Finite Group Databasebeta

← #28559317

Subgroups — #28559317

Order 15104.

622 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434691C211yesmaximal, characteristic
2#21433241C211yesmaximal, characteristic, complemented
3#21433174C211yesmaximal, characteristic, complemented
4#21423009C211yesmaximal, characteristic, complemented
5#21422997C211yesmaximal, characteristic, complemented
6#21422985C211yesmaximal, characteristic, complemented
7#21422980C211yesmaximal, characteristic, complemented
8#21422919C211yesmaximal, characteristic, complemented
9#21422907C211yesmaximal, characteristic, complemented
10#21422895C211yesmaximal, characteristic, complemented
11#21422890C211yesmaximal, characteristic, complemented
12#21422554C211yesmaximal, characteristic, complemented
13#21422546C211yesmaximal, characteristic, complemented
14M16×C59⋊C8C211yesmaximal, characteristic, complemented
15C59×#1569C211yesmaximal, characteristic, complemented
16#16837952C2211yescharacteristic
17#16837936C2211yescharacteristic
18#16837811C411yescharacteristic
19#16837791C411yescharacteristic
20#16837767C411yescharacteristic
21#16837762C411yescharacteristic
22#16837655C411yescharacteristic
23#16837655C411yescharacteristic
24#16837648C411yescharacteristic
25#16837648C411yescharacteristic
26#16837529C2211yescharacteristic
27#16837526C2211yescharacteristic
28C4×#11745660C2211yescharacteristic
29C4×#11745660C2211yescharacteristic
30C4×D472C2211yescharacteristic
31C4×D472C2211yescharacteristic
32#16837479C2211yescharacteristic
33#16837479C2211yescharacteristic
34#16837479C2211yescharacteristic
35#16837478—21no—
36#16837475—21no—
37#16837475—21no—
38#16837471—21no—
39#16837467C2211yescharacteristic
40#16837467C2211yescharacteristic
41#16837467C2211yescharacteristic
42#16837462—21no—
43#16837455C2211yescharacteristic
44#16837427—21no—
45#16837367—21no—
46#16837367—21no—
47C2×C4×C59⋊C8C2211yescharacteristic
48#16837132C2211yescharacteristic
49#16837132C2211yescharacteristic
50#16837129C2211yescharacteristic
51#16837129C2211yescharacteristic
52#16837129C2211yescharacteristic
53#16837129C2211yescharacteristic
54#16837129C2211yescharacteristic
55#16837129C2211yescharacteristic
56#16837123C2211yescharacteristic
57#16837123C2211yescharacteristic
58C8×C59⋊C8C2211yescharacteristic
59C8×C59⋊C8C2211yescharacteristic
60C59×#432C2211yescharacteristic
61C59×#423C2211yescharacteristic
62C4×M16×C59C2211yescharacteristic
63C59×#335C2211yescharacteristic
64C59×#335C2211yescharacteristic
65C59×#334C2211yescharacteristic
66C59×#334C2211yescharacteristic
67C2×#11745685—21no—
68C2×#11745660C2×C411yescharacteristic
69C2×#11745660C2×C411yescharacteristic
70C2×D472C2×C411yescharacteristic
71C2×D472C2×C411yescharacteristic
72C2×#11745659—21no—
73C2×#11745659—21no—
74C2×#11745659—21no—
75C22×C59⋊C8—21no—
76C2×C8×D59—21no—
77C2×C8×D59—21no—
78C2×C8×D59—21no—
79#14273727C2×C411yescharacteristic
