Finite Group Databasebeta

← #27420167

Subgroups — #27420167

Order 13568.

1150 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810910C211yesmaximal, characteristic, complemented
2C2×#16810900C211yesmaximal, characteristic, complemented
3C4×#13133191C211yesmaximal, characteristic
4#20276050C212yesmaximal, complemented
5#20275858C212yesmaximal, complemented
6#20275668C211yesmaximal, characteristic, complemented
7#20275664C211yesmaximal, characteristic, complemented
8#20275037C212yesmaximal
9#20274914C212yesmaximal, complemented
10#20274695C211yesmaximal, characteristic, complemented
11C2×C53×#533C211yesmaximal, characteristic, complemented
12C2×#13133191C2211yescharacteristic
13C2×C4×D212C2211yescharacteristic
14C2×C4×C22⋊D53—21no—
15#16810910—24no—
16#16810900—24no—
17C2×#13133150C2212yes—
18C2×#13133148C2212yes—
19C2×C4⋊#88322C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133137C2212yes—
22C2×#13133137C2212yes—
23C2×C22⋊#88322C2212yes—
24#16810525C2211yescharacteristic
25#16810521C2211yescharacteristic
26#16810514—21no—
27#16810502—21no—
28#16810483C2211yescharacteristic
29C4×C4⋊Dic53C2211yescharacteristic
30C4×#11719921—21no—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35#16810378C2212yes—
36C53×#533—24no—
37C53×#533—24no—
38C2×C4⋊Q8×C53C2211yescharacteristic
39C2×C4⋊Q8×C53C2211yescharacteristic
40C2×#125×C53C2212yes—
41C2×C22⋊Q8×C53—21no—
42C2×C22⋊Q8×C53—21no—
43C2×C4×Q8×C53—21no—
44C2×C4×Q8×C53—21no—
45C2×#117×C53C2212yes—
46C22×C42×C53C2211yescharacteristic
47C22×D212C2311yescharacteristic
48C22×C22⋊D53—21no—
49C22×C4×D53—21no—
50C22×#88322C2312yes—
51C22×#88322C2312yes—
52#13133191—22no—
53#13133191Q814yescomplemented
54C4×D212—22no—
55C4×D212—22no—
56C4×C22⋊D53—44no—
57C2×C4⋊Dic53C2311yescharacteristic
58C2×C4⋊Dic53C2311yescharacteristic
59C2×C4⋊Dic53C2311yescharacteristic
60C2×#11719919—21no—
61C2×C22⋊Dic53—21no—
62C2×#11719921—21no—
63C2×#11719921—21no—
64C2×#11719921—21no—
65C2×#11719921—21no—
66C2×C4×Dic53—21no—
67#13133150—24no—
68#13133148—24no—
69C4⋊#88322—22no—
70C4⋊#88322—22no—
71#13133137—24no—
72#13133137—24no—
73C22⋊#88322—24no—
74#13133124—21no—
75#13133124—21no—
76#13133124—21no—
77C22×Q8×C53—21no—
78C22×Q8×C53—21no—
79C23×C4×C53D411yescharacteristic
80C23×C4×C53D411yescharacteristic
81C23×C4×C53C2311yescharacteristic
82C4⋊Q8×C53—22no—
83C4⋊Q8×C53—22no—
84C4⋊Q8×C53D414yescomplemented
85C4⋊Q8×C53D414yescomplemented
86#125×C53—24no—
87#125×C53—24no—
88C22⋊Q8×C53—28no—
89C22⋊Q8×C53—28no—
90C4×Q8×C53—28no—
91C4×Q8×C53—28no—
92#117×C53—24no—
93#117×C53—24no—
94C2×C4⋊C4×C53—22no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53C2312yes—
99C2×C4⋊C4×C53C2312yes—
100C2×C22⋊C4×C53—22no—
101C2×C42×C53—22no—
102C2×C42×C53—22no—
103C2×C42×C53—22no—
104C2×C42×C53D411yescharacteristic
105C2×C42×C53D411yescharacteristic
106C2×C42×C53C2311yescharacteristic
107C2×C42×C53C2311yescharacteristic
108C24×C53C2×D411yescharacteristic
