Finite Group Databasebeta

← #28610491

Subgroups — #28610491

Order 15104.

926 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×#16837681C211yesmaximal, characteristic
2#21427155C211yesmaximal, characteristic, complemented
3#21426997C211yesmaximal, characteristic
4#21425055C211yesmaximal, characteristic, complemented
5#21424853C211yesmaximal, characteristic, complemented
6#21424830C211yesmaximal, characteristic, complemented
7#21424826C211yesmaximal, characteristic, complemented
8#21424670C211yesmaximal, characteristic, complemented
9#21424346C211yesmaximal, characteristic
10#21424344C211yesmaximal, characteristic
11#21423957C211yesmaximal, characteristic, complemented
12C4×#14273647C211yesmaximal, characteristic
13#21423910C211yesmaximal, characteristic
14#21423910C211yesmaximal, characteristic
15C59×#1918C211yesmaximal, characteristic, complemented
16C2×#14273726C2211yescharacteristic
17C2×C4×Dic118—21no—
18C2×C4×D236C2211yescharacteristic
19C2×#14273689C2211yescharacteristic
20C2×#14273689C2211yescharacteristic
21C2×C4⋊C4×D59—21no—
22#16837681C418yescomplemented
23C2×#14273661C2211yescharacteristic
24C2×#14273660C2211yescharacteristic
25C2×#14273660C2211yescharacteristic
26C2×Dic59⋊C8C2211yescharacteristic
27C2×#14273649C2211yescharacteristic
28C2×#14273649C2211yescharacteristic
29C2×#14273647C2211yescharacteristic
30C2×#14273647C2211yescharacteristic
31C2×C4×C59⋊C8C2211yescharacteristic
32C2×C8×Dic59C2211yescharacteristic
33C2×C8×Dic59C2211yescharacteristic
34#16837306—21no—
35#16837301C2211yescharacteristic
36#16837301C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837289—21no—
39#16837286—21no—
40#16837276—21no—
41#16837273—21no—
42#16837271C2211yescharacteristic
43#16837261—21no—
44C4⋊C4×Dic59C2211yescharacteristic
45C4⋊C4×Dic59C2211yescharacteristic
46C4×Dic59⋊C4C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C4×C4⋊C4×C59C2211yescharacteristic
56C59×#340C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#336C2211yescharacteristic
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×Dic118—44no—
70C4×D236—22no—
71C4×D236—41no—
72#14273689—22no—
73#14273689—22no—
74#14273689—41no—
75#14273689C2×C414yes—
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C2×Dic59⋊C4D411yescharacteristic
86C2×Dic59⋊C4D411yescharacteristic
87C4⋊C4×D59—44no—
88C2×(C2×D59)⋊C4—21no—
89C2×(C2×D59)⋊C4—21no—
90C2×(C2×D59)⋊C4—21no—
91C2×C4×Dic59—21no—
92C2×C4×Dic59—21no—
93C2×C4×Dic59C2311yescharacteristic
94C2×C4×Dic59C2311yescharacteristic
95#14273661—41no—
96#14273660—41no—
97#14273660C2×C414yescomplemented
98Dic59⋊C8C2×C414yescomplemented
99#14273649—22no—
100#14273649C2×C414yes—
101#14273647C2×C414yes—
102#14273647Q814yescomplemented
103C4×C59⋊C8—41no—
104C8×Dic59—22no—
105C8×Dic59—41no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C59⋊#95D411yescharacteristic
112C59⋊#95D411yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C22×C8×C59C2311yescharacteristic
115C2×C4⋊C4×C59—21no—
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C42×C59—21no—
121C2×C42×C59C2311yescharacteristic
122#107×C59C2×C414yescomplemented
123#95×C59—21no—
124#95×C59—21no—
