Finite Group Databasebeta

← #28524999

Subgroups — #28524999

Order 15104.

1462 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837340C211yesmaximal, characteristic, complemented
2C2×#16837337C211yesmaximal, characteristic
3C2×#16837293C211yesmaximal, characteristic
4#21422348C212yesmaximal, complemented
5#21422346C211yesmaximal, characteristic, complemented
6#21422342C212yesmaximal
7#21422338C211yesmaximal, characteristic
8#21422334C212yesmaximal
9#21422329C212yesmaximal, complemented
10#21422326C211yesmaximal, characteristic, complemented
11C2×C4×C4⋊C4×C59C211yesmaximal, characteristic, complemented
12C22×C4⋊Dic59C2212yes—
13C22×Dic59⋊C4C2211yescharacteristic
14C22×Dic59⋊C4C2212yes—
15C22×(C2×D59)⋊C4C2211yescharacteristic
16C22×(C2×D59)⋊C4C2211yescharacteristic
17C22×(C2×D59)⋊C4C2212yes—
18C22×(C2×D59)⋊C4C2211yescharacteristic
19C22×(C2×D59)⋊C4C2212yes—
20#16837340—24no—
21#16837337—24no—
22#16837293—24no—
23C2×C59⋊#95C2211yescharacteristic
24C2×C59⋊#95C2212yes—
25C2×C59⋊#95C2212yes—
26C2×C59⋊#95C2211yescharacteristic
27C2×C59⋊#95C2211yescharacteristic
28C2×C59⋊#95C2212yes—
29C2×C59⋊#95C2211yescharacteristic
30C2×C59⋊#95C2212yes—
31C2×C59⋊#95C2212yes—
32C2×C59⋊#95C2212yes—
33C22×C4⋊C4×C59C2211yescharacteristic
34C22×C4⋊C4×C59C2211yescharacteristic
35C22×C42×C59C2211yescharacteristic
36C22×C42×C59C2212yes—
37C4×C4⋊C4×C59—24no—
38C2×#95×C59C2212yes—
39C24×D59C2311yescharacteristic
40C23×Dic59C2311yescharacteristic
41C23×Dic59C2312yes—
42C23×Dic59C2311yescharacteristic
43C23×Dic59C2311yescharacteristic
44C23×Dic59C2312yes—
45C2×C4⋊Dic59—24no—
46C2×C4⋊Dic59—24no—
47C2×C4⋊Dic59—42no—
48C2×Dic59⋊C4—22no—
49C2×Dic59⋊C4—24no—
50C2×Dic59⋊C4—22no—
51C2×Dic59⋊C4—22no—
52C2×Dic59⋊C4—24no—
53C2×Dic59⋊C4—42no—
54C2×(C2×D59)⋊C4—22no—
55C2×(C2×D59)⋊C4—22no—
56C2×(C2×D59)⋊C4—22no—
57C2×(C2×D59)⋊C4—22no—
58C2×(C2×D59)⋊C4—24no—
59C2×(C2×D59)⋊C4—22no—
60C2×(C2×D59)⋊C4—24no—
61C2×(C2×D59)⋊C4—22no—
62C2×(C2×D59)⋊C4—24no—
63C2×(C2×D59)⋊C4—24no—
64C2×(C2×D59)⋊C4—42no—
65C2×(C2×D59)⋊C4—42no—
66C2×(C2×D59)⋊C4—42no—
67C2×(C2×D59)⋊C4Q814yes—
68C59⋊#95—22no—
69C59⋊#95—22no—
70C59⋊#95—22no—
71C59⋊#95—24no—
72C59⋊#95—22no—
73C59⋊#95—24no—
74C59⋊#95—24no—
75C59⋊#95—24no—
76C59⋊#95—42no—
77C59⋊#95—42no—
78C23×C4×C59C2312yes—
79C23×C4×C59C2312yes—
80C23×C4×C59C2311yescharacteristic
81C23×C4×C59C2311yescharacteristic
82C23×C4×C59C2311yescharacteristic
83C2×C4⋊C4×C59—22no—
84C2×C4⋊C4×C59—22no—
85C2×C4⋊C4×C59—24no—
86C2×C4⋊C4×C59—22no—
87C2×C4⋊C4×C59D414yes—
88C2×C42×C59—24no—
89C2×C42×C59—24no—
90C2×C42×C59—42no—
91C2×C42×C59—42no—
92C2×C42×C59D414yes—
93#95×C59—24no—
94C24×C59C2411yescharacteristic
95C23×D59—21no—
96C23×D59—21no—
97C23×D59—21no—
98C23×D59—22no—
99C23×D59—22no—
100C23×D59—21no—
101C23×D59—22no—
102C23×D59—22no—
