Finite Group Databasebeta

← #27383564

Subgroups — #27383564

Order 13568.

1514 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811091C211yesmaximal, characteristic
2C22×#13133146C211yesmaximal, characteristic, complemented
3C2×#16810620C211yesmaximal, characteristic
4C2×#16810617C211yesmaximal, characteristic
5#20277118C211yesmaximal, characteristic, complemented
6#20274729C212yesmaximal, complemented
7#20274724C212yesmaximal, complemented
8#20274719C212yesmaximal, complemented
9#20274714C212yesmaximal, complemented
10#20274431C211yesmaximal, characteristic, complemented
11C53×#1735C211yesmaximal, characteristic, complemented
12C22×#11719945C2211yescharacteristic
13C23×#88322C2211yescharacteristic
14#16811091—24no—
15#16811091C418yescomplemented
16C2×#13133182C2211yescharacteristic
17C2×#13133182C2211yescharacteristic
18C2×#13133179C2211yescharacteristic
19C2×#13133179C2211yescharacteristic
20C2×#13133177—21no—
21C2×#13133177—21no—
22C2×#13133176C2211yescharacteristic
23C2×#13133174—21no—
24C22×C4×Dic53C2211yescharacteristic
25C2×C42×D53—21no—
26C2×#13133146—24no—
27C2×#13133146C2212yes—
28C2×#13133146C2212yes—
29#16810620—24no—
30#16810617—24no—
31C2×#13133142C2212yes—
32C2×#13133140C2212yes—
33C2×#13133139C2212yes—
34C2×#13133137C2212yes—
35#16810492C2212yes—
36#16810488C2212yes—
37#16810481—21no—
38#16810481—21no—
39C4×#11719919—21no—
40C4×#11719919—21no—
41#16810385C2212yes—
42#16810385C2212yes—
43C23×C8×C53C2211yescharacteristic
44C2×#117×C53C2211yescharacteristic
45C2×#117×C53C2211yescharacteristic
46C53×#340C2212yes—
47C53×#340C2212yes—
48C2×#11719945—22no—
49C2×#11719945—22no—
50C2×#11719945—22no—
51C2×#11719945C2×C414yes—
52C2×#11719945D414yes—
53C2×#11719945Q814yes—
54C22×Dic106C2311yescharacteristic
55C22×D212C2311yescharacteristic
56C22×C22⋊D53—21no—
57C23×Dic53D411yescharacteristic
58C23×Dic53D411yescharacteristic
59C22×C4×D53—21no—
60C22×C4×D53—21no—
61C22×C4×D53—21no—
62C22×#88322—22no—
63C22×#88322—22no—
64C22×#88322—22no—
65C22×#88322C2312yes—
66#13133182—22no—
67#13133182—22no—
68#13133182C2×C414yes—
69#13133182C2×C414yes—
70#13133179—22no—
71#13133179—22no—
72#13133179C2×C414yes—
73#13133179C2×C414yes—
74#13133177—28no—
75#13133177—28no—
76#13133176—22no—
77#13133176—22no—
78#13133174—28no—
79C2×C4⋊Dic53C2312yes—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—22no—
83C2×#11719919—22no—
84C2×C22⋊Dic53—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53C2×C414yes—
91C2×C4×Dic53D414yes—
92C2×C4×Dic53D411yescharacteristic
93C2×C4×Dic53D411yescharacteristic
94C2×C4×Dic53C2311yescharacteristic
95C2×C4×Dic53C2311yescharacteristic
96C42×D53—28no—
97#13133146—24no—
98#13133146—24no—
99#13133142—24no—
100#13133140—24no—
101#13133139—24no—
102#13133137—24no—
103#13133124—22no—
104#13133124—22no—
105C23×C4×C53C2311yescharacteristic
106C22×C8×C53—22no—
107C22×C8×C53—22no—
108C22×C8×C53—22no—
109C22×C8×C53C2312yes—
110#117×C53—22no—
111#117×C53—22no—
112#117×C53—22no—
113#117×C53C2×C414yes—
114C2×C4⋊C4×C53C2312yes—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C42×C53—21no—
120C2×C42×C53—21no—
121C24×C53C2×D411yescharacteristic
122C23×D53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53C22×C412yes—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D412yes—
