Finite Group Databasebeta

← #28617642

Subgroups — #28617642

Order 15104.

888 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837451C211yesmaximal, characteristic, complemented
2#21432616C211yesmaximal, characteristic
3#21432613C211yesmaximal, characteristic
4C2×#16837384C211yesmaximal, characteristic
5#21425562C214yesmaximal, complemented
6#21425555C214yesmaximal, complemented
7C2×#16837173C211yesmaximal, characteristic, complemented
8C2×#16837158C211yesmaximal, characteristic, complemented
9C2×C59×#344C211yesmaximal, characteristic, complemented
10C22×#11745685C2211yescharacteristic
11C22×#11745664C2211yescharacteristic
12C22×#11745664C2211yescharacteristic
13C2×#14273686C2211yescharacteristic
14#16837451—22no—
15#16837451C2214yes—
16C2×#14273663—22no—
17#16837412C2212yes—
18#16837410C2212yes—
19#16837408C2212yes—
20#16837406C2212yes—
21#16837404—22no—
22#16837401—22no—
23#16837399—22no—
24#16837396—22no—
25#16837384—22no—
26#16837384—22no—
27C2×#14273645C2212yes—
28C2×#14273643C2212yes—
29#16837173C2214yes—
30#16837158C2214yes—
31C22×M16×C59C2211yescharacteristic
32C22×M16×C59C2211yescharacteristic
33C2×#117×C59C2211yescharacteristic
34C59×#344C2214yes—
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—
44C2×#11745685—21no—
45C2×#11745685D412yes—
46C2×#11745685D412yes—
47C22×Dic118C2311yescharacteristic
48C22×D236C2311yescharacteristic
49C22×C22⋊D59—21no—
50C22×C4×D59—21no—
51C2×#11745664—21no—
52C2×#11745664—21no—
53C2×#11745664—21no—
54C2×#11745664—21no—
55C2×#11745664—21no—
56C2×#11745664—21no—
57C2×#11745664—21no—
58C2×#11745664—21no—
59C2×#11745664—21no—
60C2×#11745664—21no—
61C2×#11745664C2312yes—
62C2×#11745664C2312yes—
63C22×C59⋊C8—21no—
64C22×C59⋊C8—21no—
65#14273686—21no—
66#14273686—21no—
67#14273686—21no—
68#14273686C2312yes—
69C2×C4⋊Dic59—21no—
70C2×C22⋊Dic59—21no—
71C2×C4×Dic59—21no—
72#14273663—28no—
73#14273652—24no—
74#14273650—24no—
75#14273649—24no—
76#14273647—24no—
77#14273645—24no—
78#14273643—24no—
79C23×C4×C59C2311yescharacteristic
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×C59D412yes—
90C2×M16×C59C2312yes—
91C2×M16×C59C2312yes—
92C22×C8×C59—21no—
93C22×C8×C59D412yes—
94#117×C59—21no—
95#117×C59—21no—
96#117×C59—21no—
97#117×C59C2312yes—
98C2×C4⋊C4×C59—21no—
99C2×C22⋊C4×C59—21no—
100C2×C42×C59—21no—
101C24×C59#4111yescharacteristic
102C23×D59—21no—
103C22×C4×C59—41no—
104C22×C4×C59C2×D411yescharacteristic
105C22×C4×C59C2×D411yescharacteristic
106C22×C4×C59C2×D411yescharacteristic
107C22×C4×C59C2×D411yescharacteristic
108C22×C4×C59#4111yescharacteristic
109C22×C4×C59#4111yescharacteristic
110C22×C4×C59#4111yescharacteristic
111C22×C4×C59#4111yescharacteristic
112C22×C4×C59#4111yescharacteristic
113C22×C4×C59#4111yescharacteristic
114C22×C4×C59#4111yescharacteristic
115C22×C4×C59#4111yescharacteristic
116C22×C4×C59#4111yescharacteristic
