Finite Group Databasebeta

← #27536993

Subgroups — #27536993

Order 13568.

1170 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#135×D53C211yesmaximal, characteristic
2C22×#13133156C211yesmaximal, characteristic, complemented
3#427×D53C211yesmaximal, characteristic, complemented
4#20276433C212yesmaximal, complemented
5#20276429C212yesmaximal, complemented
6#20276426C212yesmaximal, complemented
7#20276424C212yesmaximal, complemented
8#20276423C211yesmaximal, characteristic, complemented
9#20276422C211yesmaximal, characteristic, complemented
10#20276420C211yesmaximal, characteristic, complemented
11C53×#2138C211yesmaximal, characteristic, complemented
12C2×#41×D53—21no—
13C2×#13133240C2211yescharacteristic
14C2×#13133239C2211yescharacteristic
15C2×#13133238—21no—
16C2×#13133237—21no—
17C2×#13133236C2211yescharacteristic
18#135×D53—24no—
19#135×D53C418yescomplemented
20C2×Q16×D53C2211yescharacteristic
21C2×SD16×D53—21no—
22C2×D8×D53C2211yescharacteristic
23C22×C8×D53C2211yescharacteristic
24#117×D53—21no—
25C2×#13133156—24no—
26C2×#13133156C2212yes—
27C2×#13133156C2212yescomplemented
28C22×#11719908C2211yescharacteristic, complemented
29#16810661C2211yescharacteristic, complemented
30#16810660C2212yes—
31#16810659—21no—
32#16810658—21no—
33#107×D53C2212yescomplemented
34#106×D53—21no—
35#16810428C2212yescomplemented
36#16810427C2212yescomplemented
37#16810426C2212yescomplemented
38#16810425C2212yescomplemented
39#16810424C2212yescomplemented
40#16810423C2212yescomplemented
41C2×#135×C53C2211yescharacteristic
42C22×C16×C53C2211yescharacteristic, complemented
43C53×#427C2211yescharacteristic, complemented
44C53×#358C2212yescomplemented
45C53×#357C2212yescomplemented
46C2×#11719949—21no—
47C2×#11719945—21no—
48C2×#11719945—41no—
49C2×#11719947—21no—
50C2×#11719947—41no—
51#41×D53—28no—
52#41×D53—44no—
53C2×Q8×D53—21no—
54C2×D4×D53—21no—
55C2×D4×D53—41no—
56C22×C4×D53—41no—
57C22×C4×D53D411yescharacteristic
58#13133240—22no—
59#13133240C2×C414yes—
60#13133239—22no—
61#13133239C2×C414yescomplemented
62#13133238—28no—
63#13133237—28no—
64#13133236—22no—
65#13133236C2×C414yes—
66C2×Dic212C2311yescharacteristic
67C2×#11719925—21no—
68Q16×D53—22no—
69Q16×D53C2×C414yescomplemented
70C2×#11719913—21no—
71C2×#11719923—21no—
72C2×#11719924—21no—
73SD16×D53—28no—
74C2×D424C2311yescharacteristic
75C2×#11719922—21no—
76D8×D53—22no—
77D8×D53C2×C414yescomplemented
78C22×#88322D411yescharacteristic
79C2×C8×D53C2×C414yes—
80C2×C8×D53D414yes—
81C2×C8×D53D411yescharacteristic
82C2×C8×D53D411yescharacteristic
83C2×C8×D53C2311yescharacteristic
84C2×C8×D53C2311yescharacteristic
85#13133176—21no—
86#13133175—22no—
87#13133174—41no—
88#13133173—41no—
89C4⋊C4×D53—22no—
90C22⋊C4×D53—41no—
91C42×D53—41no—
92#13133156—24no—
93#13133156—24no—
94C2×#11719908—22no—
95C2×#11719908C2312yes—
96#13133151C2312yes—
97#13133150—22no—
98#13133149—21no—
99#13133148—22no—
100C2×#41×C53—21no—
101#135×C53—22no—
102#135×C53C2×C414yescomplemented
103C2×Q16×C53C2311yescharacteristic
104C2×SD16×C53—21no—
105C2×D8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107#117×C53—21no—
108C2×C16×C53—22no—
109C2×C16×C53C2312yes—
110#107×C53C2312yes—
111#106×C53—21no—
112C23×D53—41no—
113C23×D53D811yescharacteristic
