Finite Group Databasebeta

← #27412771

Subgroups — #27412771

Order 13568.

1102 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810666C211yesmaximal, characteristic, complemented
2C2×#16810650C211yesmaximal, characteristic, complemented
3C2×#16810622C211yesmaximal, characteristic
4C2×#16810617C211yesmaximal, characteristic
5#20277303C211yesmaximal, characteristic
6#20276107C212yesmaximal, complemented
7#20276100C212yesmaximal, complemented
8#20276093C212yesmaximal, complemented
9#20276086C212yesmaximal, complemented
10C2×#16810385C211yesmaximal, characteristic, complemented
11C53×#2317C211yesmaximal, characteristic, complemented
12C22×#11719945C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14C23×#88322C2211yescharacteristic
15C22×C4⋊Dic53C2211yescharacteristic
16#16810666—24no—
17C2×#13133150C2212yes—
18C2×#13133148C2212yes—
19#16810650—24no—
20C2×C4⋊#88322C2212yes—
21C2×C4⋊#88322C2212yes—
22#16810622—24no—
23#16810617—24no—
24C2×#13133140C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133140C2212yes—
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133137C2212yes—
30#16810529C2212yes—
31#16810525C2212yes—
32#16810500—22no—
33#16810480—21no—
34#16810480—21no—
35#16810385C2212yes—
36#16810385C2212yes—
37C2×#117×C53C2211yescharacteristic
38C2×#117×C53C2211yescharacteristic
39C22×C4⋊C4×C53C2211yescharacteristic
40C53×#389C2212yes—
41C53×#389C2212yes—
42C53×#385—22no—
43C53×#384—21no—
44C53×#382—21no—
45C2×#11719945—22no—
46C2×#11719945—22no—
47C2×#11719945—22no—
48C2×#11719945—22no—
49C2×#11719945—22no—
50C2×#11719945Q814yes—
51C22×Dic106C2311yescharacteristic
52C22×D212C2311yescharacteristic
53C22×C22⋊D53—21no—
54C23×Dic53—21no—
55C22×C4×D53—21no—
56C2×#11719917—22no—
57C2×#11719917—22no—
58C2×#11719917D414yes—
59C22×#88322—22no—
60C22×#88322—22no—
61C22×#88322—22no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2312yes—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—22no—
67C2×C4⋊Dic53—22no—
68C2×C4⋊Dic53C2312yes—
69C2×#11719921—22no—
70C2×#11719921—22no—
71#13133150—24no—
72#13133148—24no—
73C4⋊#88322—24no—
74C4⋊#88322—24no—
75#13133140—22no—
76#13133140—22no—
77#13133140—24no—
78#13133137—22no—
79#13133137—22no—
80#13133137—24no—
81#13133124—22no—
82#13133124—22no—
83C23×C4×C53—21no—
84C23×C4×C53C2311yescharacteristic
85#117×C53—22no—
86#117×C53—22no—
87#117×C53—22no—
88#117×C53D414yes—
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53—22no—
91C2×C4⋊C4×C53—22no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53—22no—
94C2×C4⋊C4×C53D411yescharacteristic
95C2×C4⋊C4×C53D411yescharacteristic
96C2×C4⋊C4×C53C2312yes—
97C2×C4⋊C4×C53C2311yescharacteristic
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C22⋊C4×C53—21no—
100C2×C22⋊C4×C53D412yes—
101C2×C42×C53—21no—
102C2×C42×C53C2312yes—
103#95×C53—22no—
104#95×C53—22no—
105C24×C53C2×D411yescharacteristic
106C23×D53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—22no—
110C22×C4×C53—42no—
111C22×C4×C53—81no—
112C22×C4×C53—81no—
113C22×C4×C53C2×D412yes—
114C22×C4×C53C2×D412yes—
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D412yes—
