Finite Group Databasebeta

← #27443147

Subgroups — #27443147

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279950C211yesmaximal, characteristic, complemented
2#20277690C211yesmaximal, characteristic
3#20276240C211yesmaximal, characteristic
4#20276233C211yesmaximal, characteristic
5#20276084C211yesmaximal, characteristic
6#20276084C211yesmaximal, characteristic
7#20275484C211yesmaximal, characteristic, complemented
8#20275481C211yesmaximal, characteristic, complemented
9#20275475C211yesmaximal, characteristic, complemented
10#20275433C211yesmaximal, characteristic, complemented
11#20275309C211yesmaximal, characteristic, complemented
12#20275288C211yesmaximal, characteristic, complemented
13#20274722C211yesmaximal, characteristic
14C4×#13133139C211yesmaximal, characteristic
15C53×#1846C211yesmaximal, characteristic, complemented
16C2×C4×D212C2211yescharacteristic
17C2×#13133150C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133146C2211yescharacteristic
21C2×#13133146C2211yescharacteristic
22C2×#13133139C2211yescharacteristic
23C2×#13133139C2211yescharacteristic
24C2×#13133137C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26#16810565—21no—
27#16810555—21no—
28#16810524C2211yescharacteristic
29#16810524C2211yescharacteristic
30#16810522C2211yescharacteristic
31#16810522C2211yescharacteristic
32#16810515—21no—
33#16810515—21no—
34#16810510—21no—
35#16810506—21no—
36#16810492C2211yescharacteristic
37#16810488C2211yescharacteristic
38#16810485—21no—
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42C4×C4⋊Dic53C2211yescharacteristic
43#16810470—21no—
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#107×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×#106×C53C2211yescharacteristic
56C2×C4×C8×C53C2211yescharacteristic
57C53×#382C2211yescharacteristic
58C53×#382C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C4×D53—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C4×D212—22no—
64C4×D212Q814yes—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53D411yescharacteristic
68C2×C4⋊Dic53D411yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719921—21no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53C2311yescharacteristic
83C2×C4×Dic53C2311yescharacteristic
84#13133150—41no—
85#13133148—41no—
86C4⋊#88322—22no—
87#13133146—22no—
88#13133146—22no—
89#13133139—22no—
90#13133139—22no—
91#13133137—41no—
92#13133137—41no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124D411yescharacteristic
101#13133124D411yescharacteristic
102C22×C8×C53C2311yescharacteristic
103C22×C8×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C42×C53C2311yescharacteristic
109#107×C53—22no—
110#107×C53—22no—
111#106×C53—41no—
112#106×C53—41no—
113C4×C8×C53—22no—
114#95×C53—21no—
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118C23×D53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2411yescharacteristic
126C22×Dic53—21no—
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53C2×D411yescharacteristic
130C22×Dic53C2×D411yescharacteristic
131C22×Dic53C2×D411yescharacteristic
132C22×Dic53C2×D411yescharacteristic
133C2×D212—21no—
134C2×D212—21no—
135C2×D212—21no—
136C2×D212—21no—
137C2×D212—21no—
138C2×D212C2×Q812yes—
139C2×C4×D53—24no—
140C2×C4×D53—81no—
141C2×C4×D53—81no—
142C2×C8×C53—21no—
143C2×C8×C53—21no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53#4112yes—
147C2×C8×C53#4112yes—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C42×C53C2×Q812yes—
157C42×C53#4112yes—
158#11719921—24no—
159#11719921—42no—
160#11719921—42no—
161#11719921—81no—
162#11719921—81no—
163C4⋊Dic53—21no—
164C4⋊Dic53—22no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—41no—
170C4⋊Dic53—42no—
171C4⋊Dic53—81no—
172C4⋊Dic53C2×Q812yes—
173#11719919—42no—
174#11719919—81no—
175C4×Dic53—21no—
176C4×Dic53—21no—
177C4×Dic53—21no—
178C4×Dic53—21no—
179C4×Dic53—81no—
180C4×Dic53—81no—