114#41×C53—24no—
115#41×C53—42no—
116C2×Q8×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53C2×D411yescharacteristic
121C2×C22⋊D53—41no—
122C2×C22⋊D53—41no—
123C2×C22⋊D53—41no—
124C22×Dic53—41no—
125C22×Dic53D811yescharacteristic
126#11719949—24no—
127#11719949—42no—
128Q8×D53—24no—
129Q8×D53—42no—
130#11719947—24no—
131#11719947—48no—
132#11719947—42no—
133D4×D53—24no—
134D4×D53—48no—
135D4×D53—42no—
136#11719945—24no—
137#11719945—42no—
138#11719945—48no—
139C2×D212—21no—
140C2×D212—41no—
141C2×C4×D53—41no—
142C2×C4×D53—41no—
143C2×C4×D53—41no—
144C2×C4×D53—48no—
145C2×C4×D53—42no—
146C2×C4×D53C22⋊C414yes—
147C2×C4×D53D811yescharacteristic
148C2×C4×D53D811yescharacteristic
149C2×C4×D53SD1614yes—
150C2×C4×D53C2×D411yescharacteristic
151C2×C4×D53C2×D411yescharacteristic
152C2×Dic106—21no—
153C2×Dic106—41no—
154Q16×C53—21no—
155Q16×C53C22×C412yes—
156SD16×C53—24no—
157D8×C53—21no—
158D8×C53C22×C412yes—
159C2×C8×C53C22×C412yes—
160C2×C8×C53C2×D411yescharacteristic
161C2×C8×C53C2×D412yes—
162C2×C8×C53C2411yescharacteristic
163C4⋊C4×C53—22no—
164C22⋊C4×C53—41no—
165C42×C53—41no—
166C22⋊Dic53—41no—
167#11719925—24no—
168#11719924—24no—
169#11719923—24no—
170#11719922—24no—
171#11719921—42no—
172C4⋊Dic53—22no—
173#11719919—42no—
174C4×Dic53—41no—
175C4×Dic53—42no—
176C2×#88322C22⋊C414yes—
177C2×#88322C2×D411yescharacteristic
178C2×#88322C2×D411yescharacteristic
179Dic212—21no—
180Dic212C22×C412yes—
181D424—21no—
182D424C22×C412yes—
183#11719913—24no—
184C8×D53C22⋊C414yes—
185C8×D53C22⋊C414yes—
186C8×D53C22×C412yes—
187C8×D53C22×C412yes—
188C8×D53C2×D412yes—
189C8×D53C2×D412yes—
190C16×C53—22no—
191#11719908—22no—
192C23×C53—41no—
193C23×C53C2×D811yescharacteristic
194C22×D53—42no—
195C22×D53—41no—
196C22×D53—44no—
197C22×D53—84no—
198C22×D53#10214yes—
199C22×D53C2×D811yescharacteristic
200C22×D53C2×D811yescharacteristic
201Q8×C53—22no—
202Q8×C53—41no—
203D4×C53—22no—
204D4×C53—44no—
205D4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—44no—
208C2×C4×C53—42no—
209C2×C4×C53C2×C22⋊C412yes—
210C2×C4×C53C2×D811yescharacteristic
211C2×C4×C53C2×SD1612yes—
212C2×C4×C53C22×D411yescharacteristic
213C22⋊D53—44no—
214C22⋊D53—44no—
215C22⋊D53—44no—
216C2×Dic53—41no—
217C2×Dic53—44no—
218C2×Dic53—42no—
219C2×Dic53#10214yes—
220C2×Dic53C2×D811yescharacteristic
221C2×Dic53C2×D811yescharacteristic
222D212—22no—
223D212—44no—
224D212—41no—
225C4×D53—44no—
226C4×D53—44no—
227C4×D53—44no—
228C4×D53—84no—
229C4×D53#10214yes—
230C4×D53#10214yes—
231C4×D53C2×C22⋊C412yes—
232C4×D53C2×C22⋊C412yes—
233C4×D53C2×SD1612yes—
234C4×D53C2×SD1612yes—
235Dic106—22no—
236Dic106—41no—
237Dic106—44no—
238C8×C53C2×C22⋊C412yes—
239C8×C53C23×C411yescharacteristic
240C8×C53C22×D411yescharacteristic
241#88322C2×C22⋊C412yes—
242#88322C2×C22⋊C412yes—
243#33380—531nomaximal, Sylow
244C22×C53—22no—
245C22×C53—42no—
246C22×C53—82no—
247C22×C53C2×#10212yes—
248C22×C53C22×D811yescharacteristic
249C2×D53—22no—
250C2×D53—22no—
251C2×D53—44no—
252C2×D53—42no—
253C2×D53—88no—
254C2×D53—82no—
255C2×D53C2×#10212yes—
256C2×D53C2×#10212yes—
257C4×C53—42no—
258C4×C53—82no—
259C4×C53C2×#10212yes—
260C4×C53C22×C22⋊C411yescharacteristic
261C4×C53C22×SD1611yescharacteristic
262Dic53—42no—
263Dic53—82no—
