Finite Group Databasebeta

← #27418003

Subgroups — #27418003

Order 13568.

1218 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811330C211yesmaximal, characteristic, complemented
2C2×#16810913C211yesmaximal, characteristic, complemented
3C2×C4×#11719923C211yesmaximal, characteristic, complemented
4#20275911C212yesmaximal, complemented
5#20275718C212yesmaximal, complemented
6#20275675C211yesmaximal, characteristic, complemented
7#20275510C211yesmaximal, characteristic, complemented
8#20275033C212yesmaximal, complemented
9#20274907C212yesmaximal, complemented
10#20274717C211yesmaximal, characteristic, complemented
11C53×#2537C211yesmaximal, characteristic
12C22×#11719923C2211yescharacteristic
13#16811330—24no—
14#16811330—24no—
15C2×#13133217C2211yescharacteristic
16C2×#13133208—21no—
17C2×#13133196C2212yes—
18C2×#13133194—22no—
19C2×C4×Dic106C2211yescharacteristic
20C2×D4×Dic53—21no—
21C22×C4⋊Dic53C2212yes—
22#16810913—24no—
23C4×#11719923—24no—
24C2×#13133148C2212yes—
25C2×#13133148C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×#13133140C2212yes—
29C2×#13133138C2212yes—
30C2×#13133138C2211yescharacteristic
31C2×C22⋊#88322C2212yes—
32C2×C22⋊#88322C2212yes—
33C2×C4×#88322C2211yescharacteristic
34#16810385C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37C2×#121×C53—21no—
38C2×#121×C53C2212yes—
39C2×C4×D4×C53C2211yescharacteristic
40C2×C4×D4×C53C2211yescharacteristic
41C53×#394—22no—
42C53×#384C2211yescharacteristic
43C53×#380—21no—
44C53×#380C2212yes—
45C22×Dic106—21no—
46C22×Dic106C2311yescharacteristic
47C23×Dic53—22no—
48C2×#11719923—22no—
49C2×#11719923—22no—
50C2×#11719923—22no—
51C22×#88322C2311yescharacteristic
52C22×#88322C2312yes—
53C22×#88322C2311yescharacteristic
54#13133217—22no—
55#13133217D414yescomplemented
56#13133208—28no—
57#13133196—24no—
58#13133196—24no—
59#13133194—216no—
60C4×Dic106—22no—
61C4×Dic106D414yescomplemented
62D4×Dic53—28no—
63C2×C4⋊Dic53—24no—
64C2×C4⋊Dic53—24no—
65C2×C4⋊Dic53—22no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2312yes—
69C2×#11719919—21no—
70C2×#11719919—22no—
71C2×C22⋊Dic53—22no—
72C2×C4×Dic53—21no—
73#13133148—24no—
74#13133148—22no—
75C4⋊#88322—22no—
76#13133140—22no—
77#13133140—42no—
78#13133138—24no—
79#13133138—22no—
80C22⋊#88322—24no—
81C22⋊#88322—42no—
82C4×#88322—22no—
83C22×D4×C53—22no—
84C22×D4×C53D411yescharacteristic
85C22×D4×C53C2311yescharacteristic
86C23×C4×C53—21no—
87C23×C4×C53C2312yes—
88#121×C53—24no—
89#121×C53—42no—
90#121×C53—44no—
91C4×D4×C53—22no—
92C4×D4×C53—22no—
93C4×D4×C53—22no—
94C4×D4×C53—22no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C4⋊C4×C53C2312yes—
100C2×C22⋊C4×C53—22no—
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53—22no—
103C2×C22⋊C4×C53D412yes—
104C2×C42×C53C2311yescharacteristic
105#95×C53—22no—
106C24×C53—21no—
107C24×C53C2×D412yes—
108C2×D4×C53—22no—
109C2×D4×C53—22no—
110C2×D4×C53—21no—
111C2×D4×C53—21no—
112C2×D4×C53—21no—
