Finite Group Databasebeta

← #28559917

Subgroups — #28559917

Order 15104.

642 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434145C211yesmaximal, characteristic
2#21433252C211yesmaximal, characteristic, complemented
3#21433133C211yesmaximal, characteristic, complemented
4#21423009C211yesmaximal, characteristic, complemented
5#21422997C211yesmaximal, characteristic, complemented
6#21422938C211yesmaximal, characteristic, complemented
7#21422931C211yesmaximal, characteristic, complemented
8#21422924C211yesmaximal, characteristic, complemented
9#21422924C211yesmaximal, characteristic, complemented
10#21422919C211yesmaximal, characteristic, complemented
11#21422907C211yesmaximal, characteristic, complemented
12#21422554C211yesmaximal, characteristic, complemented
13C2×#16837129C211yesmaximal, characteristic, complemented
14#21422389C211yesmaximal, characteristic, complemented
15C59×#1573C211yesmaximal, characteristic, complemented
16#16837936C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18#16837738C411yescharacteristic
19#16837721C411yescharacteristic
20#16837703C411yescharacteristic
21#16837703C411yescharacteristic
22#16837618C411yescharacteristic
23#16837618C411yescharacteristic
24#16837555C411yescharacteristic
25#16837538C411yescharacteristic
26#16837529C2211yescharacteristic, complemented
27#16837526C2211yescharacteristic, complemented
28C4×#11745660C2211yescharacteristic
29C4×#11745660C2211yescharacteristic
30C4×D472C2211yescharacteristic
31C4×D472C2211yescharacteristic
32#16837479C2211yescharacteristic
33#16837479C2211yescharacteristic
34#16837479C2211yescharacteristic, complemented
35#16837475—21no—
36#16837475—21no—
37#16837473—21no—
38#16837467C2211yescharacteristic
39#16837467C2211yescharacteristic
40#16837467C2211yescharacteristic, complemented
41C8×C22⋊D59—21no—
42#16837462—21no—
43#16837455C2211yescharacteristic
44#16837427—21no—
45#16837364—21no—
46C22⋊C8×D59—21no—
47C2×C4×C59⋊C8C2211yescharacteristic
48#16837132C2211yescharacteristic, complemented
49#16837132C2211yescharacteristic, complemented
50#16837129C2211yescharacteristic
51#16837129C2211yescharacteristic
52#16837129C2211yescharacteristic
53#16837129C2211yescharacteristic
54#16837129C2211yescharacteristic, complemented
55#16837129C2211yescharacteristic, complemented
56#16837124C2211yescharacteristic, complemented
57#16837124C2211yescharacteristic, complemented
58#16837123C2211yescharacteristic, complemented
59#16837123C2211yescharacteristic, complemented
60C59×#432C2211yescharacteristic
61C59×#431C2211yescharacteristic
62C2×C4⋊C8×C59C2211yescharacteristic
63C59×#335C2211yescharacteristic, complemented
64C59×#335C2211yescharacteristic, complemented
65C59×#334C2211yescharacteristic, complemented
66C59×#334C2211yescharacteristic, complemented
67C22×D236C2×C411yescharacteristic
68C22×C4×D59—21no—
69C2×#11745660C2×C411yescharacteristic
70C2×#11745660C2×C411yescharacteristic
71C2×D472C2×C411yescharacteristic
72C2×D472C2×C411yescharacteristic
73C2×#11745659—21no—
74C2×#11745659—21no—
75C2×#11745659—21no—
76C22×C59⋊C8—21no—
77C2×C8×D59—21no—
78C2×C8×D59—21no—
79C2×C8×D59—21no—
80#14273721—21no—
81#14273719C2×C411yescharacteristic
82#14273715C2×C411yescharacteristic
83#14273710—21no—
84#14273706C2×C411yescharacteristic
85#14273701C2×C411yescharacteristic
86C4×Dic118C2311yescharacteristic
87C4×D236—21no—
88C4×D236—21no—
89C4×D236C2×C411yescharacteristic
90C4×D236C2×C411yescharacteristic
91C4×D236C2311yescharacteristic
92C4×D236C2311yescharacteristic
93C4×D236C2311yescharacteristic
94C4×C22⋊D59—21no—
95#14273684—21no—
96C2×C4⋊Dic59C2×C411yescharacteristic
97C2×(C2×D59)⋊C4—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—
129C22×C8×C59—21no—
130C2×C42×C59C2311yescharacteristic
131C4⋊C8×C59—21no—
132C4⋊C8×C59C2×C411yescharacteristic
133C4⋊C8×C59C2×C411yescharacteristic
134C4⋊C8×C59C2311yescharacteristic
135C4⋊C8×C59C2311yescharacteristic
136C4⋊C8×C59C2311yescharacteristic
137C4⋊C8×C59C2311yescharacteristic
138C22⋊C8×C59—21no—
139C22⋊C8×C59C2×C411yescharacteristic
140C22⋊C8×C59C2×C411yescharacteristic
141#97×C59—21no—
