemented
8#20274975C211yesmaximal, characteristic, complemented
9#20274961C211yesmaximal, characteristic, complemented
10#20274914C211yesmaximal, characteristic, complemented
11#20274895C211yesmaximal, characteristic, complemented
12#20274742C211yesmaximal, characteristic, complemented
13#20274722C211yesmaximal, characteristic, complemented
14#20274653C211yesmaximal, characteristic
15C53×#2439C211yesmaximal, characteristic, complemented
16C22×#11719945C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×C4×D212C2211yescharacteristic
20#16810900—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27#16810617—24no—
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×#13133137C2211yescharacteristic
33C2×#13133137C2211yescharacteristic
34C2×#13133137C2211yescharacteristic
35C2×#13133137C2211yescharacteristic
36C2×C22⋊#88322C2211yescharacteristic
37#16810529C2211yescharacteristic
38#16810529C2211yescharacteristic
39#16810499—21no—
40#16810499—21no—
41#16810485—21no—
42#16810483C2211yescharacteristic
43#95×D53—21no—
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C4⋊Q8×C53C2211yescharacteristic
51C2×#117×C53C2211yescharacteristic
52C53×#396—21no—
53C53×#393—21no—
54C53×#391—21no—
55C53×#389C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#387—21no—
58C53×#382C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C4×C22⋊C4×C53C2211yescharacteristic
61C2×#11719945—22no—
62C2×#11719945—22no—
63C2×#11719945—22no—
64C2×#11719945—41no—
65C2×#11719945—41no—
66C2×#11719945Q814yescomplemented
67C22×Dic106C2311yescharacteristic
68C22×D212C2311yescharacteristic
69C22×C22⋊D53—21no—
70C22×C4×D53—21no—
71C22×C4×D53—21no—
72C22×C4×D53—21no—
73C22×#88322—22no—
74C22×#88322—22no—
75C22×#88322—41no—
76C22×#88322C2311yescharacteristic
77C22×#88322C2311yescharacteristic
78C22×#88322C2311yescharacteristic
79C22×#88322C2311yescharacteristic
80C4×D212—22no—
81C4×D212—22no—
82C4×D212—41no—
83C4×D212—41no—
84C2×C4⋊Dic53C2311yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×C22⋊Dic53—21no—
87C2×#11719921—21no—
88C2×#11719921—21no—
89#13133150—22no—
90#13133150—22no—
91#13133150—41no—
92#13133148—41no—
93C4⋊#88322—22no—
94C4⋊#88322—41no—
95#13133140—22no—
96#13133140—22no—
97#13133140—41no—
98#13133140—41no—
99#13133137—22no—
100#13133137—22no—
101#13133137—22no—
102#13133137D414yes—
103C22⋊#88322—22no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109C22×Q8×C53—21no—
110C23×C4×C53C2311yescharacteristic
111C4⋊Q8×C53—22no—
112C4⋊Q8×C53—41no—
113#117×C53—22no—
114#117×C53—41no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C22⋊C4×C53—21no—
123C2×C22⋊C4×C53—21no—
124C2×C22⋊C4×C53D411yescharacteristic
125C2×C22⋊C4×C53D411yescharacteristic
126C2×C42×C53—21no—
127C2×C42×C53C2311yescharacteristic
128C2×C42×C53C2311yescharacteristic
129#95×C53—21no—
130#95×C53—21no—
