Finite Group Databasebeta

← #28593028

Subgroups — #28593028

Order 15104.

874 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837504C211yesmaximal, characteristic
2#21426573C211yesmaximal, characteristic, complemented
3#21426560C211yesmaximal, characteristic
4#21424977C211yesmaximal, characteristic
5#21424950C211yesmaximal, characteristic
6#21424834C211yesmaximal, characteristic, complemented
7#21424829C211yesmaximal, characteristic, complemented
8#21424793C211yesmaximal, characteristic
9#21424786C211yesmaximal, characteristic, complemented
10#21424779C211yesmaximal, characteristic
11#21424772C211yesmaximal, characteristic, complemented
12#21423993C211yesmaximal, characteristic, complemented
13#21423927C211yesmaximal, characteristic, complemented
14#21423922C211yesmaximal, characteristic
15C4×#97×C59C211yesmaximal, characteristic, complemented
16C22×#11745660C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19C2×#14273684—21no—
20#16837504C418yescomplemented
21C2×#14273661C2211yescharacteristic
22C2×#14273661C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273652C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273652C2211yescharacteristic
27C2×#14273649C2211yescharacteristic
28C2×#14273649C2211yescharacteristic
29C2×#14273649C2211yescharacteristic
30#16837308C2211yescharacteristic
31#16837308C2211yescharacteristic
32#16837304C2211yescharacteristic
33#16837304C2211yescharacteristic
34#16837300C2211yescharacteristic
35#16837300C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837293—21no—
39#16837293—21no—
40#16837286—21no—
41#16837286—21no—
42C4×Dic59⋊C4—21no—
43C4×C4⋊Dic59C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45C4×(C2×D59)⋊C4—21no—
46#16837248—21no—
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53C2×#97×C59C2211yescharacteristic
54C2×#97×C59C2211yescharacteristic
55C2×#97×C59C2211yescharacteristic
56C2×#97×C59C2211yescharacteristic
57C2×C4×C8×C59C2211yescharacteristic
58C2×C4×C8×C59C2211yescharacteristic
59C43×C59C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C22×D236C2311yescharacteristic
62C22×C4×D59—21no—
63C2×#11745660—22no—
64C2×#11745660—22no—
65C2×#11745660C2×C414yes—
66C4×Dic118—22no—
67C4×Dic118C2×C414yes—
68C4×D236—22no—
69C4×D236C2×C414yes—
70#14273684—44no—
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86C2×(C2×D59)⋊C4—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90#14273661—22no—
91#14273661C2×C414yes—
92#14273659—22no—
93#14273652—22no—
94#14273652—22no—
95#14273652C2×C414yes—
96#14273649—22no—
97#14273649—22no—
98#14273649C2×C414yes—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C22×C8×C59C2311yescharacteristic
110C22×C8×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C2×C42×C59D411yescharacteristic
114C2×C42×C59D411yescharacteristic
115C2×C42×C59D411yescharacteristic
116C2×C42×C59D411yescharacteristic
117C2×C42×C59C2311yescharacteristic
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120#97×C59—22no—
121#97×C59—22no—
122#97×C59—22no—
123#97×C59C2×C414yes—
124C4×C8×C59—22no—
125C4×C8×C59—22no—
126C23×D59—21no—
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2411yescharacteristic
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C2×D236—21no—
142C2×D236—21no—
143C2×D236—21no—
144C2×D236—21no—
145C2×D236—21no—
146C2×D236C22×C412yes—
147C2×C4×D59—24no—
148C2×C4×D59—42no—
149C2×C4×D59—42no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118C22×C412yes—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59C22×C412yes—
163C2×C8×C59C22×C412yes—
164C2×C8×C59#4112yes—
165C2×C8×C59#4112yes—
166C2×C8×C59#4112yes—
167C2×C8×C59#4112yes—
168C42×C59—22no—
169C42×C59—22no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—41no—
175C42×C59—41no—
176C42×C59C22×C412yes—
177C42×C59#4112yes—
178(C2×D59)⋊C4—24no—
179(C2×D59)⋊C4—42no—
180(C2×D59)⋊C4—42no—
181(C2×D59)⋊C4—42no—
182(C2×D59)⋊C4—42no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59C22×C412yes—
194C4⋊Dic59C22×C412yes—
