Finite Group Databasebeta

← #27386784

Subgroups — #27386784

Order 13568.

1122 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810625C211yesmaximal, characteristic
2C2×#16810617C211yesmaximal, characteristic
3C2×#16810581C211yesmaximal, characteristic, complemented
4#20277331C211yesmaximal, characteristic, complemented
5#20277308C211yesmaximal, characteristic
6#20275410C212yesmaximal, complemented
7#20275403C212yesmaximal, complemented
8#20275396C212yesmaximal, complemented
9#20275389C212yesmaximal, complemented
10C2×#16810385C211yesmaximal, characteristic, complemented
11C53×#1736C211yesmaximal, characteristic, complemented
12C22×#11719945C2211yescharacteristic
13C23×#88322C2211yescharacteristic
14C2×#13133176C2211yescharacteristic
15C22×C4⋊Dic53C2211yescharacteristic
16#16810625C418yescomplemented
17#16810617—24no—
18C2×#13133142C2212yes—
19C2×#13133140C2211yescharacteristic
20C2×#13133140C2211yescharacteristic
21C2×#13133139C2212yes—
22C2×#13133137C2211yescharacteristic
23C2×#13133137C2211yescharacteristic
24#16810581—24no—
25C2×#13133127C2212yes—
26C2×C8×Dic53C2212yes—
27#16810528C2212yes—
28#16810524C2212yes—
29#16810521—21no—
30#16810521—21no—
31#16810519C2212yes—
32#16810519C2212yes—
33#16810502—22no—
34#16810481—22no—
35#16810480—21no—
36#16810480—21no—
37#16810385C2212yes—
38#16810385C2212yes—
39C22×M16×C53C2211yescharacteristic
40C2×#117×C53C2211yescharacteristic
41C22×C4⋊C4×C53C2211yescharacteristic
42C53×#340C2212yes—
43C53×#340C2212yes—
44C2×#11719945—22no—
45C2×#11719945—22no—
46C2×#11719945—22no—
47C2×#11719945—22no—
48C2×#11719945—22no—
49C2×#11719945C2×C414yes—
50C22×Dic106C2311yescharacteristic
51C22×D212C2311yescharacteristic
52C22×C22⋊D53—21no—
53C23×Dic53—21no—
54C22×C4×D53—21no—
55C22×#88322—22no—
56C22×#88322—22no—
57C22×#88322D414yes—
58C22×#88322C2311yescharacteristic
59C22×#88322C2311yescharacteristic
60#13133176—22no—
61#13133176—22no—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53—22no—
64C2×C4⋊Dic53C2×C414yes—
65C2×C4⋊Dic53D411yescharacteristic
66C2×C4⋊Dic53D411yescharacteristic
67C2×C4⋊Dic53C2312yes—
68C2×#11719919—22no—
69C2×#11719919—22no—
70C2×C22⋊Dic53D412yes—
71C2×#11719921—22no—
72C2×#11719921—22no—
73C2×C4×Dic53C2312yes—
74#13133142C2×C418yescomplemented
75#13133140—22no—
76#13133140—22no—
77#13133139C2×C418yescomplemented
78#13133137—22no—
79#13133137—22no—
80#13133127—24no—
81C8×Dic53—24no—
82#13133124—22no—
83#13133124—22no—
84#13133124—22no—
85#13133124—22no—
86C23×C4×C53—21no—
87C23×C4×C53C2311yescharacteristic
88C2×M16×C53—22no—
89C2×M16×C53—22no—
90C2×M16×C53C2×C414yes—
91C22×C8×C53C2312yes—
92#117×C53—22no—
93#117×C53—22no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53C2311yescharacteristic
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C4⋊C4×C53C2312yes—
100C2×C22⋊C4×C53—21no—
101C2×C42×C53—21no—
102C24×C53C2×D411yescharacteristic
103C23×D53—21no—
104C22×C4×C53—21no—
105C22×C4×C53—21no—
106C22×C4×C53—22no—
107C22×C4×C53—42no—
108C22×C4×C53—81no—
109C22×C4×C53—81no—
110C22×C4×C53C22×C412yes—
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D412yes—
113C22×C4×C53#4112yes—
114C22×C4×C53#4112yes—
115C22×C4×C53#4112yes—
116C22×C4×C53#4112yes—
