Finite Group Databasebeta

← #27383789

Subgroups — #27383789

Order 13568.

1330 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×#16811389C211yesmaximal, characteristic
2C2×#16810671C211yesmaximal, characteristic, complemented
3C2×#16810663C211yesmaximal, characteristic, complemented
4#20283235C211yesmaximal, characteristic
5C2×#16810622C211yesmaximal, characteristic
6#20274778C214yesmaximal, complemented
7#20274773C214yesmaximal, complemented
8#20274453C211yesmaximal, characteristic, complemented
9C2×C53×#343C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11#16811389—24no—
12#16811389C418yescomplemented
13C2×#13133225C2214yes—
14C2×#13133223—22no—
15C22×#11719917C2211yescharacteristic
16C22×#11719917C2211yescharacteristic
17C2×M16×D53—22no—
18C2×#13133176C2211yescharacteristic
19#16810671—24no—
20C2×#13133152C2214yescomplemented
21#16810663—22no—
22#16810663—22no—
23C2×#13133150C2212yescomplemented
24C2×#13133148C2212yescomplemented
25#16810633—21no—
26#16810629—21no—
27#16810625C2212yes—
28#16810625C2212yes—
29#16810622—22no—
30#16810622C2214yes—
31#16810620—22no—
32C2×#13133140—21no—
33C2×#13133137—21no—
34#16810391C2212yescomplemented
35#16810391C2212yescomplemented
36C22×M16×C53C2211yescharacteristic
37C22×M16×C53C2211yescharacteristic
38C22×C4⋊C4×C53C2211yescharacteristic
39C53×#343C2214yescomplemented
40C2×#11719945—21no—
41C2×#11719945—22no—
42C2×#11719945—21no—
43C2×#11719945—22no—
44C2×#11719945C2×C414yes—
45C2×#11719945D412yes—
46C2×#11719945Q814yes—
47C2×#11719945C2312yes—
48C22×Dic106D411yescharacteristic
49C22×D212D411yescharacteristic
50C22×C22⋊D53—21no—
51C22×C4×D53—21no—
52#13133225—28no—
53#13133225C2×C4116yescomplemented
54#13133223—216no—
55C2×#11719917—21no—
56C2×#11719917—22no—
57C2×#11719917—22no—
58C2×#11719917C2×C414yes—
59C2×#11719917D412yes—
60C2×#11719917D412yes—
61C2×#11719917D414yes—
62C2×#11719917C2312yes—
63C2×#11719912—24no—
64M16×D53—216no—
65C22×#88322—21no—
66C22×#88322C2312yes—
67C2×C8×D53—24no—
68#13133176—21no—
69#13133176—21no—
70#13133176—21no—
71#13133176—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C22⋊Dic53—21no—
75C2×C4×Dic53—21no—
76#13133152—28no—
77#13133150—24no—
78#13133148—24no—
79#13133142—22no—
80#13133142—22no—
81#13133140—24no—
82#13133139—22no—
83#13133139—22no—
84#13133137—24no—
85C23×C4×C53—21no—
86C23×C4×C53C2311yescharacteristic
87C2×M16×C53—21no—
88C2×M16×C53—22no—
89C2×M16×C53—21no—
90C2×M16×C53—22no—
91C2×M16×C53—22no—
92C2×M16×C53C2×C414yes—
93C2×M16×C53C2312yes—
94C2×M16×C53C2312yes—
95C22×C8×C53—21no—
96C22×C8×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53C2312yes—
103C24×C53#4111yescharacteristic
104C23×D53—21no—
105C22×C4×C53—22no—
106C22×C4×C53—41no—
107C22×C4×C53—41no—
108C22×C4×C53—82no—
109C22×C4×C53C22×C412yes—
110C22×C4×C53C2×D412yes—
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×Q812yes—
116C22×C4×C53#4111yescharacteristic
117C22×C4×C53#4112yes—
118C22×C4×C53C2411yescharacteristic
119C2×C22⋊D53—24no—
120C2×C22⋊D53—22no—
121C2×C22⋊D53—24no—
122C2×C22⋊D53—22no—
123C22×Dic53—21no—
124C22×Dic53—21no—
125C22×Dic53—21no—
