Finite Group Databasebeta

← #28610570

Subgroups — #28610570

Order 15104.

886 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×#16837691C211yesmaximal, characteristic
2#21427182C211yesmaximal, characteristic, complemented
3#21426997C211yesmaximal, characteristic
4#21425055C211yesmaximal, characteristic, complemented
5#21424851C211yesmaximal, characteristic, complemented
6#21424838C211yesmaximal, characteristic, complemented
7#21424683C211yesmaximal, characteristic, complemented
8#21424678C211yesmaximal, characteristic, complemented
9#21424342C211yesmaximal, characteristic
10#21424340C211yesmaximal, characteristic
11#21423964C211yesmaximal, characteristic, complemented
12#21423922C211yesmaximal, characteristic
13#21423920C211yesmaximal, characteristic
14#21423910C211yesmaximal, characteristic
15C59×#1944C211yesmaximal, characteristic, complemented
16C2×#14273726C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×#14273689C2211yescharacteristic
19C2×#14273689C2211yescharacteristic
20C2×C4⋊C4×D59—21no—
21#16837691C418yescomplemented
22C2×#14273660C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×Dic59⋊C8C2211yescharacteristic
25C2×#14273649C2211yescharacteristic
26C2×#14273649C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×#14273637C2211yescharacteristic
30C2×C8×Dic59C2211yescharacteristic
31#16837307—21no—
32#16837306—21no—
33#16837301C2211yescharacteristic
34#16837298C2211yescharacteristic
35#16837297—21no—
36#16837276—21no—
37#16837273—21no—
38#16837271C2211yescharacteristic
39#16837261—21no—
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4×Dic59⋊C4C2211yescharacteristic
43C4×C4⋊Dic59C2211yescharacteristic
44#16837253—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837157C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C2×#106×C59C2211yescharacteristic
55C2×#97×C59C2211yescharacteristic
56C59×#389C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×Dic118—21no—
61C22×D236C2311yescharacteristic
62C22×C4×D59—21no—
63C22×C4×D59—21no—
64C22×C4×D59—21no—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273726—22no—
68#14273726C2×C414yescomplemented
69C4×D236—22no—
70C4×D236—41no—
71#14273689—22no—
72#14273689—22no—
73#14273689—41no—
74#14273689C2×C414yes—
75C2×C4⋊Dic59—21no—
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4D411yescharacteristic
84C2×Dic59⋊C4D411yescharacteristic
85C4⋊C4×D59—44no—
86C2×(C2×D59)⋊C4—21no—
87C2×(C2×D59)⋊C4—21no—
88C2×(C2×D59)⋊C4—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59—21no—
91C2×C4×Dic59—21no—
92C2×C4×Dic59C2311yescharacteristic
93C2×C4×Dic59C2311yescharacteristic
94#14273660—41no—
95#14273658C2×C414yescomplemented
96Dic59⋊C8C2×C414yescomplemented
97#14273649C2×C414yes—
98#14273649Q814yescomplemented
99#14273647—22no—
100#14273647C2×C414yes—
101#14273637—41no—
102C8×Dic59—22no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95D411yescharacteristic
109C59⋊#95D411yescharacteristic
110C22×C8×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C42×C59C2311yescharacteristic
119#107×C59—41no—
120#106×C59C2×C414yescomplemented
121#97×C59—41no—
122#95×C59—21no—
123#95×C59—21no—
