Finite Group Databasebeta

← #27390010

Subgroups — #27390010

Order 13568.

1486 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810791C211yesmaximal, characteristic
2C4×#13133242C212yesmaximal, complemented
3#20277437C211yesmaximal, characteristic, complemented
4#20275113C211yesmaximal, characteristic, complemented
5#20275091C212yesmaximal, complemented
6#20274948C212yesmaximal, complemented
7#20274925C211yesmaximal, characteristic, complemented
8#20274669C211yesmaximal, characteristic, complemented
9#20274525C212yesmaximal, complemented
10C4×#13133134C211yesmaximal, characteristic, complemented
11C4×H(5,𝔽2)×C53C211yesmaximal, characteristic, complemented
12C2×#13133242C2212yes—
13C2×#13133213C2211yescharacteristic
14C2×C4×D212C2211yescharacteristic, complemented
15#16810791C418yescomplemented
16#16810733—22no—
17#16810721—22no—
18C4×#11719924—22no—
19C4×#11719922—22no—
20#16810666C2212yescomplemented
21C2×#13133144C2212yes—
22C2×#13133144C2212yes—
23#16810627C2212yescomplemented
24#16810624C2212yes—
25C2×#13133134C2211yescharacteristic
26C2×#13133134C2211yescharacteristic
27C4×#11719917C2212yescomplemented
28#16810526C2211yescharacteristic
29#16810524C2211yescharacteristic, complemented
30#16810502—22no—
31#16810499—22no—
32#16810488C2211yescharacteristic, complemented
33#16810469C2211yescharacteristic, complemented
34#16810392C2212yescomplemented
35#16810382C2212yescomplemented
36#16810382C2212yescomplemented
37C2×H(5,𝔽2)×C53C2211yescharacteristic
38C53×#520—22no—
39C53×#520C2212yes—
40C53×#518—22no—
41C53×#518—22no—
42C53×#518C2211yescharacteristic
43C4×#41×C53—22no—
44C4×#41×C53C2212yes—
45C2×C4×D4×C53—22no—
46C2×C4×D4×C53—22no—
47C2×C4×D4×C53C2211yescharacteristic
48C22×D212C2311yescharacteristic
49C22×C22⋊D53—22no—
50C22×C4×D53—21no—
51#13133242—24no—
52#13133242C2×C418yescomplemented
53C2×#11719924—22no—
54C2×#11719922—22no—
55C2×#11719917C2312yes—
56C2×#11719917C2312yes—
57#13133213—22no—
58#13133213C2×C414yescomplemented
59C4×D212—22no—
60C4×D212—22no—
61C2×C4⋊Dic53C2311yescharacteristic
62C2×C22⋊Dic53—22no—
63C2×#11719921—21no—
64C2×#11719921—22no—
65C2×C4×Dic53C2311yescharacteristic
66C2×C4×Dic53C2311yescharacteristic
67#13133150—22no—
68#13133148—22no—
69#13133144—24no—
70#13133144C2×C418yescomplemented
71#13133141—22no—
72#13133138—22no—
73#13133137—22no—
74#13133137—22no—
75#13133134—22no—
76#13133134C2×C414yescomplemented
77#13133128—22no—
78C4×#88322—22no—
79#13133124—21no—
80#13133124—22no—
81#13133124—21no—
82H(5,𝔽2)×C53—24no—
83H(5,𝔽2)×C53—22no—
84H(5,𝔽2)×C53C2×C414yescomplemented
85C2×#41×C53—22no—
86C2×#41×C53C2312yes—
87C22×D4×C53—22no—
88C22×D4×C53—22no—
89C22×D4×C53C2311yescharacteristic
90C23×C4×C53—22no—
91C23×C4×C53D412yes—
92C4×Q8×C53—22no—
93C4×Q8×C53D412yescomplemented
94C4×Q8×C53D412yescomplemented
95C4×D4×C53—24no—
96C4×D4×C53—24no—
97C4×D4×C53—24no—
98C4×D4×C53—24no—
99C4×D4×C53—22no—
100C4×D4×C53—22no—
101C4×D4×C53—22no—
102C4×D4×C53—22no—
103C4×D4×C53—22no—
104C4×D4×C53—22no—
105C4×D4×C53—24no—
106C4×D4×C53—22no—
107C4×D4×C53D412yescomplemented
108C4×D4×C53D412yescomplemented
109#117×C53—24no—
110#117×C53—22no—
111#117×C53—22no—
112#117×C53C2312yes—
113C2×C4⋊C4×C53—22no—
114C2×C4⋊C4×C53—22no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C22⋊C4×C53—22no—
117C2×C22⋊C4×C53—24no—
118C2×C22⋊C4×C53—22no—
119C2×C22⋊C4×C53D412yes—
120C2×C42×C53—22no—
121C2×C42×C53—22no—
122C2×C42×C53C2311yescharacteristic
123C24×C53—22no—