80#14273724C2×C411yescharacteristic
81#14273721—21no—
82#14273719C2×C411yescharacteristic
83#14273715—21no—
84#14273715C2×C411yescharacteristic
85#14273710—21no—
86#14273706C2×C411yescharacteristic
87#14273701—21no—
88#14273701C2×C411yescharacteristic
89#14273701C2×C411yescharacteristic
90#14273701C2×C411yescharacteristic
91C4×Dic118C2311yescharacteristic
92C4×D236—21no—
93C4×D236C2311yescharacteristic
94C4×D236C2311yescharacteristic
95C4×D236C2311yescharacteristic
96C4×C22⋊D59—21no—
97#14273684—21no—
98#14273661C2×C411yescharacteristic
99#14273661C2×C411yescharacteristic
100#14273659C2×C411yescharacteristic
101#14273659C2×C411yescharacteristic
102C4⋊(C59⋊C8)D411yescharacteristic
103C4⋊(C59⋊C8)D411yescharacteristic
104Dic59⋊C8—21no—
105Dic59⋊C8—21no—
106#14273652C2×C411yescharacteristic
107#14273652C2×C411yescharacteristic
108#14273649C2×C411yescharacteristic
109#14273649C2×C411yescharacteristic
110#14273649C2×C411yescharacteristic
111#14273649C2×C411yescharacteristic
112#14273649C2×C411yescharacteristic
113#14273649C2×C411yescharacteristic
114C22⋊(C59⋊C8)D411yescharacteristic
115C22⋊(C59⋊C8)D411yescharacteristic
116(C2×D59)⋊C8—21no—
117(C2×D59)⋊C8—21no—
118(C2×D59)⋊C8—21no—
119(C2×D59)⋊C8—21no—
120(C2×D59)⋊C8—21no—
121(C2×D59)⋊C8—21no—
122#14273637—21no—
123C4×C59⋊C8—21no—
124C4×C59⋊C8C2311yescharacteristic
125C4×C59⋊C8C2311yescharacteristic
126C4×C59⋊C8C2311yescharacteristic
127C4×C59⋊C8C2311yescharacteristic
128C8×Dic59—21no—
129C2×M16×C59—21no—
130C2×M16×C59C2×C411yescharacteristic
131C2×M16×C59C2×C411yescharacteristic
132C2×C42×C59C2311yescharacteristic
133C4⋊C8×C59—21no—
134C4⋊C8×C59C2311yescharacteristic
135C4⋊C8×C59C2311yescharacteristic
136C4⋊C8×C59C2311yescharacteristic
137C4⋊C8×C59C2311yescharacteristic
138C22⋊C8×C59—21no—
139#97×C59—21no—
140#97×C59C2×C411yescharacteristic
141#97×C59C2×C411yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C22×C4×C59C22×C411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59#4111yescharacteristic
147C2×C22⋊D59—21no—
148C2×C22⋊D59—21no—
149C2×C22⋊D59—21no—
150#11745685—81no—
151#11745685—81no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236C22×C411yescharacteristic
155C2×D236C22×C411yescharacteristic
156C2×D236C22×C411yescharacteristic
157C2×C4×D59—21no—
158C2×C4×D59—21no—
159C2×C4×D59—21no—
160C2×Dic118—21no—
161C2×Dic118C22×C411yescharacteristic
162M16×C59—22no—
163M16×C59—22no—
164M16×C59—81no—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C2×C8×C59C22×C411yescharacteristic
170C2×C8×C59C22×C411yescharacteristic
171C2×C8×C59C22×C411yescharacteristic
172C2×C8×C59C22×C411yescharacteristic
173C42×C59C22×C411yescharacteristic
174C42×C59C2×D411yescharacteristic
175C42×C59#4111yescharacteristic
176C42×C59C2411yescharacteristic
177C22⋊Dic59—21no—
178(C2×D59)⋊C4—21no—
179(C2×D59)⋊C4—21no—
180(C2×D59)⋊C4—21no—
181(C2×D59)⋊C4—21no—
182(C2×D59)⋊C4—21no—
183(C2×D59)⋊C4—21no—
184(C2×D59)⋊C4—21no—
