Finite Group Databasebeta

← #28640735

Subgroups — #28640735

Order 15104.

790 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#21429454C211yesmaximal, characteristic
3#21429081C211yesmaximal, characteristic, complemented
4#21425594C211yesmaximal, characteristic, complemented
5#21425594C211yesmaximal, characteristic, complemented
6#21425483C211yesmaximal, characteristic
7#21425469C211yesmaximal, characteristic
8#21425069C211yesmaximal, characteristic, complemented
9#21424866C211yesmaximal, characteristic, complemented
10#21424834C211yesmaximal, characteristic, complemented
11#21424824C211yesmaximal, characteristic, complemented
12#21424793C211yesmaximal, characteristic
13#21424779C211yesmaximal, characteristic
14#21423922C211yesmaximal, characteristic
15C2×C59×#446C211yesmaximal, characteristic, complemented
16C2×#14273727C2211yescharacteristic
17C2×#14273725—21no—
18C2×C4×D236C2211yescharacteristic
19#16837693—24no—
20C2×#14273661C2211yescharacteristic
21C2×#14273659C2211yescharacteristic
22C2×#14273659C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273649C2211yescharacteristic
25C2×#14273649C2211yescharacteristic
26C2×#14273649C2211yescharacteristic
27C2×#14273649C2211yescharacteristic
28#16837337—21no—
29#16837332—21no—
30#16837332—21no—
31#16837313—21no—
32#16837307—21no—
33#16837304C2211yescharacteristic
34#16837304C2211yescharacteristic
35#16837300C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837300C2211yescharacteristic
41#16837273—21no—
42#16837273—21no—
43#16837268C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53C2×C4×Q8×C59C2211yescharacteristic
54C59×#446—24no—
55C2×C4⋊C8×C59C2211yescharacteristic
56C2×C4⋊C8×C59C2211yescharacteristic
57C2×C4⋊C8×C59C2211yescharacteristic
58C2×#97×C59C2211yescharacteristic
59C2×#97×C59C2211yescharacteristic
60C2×C4×C8×C59C2211yescharacteristic
61C22×Dic118—21no—
62C22×D236C2311yescharacteristic
63C22×C4×D59—21no—
64#14273727—22no—
65#14273727—22no—
66#14273725—44no—
67C4×D236—22no—
68C4×D236—22no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59C2311yescharacteristic
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×Dic59⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86C2×(C2×D59)⋊C4—21no—
87C2×(C2×D59)⋊C4—21no—
88C2×C4×Dic59—21no—
89#14273661—22no—
90#14273659—22no—
91#14273659—22no—
92#14273659—22no—
93#14273649—22no—
94#14273649—22no—
95#14273649—22no—
96#14273649Q814yescomplemented
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C22×Q8×C59D411yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110C22×C8×C59C2311yescharacteristic
111C4×Q8×C59—22no—
112C4×Q8×C59—22no—
113C2×C4⋊C4×C59D411yescharacteristic
114C2×C4⋊C4×C59D411yescharacteristic
115C2×C4⋊C4×C59D411yescharacteristic
116C2×C42×C59C2311yescharacteristic
117C2×C42×C59C2311yescharacteristic
118C2×C42×C59C2311yescharacteristic
119C4⋊C8×C59—22no—
120C4⋊C8×C59—22no—
121C4⋊C8×C59D414yes—
122#97×C59—22no—
123#97×C59—22no—
124C4×C8×C59—22no—
125C23×D59—21no—
126C2×Q8×C59—22no—
127C2×Q8×C59—22no—
128C2×Q8×C59—41no—
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2411yescharacteristic
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C2×D236—21no—
144C2×D236—21no—
145C2×D236—21no—
146C2×D236—21no—
147C2×D236—21no—
148C2×D236C2×Q812yes—
149C2×C4×D59—42no—
150C2×C4×D59—81no—
151C2×C4×D59—81no—
152C2×Dic118—42no—
153C2×Dic118—42no—
154C2×Dic118—42no—
155C2×C8×C59—21no—
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×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59C2×D412yes—
165C2×C8×C59C2×D412yes—
166C2×C8×C59C2×Q812yes—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—22no—
169C4⋊C4×C59—22no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59C2×D412yes—
175C42×C59#4112yes—
176(C2×D59)⋊C4—42no—
177(C2×D59)⋊C4—42no—
178(C2×D59)⋊C4—42no—
179(C2×D59)⋊C4—81no—
