Finite Group Databasebeta

← #28495950

Subgroups — #28495950

Order 15104.

850 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273662C211yesmaximal, characteristic, complemented
2C2×#16837384C211yesmaximal, characteristic
3C2×#16837384C211yesmaximal, characteristic
4C2×#16837370C211yesmaximal, characteristic
5C2×#16837166C211yesmaximal, characteristic, complemented
6C2×C59⋊#338C211yesmaximal, characteristic, complemented
7C2×C59×#338C211yesmaximal, characteristic, complemented
8C22×#11745685C2211yescharacteristic
9C22×#11745664C2211yescharacteristic
10C22×#11745664C2211yescharacteristic
11C23×C59⋊C8C2211yescharacteristic
12C2×#14273662—22no—
13C2×#14273662—22no—
14C2×#14273662—22no—
15#16837384—22no—
16#16837384—22no—
17#16837384—22no—
18#16837384—22no—
19C2×#14273645—21no—
20C2×#14273645—21no—
21C2×#14273643—21no—
22C2×#14273643—21no—
23#16837370—22no—
24#16837370—22no—
25C2×(C2×D59)⋊C8C414yes—
26#16837166—22no—
27C59⋊#338—22no—
28C22×M16×C59C2211yescharacteristic
29C22×M16×C59C2211yescharacteristic
30C22×M16×C59C2211yescharacteristic
31C59×#338—22no—
32C2×#11745685—21no—
33C2×#11745685—21no—
34C2×#11745685—21no—
35C2×#11745685—21no—
36C2×#11745685—21no—
37C2×#11745685—21no—
38C2×#11745685—21no—
39C2×#11745685—21no—
40C2×#11745685—21no—
41C2×#11745685—21no—
42C2×#11745685—21no—
43C2×#11745685—21no—
44C22×Dic118D411yescharacteristic
45C22×D236D411yescharacteristic
46C22×C22⋊D59Q812yes—
47C22×C4×D59C2×C412yes—
48C2×#11745664—21no—
49C2×#11745664—21no—
50C2×#11745664—21no—
51C2×#11745664—21no—
52C2×#11745664—21no—
53C2×#11745664—21no—
54C2×#11745664—21no—
55C2×#11745664—21no—
56C2×#11745664—21no—
57C2×#11745664—21no—
58C2×#11745664D412yes—
59C2×#11745664D412yes—
60C22×C59⋊C8—21no—
61C22×C59⋊C8—21no—
62C22×C59⋊C8—21no—
63C22×C59⋊C8—21no—
64C22×C59⋊C8—21no—
65C22×C59⋊C8—21no—
66C22×C59⋊C8C2×C412yes—
67C22×C59⋊C8D412yes—
68C22×C59⋊C8D412yes—
69#14273662—28no—
70#14273645—24no—
71#14273645—24no—
72#14273643—24no—
73#14273643—24no—
74(C2×D59)⋊C8—28no—
75C23×C4×C59C2311yescharacteristic
76C2×M16×C59—21no—
77C2×M16×C59—21no—
78C2×M16×C59—21no—
79C2×M16×C59—21no—
80C2×M16×C59—21no—
81C2×M16×C59—21no—
82C2×M16×C59—21no—
83C2×M16×C59—21no—
84C2×M16×C59—21no—
85C2×M16×C59—21no—
86C2×M16×C59—21no—
87C2×M16×C59—21no—
88C2×M16×C59—21no—
89C2×M16×C59—21no—
90C2×M16×C59—21no—
91C2×M16×C59—21no—
92C2×M16×C59D412yes—
93C2×M16×C59D412yes—
94C22×C8×C59—21no—
95C22×C8×C59—21no—
96C22×C8×C59C2×C412yes—
97C24×C59C2×Q811yescharacteristic
98C23×D59C4⋊C412yes—
99C22×C4×C59C22×C411yescharacteristic
100C22×C4×C59C2×D411yescharacteristic
101C22×C4×C59C2×D411yescharacteristic
102C22×C4×C59C2×D411yescharacteristic
103C22×C4×C59C2×D411yescharacteristic
104C22×C4×C59C2×D411yescharacteristic
105C22×C4×C59C2×D411yescharacteristic
106C22×C4×C59C2×D411yescharacteristic
