Finite Group Databasebeta

← #27383575

Subgroups — #27383575

Order 13568.

1450 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D53C211yesmaximal, characteristic
2C2×#16810645C211yesmaximal, characteristic
3C2×#16810630C212yesmaximal
4#20277120C211yesmaximal, characteristic
5#20274729C212yesmaximal, complemented
6#20274728C212yesmaximal, complemented
7#20274721C211yesmaximal, characteristic, complemented
8#20274720C211yesmaximal, characteristic, complemented
9#20274717C211yesmaximal, characteristic, complemented
10#20274433C212yesmaximal
11C53×#1803C211yesmaximal, characteristic, complemented
12C22×Q8×D53C2211yescharacteristic
13C22×#11719912C2212yes—
14C2×Q8×Dic53C2211yescharacteristic
15C2×C4×Dic106—21no—
16C2×C4×Dic106C2211yescharacteristic
17C2×#13133182—21no—
18C2×#13133182C2211yescharacteristic
19C4×Q8×D53—24no—
20C4×Q8×D53C418yescomplemented
21C2×#13133175C2212yes—
22C2×C4⋊C4×D53—21no—
23C2×C4⋊C4×D53C2211yescharacteristic
24C2×C42×D53—21no—
25C2×C42×D53C2211yescharacteristic
26#16810645—24no—
27C2×C4⋊#88322C2211yescharacteristic
28C2×#13133146C2212yes—
29#16810630—28no—
30C2×#13133142C2212yes—
31C2×#13133141C2212yes—
32C2×#13133140C2212yes—
33#16810493C2211yescharacteristic
34#16810488C2212yes—
35#16810480—21no—
36#16810480—21no—
37C4⋊C4×Dic53—21no—
38C4⋊C4×Dic53—21no—
39#16810386C2212yes—
40#16810385C2212yes—
41#16810385C2212yes—
42C2×C4×Q8×C53C2211yescharacteristic
43C2×C4⋊C8×C53C2211yescharacteristic
44C2×#103×C53C2212yes—
45C53×#384C2211yescharacteristic
46C53×#340C2212yes—
47C22×Dic106—21no—
48C22×Dic106C2311yescharacteristic
49C2×Q8×D53—22no—
50C2×Q8×D53—22no—
51C2×Q8×D53—22no—
52C2×Q8×D53C2×C414yes—
53C2×Q8×D53D414yes—
54C2×Q8×D53Q814yes—
55C22×C4×D53—21no—
56C22×C4×D53—21no—
57C22×C4×D53D411yescharacteristic
58C22×C4×D53D411yescharacteristic
59C22×C4×D53C2311yescharacteristic
60C2×#11719912—24no—
61C2×#11719912—24no—
62C2×#11719912—24no—
63C22×#88322C2312yes—
64Q8×Dic53—22no—
65Q8×Dic53C2×C414yes—
66C4×Dic106—22no—
67C4×Dic106—28no—
68C4×Dic106C2×C414yes—
69#13133182—28no—
70#13133182—22no—
71#13133182C2×C414yes—
72#13133175—24no—
73#13133175—24no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53C2312yes—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C4⋊C4×D53—28no—
83C4⋊C4×D53—22no—
84C4⋊C4×D53C2×C414yes—
85C2×#11719921—22no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—22no—
90C2×C4×Dic53D411yescharacteristic
91C2×C4×Dic53D411yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93C42×D53—28no—
94C42×D53C2×C414yes—
95C42×D53D414yes—
96C4⋊#88322—22no—
97#13133146—24no—
98#13133142—24no—
99#13133141—24no—
100#13133140—24no—
101#13133124—22no—
102#13133124—22no—
103C22×Q8×C53C2311yescharacteristic
104C22×C8×C53C2312yes—
105C4×Q8×C53—22no—
106C4×Q8×C53C2×C414yes—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53C2312yes—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C42×C53—21no—
113C2×C42×C53C2311yescharacteristic
114C4⋊C8×C53—22no—
115#103×C53—24no—
116C23×D53C2×D411yescharacteristic
117C2×Q8×C53—21no—
118C2×Q8×C53—21no—
119C2×Q8×C53—21no—
120C2×Q8×C53C22×C412yes—
121C2×Q8×C53C2×D412yes—
122C2×Q8×C53C2×Q812yes—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2411yescharacteristic
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—22no—