180(C2×D59)⋊C4—81no—
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⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—42no—
194C4⋊Dic59C2×Q812yes—
195C4⋊Dic59#4112yes—
196Dic59⋊C4—42no—
197Dic59⋊C4—42no—
198Dic59⋊C4—42no—
199Dic59⋊C4—42no—
200Dic59⋊C4—81no—
201Dic59⋊C4—81no—
202C4×Dic59—42no—
203C23×C59C22×D411yescharacteristic
204C22×D59—22no—
205C22×D59—41no—
206C22×D59—41no—
207C22×D59—41no—
208C22×D59—41no—
209C22×D59—41no—
210C22×D59—41no—
211Q8×C59—44no—
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×C59—21no—
227C2×C4×C59#12112yes—
228C2×C4×C59#12112yes—
229C2×C4×C59AGL(1,ℤ/8)12yes—
230C2×C4×C59C22×D411yescharacteristic
231C2×C4×C59C22×Q811yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C592−1+411yescharacteristic
235C2×C4×C592−1+411yescharacteristic
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—
255C2×Dic59—41no—
256C2×Dic59—41no—
257D236—22no—
258D236—22no—
259D236—41no—
260D236—41no—
261C4×D59—84no—
262Dic118—84no—
263C8×C59—22no—
264C8×C59—22no—
265C8×C59—22no—
266C8×C59—41no—
267#25485—591nomaximal, Sylow
268C22×C59C2×#12111yescharacteristic
269C22×C59#54811yescharacteristic
270C22×C59D4×Q811yescharacteristic
271C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C59#57511yescharacteristic
273C22×C59#57711yescharacteristic
274C22×C59#57711yescharacteristic
275C2×D59—42no—
276C2×D59—42no—
277C2×D59—42no—
278C2×D59—81no—
279C2×D59—81no—
280C2×D59—81no—
281C2×D59—81no—
282C4×C59—22no—
283C4×C59—22no—
284C4×C59—22no—
285C4×C59—41no—
286C4×C59—41no—
287C4×C59—41no—
288C4×C59#53311yescharacteristic
289C4×C59#54111yescharacteristic
290C4×C59#54711yescharacteristic
291C4×C59D4×Q811yescharacteristic
292Dic59—42no—
293Dic59—42no—
294Dic59—81no—
295Dic59—81no—
296Dic59—81no—
297Dic59—81no—
298Dic59—81no—
299C2×#476—591no—
300C2×#446D5911yescharacteristic, complemented
301#2580—591no—
302#2546—591no—
303#2088—591no—
304#2088—591no—
305#2067—591no—
306#2065—591no—
307#1993—591no—
308#1949—591no—
309#1943—591no—
310#1941—591no—
311#1936—591no—
312#1934—591no—
313#1762—591no—
314C2×C59#242911yescharacteristic
315C2×C59#305511yescharacteristic
316C2×C59#305711yescharacteristic
317C2×C59#337111yescharacteristic
318C2×C59#337311yescharacteristic
319C2×C59#338411yescharacteristic
320C2×C59#338811yescharacteristic
321D59—82no—
322C2×C4⋊Q8—591no—
323C2×C4⋊Q8—1181no—
324C2×C4×Q8C2×D5911yescharacteristic
325C2×C4×D4—591no—
326#476—1184no—
327#446—24no—
328C2×#107—591no—
329C2×#106—591no—
330C2×#106—591no—
331C2×#106—591no—
332C2×C4⋊C8C2×D5911yescharacteristic
333C2×C4⋊C8C2×D5911yescharacteristic
334C2×C4⋊C8C2×D5911yescharacteristic
335C2×#102—591no—
336C2×#102—591no—
337C2×#102—591no—
338C2×#102—591no—
339C2×#97C2×D5911yescharacteristic
340C2×#97C2×D5911yescharacteristic
341C2×C4×C8C2×D5911yescharacteristic
342#399—1181no—
343#398—1181no—
344#398—1181no—
345#393—1181no—
346#391—1181no—
347#390—591no—
348#390—591no—
349#389—591no—
350#389—591no—
351#389—591no—
352#389—591no—
353#389—591no—
354#389—591no—
355#385—1181no—
356#385—1181no—
357#382—591no—
358C4×C4⋊C4—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367C59#2548511yescharacteristic, complemented, Sylow
368C22×Q8—1181no—
369C22×Q8D23611yescharacteristic
370C22×D4—591no—
371C23×C4—1181no—
372C22×C8C22×D5911yescharacteristic
373C22×C8C22×D5911yescharacteristic
374C22×C8C22×D5911yescharacteristic
375C22×C8C22×D5911yescharacteristic
376C4⋊Q8—1182no—
377C4⋊Q8—1182no—
378C4⋊Q8—2364no—
379C4×Q8—22no—
380C4×Q8—22no—
381C4×D4—1182no—
382C4×D4—1182no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
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⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4D23611yescharacteristic
398C2×C4⋊C4D23611yescharacteristic
399C2×C4⋊C4D23611yescharacteristic
400C2×C22⋊C4—1181no—
401C2×C22⋊C4—1181no—
402C2×C22⋊C4—1181no—
