Finite Group Databasebeta

← #28610655

Subgroups — #28610655

Order 15104.

878 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837693C211yesmaximal, characteristic
2#21427154C211yesmaximal, characteristic, complemented
3#21427007C211yesmaximal, characteristic
4#21425069C211yesmaximal, characteristic, complemented
5#21424866C211yesmaximal, characteristic, complemented
6#21424839C211yesmaximal, characteristic, complemented
7#21424824C211yesmaximal, characteristic, complemented
8#21424682C211yesmaximal, characteristic, complemented
9#21424675C211yesmaximal, characteristic, complemented
10#21424360C211yesmaximal, characteristic
11#21424346C211yesmaximal, characteristic
12#21423922C211yesmaximal, characteristic
13#21423922C211yesmaximal, characteristic
14C4×#14273649C211yesmaximal, characteristic
15C59×#1771C211yesmaximal, characteristic, complemented
16C2×#14273727C2211yescharacteristic
17C2×#14273692—21no—
18C2×C4×D236C2211yescharacteristic
19C2×C4×D236C2211yescharacteristic
20C2×C4×D236C2211yescharacteristic
21#16837693C418yescomplemented
22C2×#14273661C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273659C2211yescharacteristic
25C2×#14273659C2211yescharacteristic
26C2×#14273649C2211yescharacteristic
27C2×#14273649C2211yescharacteristic
28C2×#14273649C2211yescharacteristic
29C2×#14273649C2211yescharacteristic
30#16837307—21no—
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837300C2211yescharacteristic
35#16837287—21no—
36#16837287—21no—
37#16837273—21no—
38#16837273—21no—
39#16837268C2211yescharacteristic
40#16837261—21no—
41C4×C4⋊Dic59C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43C4×C4⋊Dic59C2211yescharacteristic
44#95×D59—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53C2×C4⋊C8×C59C2211yescharacteristic
54C2×#97×C59C2211yescharacteristic
55C2×#97×C59C2211yescharacteristic
56C2×C4×C8×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×Dic118—21no—
61C22×D236C2311yescharacteristic
62C22×C4×D59—21no—
63C22×C4×D59—21no—
64C22×C4×D59—21no—
65#14273727—22no—
66#14273727C2×C414yescomplemented
67#14273692—44no—
68C4×D236—22no—
69C4×D236—22no—
70C4×D236—22no—
71C4×D236—41no—
72C4×D236—41no—
73C4×D236C2×C414yes—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C2×Dic59⋊C4—21no—
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—
92#14273661—41no—
93#14273659—41no—
94#14273659C2×C414yescomplemented
95#14273659C2×C414yescomplemented
96#14273649—22no—
97#14273649C2×C414yes—
98#14273649C2×C414yes—
99#14273649Q814yescomplemented
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×C4⋊C4×C59D411yescharacteristic
114C2×C4⋊C4×C59D411yescharacteristic
115C2×C42×C59C2311yescharacteristic
116C2×C42×C59C2311yescharacteristic
117C2×C42×C59C2311yescharacteristic
118C4⋊C8×C59C2×C414yescomplemented
119#97×C59—22no—
120#97×C59—41no—
121C4×C8×C59—41no—
122#95×C59D411yescharacteristic
123#95×C59D411yescharacteristic
124C23×D59—21no—
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2411yescharacteristic
132C22×Dic59—21no—
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—21no—
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—81no—
151C2×C4×D59—81no—
152C2×C4×D59—81no—
153C2×C4×D59—81no—
154C2×Dic118—24no—
155C2×Dic118—42no—
156C2×Dic118—42no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59C22×C412yes—
167C2×C8×C59C22×C412yes—
168C2×C8×C59C2×Q812yes—
169C4⋊C4×C59—22no—
170C4⋊C4×C59—41no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59C22×C412yes—
176C42×C59#4112yes—
177(C2×D59)⋊C4—24no—
178(C2×D59)⋊C4—42no—
179(C2×D59)⋊C4—42no—
180C4⋊Dic59—24no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
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⋊Dic59C22×C412yes—
192C4⋊Dic59C22×C412yes—
193C4⋊Dic59C22×C412yes—
194C4⋊Dic59C2×Q812yes—
195Dic59⋊C4—42no—
196Dic59⋊C4—42no—
197Dic59⋊C4—81no—
198Dic59⋊C4—81no—
199C4×Dic59—42no—
200C4×Dic59—42no—
201C4×Dic59—42no—
202C23×C59C22×D411yescharacteristic
203C22×D59—22no—
