Finite Group Databasebeta

← #28317576

Subgroups — #28317576

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291552C211yesmaximal, characteristic
2Q8×#11719940C211yesmaximal, characteristic, complemented
3#20290362C211yesmaximal, characteristic, complemented
4#20290355C211yesmaximal, characteristic, complemented
5#20290340C211yesmaximal, characteristic, complemented
6#20290332C211yesmaximal, characteristic, complemented
7#20290328C211yesmaximal, characteristic, complemented
8#20290320C211yesmaximal, characteristic, complemented
9SD16⋊#88324C211yesmaximal, characteristic, complemented
10SD16×#88324C211yesmaximal, characteristic, complemented
11#20290279C211yesmaximal, characteristic, complemented
12#20290057C211yesmaximal, characteristic, complemented
13#20290050C211yesmaximal, characteristic
14#20289938C211yesmaximal, characteristic, complemented
15#137×Dic53C211yesmaximal, characteristic
16#16811674—21no—
17#16811674—21no—
18#16811674C2211yescharacteristic
19#16811674C2211yescharacteristic
20#16811674C2211yescharacteristic
21#16811674C2211yescharacteristic
22Q8×#88324C2211yescharacteristic
23Q8×#88324C2211yescharacteristic
24#16811654—21no—
25#16811654—21no—
26D4×#88324C2211yescharacteristic
27D4×#88324C2211yescharacteristic
28#16811644C2211yescharacteristic
29#16811643—21no—
30#16811626—21no—
31#16811626C2211yescharacteristic
32C4×#11719940—21no—
33C4×#11719940C2211yescharacteristic
34C4×#11719940C2211yescharacteristic
35C2×#13133272—21no—
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40Q8⋊#88324C2211yescharacteristic
41Q8⋊#88324C2211yescharacteristic
42Q8⋊#88324C2211yescharacteristic
43Q8⋊#88324C2211yescharacteristic
44Q8⋊#88324C2211yescharacteristic
45Q8⋊#88324C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50#16811464C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52#41×Dic53C2211yescharacteristic
53C2×Q8×Dic53C2211yescharacteristic
54#16810730C411yescharacteristic
55#16810730C411yescharacteristic
56#16810724C411yescharacteristic
57#16810724C411yescharacteristic
58Q16×Dic53C2211yescharacteristic
59Q16×Dic53C2211yescharacteristic
60SD16×Dic53C2211yescharacteristic
61SD16×Dic53C2211yescharacteristic
62#16810665C411yescharacteristic
63#16810632C411yescharacteristic
64#16810624C411yescharacteristic
65M16×Dic53C2211yescharacteristic
66C2×#137×C53C411yescharacteristic
67C2×#11719940—21no—
68C2×#11719940—21no—
69C2×#11719940D411yescharacteristic
70C2×#11719940D411yescharacteristic
71C22×#88324—21no—
72C22×#88324—21no—
73#13133284—21no—
74#13133284—21no—
75#13133284—21no—
76#13133284—21no—
77#13133284—21no—
78C4⋊#88324—21no—
79C4⋊#88324—21no—
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84C22⋊#88324—21no—
85C22⋊#88324—21no—
86C22⋊#88324—21no—
87#13133272—21no—
88#13133272—21no—
89#13133272—21no—
90#13133272—21no—
91#13133272—21no—
92#13133272—21no—
93#13133272D411yescharacteristic
94#13133272D411yescharacteristic
95C4×#88324—21no—
96C4×#88324—21no—
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C2×#11719917C2×C411yescharacteristic
102Q8×Dic53—21no—
103Q8×Dic53—21no—
104Q8×Dic53C2×C411yescharacteristic
105Q8×Dic53C2×C411yescharacteristic
106Q8×Dic53C2×C411yescharacteristic
107Q8×Dic53C2311yescharacteristic
108Q8×Dic53C2311yescharacteristic
109Q8×Dic53C2311yescharacteristic
110D4×Dic53—21no—
111D4×Dic53—21no—