264Dic53C2×#10212yes—
265Dic53C2×#10212yes—
266C22×#135—531no—
267C22×M32—531no—
268C2×#427—531no—
269#2141—532no—
270#2140—532no—
271#2139—532no—
272#2139—532no—
273#2138—531no—
274#2138—531no—
275#2138—531no—
276#2138D5311yescharacteristic, complemented
277C2×C53—22no—
278C2×C53—82no—
279C2×C53C22×#10211yescharacteristic
280C2×C53#341412yes—
281D53—22no—
282D53—22no—
283D53—82no—
284C22×#41—1061no—
285C2×#135—538no—
286C2×#135—531no—
287C2×#135—531no—
288C2×#135—531no—
289C2×#135—1061no—
290C2×#135—1064no—
291C2×#135—1061no—
292C2×#135C2×D5311yescharacteristic
293C22×Q16—531no—
294C22×SD16—1061no—
295C22×D8—531no—
296C23×C8—531no—
297C2×#117—1061no—
298C2×M32—532no—
299C2×M32—532no—
300C2×M32—1064no—
301C22×C16—531no—
302C22×C16C2×D5311yescharacteristic, complemented
303#427—531no—
304#427—532no—
305#427—1061no—
306#427—1061no—
307#427C2×D5311yescharacteristic, complemented
308C2×#107—532no—
309C2×#106—1061no—
310#358—532no—
311#358—532no—
312#358—532no—
313#358C2×D5312yescomplemented
314#357—532no—
315#357—532no—
316#357—532no—
317#357C2×D5312yescomplemented
318C53#3338011yescharacteristic, complemented, Sylow
319C2×#41—21no—
320C2×#41—1061no—
321C2×#41—1061no—
322C2×#41—1068no—
323C2×#41—1061no—
324C2×#41—2121no—
325C2×#41—2121no—
326C2×#41—2124no—
327C22×Q8—1061no—
328C22×D4—1061no—
329C22×D4—2121no—
330C23×C4—531no—
331C23×C4—2121no—
332#135—22no—
333#135—534no—
334#135—534no—
335#135—534no—
336#135—1062no—
337#135—1068no—
338#135—1062no—
339#135—1068no—
340#135—1062no—
341#135C4×D5314yescomplemented
342C2×Q16—534no—
343C2×Q16—531no—
344C2×Q16—1062no—
345C2×Q16—1061no—
346C2×Q16C22×D5311yescharacteristic
347C2×SD16—21no—
348C2×SD16—1068no—
349C2×SD16—1061no—
350C2×SD16—1061no—
351C2×SD16—1061no—
352C2×D8—534no—
353C2×D8—531no—
354C2×D8—1062no—
355C2×D8—1061no—
356C2×D8C22×D5311yescharacteristic
357C22×C8—531no—
358C22×C8—534no—
359C22×C8—531no—
360C22×C8—531no—
361C22×C8—534no—
362C22×C8—531no—
363C22×C8—531no—
364C22×C8C22×D5311yescharacteristic
365#117—21no—
366#117—1062no—
367#117—1061no—
368#117—2121no—
369#117—2121no—
370C2×C4⋊C4—1062no—
371C2×C22⋊C4—2121no—
372C2×C42—2121no—
373M32—1064no—
374M32—1064no—
375C2×C16—22no—
376C2×C16—532no—
377C2×C16—1062no—
378C2×C16C22×D5312yes—
379#107—532no—
380#107—1062no—
381#107C22×D5312yes—
382#106—21no—
383#106—1062no—
384#106—1061no—
385C24—531no—
386C24—2121no—
387#41—24no—
388#41—42no—
389#41—1064no—
390#41—1064no—
391#41—1064no—
392#41—2128no—
393#41—2128no—
394#41—2122no—
395#41—2122no—
396#41—2122no—
397C2×Q8—21no—
398C2×Q8—1064no—
399C2×Q8—1061no—
400C2×Q8—2121no—
401C2×Q8—2122no—
402C2×D4—21no—
403C2×D4—41no—
404C2×D4—1064no—
405C2×D4—1061no—
406C2×D4—2121no—
407C2×D4—2122no—
408C2×D4—2121no—
409C2×D4—2128no—
410C2×D4—2121no—
411C2×D4—2121no—
412C22×C4—41no—
413C22×C4—531no—
414C22×C4—534no—
415C22×C4—531no—
416C22×C4—531no—
417C22×C4—534no—
418C22×C4—531no—
419C22×C4—531no—
420C22×C4—2121no—
421C22×C4—2128no—
422C22×C4—2122no—
423C22×C4—2121no—
424C22×C4—2121no—
425C22×C4—2121no—
426C22×C4D4×D5311yescharacteristic
427Q16—21no—
428Q16—532no—
429Q16—1064no—
430Q16—1061no—
431Q16C2×C4×D5312yes—
432SD16—24no—
433SD16—1064no—
434SD16—1064no—
435SD16—1064no—
436D8—21no—
437D8—532no—
438D8—1064no—
439D8—1061no—