125C23×D59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2411yescharacteristic
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59C2×D411yescharacteristic
137C22×Dic59C2×D411yescharacteristic
138C22×Dic59C2×D411yescharacteristic
139C22×Dic59C2×D411yescharacteristic
140C2×D236—21no—
141C2×D236—21no—
142C2×D236—21no—
143C2×D236C22×C412yes—
144C2×D236C2×Q812yes—
145C2×D236#4112yes—
146C2×C4×D59—24no—
147C2×C4×D59—42no—
148C2×C4×D59—42no—
149C2×C4×D59—42no—
150C2×C4×D59—42no—
151C2×C4×D59—42no—
152C2×C4×D59—42no—
153C2×C4×D59—81no—
154C2×C4×D59—81no—
155C2×Dic118—24no—
156C2×Dic118—42no—
157C2×Dic118—42no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59C22×C412yes—
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×C59C22×C412yes—
171C4⋊C4×C59C22×C412yes—
172C4⋊C4×C59C2×Q812yes—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—42no—
176(C2×D59)⋊C4—24no—
177(C2×D59)⋊C4—42no—
178(C2×D59)⋊C4—42no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—42no—
185C4⋊Dic59—42no—
186C4⋊Dic59C22×C412yes—
187Dic59⋊C4—24no—
188Dic59⋊C4—22no—
189Dic59⋊C4—42no—
190Dic59⋊C4—42no—
191Dic59⋊C4—81no—
192Dic59⋊C4D814yes—
193C4×Dic59—21no—
194C4×Dic59—21no—
195C4×Dic59—42no—
196C4×Dic59—42no—
197C4×Dic59C22×C412yes—
198C4×Dic59#4112yes—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8C22×C412yes—
204C2×C59⋊C8C2×Q812yes—
205C23×C59C22×D411yescharacteristic
206C22×D59—22no—
207C22×D59—22no—
208C22×D59—22no—
209C22×D59—41no—
210C22×D59—41no—
211C22×D59—41no—
212C22×D59—41no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—21no—
218C2×C4×C59—22no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
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×C59C2×D812yes—
231C2×C4×C59C23×C411yescharacteristic
232C2×C4×C59C22×Q811yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59H(5,𝔽2)11yescharacteristic
236C2×C4×C592−1+411yescharacteristic
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—22no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
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×Dic59C4×D412yes—
254C2×Dic59C4×D412yes—
255C2×Dic59C2×D812yes—
256C2×Dic59C2×D812yes—
257C2×Dic59#13512yes—
258D236—22no—
259D236—22no—
260D236—22no—
261D236C4×Q814yescomplemented
262C4×D59—48no—
263C4×D59—84no—
264C4×D59—84no—
265Dic118—48no—
266C8×C59—22no—
267C8×C59—41no—
268C59⋊C8—22no—
269C59⋊C8—22no—
270#22962—591nomaximal, Sylow
271C22×C59C2×C4×D411yescharacteristic
272C22×C59#54611yescharacteristic
273C22×C59D4×Q811yescharacteristic
274C22×C59C22×D811yescharacteristic
275C22×C59C2×#13511yescharacteristic
276C22×C59#57711yescharacteristic
277C22×C59#57811yescharacteristic
278C2×D59—24no—
279C2×D59—42no—
280C2×D59—42no—
281C2×D59—42no—
282C2×D59—42no—
283C2×D59—42no—
284C2×D59—42no—
285C4×C59—22no—
286C4×C59—42no—
287C4×C59—42no—
288C4×C59—42no—
289C4×C59—41no—
290C4×C59—81no—
291C4×C59C2×C4×Q811yescharacteristic
292C4×C59#52011yescharacteristic
293C4×C59#55211yescharacteristic
294C4×C59#55411yescharacteristic
295Dic59—22no—
296Dic59—22no—
297Dic59—42no—
298Dic59—41no—
299Dic59—81no—
300Dic59—81no—
301Dic59C4×D814yes—
302C2×#474—591no—
303#2365—591no—