112D4×Dic53C2×C411yescharacteristic
113D4×Dic53C2311yescharacteristic
114#13133176—21no—
115#13133176C2×C411yescharacteristic
116C2×C4⋊Dic53—21no—
117C2×C4⋊Dic53C2×C411yescharacteristic
118C2×C4×Dic53—21no—
119C2×C4×Dic53—21no—
120C2×C4×Dic53C2311yescharacteristic
121#13133151C2×C411yescharacteristic
122#13133151C2×C411yescharacteristic
123#13133149C2×C411yescharacteristic
124#13133149C2×C411yescharacteristic
125#13133141C2×C411yescharacteristic
126#13133141C2×C411yescharacteristic
127#13133141C2×C411yescharacteristic
128#13133141C2×C411yescharacteristic
129#13133141C2×C411yescharacteristic
130#13133141C2×C411yescharacteristic
131#13133138C2×C411yescharacteristic
132#13133138C2×C411yescharacteristic
133#13133127C2×C411yescharacteristic
134#13133127C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C2×#41×C53C2×C411yescharacteristic
138C22×Q8×C53C2×C411yescharacteristic
139#137×C53—24no—
140C2×Q16×C53C2×C411yescharacteristic
141C2×Q16×C53C2×C411yescharacteristic
142C2×SD16×C53C2×C411yescharacteristic
143C2×SD16×C53C2×C411yescharacteristic
144C2×M16×C53C2×C411yescharacteristic
145#41×C53—22no—
146#41×C53—41no—
147C2×Q8×C53—21no—
148C2×Q8×C53—22no—
149C2×Q8×C53—41no—
150C2×Q8×C53C22×C411yescharacteristic
151C2×Q8×C53C22×C411yescharacteristic
152C2×Q8×C53C22×C411yescharacteristic
153C2×D4×C53—21no—
154C2×D4×C53C22×C411yescharacteristic
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53C22×C411yescharacteristic
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53C2×D411yescharacteristic
162C22×Dic53#4111yescharacteristic
163#11719940—22no—
164#11719940—41no—
165#11719940—42no—
166#11719940—42no—
167C2×#88324—21no—
168C2×#88324—21no—
169C2×#88324—21no—
170C2×#88324—21no—
171C2×#88324—21no—
172C2×#88324—21no—
173C2×#88324—42no—
174C2×#88324—42no—
175C2×#88324C2×D411yescharacteristic
176C2×#88324C2×D411yescharacteristic
177C2×#88324C2×D411yescharacteristic
178C2×#88324C2×D411yescharacteristic
179Q16×C53—22no—
180Q16×C53—22no—
181SD16×C53—22no—
182SD16×C53M1614yescomplemented
183M16×C53—22no—
184C2×C8×C53C22×C411yescharacteristic
185C2×C8×C53C22×C411yescharacteristic
186C22⋊Dic53—21no—
187C22⋊Dic53—21no—
188C22⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53C22×C411yescharacteristic
197C4⋊Dic53C22×C411yescharacteristic
198C4⋊Dic53C22×C411yescharacteristic
199C4⋊Dic53C22×C411yescharacteristic
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—21no—
206C4×Dic53—21no—
207C4×Dic53C22×C411yescharacteristic
208C4×Dic53C2×D411yescharacteristic
209C4×Dic53#4111yescharacteristic
210C4×Dic53C2411yescharacteristic
211#11719917—41no—
212C2×#88322C22×C411yescharacteristic
213C2×#88322C22×C411yescharacteristic
214C23×C53—21no—
215C23×C53C4×D411yescharacteristic
216Q8×C53—21no—
217Q8×C53—21no—
218Q8×C53—21no—
219Q8×C53—21no—
220Q8×C53—21no—
221Q8×C53—42no—
222Q8×C53C2×M1612yes—
223D4×C53—21no—
224D4×C53—42no—
225D4×C53C2×M1612yes—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53C4×D411yescharacteristic
234C2×C4×C53C23×C411yescharacteristic
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—81no—
245C2×Dic53—81no—
246C2×Dic53C4×D411yescharacteristic
247C2×Dic53C4×D411yescharacteristic
248C2×Dic53C22×D411yescharacteristic
249C2×Dic53C2×#4111yescharacteristic
250#88324—22no—
251#88324—21no—
252#88324—21no—
253#88324—21no—