185(C2×D59)⋊C4—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59C22×C411yescharacteristic
188C4⋊Dic59C22×C411yescharacteristic
189C4⋊Dic59C22×C411yescharacteristic
190C4⋊Dic59C22×C411yescharacteristic
191Dic59⋊C4—21no—
192Dic59⋊C4—21no—
193Dic59⋊C4—21no—
194Dic59⋊C4—21no—
195C4×Dic59—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—42no—
199C2×C59⋊C8C2×D411yescharacteristic
200C2×C59⋊C8C2×D411yescharacteristic
201C2×C59⋊C8C2×D411yescharacteristic
202C2×C59⋊C8C2×D411yescharacteristic
203D472—22no—
204D472—22no—
205#11745660—22no—
206#11745660—22no—
207#11745659—81no—
208#11745659—81no—
209#11745659—81no—
210C8×D59—81no—
211C8×D59—81no—
212C8×D59—81no—
213C23×C59C4×D411yescharacteristic
214C22×D59—21no—
215C22×D59—21no—
216C22×D59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—41no—
222C2×C4×C59C4×D411yescharacteristic
223C2×C4×C59C4×D411yescharacteristic
224C2×C4×C59C4×D411yescharacteristic
225C2×C4×C59C23×C411yescharacteristic
226C2×C4×C59C22×D411yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C22⋊D59—81no—
229C22⋊D59—81no—
230C22⋊D59—81no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236D236—21no—
237D236—21no—
238D236—21no—
239D236—21no—
240D236—21no—
241D236—21no—
242D236—81no—
243D236—81no—
244C4×D59—41no—
245C4×D59—41no—
246C4×D59—41no—
247C4×D59—41no—
248C4×D59—41no—
249C4×D59—41no—
250Dic118—21no—
251Dic118—21no—
252Dic118—81no—
253C8×C59—21no—
254C8×C59—21no—
255C8×C59—41no—
256C8×C59—41no—
257C8×C59—41no—
258C8×C59—41no—
259C8×C59#13112yes—
260C8×C59#13112yes—
261C59⋊C8—22no—
262C59⋊C8—21no—
263C59⋊C8—21no—
264C59⋊C8—21no—
265C59⋊C8—21no—
266C59⋊C8—41no—
267#17968—591nomaximal, Sylow
268C22×C59—21no—
269C22×C59—21no—
270C22×C59—21no—
271C22×C59C2×C4×D411yescharacteristic
272C2×D59—41no—
273C2×D59—41no—
274C2×D59—41no—
275C2×D59—41no—
276C2×D59—41no—
277C2×D59—41no—
278C2×D59—41no—
279C2×D59—41no—
280C2×D59—41no—
281C4×C59—21no—
282C4×C59—21no—
283C4×C59—21no—
284C4×C59—21no—
285C4×C59C2×#13111yescharacteristic
286C4×C59C2×#13111yescharacteristic
287C4×C59#57611yescharacteristic
288C4×C59#57711yescharacteristic
289Dic59—41no—
290Dic59—41no—
291Dic59—41no—
292Dic59—41no—
293Dic59—41no—
294#3171—591no—
295#2993—591no—
296#2981—591no—
297#1594—591no—
298#1592—591no—
299#1590—591no—
300#1589—591no—
301#1580—591no—
302SD16⋊C8—591no—
303C8×SD16—591no—
304C8×D8—591no—
305#1569D5911yescharacteristic, complemented
306#1494—591no—
307#1492—591no—
308C8×M16—591no—
309C2×C59—41no—
310C2×C59#293111yescharacteristic
311C2×C59#294611yescharacteristic
312C2×C59#295611yescharacteristic
313D59—81no—
314D59—81no—
315D59—81no—
316#532—591no—
317#529—591no—
318#501—591no—
319#496—591no—
320#490—591no—
321#489—591no—
322#469—591no—
323#469—591no—
324#468—591no—
325#468—591no—
326#446—591no—
327C8×Q8—591no—
328C4×SD16—591no—
329C4×SD16—591no—
330C4×D8—591no—
331C4×D8—591no—
332#436—591no—
333#436—591no—
334#436—591no—
335#436—1181no—
336#435—1181no—