403C2×C22⋊C4—1181no—
404C2×C22⋊C4—1181no—
405C2×C42—1181no—
406C2×C42C22×D5911yescharacteristic
407C2×C42C22×D5911yescharacteristic
408C2×C42C22×D5911yescharacteristic
409#107—1182no—
410#106—1182no—
411#106—1182no—
412#106—1182no—
413C4⋊C8—22no—
414C4⋊C8—22no—
415C4⋊C8C22⋊D5914yes—
416#102—594no—
417#102—1182no—
418#102—1182no—
419#102—1182no—
420#97—22no—
421#97—22no—
422C4×C8—22no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432C24—1181no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—41no—
436C2×Q8—2362no—
437C2×Q8—2362no—
438C2×Q8—2362no—
439C2×D4—592no—
440C2×D4—1181no—
441C2×D4—1181no—
442C2×D4—1181no—
443C2×D4—1181no—
444C2×D4—1181no—
445C22×C4—1181no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—1181no—
452C22×C4—4721no—
453C22×C4—4721no—
454C22×C4—2362no—
455C22×C4C2×D23611yescharacteristic
456C22×C4C2×D23611yescharacteristic
457C22×C4C2×D23611yescharacteristic
458C22×C4C2×D23611yescharacteristic
459C22×C4C2×D23611yescharacteristic
460C22×C4C2×D23611yescharacteristic
461C22×C4C23×D5911yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8Q8×D5912yes—
472C2×C8C2×C22⋊D5912yes—
473C2×C8C2×C22⋊D5912yes—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—592no—
478C4⋊C4—592no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
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—4721no—
492C4⋊C4—4721no—
493C4⋊C4—2362no—
494C4⋊C4—2362no—
495C4⋊C4—2362no—
496C4⋊C4—2362no—
497C4⋊C4—2362no—
498C22⋊C4—4721no—
499C22⋊C4—4721no—
500C22⋊C4—2362no—
501C22⋊C4—2362no—
502C22⋊C4—2362no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—2362no—
508C42#1174568512yes—
509C42C2×C22⋊D5912yes—
510C23—1182no—
511C23—2361no—
512C23—2361no—
513C23—2361no—
514C23—2361no—
515C23—2361no—
516C23—2361no—
517C23C22×D23611yescharacteristic
518Q8—44no—
519Q8—4724no—
520D4—1182no—
521D4—1182no—
522D4—2361no—
523D4—2361no—
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—21no—
538C2×C4—21no—
539C2×C4—1182no—
540C2×C4—1182no—
541C2×C4—4724no—
542C2×C4—2361no—
543C2×C4—2361no—
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×C4#1427370112yes—
562C2×C4#1427370112yes—
563C2×C4#1427375512yes—
564C2×C4C22×C22⋊D5911yescharacteristic
565C2×C4C2×Q8×D5911yescharacteristic
566C2×C4#41×D5911yescharacteristic
567C2×C4C2×#1174568511yescharacteristic
568C2×C4#1427377411yescharacteristic
569C2×C4#1427377511yescharacteristic
570C8—22no—
571C8—22no—
572C8—22no—
573C8—41no—
574C22—4721no—
575C22—4721no—
576C22—4721no—
577C22—4721no—
578C22—2362no—
579C22—2362no—
580C22—2362no—
581C22C2×#1427370111yescharacteristic
582C22#1683806411yescharacteristic
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—4721no—
598C4—4721no—
599C4—2362no—
600C4—2362no—
601C4#1683795611yescharacteristic
602C4#1683801011yescharacteristic
603C4#1683805111yescharacteristic
604C4Q8×C22⋊D5911yescharacteristic
605C2—4722no—
606C2#2142768911yescharacteristic
607C2#2143371511yescharacteristic
608C2#2143373511yescharacteristic
609C2#2143686611yescharacteristic
610C2#2143689011yescharacteristic
611C2#2143699711yescharacteristic
612C2#2143704311yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#446C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenterC2×#446D59
hypocenterC59#25485
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
3. #21429081C2semidirect
4. #21425594C2semidirect
5. #21425594C2semidirect
8. #21425069C2semidirect
9. #21424866C2semidirect
10. #21424834C2semidirect
11. #21424824C2semidirect
15. C2×C59×#446C2semidirect
96. #14273649Q8semidirect
300. C2×#446D59semidirect
367. C59#25485semidirect

Maximal subgroups

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

subgroupindex
C2×#168376932
#214294542
#214290812
#214255942
#214254832
#214254692
#214250692
#214248662
#214248342
#214248242
#214247932
#214247792
#214239222
C2×C59×#4462
#2548559