204C22×D59—22no—
205C22×D59—22no—
206C22×D59—41no—
207C22×D59—41no—
208C22×D59—41no—
209C22×D59—41no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
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×C59C4×D412yes—
226C2×C4×C59C4×D412yes—
227C2×C4×C59AGL(1,ℤ/8)12yes—
228C2×C4×C59C23×C411yescharacteristic
229C2×C4×C59C22×Q811yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59H(5,𝔽2)11yescharacteristic
233C2×C4×C592−1+411yescharacteristic
234C2×Dic59—22no—
235C2×Dic59—22no—
236C2×Dic59—22no—
237C2×Dic59—22no—
238C2×Dic59—41no—
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—
255D236—22no—
256D236—22no—
257D236—22no—
258D236C4×Q814yescomplemented
259C4×D59—84no—
260C4×D59—84no—
261C4×D59—84no—
262Dic118—48no—
263C8×C59—22no—
264C8×C59—22no—
265C8×C59—22no—
266C8×C59—41no—
267#22975—591nomaximal, Sylow
268C22×C59C2×C4×D411yescharacteristic
269C22×C59#54611yescharacteristic
270C22×C59D4×Q811yescharacteristic
271C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C59#57511yescharacteristic
273C22×C59#57711yescharacteristic
274C22×C59#57711yescharacteristic
275C2×D59—24no—
276C2×D59—42no—
277C2×D59—42no—
278C2×D59—42no—
279C2×D59—42no—
280C2×D59—42no—
281C2×D59—42no—
282C4×C59—22no—
283C4×C59—22no—
284C4×C59—22no—
285C4×C59—41no—
286C4×C59—41no—
287C4×C59—41no—
288C4×C59C2×C4×Q811yescharacteristic
289C4×C59#52011yescharacteristic
290C4×C59#55211yescharacteristic
291C4×C59#55411yescharacteristic
292Dic59—42no—
293Dic59—42no—
294Dic59—42no—
295Dic59—42no—
296Dic59—81no—
297Dic59—81no—
298Dic59—81no—
299C2×#476—591no—
300#2365—591no—
301#2349—591no—
302#1993—591no—
303#1949—591no—
304#1944—591no—
305#1941—591no—
306#1920—591no—
307#1919—591no—
308#1864—591no—
309#1862—591no—
310#1771D5911yescharacteristic, complemented
311#1762—591no—
312#1762—591no—
313C4×#102—591no—
314C2×C59#236011yescharacteristic
315C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
316C2×C59#294911yescharacteristic
317C2×C59#330611yescharacteristic
318C2×C59#331111yescharacteristic
319C2×C59#338411yescharacteristic
320C2×C59#338811yescharacteristic
321D59—44no—
322C2×C4⋊Q8—591no—
323C2×C4×Q8—1181no—
324C2×C4×D4—591no—
325C2×C4×D4—591no—
326C2×C4×D4—591no—
327#476—598no—
328C2×#107—591no—
329C2×#106—591no—
330C2×#106—591no—
331C2×#106—591no—
332C2×C4⋊C8C2×D5911yescharacteristic
333C2×#102—591no—
334C2×#102—591no—
335C2×#102—591no—
336C2×#102—591no—
337C2×#97C2×D5911yescharacteristic
338C2×#97C2×D5911yescharacteristic
339C2×C4×C8C2×D5911yescharacteristic
340#391—1181no—
341#389—591no—
342#389—591no—
343#389—591no—
344#389—591no—
345#387—1181no—
346#387—1181no—
347#385—1181no—
348#385—1181no—
349#382—591no—
350#380—1181no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—591no—
353C4×C4⋊C4—591no—
354C4×C4⋊C4C2×D5911yescharacteristic
355C2×#95—1181no—
356#340—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#336C2×D5911yescharacteristic
365#336C2×D5911yescharacteristic
366C59#2297511yescharacteristic, complemented, Sylow
367C22×Q8—1181no—
368C22×D4—591no—
369C23×C4—1181no—
370C23×C4—1181no—
371C23×C4—1181no—
372C22×C8C22×D5911yescharacteristic
373C22×C8C22×D5911yescharacteristic
374C22×C8C22×D5911yescharacteristic
375C22×C8C22×D5911yescharacteristic
376C4⋊Q8—594no—
377C4⋊Q8—1182no—
378C4×Q8—2364no—
379C4×D4—594no—
380C4×D4—1182no—
381C4×D4—1182no—
382C4×D4—1182no—
383C4×D4—2361no—
384C4×D4—2361no—
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—591no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4D23611yescharacteristic
398C2×C4⋊C4D23611yescharacteristic
399C2×C22⋊C4—1181no—
400C2×C22⋊C4—1181no—
401C2×C22⋊C4—1181no—
402C2×C42—1181no—
403C2×C42—1181no—
404C2×C42—1181no—
405C2×C42C22×D5911yescharacteristic
406C2×C42C22×D5911yescharacteristic
407C2×C42C22×D5911yescharacteristic
408#107—2361no—
409#106—594no—
410#106—594no—
411#106—2361no—
412C4⋊C8C4×D5914yescomplemented
413#102—594no—
414#102—594no—
415#102—594no—
416#102—1182no—
417#97—22no—