129C22×C4×C53C2×D412yes—
130C22×C4×C53C2×Q812yes—
131C22×C4×C53#4112yes—
132C22×C4×C53#4112yes—
133C22×C4×C53C2411yescharacteristic
134C2×C22⋊D53—24no—
135C2×C22⋊D53—24no—
136C2×C22⋊D53—24no—
137C22×Dic53—21no—
138C22×Dic53—24no—
139C22×Dic53—22no—
140C22×Dic53—42no—
141C22×Dic53C22⋊C414yes—
142C22×Dic53C22⋊C414yes—
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147#11719945—28no—
148#11719945—28no—
149#11719945—28no—
150#11719945C4⋊C4116yescomplemented
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212C22×C412yes—
155C2×D212C2×D412yes—
156C2×D212C2×Q812yes—
157C2×C4×D53—24no—
158C2×C4×D53—24no—
159C2×C4×D53—24no—
160C2×C4×D53—24no—
161C2×C4×D53—24no—
162C2×C4×D53—44no—
163C2×C4×D53—44no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106C22×C412yes—
168C2×Dic106C2×D412yes—
169C2×Dic106C2×Q812yes—
170C2×C8×C53—22no—
171C2×C8×C53—22no—
172C2×C8×C53—22no—
173C2×C8×C53—44no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53—22no—
178C4⋊C4×C53C22×C412yes—
179C4⋊C4×C53C22×C412yes—
180C22⋊C4×C53—24no—
181C22⋊C4×C53—42no—
182C42×C53—24no—
183C42×C53—42no—
184C22⋊Dic53—42no—
185#11719921—24no—
186#11719921—24no—
187C4⋊Dic53—22no—
188C4⋊Dic53—22no—
189#11719919—24no—
190#11719919—24no—
191#11719919—44no—
192#11719919—44no—
193C4×Dic53—24no—
194C4×Dic53—24no—
195C4×Dic53—42no—
196C4×Dic53C22⋊C414yes—
197C4×Dic53C22⋊C414yes—
198C4×Dic53C22×C412yes—
199C4×Dic53C22×C412yes—
200C4×Dic53C2×D412yes—
201C4×Dic53C2×D412yes—
202C2×#88322—22no—
203C2×#88322—22no—
204C2×#88322—22no—
205C2×#88322—44no—
206C23×C53—22no—
207C23×C53—22no—
208C23×C53C2×C22⋊C412yes—
209C23×C53#12112yes—
210C23×C53C22⋊Q812yes—
211C23×C53C22×D411yescharacteristic
212C22×D53—22no—
213C22×D53—22no—
214C22×D53—22no—
215C22×D53—42no—
216C22×D53—42no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53C2×C22⋊C412yes—
225C2×C4×C53C2×C4⋊C414yes—
226C2×C4×C53#11714yes—
227C2×C4×C53#12114yes—
228C2×C4×C53#12112yes—
229C2×C4×C53C22⋊Q814yes—
230C2×C4×C53C22⋊Q812yes—
231C2×C4×C53C23×C411yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C22×Q811yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C22⋊D53—216no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—28no—
245C2×Dic53—44no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53C2×C22⋊C412yes—
250C2×Dic53C2×C22⋊C412yes—
251C2×Dic53C2×C22⋊C412yes—
252C2×Dic53C2×C22⋊C412yes—
253C2×Dic53#13514yes—
254D212—22no—
255D212—22no—
256D212—22no—
257D212C2×C4⋊C414yes—
258C4×D53—216no—
259C4×D53—48no—
260C4×D53—48no—
261Dic106—22no—
262Dic106—22no—
263Dic106—22no—
264Dic106C2×C4⋊C414yes—
265C8×C53—42no—
266#88322—42no—
267#20645—531nomaximal, Sylow
268C22×C53—24no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—24no—
272C22×C53#38014yes—
273C22×C53C22×C22⋊C411yescharacteristic
274C22×C53C2×#12111yescharacteristic
275C22×C53C2×C22⋊Q811yescharacteristic
276C22×C53C2×#13512yes—
277C22×C53#57512yes—
278C2×D53—24no—
279C2×D53—44no—
280C2×D53—42no—
281C2×D53—42no—
282C2×D53—44no—
283C4×C53—42no—
284C4×C53—42no—
285C4×C53—82no—
286C4×C53#38014yes—
287C4×C53C22×C4⋊C411yescharacteristic
288C4×C53C2×#11711yescharacteristic
289C4×C53C2×#12111yescharacteristic
290C4×C53C2×C22⋊Q811yescharacteristic
291Dic53—22no—
292Dic53—22no—
293Dic53—22no—