124C23×D59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2411yescharacteristic
132C22×Dic59—21no—
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59C2×D411yescharacteristic
136C22×Dic59C2×D411yescharacteristic
137C22×Dic59C2×D411yescharacteristic
138C22×Dic59C2×D411yescharacteristic
139C2×D236—21no—
140C2×D236—21no—
141C2×D236—21no—
142C2×D236C22×C412yes—
143C2×D236C2×Q812yes—
144C2×D236#4112yes—
145C2×C4×D59—24no—
146C2×C4×D59—42no—
147C2×C4×D59—42no—
148C2×C4×D59—42no—
149C2×C4×D59—42no—
150C2×C4×D59—42no—
151C2×C4×D59—42no—
152C2×C4×D59—81no—
153C2×C4×D59—81no—
154C2×Dic118—24no—
155C2×Dic118—81no—
156C2×Dic118—81no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59C22×C412yes—
162C2×C8×C59C2×Q812yes—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—42no—
170C4⋊C4×C59—81no—
171C4⋊C4×C59C22×C412yes—
172C4⋊C4×C59C22×C412yes—
173C42×C59—21no—
174C42×C59—21no—
175(C2×D59)⋊C4—24no—
176(C2×D59)⋊C4—42no—
177(C2×D59)⋊C4—42no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—42no—
183C4⋊Dic59C22×C412yes—
184C4⋊Dic59C2×Q812yes—
185Dic59⋊C4—24no—
186Dic59⋊C4—22no—
187Dic59⋊C4—22no—
188Dic59⋊C4—42no—
189Dic59⋊C4—81no—
190Dic59⋊C4—81no—
191C4×Dic59—21no—
192C4×Dic59—21no—
193C4×Dic59—42no—
194C4×Dic59—81no—
195C4×Dic59—81no—
196C4×Dic59C22×C412yes—
197C4×Dic59#4112yes—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8C22×C412yes—
204C23×C59C22×D411yescharacteristic
205C22×D59—22no—
206C22×D59—22no—
207C22×D59—22no—
208C22×D59—41no—
209C22×D59—41no—
210C22×D59—41no—
211C22×D59—41no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
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—41no—
221C2×C4×C59—41no—
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—41no—
229C2×C4×C59—41no—
230C2×C4×C59C23×C411yescharacteristic
231C2×C4×C59C22×Q811yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59H(5,𝔽2)11yescharacteristic
235C2×C4×C592−1+411yescharacteristic
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—22no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—22no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59C4×D412yes—
254C2×Dic59C4×D412yes—
255C2×Dic59AGL(1,ℤ/8)12yes—
256C2×Dic59#13712yes—
257D236—22no—
258D236—22no—
259D236—22no—
260D236C4×Q814yes—
261C4×D59—48no—
262C4×D59—84no—
263C4×D59—84no—
264Dic118—84no—
265C8×C59—22no—
266C8×C59—22no—
267C59⋊C8—22no—
268C59⋊C8—41no—
269#22969—591nomaximal, Sylow
270C22×C59C2×C4×D411yescharacteristic
271C22×C59#54611yescharacteristic
272C22×C59D4×Q811yescharacteristic
273C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C59C2×#13711yescharacteristic
275C22×C59#57611yescharacteristic
276C22×C59#57711yescharacteristic
277C2×D59—24no—
278C2×D59—42no—
279C2×D59—42no—
280C2×D59—42no—
281C2×D59—42no—
282C2×D59—42no—
283C2×D59—42no—
284C4×C59—41no—
285C4×C59—42no—
286C4×C59—42no—
287C4×C59—41no—
288C4×C59—81no—
289C4×C59—81no—
290C4×C59C2×C4×Q811yescharacteristic
291C4×C59#52011yescharacteristic
292C4×C59#55211yescharacteristic
293C4×C59#55411yescharacteristic
294Dic59—22no—
295Dic59—41no—
296Dic59—42no—
297Dic59—42no—
298Dic59—41no—
299Dic59—81no—
300Dic59#44014yes—
301C2×#476—591no—