109C23×D53—21no—
110C2×Q8×C53—24no—
111C2×Q8×C53—24no—
112C2×Q8×C53—42no—
113C2×Q8×C53—42no—
114C2×Q8×C53—42no—
115C2×Q8×C53—42no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—41no—
127C22×C4×C53—41no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×Q812yes—
134C22×C4×C53#4112yes—
135C22×C4×C53C2411yescharacteristic
136C2×C22⋊D53—24no—
137C2×C22⋊D53—42no—
138C2×C22⋊D53—42no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212C2×Q812yes—
148C2×C4×D53—42no—
149C2×C4×D53—42no—
150C2×C4×D53—42no—
151C4⋊C4×C53—28no—
152C4⋊C4×C53—28no—
153C4⋊C4×C53—24no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53C2×D414yes—
158C4⋊C4×C53C2×D414yes—
159C22⋊C4×C53—28no—
160C42×C53—28no—
161C42×C53—22no—
162C42×C53—22no—
163C42×C53—21no—
164C42×C53—21no—
165C42×C53C2×D412yes—
166C42×C53C2×D412yes—
167C22⋊Dic53—42no—
168#11719921—24no—
169#11719921—42no—
170#11719921—42no—
171#11719921—42no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53C2×Q812yes—
178#11719919—42no—
179C4×Dic53—42no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322—22no—
186C23×C53—21no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53C22⋊Q812yes—
193C23×C53C22×D411yescharacteristic
194C22×D53—22no—
195C22×D53—41no—
196C22×D53—41no—
197C22×D53—41no—
198C22×D53—41no—
199C22×D53—41no—
200C22×D53—41no—
201Q8×C53—48no—
202Q8×C53—48no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—24no—
214C2×C4×C53—24no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—44no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—44no—
225C2×C4×C53#12012yes—
226C2×C4×C53#12012yes—
227C2×C4×C53C22⋊Q812yes—
228C2×C4×C53#13512yes—
229C2×C4×C53AGL(1,ℤ/8)12yes—
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C22×Q811yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C532−1+411yescharacteristic
235C2×C4×C532−1+411yescharacteristic
236C22⋊D53—48no—
237C2×Dic53—22no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246D212—22no—
247D212—22no—
248D212—41no—
249D212—41no—
250C4×D53—84no—
251#88322—42no—
252#88322—42no—
253#24300—531nomaximal, Sylow
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53C2×#12011yescharacteristic
262C22×C53C2×C22⋊Q811yescharacteristic
263C22×C53#53611yescharacteristic
264C22×C53C2×#13511yescharacteristic
265C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C53#57511yescharacteristic
267C22×C53#57511yescharacteristic
268C2×D53—42no—
269C2×D53—42no—
270C2×D53—42no—
271C2×D53—81no—
272C2×D53—81no—
273C2×D53—81no—
274C2×D53—81no—
275C4×C53—22no—
276C4×C53—22no—
277C4×C53—41no—
278C4×C53—44no—
279C4×C53—41no—
280C4×C53—44no—
281C4×C53#44914yes—
282C4×C53#54811yescharacteristic
283C4×C53#54811yescharacteristic
284C4×C53D4×Q811yescharacteristic
285C4×C53D4×Q811yescharacteristic
286Dic53—42no—
287Dic53—81no—
288Dic53—81no—
289C2×#533D5311yescharacteristic, complemented
290C2×#476—531no—
291C2×#474—531no—
292C4×#121—531no—
293#2059—532no—
