Finite Group Databasebeta

← #27389287

Subgroups — #27389287

Order 13568.

1218 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811152C211yesmaximal, characteristic
2C2×#16810575C211yesmaximal, characteristic, complemented
3#20277120C211yesmaximal, characteristic, complemented
4#20275684C211yesmaximal, characteristic, complemented
5#20275683C211yesmaximal, characteristic, complemented
6#20275532C211yesmaximal, characteristic, complemented
7#20275150C212yesmaximal
8#20275145C212yesmaximal, complemented
9#103×Dic53C212yesmaximal, complemented
10#102×Dic53C212yesmaximal, complemented
11C53×#1959C211yesmaximal, characteristic, complemented
12C22×#11719912C2212yes—
13#16811152—24no—
14#16811152C418yescomplemented
15C2×#13133210—21no—
16C2×#13133209C2211yescharacteristic
17C2×#13133199—21no—
18C2×#13133193C2211yescharacteristic
19C2×#13133188C2211yescharacteristic
20C2×Q8×Dic53C2211yescharacteristic
21C2×D4×Dic53C2211yescharacteristic
22C2×#13133177—21no—
23C2×#13133175—21no—
24C2×#13133173—21no—
25C2×C42×D53C2211yescharacteristic
26C2×#13133141C2212yes—
27C2×#13133138C2212yes—
28#16810575—24no—
29C2×#13133128C2211yescharacteristic
30C2×C8×Dic53C2212yes—
31#16810493C2211yescharacteristic
32#16810488—21no—
33#16810480—21no—
34#16810480C2212yes—
35C4⋊C4×Dic53—21no—
36C4⋊C4×Dic53C2212yes—
37#16810386C2212yes—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×#124×C53C2211yescharacteristic
42C2×#103×C53C2212yes—
43C2×#102×C53C2212yes—
44C2×#97×C53C2211yescharacteristic
45C53×#384C2211yescharacteristic
46C22×C22⋊D53—21no—
47C23×Dic53—21no—
48C22×C4×D53D411yescharacteristic
49C22×C4×D53D411yescharacteristic
50C22×C4×D53C2311yescharacteristic
51C2×#11719912—24no—
52C2×#11719912—24no—
53C2×#11719912—24no—
54C22×#88322C2312yes—
55#13133210—28no—
56#13133209—22no—
57#13133209C2×C414yes—
58#13133199—28no—
59#13133193—22no—
60#13133193C2×C414yes—
61#13133188—22no—
62#13133188C2×C414yes—
63Q8×Dic53—22no—
64Q8×Dic53C2×C414yes—
65D4×Dic53—22no—
66D4×Dic53C2×C414yes—
67#13133177—28no—
68#13133175—44no—
69#13133173—28no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2312yes—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×#11719921—21no—
82C2×#11719921—22no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—22no—
85C2×C4×Dic53D411yescharacteristic
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88C42×D53C2×C414yes—
89C42×D53D414yes—
90#13133141—24no—
91#13133138—24no—
92#13133128—22no—
93C8×Dic53—24no—
94#13133124—22no—
95#13133124—22no—
96C22×Q8×C53C2311yescharacteristic
97C22×D4×C53C2311yescharacteristic
98C22×C8×C53C2312yes—
99#124×C53—22no—
100#124×C53C2×C414yes—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53C2312yes—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—21no—
106C2×C42×C53C2311yescharacteristic
107#103×C53—24no—
108#102×C53—24no—
109#97×C53—22no—
110C24×C53—21no—
111C23×D53C2×D411yescharacteristic
112C2×Q8×C53—21no—
113C2×Q8×C53—22no—
114C2×Q8×C53—22no—
115C2×Q8×C53C22×C412yes—
116C2×D4×C53—21no—
117C2×D4×C53—22no—
118C2×D4×C53—22no—
119C2×D4×C53C22×C412yes—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2411yescharacteristic