117C22×C4×C53C2×D412yes—
118C22×C4×C53C2×Q812yes—
119C22×C4×C53#4112yes—
120C22×C4×C53#4112yes—
121C22×C4×C53#4112yes—
122C22×C4×C53C2411yescharacteristic
123C2×C22⋊D53—24no—
124C2×C22⋊D53—42no—
125C2×C22⋊D53—42no—
126C22×Dic53—21no—
127C22×Dic53—22no—
128C22×Dic53—42no—
129C22×Dic53—81no—
130C22×Dic53—81no—
131#11719945—28no—
132#11719945—28no—
133#11719945—44no—
134#11719945—44no—
135C2×D212—21no—
136C2×D212—21no—
137C2×D212—21no—
138C2×D212—21no—
139C2×D212—21no—
140C2×D212C2×Q812yes—
141C2×C4×D53—24no—
142C2×C4×D53—42no—
143C2×C4×D53—42no—
144C2×Dic106—21no—
145C2×Dic106—21no—
146C2×Dic106—21no—
147C2×Dic106—21no—
148C2×Dic106—21no—
149C2×Dic106C2×Q812yes—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—42no—
157C4⋊C4×C53—82no—
158C4⋊C4×C53—82no—
159C4⋊C4×C53C2×D412yes—
160C4⋊C4×C53C2×D412yes—
161C22⋊C4×C53—24no—
162C22⋊C4×C53—24no—
163C42×C53—24no—
164C42×C53—22no—
165C42×C53—22no—
166#11719921—82no—
167#11719921—82no—
168C4⋊Dic53—22no—
169C4⋊Dic53—22no—
170C4⋊Dic53—42no—
171#11719917—28no—
172C2×#88322—22no—
173C2×#88322—22no—
174C2×#88322—21no—
175C2×#88322—21no—
176C2×#88322—21no—
177C2×#88322—21no—
178C2×#88322—21no—
179C2×#88322—21no—
180C2×#88322—44no—
181C2×#88322C2×D414yes—
182C23×C53—21no—
183C23×C53—21no—
184C23×C53—21no—
185C23×C53—21no—
186C23×C53#12112yes—
187C23×C53#12112yes—
188C23×C53C22⋊Q812yes—
189C23×C53C22×D411yescharacteristic
190C22×D53—22no—
191C22×D53—41no—
192C22×D53—41no—
193C22×D53—41no—
194C22×D53—41no—
195C22×D53—41no—
196C22×D53—41no—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—84no—
213C2×C4×C53#12112yes—
214C2×C4×C53#12112yes—
215C2×C4×C53#12114yes—
216C2×C4×C53C22⋊Q812yes—
217C2×C4×C53C22⋊Q814yes—
218C2×C4×C53C22⋊Q814yes—
219C2×C4×C53#12614yes—
220C2×C4×C53C22×D411yescharacteristic
221C2×C4×C53C22×D411yescharacteristic
222C2×C4×C53C22×Q811yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C22⋊D53—48no—
227C2×Dic53—22no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—42no—
231C2×Dic53—42no—
232C2×Dic53—42no—
233C2×Dic53—84no—
234D212—22no—
235D212—22no—
236D212—41no—
237D212—41no—
238C4×D53—48no—
239Dic106—22no—
240Dic106—22no—
241Dic106—41no—
242Dic106—41no—
243#88322—24no—
244#88322—41no—
245#88322—41no—
246#23616—531nomaximal, Sylow
247C22×C53—22no—
248C22×C53—22no—
249C22×C53—22no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—22no—
253C22×C53#39614yes—
254C22×C53C2×#12111yescharacteristic
255C22×C53C2×#12111yescharacteristic
256C22×C53C2×C22⋊Q811yescharacteristic
257C22×C53#57611yescharacteristic
258C22×C53#57711yescharacteristic
259C22×C53#57711yescharacteristic
260C22×C53#57811yescharacteristic
261C2×D53—42no—
262C2×D53—42no—
263C2×D53—42no—
264C2×D53—81no—
265C2×D53—81no—
266C2×D53—81no—
267C2×D53—81no—
268C4×C53—42no—
269C4×C53—42no—
270C4×C53—42no—
271C4×C53—42no—
272C4×C53—82no—
273C4×C53#39614yes—
274C4×C53C2×#12111yescharacteristic
275C4×C53C2×C22⋊Q811yescharacteristic
276C4×C53C2×C22⋊Q811yescharacteristic
277C4×C53C2×#12611yescharacteristic
278Dic53—42no—
279Dic53—82no—
280C2×#428—531no—
281C2×#424—531no—
282C2×#417—531no—
283C2×#416—531no—
