Finite Group Databasebeta

← #27441191

Subgroups — #27441191

Order 13568.

958 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810927C211yesmaximal, characteristic, complemented
2C2×#16810617C211yesmaximal, characteristic
3#20277555C211yesmaximal, characteristic
4#20276249C211yesmaximal, characteristic, complemented
5#20276100C211yesmaximal, characteristic, complemented
6#20276086C211yesmaximal, characteristic, complemented
7#20275858C211yesmaximal, characteristic, complemented
8#20274982C211yesmaximal, characteristic, complemented
9#20274982C211yesmaximal, characteristic, complemented
10#20274914C211yesmaximal, characteristic, complemented
11#20274895C211yesmaximal, characteristic, complemented
12#20274738C211yesmaximal, characteristic, complemented
13#20274717C211yesmaximal, characteristic, complemented
14#20274653C211yesmaximal, characteristic
15C53×#2441C211yesmaximal, characteristic, complemented
16C22×#11719945C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20#16810927—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
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#16810525C2211yescharacteristic
38#16810525C2211yescharacteristic
39#16810523—21no—
40#16810523—21no—
41#16810483C2211yescharacteristic
42#16810482—21no—
43#16810470—21no—
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#125×C53C2211yescharacteristic
51C2×#117×C53C2211yescharacteristic
52C53×#399—21no—
53C53×#396—21no—
54C53×#389—21no—
55C53×#389C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#387—21no—
58C53×#384C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C4×C22⋊C4×C53C2211yescharacteristic
61C2×#11719945—22no—
62C2×#11719945—22no—
63C2×#11719945—22no—
64C2×#11719945—41no—
65C2×#11719945—41no—
66C2×#11719945Q814yes—
67C22×Dic106C2311yescharacteristic
68C22×D212C2311yescharacteristic
69C22×C22⋊D53—21no—
70C22×C4×D53—21no—
71C22×#88322—22no—
72C22×#88322—22no—
73C22×#88322—41no—
74C22×#88322C2311yescharacteristic
75C22×#88322C2311yescharacteristic
76C22×#88322C2311yescharacteristic
77C22×#88322C2311yescharacteristic
78C4×Dic106—22no—
79C4×Dic106—22no—
80C4×Dic106—41no—
81C4×Dic106—41no—
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×C22⋊Dic53—21no—
87C2×#11719921—21no—
88C2×#11719921—21no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93#13133150—22no—
94#13133148—22no—
95#13133148—41no—
96#13133148—41no—
97C4⋊#88322—22no—
98C4⋊#88322—41no—
99#13133140—22no—
100#13133140—22no—
101#13133140—22no—
102#13133140D414yes—
103#13133137—22no—
104#13133137—22no—
105#13133137—41no—
106#13133137—41no—
107C22⋊#88322—22no—
108#13133124—21no—
109C23×C4×C53C2311yescharacteristic
110#125×C53—22no—
111#125×C53—41no—
112#117×C53—22no—
113#117×C53—41no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53—21no—
119C2×C4⋊C4×C53—21no—
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C22⋊C4×C53—21no—
125C2×C22⋊C4×C53—21no—
126C2×C22⋊C4×C53D411yescharacteristic
127C2×C22⋊C4×C53D411yescharacteristic
128C2×C42×C53—21no—
129C2×C42×C53C2311yescharacteristic
130C2×C42×C53C2311yescharacteristic
131#95×C53—21no—
132#95×C53—21no—
133#95×C53—21no—
134#95×C53D411yescharacteristic
135#95×C53D411yescharacteristic
136C24×C53C2×D411yescharacteristic
137C23×D53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×Q812yes—
152C22×C4×C53#4112yes—
153C22×C4×C53C2411yescharacteristic