117C22×C4×C53C2411yescharacteristic
118C2×C22⋊D53—24no—
119C2×C22⋊D53—42no—
120C2×C22⋊D53—42no—
121C22×Dic53—21no—
122C22×Dic53—22no—
123C22×Dic53—24no—
124C22×Dic53—81no—
125C22×Dic53—81no—
126C22×Dic53C2×D412yes—
127C22×Dic53C2×D412yes—
128#11719945—28no—
129#11719945—28no—
130#11719945—44no—
131#11719945—44no—
132C2×D212—21no—
133C2×D212—21no—
134C2×D212—21no—
135C2×D212—21no—
136C2×D212—21no—
137C2×D212C22×C412yes—
138C2×C4×D53—24no—
139C2×C4×D53—42no—
140C2×C4×D53—42no—
141C2×Dic106—21no—
142C2×Dic106—21no—
143C2×Dic106—21no—
144C2×Dic106—21no—
145C2×Dic106—21no—
146C2×Dic106C22×C412yes—
147M16×C53—28no—
148C2×C8×C53—22no—
149C2×C8×C53—22no—
150C2×C8×C53C22×C414yes—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—42no—
158C22⋊C4×C53—42no—
159C42×C53—42no—
160C22⋊Dic53—24no—
161#11719921—82no—
162#11719921—82no—
163C4⋊Dic53—24no—
164C4⋊Dic53—22no—
165C4⋊Dic53—22no—
166C4⋊Dic53—22no—
167C4⋊Dic53C22×C414yes—
168#11719919—82no—
169#11719919—82no—
170C4×Dic53—22no—
171C4×Dic53—22no—
172C2×#88322—28no—
173C2×#88322—21no—
174C2×#88322—21no—
175C2×#88322—21no—
176C2×#88322—21no—
177C2×#88322C2×D412yes—
178C2×#88322C2×D412yes—
179C23×C53—21no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—21no—
183C23×C53C4×D412yes—
184C23×C53#12112yes—
185C23×C53#12412yes—
186C23×C53C22×D411yescharacteristic
187C22×D53—22no—
188C22×D53—41no—
189C22×D53—41no—
190C22×D53—41no—
191C22×D53—41no—
192C22×D53—41no—
193C22×D53—41no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—42no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—84no—
204C2×C4×C53#11714yes—
205C2×C4×C53C4×D414yes—
206C2×C4×C53C4×D412yes—
207C2×C4×C53#12112yes—
208C2×C4×C53#12314yes—
209C2×C4×C53#12412yes—
210C2×C4×C53#12614yes—
211C2×C4×C53C23×C411yescharacteristic
212C2×C4×C53C22×D411yescharacteristic
213C2×C4×C53C2×#4111yescharacteristic
214C2×C4×C53C2×#4111yescharacteristic
215C2×C4×C53C2×#4111yescharacteristic
216C2×C4×C53C2×#4111yescharacteristic
217C22⋊D53—48no—
218C2×Dic53—22no—
219C2×Dic53—24no—
220C2×Dic53—22no—
221C2×Dic53—22no—
222C2×Dic53—22no—
223C2×Dic53—22no—
224C2×Dic53—22no—
225C2×Dic53—22no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—42no—
229C2×Dic53—42no—
230C2×Dic53—84no—
231D212—22no—
232D212—22no—
233D212—41no—
234D212—41no—
235C4×D53—48no—
236Dic106—22no—
237Dic106—22no—
238Dic106—41no—
239Dic106—41no—
240C8×C53—24no—
241#88322—22no—
242#88322—22no—
243#21092—531nomaximal, Sylow
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—22no—
247C22×C53—22no—
248C22×C53—22no—
249C22×C53—22no—
250C22×C53#38814yes—
251C22×C53C2×C4×D411yescharacteristic
252C22×C53C2×#12111yescharacteristic
253C22×C53C2×#12411yescharacteristic
254C22×C53#57611yescharacteristic
255C22×C53#57711yescharacteristic
256C22×C53#57711yescharacteristic
257C22×C53#57811yescharacteristic
258C2×D53—42no—
259C2×D53—42no—
260C2×D53—42no—
261C2×D53—81no—
262C2×D53—81no—
263C2×D53—81no—
264C2×D53—81no—
265C4×C53—82no—
266C4×C53—81no—
267C4×C53—81no—
268C4×C53#38814yes—
269C4×C53C2×#11711yescharacteristic
270C4×C53C2×C4×D411yescharacteristic
271C4×C53C2×#12311yescharacteristic
272C4×C53C2×#12611yescharacteristic
273Dic53—44no—
274Dic53—42no—
275Dic53—42no—