195Dic59⋊C4—24no—
196Dic59⋊C4—42no—
197Dic59⋊C4—42no—
198Dic59⋊C4—42no—
199Dic59⋊C4—42no—
200C4×Dic59—24no—
201C4×Dic59—42no—
202C4×Dic59—42no—
203#11745660—28no—
204C23×C59C22×D411yescharacteristic
205C22×D59—22no—
206C22×D59—41no—
207C22×D59—41no—
208C22×D59—41no—
209C22×D59—41no—
210C22×D59—41no—
211C22×D59—41no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59C4×D412yes—
227C2×C4×C59C4×D412yes—
228C2×C4×C59AGL(1,ℤ/8)12yes—
229C2×C4×C59#13712yes—
230C2×C4×C59C23×C411yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×Dic59—22no—
237C2×Dic59—22no—
238C2×Dic59—22no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257D236—22no—
258D236—22no—
259D236—41no—
260D236—41no—
261C4×D59—48no—
262Dic118—22no—
263Dic118—22no—
264Dic118—41no—
265Dic118—41no—
266C8×C59—22no—
267C8×C59—22no—
268C8×C59—22no—
269C8×C59#11714yescomplemented
270#21288—591nomaximal, Sylow
271C22×C59C2×C4×D411yescharacteristic
272C22×C59#53211yescharacteristic
273C22×C59#53211yescharacteristic
274C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C59C2×#13711yescharacteristic
276C22×C59#57511yescharacteristic
277C22×C59#57511yescharacteristic
278C2×D59—42no—
279C2×D59—42no—
280C2×D59—42no—
281C2×D59—81no—
282C2×D59—81no—
283C2×D59—81no—
284C2×D59—81no—
285C4×C59—22no—
286C4×C59—22no—
287C4×C59—22no—
288C4×C59—22no—
289C4×C59—41no—
290C4×C59—41no—
291C4×C59C2×#11711yescharacteristic
292C4×C59C4×#4111yescharacteristic
293C4×C59#52911yescharacteristic
294C4×C59#52911yescharacteristic
295Dic59—42no—
296Dic59—42no—
297Dic59—42no—
298Dic59—81no—
299Dic59—81no—
300Dic59—81no—
301Dic59—81no—
302C2×#440—591no—
303#2295—591no—
304#2293—591no—
305#1973—591no—
306#1968—591no—
307#1943—591no—
308#1942—591no—
309#1936—591no—
310#1935—591no—
311#1934—591no—
312#1933—591no—
313#1776—591no—
314#1763—591no—
315#1762—591no—
316C4×#97D5911yescharacteristic, complemented
317C2×C59#229311yescharacteristic
318C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
319C2×C59C4×#13711yescharacteristic
320C2×C59#318011yescharacteristic
321C2×C59#318111yescharacteristic
322C2×C59#318211yescharacteristic
323C2×C59#318311yescharacteristic
324D59—82no—
325C22×SD16—591no—
326C2×C4×Q8—591no—
327C2×C4×D4—591no—
328C2×#117—1181no—
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×#102—591no—
338C2×#102—591no—
339C2×#97C2×D5911yescharacteristic
340C2×#97C2×D5911yescharacteristic
341C2×#97C2×D5911yescharacteristic
342C2×#97C2×D5911yescharacteristic
343C2×C4×C8C2×D5911yescharacteristic
344C2×C4×C8C2×D5911yescharacteristic
345#391—591no—
346#391—591no—
347#390—591no—
348#390—591no—
349#389—591no—
350#389—591no—
351#389—591no—
352#389—591no—
353#388—1181no—
354#388—1181no—
355#387—1181no—
356#387—1181no—
357C4×C4⋊C4—591no—
358C4×C4⋊C4—591no—
359C4×C4⋊C4—1181no—
360C4×C22⋊C4—1181no—
361#376—1181no—
362C43C2×D5911yescharacteristic
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369C59#2128811yescharacteristic, complemented, Sylow
370C22×Q8—591no—
371C22×D4—591no—
372C23×C4—1181no—
373C2×SD16—594no—
374C2×SD16—1182no—
375C2×SD16—1182no—
376C22×C8C22×D5911yescharacteristic
377C22×C8C22×D5911yescharacteristic
378C22×C8C22×D5911yescharacteristic
379C22×C8C22×D5911yescharacteristic
380C4×Q8—594no—
381C4×Q8—1182no—
382C4×D4—594no—
383C4×D4—1182no—
384#117—2364no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C22⋊C4—1181no—
397C2×C22⋊C4—1181no—
398C2×C22⋊C4—1181no—
399C2×C22⋊C4—1181no—
400C2×C22⋊C4—1181no—
401C2×C42—1181no—
402C2×C42—1181no—
403C2×C42—1181no—
404C2×C42D23611yescharacteristic
405C2×C42D23611yescharacteristic
406C2×C42D23611yescharacteristic
407C2×C42D23611yescharacteristic
408C2×C42C22×D5911yescharacteristic
409C2×C42C22×D5911yescharacteristic
410C2×C42C22×D5911yescharacteristic
411#107—594no—
412#107—1182no—
413#106—1182no—
414#103—594no—
415#103—1182no—
416#103—1182no—
417#102—594no—
418#102—1182no—
419#102—1182no—
420#97—22no—
421#97—22no—
422#97—22no—
423#97C4×D5914yes—
424C4×C8—22no—