418#97—41no—
419C4×C8—41no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95D23611yescharacteristic
430#95D23611yescharacteristic
431C24—1181no—
432C2×Q8—1184no—
433C2×Q8—2362no—
434C2×Q8—2362no—
435C2×D4—592no—
436C2×D4—592no—
437C2×D4—592no—
438C2×D4—1181no—
439C2×D4—1181no—
440C2×D4—1181no—
441C22×C4—1181no—
442C22×C4—1181no—
443C22×C4—1181no—
444C22×C4—1181no—
445C22×C4—1184no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4—4721no—
450C22×C4—4721no—
451C22×C4—4721no—
452C22×C4—4721no—
453C22×C4—2362no—
454C22×C4—2362no—
455C22×C4—2362no—
456C22×C4—2362no—
457C22×C4C2×D23611yescharacteristic
458C22×C4C2×D23611yescharacteristic
459C22×C4C2×D23611yescharacteristic
460C22×C4C2×D23611yescharacteristic
461C22×C4C2×D23611yescharacteristic
462C22×C4C2×D23611yescharacteristic
463C22×C4C23×D5911yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
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×C8Q8×D5912yes—
476C4⋊C4—22no—
477C4⋊C4—41no—
478C4⋊C4—592no—
479C4⋊C4—592no—
480C4⋊C4—592no—
481C4⋊C4—592no—
482C4⋊C4—1184no—
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—1181no—
493C4⋊C4—4721no—
494C4⋊C4—4721no—
495C4⋊C4—2362no—
496C4⋊C4—2362no—
497C22⋊C4—1184no—
498C22⋊C4—2362no—
499C22⋊C4—2362no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—2362no—
505C42—2362no—
506C42—2362no—
507C42C2×C4×D5912yes—
508C42#1174568512yes—
509C23—1182no—
510C23—1182no—
511C23—1182no—
512C23—2361no—
513C23—2361no—
514C23—2361no—
515C23—2361no—
516C23C22×D23611yescharacteristic
517Q8—2368no—
518D4—594no—
519D4—1182no—
520D4—1182no—
521D4—1182no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—1182no—
538C2×C4—1182no—
539C2×C4—1182no—
540C2×C4—1182no—
541C2×C4—4724no—
542C2×C4—4724no—
543C2×C4—4724no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
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×C4C4×D23612yes—
562C2×C4C4×D23612yes—
563C2×C4#1427375512yes—
564C2×C4C22×C4×D5911yescharacteristic
565C2×C4C2×Q8×D5911yescharacteristic
566C2×C4#41×D5911yescharacteristic
567C2×C4C2×#1174568511yescharacteristic
568C2×C4#1427377211yescharacteristic
569C2×C4#1427377411yescharacteristic
570C8—22no—
571C8—22no—
572C8—22no—
573C8—41no—
574C22—1184no—
575C22—2362no—
576C22—2362no—
577C22—2362no—
578C22—2362no—
579C22—2362no—
580C22—2362no—
581C22C2×C4×D23611yescharacteristic
582C22#1683804711yescharacteristic
583C22Q8×D23611yescharacteristic
584C22C2×#1427375511yescharacteristic
585C22#1683820211yescharacteristic
586C22#1683821411yescharacteristic
587C22#1683821411yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—4721no—
595C4—4721no—
596C4—4721no—
597C4—2362no—
598C4—2362no—
599C4—2362no—
600C4—2362no—
601C4C4×Q8×D5911yescharacteristic
602C4#1683788111yescharacteristic
603C4#1683809111yescharacteristic
604C4#1683810711yescharacteristic
605C2—2364no—
606C2#2142710311yescharacteristic
607C2C4×#1427375511yescharacteristic
608C2#2143295711yescharacteristic
609C2#2143605911yescharacteristic
610C2#2143613411yescharacteristic
611C2#2143699711yescharacteristic
612C2#2143704311yescharacteristic
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1771C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#1771D59
hypocenterC59#22975
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
2. #21427154C2semidirect
4. #21425069C2semidirect
5. #21424866C2semidirect
6. #21424839C2semidirect
7. #21424824C2semidirect
8. #21424682C2semidirect
9. #21424675C2semidirect
15. C59×#1771C2semidirect
21. #16837693C4semidirect
66. #14273727C2×C4semidirect
94. #14273659C2×C4semidirect
95. #14273659C2×C4semidirect
99. #14273649Q8semidirect
118. C4⋊C8×C59C2×C4semidirect
258. D236C4×Q8semidirect
310. #1771D59semidirect
366. C59#22975semidirect
412. C4⋊C8C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168376932
#214271542
#214270072
#214250692
#214248662
#214248392
#214248242
#214246822
#214246752
#214243602
#214243462
#214239222
C4×#142736492
C59×#17712
#2297559