294Dic53—24no—
295Dic53—82no—
296Dic53#41614yescomplemented
297C2×C4×#41—531no—
298C22×C4⋊C8—531no—
299C2×#416—531no—
300C2×#416—531no—
301#2285—531no—
302#1805—532no—
303#1804—532no—
304#1803—532no—
305#1802—532no—
306#1735—531no—
307#1735D5311yescharacteristic, complemented
308C2×C53—24no—
309C2×C53C2×#38011yescharacteristic
310C2×C53C2×#41611yescharacteristic
311C2×C53#289611yescharacteristic
312C2×C53#303512yes—
313C2×C53#307612yes—
314D53—44no—
315C22×#41—531no—
316C23×C8—531no—
317C23×C8C2×D5311yescharacteristic
318C4×#41—538no—
319C4×#41—1064no—
320C2×C4×Q8—531no—
321C2×C4×Q8—531no—
322C2×C4×D4—531no—
323C2×C4×D4—531no—
324C2×C4×D4—1061no—
325C2×C4×D4—1061no—
326C2×#117—531no—
327C2×#117—1061no—
328C2×#117C2×D5311yescharacteristic
329C2×#117C2×D5311yescharacteristic
330C22×C42—531no—
331C22×C42—1061no—
332C2×C4⋊C8—532no—
333C2×C4⋊C8—532no—
334C2×C4⋊C8—1064no—
335#416—1064no—
336#416—1064no—
337C2×#103—532no—
338C2×#103—532no—
339C2×#102—532no—
340C2×#102—532no—
341#384—532no—
342#382—532no—
343#380—1061no—
344#380—1061no—
345C4×C4⋊C4—1061no—
346C4×C4⋊C4—1061no—
347#340—532no—
348#340—532no—
349#340C2×D5312yes—
350#340C2×D5312yes—
351C53#2064511yescharacteristic, complemented, Sylow
352C2×#41—534no—
353C2×#41—534no—
354C2×#41—534no—
355C2×#41—1062no—
356C2×#41—1062no—
357C2×#41—1062no—
358C22×Q8—531no—
359C22×D4—531no—
360C22×D4—1061no—
361C23×C4—531no—
362C23×C4—531no—
363C23×C4—1061no—
364C23×C4—1061no—
365C23×C4—1061no—
366C23×C4C22×D5311yescharacteristic
367C22×C8—22no—
368C22×C8—22no—
369C22×C8—22no—
370C22×C8—532no—
371C22×C8—1062no—
372C22×C8—1062no—
373C22×C8—1062no—
374C22×C8C22×D5312yes—
375C4×Q8—534no—
376C4×Q8—534no—
377C4×Q8—1062no—
378C4×Q8—1062no—
379C4×D4—534no—
380C4×D4—534no—
381C4×D4—1062no—
382C4×D4—1062no—
383C4×D4—1068no—
384C4×D4—1068no—
385#117—22no—
386#117—22no—
387#117—22no—
388#117—1068no—
389#117—1062no—
390#117—1062no—
391#117C4×D5314yes—
392C2×C4⋊C4—532no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1062no—
396C2×C4⋊C4—1062no—
397C2×C4⋊C4C22×D5312yes—
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—534no—
408C2×C42—534no—
409C2×C42—531no—
410C2×C42—531no—
411C2×C42—531no—
412C2×C42—531no—
413C2×C42—1068no—
414C2×C42—1061no—
415C2×C42—1061no—
416C2×C42—1061no—
417C4⋊C8—1064no—
418C4⋊C8—1064no—
419#103—1064no—
420#103—1064no—
421#102—1064no—
422#102—1064no—
423#95—1062no—
424#95—1062no—
425C24—1061no—
426C24D4×D5311yescharacteristic
427#41—5316no—
428#41—1068no—
429#41—1068no—
430#41—1068no—
431C2×Q8—532no—
432C2×Q8—532no—
433C2×Q8—532no—
434C2×Q8—1061no—
435C2×Q8—1061no—
436C2×Q8—1061no—
437C2×D4—532no—
438C2×D4—532no—
439C2×D4—532no—
440C2×D4—1064no—
441C2×D4—1064no—
442C2×D4—1064no—
443C2×D4—1061no—
444C2×D4—1061no—
445C2×D4—1061no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—531no—
450C22×C4—534no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—534no—
455C22×C4—1064no—
456C22×C4—1064no—
457C22×C4—1064no—
458C22×C4—1061no—
459C22×C4—1064no—
460C22×C4—1064no—
461C22×C4—1064no—
462C22×C4—1062no—
463C22×C4—2124no—
464C22×C4—2124no—
465C22×C4—2122no—
466C22×C4C2×C4×D5312yes—
467C22×C4#1171994512yes—
468C22×C4#1171994512yes—
469C22×C4D4×D5312yes—
470C22×C4D4×D5312yes—
471C22×C4D4×D5311yescharacteristic
472C22×C4Q8×D5312yes—