284#2317D5311yescharacteristic, complemented
285#2311—531no—
286#2068—532no—
287#2067—532no—
288#2066—532no—
289#2065—532no—
290C2×#340—531no—
291C2×C53—24no—
292C2×C53C2×#39611yescharacteristic
293C2×C53#304511yescharacteristic
294C2×C53#304611yescharacteristic
295C2×C53#306211yescharacteristic
296C2×C53#306311yescharacteristic
297C2×C53#308311yescharacteristic
298C2×C53#308411yescharacteristic
299D53—82no—
300C22×#41—531no—
301C22×M16—531no—
302C23×C8—531no—
303C2×#117C2×D5311yescharacteristic
304C2×#117C2×D5311yescharacteristic
305C22×C4⋊C4—531no—
306C22×C4⋊C4C2×D5311yescharacteristic
307#428—1064no—
308C2×#107—532no—
309C2×#106—532no—
310#424—1064no—
311C2×C4⋊C8—532no—
312C2×C4⋊C8—532no—
313#417—1064no—
314#416—1064no—
315C2×#103—531no—
316C2×#103—532no—
317C2×#103—531no—
318C2×#102—531no—
319C2×#102—532no—
320C2×#102—531no—
321#391—532no—
322#390—532no—
323#389C2×D5312yes—
324#389C2×D5312yes—
325#386—1062no—
326#385—22no—
327#384—21no—
328#382—21no—
329#380—1061no—
330#380—1061no—
331#340—532no—
332#340—532no—
333C53#2361611yescharacteristic, complemented, Sylow
334C2×#41—534no—
335C2×#41—1062no—
336C2×#41—1062no—
337C2×#41—1062no—
338C2×#41—1062no—
339C2×#41—1062no—
340C22×Q8—531no—
341C22×D4—531no—
342C22×D4—1061no—
343C23×C4—21no—
344C23×C4—1061no—
345C23×C4—1061no—
346C23×C4C22×D5311yescharacteristic
347C2×M16—534no—
348C2×M16—1062no—
349C2×M16—1062no—
350C22×C8—532no—
351C22×C8—531no—
352C22×C8—531no—
353C22×C8—1062no—
354C22×C8—1062no—
355C22×C8—1062no—
356#117—22no—
357#117—22no—
358#117—22no—
359#117D21214yes—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—532no—
366C2×C4⋊C4—1062no—
367C2×C4⋊C4—1062no—
368C2×C4⋊C4—1061no—
369C2×C4⋊C4C22⋊D5311yescharacteristic
370C2×C4⋊C4C22⋊D5311yescharacteristic
371C2×C4⋊C4C22×D5312yes—
372C2×C4⋊C4C22×D5311yescharacteristic
373C2×C4⋊C4C22×D5311yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—1062no—
376C2×C22⋊C4—1062no—
377C2×C22⋊C4C22⋊D5312yes—
378C2×C42—21no—
379C2×C42C22×D5312yes—
380#107—1064no—
381#106—1064no—
382C4⋊C8—1064no—
383C4⋊C8—1064no—
384#103—1064no—
385#103—1062no—
386#103—1062no—
387#102—1064no—
388#102—1062no—
389#102—1062no—
390#95—22no—
391#95—22no—
392#95—1062no—
393#95—1062no—
394C24—1061no—
395C24C2×C22⋊D5311yescharacteristic
396#41—1068no—
397#41—1068no—
398#41—2124no—
399#41—2124no—
400C2×Q8—532no—
401C2×Q8—1061no—
402C2×Q8—1061no—
403C2×Q8—1061no—
404C2×Q8—1061no—
405C2×Q8—1061no—
406C2×D4—532no—
407C2×D4—1064no—
408C2×D4—1061no—
409C2×D4—1061no—
410C2×D4—1061no—
411C2×D4—1061no—
412C2×D4—1061no—
413C2×D4—2122no—
414C2×D4—2122no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—42no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—1061no—
422C22×C4—1064no—
423C22×C4—1062no—
424C22×C4—4241no—
425C22×C4—4241no—
426C22×C4—2122no—
427C22×C4—2122no—
428C22×C4—2122no—
429C22×C4C2×D21212yes—
430C22×C4#1171994512yes—
431C22×C4D4×D5312yes—
432C22×C4#1171994712yes—
433C22×C4Q8×D5312yes—
434C22×C4#1171994912yes—
435C22×C4C2×C22⋊D5312yes—
436C22×C4C2×C22⋊D5312yes—
437C22×C4C2×C22⋊D5311yescharacteristic
438C22×C4C23×D5311yescharacteristic