132C22×Dic53C2×D411yescharacteristic
133C22×Dic53C2×D411yescharacteristic
134C22×Dic53C2×D411yescharacteristic
135Q8×D53—28no—
136Q8×D53—28no—
137Q8×D53—28no—
138Q8×D53C4⋊C4116yescomplemented
139C2×C4×D53—24no—
140C2×C4×D53—24no—
141C2×C4×D53—24no—
142C2×C4×D53—22no—
143C2×C4×D53—22no—
144C2×C4×D53—24no—
145C2×C4×D53—41no—
146C2×C4×D53—41no—
147C2×C4×D53—42no—
148C2×C4×D53—42no—
149C2×C4×D53C22⋊C414yes—
150C2×C4×D53C22⋊C414yes—
151C2×C4×D53C22×C412yes—
152C2×C4×D53C2×D412yes—
153C2×C4×D53C2×D412yes—
154C2×C4×D53C2×Q812yes—
155C2×C4×D53#4112yes—
156C2×C4×D53#4112yes—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—24no—
161C2×Dic106—24no—
162C2×Dic106—24no—
163C2×Dic106C22×C412yes—
164C2×Dic106C2×D412yes—
165C2×Dic106C2×Q812yes—
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C2×C8×C53—22no—
169C4⋊C4×C53—24no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—42no—
174C4⋊C4×C53—42no—
175C4⋊C4×C53C22×C412yes—
176C42×C53—24no—
177C42×C53C22×C412yes—
178C42×C53C2×D412yes—
179#11719921—44no—
180C4⋊Dic53—21no—
181C4⋊Dic53—24no—
182C4⋊Dic53—22no—
183C4⋊Dic53—22no—
184C4⋊Dic53—42no—
185C4⋊Dic53—42no—
186C4⋊Dic53C22×C412yes—
187#11719919—24no—
188#11719919—24no—
189#11719919—24no—
190C4×Dic53—24no—
191C4×Dic53—24no—
192C4×Dic53—24no—
193C4×Dic53—44no—
194C4×Dic53C22⋊C414yes—
195C4×Dic53C22⋊C414yes—
196C4×Dic53C22×C412yes—
197C4×Dic53C2×D412yes—
198C2×#88322—22no—
199C2×#88322—22no—
200C2×#88322—22no—
201#11719912—48no—
202C23×C53C22×D411yescharacteristic
203C22×D53—21no—
204C22×D53—21no—
205C22×D53—21no—
206C22×D53—21no—
207C22×D53C2×C22⋊C412yes—
208C22×D53#12112yes—
209C22×D53C22⋊Q812yes—
210Q8×C53—22no—
211Q8×C53—22no—
212Q8×C53—22no—
213Q8×C53C2×C4⋊C414yes—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53C2×C22⋊C412yes—
226C2×C4×C53C2×C22⋊C412yes—
227C2×C4×C53#13712yes—
228C2×C4×C53#13712yes—
229C2×C4×C53C23×C411yescharacteristic
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C22×Q811yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—22no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53C2×C22⋊C412yes—
249C2×Dic53C2×C22⋊C412yes—
250C2×Dic53C2×C22⋊C412yes—
251C2×Dic53#12112yes—
252C2×Dic53C22⋊Q812yes—
253C4×D53—216no—
254C4×D53—28no—
255C4×D53—44no—
256C4×D53—48no—
257C4×D53C2×C4⋊C414yes—
258C4×D53#11714yes—
259C4×D53#12114yes—
260C4×D53C22⋊Q814yes—
261Dic106—22no—
262Dic106—22no—
263Dic106—22no—
264Dic106—216no—
265Dic106C2×C4⋊C414yes—
266C8×C53—42no—
267#88322—42no—
268#20647—531nomaximal, Sylow
269C22×C53C22×C22⋊C411yescharacteristic
270C22×C53C2×#12111yescharacteristic
271C22×C53C2×C22⋊Q811yescharacteristic
272C22×C53C2×#13711yescharacteristic
273C22×C53C2×#13711yescharacteristic
274C22×C53#57511yescharacteristic
275C22×C53#57511yescharacteristic
276C2×D53—22no—
277C2×D53—22no—
278C2×D53—22no—
279C2×D53—22no—
280C2×D53—22no—
281C2×D53—22no—
282C2×D53#38014yes—
283C4×C53—22no—
284C4×C53—24no—
285C4×C53—42no—
286C4×C53—82no—
287C4×C53#41914yescomplemented
288C4×C53C22×C4⋊C411yescharacteristic
289C4×C53C2×#11711yescharacteristic
290C4×C53C2×#12111yescharacteristic
291C4×C53C2×C22⋊Q811yescharacteristic
292Dic53—22no—
293Dic53—22no—
294Dic53—24no—
295Dic53—42no—
296Dic53—82no—
297Dic53#38014yes—
298C22×C4×Q8—531no—
299C2×#423—531no—