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C23×D59—21no—
145C22×C4×C59C22×C411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59#4111yescharacteristic
148C2×C22⋊D59—21no—
149C22×Dic59—21no—
150C2×D236—22no—
151C2×D236—21no—
152C2×D236—41no—
153C2×D236C22×C411yescharacteristic
154C2×D236C22×C411yescharacteristic
155C2×D236C22×C411yescharacteristic
156C2×C4×D59—21no—
157C2×C4×D59—21no—
158C2×C4×D59—21no—
159C2×C4×D59—21no—
160C2×C4×D59—21no—
161C2×C4×D59—81no—
162C2×C4×D59—81no—
163C2×Dic118C22×C411yescharacteristic
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59—81no—
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—41no—
205#11745660—22no—
206#11745660—41no—
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—
217C22×D59—42no—
218C22×D59—42no—
219C22×D59—81no—
220C2×C4×C59—22no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59C4×D411yescharacteristic
226C2×C4×C59C4×D411yescharacteristic
227C2×C4×C59C4×D411yescharacteristic
228C2×C4×C59C23×C411yescharacteristic
229C2×C4×C59C22×D411yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C22⋊D59—42no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—81no—
238D236—21no—
239D236—21no—
240D236—21no—
241D236—21no—
242D236—21no—
243D236—42no—
244D236#13112yes—
245C4×D59—41no—
246C4×D59—41no—
247C4×D59—41no—
248C4×D59—41no—
249C4×D59—41no—
250C4×D59—41no—
251C4×D59—81no—
252C4×D59—81no—
253Dic118—21no—
254Dic118—21no—
255C8×C59—21no—
256C8×C59—21no—
257C8×C59—21no—
258C8×C59—21no—
259C8×C59—41no—
260C8×C59—41no—
261C8×C59—41no—
262C8×C59—41no—
263C59⋊C8—21no—
264C59⋊C8—21no—
265C59⋊C8—21no—
266C59⋊C8—21no—
267C59⋊C8—41no—
268C59⋊C8—41no—
269#18020—591nomaximal, Sylow
270C22×C59—21no—
271C22×C59—21no—
272C22×C59#44412yes—
273C22×C59C2×C4×D411yescharacteristic
274C2×D59—22no—
275C2×D59—41no—
276C2×D59—41no—
277C2×D59—41no—
278C2×D59—41no—
279C2×D59—41no—
280C2×D59—41no—
281C2×D59—41no—
282C2×D59—41no—
283C2×D59—84no—
284C4×C59—21no—
285C4×C59—21no—
286C4×C59—21no—
287C4×C59—21no—
288C4×C59C2×#13111yescharacteristic
289C4×C59#56811yescharacteristic
290C4×C59#57611yescharacteristic
291C4×C59#57711yescharacteristic
292Dic59—41no—
293Dic59—41no—
294Dic59—41no—
295Dic59—41no—
296Dic59—41no—
297#3105—591no—
298#2994—591no—
299#2974—591no—
300#1594—591no—
301#1592—591no—
302#1583—591no—
303#1582—591no—
304#1581—591no—
305#1581—591no—
306#1580—591no—
307SD16⋊C8—591no—
308#1573D5911yescharacteristic, complemented
309#1494—591no—
310C2×D4⋊C8—591no—
311#1452—591no—
312C2×C59—22no—
313C2×C59#293311yescharacteristic
314C2×C59#294611yescharacteristic
315C2×C59C2×#44411yescharacteristic
316D59—42no—
317D59—81no—
318D59—81no—
319#529—591no—
320C2×C4×D4—591no—
321#484—591no—
322#481—591no—
323#478—591no—
324#478—591no—
325#463—591no—
326#463—591no—
327#450—591no—
328#447—591no—
329#446—591no—
330C8×Q8—591no—
331C4×SD16—591no—
332C4×SD16—591no—
333C4×D8—591no—
334C4×D8—591no—
335#436—591no—
336#436—591no—
337#436—591no—
338#435—1181no—
339#435—1181no—
340#435—1181no—
341C8×D4—591no—
342C8×D4—591no—
343C8×D4—591no—
344C8×D4—1181no—
345#433—1181no—
346#432C2×D5911yescharacteristic
347#431—591no—
348#431C2×D5911yescharacteristic
349#424—1181no—
350C2×C4⋊C8C2×D5911yescharacteristic
351#407—1181no—
352C2×C22⋊C8—1181no—
353C2×C4×C8—591no—
354#335C2×D5911yescharacteristic, complemented
355#335C2×D5911yescharacteristic, complemented
356#334C2×D5911yescharacteristic, complemented
357#334C2×D5911yescharacteristic, complemented
358Q8⋊C8—591no—
359Q8⋊C8—591no—
360D4⋊C8—591no—
361D4⋊C8—591no—
362D4⋊C8—591no—
363D4⋊C8—591no—
364D4⋊C8—591no—
365D4⋊C8—591no—
366#322—591no—
367#322—591no—
368#322—591no—
369#322—591no—
370C59#1802011yescharacteristic, complemented, Sylow
371C22×D4—591no—
372C23×C4—1181no—
373C2×SD16—591no—
374C2×SD16—591no—
375C2×D8—591no—
376C2×D8—591no—
377C2×M16—1181no—
378C2×M16—1181no—
379C2×M16—1181no—
380C22×C8—21no—
381C22×C8—1181no—
382C22×C8—1181no—
383C22×C8—1181no—
384C22×C8—1181no—
385#126—1181no—
386#125—591no—
387#124—591no—
388#123—1181no—
389C22⋊Q8—591no—
390#121—591no—
391C4×Q8—591no—
392C4×D4—591no—
393C4×D4—591no—