125C2×C22⋊D53—24no—
126C2×C22⋊D53—42no—
127C2×C22⋊D53—42no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—24no—
131C22×Dic53—22no—
132C22×Dic53—42no—
133C22×Dic53—42no—
134C22×Dic53C2×D411yescharacteristic
135C22×Dic53C2×D411yescharacteristic
136C22×Dic53C2×D411yescharacteristic
137C2×C4×D53—22no—
138C2×C4×D53—22no—
139C2×C4×D53—22no—
140C2×C4×D53—22no—
141C2×C4×D53—41no—
142C2×C4×D53—41no—
143C2×C4×D53C22⋊C414yes—
144C2×C4×D53C22⋊C414yes—
145C2×C4×D53C22×C412yes—
146C2×C4×D53C2×D412yes—
147C2×C8×C53—22no—
148C2×C8×C53—22no—
149C2×C8×C53#4114yes—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—42no—
153C4⋊C4×C53—42no—
154C22⋊C4×C53—24no—
155C22⋊C4×C53—24no—
156C42×C53C22×C412yes—
157C42×C53C2×D412yes—
158C22⋊Dic53—24no—
159C22⋊Dic53—24no—
160#11719921—24no—
161#11719921—44no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53—24no—
165C4⋊Dic53—22no—
166C4⋊Dic53—22no—
167C4⋊Dic53—42no—
168C4⋊Dic53—42no—
169C4⋊Dic53C22×C412yes—
170C4⋊Dic53C22×C412yes—
171#11719919—24no—
172#11719919—24no—
173#11719919—24no—
174C4×Dic53—24no—
175C4×Dic53—44no—
176C4×Dic53C22⋊C414yes—
177C4×Dic53C22⋊C414yes—
178C4×Dic53C22×C412yes—
179C4×Dic53C2×D412yes—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322#4114yes—
183#11719912—216no—
184C23×C53—22no—
185C23×C53—41no—
186C23×C53—41no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53—41no—
190C23×C53—41no—
191C23×C53C22×D411yescharacteristic
192C22×D53—21no—
193C22×D53—21no—
194C22×D53—21no—
195C22×D53—21no—
196C22×D53—21no—
197C22×D53—21no—
198C22×D53C2×C22⋊C412yes—
199Q8×C53—24no—
200Q8×C53—42no—
201D4×C53—24no—
202D4×C53—42no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—22no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53C2×C22⋊C412yes—
214C2×C4×C53C2×C22⋊C412yes—
215C2×C4×C53C23×C411yescharacteristic
216C2×C4×C53C22×D411yescharacteristic
217C2×C4×C53C2×#4112yes—
218C2×C4×C53C2×#4112yes—
219C22⋊D53—48no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—22no—
225C2×Dic53—22no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—48no—
231C2×Dic53—42no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234C2×Dic53C2×C22⋊C412yes—
235C2×Dic53C2×C22⋊C412yes—
236C2×Dic53C2×C22⋊C412yes—
237C2×Dic53AGL(1,ℤ/8)12yes—
238C2×Dic53#13712yes—
239C4×D53—24no—
240C4×D53—24no—
241C4×D53—28no—
242C4×D53—44no—
243C8×C53—24no—
244#88322—24no—
245#21403—531nomaximal, Sylow
246C22×C53—42no—
247C22×C53—42no—
248C22×C53—42no—
249C22×C53—81no—
250C22×C53—81no—
251C22×C53—81no—
252C22×C53—81no—
253C22×C53C22×C22⋊C411yescharacteristic
254C22×C53#53211yescharacteristic
255C22×C53#53211yescharacteristic
256C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
257C22×C53C2×#13711yescharacteristic
258C22×C53#57511yescharacteristic
259C22×C53#57511yescharacteristic
260C2×D53—22no—
261C2×D53—22no—
262C2×D53—22no—
263C2×D53—41no—
264C2×D53—41no—
265C2×D53—41no—
266C2×D53—41no—
267C4×C53—22no—
268C4×C53—22no—
269C4×C53—42no—
270C4×C53—82no—
271C4×C53C4×#4112yes—
272C4×C53#52512yes—
273Dic53—22no—
274Dic53—22no—
275Dic53—42no—
276Dic53—42no—
277Dic53—82no—
278Dic53#41714yes—
279C2×#529—531no—