337#435—1181no—
338#435—1181no—
339C8×D4—591no—
340C8×D4—591no—
341C8×D4—591no—
342#433—1181no—
343#432C2×D5911yescharacteristic
344#431—591no—
345#424—1181no—
346#423C2×D5911yescharacteristic
347#408—1181no—
348#408—1181no—
349C4×M16C2×D5911yescharacteristic
350C2×C4×C8—591no—
351#335C2×D5911yescharacteristic
352#335C2×D5911yescharacteristic
353#334C2×D5911yescharacteristic
354#334C2×D5911yescharacteristic
355Q8⋊C8—591no—
356Q8⋊C8—591no—
357D4⋊C8—591no—
358D4⋊C8—591no—
359D4⋊C8—591no—
360D4⋊C8—591no—
361D4⋊C8—591no—
362D4⋊C8—591no—
363#322—591no—
364#322—591no—
365C82—591no—
366C82—591no—
367C59#1796811yescharacteristic, complemented, Sylow
368C2×#41—1181no—
369C2×SD16—591no—
370C2×SD16—591no—
371C2×D8—591no—
372C2×D8—591no—
373C2×M16—21no—
374C2×M16—1181no—
375C2×M16—1181no—
376C2×M16—1181no—
377C2×M16C4×D5911yescharacteristic
378C2×M16C4×D5911yescharacteristic
379C22×C8—1181no—
380C22×C8—1181no—
381C22×C8—1181no—
382C22×C8—1181no—
383C4⋊Q8—591no—
384#127—591no—
385#126—1181no—
386#125—591no—
387#124—591no—
388#124—1181no—
389#123—1181no—
390C22⋊Q8—591no—
391#121—591no—
392#121—591no—
393#121—591no—
394#121—1181no—
395C4×Q8—591no—
396C4×D4—591no—
397C4×D4—591no—
398C4×D4—591no—
399C4×D4—1181no—
400C4×D4—1181no—
401#117—1181no—
402C2×C42C22×D5911yescharacteristic
403#107—591no—
404#107—591no—
405#106—591no—
406#106—591no—
407C4⋊C8—21no—
408C4⋊C8—591no—
409C4⋊C8—591no—
410C4⋊C8—1181no—
411C4⋊C8—1181no—
412C4⋊C8C22×D5911yescharacteristic
413C4⋊C8C22×D5911yescharacteristic
414C4⋊C8C22×D5911yescharacteristic
415C4⋊C8C22×D5911yescharacteristic
416#103—591no—
417#103—591no—
418#102—591no—
419#102—591no—
420#102—591no—
421#102—591no—
422#102—591no—
423#102—591no—
424C22⋊C8—21no—
425C22⋊C8—591no—
426C22⋊C8—591no—
427C22⋊C8—1181no—
428C22⋊C8—1181no—
429C22⋊C8—1181no—
430C22⋊C8—1181no—
431C22⋊C8—1181no—
432C22⋊C8—1181no—
433#97—21no—
434#97—1181no—
435#97C4×D5911yescharacteristic
436#97C4×D5911yescharacteristic
437C4×C8—591no—
438C4×C8—591no—
439C4×C8—591no—
440C4×C8—591no—
441C4×C8—1181no—
442C4×C8—1181no—
443C4×C8C22×D5911yescharacteristic
444C4×C8C22×D5911yescharacteristic
445#41—4721no—
446#41—4721no—
447C2×Q8—591no—
448C2×Q8—1181no—
449C2×D4—591no—
450C2×D4—591no—
451C2×D4—591no—
452C2×D4—1181no—
453C2×D4—1181no—
454C2×D4—1181no—
455C2×D4—1181no—
456C2×D4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4C2×C4×D5911yescharacteristic
461C22×C4D4×D5911yescharacteristic
462C22×C4#1174568911yescharacteristic
463SD16—1182no—
464SD16—1182no—
465D8—1182no—
466D8—1182no—
467M16—22no—
468M16—22no—
469M16—81no—
470M16—4721no—
471M16—4721no—
472M16—4721no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—591no—
478C2×C8—591no—
479C2×C8—591no—
480C2×C8—591no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—4721no—
484C2×C8—4721no—
485C2×C8—4721no—
486C2×C8—2362no—
487C2×C8C2×C4×D5911yescharacteristic
488C2×C8C2×C4×D5911yescharacteristic