117C22×C4×C59C2411yescharacteristic
118C2×C22⋊D59—22no—
119C2×C22⋊D59—22no—
120C2×C22⋊D59—41no—
121C2×C22⋊D59—41no—
122C2×C22⋊D59—41no—
123C2×C22⋊D59—41no—
124C22×Dic59—21no—
125C22×Dic59—41no—
126#11745685—22no—
127#11745685—22no—
128#11745685—22no—
129#11745685—22no—
130#11745685—22no—
131#11745685—22no—
132#11745685—22no—
133#11745685—22no—
134#11745685—41no—
135#11745685—41no—
136#11745685—41no—
137#11745685—41no—
138#11745685—41no—
139#11745685—41no—
140#11745685—41no—
141#11745685—41no—
142C2×D236—21no—
143C2×D236—21no—
144C2×D236—21no—
145C2×D236—21no—
146C2×D236—21no—
147C2×D236C2×D412yes—
148C2×C4×D59—22no—
149C2×C4×D59—22no—
150C2×C4×D59—41no—
151C2×C4×D59—41no—
152C2×C4×D59—41no—
153C2×C4×D59—41no—
154C2×Dic118—21no—
155C2×Dic118—21no—
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118—21no—
159C2×Dic118C2×D412yes—
160M16×C59—22no—
161M16×C59—22no—
162M16×C59—22no—
163M16×C59—22no—
164M16×C59—22no—
165M16×C59—22no—
166M16×C59—41no—
167M16×C59—41no—
168C2×C8×C59—22no—
169C2×C8×C59—22no—
170C2×C8×C59—22no—
171C2×C8×C59—41no—
172C2×C8×C59—41no—
173C2×C8×C59C2×D414yes—
174C4⋊C4×C59—22no—
175C4⋊C4×C59—41no—
176C22⋊C4×C59—22no—
177C22⋊C4×C59—41no—
178C42×C59—22no—
179C42×C59—41no—
180C22⋊Dic59—22no—
181C22⋊Dic59—41no—
182C4⋊Dic59—22no—
183C4⋊Dic59—41no—
184C4×Dic59—22no—
185C4×Dic59—41no—
186#11745664—22no—
187#11745664—22no—
188#11745664—22no—
189#11745664—22no—
190#11745664—41no—
191#11745664—41no—
192#11745664—41no—
193#11745664—41no—
194C2×C59⋊C8—22no—
195C2×C59⋊C8—22no—
196C2×C59⋊C8—41no—
197C2×C59⋊C8—41no—
198C2×C59⋊C8—41no—
199C2×C59⋊C8—41no—
200C23×C59—81no—
201C23×C59#12311yescharacteristic
202C23×C59#12311yescharacteristic
203C23×C59#12411yescharacteristic
204C23×C59#12411yescharacteristic
205C23×C59#12611yescharacteristic
206C23×C59#12611yescharacteristic
207C23×C59C2×#4111yescharacteristic
208C22×D59—22no—
209C22×D59—41no—
210C22×D59—41no—
211C22×D59—82no—
212C2×C4×C59—42no—
213C2×C4×C59—82no—
214C2×C4×C59#12111yescharacteristic
215C2×C4×C59#12111yescharacteristic
216C2×C4×C59#12111yescharacteristic
217C2×C4×C59#12111yescharacteristic
218C2×C4×C59#12311yescharacteristic
219C2×C4×C59#12311yescharacteristic
220C2×C4×C59#12311yescharacteristic
221C2×C4×C59#12311yescharacteristic
222C2×C4×C59#12311yescharacteristic
223C2×C4×C59#12311yescharacteristic
224C2×C4×C59#12311yescharacteristic
225C2×C4×C59#12311yescharacteristic
226C2×C4×C59#12311yescharacteristic
227C2×C4×C59#12311yescharacteristic
228C2×C4×C59#12411yescharacteristic
229C2×C4×C59#12411yescharacteristic
230C2×C4×C59#12611yescharacteristic
231C2×C4×C59#12611yescharacteristic
232C2×C4×C59#12611yescharacteristic
233C2×C4×C59#12611yescharacteristic
234C2×C4×C59#12611yescharacteristic
235C2×C4×C59#12611yescharacteristic
236C2×C4×C59C22×D411yescharacteristic