394C4×D4—591no—
395C4×D4—591no—
396C4×D4—591no—
397C4×D4—1181no—
398C4×D4—1181no—
399C4×D4—1181no—
400#117—1181no—
401C2×C4⋊C4—591no—
402C2×C22⋊C4—1181no—
403C2×C42C22×D5911yescharacteristic
404#107—591no—
405#107—591no—
406#106—591no—
407#106—591no—
408C4⋊C8—21no—
409C4⋊C8—591no—
410C4⋊C8—591no—
411C4⋊C8—1181no—
412C4⋊C8—1181no—
413C4⋊C8C4×D5911yescharacteristic
414C4⋊C8C4×D5911yescharacteristic
415C4⋊C8C22×D5911yescharacteristic
416C4⋊C8C22×D5911yescharacteristic
417C4⋊C8C22×D5911yescharacteristic
418C4⋊C8C22×D5911yescharacteristic
419#103—591no—
420#103—591no—
421#102—591no—
422#102—591no—
423#102—591no—
424#102—591no—
425#102—591no—
426#102—591no—
427C22⋊C8—21no—
428C22⋊C8—591no—
429C22⋊C8—591no—
430C22⋊C8—1181no—
431C22⋊C8—1181no—
432C22⋊C8—1181no—
433C22⋊C8—1181no—
434C22⋊C8—1181no—
435C22⋊C8—1181no—
436C22⋊C8C4×D5911yescharacteristic
437C22⋊C8C4×D5911yescharacteristic
438#97—21no—
439#97—1181no—
440C4×C8—591no—
441C4×C8—591no—
442C4×C8—591no—
443C4×C8—591no—
444C4×C8—1181no—
445C4×C8—1181no—
446C4×C8C22×D5911yescharacteristic
447C4×C8C22×D5911yescharacteristic
448C24—1181no—
449C2×Q8—591no—
450C2×D4—591no—
451C2×D4—591no—
452C2×D4—591no—
453C2×D4—1182no—
454C2×D4—1181no—
455C2×D4—1181no—
456C2×D4—2361no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4—1181no—
463C22×C4—4721no—
464C22×C4—4721no—
465C22×C4C2×C4×D5911yescharacteristic
466C22×C4D4×D5911yescharacteristic
467C22×C4#1174568911yescharacteristic
468SD16—1182no—
469SD16—2361no—
470D8—1182no—
471D8—2361no—
472M16—4721no—
473M16—4721no—
474M16—4721no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—81no—
480C2×C8—591no—
481C2×C8—591no—
482C2×C8—591no—
483C2×C8—591no—
484C2×C8—1181no—
485C2×C8—1181no—
486C2×C8—4721no—
487C2×C8—4721no—
488C2×C8—4721no—
489C2×C8—2362no—
490C2×C8C2×C4×D5911yescharacteristic
491C2×C8C2×C4×D5911yescharacteristic
492C2×C8C2×C4×D5911yescharacteristic
493C2×C8C2×C4×D5911yescharacteristic
494C4⋊C4—591no—
495C4⋊C4—591no—
496C4⋊C4—591no—
497C4⋊C4—591no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C4⋊C4—1181no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
503C22⋊C4—1181no—
504C22⋊C4—1181no—
505C22⋊C4—1181no—
506C22⋊C4—1181no—
507C22⋊C4—1181no—
508C22⋊C4—1181no—
509C22⋊C4—1181no—
510C22⋊C4—1181no—
511C22⋊C4—1181no—
512C42—1181no—
513C42C2×C4×D5911yescharacteristic
514C42D4×D5911yescharacteristic
515C42#1174568911yescharacteristic
516C42C23×D5911yescharacteristic
517C23—1181no—
518C23—1181no—
519C23—1181no—
520C23—4721no—
521C23—2362no—
522C23—2362no—
523C23#1427368911yescharacteristic
524Q8—1181no—
525Q8—1181no—
526D4—592no—
527D4—1181no—
528D4—1181no—
529D4—1181no—
530D4—1181no—
531D4—1181no—
532D4—2362no—
533D4—2362no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—1181no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—4721no—
545C2×C4—4721no—
546C2×C4—4721no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4#1427368911yescharacteristic
554C2×C4#1427368911yescharacteristic
555C2×C4#1427368911yescharacteristic
556C2×C4C22×C4×D5911yescharacteristic
557C2×C4C2×D4×D5911yescharacteristic
558C2×C4C2×#1174568911yescharacteristic
559C8—21no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—1181no—
568C8—1181no—
569C8—1181no—
570C8—1181no—
571C8—2361no—
572C8—2361no—
573C22—21no—
574C22—21no—
575C22—1182no—
576C22—4724no—
577C22—2361no—
578C22—2361no—
579C22—2361no—
580C22—2361no—
581C22—2361no—
582C22—2361no—
583C22—2361no—
584C22—2361no—
585C22#1683752312yes—
586C22C2×#1427368911yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—2361no—
592C4—2361no—
593C4—2361no—
594C4—2361no—
595C4—2361no—
596C4#131×D5911yescharacteristic
597C4#1683816711yescharacteristic
598C4#1683820611yescharacteristic
599C4#1683821111yescharacteristic
600C2—22no—
601C2—4721no—
602C2—4721no—
603C2—2362no—
604C2#2143282011yescharacteristic
605C2#2143292611yescharacteristic
606C2C2×#1683752311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1573C2
socleC22×C59C2×C4×D4
prime radical#16837538C4
cosocleC42×C59C24
hypercenter#1573D59
hypocenterC59#18020
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1573

