Finite Group Databasebeta

← #27393200

Subgroups — #27393200

Order 13568.

1326 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811194C211yesmaximal, characteristic
2#20277307C211yesmaximal, characteristic
3#20275148C212yesmaximal, complemented
4#20275138C212yesmaximal, complemented
5#20275025C212yesmaximal, complemented
6#20275020C212yesmaximal, complemented
7#20274811C212yesmaximal, complemented
8#20274651C212yesmaximal, complemented
9C2×C53×#408C211yesmaximal, characteristic
10C23×Dic106C2211yescharacteristic
11#16811194—24no—
12#16811194C418yescomplemented
13C2×#13133215—21no—
14C2×#13133204—21no—
15C2×#13133198—21no—
16C2×#13133198—21no—
17C2×#13133196C2212yes—
18C2×#13133196C2212yes—
19C2×#13133176C2211yescharacteristic
20C2×#13133176C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22C2×#13133142C2212yes—
23C2×#13133142C2212yes—
24C2×#13133142C2212yes—
25C2×#13133142C2212yes—
26#16810530—21no—
27#16810526—21no—
28#16810503—21no—
29#16810503—21no—
30#16810483C2212yes—
31#16810483C2212yes—
32#16810386C2212yes—
33#16810386C2212yes—
34#16810386C2212yes—
35#16810386C2212yes—
36C22×#41×C53C2211yescharacteristic
37C22×M16×C53C2211yescharacteristic
38C23×C8×C53C2211yescharacteristic
39C53×#408—24no—
40C2×C22⋊C8×C53C2212yes—
41C2×C22⋊C8×C53C2212yes—
42C22×Dic106—21no—
43C22×Dic106—22no—
44C22×Dic106—41no—
45C22×Dic106—41no—
46C22×Dic106—41no—
47C22×Dic106—41no—
48C22×Dic106C2×C414yescomplemented
49C22×Dic106C2312yes—
50C23×Dic53—21no—
51C23×Dic53—21no—
52#13133215—28no—
53#13133204—28no—
54#13133198—28no—
55#13133198—28no—
56#13133196—24no—
57#13133196—24no—
58#13133196C2×C418yes—
59#13133196C2×C418yescomplemented
60#13133176—22no—
61#13133176—41no—
62#13133176—41no—
63#13133176C2×C414yescomplemented
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—41no—
66C2×C4⋊Dic53—41no—
67C2×C4⋊Dic53C2312yes—
68C2×C4⋊Dic53C2312yes—
69C2×C4⋊Dic53C2312yes—
70C2×#11719919—22no—
71C2×#11719919—22no—
72C2×C22⋊Dic53—21no—
73C2×C22⋊Dic53—22no—
74C2×C22⋊Dic53—22no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—22no—
77C2×C22⋊Dic53—22no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53—21no—
80#13133142—42no—
81#13133142—42no—
82#13133142—42no—
83#13133142—42no—
84#13133124—22no—
85#13133124—22no—
86C2×#41×C53—41no—
87C2×#41×C53—41no—
88C2×#41×C53—41no—
89C2×#41×C53—41no—
90C2×#41×C53C2×C414yes—
91C2×#41×C53D414yes—
92C22×Q8×C53D411yescharacteristic
93C22×D4×C53D411yescharacteristic
94C22×D4×C53D411yescharacteristic
95C22×D4×C53D411yescharacteristic
96C23×C4×C53C2311yescharacteristic
97C23×C4×C53C2311yescharacteristic
98C23×C4×C53C2311yescharacteristic
99C2×M16×C53—22no—
100C2×M16×C53—41no—
101C2×M16×C53—41no—
102C22×C8×C53—22no—
103C22×C8×C53—41no—
104C22×C8×C53—41no—
105C22×C8×C53C2312yes—
106C22×C8×C53C2312yes—
107C22⋊C8×C53—24no—
108C22⋊C8×C53—24no—
109C24×C53C2×D411yescharacteristic
110C24×C53C2×D411yescharacteristic
111C24×C53C2×D411yescharacteristic
112#41×C53—44no—
113#41×C53—44no—