107C22×C4×C59#4111yescharacteristic
108C22×C4×C59#4111yescharacteristic
109C22×C4×C59#4111yescharacteristic
110C22×C4×C59#4111yescharacteristic
111C22×C4×C59#4111yescharacteristic
112C22×C4×C59#4111yescharacteristic
113C2×C22⋊D59—22no—
114C2×C22⋊D59—22no—
115C2×C22⋊D59—22no—
116C2×C22⋊D59—22no—
117C2×C22⋊D59—22no—
118C2×C22⋊D59—22no—
119C22×Dic59C4⋊C412yes—
120#11745685—41no—
121#11745685—41no—
122#11745685—41no—
123#11745685—41no—
124#11745685—41no—
125#11745685—41no—
126#11745685—41no—
127#11745685—41no—
128#11745685—41no—
129#11745685—41no—
130#11745685—41no—
131#11745685—41no—
132#11745685—41no—
133#11745685—41no—
134#11745685—41no—
135#11745685—41no—
136C2×D236—21no—
137C2×D236—21no—
138C2×D236—21no—
139C2×D236—21no—
140C2×D236—21no—
141C2×D236—21no—
142C2×C4×D59—22no—
143C2×C4×D59—22no—
144C2×C4×D59—22no—
145C2×C4×D59—22no—
146C2×C4×D59—22no—
147C2×C4×D59—22no—
148C2×Dic118—21no—
149C2×Dic118—21no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154M16×C59—22no—
155M16×C59—22no—
156M16×C59—22no—
157M16×C59—22no—
158M16×C59—41no—
159M16×C59—41no—
160M16×C59—41no—
161M16×C59—41no—
162M16×C59—41no—
163M16×C59—41no—
164M16×C59—41no—
165M16×C59—41no—
166C2×C8×C59—22no—
167C2×C8×C59—22no—
168C2×C8×C59—22no—
169C2×C8×C59—22no—
170C2×C8×C59—22no—
171C2×C8×C59—41no—
172C2×C8×C59—41no—
173C2×C8×C59—41no—
174C2×C8×C59—41no—
175#11745664—22no—
176#11745664—22no—
177#11745664—22no—
178#11745664—22no—
179#11745664—41no—
180#11745664—41no—
181#11745664—41no—
182#11745664—41no—
183C2×C59⋊C8—22no—
184C2×C59⋊C8—22no—
185C2×C59⋊C8—22no—
186C2×C59⋊C8—22no—
187C2×C59⋊C8—22no—
188C2×C59⋊C8—22no—
189C2×C59⋊C8—22no—
190C2×C59⋊C8—22no—
191C2×C59⋊C8—41no—
192C2×C59⋊C8—41no—
193C2×C59⋊C8—41no—
194C2×C59⋊C8—41no—
195C2×C59⋊C8C22⋊C414yes—
196C23×C59—42no—
197C23×C59C2×C4⋊C411yescharacteristic
198C23×C59C22⋊Q811yescharacteristic
199C23×C59C22⋊Q811yescharacteristic
200C23×C59C22⋊Q811yescharacteristic
201C23×C59C22⋊Q811yescharacteristic
202C23×C59C22⋊Q811yescharacteristic
203C23×C59C22⋊Q811yescharacteristic
204C22×D59—22no—
205C22×D59—22no—
206C22×D59—22no—
207C22×D59—44no—
208C2×C4×C59C2×C22⋊C411yescharacteristic
209C2×C4×C59#11711yescharacteristic
210C2×C4×C59C4×D411yescharacteristic
211C2×C4×C59C4×D411yescharacteristic
212C2×C4×C59C4×D411yescharacteristic
213C2×C4×C59C4×D411yescharacteristic
214C2×C4×C59#12011yescharacteristic
215C2×C4×C59#12011yescharacteristic
216C2×C4×C59#12011yescharacteristic
217C2×C4×C59#12111yescharacteristic
218C2×C4×C59#12111yescharacteristic
219C2×C4×C59#12111yescharacteristic
220C2×C4×C59#12111yescharacteristic
221C2×C4×C59#12111yescharacteristic
222C2×C4×C59#12111yescharacteristic
223C2×C4×C59#12111yescharacteristic
224C2×C4×C59#12111yescharacteristic
225C2×C4×C59#12311yescharacteristic