131#95×C53—21no—
132#95×C53—21no—
133#95×C53—21no—
134#95×C53D411yescharacteristic
135#95×C53D411yescharacteristic
136C24×C53C2×D411yescharacteristic
137C23×D53—21no—
138C2×Q8×C53—42no—
139C2×Q8×C53—42no—
140C2×Q8×C53—81no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×Q812yes—
155C22×C4×C53#4112yes—
156C22×C4×C53C2411yescharacteristic
157C2×C22⋊D53—24no—
158C2×C22⋊D53—42no—
159C2×C22⋊D53—42no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—21no—
163#11719945—28no—
164#11719945—44no—
165#11719945—44no—
166#11719945—44no—
167C2×D212—21no—
168C2×D212—21no—
169C2×D212—21no—
170C2×D212C2×D412yes—
171C2×D212C2×Q812yes—
172C2×D212#4112yes—
173C2×C4×D53—24no—
174C2×C4×D53—42no—
175C2×C4×D53—42no—
176C2×C4×D53—42no—
177C2×C4×D53—42no—
178C2×C4×D53—42no—
179C2×C4×D53—42no—
180C2×C4×D53—81no—
181C2×C4×D53—81no—
182C2×Dic106—21no—
183C2×Dic106—21no—
184C2×Dic106—21no—
185C2×Dic106—21no—
186C2×Dic106—21no—
187C2×Dic106C2×Q812yes—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—21no—
190C4⋊C4×C53—21no—
191C4⋊C4×C53—21no—
192C4⋊C4×C53—21no—
193C4⋊C4×C53—21no—
194C4⋊C4×C53—21no—
195C4⋊C4×C53—42no—
196C4⋊C4×C53—42no—
197C4⋊C4×C53—81no—
198C4⋊C4×C53C2×D412yes—
199C22⋊C4×C53—22no—
200C22⋊C4×C53—41no—
201C22⋊C4×C53—42no—
202C22⋊C4×C53—81no—
203C42×C53—21no—
204C42×C53—21no—
205C42×C53—21no—
206C42×C53—21no—
207C42×C53—42no—
208C22⋊Dic53—81no—
209#11719921—42no—
210#11719921—42no—
211C4⋊Dic53—21no—
212C4⋊Dic53—21no—
213C4⋊Dic53—21no—
214C4⋊Dic53—21no—
215C2×#88322—21no—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322—21no—
223C2×#88322—21no—
224C2×#88322—21no—
225C2×#88322—21no—
226C2×#88322—44no—
227C2×#88322C2×D412yes—
228C23×C53—21no—
229C23×C53—21no—
230C23×C53—21no—
231C23×C53—21no—
232C23×C53—21no—
233C23×C53—21no—
234C23×C53C22⋊Q812yes—
235C23×C53C22×D411yescharacteristic
236C22×D53—22no—
237C22×D53—22no—
238C22×D53—22no—
239C22×D53—41no—
240C22×D53—41no—
241C22×D53—41no—
242C22×D53—41no—
243Q8×C53—84no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—22no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—22no—
253C2×C4×C53—21no—
254C2×C4×C53—21no—
255C2×C4×C53—21no—
256C2×C4×C53—21no—
257C2×C4×C53—21no—
258C2×C4×C53—21no—
259C2×C4×C53—21no—
260C2×C4×C53—21no—
261C2×C4×C53—21no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53—41no—
265C2×C4×C53—41no—
266C2×C4×C53—41no—
267C2×C4×C53—41no—
268C2×C4×C53—41no—
269C2×C4×C53—41no—
270C2×C4×C53—41no—
271C2×C4×C53—41no—
272C2×C4×C53—41no—
273C2×C4×C53C22⋊Q812yes—
274C2×C4×C53#13512yes—
275C2×C4×C53C22×D411yescharacteristic
276C2×C4×C53C22×Q811yescharacteristic
277C2×C4×C53C2×#4111yescharacteristic
278C2×C4×C53C2×#4111yescharacteristic
279C2×C4×C53H(5,𝔽2)11yescharacteristic
280C2×C4×C53H(5,𝔽2)11yescharacteristic
281C22⋊D53—48no—
282C2×Dic53—22no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287C2×Dic53—41no—
