Finite Group Databasebeta

← #27472227

Subgroups — #27472227

Order 13568.

1072 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×#16810823C211yesmaximal, characteristic, complemented
2C2×#16810818C211yesmaximal, characteristic
3C4×#13133214C211yesmaximal, characteristic, complemented
4#20276129C212yesmaximal, complemented
5#20275792C212yesmaximal, complemented
6#20275066C212yesmaximal, complemented
7#20275052C212yesmaximal, complemented
8#20274832C211yesmaximal, characteristic, complemented
9#20274720C211yesmaximal, characteristic, complemented
10#20274715C211yesmaximal, characteristic, complemented
11C4×#124×C53C211yesmaximal, characteristic, complemented
12C22×#11719924C2212yes—
13C22×#11719922C2212yes—
14C2×#13133214C2211yescharacteristic
15C2×C4×D212—21no—
16C2×C4×D212—21no—
17C2×C4×D212C2212yes—
18#16810823—24no—
19#16810818—24no—
20C2×C4⋊#88322C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133141C2212yes—
23C2×#13133138C2212yes—
24C2×#13133137C2212yes—
25C2×#13133137C2212yes—
26#16810525—21no—
27#16810525—21no—
28#16810525C2212yes—
29#16810493C2211yescharacteristic
30#16810385C2212yes—
31#16810385C2212yes—
32#16810378C2212yes—
33#16810378C2212yes—
34C2×#124×C53C2211yescharacteristic
35C2×C4×Q8×C53C2211yescharacteristic
36C2×C4×D4×C53C2211yescharacteristic
37C53×#391C2211yescharacteristic
38C53×#390C2211yescharacteristic
39C53×#388—21no—
40C53×#388—21no—
41C53×#382C2211yescharacteristic
42C4×C22⋊C4×C53—21no—
43C4×C22⋊C4×C53—21no—
44C43×C53C2211yescharacteristic
45C22×D212—21no—
46C22×D212—21no—
47C22×D212C2312yes—
48C22×C4×D53—22no—
49C2×#11719924—24no—
50C2×#11719924—24no—
51C2×#11719924—42no—
52C2×#11719922—24no—
53C2×#11719922—24no—
54C2×#11719922—42no—
55C22×#88322C2312yes—
56C22×#88322C2312yes—
57#13133214—22no—
58#13133214D414yes—
59C4×D212—42no—
60C4×D212—42no—
61C4×D212—44no—
62C4×D212—44no—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53C2312yes—
66C2×#11719921—22no—
67C2×#11719921—22no—
68C2×#11719921—22no—
69C4⋊#88322—22no—
70C4⋊#88322—22no—
71#13133141—42no—
72#13133138—42no—
73#13133137—24no—
74#13133137—24no—
75C22×Q8×C53C2311yescharacteristic
76C22×D4×C53C2311yescharacteristic
77C23×C4×C53—21no—
78#124×C53—22no—
79#124×C53—22no—
80C4×Q8×C53—22no—
81C4×Q8×C53—22no—
82C4×D4×C53—22no—
83C4×D4×C53—22no—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53C2311yescharacteristic
86C2×C4⋊C4×C53C2311yescharacteristic
87C2×C22⋊C4×C53—21no—
88C2×C22⋊C4×C53—21no—
89C2×C22⋊C4×C53—21no—
90C2×C22⋊C4×C53—21no—
91C2×C22⋊C4×C53—21no—
92C2×C42×C53—21no—
93C2×C42×C53D411yescharacteristic
94C2×C42×C53D411yescharacteristic
95C2×C42×C53D411yescharacteristic
96C2×C42×C53D411yescharacteristic
97C2×C42×C53C2311yescharacteristic
98C2×C42×C53C2311yescharacteristic
99C2×C42×C53C2311yescharacteristic
100#95×C53—21no—
101#95×C53—21no—
102#95×C53—21no—
103#95×C53—21no—
104C24×C53—21no—
105C23×D53—22no—
106C2×Q8×C53—21no—
107C2×Q8×C53—21no—
108C2×Q8×C53—21no—
109C2×Q8×C53—21no—
110C2×Q8×C53—21no—
111C2×Q8×C53#4112yes—
112C2×D4×C53—21no—