226C2×C4×C59#12311yescharacteristic
227C2×C4×C59#12311yescharacteristic
228C2×C4×C59#12311yescharacteristic
229C2×C4×C59#12311yescharacteristic
230C2×C4×C59#12411yescharacteristic
231C2×C4×C59#12411yescharacteristic
232C2×C4×C59#12411yescharacteristic
233C2×C4×C59#12611yescharacteristic
234C2×C4×C59#12611yescharacteristic
235C2×C4×C59#12711yescharacteristic
236C22⋊D59—42no—
237C22⋊D59—42no—
238C22⋊D59—42no—
239C22⋊D59—42no—
240C2×Dic59—22no—
241C2×Dic59—22no—
242C2×Dic59—22no—
243D236—41no—
244D236—41no—
245D236—41no—
246D236—41no—
247C4×D59—42no—
248C4×D59—42no—
249C4×D59—42no—
250C4×D59—42no—
251Dic118—41no—
252Dic118—41no—
253Dic118—41no—
254Dic118—41no—
255C8×C59—42no—
256C8×C59—42no—
257C8×C59—42no—
258C59⋊C8—24no—
259C59⋊C8—42no—
260C59⋊C8—42no—
261#11099—591nomaximal, Sylow
262C22×C59—22no—
263C22×C59—22no—
264C22×C59—22no—
265C22×C59—41no—
266C22×C59—41no—
267C22×C59—41no—
268C22×C59—41no—
269C22×C59#38011yescharacteristic
270C22×C59#38511yescharacteristic
271C22×C59#38511yescharacteristic
272C22×C59#39311yescharacteristic
273C22×C59#39611yescharacteristic
274C22×C59#39611yescharacteristic
275C22×C59#39911yescharacteristic
276C2×D59—44no—
277C2×D59—42no—
278C2×D59—82no—
279C2×D59—82no—
280C4×C59#38611yescharacteristic
281C4×C59#38811yescharacteristic
282C4×C59#38811yescharacteristic
283C4×C59#39011yescharacteristic
284C4×C59#39211yescharacteristic
285C4×C59#39411yescharacteristic
286C4×C59#39411yescharacteristic
287C4×C59#39711yescharacteristic
288Dic59—42no—
289C22×#108—591no—
290C2×#413—591no—
291C2×#413—591no—
292C2×#408—591no—
293C2×#341—591no—
294C2×#338—591no—
295C2×#338D5911yescharacteristic, complemented
296C2×C59—21no—
297C2×C59—21no—
298C2×C59—21no—
299C2×C59—21no—
300C2×C59—21no—
301C2×C59—21no—
302C2×C59#143911yescharacteristic
303C2×C59#181312yes—
304D59—82no—
305C22×#41—591no—
306C22×M16—591no—
307C22×M16—591no—
308C22×M16C2×D5911yescharacteristic
309C22×M16C2×D5911yescharacteristic
310C22×M16C2×D5911yescharacteristic
311C23×C8—591no—
312C2×#108—1182no—
313C2×#108—1182no—
314C2×#108—1182no—
315#413—1182no—
316#413—1182no—
317#413—1182no—
318#413—1182no—
319C2×#101—1181no—
320C2×#101—1181no—
321C2×#100—1181no—
322C2×#100—1181no—
323#408—1182no—
324#408—1182no—
325C2×C22⋊C8—594no—
326#341—1182no—
327#338—22no—
328#338—1182no—
329C59#1109911yescharacteristic, complemented, Sylow
330C2×#41—1181no—
331C2×#41—1181no—
332C2×#41—1181no—
333C2×#41—1181no—
334C2×#41—1181no—
335C2×#41—1181no—
336C2×#41—1181no—
337C2×#41—1181no—
338C2×#41—1181no—
339C2×#41—1181no—
340C2×#41—1181no—
341C2×#41—1181no—
342C22×Q8—591no—
343C22×D4—592no—
344C22×D4—591no—
345C23×C4—592no—
346C23×C4C22×D5911yescharacteristic
347C2×M16—21no—
348C2×M16—21no—
349C2×M16—21no—
350C2×M16—21no—
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—21no—
