Finite Group Databasebeta

← #27436368

Subgroups — #27436368

Order 13568.

1170 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810871C212yesmaximal
2#20283046C211yesmaximal, characteristic
3#20276040C211yesmaximal, characteristic, complemented
4#20275958C211yesmaximal, characteristic, complemented
5#20275848C211yesmaximal, characteristic, complemented
6#20275798C211yesmaximal, characteristic, complemented
7#20275149C212yesmaximal, complemented
8#20275070C212yesmaximal, complemented
9#20274969C212yesmaximal, complemented
10#20274409C211yesmaximal, characteristic, complemented
11C2×C53×#533C211yesmaximal, characteristic, complemented
12C22×#11719924C2212yes—
13C22×#11719917C2212yes—
14C2×#13133200—21no—
15C2×#13133191C2211yescharacteristic
16#16810871—28no—
17C2×#13133150C2212yes—
18C2×#13133141C2212yes—
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×#13133137C2212yes—
22#16810525C2211yescharacteristic
23#16810525C2211yescharacteristic
24#16810521C2211yescharacteristic
25#16810521C2211yescharacteristic
26#16810514—21no—
27#16810514—21no—
28#16810502—21no—
29#16810483C2211yescharacteristic
30#16810385C2212yes—
31#16810385C2212yes—
32#16810378C2212yes—
33#16810378C2211yescharacteristic
34#16810378C2211yescharacteristic
35C53×#533—28no—
36C2×C4⋊Q8×C53C2211yescharacteristic
37C2×C4⋊Q8×C53C2211yescharacteristic
38C2×#125×C53—21no—
39C2×C22⋊Q8×C53—21no—
40C2×C22⋊Q8×C53C2212yes—
41C2×C4×Q8×C53—21no—
42C2×C4×Q8×C53C2212yes—
43C2×#117×C53—21no—
44C22×C42×C53C2211yescharacteristic
45C22×D212C2311yescharacteristic
46C22×C22⋊D53—21no—
47C2×#11719924—24no—
48C2×#11719924—24no—
49C2×#11719924—42no—
50C2×#11719917—24no—
51C2×#11719917—24no—
52C2×#11719917—42no—
53C22×#88322C2312yes—
54C22×#88322C2312yes—
55#13133200—44no—
56#13133191—24no—
57C2×C4⋊Dic53C2311yescharacteristic
58C2×C4⋊Dic53C2311yescharacteristic
59C2×C4⋊Dic53C2311yescharacteristic
60C2×#11719919—21no—
61C2×#11719919—21no—
62C2×C22⋊Dic53—21no—
63C2×#11719921—21no—
64C2×#11719921—21no—
65C2×#11719921—21no—
66C2×#11719921—21no—
67C2×#11719921—21no—
68#13133150—24no—
69#13133141—24no—
70#13133141—24no—
71#13133141—42no—
72#13133137—24no—
73#13133124—21no—
74#13133124—21no—
75#13133124—21no—
76C22×Q8×C53—21no—
77C22×Q8×C53C2312yes—
78C23×C4×C53D411yescharacteristic
79C23×C4×C53D411yescharacteristic
80C23×C4×C53C2311yescharacteristic
81C4⋊Q8×C53—24no—
82C4⋊Q8×C53—24no—
83#125×C53—28no—
84C22⋊Q8×C53—28no—
85C22⋊Q8×C53—24no—
86C22⋊Q8×C53D418yes—
87C4×Q8×C53—28no—
88C4×Q8×C53—24no—
89C4×Q8×C53—24no—
90#117×C53—28no—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—22no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53C2312yes—
97C2×C22⋊C4×C53—22no—
98C2×C42×C53—24no—
99C2×C42×C53—22no—
100C2×C42×C53D411yescharacteristic
101C2×C42×C53D411yescharacteristic
102C2×C42×C53C2311yescharacteristic
103C2×C42×C53C2311yescharacteristic
104C24×C53C2×D411yescharacteristic