425C4×C8—22no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436C24—1181no—
437C2×Q8—592no—
438C2×Q8—1181no—
439C2×Q8—1181no—
440C2×Q8—1181no—
441C2×Q8—1181no—
442C2×Q8—1181no—
443C2×D4—592no—
444C2×D4—1181no—
445C2×D4—1181no—
446C2×D4—1181no—
447C2×D4—1181no—
448C2×D4—1181no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—1181no—
452C22×C4—1184no—
453C22×C4—1181no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—2362no—
458C22×C4—2362no—
459C22×C4C2×D23611yescharacteristic
460C22×C4C2×D23611yescharacteristic
461C22×C4C2×D23611yescharacteristic
462C22×C4C2×D23611yescharacteristic
463C22×C4C2×D23611yescharacteristic
464C22×C4C2×D23611yescharacteristic
465C22×C4C23×D5911yescharacteristic
466SD16—1188no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8C2×C4×D5912yes—
474C2×C8C2×C4×D5912yes—
475C2×C8#1174568512yes—
476C2×C8#1174568512yes—
477C2×C8#1174568512yes—
478C2×C8#1174568512yes—
479C4⋊C4—592no—
480C4⋊C4—592no—
481C4⋊C4—1184no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—2362no—
493C4⋊C4—2362no—
494C4⋊C4—2362no—
495C4⋊C4—2362no—
496C22⋊C4—1184no—
497C22⋊C4—2362no—
498C22⋊C4—2362no—
499C22⋊C4—2362no—
500C22⋊C4—2362no—
501C42—22no—
502C42—22no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—41no—
508C42—41no—
509C42—1184no—
510C42—2362no—
511C42—2362no—
512C42C2×C4×D5912yes—
513C42#1174568512yes—
514C23—1182no—
515C23—2361no—
516C23—2361no—
517C23—2361no—
518C23—2361no—
519C23—2361no—
520C23—2361no—
521C23C22×D23611yescharacteristic
522Q8—1182no—
523Q8—1182no—
524Q8—2361no—
525Q8—2361no—
526D4—1182no—
527D4—1182no—
528D4—2361no—
529D4—2361no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—1182no—
545C2×C4—1182no—
546C2×C4—1182no—
547C2×C4—2368no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
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×C4C4×D23612yes—
567C2×C4C4×D23612yes—
568C2×C4#1427375512yes—
569C2×C4#1427376012yes—
570C2×C4C22×C4×D5911yescharacteristic
571C2×C4C2×#1174568511yescharacteristic
572C2×C4C2×#1174568511yescharacteristic
573C2×C4C2×#1174568511yescharacteristic
574C2×C4C2×#1174568511yescharacteristic
575C2×C4C2×#1174568511yescharacteristic
576C8—22no—
577C8—22no—
578C8—22no—
579C8#1427368414yescomplemented
580C22—4721no—
581C22—4721no—
582C22—4721no—
583C22—4721no—
584C22—2362no—
585C22—2362no—
586C22—2362no—
587C22C2×C4×D23611yescharacteristic
588C22#1683795211yescharacteristic
589C22#1683795211yescharacteristic
590C22C2×#1427375511yescharacteristic
591C22C2×#1427376011yescharacteristic
592C22#1683820211yescharacteristic
593C22#1683820211yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—4721no—
601C4—4721no—
602C4—4721no—
603C4—4721no—
604C4—2362no—
605C4—2362no—
606C4—2362no—
607C4C2×#1427368411yescharacteristic
608C4C4×#1174568511yescharacteristic
609C4#1683793611yescharacteristic
610C4#1683793611yescharacteristic
611C2—4722no—
612C2#2142656011yescharacteristic
613C2C4×#1427375511yescharacteristic
614C2C4×#1427376011yescharacteristic
615C2#2143479411yescharacteristic
616C2#2143480511yescharacteristic
617C2#2143481411yescharacteristic
618C2#2143482511yescharacteristic
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC4×#97×C59C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenterC4×#97D59
hypocenterC59#21288
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C43 ◁ C4×#97

Complements

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

NH ≅ G/Ntype
2. #21426573C2semidirect
6. #21424834C2semidirect
7. #21424829C2semidirect
9. #21424786C2semidirect
11. #21424772C2semidirect
12. #21423993C2semidirect
13. #21423927C2semidirect
15. C4×#97×C59C2semidirect
20. #16837504C4semidirect
269. C8×C59#117semidirect
316. C4×#97D59semidirect
369. C59#21288semidirect
579. C8#14273684semidirect

Maximal subgroups

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

subgroupindex
C2×#168375042
#214265732
#214265602
#214249772
#214249502
#214248342
#214248292
#214247932
#214247862
#214247792
#214247722
#214239932
#214239272
#214239222
C4×#97×C592
#2128859