439M16—1068no—
440C2×C8—534no—
441C2×C8—1062no—
442C2×C8—1062no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1061no—
448C2×C8—1061no—
449C2×C8—2124no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—1062no—
460C4⋊C4—1062no—
461C4⋊C4—2122no—
462C4⋊C4C2×D21212yes—
463C4⋊C4C2×D21212yes—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—4242no—
467C22⋊C4—4242no—
468C42—24no—
469C42—22no—
470C42—22no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—1062no—
476C23—2121no—
477C23—2121no—
478C23—2121no—
479C23—2121no—
480C23—2121no—
481C23—2121no—
482C23#1313318912yes—
483C23#1313319112yes—
484C23#1313319312yes—
485C23C22×C22⋊D5311yescharacteristic
486Q8—1062no—
487Q8—1062no—
488Q8—2121no—
489Q8—2121no—
490D4—1062no—
491D4—1062no—
492D4—2128no—
493D4—2121no—
494D4—2121no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—84no—
511C2×C4—1062no—
512C2×C4—4244no—
513C2×C4—2122no—
514C2×C4—2122no—
515C2×C4—2122no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2128no—
519C2×C4#1313318912yes—
520C2×C4#1313319112yes—
521C2×C4#1313319214yes—
522C2×C4#1313319312yes—
523C2×C4#1313319514yes—
524C2×C4#1313319714yes—
525C2×C4#1313321214yes—
526C2×C4C2×D4×D5311yescharacteristic
527C2×C4C22×D21211yescharacteristic
528C2×C4C2×Q8×D5311yescharacteristic
529C2×C4C2×#1171994711yescharacteristic
530C2×C4C2×#1171994511yescharacteristic
531C2×C4C2×#1171994911yescharacteristic
532C8—1064no—
533C8—2121no—
534C8—2121no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—4241no—
542C22—4241no—
543C22—4241no—
544C22—4241no—
545C22—2122no—
546C22—2122no—
547C22—2122no—
548C22#1681054514yes—
549C22C2×#1313318911yescharacteristic
550C22C2×#1313319111yescharacteristic
551C22C2×#1313319311yescharacteristic
552C22#1681142611yescharacteristic
553C22#1681143111yescharacteristic
554C22#1681143411yescharacteristic
555C22#1681143811yescharacteristic
556C4—42no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—82no—
561C4—4242no—
562C4—2122no—
563C4#1681054514yes—
564C4C2×#1313319211yescharacteristic
565C4C2×#1313319511yescharacteristic
566C4C2×#1313319711yescharacteristic
567C4C2×#1313321211yescharacteristic
568C2—24no—
569C2—4242no—
570C2C2×#1681054511yescharacteristic
571C2#2028425411yescharacteristic
572C2#2028426511yescharacteristic
573C2#2028439111yescharacteristic
574C2#2028440211yescharacteristic
575C2#2028456211yescharacteristic
576C2#2028457311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2317C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
hypercenter#2317D53
hypocenterC53#23616
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
1. C2×#16810666C2semidirect
2. C2×#16810650C2semidirect
6. #20276107C2semidirect
7. #20276100C2semidirect
8. #20276093C2semidirect
9. #20276086C2semidirect
10. C2×#16810385C2semidirect
11. C53×#2317C2semidirect
284. #2317D53semidirect
333. C53#23616semidirect

Maximal subgroups

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

subgroupindex
C2×#168106662
C2×#168106502
C2×#168106222
C2×#168106172
#202773032
#202761072
#202761002
#202760932
#202760862
C2×#168103852
C53×#23172
#2361653