Finite Group Databasebeta

← #28515501

Subgroups — #28515501

Order 15104.

1406 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837337C211yesmaximal, characteristic
2C2×#16837293C211yesmaximal, characteristic
3#21422364C211yesmaximal, characteristic, complemented
4#21422355C211yesmaximal, characteristic, complemented
5#21422347C212yesmaximal, complemented
6#21422344C212yesmaximal
7#21422338C211yesmaximal, characteristic
8#21422334C212yesmaximal
9#21422329C212yesmaximal, complemented
10#21422323C211yesmaximal, characteristic, complemented
11C2×C4×C4⋊C4×C59C211yesmaximal, characteristic, complemented
12C22×Dic59⋊C4C2211yescharacteristic
13C22×Dic59⋊C4C2212yes—
14C22×(C2×D59)⋊C4C2211yescharacteristic
15C22×(C2×D59)⋊C4C2211yescharacteristic
16C22×(C2×D59)⋊C4C2212yes—
17C22×(C2×D59)⋊C4C2211yescharacteristic
18C22×(C2×D59)⋊C4C2212yes—
19#16837337C418yescomplemented
20#16837293—24no—
21C2×C59⋊#95C2211yescharacteristic
22C2×C59⋊#95C2212yes—
23C2×C59⋊#95C2211yescharacteristic
24C2×C59⋊#95C2211yescharacteristic
25C2×C59⋊#95C2212yes—
26C2×C59⋊#95C2212yes—
27C2×C59⋊#95C2212yes—
28C2×C59⋊#95C2212yes—
29C2×C59⋊#95C2212yes—
30C2×C59⋊#95C2212yes—
31C2×C59⋊#95C2211yescharacteristic
32C22×C4⋊C4×C59C2211yescharacteristic
33C22×C4⋊C4×C59C2211yescharacteristic
34C22×C42×C59C2211yescharacteristic
35C22×C42×C59C2212yes—
36C4×C4⋊C4×C59—24no—
37C2×#95×C59C2212yes—
38C24×D59C2311yescharacteristic
39C23×Dic59C2311yescharacteristic
40C23×Dic59C2312yes—
41C23×Dic59C2311yescharacteristic
42C23×Dic59C2312yes—
43C23×Dic59C2311yescharacteristic
44C2×Dic59⋊C4—22no—
45C2×Dic59⋊C4—42no—
46C2×Dic59⋊C4—42no—
47C2×Dic59⋊C4—42no—
48C2×Dic59⋊C4C2×C418yes—
49C2×(C2×D59)⋊C4—22no—
50C2×(C2×D59)⋊C4—22no—
51C2×(C2×D59)⋊C4—22no—
52C2×(C2×D59)⋊C4—22no—
53C2×(C2×D59)⋊C4—24no—
54C2×(C2×D59)⋊C4—24no—
55C2×(C2×D59)⋊C4—42no—
56C2×(C2×D59)⋊C4—42no—
57C2×(C2×D59)⋊C4—42no—
58C2×(C2×D59)⋊C4—42no—
59C2×(C2×D59)⋊C4—42no—
60C2×(C2×D59)⋊C4C2×C414yes—
61C2×(C2×D59)⋊C4C2×C418yes—
62C59⋊#95—22no—
63C59⋊#95—22no—
64C59⋊#95—24no—
65C59⋊#95—24no—
66C59⋊#95—22no—
67C59⋊#95—42no—
68C59⋊#95—42no—
69C59⋊#95—42no—
70C59⋊#95—42no—
71C59⋊#95—42no—
72C59⋊#95C2×C414yes—
73C23×C4×C59C2312yes—
74C23×C4×C59C2311yescharacteristic
75C23×C4×C59C2312yes—
76C23×C4×C59C2311yescharacteristic
77C23×C4×C59C2311yescharacteristic
78C2×C4⋊C4×C59—22no—
79C2×C4⋊C4×C59—22no—
80C2×C4⋊C4×C59—24no—
81C2×C4⋊C4×C59—22no—
82C2×C4⋊C4×C59C2×C414yes—
83C2×C42×C59—24no—
84C2×C42×C59—24no—
85C2×C42×C59—42no—
86C2×C42×C59—42no—
87C2×C42×C59D414yes—
88#95×C59—24no—
89C24×C59C2411yescharacteristic
90C23×D59—21no—
91C23×D59—21no—
92C23×D59—22no—
93C23×D59—22no—
94C23×D59—22no—
95C23×D59—22no—
96C23×D59—21no—
97C23×D59—21no—
98C23×D59—21no—
99C23×D59—21no—
100C23×D59C22×C412yes—
101C22×C4×C59—22no—
102C22×C4×C59—22no—
103C22×C4×C59—22no—
104C22×C4×C59—21no—
105C22×C4×C59—21no—
106C22×C4×C59—22no—
107C22×C4×C59—22no—
108C22×C4×C59—22no—
109C22×C4×C59—22no—
110C22×C4×C59—22no—
111C22×C4×C59—22no—
112C22×C4×C59—22no—