276Dic53—42no—
277C2×#417—531no—
278C2×#416—531no—
279C2×#405—531no—
280#2316—531no—
281#2311—531no—
282#1936—532no—
283#1935—532no—
284#1934—532no—
285#1933—532no—
286#1736D5311yescharacteristic, complemented
287C2×#340—531no—
288C2×C53—24no—
289C2×C53C2×#38811yescharacteristic
290C2×C53#294611yescharacteristic
291C2×C53#294711yescharacteristic
292C2×C53#306011yescharacteristic
293C2×C53#306111yescharacteristic
294C2×C53#313611yescharacteristic
295C2×C53#313711yescharacteristic
296D53—82no—
297C22×#41—531no—
298C22×M16C2×D5311yescharacteristic
299C23×C8—531no—
300C2×#117—531no—
301C2×#117C2×D5311yescharacteristic
302C22×C4⋊C4—531no—
303C22×C4⋊C4C2×D5311yescharacteristic
304#417—538no—
305#416—1064no—
306C2×#103—532no—
307C2×#103—531no—
308C2×#103—531no—
309C2×#102—532no—
310C2×#102—531no—
311C2×#102—531no—
312#405—1064no—
313C2×#97—532no—
314C2×C4×C8—532no—
315#391—532no—
316#390—532no—
317#389—532no—
318#389—532no—
319#389—1061no—
320#389—1061no—
321#386—1062no—
322#380—1062no—
323#380—1061no—
324#380—1061no—
325#340—532no—
326#340—532no—
327#340C2×D5312yes—
328#340C2×D5312yes—
329C53#2109211yescharacteristic, complemented, Sylow
330C2×#41—534no—
331C2×#41—1062no—
332C2×#41—1062no—
333C2×#41—1062no—
334C2×#41—1062no—
335C2×#41—1062no—
336C22×Q8—531no—
337C22×D4—531no—
338C22×D4—1061no—
339C23×C4—21no—
340C23×C4—1061no—
341C23×C4—1061no—
342C23×C4C22×D5311yescharacteristic
343C2×M16—22no—
344C2×M16—22no—
345C2×M16C4×D5314yes—
346C22×C8—534no—
347C22×C8—531no—
348C22×C8—531no—
349C22×C8—1062no—
350C22×C8—1062no—
351C22×C8C22×D5312yes—
352#117—22no—
353#117—22no—
354#117—1062no—
355#117—1062no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—534no—
360C2×C4⋊C4—532no—
361C2×C4⋊C4—531no—
362C2×C4⋊C4—531no—
363C2×C4⋊C4—1061no—
364C2×C4⋊C4—1062no—
365C2×C4⋊C4—1062no—
366C2×C4⋊C4—1062no—
367C2×C4⋊C4C22×D5312yes—
368C2×C4⋊C4C22×D5311yescharacteristic
369C2×C4⋊C4C22×D5311yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—532no—
372C2×C22⋊C4—1062no—
373C2×C22⋊C4—1062no—
374C2×C42—21no—
375C2×C42—532no—
376#103—538no—
377#103—1062no—
378#103—1062no—
379#102—538no—
380#102—1062no—
381#102—1062no—
382#97—1064no—
383C4×C8—1064no—
384#95—1062no—
385#95—1062no—
386#95—1062no—
387#95—1062no—
388C24—1061no—
389C24D4×D5311yescharacteristic
390#41—1068no—
391#41—1068no—
392#41—2124no—
393#41—2124no—
394C2×Q8—532no—
395C2×Q8—1061no—
396C2×Q8—1061no—
397C2×Q8—1061no—
398C2×Q8—1061no—
399C2×Q8—1061no—
400C2×D4—532no—
401C2×D4—1064no—
402C2×D4—1061no—
403C2×D4—1061no—
404C2×D4—1061no—
405C2×D4—1061no—
406C2×D4—1061no—
407C2×D4—2122no—
408C2×D4—2122no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—42no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—532no—
416C22×C4—532no—
417C22×C4—1064no—
418C22×C4—1061no—
419C22×C4—1062no—
420C22×C4—1064no—
421C22×C4—4241no—
422C22×C4—4241no—
423C22×C4—2122no—
424C22×C4—2122no—
425C22×C4C2×C4×D5312yes—
426C22×C4#1171994512yes—
427C22×C4#1171994512yes—
428C22×C4D4×D5312yes—
429C22×C4D4×D5311yescharacteristic
430C22×C4#1171994712yes—
431C22×C4#1171994912yes—
432C22×C4C23×D5311yescharacteristic
433M16—28no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—532no—
437C2×C8—532no—