302#2369—591no—
303D4×C4⋊C4—591no—
304#1990—591no—
305#1947—591no—
306#1944—591no—
307#1944D5911yescharacteristic, complemented
308#1920—591no—
309#1920—591no—
310#1862—591no—
311#1862—591no—
312#1770—591no—
313#1762—591no—
314#1762—591no—
315C4×#102—591no—
316C2×C59#236011yescharacteristic
317C2×C59C2×#44011yescharacteristic
318C2×C59#294611yescharacteristic
319C2×C59#332511yescharacteristic
320C2×C59#333111yescharacteristic
321C2×C59#338411yescharacteristic
322C2×C59#338511yescharacteristic
323D59—44no—
324C2×C4⋊Q8—591no—
325C2×C4×D4—591no—
326C2×C4×D4—591no—
327C2×C4×D4—591no—
328C22×C4⋊C4—1181no—
329#476—598no—
330C2×#107—591no—
331C2×#107C2×D5911yescharacteristic
332C2×#106—591no—
333C2×#106C2×D5911yescharacteristic
334C2×C4⋊C8—591no—
335C2×#102—591no—
336C2×#102—591no—
337C2×#102—591no—
338C2×#102—591no—
339C2×#97—591no—
340C2×#97C2×D5911yescharacteristic
341C2×C4×C8—591no—
342#391—1181no—
343#391—1181no—
344#389—591no—
345#389—591no—
346#389—1181no—
347#389C2×D5911yescharacteristic
348#389C2×D5911yescharacteristic
349#385—1181no—
350#385—1181no—
351#384—591no—
352#380—1181no—
353C4×C4⋊C4—591no—
354C4×C4⋊C4—591no—
355C4×C4⋊C4—591no—
356C4×C4⋊C4—591no—
357C4×C4⋊C4—1181no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340C2×D5911yescharacteristic
365#340C2×D5911yescharacteristic
366#336—591no—
367#336—591no—
368C59#2296911yescharacteristic, complemented, Sylow
369C22×Q8—1181no—
370C22×D4—591no—
371C23×C4—1181no—
372C23×C4—1181no—
373C23×C4—1181no—
374C22×C8—591no—
375C22×C8—591no—
376C22×C8C22×D5911yescharacteristic
377C22×C8C22×D5911yescharacteristic
378C4⋊Q8—594no—
379C4⋊Q8—1182no—
380C4×D4—594no—
381C4×D4—1182no—
382C4×D4—1182no—
383C4×D4—1182no—
384C4×D4—2361no—
385C4×D4—2361no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—2364no—
399C2×C4⋊C4C22×D5911yescharacteristic
400C2×C4⋊C4C22×D5911yescharacteristic
401C2×C4⋊C4C22×D5911yescharacteristic
402C2×C4⋊C4C22×D5911yescharacteristic
403C2×C22⋊C4—1181no—
404C2×C22⋊C4—1181no—
405C2×C22⋊C4—1181no—
406C2×C42—591no—
407C2×C42—591no—
408C2×C42—1181no—
409C2×C42—1181no—
410C2×C42—1181no—
411C2×C42C22×D5911yescharacteristic
412#107—41no—
413#107—2361no—
414#106—594no—
415#106C4×D5914yescomplemented
416C4⋊C8—594no—
417#102—594no—
418#102—594no—
419#102—594no—
420#102—1182no—
421#97—41no—
422#97—2361no—
423C4×C8—1182no—
424#95—21no—
425#95—21no—
426#95—591no—
427#95—591no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433C24—1181no—
434C2×Q8—1184no—
435C2×Q8—4721no—
436C2×Q8—4721no—
437C2×D4—592no—
438C2×D4—592no—
439C2×D4—592no—
440C2×D4—1181no—
441C2×D4—1181no—
442C2×D4—1181no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—591no—
448C22×C4—591no—
449C22×C4—591no—
450C22×C4—591no—
451C22×C4—1184no—
452C22×C4—1181no—
453C22×C4—1181no—
454C22×C4—1181no—
455C22×C4—4721no—
456C22×C4—4721no—
457C22×C4—2362no—
458C22×C4—2362no—
459C22×C4—2362no—
460C22×C4—2362no—
461C22×C4—2362no—
462C22×C4—2362no—
463C22×C4D4×D5911yescharacteristic
464C22×C4D4×D5911yescharacteristic
465C22×C4C23×D5911yescharacteristic
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—592no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8C2×C4×D5912yes—