280C2×C4×M16—531no—
281#2285—531no—
282#1992—531no—
283#1992—531no—
284#1959—531no—
285#1959D5311yescharacteristic, complemented
286#1895—532no—
287#1894—532no—
288C4×#103—532no—
289C4×#102—532no—
290C2×C53—82no—
291C2×C53#229711yescharacteristic
292C2×C53C2×#41711yescharacteristic
293C2×C53#289611yescharacteristic
294C2×C53#317111yescharacteristic
295C2×C53#317111yescharacteristic
296C2×C53#317211yescharacteristic
297C2×C53#317211yescharacteristic
298D53—42no—
299C22×M16—532no—
300#529—538no—
301#529—1064no—
302C2×#126—1061no—
303C2×#125—531no—
304C2×#124C2×D5311yescharacteristic
305C2×#123—1061no—
306C2×C22⋊Q8—531no—
307C2×#121—531no—
308C2×C4×Q8—531no—
309C2×C4×D4—531no—
310C2×C4×D4—1061no—
311C2×#117—1061no—
312C2×#117—1061no—
313C22×C42—531no—
314C2×#103—532no—
315C2×#103C2×D5312yes—
316C2×#102—532no—
317C2×#102C2×D5312yes—
318C4×M16—1064no—
319C2×#97—531no—
320C2×#97C2×D5311yescharacteristic
321C2×C4×C8—532no—
322#384—531no—
323#384C2×D5311yescharacteristic
324#382—1061no—
325#380—532no—
326#380—1061no—
327C4×C4⋊C4—532no—
328C4×C4⋊C4—1061no—
329#340—532no—
330#340—532no—
331#340—532no—
332#340—532no—
333C53#2140311yescharacteristic, complemented, Sylow
334C22×Q8C22×D5311yescharacteristic
335C22×D4—1061no—
336C22×D4C22×D5311yescharacteristic
337C23×C4—531no—
338C23×C4—531no—
339C23×C4—531no—
340C23×C4—1061no—
341C2×M16—1064no—
342C2×M16—1064no—
343C2×M16—1064no—
344C22×C8—532no—
345C22×C8C22×D5312yes—
346#126—1068no—
347#125—534no—
348#125—1062no—
349#124—22no—
350#124C4×D5314yes—
351#123—1068no—
352C22⋊Q8—534no—
353C22⋊Q8—1062no—
354#121—534no—
355#121—1062no—
356C4×Q8—534no—
357C4×Q8—1062no—
358C4×D4—534no—
359C4×D4—1062no—
360C4×D4—1068no—
361#117—1068no—
362#117—2124no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—531no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—532no—
368C2×C4⋊C4—1061no—
369C2×C4⋊C4—1061no—
370C2×C4⋊C4—1061no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4C22×D5312yes—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—1061no—
378C2×C22⋊C4—1061no—
379C2×C22⋊C4—1061no—
380C2×C22⋊C4—1062no—
381C2×C42—531no—
382C2×C42—531no—
383C2×C42—531no—
384C2×C42—534no—
385C2×C42—534no—
386C2×C42—1061no—
387C2×C42—1062no—
388C2×C42C22×D5311yescharacteristic
389#103—24no—
390#103—1064no—
391#102—24no—
392#102—1064no—
393#97—22no—
394#97—1062no—
395C4×C8—1064no—
396#95—1062no—
397#95—1062no—
398C24—21no—
399C24—531no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8C2×C4×D5312yes—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—1064no—
408C2×D4—2122no—
409C2×D4—2122no—
410C2×D4C2×C4×D5312yes—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—531no—
414C22×C4—531no—
415C22×C4—531no—
416C22×C4—534no—
417C22×C4—532no—
418C22×C4—532no—
419C22×C4—534no—
420C22×C4—1062no—
421C22×C4—1062no—
422C22×C4—1064no—
423C22×C4—1062no—
424C22×C4—1062no—
425C22×C4—1061no—
426C22×C4—1061no—
427C22×C4—1062no—
428C22×C4—2121no—
429C22×C4—2121no—
430C22×C4—2122no—
431C22×C4—2122no—
432C22×C4D4×D5311yescharacteristic
433C22×C4D4×D5311yescharacteristic
434C22×C4C23×D5311yescharacteristic
435M16—10616no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—534no—
