Finite Group Databasebeta

← #28591816

Subgroups — #28591816

Order 15104.

862 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837502C211yesmaximal, characteristic
2#21426739C211yesmaximal, characteristic
3#21426460C211yesmaximal, characteristic, complemented
4#21424726C211yesmaximal, characteristic
5#21424719C211yesmaximal, characteristic
6#21424691C211yesmaximal, characteristic, complemented
7#21424670C211yesmaximal, characteristic, complemented
8#21424448C211yesmaximal, characteristic
9#21424448C211yesmaximal, characteristic
10#21424365C211yesmaximal, characteristic, complemented
11#21424351C211yesmaximal, characteristic, complemented
12#21423994C211yesmaximal, characteristic, complemented
13C4×#14273652C211yesmaximal, characteristic, complemented
14#21423847C211yesmaximal, characteristic, complemented
15C4×#102×C59C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×#14273692C2211yescharacteristic
19C2×#14273692C2211yescharacteristic
20#16837502C418yescomplemented
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273652C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29#16837305C2211yescharacteristic
30#16837305C2211yescharacteristic
31#16837301C2211yescharacteristic
32#16837301C2211yescharacteristic
33#16837295—21no—
34#16837295—21no—
35#16837265—21no—
36#16837262—21no—
37C4⋊C4×Dic59C2211yescharacteristic
38C4⋊C4×Dic59C2211yescharacteristic
39C4×Dic59⋊C4—21no—
40C4×Dic59⋊C4—21no—
41C4×C4⋊Dic59C2211yescharacteristic
42#16837253—21no—
43#16837248C2211yescharacteristic
44#16837247—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837156C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×C4×D4×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C2×C4×C8×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C2×#11745670—22no—
62C2×#11745670—22no—
63C2×#11745670C2×C414yes—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C4×Dic118—22no—
67C4×Dic118C2×C414yes—
68#14273692—22no—
69#14273692—22no—
70#14273692—22no—
71#14273692—22no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×C22⋊Dic59—21no—
81C2×C22⋊Dic59—21no—
82C2×C22⋊Dic59—21no—
83C2×C22⋊Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59C2311yescharacteristic
91C2×C4×Dic59C2311yescharacteristic
92#14273660—22no—
93#14273660C2×C414yes—
94#14273658—22no—
95#14273652—22no—
96#14273652—22no—
97#14273650C2×C414yes—
98#14273648C2×C414yes—
99#14273638C2×C414yes—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95D411yescharacteristic
106C59⋊#95D411yescharacteristic
107C59⋊#95D411yescharacteristic
108C59⋊#95D411yescharacteristic
109C22×D4×C59C2311yescharacteristic
110C23×C4×C59—21no—
111C22×C8×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C4×D4×C59—22no—
114C4×D4×C59C2×C414yes—
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C22⋊C4×C59—21no—
119C2×C42×C59—21no—
120C2×C42×C59C2311yescharacteristic
121#102×C59—22no—
122#102×C59—22no—
123C4×C8×C59—22no—
124#95×C59—21no—
125C24×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59C22×C412yes—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—24no—
136C22×C4×C59—81no—
137C22×C4×C59—81no—
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2411yescharacteristic
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59C2×D411yescharacteristic
146C22×Dic59C2×D411yescharacteristic
147C22×Dic59C2×D411yescharacteristic
148C22×Dic59C2×D411yescharacteristic
149C2×Dic118—21no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118C22×C412yes—
153C2×Dic118#4112yes—
154C2×Dic118#4112yes—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59C22×C412yes—
165C4⋊C4×C59#4112yes—
166C4⋊C4×C59#4112yes—
167C22⋊C4×C59—24no—
168C42×C59—81no—
169C42×C59C22×C412yes—
170C42×C59#4112yes—
171C22⋊Dic59—81no—
172C22⋊Dic59—81no—
173C22⋊Dic59—81no—
174C22⋊Dic59—81no—
175#11745670—28no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—81no—
182C4⋊Dic59—81no—