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53—44no—
116C2×D4×C53—44no—
117C2×D4×C53—82no—
118C2×D4×C53SD1614yescomplemented
119C2×D4×C53#4112yes—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—21no—
127C22×C4×C53—22no—
128C22×C4×C53—42no—
129C22×C4×C53—42no—
130C22×C4×C53—42no—
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2411yescharacteristic
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—22no—
138C22×Dic53—28no—
139C22×Dic53—82no—
140C22×Dic53—82no—
141C2×Dic106—21no—
142C2×Dic106—21no—
143C2×Dic106—21no—
144C2×Dic106—21no—
145C2×Dic106—21no—
146C2×Dic106—24no—
147C2×Dic106—81no—
148C2×Dic106—81no—
149C2×Dic106C2×D412yes—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—42no—
156C4⋊C4×C53—81no—
157C22⋊C4×C53—24no—
158C22⋊C4×C53—44no—
159C22⋊C4×C53—44no—
160C22⋊C4×C53—42no—
161C42×C53C2×D412yes—
162C42×C53C2×D412yes—
163C22⋊Dic53—28no—
164#11719923—28no—
165C4⋊Dic53—28no—
166C4⋊Dic53—24no—
167C4⋊Dic53—21no—
168C4⋊Dic53—22no—
169C4⋊Dic53C2×D412yes—
170C4⋊Dic53C2×D414yes—
171#11719919—28no—
172#11719919—24no—
173C4×Dic53—24no—
174C2×#88322—21no—
175C2×#88322—21no—
176C2×#88322—21no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—21no—
180C2×#88322—22no—
181C2×#88322—21no—
182C2×#88322#4112yes—
183C23×C53—22no—
184C23×C53—22no—
185C23×C53—22no—
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53—22no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53C2×SD1614yes—
197C23×C53C22×D411yescharacteristic
198D4×C53—22no—
199D4×C53—22no—
200D4×C53—28no—
201D4×C53—41no—
202D4×C53—41no—
203D4×C53—88no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—41no—
214C2×C4×C53—42no—
215C2×C4×C53—41no—
216C2×C4×C53—42no—
217C2×C4×C53—41no—
218C2×C4×C53—42no—
219C2×C4×C53—84no—
220C2×C4×C53#12012yes—
221C2×C4×C53#12012yes—
222C2×C4×C53C2×SD1612yes—
223C2×C4×C53#13712yes—
224C2×C4×C53C22×D411yescharacteristic
225C2×C4×C53C22×D411yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53H(5,𝔽2)11yescharacteristic
229C2×C4×C532−1+411yescharacteristic
230C2×Dic53—24no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—42no—
236C2×Dic53—41no—
237C2×Dic53—42no—
238C2×Dic53—41no—
239C2×Dic53—88no—
240Dic106—22no—
241Dic106—22no—
242Dic106—41no—
243Dic106—41no—
244Dic106—84no—
245#88322—22no—
246#88322—41no—
247#88322—42no—
248#24118—531nomaximal, Sylow
249C22×C53—24no—
250C22×C53—24no—
251C22×C53—24no—
252C22×C53—42no—
253C22×C53—42no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—81no—
260C22×C53—81no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53C2×#12011yescharacteristic
264C22×C53#54611yescharacteristic
265C22×C53#54711yescharacteristic
266C22×C53C22×SD1611yescharacteristic
267C22×C53C2×#13711yescharacteristic
268C22×C53#57611yescharacteristic
269C22×C53#57711yescharacteristic
270C4×C53—22no—
271C4×C53—42no—
272C4×C53—81no—
273C4×C53—82no—
274C4×C53#44814yes—
275C4×C53#52511yescharacteristic
276C4×C53#53711yescharacteristic
277C4×C53#54811yescharacteristic