440D8C2×C4×D5312yes—
441C2×C8—534no—
442C2×C8—531no—
443C2×C8—531no—
444C2×C8—534no—
445C2×C8—534no—
446C2×C8—532no—
447C2×C8—532no—
448C2×C8—532no—
449C2×C8—532no—
450C2×C8C2×C4×D5312yes—
451C2×C8D4×D5311yescharacteristic
452C2×C8D4×D5312yes—
453C2×C8C23×D5311yescharacteristic
454C4⋊C4—22no—
455C4⋊C4—1062no—
456C4⋊C4—2122no—
457C22⋊C4—41no—
458C22⋊C4—2122no—
459C22⋊C4—2121no—
460C42—41no—
461C42—2122no—
462C42—2121no—
463C16—22no—
464C16—1062no—
465C23—41no—
466C23—531no—
467C23—531no—
468C23—534no—
469C23—4244no—
470C23—2122no—
471C23—2124no—
472C23—2121no—
473C23D8×D5311yescharacteristic
474Q8—22no—
475Q8—41no—
476Q8—1062no—
477Q8—2124no—
478Q8—2121no—
479D4—22no—
480D4—44no—
481D4—41no—
482D4—1062no—
483D4—2124no—
484D4—2124no—
485D4—2124no—
486D4—2124no—
487D4—2121no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—534no—
492C2×C4—531no—
493C2×C4—531no—
494C2×C4—532no—
495C2×C4—532no—
496C2×C4—534no—
497C2×C4—534no—
498C2×C4—532no—
499C2×C4—532no—
500C2×C4—4244no—
501C2×C4—2124no—
502C2×C4—2124no—
503C2×C4—2124no—
504C2×C4—2122no—
505C2×C4—2121no—
506C2×C4—2124no—
507C2×C4C22⋊C4×D5312yes—
508C2×C4D8×D5311yescharacteristic
509C2×C4SD16×D5312yes—
510C2×C4C2×D4×D5311yescharacteristic
511C8—532no—
512C8—532no—
513C8C22⋊C4×D5312yes—
514C8C22×C4×D5311yescharacteristic
515C8C2×D4×D5311yescharacteristic
516C22—22no—
517C22—42no—
518C22—82no—
519C22—532no—
520C22—532no—
521C22—1062no—
522C22—1062no—
523C22—4248no—
524C22—4242no—
525C22—2124no—
526C22—2122no—
527C22#102×D5312yes—
528C22C2×D8×D5311yescharacteristic
529C4—42no—
530C4—82no—
531C4—532no—
532C4—532no—
533C4—4242no—
534C4—2122no—
535C4#102×D5312yes—
536C4C2×C22⋊C4×D5311yescharacteristic
537C4C2×SD16×D5311yescharacteristic
538C2—22no—
539C2—82no—
540C2—1062no—
541C2—1062no—
542C2—4242no—
543C2C2×#102×D5311yescharacteristic
544C2#2028784812yes—
subgroupG/H
centerC2×C4D8×D53
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC53×#2138C2
socleC22×C53C22×D8
prime radicalC2×#135×D53C2
cosocleC2×C8×C53C24
hypercenter#2138D53
hypocenterC53#33380
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
2. C22×#13133156C2semidirect
3. #427×D53C2semidirect
4. #20276433C2semidirect
5. #20276429C2semidirect
6. #20276426C2semidirect
7. #20276424C2semidirect
8. #20276423C2semidirect
9. #20276422C2semidirect
10. #20276420C2semidirect
11. C53×#2138C2semidirect
19. #135×D53C4semidirect
27. C2×#13133156C22semidirect
28. C22×#11719908C22semidirect
29. #16810661C22semidirect
33. #107×D53C22semidirect
35. #16810428C22semidirect
36. #16810427C22semidirect
37. #16810426C22semidirect
38. #16810425C22semidirect
39. #16810424C22semidirect
40. #16810423C22semidirect
42. C22×C16×C53C22semidirect
43. C53×#427C22semidirect
44. C53×#358C22semidirect
45. C53×#357C22semidirect
61. #13133239C2×C4semidirect
69. Q16×D53C2×C4semidirect
77. D8×D53C2×C4semidirect
102. #135×C53C2×C4semidirect
276. #2138D53semidirect
302. C22×C16C2×D53semidirect
307. #427C2×D53semidirect
313. #358C2×D53semidirect
317. #357C2×D53semidirect
318. C53#33380semidirect
341. #135C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#135×D532
C22×#131331562
#427×D532
#202764332
#202764292
#202764262
#202764242
#202764232
#202764222
#202764202
C53×#21382
#3338053