304D4×C4⋊C4—591no—
305#1990—591no—
306#1947—591no—
307#1942—591no—
308#1942—591no—
309#1918—591no—
310#1918D5911yescharacteristic, complemented
311#1862—591no—
312#1862—591no—
313C8×C4⋊C4—591no—
314C4×#102—591no—
315C4×#102—591no—
316C4×#102—591no—
317C2×C59#236011yescharacteristic
318C2×C59C2×C4×D811yescharacteristic
319C2×C59#294711yescharacteristic
320C2×C59#329411yescharacteristic
321C2×C59#330211yescharacteristic
322C2×C59Q8×D811yescharacteristic
323C2×C59#338111yescharacteristic
324D59—44no—
325C2×C4⋊Q8—591no—
326C2×C4×Q8—1181no—
327C2×C4×D4—591no—
328C2×C4×D4—591no—
329C2×C4×D4—591no—
330C22×C4⋊C4—1181no—
331#474—598no—
332C2×#107—591no—
333C2×#107—591no—
334C2×#107—591no—
335C2×#107C2×D5911yescharacteristic
336C2×C4⋊C8—591no—
337C2×#102—591no—
338C2×#102—591no—
339C2×#102—591no—
340C2×#102—591no—
341C2×C4×C8—591no—
342C2×C4×C8—591no—
343C2×C4×C8—591no—
344#391—1181no—
345#389—591no—
346#389—591no—
347#389—591no—
348#389C2×D5911yescharacteristic
349#387—1181no—
350#387—1181no—
351#385—1181no—
352#385—1181no—
353#384—591no—
354#380—1181no—
355C4×C4⋊C4—591no—
356C4×C4⋊C4—591no—
357C4×C4⋊C4—591no—
358C4×C4⋊C4C2×D5911yescharacteristic
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340C2×D5911yescharacteristic
365#340C2×D5911yescharacteristic
366#340C2×D5911yescharacteristic
367#336—591no—
368#336C2×D5911yescharacteristic
369C59#2296211yescharacteristic, complemented, Sylow
370C22×Q8—1181no—
371C22×D4—591no—
372C23×C4—1181no—
373C23×C4—1181no—
374C23×C4—1181no—
375C22×C8—591no—
376C22×C8—591no—
377C22×C8C22×D5911yescharacteristic
378C22×C8C22×D5911yescharacteristic
379C4⋊Q8—594no—
380C4⋊Q8—1182no—
381C4×Q8—2364no—
382C4×D4—594no—
383C4×D4—1182no—
384C4×D4—1182no—
385C4×D4—1182no—
386C4×D4—2361no—
387C4×D4—2361no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—2364no—
401C2×C4⋊C4C22×D5911yescharacteristic
402C2×C4⋊C4C22×D5911yescharacteristic
403C2×C4⋊C4C22×D5911yescharacteristic
404C2×C4⋊C4C22×D5911yescharacteristic
405C2×C22⋊C4—1181no—
406C2×C22⋊C4—1181no—
407C2×C22⋊C4—1181no—
408C2×C42—21no—
409C2×C42—591no—
410C2×C42—591no—
411C2×C42—1181no—
412C2×C42—1181no—
413C2×C42C22×D5911yescharacteristic
414#107—594no—
415#107—2361no—
416#107—2361no—
417#107C4×D5914yescomplemented
418C4⋊C8—594no—
419#102—594no—
420#102—594no—
421#102—594no—
422#102—1182no—
423C4×C8—1182no—
424C4×C8—2361no—
425C4×C8—2361no—
426#95—21no—
427#95—21no—
428#95—591no—
429#95—591no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435C24—1181no—
436C2×Q8—1184no—
437C2×Q8—2362no—
438C2×Q8—2362no—
439C2×D4—592no—
440C2×D4—592no—
441C2×D4—592no—
442C2×D4—1181no—
443C2×D4—1181no—
444C2×D4—1181no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—591no—
450C22×C4—591no—
451C22×C4—591no—
452C22×C4—591no—
453C22×C4—1184no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—4721no—
458C22×C4—4721no—
459C22×C4—2362no—
460C22×C4—2362no—
461C22×C4—2362no—
462C22×C4—2362no—
463C22×C4—2362no—
464C22×C4—2362no—
465C22×C4D4×D5911yescharacteristic
466C22×C4D4×D5911yescharacteristic
467C22×C4C23×D5911yescharacteristic
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—592no—