288C2×Dic53—41no—
289C2×Dic53—41no—
290C2×Dic53—41no—
291D212—22no—
292D212—22no—
293D212—22no—
294D212C22⋊Q814yes—
295C4×D53—48no—
296C4×D53—84no—
297C4×D53—84no—
298Dic106—22no—
299Dic106—22no—
300Dic106—41no—
301Dic106—41no—
302#88322—22no—
303#88322—41no—
304#88322—41no—
305#88322—41no—
306#25954—531nomaximal, Sylow
307C22×C53—22no—
308C22×C53—22no—
309C22×C53—22no—
310C22×C53—41no—
311C22×C53—41no—
312C22×C53—41no—
313C22×C53—41no—
314C22×C53C2×C22⋊Q811yescharacteristic
315C22×C53D4211yescharacteristic
316C22×C53#54611yescharacteristic
317C22×C53C2×#13511yescharacteristic
318C22×C53#57511yescharacteristic
319C22×C53#57611yescharacteristic
320C22×C53#57811yescharacteristic
321C2×D53—24no—
322C2×D53—42no—
323C2×D53—42no—
324C2×D53—42no—
325C2×D53—42no—
326C2×D53—42no—
327C2×D53—42no—
328C4×C53—22no—
329C4×C53—22no—
330C4×C53—41no—
331C4×C53—42no—
332C4×C53—41no—
333C4×C53—41no—
334C4×C53—81no—
335C4×C53—81no—
336C4×C53—81no—
337C4×C53C2×C22⋊Q811yescharacteristic
338C4×C53#54311yescharacteristic
339C4×C53#54611yescharacteristic
340C4×C53#55111yescharacteristic
341Dic53—42no—
342Dic53—81no—
343Dic53—81no—
344C2×#474—531no—
345C2×#416—531no—
346#2439D5311yescharacteristic, complemented
347#2339—531no—
348#2097—531no—
349#2068—531no—
350#2068—531no—
351#2024—531no—
352#1864—531no—
353#1862—531no—
354#1854—531no—
355#1850—531no—
356#1808—531no—
357#1804—531no—
358#1788—531no—
359C2×C53—42no—
360C2×C53#243711yescharacteristic
361C2×C53#307611yescharacteristic
362C2×C53#308311yescharacteristic
363C2×C53#326911yescharacteristic
364C2×C53#328011yescharacteristic
365C2×C53#330211yescharacteristic
366C2×C53#331911yescharacteristic
367D53—44no—
368C22×#41—531no—
369C23×C8—531no—
370C2×C4⋊Q8C2×D5311yescharacteristic
371C2×C4×D4—531no—
372C2×C4×D4—531no—
373C2×#117C2×D5311yescharacteristic
374#474—1064no—
375C2×#107—531no—
376C2×#107—531no—
377C2×#107—531no—
378C2×#106—531no—
379C2×C4⋊C8—531no—
380C2×C4⋊C8—531no—
381#416—1064no—
382C2×#103—531no—
383C2×#103—531no—
384C2×#103—531no—
385C2×#103—531no—
386C2×#102—531no—
387C2×#102—531no—
388C2×#102—531no—
389C2×#102—531no—
390C2×C22⋊C8—531no—
391#396—21no—
392#393—21no—
393#391—21no—
394#391—531no—
395#391—531no—
396#389C2×D5311yescharacteristic
397#389C2×D5311yescharacteristic
398#387—21no—
399#386—1061no—
400#386—1061no—
401#382C2×D5311yescharacteristic
402#381—1061no—
403#380—531no—
404#380C2×D5311yescharacteristic
405C4×C22⋊C4C2×D5311yescharacteristic
406C2×#95—1061no—
407#340—531no—
408#340—531no—
409#340—531no—
410#340—531no—
411#336—531no—
412#336—531no—
413C53#2595411yescharacteristic, complemented, Sylow
414C2×#41—534no—
415C2×#41—1062no—
416C2×#41—1062no—
417C2×#41—1062no—
418C2×#41—2121no—
419C2×#41—2121no—
420C22×Q8—21no—
421C22×Q8—531no—
422C22×D4—531no—
423C22×D4—1061no—
