Finite Group Databasebeta

← #27416845

Subgroups — #27416845

Order 13568.

1276 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810956C211yesmaximal, characteristic, complemented
2C2×#16810951C211yesmaximal, characteristic, complemented
3#20277307C211yesmaximal, characteristic
4#20275915C212yesmaximal, complemented
5#20275908C212yesmaximal, complemented
6#20275028C211yesmaximal, characteristic, complemented
7#20275028C211yesmaximal, characteristic, complemented
8#20274914C212yesmaximal, complemented
9#20274650C212yesmaximal, complemented
10#20274627C211yesmaximal, characteristic
11C2×C53×#536C211yesmaximal, characteristic
12C2×#13133176C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14#16810956—24no—
15#16810951—24no—
16C2×#13133150C2212yes—
17C2×#13133148C2212yes—
18C2×#13133141C2212yes—
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×#13133141C2212yes—
22C2×C22⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2212yes—
25C2×C22⋊#88322C2212yes—
26#16810530—21no—
27#16810526—21no—
28#16810503—21no—
29#16810503—21no—
30#16810483C2212yes—
31#16810483C2212yes—
32#16810385C2212yes—
33#16810385C2212yes—
34C22×#41×C53C2211yescharacteristic
35C23×Q8×C53C2211yescharacteristic
36C53×#536—24no—
37C53×#536—24no—
38C2×C4⋊Q8×C53—21no—
39C2×#124×C53—21no—
40C2×#123×C53—21no—
41C2×#123×C53—21no—
42C2×C22⋊Q8×C53C2211yescharacteristic
43C2×C22⋊Q8×C53C2211yescharacteristic
44C2×C22⋊Q8×C53C2211yescharacteristic
45C2×C22⋊Q8×C53C2211yescharacteristic
46C2×#117×C53C2211yescharacteristic
47C23×Dic53—21no—
48C22×#88322C2312yes—
49C22×#88322C2312yes—
50#13133176—42no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53C2312yes—
55C2×C4⋊Dic53C2312yes—
56C2×C22⋊Dic53—21no—
57C2×C22⋊Dic53—22no—
58C2×C22⋊Dic53—22no—
59C2×C4×Dic53—21no—
60#13133150—24no—
61#13133148—24no—
62#13133141—24no—
63#13133141—24no—
64#13133141—42no—
65#13133141—42no—
66C22⋊#88322—22no—
67C22⋊#88322—22no—
68C22⋊#88322—24no—
69C22⋊#88322—42no—
70#13133124—22no—
71#13133124—22no—
72C2×#41×C53—22no—
73C2×#41×C53—22no—
74C2×#41×C53—24no—
75C2×#41×C53—42no—
76C22×Q8×C53—22no—
77C22×Q8×C53—22no—
78C22×Q8×C53—21no—
79C22×Q8×C53—24no—
80C22×Q8×C53—42no—
81C22×Q8×C53D411yescharacteristic
82C22×Q8×C53C2311yescharacteristic
83C22×Q8×C53C2311yescharacteristic
84C22×D4×C53D411yescharacteristic
85C22×D4×C53D411yescharacteristic
86C22×D4×C53D411yescharacteristic
87C23×C4×C53—21no—
88C23×C4×C53C2311yescharacteristic
89C23×C4×C53C2311yescharacteristic
90C23×C4×C53C2311yescharacteristic
91C4⋊Q8×C53—28no—
92#124×C53—28no—
93#123×C53—28no—
94#123×C53—28no—
95C22⋊Q8×C53—22no—
96C22⋊Q8×C53—22no—
97C22⋊Q8×C53—22no—
98C22⋊Q8×C53—22no—
99C22⋊Q8×C53—22no—
100C22⋊Q8×C53—22no—
101C22⋊Q8×C53D414yes—
102C22⋊Q8×C53D414yes—
103#117×C53—22no—
104#117×C53—22no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C42×C53—21no—
117C24×C53C2×D411yescharacteristic
118C24×C53C2×D411yescharacteristic
119C24×C53C2×D411yescharacteristic
