Finite Group Databasebeta

← #27291681

Subgroups — #27291681

Order 13568.

850 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133143C211yesmaximal, characteristic
2#20283232C211yesmaximal, characteristic
3C2×#16810605C211yesmaximal, characteristic
4C2×#16810382C211yesmaximal, characteristic, complemented
5C2×#16810381C211yesmaximal, characteristic, complemented
6#20274408C211yesmaximal, characteristic, complemented
7C53×#1742C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9C22×#11719917C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C22×C4×Dic53C2211yescharacteristic
12C2×#13133143—22no—
13C2×#13133143—22no—
14C2×#13133143—22no—
15#16810631—21no—
16#16810627—21no—
17#16810622—22no—
18#16810617—22no—
19#16810605—22no—
20#16810605C414yescomplemented
21C2×#13133135—21no—
22C2×#13133133—21no—
23#16810382—22no—
24#16810381—22no—
25#16810377—22no—
26C22×M16×C53C2211yescharacteristic
27C22×M16×C53C2211yescharacteristic
28C2×#117×C53C2211yescharacteristic
29C53×#343—22no—
30C2×#11719945—21no—
31C2×#11719945—21no—
32C2×#11719945—21no—
33C2×#11719945—21no—
34C2×#11719945—21no—
35C2×#11719945—21no—
36C2×#11719945—21no—
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945C2×C412yes—
41C2×#11719945D412yes—
42C22×Dic106D411yescharacteristic
43C22×D212D411yescharacteristic
44C22×C22⋊D53—21no—
45C23×Dic53—21no—
46C22×C4×D53—21no—
47C2×#11719917—21no—
48C2×#11719917—21no—
49C2×#11719917—21no—
50C2×#11719917—21no—
51C2×#11719917—21no—
52C2×#11719917—21no—
53C2×#11719917—21no—
54C2×#11719917—21no—
55C2×#11719917—21no—
56C2×#11719917—21no—
57C2×#11719917—21no—
58C2×#11719917C2×C412yes—
59C22×#88322—21no—
60C22×#88322D412yes—
61C2×C4×Dic53—21no—
62C2×C4×Dic53—21no—
63C2×C4×Dic53—21no—
64C2×C4×Dic53—21no—
65C2×C4×Dic53—21no—
66C2×C4×Dic53D412yes—
67#13133143—28no—
68#13133140—24no—
69#13133137—24no—
70#13133135—24no—
71#13133133—24no—
72C23×C4×C53C2311yescharacteristic
73C2×M16×C53—21no—
74C2×M16×C53—21no—
75C2×M16×C53—21no—
76C2×M16×C53—21no—
77C2×M16×C53—21no—
78C2×M16×C53—21no—
79C2×M16×C53—21no—
80C2×M16×C53—21no—
81C2×M16×C53—21no—
82C2×M16×C53—21no—
83C2×M16×C53C2×C412yes—
84C2×M16×C53D412yes—
85C22×C8×C53—21no—
86C22×C8×C53—21no—
87#117×C53—21no—
88#117×C53—21no—
89#117×C53—21no—
90#117×C53—21no—
91C2×C4⋊C4×C53D412yes—
92C2×C22⋊C4×C53—21no—
93C2×C42×C53—21no—
94C24×C53#4111yescharacteristic
95C23×D53—21no—
96C22×C4×C53—22no—
97C22×C4×C53C22×C411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53#4111yescharacteristic
105C22×C4×C53#4111yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C22×C4×C53#4111yescharacteristic
109C22×C4×C53#4111yescharacteristic
110C22×C4×C53#4111yescharacteristic
111C2×C22⋊D53—22no—
112C2×C22⋊D53—22no—
113C2×C22⋊D53—41no—
114C2×C22⋊D53—41no—
115C2×C22⋊D53—41no—
116C2×C22⋊D53—41no—
117C22×Dic53—22no—
118C22×Dic53—21no—
119C22×Dic53—44no—
120C22×Dic53—41no—
121C22×Dic53—41no—
122#11719945—22no—
123#11719945—22no—
124#11719945—22no—
125#11719945—22no—
126#11719945—22no—
127#11719945—22no—
128#11719945—22no—
129#11719945—22no—
130#11719945—41no—
131#11719945—41no—
132#11719945—41no—
133#11719945—41no—
134#11719945—41no—
135#11719945—41no—
136#11719945—41no—
137#11719945—41no—
138C2×D212—21no—
139C2×D212—21no—
140C2×D212—21no—
141C2×D212—21no—
142C2×D212—21no—
143C2×D212C22⋊C412yes—
144C2×C4×D53—22no—
145C2×C4×D53—22no—
146C2×C4×D53—41no—
147C2×C4×D53—41no—
148C2×C4×D53—41no—
