Finite Group Databasebeta

← #27383787

Subgroups — #27383787

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×#16810672C211yesmaximal, characteristic, complemented
3#20283232C211yesmaximal, characteristic
4C2×#16810622C211yesmaximal, characteristic
5#20274780C214yesmaximal, complemented
6#20274775C214yesmaximal, complemented
7#20274454C211yesmaximal, characteristic, complemented
8C2×#16810392C211yesmaximal, characteristic, complemented
9C2×C53×#428C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11#16811389—24no—
12#16811389C418yescomplemented
13C2×#13133225—22no—
14C2×#13133223C2214yes—
15C22×#11719917C2211yescharacteristic
16C22×#11719917C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18C2×M16×D53—22no—
19#16810672—24no—
20C2×#13133153C2214yescomplemented
21#16810631—21no—
22#16810627—21no—
23#16810622—22no—
24#16810622C2212yes—
25#16810622C2214yes—
26#16810622C2212yes—
27#16810617—22no—
28C2×#13133140—21no—
29C2×#13133137—21no—
30#16810392C2214yescomplemented
31#16810392C2212yescomplemented
32#16810392C2212yescomplemented
33C22×M16×C53C2211yescharacteristic
34C2×#117×C53C2211yescharacteristic
35C22×C4⋊C4×C53C2211yescharacteristic
36C53×#428—22no—
37C53×#428—22no—
38C2×#107×C53C2212yescomplemented
39C2×#106×C53C2212yescomplemented
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—216no—
53#13133223—28no—
54#13133223C2×C4116yescomplemented
55C2×#11719917—21no—
56C2×#11719917—22no—
57C2×#11719917—21no—
58C2×#11719917—21no—
59C2×#11719917—22no—
60C2×#11719917—22no—
61C2×#11719917—22no—
62C2×#11719917—22no—
63C2×#11719917C2×C414yes—
64C2×#11719917C2312yes—
65C2×#11719917C2312yes—
66C2×#11719917C2312yes—
67C2×#11719912—24no—
68M16×D53—216no—
69C22×#88322—21no—
70C22×#88322—21no—
71C22×#88322—21no—
72C2×C8×D53—24no—
73#13133153—28no—
74#13133140—24no—
75#13133140—22no—
76#13133140—22no—
77#13133137—24no—
78#13133137—22no—
79#13133137—22no—
80C23×C4×C53—21no—
81C23×C4×C53C2311yescharacteristic
82C2×M16×C53C2×C414yes—
83C2×M16×C53D414yes—
84C2×M16×C53D412yes—
85C2×M16×C53D412yes—
86C22×C8×C53C2312yes—
87#117×C53—21no—
88#117×C53—21no—
89#117×C53—21no—
90#117×C53—21no—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53C2312yes—
97C2×C4⋊C4×C53C2311yescharacteristic
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C22⋊C4×C53—21no—
100C2×C42×C53—21no—
101#107×C53—24no—
102#106×C53—24no—
103C24×C53#4111yescharacteristic
104C23×D53—21no—
105C22×C4×C53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—22no—
108C22×C4×C53—41no—
109C22×C4×C53—41no—
110C22×C4×C53—82no—
111C22×C4×C53C22×C412yes—
112C22×C4×C53C2×D412yes—
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×Q812yes—
118C22×C4×C53#4111yescharacteristic
119C22×C4×C53#4112yes—
120C22×C4×C53C2411yescharacteristic
121C2×C22⋊D53—24no—
122C2×C22⋊D53—22no—
123C2×C22⋊D53—24no—
124C2×C22⋊D53—22no—
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×C53C22⋊C418yes—
149M16×C53C22⋊C418yes—
150C2×C8×C53C22×C414yes—
151C2×C8×C53C2×D414yes—
152C2×C8×C53C2×D414yes—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—41no—
160C4⋊C4×C53—41no—
161C22⋊C4×C53—41no—
162C22⋊C4×C53—41no—
163C42×C53—41no—
164C42×C53—41no—
165#11719917—24no—
166#11719917—22no—
167#11719917—22no—
168#11719917—22no—
169#11719917—22no—
170#11719917—22no—
171#11719917—42no—
172#11719917—42no—
173#11719917C22×C414yes—
174C2×#88322—24no—
175C2×#88322—22no—