362C2×M16—21no—
363C2×M16—592no—
364C2×M16—592no—
365C2×M16—1181no—
366C2×M16—1181no—
367C2×M16—1181no—
368C2×M16—1181no—
369C2×M16—1181no—
370C2×M16—1181no—
371C2×M16—1181no—
372C2×M16—1181no—
373C2×M16—1181no—
374C2×M16—1181no—
375C2×M16D23612yes—
376C2×M16C22⋊D5912yes—
377C22×C8—21no—
378C22×C8—21no—
379C22×C8—592no—
380C22×C8—592no—
381C22×C8—592no—
382C22×C8—1181no—
383C22×C8—1181no—
384C22×C8—1181no—
385C22×C8—1181no—
386C22×C8—1181no—
387C22×C8—1181no—
388C22×C8C4×D5912yes—
389#108—1188no—
390#101—1184no—
391#101—1184no—
392#100—1184no—
393#100—1184no—
394C22⋊C8—1188no—
395C24—592no—
396C24Q8×D5911yescharacteristic
397#41—2361no—
398#41—2361no—
399#41—2361no—
400#41—2361no—
401#41—2361no—
402#41—2361no—
403#41—2361no—
404#41—2361no—
405#41—2361no—
406#41—2361no—
407#41—2361no—
408#41—2361no—
409#41—2361no—
410#41—2361no—
411#41—2361no—
412#41—2361no—
413C2×Q8—1181no—
414C2×Q8—1181no—
415C2×Q8—1181no—
416C2×Q8—1181no—
417C2×Q8—1181no—
418C2×Q8—1181no—
419C2×D4—1182no—
420C2×D4—1182no—
421C2×D4—1182no—
422C2×D4—1182no—
423C2×D4—1182no—
424C2×D4—1182no—
425C2×D4—1181no—
426C2×D4—1181no—
427C2×D4—1181no—
428C2×D4—1181no—
429C2×D4—1181no—
430C2×D4—1181no—
431C22×C4—592no—
432C22×C4—1182no—
433C22×C4—1182no—
434C22×C4—1182no—
435C22×C4—1182no—
436C22×C4—1182no—
437C22×C4—1182no—
438C22×C4C2×C4×D5911yescharacteristic
439C22×C4C2×D23611yescharacteristic
440C22×C4#1174568511yescharacteristic
441C22×C4#1174568511yescharacteristic
442C22×C4#1174568511yescharacteristic
443C22×C4#1174568511yescharacteristic
444C22×C4D4×D5911yescharacteristic
445C22×C4D4×D5911yescharacteristic
446C22×C4D4×D5911yescharacteristic
447C22×C4D4×D5911yescharacteristic
448C22×C4D4×D5911yescharacteristic
449C22×C4#1174568711yescharacteristic
450C22×C4#1174568911yescharacteristic
451C22×C4C2×C22⋊D5911yescharacteristic
452M16—22no—
453M16—22no—
454M16—22no—
455M16—22no—
456M16—41no—
457M16—41no—
458M16—41no—
459M16—41no—
460M16—41no—
461M16—41no—
462M16—41no—
463M16—41no—
464M16—1182no—
465M16—1182no—
466M16—1182no—
467M16—1182no—
468M16—2361no—
469M16—2361no—
470M16—2361no—
471M16—2361no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—594no—
482C2×C8—1182no—
483C2×C8—1182no—
484C2×C8—1182no—
485C2×C8—1182no—
486C2×C8—1182no—
487C2×C8—1182no—
488C2×C8—1182no—
489C2×C8—1182no—
490C2×C8—2361no—
491C2×C8—2361no—
492C2×C8—2361no—
493C2×C8—2361no—
494C23—42no—
495C23—1182no—
496C23—1182no—
497C23—1182no—
498C23—2364no—
499C23C4⋊C4×D5911yescharacteristic
500C23#1427370311yescharacteristic
501C23#1427370511yescharacteristic
502C23#1427370511yescharacteristic
503C23#1427370511yescharacteristic
504C23#1427370511yescharacteristic
505C23#1427370711yescharacteristic
506Q8—2361no—
507Q8—2361no—
508Q8—2361no—
509Q8—2361no—
510D4—2362no—
511D4—2362no—
512D4—2362no—
513D4—2362no—