103C23×D59—21no—
104C23×D59—21no—
105C23×D59C2×Q812yes—
106C22×C4×C59—22no—
107C22×C4×C59—22no—
108C22×C4×C59—22no—
109C22×C4×C59—22no—
110C22×C4×C59—22no—
111C22×C4×C59—21no—
112C22×C4×C59—21no—
113C22×C4×C59—22no—
114C22×C4×C59—22no—
115C22×C4×C59—22no—
116C22×C4×C59—22no—
117C22×C4×C59—22no—
118C22×C4×C59—22no—
119C22×C4×C59—22no—
120C22×C4×C59—22no—
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59C2×D414yes—
126C22×C4×C59C2×D412yes—
127C22×C4×C59C2×D412yes—
128C22×C4×C59C2×D412yes—
129C22×C4×C59C2×D412yes—
130C22×C4×C59C2×Q812yes—
131C22×C4×C59#4114yes—
132C22×C4×C59#4112yes—
133C22×C4×C59#4112yes—
134C22×C4×C59#4114yes—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—22no—
138C22×Dic59—22no—
139C22×Dic59—22no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—22no—
143C22×Dic59—21no—
144C22×Dic59—22no—
145C22×Dic59—22no—
146C22×Dic59—21no—
147C22×Dic59—22no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—22no—
153C22×Dic59—22no—
154C22×Dic59—21no—
155C22×Dic59—22no—
156C22×Dic59—22no—
157C22×Dic59—21no—
158C22×Dic59—22no—
159C22×Dic59—22no—
160C22×Dic59—22no—
161C22×Dic59—22no—
162C22×Dic59—22no—
163C22×Dic59—22no—
164C22×Dic59C2×Q812yes—
165C22×Dic59#4114yes—
166C4⋊C4×C59—28no—
167C4⋊C4×C59—28no—
168C42×C59—28no—
169C42×C59—48no—
170(C2×D59)⋊C4—28no—
171(C2×D59)⋊C4—48no—
172(C2×D59)⋊C4—48no—
173(C2×D59)⋊C4—44no—
174(C2×D59)⋊C4—44no—
175C4⋊Dic59—48no—
176Dic59⋊C4—48no—
177Dic59⋊C4—44no—
178C23×C59C22×D411yescharacteristic
179C23×C59C22×D411yescharacteristic
180C23×C59C22×Q811yescharacteristic
181C23×C59C2×#4112yes—
182C23×C59C2×#4111yescharacteristic
183C23×C59C2×#4112yes—
184C23×C59C2×#4112yes—
185C23×C59C2×#4111yescharacteristic
186C23×C59H(5,𝔽2)12yes—
187C23×C59H(5,𝔽2)11yescharacteristic
188C23×C592−1+411yescharacteristic
189C22×D59—22no—
190C22×D59—24no—
191C22×D59—22no—
192C22×D59—24no—
193C22×D59—22no—
194C22×D59—41no—
195C22×D59—41no—
196C22×D59—41no—
197C22×D59—42no—
198C22×D59—42no—
199C22×D59—41no—
200C22×D59—42no—
201C22×D59—41no—
202C22×D59—41no—
203C22×D59—42no—
204C22×D59—42no—
205C22×D59—42no—
206C22×D59—41no—
207C22×D59—42no—
208C22×D59—42no—
209C22×D59—42no—
210C22×D59—41no—
211C22×D59—42no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—22no—
215C2×C4×C59—24no—
216C2×C4×C59—22no—
217C2×C4×C59—24no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—24no—
221C2×C4×C59—24no—
222C2×C4×C59—24no—
223C2×C4×C59—24no—
224C2×C4×C59—24no—
225C2×C4×C59—24no—
226C2×C4×C59—24no—
227C2×C4×C59—24no—
228C2×C4×C59—24no—
229C2×C4×C59—24no—
230C2×C4×C59—42no—
231C2×C4×C59—42no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—42no—
235C2×C4×C59—42no—
236C2×C4×C59—42no—
237C2×C4×C59—42no—
238C2×C4×C59#12114yes—
239C2×C4×C59C22⋊Q814yes—
240C2×C4×C59#12314yes—
