Finite Group Databasebeta

← #27465591

Subgroups — #27465591

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×Dic53C211yesmaximal, characteristic
2#20277110C211yesmaximal, characteristic
3#20276151C211yesmaximal, characteristic
4#20276143C211yesmaximal, characteristic
5#20275915C211yesmaximal, characteristic
6#20275486C211yesmaximal, characteristic, complemented
7#20275484C211yesmaximal, characteristic, complemented
8#20275358C211yesmaximal, characteristic, complemented
9#20275352C211yesmaximal, characteristic, complemented
10#20275295C211yesmaximal, characteristic, complemented
11#20275289C211yesmaximal, characteristic, complemented
12#20274728C211yesmaximal, characteristic, complemented
13#20274728C211yesmaximal, characteristic, complemented
14#20274693C211yesmaximal, characteristic
15C53×#2035C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C22×C4×Dic53C2211yescharacteristic
18C2×#13133151C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133146C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26#16810530C2211yescharacteristic
27#16810522C2211yescharacteristic
28#16810509—21no—
29#16810498—21no—
30#16810490—21no—
31#16810488C2211yescharacteristic
32#16810488C2211yescharacteristic
33#16810484—21no—
34#16810483C2211yescharacteristic
35#16810483C2211yescharacteristic
36C4⋊C4×Dic53—21no—
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C22⋊C4×Dic53—21no—
41C22⋊C4×Dic53—21no—
42#16810468—21no—
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C22⋊Q8×C53C2211yescharacteristic
53C2×#106×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C23×Dic53D411yescharacteristic
60C23×Dic53D411yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63Q8×Dic53—22no—
64Q8×Dic53—22no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C22⋊Dic53—21no—
72C2×C22⋊Dic53—21no—
73C2×C4×Dic53—22no—
74C2×C4×Dic53—22no—
75C2×C4×Dic53—21no—
76C2×C4×Dic53—21no—
77C2×C4×Dic53—21no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53D411yescharacteristic
80C2×C4×Dic53D411yescharacteristic
81C2×C4×Dic53C2311yescharacteristic
82C2×C4×Dic53C2311yescharacteristic
83#13133151—41no—
84#13133150—22no—
85#13133148—41no—
86#13133146—22no—
87#13133146—22no—
88#13133141—22no—
89#13133141—41no—
90C22⋊#88322—41no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103C22×Q8×C53C2311yescharacteristic
104C23×C4×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22⋊Q8×C53—22no—
108C22⋊Q8×C53—22no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115#106×C53—22no—
116#103×C53—22no—
117#103×C53—41no—
118C22⋊C8×C53—41no—
119#95×C53—21no—
120C24×C53C2×D411yescharacteristic
121C2×Q8×C53—21no—
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×Q8×C53—21no—
125C2×Q8×C53—21no—
126C2×Q8×C53#4112yes—
127C22×C4×C53—21no—
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×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—41no—
148C22×Dic53—41no—
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C4⋊C4×C53—24no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53#4112yes—
166C22⋊C4×C53—24no—
167C22⋊C4×C53—42no—
168C22⋊Dic53—42no—
169C22⋊Dic53—42no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—42no—
179C4⋊Dic53—42no—
180C4×Dic53—21no—
181C4×Dic53—21no—
182C4×Dic53—22no—
183C4×Dic53—22no—
184C4×Dic53—42no—
185C4×Dic53—42no—
186C4×Dic53—42no—
187C4×Dic53—42no—
188C4×Dic53#4112yes—
189C4×Dic53#4112yes—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53#13712yes—
203C23×C53C22×D411yescharacteristic
204Q8×C53—22no—
205Q8×C53—22no—
206Q8×C53—41no—
207Q8×C53—41no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—21no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53H(5,𝔽2)11yescharacteristic
232C2×C4×C532−1+411yescharacteristic
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—44no—
243C2×Dic53—44no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53#12312yes—
257C2×Dic53#12312yes—
258C2×Dic53#13512yes—
259C8×C53—41no—
260C8×C53—41no—
261#88322—41no—
262#88322—41no—
263#27635—531nomaximal, Sylow
264C22×C53—22no—
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53C2×#12311yescharacteristic
272C22×C53#52511yescharacteristic
273C22×C53#53711yescharacteristic
274C22×C53C2×#13511yescharacteristic
275C22×C53C2×#13711yescharacteristic
276C22×C53#57511yescharacteristic
277C22×C53#57511yescharacteristic
278C4×C53—42no—
279C4×C53—41no—
280C4×C53—42no—
281C4×C53—81no—
282C4×C53—81no—
283C4×C53#55211yescharacteristic
284C4×C53#55311yescharacteristic
285C4×C53#55511yescharacteristic
286C4×C53#55611yescharacteristic
287Dic53—22no—
288Dic53—22no—
289Dic53—22no—
290Dic53—41no—
291Dic53—81no—
292Dic53—81no—
293Dic53—81no—
294Dic53—81no—
295C4×C22⋊Q8—531no—
296#2283—531no—
297#2077—531no—
298#2076—531no—
299#2036—531no—
300#2035D5311yescharacteristic
301#1949—531no—
302#1949—531no—
303#1928—531no—