154C2×C22⋊D53—24no—
155C2×C22⋊D53—42no—
156C2×C22⋊D53—42no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160#11719945—28no—
161#11719945—44no—
162#11719945—44no—
163#11719945—44no—
164C2×D212—21no—
165C2×D212—21no—
166C2×D212—21no—
167C2×D212—21no—
168C2×D212—21no—
169C2×D212C2×Q812yes—
170C2×C4×D53—24no—
171C2×C4×D53—42no—
172C2×C4×D53—42no—
173C2×Dic106—21no—
174C2×Dic106—21no—
175C2×Dic106—21no—
176C2×Dic106C2×D412yes—
177C2×Dic106C2×Q812yes—
178C2×Dic106#4112yes—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53—42no—
187C4⋊C4×C53—42no—
188C4⋊C4×C53—42no—
189C4⋊C4×C53—42no—
190C4⋊C4×C53—81no—
191C4⋊C4×C53—81no—
192C4⋊C4×C53C2×D412yes—
193C22⋊C4×C53—22no—
194C22⋊C4×C53—41no—
195C22⋊C4×C53—42no—
196C22⋊C4×C53—81no—
197C42×C53—21no—
198C42×C53—21no—
199C42×C53—21no—
200C42×C53—21no—
201C42×C53—42no—
202C22⋊Dic53—81no—
203#11719921—42no—
204#11719921—42no—
205#11719921—81no—
206#11719921—81no—
207C4⋊Dic53—21no—
208C4⋊Dic53—21no—
209C4⋊Dic53—21no—
210C4⋊Dic53—21no—
211#11719919—42no—
212#11719919—42no—
213C4×Dic53—42no—
214C4×Dic53—42no—
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—41no—
238C22×D53—41no—
239C22×D53—41no—
240C22×D53—41no—
241C22×D53—41no—
242C22×D53—41no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—22no—
248C2×C4×C53—22no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—21no—
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—41no—
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×C53C22⋊Q812yes—
273C2×C4×C53#13512yes—
274C2×C4×C53C22×D411yescharacteristic
275C2×C4×C53C22×Q811yescharacteristic
276C2×C4×C53C2×#4111yescharacteristic
277C2×C4×C53C2×#4111yescharacteristic
278C2×C4×C53H(5,𝔽2)11yescharacteristic
279C2×C4×C53H(5,𝔽2)11yescharacteristic
280C22⋊D53—48no—
281C2×Dic53—22no—
282C2×Dic53—22no—
283C2×Dic53—22no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287C2×Dic53—41no—
288C2×Dic53—41no—
289C2×Dic53—41no—
290D212—22no—
291D212—22no—
292D212—41no—
293D212—41no—
294C4×D53—48no—
295Dic106—22no—
296Dic106—22no—
297Dic106—22no—
298Dic106C22⋊Q814yes—
299#88322—22no—
300#88322—41no—
301#88322—41no—
302#88322—41no—
303#25953—531nomaximal, Sylow
304C22×C53—22no—
305C22×C53—22no—
306C22×C53—22no—
307C22×C53—41no—
308C22×C53—41no—
309C22×C53—41no—
310C22×C53—41no—
311C22×C53C2×C22⋊Q811yescharacteristic
312C22×C53D4211yescharacteristic
313C22×C53#54611yescharacteristic
314C22×C53C2×#13511yescharacteristic
315C22×C53#57511yescharacteristic
316C22×C53#57711yescharacteristic
317C22×C53#57711yescharacteristic
318C2×D53—42no—
319C2×D53—42no—
320C2×D53—42no—
321C2×D53—81no—
322C2×D53—81no—
323C2×D53—81no—
324C2×D53—81no—
325C4×C53—22no—
326C4×C53—22no—
327C4×C53—41no—
328C4×C53—41no—
329C4×C53—42no—
330C4×C53—41no—
331C4×C53—81no—
332C4×C53—81no—
333C4×C53—81no—
334C4×C53C2×C22⋊Q811yescharacteristic
335C4×C53#54311yescharacteristic
336C4×C53#54611yescharacteristic
337C4×C53#55111yescharacteristic
338Dic53—24no—
339Dic53—81no—
340Dic53—81no—
341C2×#479—531no—
342C2×#416—531no—
343#2441D5311yescharacteristic, complemented
344#2341—531no—
345#2094—531no—
346#2067—531no—
347#2065—531no—
348#2025—531no—
349#1865—531no—
350#1865—531no—
351#1854—531no—
352#1850—531no—
353#1807—531no—
354#1803—531no—
355#1788—531no—
356C2×C53—42no—
357C2×C53#243711yescharacteristic
358C2×C53#307611yescharacteristic
