Finite Group Databasebeta

← #27423009

Subgroups — #27423009

Order 13568.

976 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810773C211yesmaximal, characteristic
2#20283373C211yesmaximal, characteristic, complemented
3C2×#16810627C211yesmaximal, characteristic
4#20276072C212yesmaximal, complemented
5#20275975C212yesmaximal, complemented
6#20275617C212yesmaximal, complemented
7#20275376C212yesmaximal, complemented
8#20275184C211yesmaximal, characteristic, complemented
9#20274919C211yesmaximal, characteristic, complemented
10#20274635C211yesmaximal, characteristic
11C53×#3095C211yesmaximal, characteristic, complemented
12C23×D212C2211yescharacteristic
13C22×#11719924C2212yes—
14C22×#11719917C2211yescharacteristic
15#16810773—21no—
16#16810773—21no—
17#16810773—22no—
18#16810684—22no—
19#16810680—21no—
20#16810676—21no—
21C2×#13133152C2212yescomplemented
22#16810650C2212yescomplemented
23#16810627—21no—
24#16810627—21no—
25#16810627C2212yes—
26#16810627C2212yes—
27C2×#13133137—21no—
28C2×#13133137—21no—
29C2×#13133137C2212yes—
30C2×#13133136C2212yescomplemented
31C2×#13133133C2212yes—
32C2×#13133133C2212yes—
33#16810587C2211yescharacteristic
34C2×C22⋊#88322C2211yescharacteristic
35#16810579C2212yescomplemented
36#16810392C2212yescomplemented
37#16810387C2212yescomplemented
38C2×C22⋊Q8×C53C2211yescharacteristic
39C2×#117×C53C2211yescharacteristic
40C53×#485—21no—
41C53×#485—21no—
42C53×#484—21no—
43C53×#483—21no—
44C53×#428C2211yescharacteristic, complemented
45C53×#428C2211yescharacteristic, complemented
46C53×#419C2211yescharacteristic, complemented
47C53×#419C2211yescharacteristic, complemented
48C53×#407C2211yescharacteristic
49C24×D53—21no—
50C22×D212—21no—
51C22×D212—21no—
52C22×D212—24no—
53C22×D212—21no—
54C22×D212—21no—
55C22×D212—21no—
56C22×D212—42no—
57C22×D212C2312yes—
58C2×#11719924—22no—
59C2×#11719924—22no—
60C2×#11719924—24no—
61C2×#11719924—42no—
62C2×#11719917—21no—
63C2×#11719917—42no—
64C2×#11719917C2312yes—
65C2×#11719917C2312yes—
66C22×#88322—21no—
67C22×#88322C2312yes—
68#13133152—42no—
69C4⋊#88322—22no—
70C4⋊#88322—22no—
71#13133137—22no—
72#13133137—22no—
73#13133137—22no—
74#13133137—22no—
75#13133137—22no—
76#13133137—22no—
77#13133136—42no—
78#13133133—24no—
79#13133133—42no—
80C22⋊#88322—21no—
81C22⋊#88322—21no—
82C22⋊#88322D411yescharacteristic
83C22⋊#88322D411yescharacteristic
84C22⋊#88322D412yes—
85#13133128—22no—
86C4×#88322—22no—
87C22×Q8×C53C2311yescharacteristic
88C23×C4×C53C2311yescharacteristic
89C2×M16×C53C2311yescharacteristic
90C2×M16×C53C2311yescharacteristic
91C22⋊Q8×C53—21no—
92C22⋊Q8×C53—21no—
93C22⋊Q8×C53—21no—
94C22⋊Q8×C53—21no—
95#117×C53—21no—
96#117×C53—21no—
97#117×C53D411yescharacteristic
98#117×C53D411yescharacteristic
99#117×C53C2311yescharacteristic
100#117×C53C2311yescharacteristic
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C42×C53—21no—
108#107×C53—21no—
109#107×C53—21no—
110#106×C53—21no—
111#106×C53—21no—
112#103×C53—21no—
113#103×C53—21no—
114#103×C53—21no—
115#103×C53—21no—
116C22⋊C8×C53—21no—
117C22⋊C8×C53—21no—
