Finite Group Databasebeta

← #27411438

Subgroups — #27411438

Order 13568.

1250 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810622C211yesmaximal, characteristic
2C2×#16810617C211yesmaximal, characteristic
3#20277303C211yesmaximal, characteristic
4#20275863C212yesmaximal, complemented
5#20275858C212yesmaximal, complemented
6#20275853C212yesmaximal, complemented
7#20275848C212yesmaximal, complemented
8#20274436C211yesmaximal, characteristic, complemented
9C2×#16810385C211yesmaximal, characteristic, complemented
10#20274412C211yesmaximal, characteristic, complemented
11C2×C53×#544C211yesmaximal, characteristic, complemented
12C22×#11719945C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14C23×#88322C2211yescharacteristic
15C22×C4⋊Dic53C2211yescharacteristic
16#16810622—24no—
17#16810617—24no—
18C2×#13133140C2211yescharacteristic
19C2×#13133140C2211yescharacteristic
20C2×#13133140C2212yes—
21C2×#13133137C2211yescharacteristic
22C2×#13133137C2211yescharacteristic
23C2×#13133137C2212yes—
24#16810529C2212yes—
25#16810525C2212yes—
26#16810500—22no—
27#16810480—21no—
28#16810480—21no—
29#16810385C2212yes—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
34#16810378C2212yes—
35C53×#544—24no—
36C53×#544—24no—
37C2×C4⋊Q8×C53—21no—
38C2×C4⋊Q8×C53C2212yes—
39C2×#125×C53—21no—
40C2×#125×C53C2212yes—
41C2×C22⋊Q8×C53—22no—
42C2×#117×C53C2211yescharacteristic
43C2×#117×C53C2211yescharacteristic
44C22×C4⋊C4×C53C2211yescharacteristic
45C2×#11719945—22no—
46C2×#11719945—22no—
47C2×#11719945—22no—
48C2×#11719945—22no—
49C2×#11719945—22no—
50C2×#11719945Q814yescomplemented
51C22×Dic106C2311yescharacteristic
52C22×D212C2311yescharacteristic
53C22×C22⋊D53—21no—
54C23×Dic53—21no—
55C22×C4×D53—21no—
56C2×#11719917—22no—
57C2×#11719917—22no—
58C2×#11719917—22no—
59C22×#88322—22no—
60C22×#88322—22no—
61C22×#88322—22no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2312yes—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—22no—
67C2×C4⋊Dic53—22no—
68C2×C4⋊Dic53C2312yes—
69C2×#11719921—22no—
70C2×#11719921—22no—
71#13133140—22no—
72#13133140—22no—
73#13133140—24no—
74#13133137—22no—
75#13133137—22no—
76#13133137—24no—
77#13133124—22no—
78#13133124—22no—
79C22×Q8×C53—22no—
80C23×C4×C53—21no—
81C23×C4×C53C2311yescharacteristic
82C4⋊Q8×C53—28no—
83C4⋊Q8×C53—24no—
84C4⋊Q8×C53—24no—
85#125×C53—28no—
86#125×C53—24no—
87#125×C53—24no—
88C22⋊Q8×C53—216no—
89#117×C53—22no—
90#117×C53—22no—
91#117×C53—22no—
92#117×C53D414yes—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—22no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53—22no—
98C2×C4⋊C4×C53D414yes—
99C2×C4⋊C4×C53D411yescharacteristic
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C4⋊C4×C53C2312yes—
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53D412yes—
106C2×C42×C53—21no—
107C2×C42×C53C2312yes—
108C24×C53C2×D411yescharacteristic
109C23×D53—21no—
110C2×Q8×C53—28no—
111C2×Q8×C53—82no—
112C2×Q8×C53—82no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—22no—
116C22×C4×C53—24no—
117C22×C4×C53—81no—
118C22×C4×C53—81no—
119C22×C4×C53C2×D412yes—
120C22×C4×C53C2×D412yes—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D412yes—