359C2×C53#308411yescharacteristic
360C2×C53#327111yescharacteristic
361C2×C53#328411yescharacteristic
362C2×C53#330111yescharacteristic
363C2×C53#331711yescharacteristic
364D53—82no—
365C22×#41—531no—
366C23×C8—531no—
367C2×#125C2×D5311yescharacteristic
368C2×C4×Q8—531no—
369C2×C4×Q8—531no—
370C2×#117C2×D5311yescharacteristic
371#479—1064no—
372C2×#107—531no—
373C2×#106—531no—
374C2×#106—531no—
375C2×#106—531no—
376C2×C4⋊C8—531no—
377C2×C4⋊C8—531no—
378#416—1064no—
379C2×#103—531no—
380C2×#103—531no—
381C2×#103—531no—
382C2×#103—531no—
383C2×#102—531no—
384C2×#102—531no—
385C2×#102—531no—
386C2×#102—531no—
387C2×C22⋊C8—531no—
388#399—21no—
389#396—21no—
390#390—531no—
391#390—531no—
392#390—1061no—
393#390—1061no—
394#389—21no—
395#389C2×D5311yescharacteristic
396#389C2×D5311yescharacteristic
397#387—21no—
398#384C2×D5311yescharacteristic
399#380—531no—
400#380—1061no—
401#380C2×D5311yescharacteristic
402C4×C22⋊C4—1061no—
403C4×C22⋊C4C2×D5311yescharacteristic
404#340—531no—
405#340—531no—
406#340—531no—
407#340—531no—
408#336—531no—
409#336—531no—
410C53#2595311yescharacteristic, complemented, Sylow
411C2×#41—534no—
412C2×#41—1062no—
413C2×#41—1062no—
414C2×#41—1062no—
415C2×#41—2121no—
416C2×#41—2121no—
417C22×Q8—531no—
418C22×D4—531no—
419C22×D4—1061no—
420C23×C4—1061no—
421C23×C4C22×D5311yescharacteristic
422C22×C8—531no—
423C22×C8—531no—
424C22×C8—531no—
425C22×C8—531no—
426C22×C8—1062no—
427C22×C8—1062no—
428C22×C8—2121no—
429#125—22no—
430#125—41no—
431C4×Q8—1062no—
432C4×Q8—1062no—
433C4×Q8—2121no—
434C4×Q8—2121no—
435#117—22no—
436#117—41no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—531no—
444C2×C4⋊C4—531no—
445C2×C4⋊C4—1061no—
446C2×C4⋊C4—1061no—
447C2×C4⋊C4C22×D5311yescharacteristic
448C2×C4⋊C4C22×D5311yescharacteristic
449C2×C4⋊C4C22×D5311yescharacteristic
450C2×C4⋊C4C22×D5311yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—1061no—
454C2×C22⋊C4—1061no—
455C2×C22⋊C4—1061no—
456C2×C22⋊C4—1061no—
457C2×C22⋊C4—1061no—
458C2×C22⋊C4C22⋊D5311yescharacteristic
459C2×C22⋊C4C22⋊D5311yescharacteristic
460C2×C42—21no—
461C2×C42—1061no—
462C2×C42—1061no—
463C2×C42C22×D5311yescharacteristic
464C2×C42C22×D5311yescharacteristic
465#107—1062no—
466#106—1062no—
467#106—2121no—
468#106—2121no—
469C4⋊C8—1062no—
470C4⋊C8—2121no—
471#103—534no—
472#103—1062no—
473#103—1062no—
474#103—1062no—
475#102—1062no—
476#102—1062no—
477#102—2121no—
478#102—2121no—
479C22⋊C8—1062no—
480#95—21no—
481#95—21no—
482#95—21no—
483#95—1061no—
484#95C22⋊D5311yescharacteristic
485#95C22⋊D5311yescharacteristic
486C24—1061no—
487C24C2×C22⋊D5311yescharacteristic
488#41—1068no—
489#41—2124no—
490#41—2124no—
491#41—2124no—
492C2×Q8—532no—
493C2×Q8—532no—
494C2×Q8—532no—
495C2×Q8—1061no—
496C2×Q8—1061no—
497C2×Q8—1061no—
498C2×D4—532no—
499C2×D4—1064no—
500C2×D4—1061no—
501C2×D4—1061no—
502C2×D4—1061no—
503C2×D4—1061no—
504C2×D4—1061no—
505C2×D4—2122no—
506C2×D4—2122no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—1061no—
516C22×C4—1064no—
517C22×C4—1061no—
518C22×C4—1061no—
519C22×C4—2122no—
520C22×C4—2122no—
521C22×C4Q8×D5312yes—
522C22×C4#1171994912yes—
523C22×C4C2×C22⋊D5311yescharacteristic
524C22×C4C2×C22⋊D5311yescharacteristic
525C22×C4C2×C22⋊D5311yescharacteristic
526C22×C4C2×C22⋊D5311yescharacteristic
527C22×C4C2×C22⋊D5311yescharacteristic
528C22×C4C23×D5311yescharacteristic