114#41×C53—44no—
115#41×C53—44no—
116C2×Q8×C53—41no—
117C2×Q8×C53—41no—
118C2×Q8×C53C22⋊C414yes—
119C2×D4×C53—41no—
120C2×D4×C53—41no—
121C2×D4×C53—41no—
122C2×D4×C53—41no—
123C2×D4×C53—41no—
124C2×D4×C53—41no—
125C2×D4×C53C22⋊C414yes—
126C2×D4×C53C22⋊C414yes—
127C2×D4×C53C22⋊C414yes—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53C22×C412yes—
141C22×C4×C53C22×C412yes—
142C22×C4×C53C22×C412yes—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D412yes—
147C22×C4×C53C2×D412yes—
148C22×C4×C53C2×D412yes—
149C22×C4×C53C2411yescharacteristic
150C22×Dic53—24no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—42no—
156C22×Dic53—42no—
157C22×Dic53—42no—
158C22×Dic53—82no—
159C2×Dic106—22no—
160C2×Dic106—22no—
161C2×Dic106—22no—
162C2×Dic106—22no—
163C2×Dic106—24no—
164C2×Dic106—22no—
165C2×Dic106—44no—
166C2×Dic106—44no—
167C2×Dic106—42no—
168C2×Dic106—42no—
169C2×Dic106—44no—
170C2×Dic106—44no—
171C2×Dic106C22×C414yes—
172M16×C53—44no—
173C2×C8×C53—22no—
174C2×C8×C53—22no—
175C2×C8×C53—22no—
176C2×C8×C53—22no—
177C2×C8×C53—22no—
178C2×C8×C53—22no—
179C2×C8×C53—44no—
180C22⋊Dic53—24no—
181C22⋊Dic53—28no—
182C22⋊Dic53—28no—
183C22⋊Dic53—42no—
184C22⋊Dic53—82no—
185C22⋊Dic53—82no—
186C4⋊Dic53—22no—
187C4⋊Dic53—22no—
188C4⋊Dic53—22no—
189C4⋊Dic53—22no—
190C4⋊Dic53—22no—
191C4⋊Dic53—42no—
192C4⋊Dic53C22×C414yes—
193#11719919—28no—
194#11719919—28no—
195C4×Dic53—24no—
196C4×Dic53—42no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—22no—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53—22no—
207C23×C53—22no—
208C23×C53—22no—
209C23×C53C2×C22⋊C412yes—
210C23×C53C2×C22⋊C412yes—
211C23×C53C2×C22⋊C412yes—
212C23×C53C22×D411yescharacteristic
213Q8×C53—44no—
214D4×C53—44no—
215D4×C53—44no—
216D4×C53—44no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53C2×C22⋊C412yes—
236C2×C4×C53C2×C22⋊C412yes—
237C2×C4×C53C2×C22⋊C412yes—
238C2×C4×C53C23×C411yescharacteristic
239C2×C4×C53C22×D411yescharacteristic
240C2×C4×C53H(5,𝔽2)11yescharacteristic
241C2×C4×C53H(5,𝔽2)11yescharacteristic
242C2×C4×C53H(5,𝔽2)11yescharacteristic
243C2×C4×C53H(5,𝔽2)11yescharacteristic
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—48no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—84no—
258Dic106—24no—
259Dic106—24no—
260Dic106—48no—
261Dic106—42no—
262Dic106—42no—
263C8×C53—42no—
264C8×C53—42no—
265#21771—531nomaximal, Sylow
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—24no—
270C22×C53—24no—
271C22×C53—24no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53C22×C22⋊C411yescharacteristic
282C22×C53#53411yescharacteristic
283C22×C53#53411yescharacteristic
284C22×C53#57712yes—
285C22×C53#57812yes—
286C4×C53—22no—
287C4×C53—22no—
288C4×C53—22no—
289C4×C53#51811yescharacteristic
290C4×C53#51811yescharacteristic
291C4×C53#53411yescharacteristic
292C4×C53#53411yescharacteristic
293Dic53—44no—
294Dic53—44no—
295Dic53—82no—
296Dic53—82no—
297C2×#536—531no—
298C2×#408D5311yescharacteristic