278C4×C53#55011yescharacteristic
279Dic53—42no—
280Dic53—44no—
281Dic53—42no—
282C2×#554—531no—
283C2×#476—531no—
284C2×C4×SD16—531no—
285#2537D5311yescharacteristic
286#2035—532no—
287#2000—532no—
288#1990—531no—
289#1954—531no—
290#1874—532no—
291#1853—532no—
292#1803—531no—
293C2×C53—44no—
294C2×C53—82no—
295C2×C53#240711yescharacteristic
296C2×C53C2×#44811yescharacteristic
297C2×C53#301211yescharacteristic
298C2×C53#329511yescharacteristic
299C2×C53#333111yescharacteristic
300C2×C53#333511yescharacteristic
301C2×C53#335711yescharacteristic
302C22×SD16—531no—
303#554—1064no—
304#554—1064no—
305C2×C4⋊Q8—531no—
306C2×#125—1061no—
307C2×C22⋊Q8—532no—
308C2×C22⋊Q8—1062no—
309C2×#121—21no—
310C2×#121C2×D5312yes—
311C2×C4×Q8—531no—
312C2×C4×D4—1061no—
313C2×C4×D4C2×D5311yescharacteristic
314C2×C4×D4C2×D5311yescharacteristic
315C22×C4⋊C4—532no—
316#476—1064no—
317C4×SD16—1064no—
318C2×#106—531no—
319C2×#106—532no—
320C2×C4⋊C8—531no—
321C2×#103—531no—
322C2×#103—532no—
323C2×#102—531no—
324C2×#102—532no—
325C2×C22⋊C8—532no—
326C2×C22⋊C8—532no—
327C2×C4×C8—531no—
328#394—22no—
329#384C2×D5311yescharacteristic
330#380—21no—
331#380C2×D5312yes—
332#340—532no—
333#340—532no—
334#340—532no—
335C53#2411811yescharacteristic, complemented, Sylow
336C22×Q8—531no—
337C22×Q8—1061no—
338C22×D4—22no—
339C22×D4C22⋊D5311yescharacteristic
340C22×D4C22×D5311yescharacteristic
341C23×C4—21no—
342C23×C4—1062no—
343C23×C4C22×D5312yes—
344C2×SD16—1062no—
345C2×SD16—1062no—
346C2×SD16—1062no—
347C22×C8—531no—
348C22×C8—531no—
349C22×C8—532no—
350C4⋊Q8—534no—
351C4⋊Q8—1062no—
352#125—1068no—
353C22⋊Q8—10616no—
354C22⋊Q8—1064no—
355C22⋊Q8—1064no—
356#121—24no—
357#121—42no—
358#121—44no—
359C4×Q8—534no—
360C4×Q8—1062no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—1068no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—532no—
370C2×C4⋊C4—1064no—
371C2×C4⋊C4—1064no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1062no—
376C2×C4⋊C4C22⋊D5311yescharacteristic
377C2×C4⋊C4C22×D5311yescharacteristic
378C2×C4⋊C4C22×D5312yes—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—1062no—
383C2×C22⋊C4C22⋊D5312yes—
384C2×C42—1061no—
385C2×C42C22×D5311yescharacteristic
386#106—1062no—
387#106—1064no—
388C4⋊C8—1062no—
389#103—1062no—
390#103—2122no—
391#102—1062no—
392#102—1064no—
393C22⋊C8—1064no—
394C22⋊C8—2122no—
395C4×C8—1062no—
396#95—22no—
397C24—21no—
398C24C2×C22⋊D5312yes—
399C2×Q8—532no—
400C2×Q8—1064no—
401C2×Q8—1061no—
402C2×Q8—1061no—
403C2×Q8—1061no—
404C2×Q8—1061no—
405C2×Q8—1061no—
406C2×Q8—4241no—
407C2×Q8—4241no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—82no—
418C2×D4#1171992314yescomplemented
419C2×D4#1171994712yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—1068no—
432C22×C4—1061no—
433C22×C4—1061no—
434C22×C4—1062no—
435C22×C4—4242no—
436C22×C4—4242no—
437C22×C4C2×C22⋊D5312yes—
438C22×C4C2×C22⋊D5311yescharacteristic
439C22×C4C2×C22⋊D5311yescharacteristic
440C22×C4C23×D5311yescharacteristic