123C22×C4×C53C2×D412yes—
124C22×C4×C53C2×Q812yes—
125C22×C4×C53#4112yes—
126C22×C4×C53#4112yes—
127C22×C4×C53#4112yes—
128C22×C4×C53C2411yescharacteristic
129C2×C22⋊D53—24no—
130C2×C22⋊D53—42no—
131C2×C22⋊D53—42no—
132C22×Dic53—21no—
133C22×Dic53—22no—
134C22×Dic53—42no—
135C22×Dic53—81no—
136C22×Dic53—81no—
137#11719945—28no—
138#11719945—28no—
139#11719945—44no—
140#11719945—44no—
141C2×D212—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212C2×Q812yes—
147C2×C4×D53—24no—
148C2×C4×D53—42no—
149C2×C4×D53—42no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106C2×Q812yes—
156C4⋊C4×C53—28no—
157C4⋊C4×C53—28no—
158C4⋊C4×C53—28no—
159C4⋊C4×C53—24no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—22no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53C2×D414yes—
166C4⋊C4×C53C2×D412yes—
167C4⋊C4×C53C2×D412yes—
168C22⋊C4×C53—24no—
169C22⋊C4×C53—24no—
170C42×C53—24no—
171C42×C53—22no—
172C42×C53—22no—
173#11719921—82no—
174#11719921—82no—
175C4⋊Dic53—22no—
176C4⋊Dic53—22no—
177C4⋊Dic53—42no—
178#11719917—44no—
179C2×#88322—21no—
180C2×#88322—21no—
181C2×#88322—22no—
182C2×#88322—21no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322—44no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53#12012yes—
194C23×C53C22⋊Q812yes—
195C23×C53#12412yes—
196C23×C53C22×D411yescharacteristic
197C22×D53—22no—
198C22×D53—41no—
199C22×D53—41no—
200C22×D53—41no—
201C22×D53—41no—
202C22×D53—41no—
203C22×D53—41no—
204Q8×C53—88no—
205C2×C4×C53—24no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—42no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—42no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—84no—
221C2×C4×C53#12012yes—
222C2×C4×C53C22⋊Q814yes—
223C2×C4×C53C22⋊Q814yes—
224C2×C4×C53C22⋊Q812yes—
225C2×C4×C53#12414yes—
226C2×C4×C53#12414yes—
227C2×C4×C53#12412yes—
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C22×Q811yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C22⋊D53—48no—
235C2×Dic53—22no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—84no—
242D212—22no—
243D212—22no—
244D212—41no—
245D212—41no—
246C4×D53—48no—
247Dic106—22no—
248Dic106—22no—
249Dic106—41no—
250Dic106—41no—
251#88322—41no—
252#88322—41no—
253#88322—42no—
254#23458—531nomaximal, Sylow
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—22no—
261C22×C53#39314yes—
262C22×C53C2×#12011yescharacteristic
263C22×C53C2×C22⋊Q811yescharacteristic
264C22×C53C2×#12411yescharacteristic
265C22×C53#57611yescharacteristic
266C22×C53#57711yescharacteristic
267C22×C53#57711yescharacteristic
268C22×C53#57811yescharacteristic
269C2×D53—42no—
270C2×D53—42no—
271C2×D53—42no—
272C2×D53—81no—
273C2×D53—81no—
274C2×D53—81no—
275C2×D53—81no—
276C4×C53—42no—
277C4×C53—44no—
278C4×C53—42no—
279C4×C53—42no—
280C4×C53—42no—
281C4×C53#39314yes—
282C4×C53C2×C22⋊Q811yescharacteristic
283C4×C53C2×C22⋊Q811yescharacteristic
284C4×C53C2×#12411yescharacteristic
285C4×C53C2×#12411yescharacteristic
286Dic53—42no—
287Dic53—82no—
288C2×#544D5311yescharacteristic, complemented
289C2×#417—531no—
290C2×#416—531no—
291#2311—531no—
292#2026—532no—