181C4×Dic53—81no—
182C2×#88322—21no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C23×C53C22×D411yescharacteristic
189C22×D53—22no—
190C22×D53—41no—
191C22×D53—41no—
192C22×D53—41no—
193C22×D53—41no—
194C22×D53—41no—
195C22×D53—41no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53C22⋊Q812yes—
213C2×C4×C53C22⋊Q812yes—
214C2×C4×C53C22×Q811yescharacteristic
215C2×C4×C53C2×#4111yescharacteristic
216C2×C4×C53C2×#4111yescharacteristic
217C2×C4×C53C2×#4111yescharacteristic
218C2×C4×C53H(5,𝔽2)11yescharacteristic
219C2×C4×C532−1+411yescharacteristic
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—22no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53#13512yes—
241D212—22no—
242D212—22no—
243D212—41no—
244D212—41no—
245C4×D53—84no—
246C8×C53—22no—
247C8×C53—22no—
248#88322—41no—
249#88322—41no—
250#26094—531nomaximal, Sylow
251C22×C53C2×C22⋊Q811yescharacteristic
252C22×C53#54711yescharacteristic
253C22×C53#55011yescharacteristic
254C22×C53C2×#13511yescharacteristic
255C22×C53#57511yescharacteristic
256C22×C53#57611yescharacteristic
257C22×C53#57711yescharacteristic
258C2×D53—42no—
259C2×D53—42no—
260C2×D53—42no—
261C2×D53—81no—
262C2×D53—81no—
263C2×D53—81no—
264C2×D53—81no—
265C4×C53—22no—
266C4×C53—22no—
267C4×C53—81no—
268C4×C53—81no—
269C4×C53—81no—
270C4×C53—81no—
271C4×C53#53311yescharacteristic
272C4×C53#54411yescharacteristic
273C4×C53#55211yescharacteristic
274C4×C53#55311yescharacteristic
275Dic53—22no—
276Dic53—42no—
277Dic53—41no—
278Dic53—41no—
279Dic53—41no—
280Dic53—81no—
281Dic53—81no—
282#2570—531no—
283#2357—531no—
284#2093—531no—
285#2092—531no—
286#2065—531no—
287#2065—531no—
288#1949—531no—
289#1949—531no—
290#1948—531no—
291#1939—531no—
292#1921—531no—
293#1918—531no—
294#1846D5311yescharacteristic, complemented
295#1804—531no—
296C4×#102—531no—
297C2×C53#244111yescharacteristic
298C2×C53#307611yescharacteristic
299C2×C53#308111yescharacteristic
300C2×C53#333711yescharacteristic
301C2×C53#335511yescharacteristic
302C2×C53#339611yescharacteristic
303C2×C53#340111yescharacteristic
304D53—82no—
305C2×C4×D4—531no—
306C2×#107—531no—
307C2×#107C2×D5311yescharacteristic
308C2×#107C2×D5311yescharacteristic
309C2×#106—531no—
310C2×#106C2×D5311yescharacteristic
311C2×#106C2×D5311yescharacteristic
312C2×C4⋊C8—531no—
313C2×C4⋊C8—531no—
314C2×C4⋊C8—531no—
315C2×#102—531no—
316C2×#102—531no—
317C2×#102—531no—
318C2×#102—531no—
319C2×C4×C8C2×D5311yescharacteristic
320#400—1061no—
321#398—1061no—
322#390—531no—
323#390—531no—
324#389—531no—
325#389—531no—
326#388—1061no—
327#388—1061no—
328#387—1061no—
329#387—1061no—
330#384—531no—
331#382—531no—
332#382C2×D5311yescharacteristic
333#382C2×D5311yescharacteristic
334#381—1061no—
335C4×C4⋊C4—531no—
336C4×C4⋊C4—531no—
337C4×C4⋊C4—531no—
338C4×C4⋊C4—531no—
339C4×C22⋊C4—1061no—
340#340—531no—
341#340—531no—
342#340—531no—
343#340—531no—
344#340—531no—
345#340—531no—
346#336—531no—
347#336—531no—
348C53#2609411yescharacteristic, complemented, Sylow
349C22×D4—531no—
350C23×C4—1061no—
351C22×C8—531no—
352C22×C8—531no—
353C22×C8C22×D5311yescharacteristic
354C22×C8C22×D5311yescharacteristic
355C4×D4—534no—
356C4×D4—1062no—
357C2×C4⋊C4—531no—
358C2×C4⋊C4—531no—
359C2×C4⋊C4—531no—
360C2×C4⋊C4—531no—
361C2×C4⋊C4—531no—
362C2×C4⋊C4—1061no—
363C2×C4⋊C4—1061no—
364C2×C4⋊C4—1061no—
365C2×C4⋊C4—1061no—
366C2×C4⋊C4C22×D5311yescharacteristic
367C2×C4⋊C4C22×D5311yescharacteristic
368C2×C4⋊C4C22×D5311yescharacteristic
369C2×C4⋊C4C22×D5311yescharacteristic
370C2×C22⋊C4—1061no—
371C2×C22⋊C4—1061no—
372C2×C22⋊C4—1061no—
373C2×C22⋊C4—1061no—
374C2×C22⋊C4—1061no—
375C2×C42—531no—
376C2×C42—531no—
377C2×C42—1061no—
378C2×C42—1061no—
379C2×C42—1061no—
380C2×C42C22×D5311yescharacteristic
381#107—22no—
382#107—22no—
383#107—2121no—
384#106—41no—
385#106—41no—
386#106—2121no—
387C4⋊C8—1062no—
388C4⋊C8—1062no—
389C4⋊C8—1062no—
390#102—1062no—