176C2×#88322—22no—
177C2×#88322—22no—
178C2×#88322—42no—
179C2×#88322—42no—
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—41no—
195C2×C4×C53—41no—
196C2×C4×C53—82no—
197C2×C4×C53—82no—
198C2×C4×C53—81no—
199C2×C4×C53—81no—
200C2×C4×C53—82no—
201C2×C4×C53C2×C22⋊C412yes—
202C2×C4×C53C2×C22⋊C412yes—
203C2×C4×C53C2×C4⋊C412yes—
204C2×C4×C53C2×C4⋊C412yes—
205C2×C4×C53#11712yes—
206C2×C4×C53#12111yescharacteristic
207C2×C4×C53#12111yescharacteristic
208C2×C4×C53#12112yes—
209C2×C4×C53#12112yes—
210C2×C4×C53C22⋊Q812yes—
211C2×C4×C53C22⋊Q812yes—
212C2×C4×C53C22⋊Q812yes—
213C2×C4×C53C23×C411yescharacteristic
214C2×C4×C53C22×D411yescharacteristic
215C2×C4×C53C22×D411yescharacteristic
216C2×C4×C53C22×D411yescharacteristic
217C2×C4×C53C22×Q811yescharacteristic
218C2×C4×C53C2×#4111yescharacteristic
219C22⋊D53—24no—
220C22⋊D53—24no—
221C22⋊D53—28no—
222C2×Dic53—22no—
223C2×Dic53—24no—
224D212—22no—
225D212—24no—
226D212C2×C4⋊C414yes—
227C4×D53—24no—
228C4×D53—24no—
229C4×D53—28no—
230Dic106—22no—
231Dic106—24no—
232Dic106C2×C4⋊C414yes—
233C8×C53C2×C22⋊C418yes—
234#88322—24no—
235#88322—42no—
236#88322—42no—
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—81no—
256C4×C53—82no—
257C4×C53—81no—
258C4×C53#38012yes—
259C4×C53#38012yes—
260C4×C53C22×C22⋊C411yescharacteristic
261C4×C53C22×C4⋊C411yescharacteristic
262C4×C53C2×#12111yescharacteristic
263C4×C53C2×C22⋊Q811yescharacteristic
264Dic53—24no—
265C2×#568—531no—
266C2×#430—531no—
267C2×#428D5311yescharacteristic, complemented
268#2896—531no—
269C2×#417—531no—
270#1816—534no—
271#1815—534no—
272#1742—531no—
273C2×#343—531no—
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—531no—
290C22×M16—1062no—
291C22×M16C2×D5311yescharacteristic
292C2×#117C2×D5311yescharacteristic
293C22×C4⋊C4C2×D5311yescharacteristic
294#430—1064no—
295C2×#108—534no—
296#428—22no—
297#428—22no—
298C2×#107C2×D5312yescomplemented
299C2×#106C2×D5312yescomplemented
300#419—1061no—
301#418—1061no—
302#417—532no—
303#417—534no—
304#417—532no—
305#417—1062no—
306#416—1062no—
307C2×#103—1061no—
308C2×#102—1061no—
309#343—532no—
310#343—534no—
311#343—532no—
312C53#2067911yescharacteristic, complemented, Sylow
313C2×#41—534no—
314C2×#41—532no—
315C2×#41—534no—
316C2×#41—532no—
317C2×#41—1062no—
318C2×#41—1061no—
319C2×#41—1062no—
320C2×#41—1061no—
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—534no—
331C2×M16—532no—
332C2×M16—532no—
333C2×M16—532no—
334C2×M16—1062no—
335C2×M16—1061no—
336C2×M16—10616no—
337C2×M16—1061no—
338C2×M16—1061no—
339C2×M16—1064no—
340C2×M16—1062no—
341C2×M16—1062no—
342C2×M16—1062no—
343C2×M16—1062no—
344C2×M16C4×D5314yes—
345C2×M16D21214yes—
346C2×M16C22⋊D5312yes—
347C2×M16C22⋊D5312yes—
348C22×C8—1061no—
349C22×C8—1064no—
350C22×C8—1061no—
351C22×C8—1061no—
352C22×C8C22×D5312yes—
353#117—21no—
354#117—21no—
355#117—21no—
356#117—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4C22×D5312yes—
363C2×C4⋊C4C22×D5311yescharacteristic
364C2×C4⋊C4C22×D5311yescharacteristic
365C2×C22⋊C4—21no—
366C2×C42—21no—
367#108—1068no—
368#107—24no—
369#106—24no—
370#103—1064no—
371#103—1062no—
372#103—1062no—
373#102—1064no—
374#102—1062no—
375#102—1062no—
376C24—1061no—
377C24#1171994711yescharacteristic
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—1061no—
399C2×D4—1062no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—82no—
406C22×C4—1061no—
407C22×C4—1064no—