126#11719945—24no—
127#11719945—24no—
128#11719945—24no—
129#11719945—24no—
130#11719945—24no—
131#11719945—22no—
132#11719945—22no—
133#11719945C4⋊C418yes—
134#11719945C22×C414yes—
135#11719945C2×Q814yes—
136C2×D212—22no—
137C2×D212—21no—
138C2×D212C4⋊C414yes—
139C2×D212C2×D412yes—
140C2×C4×D53—24no—
141C2×C4×D53—22no—
142C2×C4×D53—24no—
143C2×C4×D53—22no—
144C2×Dic106—22no—
145C2×Dic106—21no—
146C2×Dic106C4⋊C414yes—
147C2×Dic106C2×D412yes—
148M16×C53—24no—
149M16×C53—22no—
150M16×C53—22no—
151M16×C53—22no—
152M16×C53—42no—
153M16×C53C22×C414yes—
154C2×C8×C53—24no—
155C2×C8×C53—22no—
156C2×C8×C53—22no—
157C2×C8×C53—42no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—41no—
161C4⋊C4×C53—41no—
162C22⋊Dic53—41no—
163C22⋊Dic53—41no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4×Dic53—41no—
169C4×Dic53—41no—
170#11719917—22no—
171#11719917—22no—
172#11719917—42no—
173#11719917C22⋊C418yes—
174#11719917C22⋊C418yes—
175C2×#88322—22no—
176C2×#88322—42no—
177C2×#88322C22×C414yes—
178C2×#88322C2×D414yes—
179C2×#88322C2×D414yes—
180#11719912—216no—
181C8×D53—216no—
182C23×C53—41no—
183C23×C53—41no—
184C23×C53#11712yes—
185C23×C53#12111yescharacteristic
186C23×C53#12111yescharacteristic
187C23×C53C22⋊Q812yes—
188C23×C53C2×#4111yescharacteristic
189C22×D53—22no—
190C22×D53—24no—
191C22×D53—81no—
192C22×D53—81no—
193C2×C4×C53—42no—
194C2×C4×C53—82no—
195C2×C4×C53—82no—
196C2×C4×C53—82no—
197C2×C4×C53C2×C22⋊C412yes—
198C2×C4×C53C2×C22⋊C412yes—
199C2×C4×C53C2×C4⋊C412yes—
200C2×C4×C53C2×C4⋊C412yes—
201C2×C4×C53#11712yes—
202C2×C4×C53#12112yes—
203C2×C4×C53#12112yes—
204C2×C4×C53#12111yescharacteristic
205C2×C4×C53#12111yescharacteristic
206C2×C4×C53C22⋊Q812yes—
207C2×C4×C53C22⋊Q812yes—
208C2×C4×C53C22⋊Q812yes—
209C2×C4×C53C23×C411yescharacteristic
210C2×C4×C53C22×D411yescharacteristic
211C2×C4×C53C22×D411yescharacteristic
212C2×C4×C53C22×D411yescharacteristic
213C2×C4×C53C22×Q811yescharacteristic
214C2×C4×C53C2×#4111yescharacteristic
215C22⋊D53—24no—
216C22⋊D53—24no—
217C22⋊D53—28no—
218C2×Dic53—22no—
219C2×Dic53—24no—
220C2×Dic53—41no—
221C2×Dic53—41no—
222C2×Dic53—81no—
223C2×Dic53—81no—
224D212—22no—
225D212—24no—
226D212C2×C4⋊C414yes—
227C4×D53—24no—
228C4×D53—24no—
229C4×D53—28no—
230Dic106—22no—
231Dic106—24no—
232Dic106C2×C4⋊C414yes—
233C8×C53—24no—
234C8×C53—42no—
235#88322—42no—
236#88322C2×C22⋊C418yes—
237#20679—531nomaximal, Sylow
238C22×C53—21no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—21no—
243C22×C53—21no—
244C22×C53—42no—
245C22×C53—42no—
246C22×C53#38012yes—
247C22×C53#38012yes—
248C22×C53C2×#11711yescharacteristic
249C22×C53C2×#12111yescharacteristic
250C22×C53C2×C22⋊Q811yescharacteristic
251C2×D53—24no—
252C2×D53—42no—
253C2×D53—42no—
254C2×D53—84no—
255C4×C53—82no—
256C4×C53#38012yes—
257C4×C53#38012yes—
258C4×C53C22×C22⋊C411yescharacteristic
259C4×C53C22×C4⋊C411yescharacteristic
260C4×C53C2×#12111yescharacteristic
261C4×C53C2×C22⋊Q811yescharacteristic
262Dic53—24no—
263Dic53—81no—
264Dic53—81no—
265C2×#568—531no—
266C2×#430—531no—
267C2×#428—531no—
268#2896—531no—
269C2×#417—531no—
270#1816—534no—