477C2×C8Q8×D5912yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—81no—
486C4⋊C4—592no—
487C4⋊C4—592no—
488C4⋊C4—1182no—
489C4⋊C4—1182no—
490C4⋊C4—1184no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—4721no—
496C4⋊C4—4721no—
497C4⋊C4—2362no—
498C4⋊C4—2362no—
499C4⋊C4C2×C4×D5912yes—
500C4⋊C4C2×C4×D5912yes—
501C22⋊C4—1184no—
502C22⋊C4—2362no—
503C22⋊C4—2362no—
504C42—21no—
505C42—21no—
506C42—592no—
507C42—592no—
508C42—1181no—
509C42—1181no—
510C42—4721no—
511C42—4721no—
512C42—2362no—
513C23—1182no—
514C23—1182no—
515C23—1182no—
516C23—2361no—
517C23—2361no—
518C23—2361no—
519C23—2361no—
520C23C2×D4×D5911yescharacteristic
521Q8—4724no—
522D4—594no—
523D4—1182no—
524D4—1182no—
525D4—1182no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—592no—
545C2×C4—592no—
546C2×C4—592no—
547C2×C4—592no—
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—1182no—
557C2×C4—1182no—
558C2×C4—4724no—
559C2×C4—4724no—
560C2×C4—2368no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4C22×C4×D5911yescharacteristic
569C2×C4C2×Q8×D5911yescharacteristic
570C2×C4#41×D5911yescharacteristic
571C2×C4#41×D5911yescharacteristic
572C2×C4#1427377211yescharacteristic
573C2×C4#1427377511yescharacteristic
574C8—22no—
575C8—22no—
576C8—1182no—
577C8—2361no—
578C22—1184no—
579C22—2362no—
580C22—2362no—
581C22—2362no—
582C22—2362no—
583C22—2362no—
584C22—2362no—
585C22C4×D4×D5911yescharacteristic
586C22#1683804111yescharacteristic
587C22#1683806711yescharacteristic
588C22AGL(1,ℤ/8)×D5911yescharacteristic
589C22#137×D5911yescharacteristic
590C22#1683820611yescharacteristic
591C22#1683821111yescharacteristic
592C4—42no—
593C4—42no—
594C4—41no—
595C4—41no—
596C4—81no—
597C4—81no—
598C4—594no—
599C4—1182no—
600C4—4721no—
601C4—2362no—
602C4—2362no—
603C4—2361no—
604C4—2361no—
605C4C4×Q8×D5911yescharacteristic
606C4#1683788311yescharacteristic
607C4#1683808411yescharacteristic
608C4#1683810611yescharacteristic
609C2—2364no—
610C2#2142709811yescharacteristic
611C2#440×D5911yescharacteristic
612C2#2143292711yescharacteristic
613C2#2143633311yescharacteristic
614C2#2143640711yescharacteristic
615C2#2143699211yescharacteristic
616C2#2143700311yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1944C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#1944D59
hypocenterC59#22969
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1944

Complements

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

NH ≅ G/Ntype
2. #21427182C2semidirect
4. #21425055C2semidirect
5. #21424851C2semidirect
6. #21424838C2semidirect
7. #21424683C2semidirect
8. #21424678C2semidirect
11. #21423964C2semidirect
15. C59×#1944C2semidirect
21. #16837691C4semidirect
68. #14273726C2×C4semidirect
95. #14273658C2×C4semidirect
96. Dic59⋊C8C2×C4semidirect
98. #14273649Q8semidirect
120. #106×C59C2×C4semidirect
307. #1944D59semidirect
368. C59#22969semidirect
415. #106C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168376912
#214271822
#214269972
#214250552
#214248512
#214248382
#214246832
#214246782
#214243422
#214243402
#214239642
#214239222
#214239202
#214239102
C59×#19442
#2296959