294#2025—532no—
295#1988—531no—
296#1987—531no—
297#1875—532no—
298#1854—532no—
299#1797—531no—
300C2×C53—42no—
301C2×C53#240911yescharacteristic
302C2×C53C2×#44911yescharacteristic
303C2×C53#300511yescharacteristic
304C2×C53#307611yescharacteristic
305C2×C53#307711yescharacteristic
306C2×C53#323311yescharacteristic
307C2×C53#323511yescharacteristic
308D53—82no—
309#533—24no—
310#533—24no—
311C2×C4⋊Q8C2×D5311yescharacteristic
312C2×C4⋊Q8C2×D5311yescharacteristic
313C2×#125C2×D5312yes—
314C2×C22⋊Q8—21no—
315C2×C22⋊Q8—21no—
316C2×#121—531no—
317C2×C4×Q8—21no—
318C2×C4×Q8—21no—
319C2×C4×D4—531no—
320C2×C4×D4—1061no—
321C2×#117C2×D5312yes—
322C22×C42C2×D5311yescharacteristic
323#476—1064no—
324#474—1064no—
325C2×#107—532no—
326C2×#106—532no—
327C2×C4⋊C8—531no—
328C2×C4⋊C8—531no—
329C2×#102—532no—
330C2×#102—532no—
331C2×C22⋊C8—532no—
332#390—531no—
333#389—531no—
334#388—1061no—
335#386—1061no—
336#380—531no—
337C4×C4⋊C4—531no—
338C4×C22⋊C4—1061no—
339#340—532no—
340#340—532no—
341#340—532no—
342#340—532no—
343#336—532no—
344C53#2430011yescharacteristic, complemented, Sylow
345C22×Q8—21no—
346C22×Q8—21no—
347C22×D4—531no—
348C22×D4—1061no—
349C23×C4—1061no—
350C23×C4C22⋊D5311yescharacteristic
351C23×C4C22⋊D5311yescharacteristic
352C23×C4C22×D5311yescharacteristic
353C22×C8—532no—
354C22×C8—532no—
355C4⋊Q8—22no—
356C4⋊Q8—22no—
357C4⋊Q8C22⋊D5314yescomplemented
358C4⋊Q8C22⋊D5314yescomplemented
359#125—24no—
360#125—24no—
361C22⋊Q8—28no—
362C22⋊Q8—28no—
363#121—534no—
364#121—1062no—
365C4×Q8—28no—
366C4×Q8—28no—
367C4×D4—1062no—
368C4×D4—1062no—
369C4×D4—2124no—
370#117—24no—
371#117—24no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4C22×D5312yes—
381C2×C4⋊C4C22×D5312yes—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—1061no—
384C2×C22⋊C4—1061no—
385C2×C22⋊C4—1061no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—1061no—
392C2×C42C22⋊D5311yescharacteristic
393C2×C42C22⋊D5311yescharacteristic
394C2×C42C22×D5311yescharacteristic
395C2×C42C22×D5311yescharacteristic
396#107—1064no—
397#106—1064no—
398C4⋊C8—1062no—
399C4⋊C8—1062no—
400#102—1064no—
401#102—1064no—
402C22⋊C8—1064no—
403#95—1061no—
404#95—1061no—
405#95—1061no—
406C24—1061no—
407C24C2×C22⋊D5311yescharacteristic
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×D4—532no—
415C2×D4—1064no—
416C2×D4—1061no—
417C2×D4—1061no—
418C2×D4—1061no—
419C2×D4—1061no—
420C2×D4—1061no—
421C2×D4—2122no—
422C2×D4—2122no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—2122no—
439C22×C4—2122no—
440C22×C4—2122no—
441C22×C4Q8×D5312yes—
442C22×C4#1171994912yes—
443C22×C4C2×C22⋊D5311yescharacteristic
444C22×C4C2×C22⋊D5311yescharacteristic
445C22×C4C2×C22⋊D5311yescharacteristic
446C22×C4C2×C22⋊D5311yescharacteristic
447C22×C4C2×C22⋊D5311yescharacteristic
448C22×C4C23×D5311yescharacteristic
449C2×C8—1062no—
450C2×C8—1062no—
451C2×C8—1062no—
452C2×C8—1062no—
453C2×C8—1062no—