149C2×C4×D53—41no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106C22⋊C412yes—
156M16×C53—22no—
157M16×C53—22no—
158M16×C53—22no—
159M16×C53—22no—
160M16×C53—41no—
161M16×C53—41no—
162M16×C53—41no—
163M16×C53—41no—
164C2×C8×C53—22no—
165C2×C8×C53—22no—
166C2×C8×C53—41no—
167C2×C8×C53—41no—
168C2×C8×C53—41no—
169C2×C8×C53—41no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—22no—
172C22⋊C4×C53—41no—
173C22⋊C4×C53—41no—
174C42×C53—41no—
175C42×C53—41no—
176C4×Dic53—22no—
177C4×Dic53—22no—
178C4×Dic53—41no—
179C4×Dic53—41no—
180#11719917—22no—
181#11719917—22no—
182#11719917—41no—
183#11719917—41no—
184#11719917—41no—
185#11719917—41no—
186#11719917—41no—
187#11719917—41no—
188C2×#88322—22no—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—41no—
193C2×#88322—41no—
194C23×C53—42no—
195C23×C53#11711yescharacteristic
196C23×C53#12111yescharacteristic
197C23×C53#12311yescharacteristic
198C23×C53#12311yescharacteristic
199C23×C53#12311yescharacteristic
200C23×C53#12411yescharacteristic
201C23×C53#12611yescharacteristic
202C22×D53—22no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—82no—
206C2×C4×C53—42no—
207C2×C4×C53—82no—
208C2×C4×C53C2×C22⋊C411yescharacteristic
209C2×C4×C53#11711yescharacteristic
210C2×C4×C53C4×D411yescharacteristic
211C2×C4×C53C4×D411yescharacteristic
212C2×C4×C53C4×D411yescharacteristic
213C2×C4×C53C4×D411yescharacteristic
214C2×C4×C53#12011yescharacteristic
215C2×C4×C53#12011yescharacteristic
216C2×C4×C53#12111yescharacteristic
217C2×C4×C53#12111yescharacteristic
218C2×C4×C53#12111yescharacteristic
219C2×C4×C53#12111yescharacteristic
220C2×C4×C53#12111yescharacteristic
221C2×C4×C53#12111yescharacteristic
222C2×C4×C53#12111yescharacteristic
223C2×C4×C53#12311yescharacteristic
224C2×C4×C53#12311yescharacteristic
225C2×C4×C53#12311yescharacteristic
226C2×C4×C53#12311yescharacteristic
227C2×C4×C53#12311yescharacteristic
228C2×C4×C53#12311yescharacteristic
229C2×C4×C53#12311yescharacteristic
230C2×C4×C53#12411yescharacteristic
231C2×C4×C53#12411yescharacteristic
232C2×C4×C53#12411yescharacteristic
233C2×C4×C53#12611yescharacteristic
234C2×C4×C53#12611yescharacteristic
235C2×C4×C53#12611yescharacteristic
236C22⋊D53—42no—
237C22⋊D53—42no—
238C22⋊D53—42no—
239C22⋊D53—42no—
240C2×Dic53—24no—
241C2×Dic53—22no—
242C2×Dic53—42no—
243C2×Dic53—44no—
244C2×Dic53—44no—
245C2×Dic53—41no—
246C2×Dic53—42no—
247C2×Dic53—41no—
248D212—22no—
249D212—22no—
250D212—41no—
251D212—41no—
252C4×D53—42no—
253C4×D53—42no—
254C4×D53—42no—
255C4×D53—42no—
256Dic106—22no—
257Dic106—22no—
258Dic106—41no—
259Dic106—41no—
260C8×C53—42no—
261C8×C53—42no—
262#88322—42no—
263#88322—42no—
264#11151—531nomaximal, Sylow
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53#38111yescharacteristic
273C22×C53#38811yescharacteristic
274C22×C53#38811yescharacteristic
275C22×C53#39411yescharacteristic
276C22×C53#39411yescharacteristic
277C22×C53#39711yescharacteristic
278C22×C53#39711yescharacteristic
279C2×D53—42no—
280C2×D53—82no—
281C2×D53—82no—
282C2×D53—82no—
283C4×C53—82no—
284C4×C53#38611yescharacteristic
285C4×C53#38611yescharacteristic
286C4×C53#38811yescharacteristic
287C4×C53#38811yescharacteristic
288C4×C53#39411yescharacteristic
289C4×C53#39411yescharacteristic
290C4×C53#39711yescharacteristic
291C4×C53#39711yescharacteristic
292Dic53—44no—
293Dic53—42no—
294C22×#104—531no—
295#2896—531no—
296C2×#413—531no—
297#1742D5311yescharacteristic, complemented