124C24×C53C2×D412yes—
125C23×D53—21no—
126#41×C53—28no—
127#41×C53—28no—
128#41×C53—22no—
129#41×C53—22no—
130#41×C53C22×C414yes—
131#41×C53#4114yes—
132C2×Q8×C53C2×D412yes—
133C2×D4×C53—24no—
134C2×D4×C53—28no—
135C2×D4×C53—28no—
136C2×D4×C53—28no—
137C2×D4×C53—22no—
138C2×D4×C53—28no—
139C2×D4×C53—22no—
140C2×D4×C53—22no—
141C2×D4×C53C22×C412yes—
142C2×D4×C53C2×D412yes—
143C2×D4×C53#4112yes—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53—22no—
147C22×C4×C53—22no—
148C22×C4×C53—24no—
149C22×C4×C53—22no—
150C22×C4×C53—22no—
151C22×C4×C53—22no—
152C22×C4×C53—22no—
153C22×C4×C53—82no—
154C22×C4×C53—84no—
155C22×C4×C53C2×D412yes—
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2411yescharacteristic
159C2×C22⋊D53—28no—
160C2×C22⋊D53—82no—
161C2×C22⋊D53—82no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C2×D212—22no—
166C2×D212—22no—
167C2×D212—21no—
168C2×D212C22×C412yescomplemented
169C2×C4×D53—42no—
170C2×C4×D53—82no—
171C4⋊C4×C53—24no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—22no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53C2×D412yes—
176C4⋊C4×C53C2×D412yes—
177C22⋊C4×C53—24no—
178C22⋊C4×C53—24no—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—22no—
181C22⋊C4×C53—22no—
182C22⋊C4×C53—24no—
183C42×C53—24no—
184C42×C53—22no—
185C42×C53—22no—
186C42×C53—22no—
187C42×C53C2×D412yes—
188C42×C53C2×D412yes—
189C22⋊Dic53—82no—
190#11719924—28no—
191#11719922—28no—
192#11719921—41no—
193#11719921—41no—
194#11719921—82no—
195C4⋊Dic53—22no—
196C4×Dic53—21no—
197C4×Dic53—22no—
198C4×Dic53C22×C412yescomplemented
199#11719917—22no—
200#11719917—22no—
201#11719917—22no—
202#11719917C22×C414yescomplemented
203C2×#88322—22no—
204C2×#88322—22no—
205C23×C53—28no—
206C23×C53—22no—
207C23×C53—22no—
208C23×C53—22no—
209C23×C53—42no—
210C23×C53—42no—
211C23×C53—82no—
212C23×C53—82no—
213C23×C53C4×D414yes—
214C23×C53C22×D411yescharacteristic
215C22×D53—22no—
216C22×D53—42no—
217C22×D53—42no—
218C22×D53—41no—
219C22×D53—41no—
220Q8×C53—22no—
221Q8×C53C4×D414yes—
222D4×C53—216no—
223D4×C53—24no—
224D4×C53—24no—
225D4×C53—24no—
226D4×C53—22no—
227D4×C53—24no—
228D4×C53C4×D414yes—
229C2×C4×C53—28no—
230C2×C4×C53—22no—
231C2×C4×C53—24no—
232C2×C4×C53—24no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—44no—
239C2×C4×C53—44no—
240C2×C4×C53—42no—
241C2×C4×C53—82no—
242C2×C4×C53—82no—
243C2×C4×C53—82no—
244C2×C4×C53—84no—
245C2×C4×C53—82no—
246C2×C4×C53#12012yes—
247C2×C4×C53#12011yescharacteristic
248C2×C4×C53#12011yescharacteristic
249C2×C4×C53C23×C411yescharacteristic
250C2×C4×C53C22×D412yes—
251C2×C4×C53C2×#4112yes—
252C2×C4×C53C2×#4111yescharacteristic
253C22⋊D53—88no—
254C2×Dic53—22no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—42no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262D212—24no—
263D212—42no—
264C4×D53—84no—
265#88322—24no—
266#88322—42no—
267#21473—531nomaximal, Sylow
268C22×C53—21no—
269C22×C53—21no—
270C22×C53—24no—
271C22×C53—24no—
272C22×C53—44no—
273C22×C53—42no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—82no—
278C22×C53—88no—
279C22×C53C2×C4×D411yescharacteristic
280C22×C53C2×#12011yescharacteristic
281C22×C53#52511yescharacteristic
282C2×D53—44no—
283C2×D53—42no—
284C2×D53—82no—
285C2×D53—81no—
286C2×D53—81no—
287C4×C53—21no—
288C4×C53—21no—
289C4×C53—22no—
290C4×C53—24no—
291C4×C53—42no—
292C4×C53—42no—
293C4×C53—42no—
294C4×C53C2×C4×D412yes—
295C4×C53#52512yes—
296Dic53—42no—
297Dic53—81no—
298Dic53—81no—