237C2×C4×C59C22×D411yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59C2×#4111yescharacteristic
242C22⋊D59—42no—
243C22⋊D59—42no—
244C22⋊D59—42no—
245C22⋊D59—42no—
246C2×Dic59—22no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—42no—
250C2×Dic59—82no—
251D236—22no—
252D236—22no—
253D236—41no—
254D236—41no—
255C4×D59—42no—
256C4×D59—42no—
257C4×D59—42no—
258C4×D59—42no—
259Dic118—22no—
260Dic118—22no—
261Dic118—41no—
262Dic118—41no—
263C8×C59—24no—
264C8×C59—42no—
265C59⋊C8—42no—
266C59⋊C8—42no—
267#23652—591nomaximal, Sylow
268C22×C59—41no—
269C22×C59—41no—
270C22×C59—41no—
271C22×C59—41no—
272C22×C59—41no—
273C22×C59—41no—
274C22×C59—41no—
275C22×C59#39711yescharacteristic
276C22×C59#39711yescharacteristic
277C22×C59#39711yescharacteristic
278C22×C59#39711yescharacteristic
279C22×C59C2×#12311yescharacteristic
280C22×C59C2×#12411yescharacteristic
281C22×C59C2×#12611yescharacteristic
282C2×D59—42no—
283C2×D59—82no—
284C2×D59—82no—
285C2×D59—82no—
286C4×C59—82no—
287C4×C59#39711yescharacteristic
288C4×C59#39711yescharacteristic
289C4×C59#39711yescharacteristic
290C4×C59#39711yescharacteristic
291C4×C59C2×#12111yescharacteristic
292C4×C59C2×#12311yescharacteristic
293C4×C59C2×#12311yescharacteristic
294C4×C59C2×#12611yescharacteristic
295Dic59—42no—
296Dic59—82no—
297C2×#430—591no—
298#2896—591no—
299#2896—591no—
300C2×#413—591no—
301#2082—594no—
302#2081—594no—
303C2×#344—591no—
304C2×#344D5911yescharacteristic, complemented
305C2×H(3,ℤ/4)—591no—
306C2×C59—21no—
307C2×C59—21no—
308C2×C59—21no—
309C2×C59—21no—
310C2×C59—21no—
311C2×C59—21no—
312C2×C59—21no—
313C2×C59C2×#39711yescharacteristic
314D59—82no—
315C22×#41—591no—
316C22×M16—591no—
317C22×M16—591no—
318C22×M16C2×D5911yescharacteristic
319C22×M16C2×D5911yescharacteristic
320C2×#117—591no—
321C2×#117C2×D5911yescharacteristic
322#430—594no—
323#430—1182no—
324C2×#108—1182no—
325#419—592no—
326#419—592no—
327#418—592no—
328#418—592no—
329#417—1182no—
330#417—1182no—
331#416—1182no—
332#416—1182no—
333#413—1182no—
334#413—1182no—
335C2×#101—592no—
336C2×#100—592no—
337#344—594no—
338#344C2×D5914yes—
339H(3,ℤ/4)—594no—
340C59#2365211yescharacteristic, complemented, Sylow
341C2×#41—592no—
342C2×#41—592no—
343C2×#41—1181no—
344C2×#41—1181no—
345C2×#41—1181no—
346C2×#41—1181no—
347C2×#41—1181no—
348C2×#41—1181no—
349C2×#41—1181no—
350C2×#41—1181no—
351C2×#41—1181no—
352C2×#41—1181no—
353C22×Q8—591no—
354C22×D4—591no—
355C22×D4—1181no—
356C23×C4—1181no—
357C23×C4C22×D5911yescharacteristic
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—21no—
367C2×M16—592no—
368C2×M16—592no—
369C2×M16—1181no—
370C2×M16—1181no—
371C2×M16—1181no—
372C2×M16—1181no—
373C2×M16—1181no—
374C2×M16—1181no—
375C2×M16—1181no—
376C2×M16—1181no—
377C2×M16—1181no—
378C2×M16—1181no—
379C2×M16D23612yes—
380C2×M16C22×D5912yes—
381C2×M16C22×D5912yes—
382C22×C8—21no—