473C22×C4C23×D5311yescharacteristic
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—44no—
478C2×C8—1062no—
479C2×C8—1062no—
480C2×C8—1062no—
481C2×C8—2124no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—1064no—
487C4⋊C4—1064no—
488C4⋊C4—1062no—
489C4⋊C4—1062no—
490C4⋊C4—2124no—
491C4⋊C4—2124no—
492C4⋊C4C2×C4×D5312yes—
493C4⋊C4C2×C4×D5312yes—
494C22⋊C4—24no—
495C22⋊C4—42no—
496C22⋊C4—1064no—
497C22⋊C4—1064no—
498C22⋊C4—2122no—
499C42—24no—
500C42—42no—
501C42—534no—
502C42—534no—
503C42—532no—
504C42—532no—
505C42—532no—
506C42—532no—
507C42—1064no—
508C42—1064no—
509C42—2122no—
510C23—22no—
511C23—22no—
512C23—1062no—
513C23—1062no—
514C23—1062no—
515C23—2122no—
516C23—2122no—
517C23C22⋊C4×D5312yes—
518C23#1313319012yes—
519C23#1313319512yes—
520C23C2×D4×D5311yescharacteristic
521Q8—534no—
522Q8—1062no—
523Q8—1062no—
524Q8—1062no—
525D4—534no—
526D4—10616no—
527D4—1062no—
528D4—1062no—
529D4—1062no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—532no—
538C2×C4—534no—
539C2×C4—532no—
540C2×C4—532no—
541C2×C4—532no—
542C2×C4—1068no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—1062no—
547C2×C4—1062no—
548C2×C4—1062no—
549C2×C4—10616no—
550C2×C4—2128no—
551C2×C4—2128no—
552C2×C4—2124no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4C22⋊C4×D5312yes—
557C2×C4C4⋊C4×D5314yes—
558C2×C4#1313317414yes—
559C2×C4#1313319014yes—
560C2×C4#1313319012yes—
561C2×C4#1313319514yes—
562C2×C4#1313319512yes—
563C2×C4C22×C4×D5311yescharacteristic
564C2×C4C2×D4×D5311yescharacteristic
565C2×C4C2×D4×D5311yescharacteristic
566C2×C4C2×Q8×D5311yescharacteristic
567C2×C4C2×#1171994511yescharacteristic
568C2×C4C2×#1171994511yescharacteristic
569C8—42no—
570C8—2122no—
571C22—24no—
572C22—22no—
573C22—24no—
574C22—22no—
575C22—1064no—
576C22—2124no—
577C22—2122no—
578C22—2124no—
579C22—2122no—
580C22#1681048214yes—
581C22C2×C22⋊C4×D5311yescharacteristic
582C22C2×#1313319011yescharacteristic
583C22C2×#1313319511yescharacteristic
584C22#135×D5312yes—
585C22#1681142012yes—
586C4—42no—
587C4—42no—
588C4—82no—
589C4—534no—
590C4—1062no—
591C4—1062no—
592C4—1062no—
593C4—1064no—
594C4—4242no—
595C4#1681048214yes—
596C4C2×C4⋊C4×D5311yescharacteristic
597C4C2×#1313317411yescharacteristic
598C4C2×#1313319011yescharacteristic
599C4C2×#1313319511yescharacteristic
600C2—24no—
601C2—2124no—
602C2C2×#1681048211yescharacteristic
603C2#416×D5311yescharacteristic
604C2#2028323011yescharacteristic
605C2#2028416812yes—
606C2#2028450112yes—
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1735C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenter#1735D53
hypocenterC53#20645
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
2. C22×#13133146C2semidirect
5. #20277118C2semidirect
6. #20274729C2semidirect
7. #20274724C2semidirect
8. #20274719C2semidirect
9. #20274714C2semidirect
10. #20274431C2semidirect
11. C53×#1735C2semidirect
15. #16811091C4semidirect
150. #11719945C4⋊C4semidirect
296. Dic53#416semidirect
307. #1735D53semidirect
351. C53#20645semidirect

Maximal subgroups

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

subgroupindex
C2×#168110912
C22×#131331462
C2×#168106202
C2×#168106172
#202771182
#202747292
#202747242
#202747192
#202747142
#202744312
C53×#17352
#2064553