183C4⋊Dic59C22×C412yes—
184Dic59⋊C4—24no—
185Dic59⋊C4—24no—
186Dic59⋊C4—24no—
187C4×Dic59—24no—
188C4×Dic59—24no—
189C4×Dic59—24no—
190C4×Dic59—21no—
191C4×Dic59—21no—
192C4×Dic59—21no—
193C4×Dic59—21no—
194C4×Dic59—81no—
195C4×Dic59—81no—
196C4×Dic59—81no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8C22×C412yes—
202C2×C59⋊C8C22×C412yes—
203C23×C59—22no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59C22×D411yescharacteristic
211D4×C59—22no—
212D4×C59—22no—
213D4×C59—41no—
214D4×C59—41no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—84no—
229C2×C4×C59C4×D412yes—
230C2×C4×C59C4×D412yes—
231C2×C4×C59C23×C411yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×Dic59—22no—
238C2×Dic59—22no—
239C2×Dic59—22no—
240C2×Dic59—22no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—21no—
252C2×Dic59—21no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261Dic118—22no—
262Dic118—22no—
263Dic118—22no—
264Dic118#11714yes—
265C8×C59—41no—
266C8×C59—41no—
267C59⋊C8—22no—
268C59⋊C8—22no—
269#21128—591nomaximal, Sylow
270C22×C59—42no—
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—81no—
274C22×C59—81no—
275C22×C59—81no—
276C22×C59—81no—
277C22×C59C2×C4×D411yescharacteristic
278C22×C59#52511yescharacteristic
279C22×C59#52511yescharacteristic
280C22×C59#57611yescharacteristic
281C22×C59#57711yescharacteristic
282C22×C59#57711yescharacteristic
283C22×C59#57811yescharacteristic
284C4×C59—22no—
285C4×C59—22no—
286C4×C59—42no—
287C4×C59—42no—
288C4×C59—42no—
289C4×C59C2×#11711yescharacteristic
290C4×C59C4×#4111yescharacteristic
291C4×C59#52911yescharacteristic
292C4×C59#52911yescharacteristic
293Dic59—24no—
294Dic59—41no—
295Dic59—41no—
296Dic59—41no—
297Dic59—41no—
298Dic59—42no—
299Dic59—81no—
300Dic59—81no—
301C2×#440—591no—
302#2319—591no—
303#2276—591no—
304#1926—591no—
305#1925—591no—
306#1921—591no—
307#1918—591no—
308#1880—591no—
309#1880—591no—
310#1865—591no—
311#1863—591no—
312#1776—591no—
313C4×#103—591no—
314C4×#102D5911yescharacteristic
315#1746—591no—
316C2×C59—82no—
317C2×C59#229011yescharacteristic
318C2×C59#294611yescharacteristic
319C2×C59#294711yescharacteristic
320C2×C59#311911yescharacteristic
321C2×C59#312111yescharacteristic
322C2×C59#312411yescharacteristic
323C2×C59#312611yescharacteristic
324C22×SD16—591no—
325C2×C4×Q8—591no—
326C2×C4×Q8—591no—
327C2×C4×Q8—591no—
328C2×C4×D4C2×D5911yescharacteristic
329#440—598no—
330C2×#107—591no—
331C2×#107—591no—
332C2×#106—591no—
333C2×#103—591no—
334C2×#103—591no—
335C2×#103—591no—
336C2×#102—591no—
337C2×#102C2×D5911yescharacteristic
338C2×#102C2×D5911yescharacteristic
339C2×#97—591no—
340C2×C4×C8C2×D5911yescharacteristic
341#390—591no—
342#390—591no—
343#389—591no—
344#389—591no—
345#388—1181no—
346#388—1181no—
347#381—1181no—
348#380—1181no—
349C4×C4⋊C4—591no—
350C4×C4⋊C4—591no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—1181no—
353C4×C4⋊C4—1181no—
354C4×C4⋊C4—1181no—
355C4×C4⋊C4C2×D5911yescharacteristic
356#376—591no—
357#376—1181no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340C2×D5911yescharacteristic
363#340C2×D5911yescharacteristic
364#336—591no—
365#336—591no—
366#336—591no—
367#336—591no—
368C59#2112811yescharacteristic, complemented, Sylow
369C22×Q8—591no—
370C22×D4C22×D5911yescharacteristic
371C23×C4—21no—
372C2×SD16—594no—
373C2×SD16—1182no—
374C2×SD16—1182no—
375C22×C8—591no—
376C22×C8—591no—
377C22×C8C22×D5911yescharacteristic
378C22×C8C22×D5911yescharacteristic
379C4×Q8—594no—
380C4×Q8—1182no—
381C4×Q8—1182no—
382C4×Q8—1182no—
383C4×Q8—1182no—
384C4×Q8—1182no—
385C4×D4—22no—
386C4×D4C4×D5914yes—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4C22×D5911yescharacteristic
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—1181no—
400C2×C22⋊C4—1181no—
401C2×C22⋊C4—1181no—
402C2×C22⋊C4—1181no—
403C2×C42—21no—
404C2×C42—591no—
405C2×C42—591no—
406C2×C42—1181no—