441SD16—1068no—
442C2×C8—532no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1062no—
447C2×C8—1062no—
448C2×C8—1061no—
449C2×C8—1062no—
450C2×C8—1061no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—81no—
458C4⋊C4—532no—
459C4⋊C4—534no—
460C4⋊C4—1068no—
461C4⋊C4—1068no—
462C4⋊C4—1064no—
463C4⋊C4—1064no—
464C4⋊C4—1061no—
465C4⋊C4—1062no—
466C22⋊C4—24no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—42no—
470C22⋊C4—1068no—
471C42—1064no—
472C42D4×D5312yes—
473C42D4×D5312yes—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23C2×#1171992314yes—
488C23C22×C22⋊D5311yescharacteristic
489Q8—1062no—
490Q8—1062no—
491Q8—4244no—
492Q8—2121no—
493Q8—2121no—
494D4—28no—
495D4—22no—
496D4—22no—
497D4—41no—
498D4—41no—
499D4—88no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—84no—
516C2×C4—1062no—
517C2×C4—1062no—
518C2×C4—1064no—
519C2×C4—4248no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2122no—
524C2×C4—2121no—
525C2×C4—2122no—
526C2×C4#1313318512yes—
527C2×C4#1313318512yes—
528C2×C4C2×#1171992312yes—
529C2×C4#1313324712yes—
530C2×C4C2×D4×D5311yescharacteristic
531C2×C4C2×D4×D5311yescharacteristic
532C2×C4#41×D5311yescharacteristic
533C2×C4C2×#1171994711yescharacteristic
534C2×C4#1313326211yescharacteristic
535C2×C4#1313326411yescharacteristic
536C8—1062no—
537C8—2122no—
538C8—2121no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×#1313318511yescharacteristic
554C22#1681126711yescharacteristic
555C22#1681127211yescharacteristic
556C22C22×#1171992311yescharacteristic
557C22C2×#1313324711yescharacteristic
558C22#1681142611yescharacteristic
559C22#1681143111yescharacteristic
560C4—22no—
561C4—42no—
562C4—82no—
563C4—81no—
564C4—2122no—
565C4—2122no—
566C4—2124no—
567C4#1681076114yes—
568C4#1681113311yescharacteristic
569C4#1681120211yescharacteristic
570C4#1681127911yescharacteristic
571C4#1681129111yescharacteristic
572C2—44no—
573C2—82no—
574C2#2027805611yescharacteristic
575C2C2×#1681076111yescharacteristic
576C2#2028401611yescharacteristic
577C2#2028653911yescharacteristic
578C2#2028703111yescharacteristic
579C2#2028707211yescharacteristic
580C2#2028733511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC53×#2537C2
socleC23×C53C22×D4
hypercenter#2537D53
hypocenterC53#24118
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. C2×#16811330C2semidirect
2. C2×#16810913C2semidirect
3. C2×C4×#11719923C2semidirect
4. #20275911C2semidirect
5. #20275718C2semidirect
6. #20275675C2semidirect
7. #20275510C2semidirect
8. #20275033C2semidirect
9. #20274907C2semidirect
10. #20274717C2semidirect
55. #13133217D4semidirect
61. C4×Dic106D4semidirect
118. C2×D4×C53SD16semidirect
335. C53#24118semidirect
418. C2×D4#11719923semidirect

Maximal subgroups

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

subgroupindex
C2×#168113302
C2×#168109132
C2×C4×#117199232
#202759112
#202757182
#202756752
#202755102
#202750332
#202749072
#202747172
C53×#25372
#2411853