271#1815—534no—
272#1742—531no—
273C2×#343D5311yescharacteristic, complemented
274C2×C53—22no—
275C2×C53—21no—
276C2×C53—21no—
277C2×C53—22no—
278C2×C53C2×#38011yescharacteristic
279C2×C53#306811yescharacteristic
280C2×C53#306911yescharacteristic
281D53—44no—
282C22×#41—531no—
283#568—538no—
284#568—1064no—
285C2×#131—534no—
286C2×#131—1062no—
287C22×M16—531no—
288C22×M16—531no—
289C22×M16—1062no—
290C22×M16C2×D5311yescharacteristic
291C22×M16C2×D5311yescharacteristic
292C2×#117—531no—
293C22×C4⋊C4C2×D5311yescharacteristic
294#430—1064no—
295C2×#108—534no—
296#428—1062no—
297#428—1062no—
298C2×#107—532no—
299C2×#106—532no—
300#419—1061no—
301#418—1061no—
302#417—532no—
303#417—532no—
304#417—534no—
305#417—1062no—
306#416—1062no—
307C2×#103—1061no—
308C2×#102—1061no—
309#343—532no—
310#343—532no—
311#343C2×D5314yescomplemented
312C53#2067911yescharacteristic, complemented, Sylow
313C2×#41—534no—
314C2×#41—532no—
315C2×#41—534no—
316C2×#41—532no—
317C2×#41—1061no—
318C2×#41—1062no—
319C2×#41—1061no—
320C2×#41—1062no—
321C22×Q8—531no—
322C22×D4—531no—
323C22×D4—1061no—
324C23×C4—21no—
325C23×C4—1061no—
326C23×C4C22×D5311yescharacteristic
327#131—5316no—
328#131—10616no—
329#131—1068no—
330C2×M16—21no—
331C2×M16—22no—
332C2×M16—21no—
333C2×M16—22no—
334C2×M16—22no—
335C2×M16—534no—
336C2×M16—532no—
337C2×M16—532no—
338C2×M16—534no—
339C2×M16—532no—
340C2×M16—1064no—
341C2×M16—10616no—
342C2×M16—1061no—
343C2×M16—1062no—
344C2×M16—1062no—
345C2×M16C4×D5314yes—
346C2×M16C22×D5312yes—
347C2×M16C22×D5312yes—
348C22×C8—21no—
349C22×C8—21no—
350C22×C8—532no—
351C22×C8—1064no—
352C22×C8—1061no—
353#117—1061no—
354#117—1061no—
355#117—1061no—
356#117—1061no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—531no—
363C2×C4⋊C4—531no—
364C2×C4⋊C4C22×D5312yes—
365C2×C22⋊C4—1061no—
366C2×C42—1061no—
367#108—1068no—
368#107—1064no—
369#106—1064no—
370#103—1064no—
371#103—1062no—
372#103—1062no—
373#102—1064no—
374#102—1062no—
375#102—1062no—
376C24—1061no—
377C24#1171994511yescharacteristic
378#41—538no—
379#41—534no—
380#41—534no—
381#41—1064no—
382#41—1064no—
383#41—1064no—
384#41—1064no—
385#41—1064no—
386#41—1062no—
387#41—1062no—
388C2×Q8—534no—
389C2×Q8—532no—
390C2×Q8—1061no—
391C2×Q8—1062no—
392C2×D4—534no—
393C2×D4—532no—
394C2×D4—1064no—
395C2×D4—1062no—
396C2×D4—1064no—
397C2×D4—1062no—
398C2×D4—1062no—
399C2×D4—1061no—
400C22×C4—22no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—82no—
404C22×C4—1061no—
405C22×C4—1064no—
406C22×C4—1062no—
407C22×C4—1064no—
408C22×C4—1062no—
409C22×C4—1061no—
410C22×C4—1061no—
411C22×C4C2×C4×D5312yes—
412C22×C4#1171994511yescharacteristic
413C22×C4#1171994512yes—
414C22×C4D4×D5312yes—
415C22×C4D4×D5311yescharacteristic
416C22×C4D4×D5311yescharacteristic
417C22×C4D4×D5311yescharacteristic
418C22×C4D4×D5311yescharacteristic
419C22×C4Q8×D5312yes—
420C22×C4C23×D5311yescharacteristic
421M16—24no—
422M16—22no—
423M16—22no—
424M16—22no—
425M16—42no—
426M16—538no—
427M16—538no—
428M16—10616no—
429M16—1062no—
430M16—1062no—
431M16—2122no—
432M16C2×C4×D5314yes—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—42no—