408C22×C4—1062no—
409C22×C4—1064no—
410C22×C4—1062no—
411C22×C4C2×C4×D5312yes—
412C22×C4C2×D21212yes—
413C22×C4D4×D5311yescharacteristic
414C22×C4D4×D5311yescharacteristic
415C22×C4#1171994711yescharacteristic
416C22×C4Q8×D5312yes—
417C22×C4#1171994912yes—
418C22×C4C2×C22⋊D5311yescharacteristic
419C22×C4C2×C22⋊D5311yescharacteristic
420C22×C4C23×D5311yescharacteristic
421M16—534no—
422M16—1064no—
423M16—1062no—
424M16—10616no—
425M16—1062no—
426M16—1062no—
427M16—1062no—
428M16—1062no—
429M16—2122no—
430M16—2122no—
431M16#1171992118yes—
432M16#1171992118yes—
433C2×C8—1064no—
434C2×C8—1062no—
435C2×C8—10616no—
436C2×C8—1062no—
437C2×C8—1062no—
438C2×C8—2122no—
439C2×C8—2122no—
440C2×C8C2×C4×D5314yes—
441C2×C8C2×D21214yes—
442C2×C8C2×C22⋊D5314yes—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C42—41no—
454C42—41no—
455C23—41no—
456C23—41no—
457C23—1064no—
458C23—1062no—
459C23—4241no—
460C23—4241no—
461C23#1313317512yes—
462C23#1313318811yescharacteristic
463C23#1313318811yescharacteristic
464C23#1313319712yes—
465C23C2×#1171994711yescharacteristic
466Q8—534no—
467Q8—1062no—
468Q8—1064no—
469D4—534no—
470D4—1064no—
471D4—1064no—
472D4—1068no—
473D4—1062no—
474D4—1064no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—1062no—
484C2×C4—1064no—
485C2×C4—1064no—
486C2×C4—1064no—
487C2×C4—1068no—
488C2×C4C2×#1171992112yes—
489C2×C4C2×#1171992112yes—
490C2×C4C4⋊C4×D5312yes—
491C2×C4C4⋊C4×D5312yes—
492C2×C4#1313317512yes—
493C2×C4#1313318811yescharacteristic
494C2×C4#1313318811yescharacteristic
495C2×C4#1313319212yes—
496C2×C4#1313319212yes—
497C2×C4#1313319312yes—
498C2×C4#1313319312yes—
499C2×C4#1313319712yes—
500C2×C4C22×C4×D5311yescharacteristic
501C2×C4C2×D4×D5311yescharacteristic
502C2×C4C22×C22⋊D5311yescharacteristic
503C2×C4C22×D21211yescharacteristic
504C2×C4C2×Q8×D5311yescharacteristic
505C2×C4C2×#1171994911yescharacteristic
506C8—1064no—
507C8—2122no—
508C8—2122no—
509C8C2×#1171992118yes—
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#1681048012yes—
523C22#1681048012yes—
524C22C2×#1313317511yescharacteristic
525C22C2×#1313318811yescharacteristic
526C22C2×#1313319711yescharacteristic
527C4—82no—
528C4—81no—
529C4—81no—
530C4—1064no—
531C4#1681048012yes—
532C4#1681048012yes—
533C4C22×#1171992111yescharacteristic
534C4C2×C4⋊C4×D5311yescharacteristic
535C4C2×#1313319211yescharacteristic
536C4C2×#1313319311yescharacteristic
537C2—22no—
538C2—21no—
539C2—21no—
540C2—22no—
541C2—2124no—
542C2C2×#1681048011yescharacteristic
543C2#2028443911yescharacteristic
544C2#2028444811yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#428C2
socleC22×C53C2×#121
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenterC2×#428D53
hypocenterC53#20679
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#16810672C2semidirect
5. #20274780C2semidirect
6. #20274775C2semidirect
7. #20274454C2semidirect
8. C2×#16810392C2semidirect
9. C2×C53×#428C2semidirect
12. #16811389C4semidirect
20. C2×#13133153C22semidirect
30. #16810392C22semidirect
31. #16810392C22semidirect
32. #16810392C22semidirect
38. C2×#107×C53C22semidirect
39. C2×#106×C53C22semidirect
54. #13133223C2×C4semidirect
267. C2×#428D53semidirect
298. C2×#107C2×D53semidirect
299. C2×#106C2×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×#168106722
#202832322
C2×#168106222
#202747802
#202747752
#202744542
C2×#168103922
C2×C53×#4282
#2067953