529C2×C8—532no—
530C2×C8—1061no—
531C2×C8—1061no—
532C2×C8—1061no—
533C2×C8—1061no—
534C2×C8—1061no—
535C2×C8—1061no—
536C2×C8—1061no—
537C2×C8—1061no—
538C2×C8—1061no—
539C2×C8—1061no—
540C2×C8—1061no—
541C2×C8—2124no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—81no—
554C4⋊C4—81no—
555C4⋊C4—1061no—
556C4⋊C4—1061no—
557C4⋊C4—1061no—
558C4⋊C4—1061no—
559C4⋊C4—2122no—
560C4⋊C4—2122no—
561C4⋊C4D4×D5312yes—
562C22⋊C4—22no—
563C22⋊C4—41no—
564C22⋊C4—42no—
565C22⋊C4—81no—
566C22⋊C4—4241no—
567C22⋊C4—4241no—
568C22⋊C4—4241no—
569C22⋊C4—2122no—
570C22⋊C4—2122no—
571C42—21no—
572C42—21no—
573C42—21no—
574C42—21no—
575C42—42no—
576C42—2122no—
577C42—2122no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—1062no—
585C23—2121no—
586C23—2121no—
587C23—2121no—
588C23—2121no—
589C23—2121no—
590C23—2121no—
591C23#1313319312yes—
592C23C22×C22⋊D5311yescharacteristic
593Q8—534no—
594Q8—1062no—
595Q8—1062no—
596Q8—1062no—
597D4—1062no—
598D4—1062no—
599D4—2128no—
600D4—2121no—
601D4—2121no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—1062no—
632C2×C4—1062no—
633C2×C4—1062no—
634C2×C4—2128no—
635C2×C4—2121no—
636C2×C4—2121no—
637C2×C4—2121no—
638C2×C4—2121no—
639C2×C4—2121no—
640C2×C4—2121no—
641C2×C4#1313319312yes—
642C2×C4#1313323712yes—
643C2×C4C2×D4×D5311yescharacteristic
644C2×C4C2×Q8×D5311yescharacteristic
645C2×C4#41×D5311yescharacteristic
646C2×C4C2×#1171994911yescharacteristic
647C2×C4#1313326211yescharacteristic
648C2×C4#1313326311yescharacteristic
649C8—1062no—
650C8—2121no—
651C8—2121no—
652C8—2121no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—4241no—
661C22—4241no—
662C22—4241no—
663C22—4241no—
664C22—2122no—
665C22—2122no—
666C22—2122no—
667C22C2×#1313319311yescharacteristic
668C22D4×C22⋊D5311yescharacteristic
669C22#1681126111yescharacteristic
670C22C2×#1313323711yescharacteristic
671C22#1681142111yescharacteristic
672C22#1681143111yescharacteristic
673C22#1681143411yescharacteristic
674C4—22no—
675C4—22no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—42no—
680C4—81no—
681C4—81no—
682C4—81no—
683C4—1064no—
684C4—4241no—
685C4—4241no—
686C4C22⋊Q8×D5311yescharacteristic
687C4#1681124811yescharacteristic
688C4#1681126411yescharacteristic
689C4#1681130011yescharacteristic
690C2—42no—
691C2—4242no—
692C2#2027837811yescharacteristic
693C2#2028449711yescharacteristic
694C2#2028457311yescharacteristic
695C2#2028629611yescharacteristic
696C2#2028644111yescharacteristic
697C2#2028661111yescharacteristic
698C2#2028683511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2441C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
hypercenter#2441D53
hypocenterC53#25953
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
1. C2×#16810927C2semidirect
4. #20276249C2semidirect
5. #20276100C2semidirect
6. #20276086C2semidirect
7. #20275858C2semidirect
8. #20274982C2semidirect
9. #20274982C2semidirect
10. #20274914C2semidirect
11. #20274895C2semidirect
12. #20274738C2semidirect
13. #20274717C2semidirect
15. C53×#2441C2semidirect
343. #2441D53semidirect
410. C53#25953semidirect

Maximal subgroups

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

subgroupindex
C2×#168109272
C2×#168106172
#202775552
#202762492
#202761002
#202760862
#202758582
#202749822
#202749142
#202748952
#202747382
#202747172
#202746532
C53×#24412
#2595353