437C2×C8—534no—
438C2×C8—534no—
439C2×C8—534no—
440C2×C8—10616no—
441C2×C8—1062no—
442C2×C8—2122no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—1061no—
448C4⋊C4—1061no—
449C4⋊C4—1061no—
450C4⋊C4—1061no—
451C22⋊C4—2121no—
452C22⋊C4—2121no—
453C42—2121no—
454C42—2121no—
455C23—41no—
456C23—41no—
457C23—1064no—
458C23—1062no—
459C23—4241no—
460C23—4241no—
461C23#1313317412yes—
462C23#1313319011yescharacteristic
463C23#1313319011yescharacteristic
464C23#1313319512yes—
465C23C2×#1171994511yescharacteristic
466Q8—534no—
467Q8—1062no—
468Q8—1064no—
469D4—534no—
470D4—1064no—
471D4—1064no—
472D4—1068no—
473D4—1062no—
474D4—1064no—
475C2×C4—42no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—1062no—
480C2×C4—1064no—
481C2×C4—1064no—
482C2×C4—1064no—
483C2×C4—1068no—
484C2×C4—4241no—
485C2×C4—4241no—
486C2×C4—2121no—
487C2×C4—2121no—
488C2×C4C22⋊C4×D5312yes—
489C2×C4C22⋊C4×D5312yes—
490C2×C4C4⋊C4×D5312yes—
491C2×C4C4⋊C4×D5312yes—
492C2×C4#1313317412yes—
493C2×C4#1313319012yes—
494C2×C4#1313319012yes—
495C2×C4#1313319011yescharacteristic
496C2×C4#1313319011yescharacteristic
497C2×C4#1313319512yes—
498C2×C4#1313319512yes—
499C2×C4#1313319512yes—
500C2×C4C22×C4×D5311yescharacteristic
501C2×C4C2×D4×D5311yescharacteristic
502C2×C4C2×D4×D5311yescharacteristic
503C2×C4C2×D4×D5311yescharacteristic
504C2×C4C2×Q8×D5311yescharacteristic
505C2×C4C2×#1171994511yescharacteristic
506C8—24no—
507C8—42no—
508C8—538no—
509C8—2122no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—42no—
517C22—42no—
518C22—1064no—
519C22—4244no—
520C22—2122no—
521C22—2122no—
522C22#1681048212yes—
523C22#1681048212yes—
524C22C2×#1313317411yescharacteristic
525C22C2×#1313319011yescharacteristic
526C22C2×#1313319511yescharacteristic
527C4—82no—
528C4—1064no—
529C4—4241no—
530C4—4241no—
531C4#1681048212yes—
532C4#1681048212yes—
533C4C2×C22⋊C4×D5311yescharacteristic
534C4C2×C4⋊C4×D5311yescharacteristic
535C4C2×#1313319011yescharacteristic
536C4C2×#1313319511yescharacteristic
537C2—22no—
538C2—21no—
539C2—21no—
540C2—22no—
541C2—2124no—
542C2C2×#1681048211yescharacteristic
543C2#2028444211yescharacteristic
544C2#2028445111yescharacteristic
subgroupG/H
centerC22C2×#13133190
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#343C2
socleC22×C53C2×#121
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenterC2×#343D53
hypocenterC53#20679
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
2. C2×#16810671C2semidirect
3. C2×#16810663C2semidirect
6. #20274778C2semidirect
7. #20274773C2semidirect
8. #20274453C2semidirect
9. C2×C53×#343C2semidirect
12. #16811389C4semidirect
20. C2×#13133152C22semidirect
23. C2×#13133150C22semidirect
24. C2×#13133148C22semidirect
34. #16810391C22semidirect
35. #16810391C22semidirect
39. C53×#343C22semidirect
53. #13133225C2×C4semidirect
273. C2×#343D53semidirect
311. #343C2×D53semidirect
312. C53#20679semidirect

Maximal subgroups

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

subgroupindex
C2×#168113892
C2×#168106712
C2×#168106632
#202832352
C2×#168106222
#202747782
#202747732
#202744532
C2×C53×#3432
#2067953