105C23×D53—21no—
106C2×Q8×C53—24no—
107C2×Q8×C53—22no—
108C2×Q8×C53—22no—
109C2×Q8×C53—22no—
110C2×Q8×C53—22no—
111C2×Q8×C53—22no—
112C2×Q8×C53—42no—
113C2×Q8×C53—42no—
114C2×Q8×C53C2×D414yes—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—41no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D414yes—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2411yescharacteristic
133C2×C22⋊D53—42no—
134C2×C22⋊D53—42no—
135C2×C22⋊D53—42no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C2×D212—22no—
140C2×D212—22no—
141C2×D212—21no—
142C2×D212—21no—
143C4⋊C4×C53—24no—
144C4⋊C4×C53—24no—
145C4⋊C4×C53—28no—
146C4⋊C4×C53—24no—
147C4⋊C4×C53—28no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53C2×D414yes—
150C22⋊C4×C53—28no—
151C42×C53—28no—
152C42×C53—22no—
153C42×C53—22no—
154C42×C53—22no—
155C42×C53—22no—
156C22⋊Dic53—42no—
157#11719924—48no—
158#11719921—42no—
159#11719921—42no—
160#11719921—42no—
161#11719921—42no—
162#11719921—42no—
163C4⋊Dic53—22no—
164C4⋊Dic53—22no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167#11719919—42no—
168#11719919—42no—
169#11719917—48no—
170C2×#88322—22no—
171C2×#88322—22no—
172C2×#88322—22no—
173C2×#88322—22no—
174C2×#88322—22no—
175C2×#88322#4114yes—
176C23×C53—21no—
177C23×C53—21no—
178C23×C53—21no—
179C23×C53—21no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53#12012yes—
183C23×C53C22×D411yescharacteristic
184C22×D53—41no—
185C22×D53—41no—
186C22×D53—41no—
187C22×D53—41no—
188C22×D53—41no—
189C22×D53—41no—
190C22×D53—41no—
191Q8×C53—24no—
192Q8×C53—24no—
193Q8×C53—48no—
194Q8×C53—42no—
195Q8×C53—42no—
196C2×C4×C53—28no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—21no—
208C2×C4×C53—24no—
209C2×C4×C53—24no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—44no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53#12012yes—
218C2×C4×C53AGL(1,ℤ/8)12yes—
219C2×C4×C53#13712yes—
220C2×C4×C53#13712yes—
221C2×C4×C53C22×D412yes—
222C2×C4×C53C2×#4112yes—
223C2×C4×C53H(5,𝔽2)11yescharacteristic
224C2×C4×C532−1+411yescharacteristic
225C22⋊D53—84no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235D212—42no—
236D212—42no—
237#88322—24no—
238#88322—42no—
239#25503—531nomaximal, Sylow
240C22×C53—22no—
241C22×C53—22no—
242C22×C53—22no—
243C22×C53—41no—
244C22×C53—41no—
245C22×C53—41no—
246C22×C53—41no—
247C22×C53C2×#12011yescharacteristic
248C22×C53#53211yescharacteristic
249C22×C53#53711yescharacteristic
250C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
251C22×C53C2×#13711yescharacteristic
252C22×C53C2×#13711yescharacteristic
253C22×C53#57511yescharacteristic
254C2×D53—81no—
255C2×D53—81no—
256C2×D53—81no—
257C2×D53—81no—
258C2×D53—81no—
259C2×D53—81no—
260C2×D53—81no—
261C4×C53—22no—
262C4×C53—22no—
263C4×C53—22no—
264C4×C53—22no—
265C4×C53—44no—
266C4×C53—41no—
267C4×C53—44no—
268C4×C53#54612yes—
269C4×C53#54712yes—
270Dic53—81no—