298C2×#339—531no—
299C2×#338—531no—
300#1728—531no—
301C2×C53—21no—
302C2×C53—21no—
303C2×C53—21no—
304C2×C53—21no—
305C2×C53—21no—
306C2×C53—21no—
307C2×C53#144011yescharacteristic
308C2×C53#205012yes—
309D53—82no—
310C22×#41—531no—
311C22×M16—531no—
312C22×M16—531no—
313C22×M16C2×D5311yescharacteristic
314C22×M16C2×D5311yescharacteristic
315C2×#117C2×D5311yescharacteristic
316C22×C42—531no—
317C2×#104—1062no—
318C2×#104—1062no—
319C2×#104—1062no—
320#419—1061no—
321#418—1061no—
322#417—1062no—
323#416—1062no—
324#413—534no—
325#413—1062no—
326C2×#101—1061no—
327C2×#100—1061no—
328#343—22no—
329#339—1062no—
330#338—1062no—
331#336—1062no—
332C53#1115111yescharacteristic, complemented, Sylow
333C2×#41—532no—
334C2×#41—532no—
335C2×#41—1061no—
336C2×#41—1061no—
337C2×#41—1061no—
338C2×#41—1061no—
339C2×#41—1061no—
340C2×#41—1061no—
341C2×#41—1061no—
342C2×#41—1061no—
343C2×#41—1061no—
344C2×#41—1061no—
345C22×Q8—531no—
346C22×D4—531no—
347C22×D4—1061no—
348C23×C4—1061no—
349C23×C4—1061no—
350C23×C4C22×D5311yescharacteristic
351C2×M16—21no—
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—532no—
362C2×M16—1061no—
363C2×M16—1061no—
364C2×M16—1061no—
365C2×M16—1061no—
366C2×M16—1061no—
367C2×M16—1061no—
368C2×M16—1061no—
369C2×M16—1061no—
370C2×M16—1061no—
371C2×M16—1061no—
372C2×M16—1061no—
373C2×M16C4×D5312yes—
374C2×M16D21212yes—
375C22×C8—21no—
376C22×C8—21no—
377C22×C8—532no—
378C22×C8—1061no—
379#117—21no—
380#117—21no—
381#117—21no—
382#117—21no—
383C2×C4⋊C4C22⋊D5312yes—
384C2×C22⋊C4—21no—
385C2×C42—21no—
386C2×C42—532no—
387C2×C42—1061no—
388C2×C42—1061no—
389C2×C42—1061no—
390C2×C42—1061no—
391C2×C42—1061no—
392#104—1068no—
393#103—1064no—
394#102—1064no—
395#101—1064no—
396#100—1064no—
397C24—1061no—
398C24#1171994511yescharacteristic
399#41—1062no—
400#41—1062no—
401#41—1062no—
402#41—1062no—
403#41—1062no—
404#41—1062no—
405#41—1062no—
406#41—1062no—
407#41—2121no—
408#41—2121no—
409#41—2121no—
410#41—2121no—
411#41—2121no—
412#41—2121no—
413#41—2121no—
414#41—2121no—
415C2×Q8—532no—
416C2×Q8—1061no—
417C2×Q8—1061no—
418C2×Q8—1061no—
419C2×Q8—1061no—
420C2×Q8—1061no—
421C2×D4—532no—
422C2×D4—1062no—
423C2×D4—1062no—
424C2×D4—1061no—
425C2×D4—1061no—
426C2×D4—1061no—
427C2×D4—1061no—
428C2×D4—1061no—
429C2×D4—2121no—
430C2×D4—2121no—
431C2×D4—2121no—
432C2×D4—2121no—
433C22×C4—22no—
434C22×C4—1062no—
435C22×C4—1061no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—2124no—
439C22×C4—2121no—
440C22×C4—2121no—
441C22×C4—2121no—
442C22×C4—2121no—
443C22×C4—2121no—
444C22×C4—2121no—
445C22×C4C2×C4×D5311yescharacteristic
446C22×C4C2×D21211yescharacteristic
447C22×C4#1171994511yescharacteristic
448C22×C4#1171994511yescharacteristic
449C22×C4#1171994511yescharacteristic
450C22×C4D4×D5311yescharacteristic
451C22×C4D4×D5311yescharacteristic
452C22×C4D4×D5311yescharacteristic
453C22×C4D4×D5311yescharacteristic
454C22×C4#1171994711yescharacteristic
455C22×C4#1171994711yescharacteristic
456C22×C4#1171994911yescharacteristic
457C22×C4#1171994911yescharacteristic
458C22×C4C2×C22⋊D5311yescharacteristic
459M16—22no—
460M16—22no—
461M16—22no—
462M16—22no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—1062no—
468M16—1062no—
469M16—2121no—
470M16—2121no—
471M16—2121no—
472M16—2121no—
473M16—2121no—
474M16—2121no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—1062no—
482C2×C8—1062no—
483C2×C8—1062no—