120#41×C53—48no—
121#41×C53—48no—
122C2×Q8×C53—24no—
123C2×Q8×C53—22no—
124C2×Q8×C53—22no—
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—22no—
128C2×Q8×C53—22no—
129C2×Q8×C53—22no—
130C2×Q8×C53—22no—
131C2×Q8×C53—41no—
132C2×Q8×C53—48no—
133C2×Q8×C53—48no—
134C2×Q8×C53—42no—
135C2×Q8×C53—42no—
136C2×Q8×C53C2×D412yes—
137C2×Q8×C53C2×D412yes—
138C2×D4×C53—22no—
139C2×D4×C53—22no—
140C2×D4×C53—22no—
141C2×D4×C53—22no—
142C2×D4×C53—22no—
143C2×D4×C53—22no—
144C2×D4×C53—41no—
145C2×D4×C53—41no—
146C2×D4×C53—41no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—22no—
150C22×C4×C53—22no—
151C22×C4×C53—22no—
152C22×C4×C53—24no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53—21no—
158C22×C4×C53—21no—
159C22×C4×C53—22no—
160C22×C4×C53—22no—
161C22×C4×C53—42no—
162C22×C4×C53—42no—
163C22×C4×C53C2×D411yescharacteristic
164C22×C4×C53C2×D411yescharacteristic
165C22×C4×C53C2×D412yes—
166C22×C4×C53C2×D412yes—
167C22×C4×C53C2×D411yescharacteristic
168C22×C4×C53#4114yes—
169C22×C4×C53C2411yescharacteristic
170C22×Dic53—22no—
171C22×Dic53—22no—
172C22×Dic53—42no—
173C22×Dic53—82no—
174C4⋊C4×C53—24no—
175C4⋊C4×C53—24no—
176C4⋊C4×C53—24no—
177C4⋊C4×C53—24no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53C2×D412yes—
181C4⋊C4×C53C2×D412yes—
182C22⋊C4×C53—24no—
183C22⋊C4×C53—24no—
184C22⋊C4×C53—24no—
185C22⋊C4×C53—24no—
186C22⋊C4×C53—24no—
187C42×C53—24no—
188C22⋊Dic53—42no—
189C22⋊Dic53—82no—
190C22⋊Dic53—82no—
191C4⋊Dic53—22no—
192C4⋊Dic53—22no—
193C4⋊Dic53—22no—
194C4⋊Dic53—22no—
195C4⋊Dic53—42no—
196C4×Dic53—42no—
197C2×#88322—22no—
198C2×#88322—22no—
199C2×#88322—22no—
200C2×#88322—22no—
201C2×#88322—22no—
202C2×#88322—22no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—22no—
208C23×C53—22no—
209C23×C53—22no—
210C23×C53—22no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—22no—
215C23×C53—22no—
216C23×C53#12012yes—
217C23×C53#12312yes—
218C23×C53C22×D411yescharacteristic
219Q8×C53—24no—
220Q8×C53—24no—
221Q8×C53—44no—
222Q8×C53—48no—
223Q8×C53—42no—
224Q8×C53—42no—
225D4×C53—44no—
226D4×C53—44no—
227D4×C53—44no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—24no—
233C2×C4×C53—24no—
234C2×C4×C53—24no—
235C2×C4×C53—24no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—21no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—48no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53#12012yes—
253C2×C4×C53#12312yes—
254C2×C4×C53C22×D411yescharacteristic
255C2×C4×C53C22×D411yescharacteristic
256C2×C4×C53C2×#4112yes—
257C2×C4×C53H(5,𝔽2)12yes—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—84no—
265#88322—42no—
266#88322—42no—
267#24014—531nomaximal, Sylow
268C22×C53—24no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—42no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—42no—
279C22×C53—41no—
280C22×C53—42no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53C2×#12011yescharacteristic
284C22×C53C2×#12311yescharacteristic