271Dic53—81no—
272Dic53—81no—
273C2×#533D5311yescharacteristic, complemented
274C2×#468—532no—
275#2854—531no—
276#2057—531no—
277#2043—531no—
278#2023—531no—
279#2013—531no—
280#1894—532no—
281#1880—532no—
282#1863—532no—
283#1728—531no—
284C2×C53—42no—
285C2×C53#243011yescharacteristic
286C2×C53#300411yescharacteristic
287C2×C53#300611yescharacteristic
288C2×C53#317211yescharacteristic
289C2×C53#318211yescharacteristic
290C2×C53#325211yescharacteristic
291C2×C53#325411yescharacteristic
292D53—161no—
293C22×SD16—532no—
294C22×M16—532no—
295#533—28no—
296C2×C4⋊Q8C2×D5311yescharacteristic
297C2×C4⋊Q8C2×D5311yescharacteristic
298C2×#125—21no—
299C2×#123—1061no—
300C2×C22⋊Q8—21no—
301C2×C22⋊Q8C2×D5312yes—
302C2×#121—531no—
303C2×C4×Q8—21no—
304C2×C4×Q8C2×D5312yes—
305C2×#117—21no—
306C22×C42C2×D5311yescharacteristic
307#468—1068no—
308C2×#107—532no—
309C2×#103—532no—
310C2×#103—532no—
311C2×#103—532no—
312C2×#102—532no—
313#390—531no—
314#390—531no—
315#389—531no—
316#389—531no—
317#388—1061no—
318#388—1061no—
319#386—1061no—
320#380—531no—
321#340—532no—
322#340—532no—
323#336—531no—
324#336—532no—
325#336—531no—
326C53#2550311yescharacteristic, complemented, Sylow
327C22×Q8—21no—
328C22×Q8C22×D5312yes—
329C22×D4—531no—
330C22×D4—1061no—
331C23×C4C22⋊D5311yescharacteristic
332C23×C4C22⋊D5311yescharacteristic
333C23×C4C22×D5311yescharacteristic
334C2×SD16—1064no—
335C2×SD16—1064no—
336C2×SD16—2122no—
337C2×M16—1064no—
338C2×M16—1064no—
339C2×M16—2122no—
340C22×C8—532no—
341C22×C8—532no—
342C4⋊Q8—24no—
343C4⋊Q8—24no—
344#125—28no—
345#123—2124no—
346C22⋊Q8—28no—
347C22⋊Q8—24no—
348C22⋊Q8C22⋊D5318yes—
349#121—1064no—
350C4×Q8—28no—
351C4×Q8—24no—
352C4×Q8—24no—
353#117—28no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—531no—
360C2×C4⋊C4—531no—
361C2×C4⋊C4—531no—
362C2×C4⋊C4—1061no—
363C2×C4⋊C4—1061no—
364C2×C4⋊C4C22×D5312yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—1061no—
367C2×C22⋊C4—1061no—
368C2×C22⋊C4—1061no—
369C2×C22⋊C4—1061no—
370C2×C22⋊C4—1061no—
371C2×C22⋊C4—1061no—
372C2×C42—22no—
373C2×C42—24no—
374C2×C42C22⋊D5311yescharacteristic
375C2×C42C22⋊D5311yescharacteristic
376C2×C42C22×D5311yescharacteristic
377C2×C42C22×D5311yescharacteristic
378#107—1064no—
379#103—1064no—
380#103—1064no—
381#103—2122no—
382#102—1064no—
383#95—1061no—
384#95—1061no—
385#95—1061no—
386C24—1061no—
387C24C2×C22⋊D5311yescharacteristic
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8C2×C22⋊D5314yes—
397C2×D4—1062no—
398C2×D4—1062no—
399C2×D4—1061no—
400C2×D4—1061no—
401C2×D4—2122no—
402C2×D4—2122no—
403C2×D4—2122no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—41no—
415C22×C4—1061no—
416C22×C4—1061no—
417C22×C4—1061no—
418C22×C4C2×C22⋊D5311yescharacteristic
419C22×C4C2×C22⋊D5311yescharacteristic
420C22×C4C2×C22⋊D5314yes—
421C22×C4C2×C22⋊D5311yescharacteristic
422C22×C4C2×C22⋊D5311yescharacteristic