424C23×C4—1061no—
425C23×C4—1061no—
426C23×C4—1061no—
427C23×C4C22×D5311yescharacteristic
428C22×C8—531no—
429C22×C8—531no—
430C22×C8—531no—
431C22×C8—531no—
432C22×C8—1062no—
433C22×C8—1062no—
434C22×C8—2121no—
435C4⋊Q8—22no—
436C4⋊Q8—41no—
437C4×D4—1062no—
438C4×D4—1062no—
439C4×D4—2121no—
440C4×D4—2121no—
441#117—22no—
442#117—41no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—21no—
446C2×C4⋊C4—531no—
447C2×C4⋊C4—531no—
448C2×C4⋊C4C22×D5311yescharacteristic
449C2×C4⋊C4C22×D5311yescharacteristic
450C2×C4⋊C4C22×D5311yescharacteristic
451C2×C4⋊C4C22×D5311yescharacteristic
452C2×C22⋊C4—21no—
453C2×C22⋊C4—21no—
454C2×C22⋊C4—1061no—
455C2×C22⋊C4—1061no—
456C2×C22⋊C4—1061no—
457C2×C22⋊C4C22⋊D5311yescharacteristic
458C2×C22⋊C4C22⋊D5311yescharacteristic
459C2×C42—21no—
460C2×C42C22×D5311yescharacteristic
461C2×C42C22×D5311yescharacteristic
462#107—1062no—
463#107—1062no—
464#107—2121no—
465#106—2121no—
466C4⋊C8—1062no—
467C4⋊C8—2121no—
468#103—1062no—
469#103—1062no—
470#103—2121no—
471#103—2121no—
472#102—534no—
473#102—1062no—
474#102—1062no—
475#102—1062no—
476C22⋊C8—1062no—
477#95—21no—
478#95—21no—
479#95—21no—
480#95—21no—
481#95—21no—
482#95—1061no—
483#95—1061no—
484#95—1061no—
485#95—1061no—
486#95—1061no—
487#95C22⋊D5311yescharacteristic
488#95C22⋊D5311yescharacteristic
489C24—1061no—
490C24C2×C22⋊D5311yescharacteristic
491#41—1068no—
492#41—2124no—
493#41—2124no—
494#41—2124no—
495C2×Q8—42no—
496C2×Q8—42no—
497C2×Q8—81no—
498C2×Q8—532no—
499C2×Q8—1061no—
500C2×Q8—1061no—
501C2×Q8—1061no—
502C2×Q8—1061no—
503C2×Q8—1061no—
504C2×D4—532no—
505C2×D4—532no—
506C2×D4—532no—
507C2×D4—1064no—
508C2×D4—1061no—
509C2×D4—1061no—
510C2×D4—1061no—
511C2×D4—2122no—
512C2×D4—2122no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—1061no—
522C22×C4—1064no—
523C22×C4—1061no—
524C22×C4—1061no—
525C22×C4—4241no—
526C22×C4—4241no—
527C22×C4—2122no—
528C22×C4—2122no—
529C22×C4—2122no—
530C22×C4—2122no—
531C22×C4—2122no—
532C22×C4—2122no—
533C22×C4Q8×D5312yes—
534C22×C4#1171994912yes—
535C22×C4C2×C22⋊D5311yescharacteristic
536C22×C4C2×C22⋊D5311yescharacteristic
537C22×C4C2×C22⋊D5311yescharacteristic
538C22×C4C2×C22⋊D5311yescharacteristic
539C22×C4C2×C22⋊D5311yescharacteristic
540C22×C4C23×D5311yescharacteristic
541C2×C8—532no—
542C2×C8—1061no—
543C2×C8—1061no—
544C2×C8—1061no—
545C2×C8—1061no—
546C2×C8—1061no—
547C2×C8—1061no—
548C2×C8—1061no—
549C2×C8—1061no—
550C2×C8—1061no—
551C2×C8—1061no—
552C2×C8—1061no—
553C2×C8—2124no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—81no—
564C4⋊C4—1061no—
565C4⋊C4—1061no—
566C4⋊C4—1061no—
567C4⋊C4—1061no—
568C4⋊C4D4×D5312yes—
569C22⋊C4—22no—
570C22⋊C4—41no—
571C22⋊C4—42no—
572C22⋊C4—81no—
573C22⋊C4—4241no—
574C22⋊C4—2122no—
575C22⋊C4—2122no—
576C42—21no—
577C42—21no—
578C42—21no—
579C42—21no—