285C22×C53#53411yescharacteristic
286C22×C53#57712yes—
287C22×C53#57812yes—
288C4×C53—22no—
289C4×C53—22no—
290C4×C53—41no—
291C4×C53—42no—
292C4×C53—42no—
293C4×C53—42no—
294C4×C53—42no—
295C4×C53#54612yes—
296C4×C53#54612yes—
297Dic53—82no—
298Dic53—82no—
299C2×#536D5311yescharacteristic
300C2×#484—531no—
301C2×#483—531no—
302#2311—531no—
303#2036—532no—
304#2035—532no—
305#1873—531no—
306#1873—531no—
307#1854—532no—
308#1787—532no—
309#1780—531no—
310C2×C53—42no—
311C2×C53—42no—
312C2×C53—81no—
313C2×C53#240511yescharacteristic
314C2×C53#301011yescharacteristic
315C2×C53#301111yescharacteristic
316C2×C53#309712yes—
317C2×C53#320112yes—
318C22×#41C2×D5311yescharacteristic
319C23×Q8C2×D5311yescharacteristic
320#536—24no—
321#536—24no—
322C2×C4⋊Q8—21no—
323C2×#124—21no—
324C2×#123—21no—
325C2×#123—21no—
326C2×C22⋊Q8C2×D5311yescharacteristic
327C2×C22⋊Q8C2×D5311yescharacteristic
328C2×C22⋊Q8C2×D5311yescharacteristic
329C2×C22⋊Q8C2×D5311yescharacteristic
330C2×#117—531no—
331C2×#117C2×D5311yescharacteristic
332C22×C4⋊C4—531no—
333#484—1064no—
334#483—1064no—
335C2×#107—532no—
336C2×#106—532no—
337C2×#103—532no—
338C2×#103—532no—
339C2×#103—532no—
340C2×#103—532no—
341C2×C22⋊C8—531no—
342C2×C22⋊C8—531no—
343C2×C22⋊C8—532no—
344C2×C22⋊C8—532no—
345#391—1061no—
346#390—1061no—
347#386—1061no—
348#386—1061no—
349#380—532no—
350#380—532no—
351#340—532no—
352#340—532no—
353C53#2401411yescharacteristic, complemented, Sylow
354C2×#41—24no—
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—42no—
358C22×Q8—22no—
359C22×Q8—22no—
360C22×Q8—21no—
361C22×Q8—24no—
362C22×Q8—42no—
363C22×Q8C22⋊D5311yescharacteristic
364C22×Q8C22×D5311yescharacteristic
365C22×Q8C22×D5311yescharacteristic
366C22×D4C22⋊D5311yescharacteristic
367C22×D4C22⋊D5311yescharacteristic
368C22×D4C22⋊D5311yescharacteristic
369C23×C4—21no—
370C23×C4—1061no—
371C23×C4C22×D5311yescharacteristic
372C23×C4C22×D5311yescharacteristic
373C23×C4C22×D5311yescharacteristic
374C22×C8—532no—
375C22×C8—532no—
376C4⋊Q8—28no—
377#124—28no—
378#123—28no—
379#123—28no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8C22⋊D5314yes—
387C22⋊Q8C22⋊D5314yes—
388#117—22no—
389#117—22no—
390#117—2122no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—532no—
396C2×C4⋊C4—532no—
397C2×C4⋊C4—1062no—
398C2×C4⋊C4—1062no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—1062no—
408C2×C22⋊C4—1062no—
409C2×C22⋊C4—1061no—
410C2×C42—21no—
411C2×C42—1061no—
412#107—1064no—
413#106—1064no—
414#103—1064no—
415#103—1064no—
416#103—2122no—
417#103—2122no—
418C22⋊C8—1064no—
419C22⋊C8—1062no—
420C22⋊C8—1062no—
421C22⋊C8—2122no—
422#95—1062no—
423#95—1062no—
424C24C2×C22⋊D5311yescharacteristic
425C24C2×C22⋊D5311yescharacteristic
426C24C2×C22⋊D5311yescharacteristic
427#41—48no—
428#41—48no—
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—41no—
439C2×Q8—48no—
440C2×Q8—48no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8C2×C22⋊D5312yes—