Complements

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

NH ≅ G/Ntype
2. #21433252C2semidirect
3. #21433133C2semidirect
4. #21423009C2semidirect
5. #21422997C2semidirect
6. #21422938C2semidirect
7. #21422931C2semidirect
8. #21422924C2semidirect
9. #21422924C2semidirect
10. #21422919C2semidirect
11. #21422907C2semidirect
12. #21422554C2semidirect
13. C2×#16837129C2semidirect
14. #21422389C2semidirect
15. C59×#1573C2semidirect
26. #16837529C22semidirect
27. #16837526C22semidirect
34. #16837479C22semidirect
40. #16837467C22semidirect
48. #16837132C22semidirect
49. #16837132C22semidirect
54. #16837129C22semidirect
55. #16837129C22semidirect
56. #16837124C22semidirect
57. #16837124C22semidirect
58. #16837123C22semidirect
59. #16837123C22semidirect
63. C59×#335C22semidirect
64. C59×#335C22semidirect
65. C59×#334C22semidirect
66. C59×#334C22semidirect
308. #1573D59semidirect
354. #335C2×D59semidirect
355. #335C2×D59semidirect
356. #334C2×D59semidirect
357. #334C2×D59semidirect
370. C59#18020semidirect

Maximal subgroups

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

subgroupindex
#214341452
#214332522
#214331332
#214230092
#214229972
#214229382
#214229312
#214229242
#214229192
#214229072
#214225542
C2×#168371292
#214223892
C59×#15732
#1802059