241C2×Dic59—22no—
242C2×Dic59—24no—
243C2×Dic59—24no—
244C2×Dic59—22no—
245C2×Dic59—24no—
246C2×Dic59—22no—
247C2×Dic59—41no—
248C2×Dic59—42no—
249C2×Dic59—41no—
250C2×Dic59—42no—
251C2×Dic59—41no—
252C2×Dic59—42no—
253C2×Dic59—42no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—42no—
260C2×Dic59—42no—
261C2×Dic59—41no—
262C2×Dic59—42no—
263C2×Dic59—42no—
264C2×Dic59—41no—
265C2×Dic59—42no—
266C2×Dic59—42no—
267C2×Dic59—42no—
268C2×Dic59—42no—
269C2×Dic59—42no—
270C2×Dic59—42no—
271C2×Dic59—42no—
272#15103—591nomaximal, Sylow
273C22×C59C2×#12111yescharacteristic
274C22×C59C2×C22⋊Q811yescharacteristic
275C22×C59C2×#12311yescharacteristic
276C22×C59#52511yescharacteristic
277C22×C59#52912yes—
278C22×C59#52912yes—
279C22×C59#53211yescharacteristic
280C22×C59#53212yes—
281C22×C59#53312yes—
282C22×C59#53511yescharacteristic
283C22×C59#53711yescharacteristic
284C22×C59#53812yes—
285C22×C59#53912yes—
286C22×C59#54612yes—
287C22×C59D4×Q811yescharacteristic
288C22×C59#55011yescharacteristic
289C22×C59#55112yes—
290C22×C59#55212yes—
291C22×C59#55312yes—
292C22×C59#55412yes—
293C22×C59#55411yescharacteristic
294C22×C59#55611yescharacteristic
295C22×C59#55912yes—
296C2×D59—44no—
297C2×D59—42no—
298C2×D59—42no—
299C2×D59—42no—
300C2×D59—44no—
301C2×D59—82no—
302C2×D59—81no—
303C2×D59—81no—
304C2×D59—81no—
305C2×D59—81no—
306C2×D59—82no—
307C4×C59—24no—
308C4×C59—24no—
309C4×C59—24no—
310C4×C59—44no—
311C4×C59—44no—
312Dic59—42no—
313Dic59—44no—
314Dic59—81no—
315Dic59—81no—
316Dic59—82no—
317C2×#400—591no—
318C2×#399—591no—
319C2×#388—591no—
320C2×C4×C4⋊C4D5911yescharacteristic, complemented
321#1441—592no—
322#1441—591no—
323#1439—592no—
324#1439—591no—
325#1437—592no—
326#1436—592no—
327C4⋊#95—591no—
328C2×C59#239611yescharacteristic
329C2×C59#248711yescharacteristic
330C2×C59#249811yescharacteristic
331C2×C59#253811yescharacteristic
332C2×C59#254411yescharacteristic
333C2×C59#255412yes—
334C2×C59#255612yes—
335C2×C59#258912yes—
336C2×C59#259612yes—
337C2×C59#263611yescharacteristic
338C2×C59#285411yescharacteristic
339D59—82no—
340C22×C4⋊C4—591no—
341C22×C4⋊C4—592no—
342C22×C4⋊C4—592no—
343C22×C4⋊C4C2×D5911yescharacteristic
344C22×C4⋊C4C2×D5911yescharacteristic
345C22×C22⋊C4—591no—
346C22×C22⋊C4—591no—
347C22×C22⋊C4—591no—
348C22×C22⋊C4—592no—
349C22×C22⋊C4—592no—
350C22×C42C2×D5911yescharacteristic
351C22×C42C2×D5912yes—
352#400—1184no—
353#399—1184no—
354#388—1184no—
355C4×C4⋊C4—24no—
356C2×#95—591no—
357C2×#95—591no—
358C2×#95—592no—
359C2×#95—592no—
360C2×#95—592no—
361C2×#95—591no—
362C2×#95—592no—
363C2×#95—591no—
364C2×#95—592no—
365C2×#95—592no—
366C2×#95C2×D5912yes—
367C59#1510311yescharacteristic, complemented, Sylow
368C25—591no—
369C23×C4—592no—
370C23×C4—591no—
371C23×C4—591no—
372C23×C4—591no—