439C2×C8—1062no—
440C2×C8—1062no—
441C2×C8#1171994714yes—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—532no—
447C4⋊C4—532no—
448C4⋊C4—1064no—
449C4⋊C4—1064no—
450C4⋊C4—1064no—
451C4⋊C4—1064no—
452C4⋊C4—1061no—
453C4⋊C4—1061no—
454C4⋊C4—1062no—
455C4⋊C4—1062no—
456C4⋊C4—2122no—
457C4⋊C4—2122no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—1064no—
461C22⋊C4—1064no—
462C22⋊C4—1064no—
463C22⋊C4—2124no—
464C42—534no—
465C42—534no—
466C42—532no—
467C42—532no—
468C42—1064no—
469C42—2124no—
470C42C2×C4×D5312yes—
471C42D4×D5312yes—
472C23—22no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—532no—
480C23—1061no—
481C23—1061no—
482C23—1061no—
483C23—1061no—
484C23—1061no—
485C23—1061no—
486C23C2×D4×D5311yescharacteristic
487Q8—24no—
488Q8—42no—
489D4—24no—
490D4—42no—
491D4—2128no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—532no—
503C2×C4—532no—
504C2×C4—532no—
505C2×C4—532no—
506C2×C4—532no—
507C2×C4—1068no—
508C2×C4—1061no—
509C2×C4—1061no—
510C2×C4—1061no—
511C2×C4—1061no—
512C2×C4—1064no—
513C2×C4—1064no—
514C2×C4—1062no—
515C2×C4—1062no—
516C2×C4—2124no—
517C2×C4—2128no—
518C2×C4—2122no—
519C2×C4—2122no—
520C2×C4—2122no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4C22⋊C4×D5312yes—
526C2×C4C22⋊C4×D5312yes—
527C2×C4C22×C4×D5311yescharacteristic
528C2×C4C2×D4×D5311yescharacteristic
529C2×C4#41×D5312yes—
530C2×C4C2×#1171994712yes—
531C8—24no—
532C8—1064no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—1062no—
541C22—1062no—
542C22—1062no—
543C22—2121no—
544C22—2121no—
545C22—2121no—
546C22—2121no—
547C22C2×C22⋊C4×D5311yescharacteristic
548C22#1681117111yescharacteristic
549C22#1681117211yescharacteristic
550C22AGL(1,ℤ/8)×D5311yescharacteristic
551C22#137×D5311yescharacteristic
552C22#1681141911yescharacteristic
553C22#1681142211yescharacteristic
554C4—22no—
555C4—22no—
556C4—42no—
557C4—82no—
558C4—534no—
559C4—1062no—
560C4—1062no—
561C4—4242no—
562C4—2122no—
563C4—2122no—
564C4C4×#1171994712yes—
565C4#1681113312yes—
566C2—82no—
567C2—2122no—
568C2#2027720211yescharacteristic
569C2#417×D5311yescharacteristic
570C2#2028323111yescharacteristic
571C2#2028530711yescharacteristic
572C2#2028531311yescharacteristic
573C2#2028531811yescharacteristic
574C2#2028532411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1959C2
socleC23×C53C22×D4
prime radicalC2×#13133188C22
cosocleC22×C4×C53C24
hypercenter#1959D53
hypocenterC53#21403
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1959

Complements

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

NH ≅ G/Ntype
2. C2×#16810575C2semidirect
3. #20277120C2semidirect
4. #20275684C2semidirect
5. #20275683C2semidirect
6. #20275532C2semidirect
8. #20275145C2semidirect
9. #103×Dic53C2semidirect
10. #102×Dic53C2semidirect
11. C53×#1959C2semidirect
14. #16811152C4semidirect
285. #1959D53semidirect
333. C53#21403semidirect

Maximal subgroups

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

subgroupindex
C2×#168111522
C2×#168105752
#202771202
#202756842
#202756832
#202755322
#202751502
#202751452
#103×Dic532
#102×Dic532
C53×#19592
#2140353