407C2×C42—1181no—
408C2×C42—1181no—
409C2×C42—1181no—
410C2×C42—1181no—
411C2×C42—1181no—
412C2×C42C22×D5911yescharacteristic
413#107—594no—
414#107—1182no—
415#106—1182no—
416#103—594no—
417#103—1182no—
418#103—1182no—
419#102—22no—
420#102—22no—
421#102—594no—
422#97—594no—
423C4×C8—22no—
424#95—21no—
425#95—591no—
426#95—591no—
427#95—591no—
428#95—591no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434C24—21no—
435C2×Q8—592no—
436C2×Q8—592no—
437C2×Q8—592no—
438C2×Q8—1181no—
439C2×Q8—1181no—
440C2×Q8—1181no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4C2×C4×D5912yes—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4—591no—
454C22×C4—591no—
455C22×C4—591no—
456C22×C4—591no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4D4×D5911yescharacteristic
462C22×C4D4×D5911yescharacteristic
463C22×C4C23×D5911yescharacteristic
464SD16—1188no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—592no—
472C2×C8—592no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—592no—
481C4⋊C4—1184no—
482C4⋊C4—1184no—
483C4⋊C4—1184no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—4721no—
490C4⋊C4—4721no—
491C4⋊C4C2×C4×D5912yes—
492C4⋊C4#1174568512yes—
493C4⋊C4#1174568512yes—
494C22⋊C4—24no—
495C22⋊C4—4721no—
496C22⋊C4—4721no—
497C22⋊C4—4721no—
498C22⋊C4—4721no—
499C42—81no—
500C42—1184no—
501C42—1184no—
502C42—1184no—
503C42—1181no—
504C42—1181no—
505C42—1181no—
506C42—1181no—
507C42—4721no—
508C42—4721no—
509C42—4721no—
510C42C2×C4×D5912yes—
511C42#1174568712yes—
512C23—22no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23C2×D4×D5911yescharacteristic
520Q8—594no—
521Q8—1182no—
522Q8—1182no—
523Q8—1182no—
524D4—22no—
525D4—22no—
526D4—41no—
527D4—41no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—84no—
542C2×C4—1182no—
543C2×C4—1182no—
544C2×C4—1182no—
545C2×C4—1181no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1181no—
557C2×C4—1182no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4(C22⋊D59)⋊C412yes—
567C2×C4(C22⋊D59)⋊C412yes—
568C2×C4C22×C4×D5911yescharacteristic
569C2×C4#41×D5911yescharacteristic
570C2×C4#41×D5911yescharacteristic
571C2×C4C2×#1174568711yescharacteristic
572C2×C4C2×#1174568511yescharacteristic
573C2×C4C2×#1174568511yescharacteristic
574C8—41no—
575C8—41no—
576C8—1182no—
577C8—1182no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×(C22⋊D59)⋊C411yescharacteristic
586C22#1683791111yescharacteristic
587C22#1683791111yescharacteristic
588C22#1683820311yescharacteristic
589C22#1683820811yescharacteristic
590C22#1683820911yescharacteristic
591C22#1683821611yescharacteristic
592C4—22no—
593C4—22no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—1184no—
598C4—4721no—
599C4—4721no—
600C4—2361no—
601C4—2362no—
602C4—2361no—
603C4—2361no—
604C4—2361no—
605C4#117×D5911yescharacteristic
606C4C4×#1174568511yescharacteristic
607C4#1683793411yescharacteristic
608C4#1683793411yescharacteristic
609C2—82no—
610C2#2142654211yescharacteristic
611C2#2143292211yescharacteristic
612C2#2143294211yescharacteristic
613C2#2143429111yescharacteristic
614C2#2143432111yescharacteristic
615C2#2143435411yescharacteristic
616C2#2143438411yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC4×#102×C59C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenterC4×#102D59
hypocenterC59#21128
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
3. #21426460C2semidirect
6. #21424691C2semidirect
7. #21424670C2semidirect
10. #21424365C2semidirect
11. #21424351C2semidirect
12. #21423994C2semidirect
13. C4×#14273652C2semidirect
14. #21423847C2semidirect
20. #16837502C4semidirect
368. C59#21128semidirect

Maximal subgroups

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

subgroupindex
C2×#168375022
#214267392
#214264602
#214247262
#214247192
#214246912
#214246702
#214244482
#214243652
#214243512
#214239942
C4×#142736522
#214238472
C4×#102×C592
#2112859