373C23×C4—592no—
374C23×C4C22×D5912yes—
375C23×C4C22×D5912yes—
376C23×C4C22×D5911yescharacteristic
377C23×C4C22×D5911yescharacteristic
378C23×C4C22×D5911yescharacteristic
379C2×C4⋊C4—24no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—1184no—
384C2×C4⋊C4—1184no—
385C2×C4⋊C4—1184no—
386C2×C4⋊C4—1182no—
387C2×C4⋊C4—1182no—
388C2×C4⋊C4—1184no—
389C2×C4⋊C4—1182no—
390C2×C4⋊C4—2362no—
391C2×C4⋊C4—2362no—
392C2×C4⋊C4D23614yes—
393C2×C22⋊C4—594no—
394C2×C22⋊C4—1182no—
395C2×C22⋊C4—1182no—
396C2×C22⋊C4—1184no—
397C2×C22⋊C4—1184no—
398C2×C22⋊C4—1182no—
399C2×C22⋊C4—1184no—
400C2×C22⋊C4—1182no—
401C2×C22⋊C4—1184no—
402C2×C22⋊C4—1182no—
403C2×C22⋊C4—1182no—
404C2×C22⋊C4—2362no—
405C2×C22⋊C4—2362no—
406C2×C22⋊C4—2362no—
407C2×C42—24no—
408C2×C42—24no—
409C2×C42—42no—
410C2×C42—42no—
411C2×C42C22⋊D5914yes—
412#95—24no—
413#95—1182no—
414#95—1184no—
415#95—1182no—
416#95—1184no—
417#95—1184no—
418#95—1184no—
419#95—1182no—
420#95—1182no—
421#95—2362no—
422#95—2362no—
423C24—592no—
424C24—1181no—
425C24—1181no—
426C24—1182no—
427C24—1182no—
428C24—1182no—
429C24—1181no—
430C24—1181no—
431C24—1182no—
432C24—1181no—
433C24—1181no—
434C24C23×D5911yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—592no—
455C22×C4—594no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1182no—
459C22×C4—1182no—
460C22×C4—1182no—
461C22×C4—1181no—
462C22×C4—1182no—
463C22×C4—1181no—
464C22×C4—1182no—
465C22×C4—1181no—
466C22×C4—1182no—
467C22×C4—1181no—
468C22×C4—1181no—
469C22×C4—1181no—
470C22×C4—1182no—
471C22×C4—1181no—
472C22×C4—1181no—
473C22×C4—1182no—
474C22×C4—1182no—
475C22×C4—1182no—
476C22×C4—1181no—
477C22×C4—1182no—
478C22×C4—1182no—
479C22×C4—1181no—
480C22×C4—1182no—
481C22×C4—1182no—
482C22×C4—1182no—
483C22×C4—1182no—
484C22×C4—1182no—
485C22×C4C2×D23614yes—
486C22×C4C2×D23612yes—
487C22×C4#1174568514yes—
488C22×C4#1174568512yes—
489C22×C4#1174568514yes—
490C22×C4Q8×D5912yes—
491C22×C4#1174568912yes—
492C22×C4C2×C22⋊D5912yes—
493C22×C4C2×C22⋊D5912yes—
494C22×C4C2×C22⋊D5912yes—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—2368no—
498C4⋊C4—2368no—
499C4⋊C4—2364no—
500C22⋊C4—1188no—
501C22⋊C4—2368no—
502C22⋊C4—2368no—
503C22⋊C4—2364no—
504C22⋊C4—2364no—
505C42—28no—
506C42—48no—
507C23—1182no—
508C23—1184no—
509C23—1182no—
510C23—1184no—
511C23—1182no—
512C23—2361no—
513C23—2361no—
514C23—2361no—
515C23—2362no—
516C23—2362no—
517C23—2361no—
518C23—2362no—
519C23—2361no—
520C23—2361no—
521C23—2362no—
522C23—2362no—
523C23—2362no—
524C23—2361no—
525C23—2362no—
526C23—2362no—
527C23—2362no—
528C23—2361no—
529C23—2362no—
530C23C22×C22⋊D5911yescharacteristic
531C23C22×D23611yescharacteristic
532C23C2×Q8×D5911yescharacteristic
533C23#41×D5912yes—
534C23C2×#1174568511yescharacteristic
535C23C2×#1174568512yes—