514D4—2361no—
515D4—2361no—
516D4—2361no—
517D4—2361no—
518C2×C4—1182no—
519C2×C4—1182no—
520C2×C4—1182no—
521C2×C4—2362no—
522C2×C4—2362no—
523C2×C4—2362no—
524C2×C4—2362no—
525C2×C4C22⋊C4×D5911yescharacteristic
526C2×C4#1427368411yescharacteristic
527C2×C4(C22⋊D59)⋊C411yescharacteristic
528C2×C4C4×C22⋊D5911yescharacteristic
529C2×C4#1427368911yescharacteristic
530C2×C4C4×D23611yescharacteristic
531C2×C4#1427369511yescharacteristic
532C2×C4#1427369711yescharacteristic
533C2×C4#1427369711yescharacteristic
534C2×C4#1427369911yescharacteristic
535C2×C4#1427370011yescharacteristic
536C2×C4#1427370011yescharacteristic
537C2×C4#1427370011yescharacteristic
538C2×C4#1427370011yescharacteristic
539C2×C4#1427370011yescharacteristic
540C2×C4#1427370111yescharacteristic
541C2×C4#1427370211yescharacteristic
542C2×C4#1427370911yescharacteristic
543C2×C4#1427371011yescharacteristic
544C2×C4#1427371211yescharacteristic
545C2×C4#1427371211yescharacteristic
546C2×C4#1427371211yescharacteristic
547C2×C4#1427371311yescharacteristic
548C2×C4#1427371311yescharacteristic
549C2×C4#1427371511yescharacteristic
550C2×C4#1427372111yescharacteristic
551C2×C4#1427372211yescharacteristic
552C2×C4#1427372311yescharacteristic
553C8—42no—
554C8—42no—
555C8—42no—
556C8—1184no—
557C8—2362no—
558C8—2362no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—4722no—
567C22—4722no—
568C22—2362no—
569C22—2364no—
570C22#1683726111yescharacteristic
571C22#1683727311yescharacteristic
572C22#1683727611yescharacteristic
573C22#1683731511yescharacteristic
574C22#1683732411yescharacteristic
575C22#1683732611yescharacteristic
576C22#1683733711yescharacteristic
577C4—2362no—
578C4#1683728011yescharacteristic
579C4#1683729111yescharacteristic
580C4#1683729311yescharacteristic
581C4#1683730211yescharacteristic
582C4#1683731111yescharacteristic
583C4#1683731711yescharacteristic
584C4#1683732011yescharacteristic
585C4#1683733111yescharacteristic
586C2—21no—
587C2—21no—
588C2—21no—
589C2—21no—
590C2—21no—
591C2—21no—
592C2—4722no—
593C2#2142234011yescharacteristic
594C2#2142414512yes—
subgroupG/H
centerC2×C4#14273684
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC2×C59×#338C2
socleC22×C59#380
prime radicalC22×#11745685C22
cosocleC23×C4×C59C23
hypercenterC2×#338D59
hypocenterC59#11099
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×M16 ◁ C2×#338

Complements

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

NH ≅ G/Ntype
1. C22×#14273662C2semidirect
5. C2×#16837166C2semidirect
6. C2×C59⋊#338C2semidirect
7. C2×C59×#338C2semidirect
295. C2×#338D59semidirect
329. C59#11099semidirect

Maximal subgroups

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

subgroupindex
C22×#142736622
C2×#168373842
C2×#168373702
C2×#168371662
C2×C59⋊#3382
C2×C59×#3382
#1109959