484C2×C8—1062no—
485C2×C8—2121no—
486C2×C8—2121no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C42—41no—
492C42—41no—
493C42—1062no—
494C42—1062no—
495C42—2121no—
496C42—2121no—
497C23—42no—
498C23—1062no—
499C23—4242no—
500C23—2121no—
501C23—2121no—
502C23#1313317411yescharacteristic
503C23#1313319011yescharacteristic
504C23#1313319911yescharacteristic
505C23#1313320011yescharacteristic
506C23#1313320211yescharacteristic
507C23#1313320511yescharacteristic
508C23#1313321211yescharacteristic
509Q8—1062no—
510Q8—1062no—
511Q8—2121no—
512Q8—2121no—
513D4—1062no—
514D4—1062no—
515D4—2122no—
516D4—2122no—
517D4—2122no—
518D4—2122no—
519D4—2121no—
520D4—2121no—
521C2×C4—42no—
522C2×C4—82no—
523C2×C4—1064no—
524C2×C4—1062no—
525C2×C4—2122no—
526C2×C4—2124no—
527C2×C4—2124no—
528C2×C4—2121no—
529C2×C4—2122no—
530C2×C4—2121no—
531C2×C4—2122no—
532C2×C4—2122no—
533C2×C4—2122no—
534C2×C4—2122no—
535C2×C4C22⋊C4×D5311yescharacteristic
536C2×C4#1313317511yescharacteristic
537C2×C4#1313317711yescharacteristic
538C2×C4C4×C22⋊D5311yescharacteristic
539C2×C4#1313317911yescharacteristic
540C2×C4C4×D21211yescharacteristic
541C2×C4#1313318511yescharacteristic
542C2×C4#1313318711yescharacteristic
543C2×C4#1313318811yescharacteristic
544C2×C4#1313318911yescharacteristic
545C2×C4#1313319011yescharacteristic
546C2×C4#1313319011yescharacteristic
547C2×C4#1313319111yescharacteristic
548C2×C4#1313319211yescharacteristic
549C2×C4#1313319211yescharacteristic
550C2×C4#1313319911yescharacteristic
551C2×C4#1313319911yescharacteristic
552C2×C4#1313320111yescharacteristic
553C2×C4#1313320211yescharacteristic
554C2×C4#1313320211yescharacteristic
555C2×C4#1313320211yescharacteristic
556C2×C4#1313320211yescharacteristic
557C2×C4#1313320311yescharacteristic
558C2×C4#1313320311yescharacteristic
559C2×C4#1313320611yescharacteristic
560C2×C4#1313321111yescharacteristic
561C2×C4#1313321211yescharacteristic
562C2×C4#1313321211yescharacteristic
563C8—42no—
564C8—42no—
565C8—2122no—
566C8—2122no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—4242no—
575C22—4242no—
576C22—4242no—
577C22—2122no—
578C22#1681048511yescharacteristic
579C22#1681051211yescharacteristic
580C22#1681051411yescharacteristic
581C22#1681053811yescharacteristic
582C22#1681054111yescharacteristic
583C22#1681054911yescharacteristic
584C22#1681055011yescharacteristic
585C4—82no—
586C4—2124no—
587C4—2122no—
588C4#1681049911yescharacteristic
589C4#1681050111yescharacteristic
590C4#1681051311yescharacteristic
591C4#1681051511yescharacteristic
592C4#1681053911yescharacteristic
593C4#1681054011yescharacteristic
594C4#1681055211yescharacteristic
595C4#1681055211yescharacteristic
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—4242no—
603C2#2027296311yescharacteristic
604C2#2027600012yes—
subgroupG/H
centerC2×C4#13133203
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1742C2
socleC22×C53#394
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenter#1742D53
hypocenterC53#11151
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
4. C2×#16810382C2semidirect
5. C2×#16810381C2semidirect
6. #20274408C2semidirect
7. C53×#1742C2semidirect
20. #16810605C4semidirect
297. #1742D53semidirect
332. C53#11151semidirect

Maximal subgroups

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

subgroupindex
C22×#131331432
#202832322
C2×#168106052
C2×#168103822
C2×#168103812
#202744082
C53×#17422
#1115153