444C2×Q8C2×C22⋊D5312yes—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—41no—
452C2×D4—41no—
453C2×D4—41no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—4242no—
473C22×C4—2122no—
474C22×C4#1171994714yes—
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C2×C22⋊D5311yescharacteristic
477C22×C4C2×C22⋊D5312yes—
478C22×C4C2×C22⋊D5312yes—
479C22×C4C2×C22⋊D5311yescharacteristic
480C22×C4C23×D5311yescharacteristic
481C2×C8—1062no—
482C2×C8—1062no—
483C2×C8—1062no—
484C2×C8—1062no—
485C2×C8—1062no—
486C2×C8—1062no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—1062no—
494C4⋊C4—1062no—
495C4⋊C4—1062no—
496C4⋊C4—1062no—
497C4⋊C4—2122no—
498C4⋊C4C2×C22⋊D5312yes—
499C4⋊C4C2×C22⋊D5312yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—4242no—
506C22⋊C4—4242no—
507C22⋊C4—2122no—
508C42—24no—
509C42—2122no—
510C23—21no—
511C23—21no—
512C23—22no—
513C23—22no—
514C23—21no—
515C23—21no—
516C23—22no—
517C23—22no—
518C23—21no—
519C23—21no—
520C23—22no—
521C23—21no—
522C23—22no—
523C23#1313318612yes—
524C23#1313319812yes—
525C23C22×C22⋊D5311yescharacteristic
526Q8—24no—
527Q8—24no—
528Q8—44no—
529Q8—48no—
530Q8—42no—
531Q8—42no—
532D4—44no—
533D4—44no—
534D4—44no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—48no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—4244no—
560C2×C4—2122no—
561C2×C4—2122no—
562C2×C4—2122no—
563C2×C4—2122no—
564C2×C4—2122no—
565C2×C4—2122no—
566C2×C4#1313318612yes—
567C2×C4#1313319812yes—
568C2×C4C22×C22⋊D5311yescharacteristic
569C2×C4C22×C22⋊D5311yescharacteristic
570C2×C4C2×#1171994712yes—
571C2×C4#1313326212yes—
572C8—2122no—
573C8—2122no—
574C22—24no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—41no—
581C22—41no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—42no—
589C22C2×#1313318611yescharacteristic
590C22C2×#1313319811yescharacteristic
591C22#1681118211yescharacteristic
592C22#1681143112yes—
593C22#1681143812yes—
594C4—22no—
595C4—22no—
596C4—41no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—4242no—
602C4—4242no—
603C4#1681126712yes—
604C4#1681126712yes—
605C2—42no—
606C2—42no—
607C2—81no—
608C2#2027803811yescharacteristic
609C2#2028400111yescharacteristic
610C2#2028401011yescharacteristic
611C2#2028467812yes—
612C2#2028559712yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#536C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
hypercenterC2×#536D53
hypocenterC53#24014
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×#16810956C2semidirect
2. C2×#16810951C2semidirect
4. #20275915C2semidirect
5. #20275908C2semidirect
6. #20275028C2semidirect
7. #20275028C2semidirect
8. #20274914C2semidirect
9. #20274650C2semidirect
353. C53#24014semidirect

Maximal subgroups

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

subgroupindex
C2×#168109562
C2×#168109512
#202773072
#202759152
#202759082
#202750282
#202749142
#202746502
#202746272
C2×C53×#5362
#2401453