299C4×H(5,𝔽2)D5311yescharacteristic, complemented
300C2×#453—531no—
301C4×AGL(1,ℤ/8)—532no—
302#2328—531no—
303#1887—531no—
304#1883—532no—
305#1860—532no—
306#1856—531no—
307#1792—531no—
308#1759—532no—
309C4×#100—531no—
310C2×C53—21no—
311C2×C53—21no—
312C2×C53—44no—
313C2×C53—44no—
314C2×C53C4×#12011yescharacteristic
315C2×C53#301911yescharacteristic
316C2×C53#302111yescharacteristic
317D53—82no—
318C2×H(5,𝔽2)C2×D5311yescharacteristic
319C2×AGL(1,ℤ/8)—532no—
320C2×#127—531no—
321#520—22no—
322#520C2×D5312yes—
323#518—22no—
324#518—22no—
325#518C2×D5311yescharacteristic
326C4×#41—22no—
327C4×#41C2×D5312yes—
328C2×C4×D4—22no—
329C2×C4×D4—22no—
330C2×C4×D4—531no—
331C2×C4×D4C2×D5311yescharacteristic
332#453—538no—
333#442—1062no—
334#440—1062no—
335C4×SD16—1062no—
336C4×D8—1062no—
337#428—532no—
338C2×#104—532no—
339C2×#104—532no—
340#418—532no—
341#417—532no—
342C2×#100—531no—
343C2×#100—531no—
344C4×M16—532no—
345#390—531no—
346#390—531no—
347#386—1062no—
348#386—1062no—
349#382—531no—
350#376—531no—
351#343—532no—
352#339—532no—
353#339—532no—
354C53#2147311yescharacteristic, complemented, Sylow
355H(5,𝔽2)—24no—
356H(5,𝔽2)—22no—
357H(5,𝔽2)C4×D5314yescomplemented
358C2×#41—22no—
359C2×#41C22×D5312yes—
360C22×D4—22no—
361C22×D4—22no—
362C22×D4—531no—
363C22×D4—1062no—
364C22×D4C22×D5311yescharacteristic
365C23×C4—22no—
366C23×C4—1061no—
367C23×C4C22⋊D5312yes—
368AGL(1,ℤ/8)—538no—
369AGL(1,ℤ/8)—1064no—
370C2×SD16—1062no—
371C2×D8—1062no—
372C2×M16—532no—
373C2×M16—532no—
374#127—534no—
375#127—1062no—
376C4×Q8—22no—
377C4×Q8C22⋊D5312yescomplemented
378C4×Q8C22⋊D5312yescomplemented
379C4×D4—24no—
380C4×D4—24no—
381C4×D4—24no—
382C4×D4—24no—
383C4×D4—22no—
384C4×D4—22no—
385C4×D4—24no—
386C4×D4—22no—
387C4×D4—22no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—1062no—
392C4×D4—1062no—
393C4×D4C22⋊D5312yescomplemented
394C4×D4C22⋊D5312yescomplemented
395#117—24no—
396#117—22no—
397#117—22no—
398#117C22×D5312yes—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C22⋊C4—22no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—1062no—
407C2×C22⋊C4—1062no—
408C2×C22⋊C4—1061no—
409C2×C22⋊C4C22⋊D5312yes—
410C2×C42—22no—
411C2×C42—22no—
412C2×C42—531no—
413C2×C42—531no—
414C2×C42C22×D5311yescharacteristic
415#107—1062no—
416#106—1062no—
417#104—538no—
418#104—1064no—
419#103—1062no—
420#102—1062no—
421#102—1062no—
422#102—1062no—
423#100—534no—
424#100—1062no—
425#97—1062no—
426C4×C8—1062no—
427#95—1061no—
428#95—1061no—
429#95—1062no—
430C24—22no—
431C24—1061no—
432C24C2×C22⋊D5312yes—
433#41—28no—
434#41—28no—
435#41—22no—
436#41—22no—
437#41C2×C4×D5314yes—
438#41#1171994514yes—
439C2×Q8C2×C22⋊D5312yes—
440C2×D4—24no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—28no—
444C2×D4—22no—
445C2×D4—28no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—532no—
449C2×D4—1068no—
450C2×D4—1062no—
451C2×D4—1062no—
452C2×D4—1061no—
453C2×D4—4242no—
454C2×D4—4242no—
455C2×D4C2×C4×D5312yes—
456C2×D4#1171994512yes—
457C2×D4C2×C22⋊D5312yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—82no—
468C22×C4—84no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—4242no—
473C22×C4—2122no—
474C22×C4C2×C22⋊D5312yes—
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C2×C22⋊D5311yescharacteristic
477C22×C4C23×D5311yescharacteristic
478SD16—1068no—
479D8—1068no—
480M16—534no—
481M16—1062no—
482M16—1062no—
483M16—1062no—
484C2×C8—1062no—
485C2×C8—1062no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—1062no—