118C24×C53#4111yescharacteristic
119C23×D53—22no—
120C23×D53—44no—
121C23×D53—41no—
122C23×D53—41no—
123C23×D53—82no—
124C23×D53—82no—
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—22no—
128C2×Q8×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53#4111yescharacteristic
140C22×C4×C53C2411yescharacteristic
141C2×D212—24no—
142C2×D212—24no—
143C2×D212—22no—
144C2×D212—22no—
145C2×D212—22no—
146C2×D212—22no—
147C2×D212—22no—
148C2×D212—44no—
149C2×D212—44no—
150C2×D212—44no—
151C2×D212—42no—
152C2×D212—42no—
153C2×D212—41no—
154C2×D212—41no—
155C2×D212#4114yes—
156M16×C53—22no—
157M16×C53—22no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—41no—
165C4⋊C4×C53—41no—
166C4⋊C4×C53C2×D411yescharacteristic
167C4⋊C4×C53C2×D411yescharacteristic
168C4⋊C4×C53C2×D411yescharacteristic
169C4⋊C4×C53C2×D411yescharacteristic
170C22⋊C4×C53—21no—
171C22⋊C4×C53—21no—
172C22⋊C4×C53—21no—
173C22⋊C4×C53—21no—
174C22⋊C4×C53—41no—
175C22⋊C4×C53—41no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180#11719924—44no—
181#11719924—44no—
182#11719917—22no—
183#11719917—22no—
184#11719917—22no—
185#11719917—22no—
186#11719917—42no—
187C2×#88322—22no—
188C2×#88322—22no—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—42no—
192C2×#88322C2×D412yes—
193C2×#88322C2×D412yes—
194C23×C53—22no—
195C23×C53—22no—
196C23×C53—41no—
197C23×C53#12111yescharacteristic
198C23×C53#12111yescharacteristic
199C23×C53C2×#4111yescharacteristic
200C22×D53—24no—
201C22×D53—44no—
202C22×D53—44no—
203C22×D53—42no—
204C22×D53—42no—
205C22×D53—82no—
206C22×D53—82no—
207C22×D53—82no—
208C22×D53—816no—
209C22×D53—82no—
210C22×D53—82no—
211C22×D53—82no—
212Q8×C53—41no—
213Q8×C53—41no—
214Q8×C53—42no—
215C2×C4×C53—22no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—42no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—42no—
230C2×C4×C53—81no—
231C2×C4×C53—81no—
232C2×C4×C53#12111yescharacteristic
233C2×C4×C53#12111yescharacteristic
234C2×C4×C53C22×D411yescharacteristic
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C2×#4112yes—
237C2×C4×C53H(5,𝔽2)12yes—
238D212—24no—
239D212—24no—
240D212—44no—
241D212—44no—
242D212—42no—
243D212—42no—
244C8×C53—41no—
245C8×C53—41no—
246#88322—22no—
247#88322—22no—
248#88322—42no—
249#88322—42no—
250#24521—531nomaximal, Sylow
251C22×C53—21no—
252C22×C53—21no—
253C22×C53—21no—
254C22×C53—21no—
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—21no—
258C22×C53—44no—
259C22×C53#47112yes—
260C22×C53C2×#12111yescharacteristic
261C22×C53#55112yes—
262C2×D53—44no—
263C2×D53—84no—
264C2×D53—88no—
265C2×D53—88no—
266C2×D53—82no—
267C2×D53—82no—
268C2×D53—84no—
269C4×C53—21no—
270C4×C53—21no—
271C4×C53—41no—
272C4×C53—41no—
273C4×C53—41no—
274C4×C53—41no—
275C4×C53—81no—
276C4×C53—81no—
277C4×C53#52511yescharacteristic
278C4×C53#53511yescharacteristic
279C4×C53#54612yes—
280#3095D5311yescharacteristic, complemented
281C2×#450—531no—
282#2924—531no—
283C2×#418—531no—
284#2063—532no—
285#2046—532no—
286#1977—532no—
287#1931—532no—
288#1901—531no—
289#1855—531no—