299#2311—531no—
300#1894—532no—
301#1892—532no—
302#1873—532no—
303#1872—532no—
304#1825—532no—
305#1787—532no—
306C2×C53—42no—
307C2×C53—42no—
308C2×C53—42no—
309C2×C53#230811yescharacteristic
310C2×C53#289812yes—
311C2×C53#320011yescharacteristic
312C2×C53#320112yes—
313C2×C53#320211yescharacteristic
314C22×#41C2×D5311yescharacteristic
315C23×Q8—531no—
316C22×M16C2×D5311yescharacteristic
317C23×C8C2×D5311yescharacteristic
318#536—538no—
319#536—1064no—
320C2×C4⋊Q8—1061no—
321C2×#124—1061no—
322C2×#123—1061no—
323C2×#123—1061no—
324C2×C22⋊Q8—532no—
325C2×C22⋊Q8—532no—
326C2×#117—531no—
327C2×#117—531no—
328C22×C4⋊C4—531no—
329C2×#103—532no—
330C2×#103—532no—
331C2×#103—532no—
332C2×#103—532no—
333#408—24no—
334C2×C22⋊C8C2×D5312yes—
335C2×C22⋊C8C2×D5312yes—
336#391—1061no—
337#390—1061no—
338#386—1061no—
339#386—1061no—
340#380—532no—
341#380—532no—
342#340—532no—
343#340—532no—
344#340—532no—
345#340—532no—
346C53#2177111yescharacteristic, complemented, Sylow
347C2×#41—41no—
348C2×#41—41no—
349C2×#41—41no—
350C2×#41—41no—
351C2×#41C4×D5314yes—
352C2×#41C22⋊D5314yes—
353C22×Q8—534no—
354C22×Q8—532no—
355C22×Q8—1061no—
356C22×Q8—1062no—
357C22×Q8—2121no—
358C22×Q8—2121no—
359C22×Q8—2121no—
360C22×Q8—2121no—
361C22×Q8D21211yescharacteristic
362C22×D4D21211yescharacteristic
363C22×D4D21211yescharacteristic
364C22×D4D21211yescharacteristic
365C23×C4—1061no—
366C23×C4—1061no—
367C23×C4C22×D5311yescharacteristic
368C23×C4C22×D5311yescharacteristic
369C23×C4C22×D5311yescharacteristic
370C2×M16—22no—
371C2×M16—41no—
372C2×M16—41no—
373C22×C8—22no—
374C22×C8—41no—
375C22×C8—41no—
376C22×C8C22×D5312yes—
377C22×C8C22×D5312yes—
378C4⋊Q8—1068no—
379#124—1068no—
380#123—1068no—
381#123—1068no—
382C22⋊Q8—538no—
383C22⋊Q8—538no—
384C22⋊Q8—1064no—
385C22⋊Q8—1064no—
386#117—534no—
387#117—1062no—
388#117—2121no—
389#117—2121no—
390C2×C4⋊C4—532no—
391C2×C4⋊C4—532no—
392C2×C4⋊C4—532no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—2121no—
397C2×C4⋊C4—2121no—
398C2×C22⋊C4—1061no—
399C2×C22⋊C4—1062no—
400C2×C22⋊C4—1062no—
401C2×C22⋊C4—1062no—
402C2×C22⋊C4—1062no—
403C2×C22⋊C4—1061no—
404C2×C42—1061no—
405C2×C42—1061no—
406#103—2122no—
407#103—2122no—
408#103—2122no—
409#103—2122no—
410C22⋊C8—24no—
411C22⋊C8—24no—
412#95—1062no—
413#95—1062no—
414C24C2×D21211yescharacteristic
415C24C2×D21211yescharacteristic
416C24C2×D21211yescharacteristic
417#41—44no—
418#41—44no—
419#41—44no—
420#41—44no—
421C2×Q8—41no—
422C2×Q8—41no—
423C2×Q8—534no—
424C2×Q8—1064no—
425C2×Q8—1062no—
426C2×Q8—1062no—
427C2×Q8—1062no—
428C2×Q8—1062no—
429C2×Q8—1062no—
430C2×Q8—2124no—
431C2×Q8—2124no—
432C2×Q8—2122no—
433C2×Q8—2122no—
434C2×Q8—2124no—
435C2×Q8—2124no—
436C2×Q8#1171992114yes—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—41no—
443C2×D4#1171992114yes—
444C2×D4#1171992114yes—
445C2×D4#1171992114yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—1064no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—4242no—