391#102—1062no—
392#102—2121no—
393#102—2121no—
394C4×C8—22no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—531no—
400#95—531no—
401#95—1061no—
402#95—1061no—
403#95—1061no—
404#95—1061no—
405#95—1061no—
406#95—1061no—
407#95—1061no—
408C24—1061no—
409C2×D4—532no—
410C2×D4—1061no—
411C2×D4—1061no—
412C2×D4—1061no—
413C2×D4—1061no—
414C2×D4—1061no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—531no—
420C22×C4—531no—
421C22×C4—531no—
422C22×C4—531no—
423C22×C4—1061no—
424C22×C4—1064no—
425C22×C4—1061no—
426C22×C4—1061no—
427C22×C4—4241no—
428C22×C4—4241no—
429C22×C4D4×D5311yescharacteristic
430C22×C4D4×D5311yescharacteristic
431C22×C4C23×D5311yescharacteristic
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—1061no—
437C2×C8—1061no—
438C2×C8—1061no—
439C2×C8—1061no—
440C2×C8—1061no—
441C2×C8—1061no—
442C2×C8#1171994712yes—
443C2×C8#1171994912yes—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—532no—
453C4⋊C4—1062no—
454C4⋊C4—1061no—
455C4⋊C4—1061no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—4241no—
460C4⋊C4—4241no—
461C4⋊C4—2121no—
462C4⋊C4—2122no—
463C4⋊C4—2122no—
464C22⋊C4—1064no—
465C22⋊C4—4241no—
466C22⋊C4—4241no—
467C22⋊C4—2122no—
468C22⋊C4—2122no—
469C42—1061no—
470C42—1061no—
471C42—1061no—
472C42—1061no—
473C42—4241no—
474C42—4241no—
475C42—4241no—
476C42#1171994512yes—
477C42Q8×D5312yes—
478C23—1062no—
479C23—2121no—
480C23—2121no—
481C23—2121no—
482C23—2121no—
483C23—2121no—
484C23—2121no—
485C23C2×D4×D5311yescharacteristic
486D4—1062no—
487D4—1062no—
488D4—2121no—
489D4—2121no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—532no—
507C2×C4—1061no—
508C2×C4—1061no—
509C2×C4—1061no—
510C2×C4—1061no—
511C2×C4—1061no—
512C2×C4—1062no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—4244no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4—2121no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4#1313319512yes—
529C2×C4#1313319512yes—
530C2×C4C2×Q8×D5311yescharacteristic
531C2×C4C2×#1171994711yescharacteristic
532C2×C4C2×#1171994511yescharacteristic
533C2×C4C2×#1171994911yescharacteristic
534C2×C4#1313326211yescharacteristic
535C2×C4#1313326511yescharacteristic
536C8—22no—
537C8—22no—
538C8—2121no—
539C8—2121no—
540C22—4241no—
541C22—4241no—
542C22—4241no—
543C22—4241no—
544C22—2122no—
545C22—2122no—
546C22—2122no—
547C22C2×#1313319511yescharacteristic
548C22#1681127611yescharacteristic
549C22#1681129311yescharacteristic
550C22#135×D5311yescharacteristic
551C22#1681142011yescharacteristic
552C22#1681142511yescharacteristic
553C22#1681143011yescharacteristic
554C4—22no—
555C4—22no—
556C4—81no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4—1062no—
561C4—4241no—
562C4—4241no—
563C4—2121no—
564C4—2121no—
565C4—2121no—
566C4—2122no—
567C4#1681117811yescharacteristic
568C4#1681125011yescharacteristic
569C4#1681130811yescharacteristic
570C4#1681132111yescharacteristic
571C2—4242no—
572C2#2027843111yescharacteristic
573C2#2028450111yescharacteristic
574C2#2028454911yescharacteristic
575C2#2028709311yescharacteristic
576C2#2028731911yescharacteristic
577C2#2028772411yescharacteristic
578C2#2028778711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1846C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#1846D53
hypocenterC53#26094
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1846

Complements

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

NH ≅ G/Ntype
1. #20279950C2semidirect
7. #20275484C2semidirect
8. #20275481C2semidirect
9. #20275475C2semidirect
10. #20275433C2semidirect
11. #20275309C2semidirect
12. #20275288C2semidirect
15. C53×#1846C2semidirect
294. #1846D53semidirect
348. C53#26094semidirect

Maximal subgroups

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

subgroupindex
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C4×#131331392
C53×#18462
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