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53#4112yes—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—42no—
122C22×C4×C53—42no—
123C22×C4×C53—42no—
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2411yescharacteristic
131C22×Dic53—22no—
132C2×D212—22no—
133C2×D212—22no—
134C2×D212—22no—
135C2×D212—22no—
136C2×D212—24no—
137C2×D212—24no—
138C2×D212—22no—
139C2×D212—42no—
140C2×D212—42no—
141C2×D212—42no—
142C2×D212—42no—
143C2×D212C2×D414yes—
144C2×C4×D53—44no—
145C2×C4×D53—44no—
146C2×C4×D53—44no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—42no—
152C22⋊C4×C53—24no—
153C22⋊C4×C53—24no—
154C22⋊C4×C53—42no—
155C22⋊C4×C53—42no—
156C22⋊C4×C53—42no—
157C42×C53—22no—
158C42×C53—22no—
159C42×C53—22no—
160C42×C53—21no—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53—21no—
164C42×C53—41no—
165C42×C53—42no—
166C42×C53C2×D412yes—
167C42×C53#4112yes—
168#11719924—48no—
169#11719922—48no—
170#11719921—44no—
171#11719921—44no—
172#11719921—44no—
173C4⋊Dic53—22no—
174C4⋊Dic53—22no—
175C4⋊Dic53—42no—
176C4⋊Dic53—42no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322—22no—
183C23×C53—22no—
184C23×C53—41no—
185C23×C53—41no—
186C23×C53—41no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53—41no—
190C23×C53C22×D411yescharacteristic
191C22×D53—24no—
192C22×D53—42no—
193C22×D53—42no—
194C22×D53—42no—
195C22×D53—42no—
196C22×D53—42no—
197C22×D53—42no—
198Q8×C53—22no—
199Q8×C53—22no—
200Q8×C53—41no—
201Q8×C53—41no—
202D4×C53—22no—
203D4×C53—22no—
204D4×C53—41no—
205D4×C53—41no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—22no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
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×C53—41no—
228C2×C4×C53—84no—
229C2×C4×C53#12112yes—
230C2×C4×C53#12112yes—
231C2×C4×C53#13512yes—
232C2×C4×C53AGL(1,ℤ/8)12yes—
233C2×C4×C53AGL(1,ℤ/8)12yes—
234C2×C4×C53C22×D411yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53H(5,𝔽2)11yescharacteristic
239C2×C4×C53H(5,𝔽2)11yescharacteristic
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243D212—24no—
244D212—24no—
245D212—48no—
246D212—42no—
247D212—42no—
248D212—48no—
249C4×D53—88no—
250#88322—42no—
251#88322—42no—
252#28158—531nomaximal, Sylow
253C22×C53—42no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53—81no—
257C22×C53—81no—
258C22×C53—81no—
259C22×C53—81no—
260C22×C53C2×#12111yescharacteristic
261C22×C53#52511yescharacteristic
262C22×C53#53511yescharacteristic
263C22×C53C2×#13511yescharacteristic
264C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C53#57511yescharacteristic
267C2×D53—44no—
268C2×D53—44no—
269C2×D53—44no—
270C2×D53—82no—
271C2×D53—82no—
272C2×D53—82no—
273C2×D53—82no—
274C4×C53—22no—
275C4×C53—22no—
276C4×C53—41no—
277C4×C53—41no—
278C4×C53—41no—
279C4×C53—42no—
280C4×C53—81no—
281C4×C53—81no—
282C4×C53#46314yes—
283C4×C53#54611yescharacteristic
284C4×C53#54611yescharacteristic
285C4×C53#55311yescharacteristic
286C4×C53#55311yescharacteristic
287Dic53—82no—
288C2×#460—531no—
289C2×#459—531no—
290C4×#127—531no—
291C4×#124D5311yescharacteristic, complemented
292#2073—532no—