423C22×C4C2×C22⋊D5311yescharacteristic
424C22×C4C23×D5311yescharacteristic
425SD16—2128no—
426M16—2128no—
427C2×C8—534no—
428C2×C8—1062no—
429C2×C8—1062no—
430C2×C8—1062no—
431C2×C8—1062no—
432C2×C8—1062no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—28no—
436C4⋊C4—24no—
437C4⋊C4—28no—
438C4⋊C4—22no—
439C4⋊C4—1062no—
440C4⋊C4—1062no—
441C4⋊C4—1061no—
442C4⋊C4—1061no—
443C4⋊C4—2122no—
444C4⋊C4—2122no—
445C4⋊C4C2×C22⋊D5314yes—
446C22⋊C4—28no—
447C22⋊C4—2122no—
448C22⋊C4—2122no—
449C22⋊C4—2122no—
450C22⋊C4—2122no—
451C22⋊C4—2122no—
452C22⋊C4—2122no—
453C42—28no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—2121no—
465C23—2121no—
466C23—2121no—
467C23—2121no—
468C23—2121no—
469C23—2121no—
470C23—2121no—
471C23#1313318612yes—
472C23C22×C22⋊D5311yescharacteristic
473Q8—24no—
474Q8—24no—
475Q8—48no—
476Q8—42no—
477Q8—42no—
478D4—4244no—
479D4—2122no—
480D4—2122no—
481C2×C4—28no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—2121no—
503C2×C4—2121no—
504C2×C4—2121no—
505C2×C4—2121no—
506C2×C4—2121no—
507C2×C4—2121no—
508C2×C4—2121no—
509C2×C4—2121no—
510C2×C4—2121no—
511C2×C4#1313318612yes—
512C2×C4#1313324212yes—
513C2×C4#1313324812yes—
514C2×C4#1313324812yes—
515C2×C4C22×C22⋊D5312yes—
516C2×C4#41×D5312yes—
517C2×C4#1313326311yescharacteristic
518C2×C4#1313326411yescharacteristic
519C8—1064no—
520C8—2122no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—4241no—
529C22—4241no—
530C22—4241no—
531C22—4241no—
532C22—4241no—
533C22—4241no—
534C22—4241no—
535C22C2×#1313318611yescharacteristic
536C22#1681116911yescharacteristic
537C22#1681120111yescharacteristic
538C22C2×#1313324211yescharacteristic
539C22C2×#1313324811yescharacteristic
540C22C2×#1313324811yescharacteristic
541C22#1681142111yescharacteristic
542C4—22no—
543C4—22no—
544C4—22no—
545C4—22no—
546C4—44no—
547C4—41no—
548C4—44no—
549C4—4241no—
550C4—4241no—
551C4—4241no—
552C4#1681126112yes—
553C4#1681127212yes—
554C2—42no—
555C2—8481no—
556C2#2027831911yescharacteristic
557C2#2028394911yescharacteristic
558C2#2028396711yescharacteristic
559C2#2028531511yescharacteristic
560C2#2028543411yescharacteristic
561C2#2028608211yescharacteristic
562C2#2028609811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#533C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenterC2×#533D53
hypocenterC53#25503
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
3. #20276040C2semidirect
4. #20275958C2semidirect
5. #20275848C2semidirect
6. #20275798C2semidirect
7. #20275149C2semidirect
8. #20275070C2semidirect
9. #20274969C2semidirect
10. #20274409C2semidirect
11. C2×C53×#533C2semidirect
273. C2×#533D53semidirect
326. C53#25503semidirect

Maximal subgroups

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

subgroupindex
C2×#168108712
#202830462
#202760402
#202759582
#202758482
#202757982
#202751492
#202750702
#202749692
#202744092
C2×C53×#5332
#2550353