254#88324—41no—
255#88324—41no—
256#88324—41no—
257#88324—41no—
258#88324—41no—
259#88324#13712yescomplemented
260C8×C53—21no—
261C8×C53C2×M1612yes—
262#88322—21no—
263#88322—21no—
264#18047—531nomaximal, Sylow
265C22×C53—21no—
266C22×C53—22no—
267C22×C53—21no—
268C22×C53—21no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53C2×C4×D411yescharacteristic
272C4×C53—21no—
273C4×C53—22no—
274C4×C53—41no—
275C4×C53—41no—
276C4×C53C22×M1611yescharacteristic
277C4×C53#56811yescharacteristic
278Dic53—21no—
279Dic53—21no—
280Dic53—21no—
281Dic53—41no—
282Dic53—41no—
283Dic53—41no—
284Dic53—41no—
285Dic53C2×#13711yescharacteristic
286Dic53#57511yescharacteristic
287#2970—531no—
288Q8×M16—531no—
289C4×#137—531no—
290#1593—531no—
291#1591—531no—
292#1587—531no—
293#1585—531no—
294#1584—531no—
295#1582—531no—
296SD16⋊C8—531no—
297C8×SD16—531no—
298#1569—531no—
299#1491—531no—
300#1489—531no—
301#1455—531no—
302C2×C53—41no—
303C2×C53—42no—
304C2×C53D4×M1611yescharacteristic
305C2×C53C4×#13711yescharacteristic
306C2×C53#295611yescharacteristic
307C2×#137#615711yescharacteristic
308C4×#41—531no—
309C2×C4×Q8—531no—
310#446—531no—
311#446—531no—
312#446—531no—
313#446—531no—
314#446—1061no—
315#446—1061no—
316C8×Q8—531no—
317C8×Q8—531no—
318#441—531no—
319#441—531no—
320#440—531no—
321#440—531no—
322C4×Q16—531no—
323C4×Q16—531no—
324C4×SD16—531no—
325C4×SD16—531no—
326#435—1061no—
327#435—1061no—
328C8×D4—531no—
329C8×D4—531no—
330#432—531no—
331#432—1061no—
332#428—531no—
333#423—531no—
334#423—1061no—
335#419—531no—
336#417—531no—
337C4×M16—531no—
338C4×M16—531no—
339C4×M16—531no—
340C4×M16—1061no—
341C2×#97—1061no—
342#335—531no—
343#335—531no—
344#334—531no—
345#334—531no—
346Q8⋊C8—531no—
347Q8⋊C8—531no—
348Q8⋊C8—531no—
349Q8⋊C8—531no—
350Q8⋊C8—531no—
351Q8⋊C8—531no—
352D4⋊C8—531no—
353D4⋊C8—531no—
354#322—531no—
355#322—531no—
356#322—531no—
357C82—531no—
358C53#1804711yescharacteristic, complemented, Sylow
359C2×#41C2×#615711yescharacteristic
360C22×Q8C2×#615711yescharacteristic
361#137—24no—
362C2×Q16C2×#615711yescharacteristic
363C2×Q16C2×#615711yescharacteristic
364C2×SD16C2×#615711yescharacteristic
365C2×SD16C2×#615711yescharacteristic
366C2×M16—531no—
367C2×M16—531no—
368C2×M16—531no—
369C2×M16—1061no—
370C2×M16—1061no—
371C2×M16C2×#615711yescharacteristic
372C22×C8—1061no—
373C22×C8—1061no—
374C4×Q8—531no—
375C4×Q8—531no—
376C4×Q8—531no—
377C4×Q8—531no—
378C4×Q8—531no—
379C4×Q8—531no—
380C4×Q8—1061no—
381C4×Q8—1061no—
382C4×D4—531no—
383C4×D4—531no—
384C4×D4—1061no—
385C4×D4—1061no—
386#117—531no—
387#117—1061no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—1061no—
390C2×C42—531no—
391C2×C42—1061no—
392C2×C42—1061no—
393#107—531no—
394#107—531no—
395#106—531no—
396#106—531no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—531no—
400C4⋊C8—531no—
401C4⋊C8—1061no—
402C4⋊C8—1061no—
403C4⋊C8—1061no—
404C4⋊C8—1061no—
405C4⋊C8—1061no—
406C4⋊C8—1061no—
407C4⋊C8—1061no—
408#103—531no—
409#103—531no—
410#103—531no—
411#103—531no—
412#103—531no—
413#103—531no—
414#102—531no—
415#102—531no—
416C22⋊C8—1061no—
417C22⋊C8—1061no—
418C22⋊C8—1061no—
419#97—531no—
420#97—531no—
421#97—531no—
422#97—531no—
423#97—1061no—
424#97—1061no—
425#97—1061no—
426#97—1061no—
427#97—1061no—
428#97—1061no—
429C4×C8—531no—
430C4×C8—531no—
431C4×C8—531no—
432C4×C8—531no—