293#2012—532no—
294#1879—532no—
295#1877—532no—
296#1831—531no—
297#1803—531no—
298#1802—531no—
299C2×C53—82no—
300C2×C53#253511yescharacteristic
301C2×C53C2×#46311yescharacteristic
302C2×C53#303611yescharacteristic
303C2×C53#310611yescharacteristic
304C2×C53#311511yescharacteristic
305C2×C53#321211yescharacteristic
306C2×C53#321711yescharacteristic
307D53—84no—
308C22×SD16—532no—
309C22×D8—532no—
310C2×#127—531no—
311C2×#124C2×D5311yescharacteristic
312C2×C4×Q8C2×D5311yescharacteristic
313C2×C4×D4—532no—
314C2×C4×D4—1061no—
315C2×C4×D4—1061no—
316C2×C4×D4C2×D5311yescharacteristic
317#460—1064no—
318#459—1064no—
319C2×C4⋊C8—531no—
320C2×C4⋊C8—531no—
321C2×#103—532no—
322C2×#102—532no—
323C2×#102—532no—
324C2×#102—532no—
325#391C2×D5311yescharacteristic
326#390—532no—
327#390—1061no—
328#390—1061no—
329#390C2×D5311yescharacteristic
330#388—21no—
331#388—21no—
332#384—531no—
333#382C2×D5311yescharacteristic
334C4×C22⋊C4—21no—
335C4×C22⋊C4—21no—
336C43C2×D5311yescharacteristic
337#340—532no—
338#340—532no—
339#336—532no—
340#336—532no—
341C53#2815811yescharacteristic, complemented, Sylow
342C22×Q8C22×D5311yescharacteristic
343C22×D4—532no—
344C22×D4—1061no—
345C22×D4—1061no—
346C22×D4C22×D5311yescharacteristic
347C23×C4—21no—
348C23×C4—1062no—
349C2×SD16—1064no—
350C2×SD16—1064no—
351C2×SD16—2122no—
352C2×D8—1064no—
353C2×D8—1064no—
354C2×D8—2122no—
355C22×C8—532no—
356C22×C8—532no—
357#127—534no—
358#127—1062no—
359#124—22no—
360#124—22no—
361C4×Q8—22no—
362C4×Q8—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—2122no—
366C4×D4—2122no—
367C4×D4—2124no—
368C4×D4—2124no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—532no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4C22×D5311yescharacteristic
374C2×C4⋊C4C22×D5311yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—1062no—
381C2×C22⋊C4—1062no—
382C2×C22⋊C4—1062no—
383C2×C42—21no—
384C2×C42C22⋊D5311yescharacteristic
385C2×C42C22⋊D5311yescharacteristic
386C2×C42C22⋊D5311yescharacteristic
387C2×C42C22⋊D5311yescharacteristic
388C2×C42C22×D5311yescharacteristic
389C2×C42C22×D5311yescharacteristic
390C2×C42C22×D5311yescharacteristic
391C4⋊C8—1062no—
392C4⋊C8—1062no—
393#103—2122no—
394#102—1064no—
395#102—1064no—
396#102—2122no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—21no—
401C24—21no—
402C24—1062no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8#1171994512yes—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—534no—
415C2×D4—1064no—
416C2×D4—1064no—
417C2×D4—1062no—
418C2×D4—1062no—
419C2×D4—1062no—
420C2×D4—1062no—
421C2×D4—1062no—
422C2×D4—2122no—
423C2×D4—2122no—
424C2×D4—2122no—
425C2×D4—2122no—
426C2×D4#1171994512yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—1062no—
434C22×C4—2124no—
435C22×C4—2124no—
436C22×C4—2124no—
437C22×C4C2×C22⋊D5311yescharacteristic
438C22×C4C2×C22⋊D5311yescharacteristic
439C22×C4C2×C22⋊D5311yescharacteristic
440C22×C4C2×C22⋊D5311yescharacteristic
441C22×C4C2×C22⋊D5311yescharacteristic
442C22×C4C2×C22⋊D5311yescharacteristic
443C22×C4C23×D5311yescharacteristic