536C23C2×#1174568512yes—
537C23C2×#1174568911yescharacteristic
538C23#1427377212yes—
539C23#1427377311yescharacteristic
540C23#1427377411yescharacteristic
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—1182no—
568C2×C4—1184no—
569C2×C4—1184no—
570C2×C4—1182no—
571C2×C4—1184no—
572C2×C4—1182no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4—2362no—
578C2×C4—2362no—
579C2×C4—2361no—
580C2×C4—2362no—
581C2×C4—2362no—
582C2×C4—2361no—
583C2×C4—2362no—
584C2×C4—2362no—
585C2×C4—2362no—
586C2×C4—2362no—
587C2×C4—2362no—
588C2×C4—2362no—
589C2×C4—2362no—
590C2×C4—2361no—
591C2×C4—2362no—
592C2×C4—2361no—
593C2×C4—2362no—
594C2×C4—2361no—
595C2×C4—2362no—
596C2×C4—2362no—
597C2×C4—2361no—
598C2×C4#1427370114yes—
599C2×C4#1427370314yes—
600C2×C4#1427371014yes—
601C22—4721no—
602C22—4721no—
603C22—4721no—
604C22—4722no—
605C22—4722no—
606C22—4721no—
607C22—2362no—
608C22—2364no—
609C22—2362no—
610C22—2364no—
611C22—2362no—
612C22C2×#1427370111yescharacteristic
613C22C2×#1427370311yescharacteristic
614C22C2×#1427371011yescharacteristic
615C22#1683791311yescharacteristic
616C22#1683793612yes—
617C22#1683793912yes—
618C22#1683794811yescharacteristic
619C22#1683795212yes—
620C22#1683795612yes—
621C22#1683796911yescharacteristic
622C22#1683798011yescharacteristic
623C22#1683798512yes—
624C22#1683799412yes—
625C22#1683804712yes—
626C22Q8×D23611yescharacteristic
627C22#1683807411yescharacteristic
628C22#1683807612yes—
629C22#1683808512yes—
630C22#1683810012yes—
631C22#1683810712yes—
632C22#1683811211yescharacteristic
633C22#1683812511yescharacteristic
634C22#1683813512yes—
635C4—24no—
636C4—24no—
637C4—24no—
638C4—44no—
639C4—44no—
640C4—4721no—
641C4—4721no—
642C4—4722no—
643C4—2362no—
644C4—2364no—
645C2—4722no—
646C2#2142736611yescharacteristic
647C2#2142840711yescharacteristic
648C2#2142853411yescharacteristic
649C2#2142901411yescharacteristic
650C2#2142906311yescharacteristic
651C2#2142915012yes—
652C2#2142917412yes—
653C2#2142954712yes—
654C2#2142963212yes—
655C2#2142999111yescharacteristic
656C2#2143242711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D59
commutator, socle, cosocleC24×C59C24
FittingC2×C4×C4⋊C4×C59C2
prime radicalC24×D59C23
hypercenterC2×C4×C4⋊C4D59
hypocenterC59#15103
LCSG ▷ C24×C59 ▷ C59
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16837340C2semidirect
4. #21422348C2semidirect
5. #21422346C2semidirect
9. #21422329C2semidirect
10. #21422326C2semidirect
11. C2×C4×C4⋊C4×C59C2semidirect
320. C2×C4×C4⋊C4D59semidirect
367. C59#15103semidirect

Maximal subgroups

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

subgroupindex
C2×#168373402
C2×#168373372
C2×#168372932
#214223482
#214223462
#214223422
#214223382
#214223342
#214223292
#214223262
C2×C4×C4⋊C4×C592
#1510359