438C2×C8—1068no—
439C2×C8—1061no—
440C2×C8—1061no—
441C2×C8—1061no—
442C2×C8—1061no—
443C2×C8C2×C4×D5314yes—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—42no—
451C4⋊C4—534no—
452C4⋊C4—1064no—
453C4⋊C4—1062no—
454C4⋊C4—1062no—
455C4⋊C4—1062no—
456C4⋊C4—4242no—
457C4⋊C4—4242no—
458C22⋊C4—42no—
459C22⋊C4—1064no—
460C22⋊C4—4242no—
461C22⋊C4—4242no—
462C42—42no—
463C42—1062no—
464C42—1062no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—1062no—
470C23—2121no—
471C23—2121no—
472C23—2121no—
473C23—2121no—
474C23—2121no—
475C23—2121no—
476C23#1313317912yes—
477C23#1313319012yes—
478C23#1313320312yes—
479C23C2×D4×D5311yescharacteristic
480Q8—1062no—
481Q8—1062no—
482Q8—2121no—
483Q8—2121no—
484D4—1062no—
485D4—1062no—
486D4—2128no—
487D4—2121no—
488D4—2121no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—84no—
499C2×C4—1064no—
500C2×C4—1062no—
501C2×C4—1062no—
502C2×C4—1062no—
503C2×C4—1062no—
504C2×C4—1062no—
505C2×C4—1062no—
506C2×C4—1062no—
507C2×C4—4244no—
508C2×C4—2121no—
509C2×C4—2121no—
510C2×C4—2122no—
511C2×C4—2122no—
512C2×C4—2128no—
513C2×C4#1313317414yes—
514C2×C4#1313317714yes—
515C2×C4#1313317912yes—
516C2×C4#1313319012yes—
517C2×C4#1313320214yes—
518C2×C4#1313320312yes—
519C2×C4#1313321214yes—
520C2×C4C22×C4×D5311yescharacteristic
521C2×C4C2×D4×D5311yescharacteristic
522C2×C4C2×#1171994711yescharacteristic
523C2×C4C2×#1171994511yescharacteristic
524C2×C4C2×#1171994511yescharacteristic
525C2×C4C2×#1171994911yescharacteristic
526C8—24no—
527C8—1062no—
528C8—1062no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—4241no—
536C22—4241no—
537C22—4241no—
538C22—4241no—
539C22—2122no—
540C22—2122no—
541C22—2122no—
542C22#1681051214yes—
543C22C2×#1313317911yescharacteristic
544C22C2×#1313319011yescharacteristic
545C22C2×#1313320311yescharacteristic
546C22#1681142711yescharacteristic
547C22#1681143211yescharacteristic
548C22#1681143311yescharacteristic
549C22#1681143911yescharacteristic
550C4—82no—
551C4—81no—
552C4—81no—
553C4—2122no—
554C4—2124no—
555C4—2122no—
556C4—2122no—
557C4#1681051214yes—
558C4C2×#1313317411yescharacteristic
559C4C2×#1313317711yescharacteristic
560C4C2×#1313320211yescharacteristic
561C4C2×#1313321211yescharacteristic
562C2—24no—
563C2—4242no—
564C2C2×#1681051211yescharacteristic
565C2#2028354511yescharacteristic
566C2#2028355611yescharacteristic
567C2#2028437611yescharacteristic
568C2#2028438411yescharacteristic
569C2#2028506511yescharacteristic
570C2#2028507611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1736C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenter#1736D53
hypocenterC53#21092
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
3. C2×#16810581C2semidirect
4. #20277331C2semidirect
6. #20275410C2semidirect
7. #20275403C2semidirect
8. #20275396C2semidirect
9. #20275389C2semidirect
10. C2×#16810385C2semidirect
11. C53×#1736C2semidirect
16. #16810625C4semidirect
74. #13133142C2×C4semidirect
77. #13133139C2×C4semidirect
286. #1736D53semidirect
329. C53#21092semidirect

Maximal subgroups

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

subgroupindex
C2×#168106252
C2×#168106172
C2×#168105812
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