290#1783—531no—
291C2×C53—22no—
292C2×C53—22no—
293C2×C53—41no—
294C2×C53#241111yescharacteristic
295C2×C53C2×#47111yescharacteristic
296C2×C53#306811yescharacteristic
297D53—84no—
298C23×D4—531no—
299C22×SD16—532no—
300C22×M16—531no—
301C2×C22⋊Q8C2×D5311yescharacteristic
302C2×#117C2×D5311yescharacteristic
303#485—21no—
304#485—21no—
305#484—21no—
306#483—21no—
307#450—1061no—
308#450—1061no—
309#450—1062no—
310#433—1062no—
311#432—1061no—
312#431—1061no—
313C2×#108—532no—
314#428C2×D5311yescharacteristic, complemented
315#428C2×D5311yescharacteristic, complemented
316#424—532no—
317#419C2×D5311yescharacteristic, complemented
318#419C2×D5311yescharacteristic, complemented
319#418—532no—
320#418—532no—
321#418—1061no—
322#418—1061no—
323C2×#102—532no—
324C2×#102—1061no—
325C2×#102—1061no—
326C2×#101—532no—
327C2×#100—532no—
328C2×#100—532no—
329#407—531no—
330#407C2×D5311yescharacteristic
331C2×C22⋊C8—531no—
332#405—532no—
333#343—532no—
334#341—532no—
335C53#2452111yescharacteristic, complemented, Sylow
336C25—1061no—
337C22×Q8C22×D5311yescharacteristic
338C22×D4—532no—
339C22×D4—1064no—
340C22×D4—1061no—
341C22×D4—1061no—
342C22×D4—1061no—
343C22×D4—1061no—
344C22×D4—1061no—
345C22×D4—2122no—
346C23×C4C22×D5311yescharacteristic
347C2×SD16—1062no—
348C2×SD16—1062no—
349C2×SD16—1064no—
350C2×SD16—2122no—
351C2×M16—532no—
352C2×M16—532no—
353C2×M16—1061no—
354C2×M16—2122no—
355C2×M16C22×D5311yescharacteristic
356C2×M16C22×D5311yescharacteristic
357C22×C8—532no—
358C22×C8—1061no—
359C22⋊Q8—21no—
360C22⋊Q8—21no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363#117—21no—
364#117—21no—
365#117C22⋊D5311yescharacteristic
366#117C22⋊D5311yescharacteristic
367#117C22×D5311yescharacteristic
368#117C22×D5311yescharacteristic
369C2×C4⋊C4—21no—
370C2×C4⋊C4C22⋊D5311yescharacteristic
371C2×C4⋊C4C22⋊D5311yescharacteristic
372C2×C4⋊C4C22×D5311yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C42—21no—
376#108—2122no—
377#107—21no—
378#107—21no—
379#106—21no—
380#106—21no—
381C4⋊C8—1062no—
382C4⋊C8—1062no—
383#103—21no—
384#103—21no—
385#103—21no—
386#103—21no—
387#102—1062no—
388#102—1062no—
389#102—1062no—
390#102—1062no—
391#102—1062no—
392#102—1062no—
393#101—2122no—
394#100—1064no—
395#100—2122no—
396C22⋊C8—21no—
397C22⋊C8—21no—
398C22⋊C8—531no—
399C22⋊C8—531no—
400C22⋊C8—532no—
401C22⋊C8—1061no—
402C22⋊C8—1061no—
403#97—1062no—
404C4×C8—1062no—
405C24—1062no—
406C24—4242no—
407C24—4242no—
408C24—2124no—
409C24—2121no—
410C24—2121no—
411C24#1171994711yescharacteristic
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×D4—534no—
417C2×D4—1064no—
418C2×D4—1064no—
419C2×D4—1062no—
420C2×D4—1062no—
421C2×D4—1062no—
422C2×D4—1062no—
423C2×D4—1062no—
424C2×D4—2124no—
425C2×D4—2124no—
426C2×D4—2124no—
427C2×D4—2122no—
428C2×D4—2122no—
429C2×D4—2121no—
430C2×D4—2121no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4D4×D5311yescharacteristic
438C22×C4D4×D5311yescharacteristic
439C22×C4#1171994711yescharacteristic
440C22×C4C2×C22⋊D5311yescharacteristic
441C22×C4C2×C22⋊D5311yescharacteristic