383C22×C8—1181no—
384C22×C8—1181no—
385C22×C8D23612yes—
386#117—21no—
387#117—21no—
388#117—21no—
389#117—592no—
390#117—1181no—
391#117—1181no—
392#117—1181no—
393#117C22×D5912yes—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—1181no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—1181no—
398C2×C42—21no—
399C2×C42—1181no—
400#108—1188no—
401#103—1184no—
402#103—1184no—
403#102—1184no—
404#102—1184no—
405#101—1184no—
406#100—1184no—
407C24—1181no—
408C24#1174568511yescharacteristic
409#41—1182no—
410#41—1182no—
411#41—1182no—
412#41—1182no—
413#41—1182no—
414#41—1182no—
415#41—1182no—
416#41—1182no—
417#41—2361no—
418#41—2361no—
419#41—2361no—
420#41—2361no—
421#41—2361no—
422#41—2361no—
423#41—2361no—
424#41—2361no—
425C2×Q8—592no—
426C2×Q8—1181no—
427C2×Q8—1181no—
428C2×Q8—1181no—
429C2×Q8—1181no—
430C2×Q8—1181no—
431C2×D4—592no—
432C2×D4—1182no—
433C2×D4—1182no—
434C2×D4—1181no—
435C2×D4—1181no—
436C2×D4—1181no—
437C2×D4—1181no—
438C2×D4—1181no—
439C2×D4—2361no—
440C2×D4—2361no—
441C2×D4—2361no—
442C2×D4—2361no—
443C22×C4—41no—
444C22×C4—1181no—
445C22×C4—1182no—
446C22×C4—1182no—
447C22×C4—2361no—
448C22×C4—2361no—
449C22×C4—2361no—
450C22×C4—2361no—
451C22×C4—2361no—
452C22×C4C2×D23611yescharacteristic
453C22×C4C2×D23611yescharacteristic
454C22×C4#1174568511yescharacteristic
455C22×C4#1174568511yescharacteristic
456C22×C4#1174568511yescharacteristic
457C22×C4D4×D5911yescharacteristic
458C22×C4D4×D5911yescharacteristic
459C22×C4#1174568711yescharacteristic
460C22×C4#1174568711yescharacteristic
461C22×C4#1174568711yescharacteristic
462C22×C4#1174568711yescharacteristic
463C22×C4#1174568911yescharacteristic
464C22×C4#1174568911yescharacteristic
465C22×C4C23×D5911yescharacteristic
466M16—22no—
467M16—22no—
468M16—22no—
469M16—22no—
470M16—22no—
471M16—22no—
472M16—41no—
473M16—41no—
474M16—1182no—
475M16—1182no—
476M16—1182no—
477M16—1182no—
478M16—2361no—
479M16—2361no—
480M16—2361no—
481M16—2361no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—1182no—
488C2×C8—1182no—
489C2×C8—2361no—
490C2×C8—2361no—
491C2×C8—2361no—
492C2×C8—2361no—
493C2×C8C2×D23614yes—
494C4⋊C4—22no—
495C4⋊C4—41no—
496C4⋊C4—1182no—
497C4⋊C4—2361no—
498C22⋊C4—22no—
499C22⋊C4—41no—
500C22⋊C4—1182no—
501C22⋊C4—2361no—
502C42—22no—
503C42—41no—
504C42—1182no—
505C42—2361no—
506C23—81no—
507C23—1182no—
508C23—4722no—
509C23—2361no—
510C23—2361no—
511C23#1427370911yescharacteristic
512C23#1427370911yescharacteristic
513C23#1427371511yescharacteristic
514C23#1427371511yescharacteristic
515C23#1427372211yescharacteristic
516C23#1427372211yescharacteristic
517C23C2×#1174568511yescharacteristic
518Q8—1182no—
519Q8—1182no—
520Q8—2361no—
521Q8—2361no—
522D4—1182no—
523D4—1182no—
524D4—2362no—
525D4—2362no—
526D4—2362no—
527D4—2362no—
528D4—2361no—
529D4—2361no—
530C2×C4—42no—