464C22×C4—2122no—
465C22×C4—2122no—
466C22×C4—2122no—
467C22×C4C2×C4×D5312yes—
468C22×C4C2×C4×D5312yes—
469C22×C4C2×C4×D5312yes—
470C22×C4C2×D21211yescharacteristic
471C22×C4C2×D21211yescharacteristic
472C22×C4C2×D21211yescharacteristic
473C22×C4C2×C22⋊D5312yes—
474C22×C4C2×C22⋊D5312yes—
475C22×C4C2×C22⋊D5312yes—
476C22×C4C23×D5311yescharacteristic
477M16—44no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—44no—
485C4⋊C4—534no—
486C4⋊C4—1068no—
487C4⋊C4—1068no—
488C4⋊C4—1062no—
489C4⋊C4—1062no—
490C4⋊C4—1062no—
491C4⋊C4—1062no—
492C4⋊C4—1062no—
493C4⋊C4—2122no—
494C22⋊C4—1064no—
495C22⋊C4—1068no—
496C22⋊C4—1068no—
497C22⋊C4—4242no—
498C22⋊C4—4242no—
499C22⋊C4—2122no—
500C42—1064no—
501C42—2122no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23C2×#1171992112yes—
515C23C2×#1171992112yes—
516C23C2×#1171992112yes—
517C23C22×D21211yescharacteristic
518Q8—44no—
519Q8—1064no—
520Q8—1064no—
521Q8—2128no—
522Q8—2122no—
523Q8—2122no—
524D4—44no—
525D4—44no—
526D4—44no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—1064no—
546C2×C4—1064no—
547C2×C4—4244no—
548C2×C4—2122no—
549C2×C4—2122no—
550C2×C4—2122no—
551C2×C4—2122no—
552C2×C4—2122no—
553C2×C4—2122no—
554C2×C4—2128no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4C2×#1171992112yes—
560C2×C4C2×#1171992112yes—
561C2×C4C2×#1171992112yes—
562C2×C4C22×C4×D5311yescharacteristic
563C2×C4C22×C22⋊D5311yescharacteristic
564C2×C4#1313326211yescharacteristic
565C2×C4#1313326211yescharacteristic
566C2×C4#1313326211yescharacteristic
567C2×C4#1313326211yescharacteristic
568C8—42no—
569C8—42no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C22×#1171992111yescharacteristic
586C22#1681118511yescharacteristic
587C22#1681118511yescharacteristic
588C22#1681143512yes—
589C22#1681144012yes—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—4242no—
594C4—4242no—
595C4—2124no—
596C4—2124no—
597C4#1681109611yescharacteristic
598C4#1681109611yescharacteristic
599C4#1681118211yescharacteristic
600C4#1681118211yescharacteristic
601C2—42no—
602C2—42no—
603C2—42no—
604C2#2027729011yescharacteristic
605C2#2028324912yes—
606C2#2028558811yescharacteristic
607C2#2028559912yes—
608C2#2028561011yescharacteristic
subgroupG/H
centerC23C22×D212
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#408C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
cosocleC22×C4×C53C24
hypercenterC2×#408D53
hypocenterC53#21771
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
3. #20275148C2semidirect
4. #20275138C2semidirect
5. #20275025C2semidirect
6. #20275020C2semidirect
7. #20274811C2semidirect
8. #20274651C2semidirect
12. #16811194C4semidirect
48. C22×Dic106C2×C4semidirect
59. #13133196C2×C4semidirect
63. #13133176C2×C4semidirect
346. C53#21771semidirect

Maximal subgroups

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

subgroupindex
C2×#168111942
#202773072
#202751482
#202751382
#202750252
#202750202
#202748112
#202746512
C2×C53×#4082
#2177153