580C42—42no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—1062no—
588C23—1062no—
589C23—1062no—
590C23—2121no—
591C23—2121no—
592C23—2121no—
593C23—2121no—
594C23#1313319312yes—
595C23C22×C22⋊D5311yescharacteristic
596Q8—84no—
597Q8—1062no—
598Q8—1062no—
599Q8—2121no—
600Q8—2121no—
601D4—534no—
602D4—1062no—
603D4—1062no—
604D4—1062no—
605D4—2128no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—1062no—
636C2×C4—4244no—
637C2×C4—4244no—
638C2×C4—2128no—
639C2×C4—2121no—
640C2×C4—2121no—
641C2×C4—2121no—
642C2×C4—2121no—
643C2×C4—2121no—
644C2×C4—2121no—
645C2×C4—2121no—
646C2×C4—2121no—
647C2×C4#1313319312yes—
648C2×C4#1313323712yes—
649C2×C4C2×D4×D5311yescharacteristic
650C2×C4C2×Q8×D5311yescharacteristic
651C2×C4#41×D5311yescharacteristic
652C2×C4C2×#1171994911yescharacteristic
653C2×C4#1313326211yescharacteristic
654C2×C4#1313326311yescharacteristic
655C8—1062no—
656C8—2121no—
657C8—2121no—
658C8—2121no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—1064no—
667C22—2122no—
668C22—2122no—
669C22—2122no—
670C22—2122no—
671C22—2122no—
672C22—2122no—
673C22C2×#1313319311yescharacteristic
674C22D4×C22⋊D5311yescharacteristic
675C22#1681126111yescharacteristic
676C22C2×#1313323711yescharacteristic
677C22#1681142111yescharacteristic
678C22#1681142611yescharacteristic
679C22#1681143811yescharacteristic
680C4—22no—
681C4—22no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—42no—
686C4—81no—
687C4—81no—
688C4—81no—
689C4—4241no—
690C4—4241no—
691C4—2122no—
692C4C22⋊Q8×D5311yescharacteristic
693C4#1681124811yescharacteristic
694C4#1681126411yescharacteristic
695C4#1681130011yescharacteristic
696C2—42no—
697C2—2124no—
698C2#2027837811yescharacteristic
699C2#2028449711yescharacteristic
700C2#2028456211yescharacteristic
701C2#2028627111yescharacteristic
702C2#2028639611yescharacteristic
703C2#2028662611yescharacteristic
704C2#2028686511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2439C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
hypercenter#2439D53
hypocenterC53#25954
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2439

Complements

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

NH ≅ G/Ntype
1. C2×#16810900C2semidirect
4. #20276266C2semidirect
5. #20276107C2semidirect
6. #20276107C2semidirect
7. #20275853C2semidirect
8. #20274975C2semidirect
9. #20274961C2semidirect
10. #20274914C2semidirect
11. #20274895C2semidirect
12. #20274742C2semidirect
13. #20274722C2semidirect
15. C53×#2439C2semidirect
66. C2×#11719945Q8semidirect
346. #2439D53semidirect
413. C53#25954semidirect

Maximal subgroups

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

subgroupindex
C2×#168109002
C2×#168106172
#202775462
#202762662
#202761072
#202758532
#202749752
#202749612
#202749142
#202748952
#202747422
#202747222
#202746532
C53×#24392
#2595453