454C2×C8—1062no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—532no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—2122no—
468C4⋊C4C2×C22⋊D5314yes—
469C4⋊C4C2×C22⋊D5314yes—
470C22⋊C4—28no—
471C22⋊C4—1064no—
472C22⋊C4—2122no—
473C22⋊C4—2122no—
474C22⋊C4—2122no—
475C22⋊C4—2122no—
476C42—28no—
477C42—22no—
478C42—22no—
479C42—21no—
480C42—21no—
481C42—2122no—
482C42C2×C22⋊D5312yes—
483C42C2×C22⋊D5312yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—1062no—
491C23—2121no—
492C23—2121no—
493C23—2121no—
494C23—2121no—
495C23—2121no—
496C23—2121no—
497C23#1313319312yes—
498C23C22×C22⋊D5311yescharacteristic
499Q8—48no—
500Q8—48no—
501D4—1062no—
502D4—1062no—
503D4—2128no—
504D4—2121no—
505D4—2121no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—1062no—
529C2×C4—4244no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4#1313318612yes—
539C2×C4#1313318612yes—
540C2×C4#1313319312yes—
541C2×C4#1313323712yes—
542C2×C4#1313324212yes—
543C2×C4C22×C22⋊D5311yescharacteristic
544C2×C4C22×C22⋊D5311yescharacteristic
545C2×C4C2×Q8×D5311yescharacteristic
546C2×C4C2×#1171994911yescharacteristic
547C2×C4#1313326511yescharacteristic
548C2×C4#1313326511yescharacteristic
549C8—2122no—
550C8—2122no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—4241no—
559C22—4241no—
560C22—4241no—
561C22—4241no—
562C22—2122no—
563C22—2122no—
564C22—2122no—
565C22C2×#1313318611yescharacteristic
566C22C2×#1313319311yescharacteristic
567C22#1681119711yescharacteristic
568C22C2×#1313323711yescharacteristic
569C22C2×#1313324211yescharacteristic
570C22#1681142111yescharacteristic
571C22#1681142111yescharacteristic
572C4—22no—
573C4—22no—
574C4—41no—
575C4—44no—
576C4—41no—
577C4—44no—
578C4—4241no—
579C4—4241no—
580C4—2122no—
581C4#1681076814yes—
582C4#1681128411yescharacteristic
583C4#1681128411yescharacteristic
584C4Q8×C22⋊D5311yescharacteristic
585C4Q8×C22⋊D5311yescharacteristic
586C2—42no—
587C2—4242no—
588C2#2027807411yescharacteristic
589C2C2×#1681076811yescharacteristic
590C2#2028395711yescharacteristic
591C2#2028449711yescharacteristic
592C2#2028450811yescharacteristic
593C2#2028589411yescharacteristic
594C2#2028591311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#533C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenterC2×#533D53
hypocenterC53#24300
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×#16810910C2semidirect
2. C2×#16810900C2semidirect
4. #20276050C2semidirect
5. #20275858C2semidirect
6. #20275668C2semidirect
7. #20275664C2semidirect
9. #20274914C2semidirect
10. #20274695C2semidirect
11. C2×C53×#533C2semidirect
53. #13133191Q8semidirect
84. C4⋊Q8×C53D4semidirect
85. C4⋊Q8×C53D4semidirect
289. C2×#533D53semidirect
344. C53#24300semidirect
357. C4⋊Q8C22⋊D53semidirect
358. C4⋊Q8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109102
C2×#168109002
C4×#131331912
#202760502
#202758582
#202756682
#202756642
#202750372
#202749142
#202746952
C2×C53×#5332
#2430053