474C2×C8—592no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8C2×C4×D5912yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—594no—
487C4⋊C4—592no—
488C4⋊C4—1182no—
489C4⋊C4—1184no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—4721no—
496C4⋊C4—2362no—
497C4⋊C4—2362no—
498C4⋊C4—2362no—
499C4⋊C4—2362no—
500C4⋊C4C2×C4×D5912yes—
501C4⋊C4C2×C4×D5912yes—
502C4⋊C4Q8×D5912yes—
503C22⋊C4—1184no—
504C22⋊C4—2362no—
505C22⋊C4—2362no—
506C42—21no—
507C42—21no—
508C42—42no—
509C42—592no—
510C42—592no—
511C42—1181no—
512C42—1181no—
513C42—2362no—
514C42—2362no—
515C23—1182no—
516C23—1182no—
517C23—1182no—
518C23—2361no—
519C23—2361no—
520C23—2361no—
521C23—2361no—
522C23C2×D4×D5911yescharacteristic
523Q8—2368no—
524D4—594no—
525D4—1182no—
526D4—1182no—
527D4—1182no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
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—41no—
545C2×C4—592no—
546C2×C4—592no—
547C2×C4—592no—
548C2×C4—592no—
549C2×C4—592no—
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—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×C4—2361no—
569C2×C4D8×D5912yes—
570C2×C4C22×C4×D5911yescharacteristic
571C2×C4C2×Q8×D5911yescharacteristic
572C2×C4#41×D5911yescharacteristic
573C2×C4#41×D5911yescharacteristic
574C2×C4#1427377311yescharacteristic
575C2×C4#1427377411yescharacteristic
576C8—22no—
577C8—41no—
578C8—1182no—
579C8—1182no—
580C22—1184no—
581C22—2362no—
582C22—2362no—
583C22—2362no—
584C22—2362no—
585C22—2362no—
586C22—2362no—
587C22C4×D4×D5911yescharacteristic
588C22#1683804411yescharacteristic
589C22#1683806511yescharacteristic
590C22C2×D8×D5911yescharacteristic
591C22#135×D5911yescharacteristic
592C22#1683821111yescharacteristic
593C22#1683821811yescharacteristic
594C4—22no—
595C4—42no—
596C4—42no—
597C4—42no—
598C4—41no—
599C4—81no—
600C4—594no—
601C4—1182no—
602C4—1182no—
603C4—4721no—
604C4—4721no—
605C4—2362no—
606C4—2361no—
607C4C4×Q8×D5911yescharacteristic
608C4#1683788211yescharacteristic
609C4#1683808811yescharacteristic
610C4#1683810811yescharacteristic
611C2—2364no—
612C2#2142710411yescharacteristic
613C2C4×D8×D5911yescharacteristic
614C2#2143293811yescharacteristic
615C2#2143590711yescharacteristic
616C2#2143601111yescharacteristic
617C2#2143694611yescharacteristic
618C2#2143697411yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1918C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#1918D59
hypocenterC59#22962
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
2. #21427155C2semidirect
4. #21425055C2semidirect
5. #21424853C2semidirect
6. #21424830C2semidirect
7. #21424826C2semidirect
8. #21424670C2semidirect
11. #21423957C2semidirect
15. C59×#1918C2semidirect
22. #16837681C4semidirect
68. #14273726C2×C4semidirect
97. #14273660C2×C4semidirect
98. Dic59⋊C8C2×C4semidirect
102. #14273647Q8semidirect
122. #107×C59C2×C4semidirect
261. D236C4×Q8semidirect
310. #1918D59semidirect
369. C59#22962semidirect
417. #107C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168376812
#214271552
#214269972
#214250552
#214248532
#214248302
#214248262
#214246702
#214243462
#214243442
#214239572
C4×#142736472
#214239102
C59×#19182
#2296259