489C2×C8C2×C4×D5911yescharacteristic
490C2×C8C2×C4×D5911yescharacteristic
491C4⋊C4—591no—
492C4⋊C4—591no—
493C4⋊C4—591no—
494C4⋊C4—591no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C22⋊C4—1181no—
501C22⋊C4—1181no—
502C22⋊C4—1181no—
503C22⋊C4—1181no—
504C22⋊C4—1181no—
505C22⋊C4—1181no—
506C22⋊C4—1181no—
507C22⋊C4—1181no—
508C22⋊C4—1181no—
509C42—1181no—
510C42C2×C4×D5911yescharacteristic
511C42D4×D5911yescharacteristic
512C42#1174568911yescharacteristic
513C42C23×D5911yescharacteristic
514C23—1181no—
515C23—1181no—
516C23—1181no—
517C23#1427368911yescharacteristic
518Q8—1181no—
519Q8—1181no—
520Q8—4721no—
521D4—1181no—
522D4—1181no—
523D4—1181no—
524D4—1181no—
525D4—1181no—
526D4—1181no—
527D4—4721no—
528D4—4721no—
529D4—4721no—
530D4—4721no—
531D4—4721no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—1181no—
538C2×C4—1181no—
539C2×C4—1181no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4#1427368911yescharacteristic
549C2×C4#1427368911yescharacteristic
550C2×C4#1427368911yescharacteristic
551C2×C4C22×C4×D5911yescharacteristic
552C2×C4C2×D4×D5911yescharacteristic
553C2×C4C2×#1174568911yescharacteristic
554C8—21no—
555C8—21no—
556C8—41no—
557C8—41no—
558C8—41no—
559C8—41no—
560C8—1182no—
561C8—1181no—
562C8—1181no—
563C8—1181no—
564C8—1181no—
565C8—2361no—
566C8#1427373512yes—
567C8#1427373512yes—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—2361no—
572C22—2361no—
573C22—2361no—
574C22—2361no—
575C22—2361no—
576C22—2361no—
577C22—2361no—
578C22—2361no—
579C22—2361no—
580C22C2×#1427368911yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—2361no—
586C4—2361no—
587C4—2361no—
588C4—2361no—
589C4—2361no—
590C4C2×#1427373511yescharacteristic
591C4C2×#1427373511yescharacteristic
592C4#1683820611yescharacteristic
593C4#1683821111yescharacteristic
594C2—41no—
595C2—4721no—
596C2—4721no—
597C2—4721no—
598C2#2143280711yescharacteristic
599C2#2143292611yescharacteristic
600C2#2143300511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1569C2
socleC22×C59C2×C4×D4
prime radical#21434691C2
cosocleC42×C59C24
hypercenter#1569D59
hypocenterC59#17968
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1569

Complements

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

NH ≅ G/Ntype
2. #21433241C2semidirect
3. #21433174C2semidirect
4. #21423009C2semidirect
5. #21422997C2semidirect
6. #21422985C2semidirect
7. #21422980C2semidirect
8. #21422919C2semidirect
9. #21422907C2semidirect
10. #21422895C2semidirect
11. #21422890C2semidirect
12. #21422554C2semidirect
13. #21422546C2semidirect
14. M16×C59⋊C8C2semidirect
15. C59×#1569C2semidirect
305. #1569D59semidirect
367. C59#17968semidirect

Maximal subgroups

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

subgroupindex
#214346912
#214332412
#214331742
#214230092
#214229972
#214229852
#214229802
#214229192
#214229072
#214228952
#214228902
#214225542
#214225462
M16×C59⋊C82
C59×#15692
#1796859