444SD16—2128no—
445D8—2128no—
446C2×C8—1062no—
447C2×C8—1062no—
448C2×C8—1062no—
449C2×C8—1062no—
450C2×C8—1062no—
451C2×C8—1062no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—42no—
457C4⋊C4—1062no—
458C4⋊C4—1062no—
459C4⋊C4—2122no—
460C4⋊C4—2122no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—2124no—
467C22⋊C4—2124no—
468C22⋊C4—2124no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—41no—
477C42—42no—
478C42D4×D5312yes—
479C42#1171994712yes—
480C23—22no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—1064no—
488C23—2122no—
489C23—2122no—
490C23—2122no—
491C23—2122no—
492C23—2122no—
493C23—2122no—
494C23C22×C22⋊D5311yescharacteristic
495Q8—22no—
496Q8—22no—
497Q8—41no—
498Q8—41no—
499D4—22no—
500D4—22no—
501D4—41no—
502D4—41no—
503D4—1064no—
504D4—1064no—
505D4—2128no—
506D4—2122no—
507D4—2122no—
508D4—2128no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—84no—
532C2×C4—4248no—
533C2×C4—2122no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4#1313318812yes—
537C2×C4#1313318812yes—
538C2×C4#1313323712yes—
539C2×C4#1313324212yes—
540C2×C4#1313324212yes—
541C2×C4C2×D4×D5311yescharacteristic
542C2×C4C2×#1171994711yescharacteristic
543C2×C4C2×#1171994511yescharacteristic
544C2×C4C2×#1171994511yescharacteristic
545C2×C4#1313326311yescharacteristic
546C2×C4#1313326311yescharacteristic
547C8—2122no—
548C8—2122no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—4242no—
557C22—4242no—
558C22—4242no—
559C22—4242no—
560C22—2124no—
561C22—2124no—
562C22—2124no—
563C22C2×#1313318811yescharacteristic
564C22#1681113411yescharacteristic
565C22#1681118711yescharacteristic
566C22C2×#1313323711yescharacteristic
567C22C2×#1313324211yescharacteristic
568C22C2×#1313324211yescharacteristic
569C22#1681142111yescharacteristic
570C4—22no—
571C4—22no—
572C4—42no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—81no—
577C4—81no—
578C4—4242no—
579C4#1681083514yes—
580C4#1681126911yescharacteristic
581C4#1681126911yescharacteristic
582C4#1681131911yescharacteristic
583C4#1681131911yescharacteristic
584C2—82no—
585C2—4244no—
586C2#2027961611yescharacteristic
587C2C2×#1681083511yescharacteristic
588C2#2028417511yescharacteristic
589C2#2028477511yescharacteristic
590C2#2028486611yescharacteristic
591C2#2028568711yescharacteristic
592C2#2028573011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC4×#124×C53C2
socleC23×C53C22×D4
prime radicalC2×#16810818C2
hypercenterC4×#124D53
hypocenterC53#28158
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
1. C2×#16810823C2semidirect
3. C4×#13133214C2semidirect
4. #20276129C2semidirect
5. #20275792C2semidirect
6. #20275066C2semidirect
7. #20275052C2semidirect
8. #20274832C2semidirect
9. #20274720C2semidirect
10. #20274715C2semidirect
11. C4×#124×C53C2semidirect
291. C4×#124D53semidirect
341. C53#28158semidirect

Maximal subgroups

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

subgroupindex
C2×#168108232
C2×#168108182
C4×#131332142
#202761292
#202757922
#202750662
#202750522
#202748322
#202747202
#202747152
C4×#124×C532
#2815853