531C2×C4—82no—
532C2×C4—1182no—
533C2×C4—4722no—
534C2×C4—2362no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2362no—
538C2×C4—2362no—
539C2×C4—2362no—
540C2×C4—2362no—
541C2×C4#1427370211yescharacteristic
542C2×C4#1427370211yescharacteristic
543C2×C4#1427370211yescharacteristic
544C2×C4#1427370211yescharacteristic
545C2×C4#1427370911yescharacteristic
546C2×C4#1427370911yescharacteristic
547C2×C4#1427370911yescharacteristic
548C2×C4#1427370911yescharacteristic
549C2×C4#1427370911yescharacteristic
550C2×C4#1427370911yescharacteristic
551C2×C4#1427371111yescharacteristic
552C2×C4#1427371111yescharacteristic
553C2×C4#1427371111yescharacteristic
554C2×C4#1427371111yescharacteristic
555C2×C4#1427371511yescharacteristic
556C2×C4#1427371511yescharacteristic
557C2×C4#1427372211yescharacteristic
558C2×C4#1427372211yescharacteristic
559C2×C4#1427372211yescharacteristic
560C2×C4#1427372211yescharacteristic
561C2×C4#1427372211yescharacteristic
562C2×C4#1427372211yescharacteristic
563C2×C4C2×D4×D5911yescharacteristic
564C2×C4C22×D23611yescharacteristic
565C2×C4C2×#1174568711yescharacteristic
566C2×C4C2×#1174568711yescharacteristic
567C2×C4C2×#1174568511yescharacteristic
568C2×C4C2×#1174568911yescharacteristic
569C8—24no—
570C8—42no—
571C8—2362no—
572C8—2362no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—4722no—
581C22—4722no—
582C22—4722no—
583C22—2362no—
584C22#1683732911yescharacteristic
585C22#1683732911yescharacteristic
586C22#1683732911yescharacteristic
587C22#1683732911yescharacteristic
588C22C2×#1427370911yescharacteristic
589C22C2×#1427371511yescharacteristic
590C22C2×#1427372211yescharacteristic
591C4—82no—
592C4—4722no—
593C4—2362no—
594C4#1683732911yescharacteristic
595C4#1683732911yescharacteristic
596C4#1683732911yescharacteristic
597C4#1683732911yescharacteristic
598C4C2×#1427370211yescharacteristic
599C4C2×#1427370911yescharacteristic
600C4C2×#1427371111yescharacteristic
601C4C2×#1427372211yescharacteristic
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—21no—
608C2—21no—
609C2—4722no—
610C2C2×#1683732911yescharacteristic
subgroupG/H
centerC4#16837329
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#344C2
socleC2×C59C2×#397
prime radicalC22×#11745685C22
hypercenterC2×#344D59
hypocenterC59#23652
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#344

Complements

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

NH ≅ G/Ntype
1. C2×#16837451C2semidirect
5. #21425562C2semidirect
6. #21425555C2semidirect
7. C2×#16837173C2semidirect
8. C2×#16837158C2semidirect
9. C2×C59×#344C2semidirect
304. C2×#344D59semidirect
340. C59#23652semidirect

Maximal subgroups

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

subgroupindex
C2×#168374512
#214326162
#214326132
C2×#168373842
#214255622
#214255552
C2×#168371732
C2×#168371582
C2×C59×#3442
#2365259