491C4⋊C4C2×C22⋊D5312yes—
492C4⋊C4C2×C22⋊D5312yes—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—24no—
499C22⋊C4—4242no—
500C22⋊C4—4242no—
501C22⋊C4—2121no—
502C22⋊C4—2121no—
503C42—24no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—532no—
508C42—1061no—
509C42—1062no—
510C42C2×C22⋊D5312yes—
511C42C2×C22⋊D5312yes—
512C23—28no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—42no—
517C23—42no—
518C23—82no—
519C23—82no—
520C23—1062no—
521C23—2122no—
522C23—2122no—
523C23—2121no—
524C23—2121no—
525C23C4×C22⋊D5314yes—
526C23C22×C22⋊D5311yescharacteristic
527Q8—22no—
528Q8C4×C22⋊D5314yes—
529D4—216no—
530D4—24no—
531D4—24no—
532D4—24no—
533D4—24no—
534D4—22no—
535D4—1064no—
536D4—4248no—
537D4—2122no—
538D4C4×C22⋊D5314yes—
539C2×C4—28no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—84no—
555C2×C4—82no—
556C2×C4—1062no—
557C2×C4—4244no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2122no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4#1313318611yescharacteristic
566C2×C4#1313318612yes—
567C2×C4#1313318611yescharacteristic
568C2×C4C22×C4×D5311yescharacteristic
569C2×C4C22×C22⋊D5312yes—
570C2×C4C2×#1171994512yes—
571C2×C4C2×#1171994511yescharacteristic
572C8—1064no—
573C8—2122no—
574C22—21no—
575C22—21no—
576C22—24no—
577C22—24no—
578C22—44no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—82no—
584C22—88no—
585C22—4242no—
586C22—4241no—
587C22—4241no—
588C22—2124no—
589C22—2122no—
590C22C2×C4×C22⋊D5311yescharacteristic
591C22C2×#1313318611yescharacteristic
592C22#1681113411yescharacteristic
593C4—21no—
594C4—21no—
595C4—22no—
596C4—24no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—4241no—
601C4—4241no—
602C4—2122no—
603C4C2×C4×C22⋊D5312yes—
604C4#1681113412yes—
605C2—21no—
606C2—21no—
607C2—44no—
608C2—44no—
609C2—4242no—
610C2C4×#1313318611yescharacteristic
611C2#2028407211yescharacteristic
612C2#2028408711yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×H(5,𝔽2)×C53C2
socleC22×C53C2×#120
prime radicalC2×#16810791C2
cosocleC22×C4×C53C24
hypercenterC4×H(5,𝔽2)D53
hypocenterC53#21473
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. C4×#13133242C2semidirect
3. #20277437C2semidirect
4. #20275113C2semidirect
5. #20275091C2semidirect
6. #20274948C2semidirect
7. #20274925C2semidirect
8. #20274669C2semidirect
9. #20274525C2semidirect
10. C4×#13133134C2semidirect
11. C4×H(5,𝔽2)×C53C2semidirect
14. C2×C4×D212C22semidirect
15. #16810791C4semidirect
20. #16810666C22semidirect
23. #16810627C22semidirect
27. C4×#11719917C22semidirect
29. #16810524C22semidirect
32. #16810488C22semidirect
33. #16810469C22semidirect
34. #16810392C22semidirect
35. #16810382C22semidirect
36. #16810382C22semidirect
52. #13133242C2×C4semidirect
58. #13133213C2×C4semidirect
70. #13133144C2×C4semidirect
76. #13133134C2×C4semidirect
84. H(5,𝔽2)×C53C2×C4semidirect
93. C4×Q8×C53D4semidirect
94. C4×Q8×C53D4semidirect
107. C4×D4×C53D4semidirect
108. C4×D4×C53D4semidirect
168. C2×D212C22×C4semidirect
198. C4×Dic53C22×C4semidirect
202. #11719917C22×C4semidirect
299. C4×H(5,𝔽2)D53semidirect
354. C53#21473semidirect
357. H(5,𝔽2)C4×D53semidirect
377. C4×Q8C22⋊D53semidirect
378. C4×Q8C22⋊D53semidirect
393. C4×D4C22⋊D53semidirect
394. C4×D4C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168107912
C4×#131332422
#202774372
#202751132
#202750912
#202749482
#202749252
#202746692
#202745252
C4×#131331342
C4×H(5,𝔽2)×C532
#2147353