293#2025—532no—
294#2024—532no—
295#2023—532no—
296#1736—531no—
297C2×#340—531no—
298#1729—531no—
299C2×C53—24no—
300C2×C53C2×#39311yescharacteristic
301C2×C53#301211yescharacteristic
302C2×C53#301311yescharacteristic
303C2×C53#308311yescharacteristic
304C2×C53#308411yescharacteristic
305C2×C53#313611yescharacteristic
306C2×C53#313711yescharacteristic
307D53—82no—
308C22×#41—531no—
309C22×M16—531no—
310C23×C8—531no—
311#544—24no—
312#544—24no—
313C2×C4⋊Q8—21no—
314C2×C4⋊Q8C2×D5312yes—
315C2×#125—21no—
316C2×#125C2×D5312yes—
317C2×C22⋊Q8—22no—
318C2×#117C2×D5311yescharacteristic
319C2×#117C2×D5311yescharacteristic
320C22×C4⋊C4—531no—
321C22×C4⋊C4C2×D5311yescharacteristic
322#417—1064no—
323#416—1064no—
324C2×#103—531no—
325C2×#103—531no—
326C2×#103—532no—
327C2×#102—531no—
328C2×#102—531no—
329C2×#102—532no—
330#391—532no—
331#390—532no—
332#386—1062no—
333#380—1061no—
334#380—1061no—
335#340—532no—
336#340—532no—
337#340—532no—
338#340—532no—
339#336—532no—
340#336—532no—
341C53#2345811yescharacteristic, complemented, Sylow
342C2×#41—534no—
343C2×#41—1062no—
344C2×#41—1062no—
345C2×#41—1062no—
346C2×#41—1062no—
347C2×#41—1062no—
348C22×Q8—22no—
349C22×Q8—531no—
350C22×D4—531no—
351C22×D4—1061no—
352C23×C4—21no—
353C23×C4—1061no—
354C23×C4—1061no—
355C23×C4C22×D5311yescharacteristic
356C2×M16—1062no—
357C2×M16—1062no—
358C2×M16—1062no—
359C22×C8—531no—
360C22×C8—531no—
361C22×C8—532no—
362C22×C8—1062no—
363C22×C8—1062no—
364C22×C8—1062no—
365C4⋊Q8—28no—
366C4⋊Q8—24no—
367C4⋊Q8—24no—
368#125—28no—
369#125—24no—
370#125—24no—
371C22⋊Q8—216no—
372#117—22no—
373#117—22no—
374#117—22no—
375#117C22⋊D5314yes—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—532no—
382C2×C4⋊C4—1062no—
383C2×C4⋊C4—1062no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4C22⋊D5311yescharacteristic
386C2×C4⋊C4C22⋊D5311yescharacteristic
387C2×C4⋊C4C22⋊D5314yes—
388C2×C4⋊C4C22×D5312yes—
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1062no—
393C2×C22⋊C4—1062no—
394C2×C22⋊C4C22⋊D5312yes—
395C2×C42—21no—
396C2×C42C22×D5312yes—
397#103—1062no—
398#103—1062no—
399#103—1064no—
400#102—1062no—
401#102—1062no—
402#102—1064no—
403#95—1062no—
404#95—1062no—
405C24—1061no—
406C24C2×C22⋊D5311yescharacteristic
407#41—1068no—
408#41—1068no—
409#41—2124no—
410#41—2124no—
411C2×Q8—28no—
412C2×Q8—82no—
413C2×Q8—82no—
414C2×Q8—532no—
415C2×Q8—1061no—
416C2×Q8—1061no—
417C2×Q8—1061no—
418C2×Q8—1061no—
419C2×Q8—1061no—
420C2×D4—532no—
421C2×D4—1064no—
422C2×D4—1061no—
423C2×D4—1061no—
424C2×D4—1061no—
425C2×D4—1061no—
426C2×D4—1061no—
427C2×D4—2122no—
428C2×D4—2122no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—81no—
434C22×C4—81no—
435C22×C4—1061no—
436C22×C4—1064no—
437C22×C4—1062no—
438C22×C4—4241no—
439C22×C4—4241no—
440C22×C4—2122no—
441C22×C4—2122no—
442C22×C4—2122no—
443C22×C4Q8×D5312yes—
444C22×C4#1171994912yes—
445C22×C4#1171994912yes—
446C22×C4#1171994912yes—
447C22×C4C2×C22⋊D5312yes—
448C22×C4C2×C22⋊D5312yes—
449C22×C4C2×C22⋊D5311yescharacteristic
450C22×C4C2×C22⋊D5312yes—
451C22×C4C2×C22⋊D5312yes—
452C22×C4C23×D5311yescharacteristic