300C2×#419—532no—
301#2285—531no—
302#1805—532no—
303#1805—532no—
304#1803—531no—
305#1803—531no—
306#1803—531no—
307#1803D5311yescharacteristic, complemented
308#1736—532no—
309C2×C53C2×#38011yescharacteristic
310C2×C53C2×#41911yescharacteristic
311C2×C53#289611yescharacteristic
312C2×C53#303711yescharacteristic
313C2×C53#303711yescharacteristic
314C2×C53#307811yescharacteristic
315C2×C53#307811yescharacteristic
316D53—24no—
317C23×Q8—531no—
318C22×M16—532no—
319C2×C4×Q8—531no—
320C2×C4×Q8—538no—
321C2×C4×Q8—531no—
322C2×C4×Q8—531no—
323C2×C4×Q8—1061no—
324C2×C4×Q8—1061no—
325C2×C4×Q8—1064no—
326C2×C4×Q8C2×D5311yescharacteristic
327C2×#117—532no—
328C22×C4⋊C4—531no—
329C22×C4⋊C4—1061no—
330C22×C42—531no—
331C22×C42—1061no—
332#423—1064no—
333C2×C4⋊C8—532no—
334C2×C4⋊C8—531no—
335C2×C4⋊C8C2×D5311yescharacteristic
336#419—1068no—
337C2×#103—532no—
338C2×#103—532no—
339C2×#103—532no—
340C2×#103C2×D5312yes—
341#384—531no—
342#384C2×D5311yescharacteristic
343#382—532no—
344#380—1061no—
345#380—1061no—
346C4×C4⋊C4—1061no—
347C4×C4⋊C4—1061no—
348#340—532no—
349#340—532no—
350#340—532no—
351#340C2×D5312yes—
352C53#2064711yescharacteristic, complemented, Sylow
353C22×Q8—534no—
354C22×Q8—534no—
355C22×Q8—534no—
356C22×Q8—531no—
357C22×Q8—1061no—
358C22×Q8—1062no—
359C22×Q8—1062no—
360C22×Q8—1062no—
361C22×Q8C22×D5311yescharacteristic
362C23×C4—531no—
363C23×C4—531no—
364C23×C4—531no—
365C23×C4—1061no—
366C23×C4—1061no—
367C2×M16—1064no—
368C2×M16—1064no—
369C2×M16—1064no—
370C22×C8—532no—
371C22×C8C22×D5312yes—
372C4×Q8—22no—
373C4×Q8—534no—
374C4×Q8—534no—
375C4×Q8—534no—
376C4×Q8—1068no—
377C4×Q8—1068no—
378C4×Q8—1062no—
379C4×Q8—1062no—
380C4×Q8—1062no—
381C4×Q8C4×D5314yes—
382#117—1064no—
383#117—1064no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—534no—
389C2×C4⋊C4—532no—
390C2×C4⋊C4—1062no—
391C2×C4⋊C4—1068no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4C22×D5312yes—
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C22⋊C4—1062no—
401C2×C42—21no—
402C2×C42—531no—
403C2×C42—531no—
404C2×C42—531no—
405C2×C42—534no—
406C2×C42—534no—
407C2×C42—1068no—
408C2×C42—1061no—
409C2×C42—1061no—
410C2×C42—1061no—
411C2×C42—1062no—
412C2×C42C22×D5311yescharacteristic
413C4⋊C8—22no—
414C4⋊C8—1064no—
415C4⋊C8—1062no—
416#103—24no—
417#103—1064no—
418#103—1064no—
419#103—1064no—
420#95—1062no—
421#95—1062no—
422C24—531no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—5316no—
427C2×Q8—532no—
428C2×Q8—532no—
429C2×Q8—532no—
430C2×Q8—1064no—
431C2×Q8—1064no—
432C2×Q8—1064no—
433C2×Q8—1068no—
434C2×Q8—1068no—
435C2×Q8—1068no—
436C2×Q8—1061no—
437C2×Q8—1061no—
438C2×Q8—1061no—
439C2×Q8C2×C4×D5312yes—
440C2×Q8D4×D5312yes—
441C2×Q8Q8×D5312yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—531no—
446C22×C4—532no—
447C22×C4—532no—
448C22×C4—532no—
449C22×C4—532no—
450C22×C4—532no—
451C22×C4—532no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—534no—
455C22×C4—534no—
456C22×C4—1064no—
457C22×C4—1064no—
458C22×C4—1064no—
459C22×C4—1064no—
460C22×C4—1061no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1061no—
464C22×C4—1062no—
465C22×C4—2121no—
466C22×C4—2121no—
467C22×C4—2122no—
468C22×C4—2122no—
469C22×C4D4×D5311yescharacteristic
470C22×C4D4×D5311yescharacteristic