442C22×C4C23×D5311yescharacteristic
443SD16—2124no—
444SD16—2124no—
445M16—22no—
446M16—22no—
447M16—1062no—
448M16—1062no—
449M16—1062no—
450M16—1062no—
451M16—2122no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—532no—
455C2×C8—532no—
456C2×C8—1062no—
457C2×C8—1062no—
458C2×C8—1062no—
459C2×C8—1062no—
460C2×C8—2122no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4C2×C22⋊D5311yescharacteristic
468C4⋊C4C2×C22⋊D5311yescharacteristic
469C4⋊C4C2×C22⋊D5311yescharacteristic
470C4⋊C4C2×C22⋊D5311yescharacteristic
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C23—22no—
482C23—22no—
483C23—41no—
484C23—1064no—
485C23—4242no—
486C23—4242no—
487C23—4242no—
488C23—4242no—
489C23—4242no—
490C23—42416no—
491C23—4242no—
492C23—2124no—
493C23—2124no—
494C23—2122no—
495C23—2122no—
496C23#1313318911yescharacteristic
497C23#1313318911yescharacteristic
498C23C2×#1171994711yescharacteristic
499Q8—42no—
500Q8—41no—
501Q8—41no—
502D4—1064no—
503D4—1064no—
504D4—2124no—
505D4—2124no—
506D4—2122no—
507D4—2122no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4#1313318911yescharacteristic
526C2×C4#1313318911yescharacteristic
527C2×C4C2×D4×D5311yescharacteristic
528C2×C4C22×C22⋊D5311yescharacteristic
529C2×C4#41×D5312yes—
530C2×C4#1313326312yes—
531C8—41no—
532C8—41no—
533C8—1062no—
534C8—1062no—
535C8—2122no—
536C8—2122no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—44no—
545C22—4244no—
546C22—4244no—
547C22—4248no—
548C22—4248no—
549C22—4242no—
550C22—4242no—
551C22—2124no—
552C22#1681088812yes—
553C22C2×#1313318911yescharacteristic
554C22#1681129612yes—
555C4—21no—
556C4—21no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—81no—
562C4—81no—
563C4#1681113011yescharacteristic
564C4#1681118711yescharacteristic
565C4#1681126512yes—
566C2—22no—
567C2—22no—
568C2—41no—
569C2—4244no—
570C2#2027810111yescharacteristic
571C2C2×#1681088811yescharacteristic
572C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#3095C2
socleC22×C53C2×#121
prime radicalC23×D212C22
hypercenter#3095D53
hypocenterC53#24521
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3095

Complements

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

NH ≅ G/Ntype
2. #20283373C2semidirect
4. #20276072C2semidirect
5. #20275975C2semidirect
6. #20275617C2semidirect
7. #20275376C2semidirect
8. #20275184C2semidirect
9. #20274919C2semidirect
11. C53×#3095C2semidirect
21. C2×#13133152C22semidirect
22. #16810650C22semidirect
30. C2×#13133136C22semidirect
35. #16810579C22semidirect
36. #16810392C22semidirect
37. #16810387C22semidirect
44. C53×#428C22semidirect
45. C53×#428C22semidirect
46. C53×#419C22semidirect
47. C53×#419C22semidirect
280. #3095D53semidirect
314. #428C2×D53semidirect
315. #428C2×D53semidirect
317. #419C2×D53semidirect
318. #419C2×D53semidirect
335. C53#24521semidirect

Maximal subgroups

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

subgroupindex
C2×#168107732
#202833732
C2×#168106272
#202760722
#202759752
#202756172
#202753762
#202751842
#202749192
#202746352
C53×#30952
#2452153