453M16—2124no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1062no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1062no—
462C2×C8—1062no—
463C2×C8—2124no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—1062no—
474C4⋊C4—1062no—
475C4⋊C4—2122no—
476C4⋊C4C2×C22⋊D5314yes—
477C4⋊C4C2×C22⋊D5312yes—
478C4⋊C4C2×C22⋊D5312yes—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—4242no—
482C22⋊C4—4242no—
483C42—24no—
484C42—22no—
485C42—22no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—1062no—
491C23—2121no—
492C23—2121no—
493C23—2121no—
494C23—2121no—
495C23—2121no—
496C23—2121no—
497C23#1313318612yes—
498C23#1313319312yes—
499C23#1313320612yes—
500C23C22×C22⋊D5311yescharacteristic
501Q8—88no—
502Q8—1062no—
503Q8—1062no—
504Q8—2121no—
505Q8—2121no—
506D4—1062no—
507D4—1062no—
508D4—2128no—
509D4—2121no—
510D4—2121no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—84no—
527C2×C4—1062no—
528C2×C4—4244no—
529C2×C4—2122no—
530C2×C4—2122no—
531C2×C4—2122no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2128no—
535C2×C4#1313318612yes—
536C2×C4#1313319312yes—
537C2×C4#1313319314yes—
538C2×C4#1313319314yes—
539C2×C4#1313320612yes—
540C2×C4#1313320614yes—
541C2×C4#1313320614yes—
542C2×C4C22×C22⋊D5311yescharacteristic
543C2×C4C22×C22⋊D5311yescharacteristic
544C2×C4C2×Q8×D5311yescharacteristic
545C2×C4C2×#1171994911yescharacteristic
546C2×C4C2×#1171994911yescharacteristic
547C2×C4C2×#1171994911yescharacteristic
548C8—2121no—
549C8—2121no—
550C8—2122no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—4241no—
558C22—4241no—
559C22—4241no—
560C22—4241no—
561C22—2122no—
562C22—2122no—
563C22—2122no—
564C22#1681053414yes—
565C22C2×#1313318611yescharacteristic
566C22C2×#1313319311yescharacteristic
567C22C2×#1313320611yescharacteristic
568C22#1681142611yescharacteristic
569C22#1681143111yescharacteristic
570C22#1681143411yescharacteristic
571C22#1681143811yescharacteristic
572C4—42no—
573C4—44no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—4242no—
578C4—2122no—
579C4#1681053414yes—
580C4C2×#1313319311yescharacteristic
581C4C2×#1313319311yescharacteristic
582C4C2×#1313320611yescharacteristic
583C4C2×#1313320611yescharacteristic
584C2—24no—
585C2—4242no—
586C2C2×#1681053411yescharacteristic
587C2#2028401911yescharacteristic
588C2#2028403011yescharacteristic
589C2#2028456211yescharacteristic
590C2#2028457311yescharacteristic
591C2#2028507411yescharacteristic
592C2#2028508511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#544C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
hypercenterC2×#544D53
hypocenterC53#23458
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
4. #20275863C2semidirect
5. #20275858C2semidirect
6. #20275853C2semidirect
7. #20275848C2semidirect
8. #20274436C2semidirect
9. C2×#16810385C2semidirect
10. #20274412C2semidirect
11. C2×C53×#544C2semidirect
50. C2×#11719945Q8semidirect
288. C2×#544D53semidirect
341. C53#23458semidirect

Maximal subgroups

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

subgroupindex
C2×#168106222
C2×#168106172
#202773032
#202758632
#202758582
#202758532
#202758482
#202744362
C2×#168103852
#202744122
C2×C53×#5442
#2345853