471C22×C4C23×D5311yescharacteristic
472M16—2128no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—1062no—
477C2×C8—1062no—
478C2×C8—1062no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—532no—
486C4⋊C4—1064no—
487C4⋊C4—1064no—
488C4⋊C4—1064no—
489C4⋊C4—1064no—
490C4⋊C4—1061no—
491C4⋊C4—1062no—
492C4⋊C4—1062no—
493C4⋊C4—2122no—
494C4⋊C4—2122no—
495C4⋊C4C2×C4×D5312yes—
496C22⋊C4—2124no—
497C42—24no—
498C42—534no—
499C42—534no—
500C42—532no—
501C42—532no—
502C42—1064no—
503C42—1064no—
504C42—1064no—
505C42—2124no—
506C42C2×C4×D5312yes—
507C42D4×D5312yes—
508C23—532no—
509C23—532no—
510C23—532no—
511C23—1061no—
512C23—1061no—
513C23—1061no—
514C23—1061no—
515C23C2×D4×D5311yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—22no—
519Q8—534no—
520Q8—10616no—
521Q8—1062no—
522Q8—1062no—
523Q8—1062no—
524Q8C4⋊C4×D5314yes—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—532no—
537C2×C4—532no—
538C2×C4—532no—
539C2×C4—534no—
540C2×C4—534no—
541C2×C4—534no—
542C2×C4—534no—
543C2×C4—532no—
544C2×C4—532no—
545C2×C4—10616no—
546C2×C4—1062no—
547C2×C4—1062no—
548C2×C4—1062no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1068no—
554C2×C4—1062no—
555C2×C4—2124no—
556C2×C4—2128no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4—2122no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4C22⋊C4×D5312yes—
563C2×C4C22⋊C4×D5312yes—
564C2×C4#1313324612yes—
565C2×C4#1313324912yes—
566C2×C4C22×C4×D5311yescharacteristic
567C2×C4C2×D4×D5311yescharacteristic
568C2×C4C2×D4×D5311yescharacteristic
569C2×C4C2×Q8×D5311yescharacteristic
570C2×C4#41×D5311yescharacteristic
571C2×C4#41×D5311yescharacteristic
572C8—42no—
573C8—2122no—
574C22—534no—
575C22—1062no—
576C22—1062no—
577C22—1062no—
578C22—1062no—
579C22—1062no—
580C22—1062no—
581C22C2×C22⋊C4×D5311yescharacteristic
582C22#121×D5311yescharacteristic
583C22C22⋊Q8×D5311yescharacteristic
584C22C2×#1313324611yescharacteristic
585C22C2×#1313324911yescharacteristic
586C22#1681141911yescharacteristic
587C22#1681142211yescharacteristic
588C4—24no—
589C4—22no—
590C4—42no—
591C4—82no—
592C4—534no—
593C4—1062no—
594C4—1064no—
595C4—1062no—
596C4—4242no—
597C4—2122no—
598C4#1681063014yescomplemented
599C4C2×C4⋊C4×D5311yescharacteristic
600C4#117×D5311yescharacteristic
601C4#121×D5311yescharacteristic
602C4C22⋊Q8×D5311yescharacteristic
603C2—1064no—
604C2#380×D5311yescharacteristic
605C2C2×#1681063011yescharacteristic
606C2#2028323111yescharacteristic
607C2#2028418211yescharacteristic
608C2#2028418811yescharacteristic
609C2#2028451511yescharacteristic
610C2#2028452111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1803C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1803D53
hypocenterC53#20647
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
5. #20274729C2semidirect
6. #20274728C2semidirect
7. #20274721C2semidirect
8. #20274720C2semidirect
9. #20274717C2semidirect
11. C53×#1803C2semidirect
20. C4×Q8×D53C4semidirect
138. Q8×D53C4⋊C4semidirect
287. C4×C53#419semidirect
307. #1803D53semidirect
352. C53#20647semidirect
598. C4#16810630semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D532
C2×#168106452
C2×#168106302
#202771202
#202747292
#202747282
#202747212
#202747202
#202747172
#202744332
C53×#18032
#2064753