304#1927—531no—
305#1919—531no—
306#1918—531no—
307#1805—531no—
308#1805—531no—
309#1797—531no—
310C2×C53—42no—
311C2×C53#248511yescharacteristic
312C2×C53#309111yescharacteristic
313C2×C53#309511yescharacteristic
314C2×C53#310211yescharacteristic
315C2×C53#311711yescharacteristic
316C2×C53#324811yescharacteristic
317C2×C53#325611yescharacteristic
318C2×C22⋊Q8C2×D5311yescharacteristic
319C2×C4×Q8—531no—
320C22×C42—531no—
321C2×#107—531no—
322C2×#107—531no—
323C2×#106—531no—
324C2×#106C2×D5311yescharacteristic
325C2×C4⋊C8—531no—
326C2×C4⋊C8—531no—
327C2×#103—531no—
328C2×#103—531no—
329C2×#103C2×D5311yescharacteristic
330C2×#103C2×D5311yescharacteristic
331C2×C22⋊C8—531no—
332C2×C22⋊C8C2×D5311yescharacteristic
333#391—531no—
334#389—531no—
335#387—1061no—
336#385—1061no—
337#383—1061no—
338#382—531no—
339#382—531no—
340#381—1061no—
341#380—531no—
342#380—531no—
343#380C2×D5311yescharacteristic
344C4×C4⋊C4—531no—
345C4×C4⋊C4—531no—
346C4×C4⋊C4—531no—
347C4×C4⋊C4—1061no—
348C4×C22⋊C4—1061no—
349C4×C22⋊C4—1061no—
350#376—1061no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340C2×D5311yescharacteristic
359#336—531no—
360#336—531no—
361C53#2763511yescharacteristic, complemented, Sylow
362C22×Q8C22×D5311yescharacteristic
363C23×C4—531no—
364C23×C4—531no—
365C23×C4C22×D5311yescharacteristic
366C22×C8—531no—
367C22×C8—531no—
368C22×C8C22×D5311yescharacteristic
369C22×C8C22×D5311yescharacteristic
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C4×Q8—1062no—
373C4×Q8—1062no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C42—531no—
389C2×C42—531no—
390C2×C42—531no—
391C2×C42—531no—
392C2×C42—1062no—
393C2×C42—1062no—
394C2×C42—1061no—
395C2×C42—1061no—
396C2×C42—1061no—
397C2×C42—1061no—
398#107—1062no—
399#107—2121no—
400#106—22no—
401#106—2121no—
402C4⋊C8—1062no—
403C4⋊C8—1062no—
404#103—22no—
405#103—41no—
406#103—1062no—
407#103—2121no—
408C22⋊C8—41no—
409C22⋊C8—2121no—
410#95—21no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420#95—1061no—
421#95—1061no—
422#95—1061no—
423C24D4×D5311yescharacteristic
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8#1171994712yes—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—531no—
441C22×C4—531no—
442C22×C4—531no—
443C22×C4—531no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1062no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—2121no—
453C22×C4—2121no—
454C22×C4D4×D5311yescharacteristic
455C22×C4C23×D5311yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—2122no—
483C4⋊C4—2122no—
484C4⋊C4#1171994712yes—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—2122no—
488C22⋊C4—2122no—
489C42—532no—
490C42—532no—
491C42—1062no—
492C42—1062no—
493C42—1061no—
494C42—1061no—
495C42—2122no—
496C42—2122no—
497C42—2122no—
498C42—2122no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23#1313324612yes—
506C23C2×D4×D5311yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—532no—
531C2×C4—532no—
532C2×C4—532no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—2124no—
543C2×C4—2124no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4#41×D5311yescharacteristic
557C2×C4#41×D5311yescharacteristic
558C2×C4C2×#1171994711yescharacteristic
559C2×C4C2×#1171994711yescharacteristic
560C2×C4#1313326211yescharacteristic
561C2×C4#1313326411yescharacteristic
562C8—41no—
563C8—41no—
564C8—2121no—
565C8—2121no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22#123×D5311yescharacteristic
574C22#1681112911yescharacteristic
575C22#1681120011yescharacteristic
576C22#135×D5311yescharacteristic
577C22C2×#1313324611yescharacteristic
578C22#1681142211yescharacteristic
579C22#1681142211yescharacteristic
580C4—42no—
581C4—41no—
582C4—42no—
583C4—81no—
584C4—81no—
585C4—1062no—
586C4—1062no—
587C4—1062no—
588C4—4241no—
589C4—4241no—
590C4—4241no—
591C4—4241no—
592C4—2121no—
593C4#1681131011yescharacteristic
594C4#1681131611yescharacteristic
595C4#1681133711yescharacteristic
596C4#1681134211yescharacteristic
597C2—42no—
598C2#2027900511yescharacteristic
599C2#2028463311yescharacteristic
600C2#2028465811yescharacteristic
601C2#2028473411yescharacteristic
602C2#2028488111yescharacteristic
603C2#2028604511yescharacteristic
604C2#2028611711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2035C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2035D53
hypocenterC53#27635
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
6. #20275486C2semidirect
7. #20275484C2semidirect
8. #20275358C2semidirect
9. #20275352C2semidirect
10. #20275295C2semidirect
11. #20275289C2semidirect
12. #20274728C2semidirect
13. #20274728C2semidirect
361. C53#27635semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×Dic532
#202771102
#202761512
#202761432
#202759152
#202754862
#202754842
#202753582
#202753522
#202752952
#202752892
#202747282
#202746932
C53×#20352
#2763553