433C4×C8—531no—
434C4×C8—531no—
435C4×C8—1061no—
436C4×C8—1061no—
437#41—22no—
438#41—41no—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—41no—
442C2×Q8C22×#615711yescharacteristic
443C2×Q8C22×#615711yescharacteristic
444C2×Q8C22×#615711yescharacteristic
445C2×D4—21no—
446C2×D4C22×#615711yescharacteristic
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—531no—
450C22×C4—531no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4C22×#615711yescharacteristic
455Q16—22no—
456Q16—22no—
457SD16—22no—
458SD16#1171994014yescomplemented
459M16—22no—
460M16—1062no—
461M16—2121no—
462M16—2121no—
463M16—2122no—
464M16—2122no—
465C2×C8—531no—
466C2×C8—531no—
467C2×C8—531no—
468C2×C8—531no—
469C2×C8—531no—
470C2×C8—531no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—2122no—
478C2×C8—2122no—
479C2×C8C22×#615711yescharacteristic
480C2×C8C22×#615711yescharacteristic
481C4⋊C4—531no—
482C4⋊C4—531no—
483C4⋊C4—531no—
484C4⋊C4—531no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C22⋊C4—1061no—
493C22⋊C4—1061no—
494C22⋊C4—1061no—
495C42—531no—
496C42—531no—
497C42—531no—
498C42—531no—
499C42—1061no—
500C42—1061no—
501C42—1061no—
502C42—1061no—
503C42—1061no—
504C42—1061no—
505C42—1061no—
506C23—21no—
507C23D4×#615711yescharacteristic
508Q8—21no—
509Q8—21no—
510Q8—21no—
511Q8—21no—
512Q8—21no—
513Q8—42no—
514Q8C2×#1171994012yes—
515D4—21no—
516D4—42no—
517D4C2×#1171994012yes—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—531no—
526C2×C4—531no—
527C2×C4—531no—
528C2×C4—531no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—4241no—
536C2×C4—4241no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4D4×#615711yescharacteristic
541C2×C4C23×#615711yescharacteristic
542C8—21no—
543C8—532no—
544C8—1061no—
545C8—1061no—
546C8—1061no—
547C8—1062no—
548C8—1061no—
549C8—1061no—
550C8—2121no—
551C8—2121no—
552C8—2121no—
553C8—2121no—
554C8—2121no—
555C8C2×#1171994012yes—
556C22—21no—
557C22—22no—
558C22—21no—
559C22—21no—
560C22—41no—
561C22—41no—
562C22C2×D4×#615711yescharacteristic
563C4—21no—
564C4—22no—
565C4—41no—
566C4—41no—
567C4—531no—
568C4—531no—
569C4—1061no—
570C4—1061no—
571C4—1061no—
572C4—2121no—
573C4—2121no—
574C4—2121no—
575C4—2121no—
576C4C22×#1171994011yescharacteristic
577C4#1681170111yescharacteristic
578C2—41no—
579C2—42no—
580C2D4×#1171994011yescharacteristic
581C2#137×#615711yescharacteristic
582C2#2029149011yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC2×#137×C53C4
socleC22×C53C2×C4×D4
prime radicalC2×#41×C53C2×C4
cosocleC4×Dic53C24
hypercenterC2×#137#6157
hypocenterC53#18047
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. Q8×#11719940C2semidirect
3. #20290362C2semidirect
4. #20290355C2semidirect
5. #20290340C2semidirect
6. #20290332C2semidirect
7. #20290328C2semidirect
8. #20290320C2semidirect
9. SD16⋊#88324C2semidirect
10. SD16×#88324C2semidirect
11. #20290279C2semidirect
12. #20290057C2semidirect
14. #20289938C2semidirect
182. SD16×C53M16semidirect
259. #88324#137semidirect
358. C53#18047semidirect
458. SD16#11719940semidirect

Maximal subgroups

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

subgroupindex
#202915522
Q8×#117199402
#202903622
#202903552
#202903402
#202903322
#202903282
#202903202
SD16⋊#883242
SD16×#883242
#202902792
#202900572
#202900502
#202899382
#137×Dic532
#1804753