113C22×C4×C59—22no—
114C22×C4×C59—22no—
115C22×C4×C59—21no—
116C22×C4×C59—22no—
117C22×C4×C59—22no—
118C22×C4×C59—22no—
119C22×C4×C59—22no—
120C22×C4×C59—21no—
121C22×C4×C59—22no—
122C22×C4×C59C22×C414yes—
123C22×C4×C59C22×C412yes—
124C22×C4×C59C2×D412yes—
125C22×C4×C59C2×D412yes—
126C22×C4×C59C2×D412yes—
127C22×C4×C59#4114yes—
128C22×C4×C59#4112yes—
129C22×C4×C59#4114yes—
130C22×Dic59—21no—
131C22×Dic59—22no—
132C22×Dic59—22no—
133C22×Dic59—21no—
134C22×Dic59—22no—
135C22×Dic59—22no—
136C22×Dic59—22no—
137C22×Dic59—22no—
138C22×Dic59—22no—
139C22×Dic59—22no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—22no—
143C22×Dic59—22no—
144C22×Dic59—22no—
145C22×Dic59—22no—
146C22×Dic59—22no—
147C22×Dic59—21no—
148C22×Dic59—22no—
149C22×Dic59—22no—
150C22×Dic59—22no—
151C22×Dic59—22no—
152C22×Dic59—22no—
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—22no—
157C22×Dic59C22×C414yes—
158C22×Dic59C22×C412yes—
159C4⋊C4×C59—28no—
160C4⋊C4×C59—28no—
161C42×C59—28no—
162C42×C59—48no—
163(C2×D59)⋊C4—28no—
164(C2×D59)⋊C4—48no—
165(C2×D59)⋊C4—44no—
166(C2×D59)⋊C4—48no—
167(C2×D59)⋊C4—44no—
168Dic59⋊C4—48no—
169Dic59⋊C4—44no—
170C23×C59C23×C411yescharacteristic
171C23×C59C22×D411yescharacteristic
172C23×C59C2×#4111yescharacteristic
173C23×C59C2×#4112yes—
174C23×C59C2×#4112yes—
175C23×C59H(5,𝔽2)12yes—
176C23×C59H(5,𝔽2)12yes—
177C23×C59H(5,𝔽2)11yescharacteristic
178C23×C592−1+411yescharacteristic
179C23×C592−1+411yescharacteristic
180C23×C592−1+411yescharacteristic
181C22×D59—22no—
182C22×D59—24no—
183C22×D59—24no—
184C22×D59—22no—
185C22×D59—22no—
186C22×D59—41no—
187C22×D59—41no—
188C22×D59—42no—
189C22×D59—41no—
190C22×D59—41no—
191C22×D59—41no—
192C22×D59—41no—
193C22×D59—42no—
194C22×D59—42no—
195C22×D59—42no—
196C22×D59—42no—
197C22×D59—42no—
198C22×D59—42no—
199C22×D59—41no—
200C22×D59—42no—
201C22×D59—41no—
202C22×D59—42no—
203C22×D59—42no—
204C2×C4×C59—24no—
205C2×C4×C59—22no—
206C2×C4×C59—22no—
207C2×C4×C59—24no—
208C2×C4×C59—22no—
209C2×C4×C59—22no—
210C2×C4×C59—24no—
211C2×C4×C59—24no—
212C2×C4×C59—24no—
213C2×C4×C59—24no—
214C2×C4×C59—24no—
215C2×C4×C59—22no—
216C2×C4×C59—24no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
219C2×C4×C59—24no—
220C2×C4×C59—24no—
221C2×C4×C59—42no—
222C2×C4×C59—42no—
223C2×C4×C59—42no—
224C2×C4×C59—42no—
225C2×C4×C59—42no—
226C2×C4×C59—42no—
227C2×C4×C59—42no—
228C2×C4×C59—42no—
229C2×C4×C59—42no—
230C2×C4×C59—42no—
231C2×C4×C59C4×D414yes—
232C2×C4×C59#12314yes—
233C2×Dic59—22no—
234C2×Dic59—24no—
235C2×Dic59—24no—
236C2×Dic59—24no—
237C2×Dic59—24no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—42no—
241C2×Dic59—42no—
242C2×Dic59—42no—
243C2×Dic59—42no—
244C2×Dic59—42no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—42no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—42no—
253C2×Dic59—42no—
254C2×Dic59—42no—
255C2×Dic59—42no—
256C2×Dic59—42no—
257C2×Dic59—42no—
258C2×Dic59—42no—
259C2×Dic59—42no—
260C2×Dic59—42no—
261C2×Dic59—42no—
262#14305—591nomaximal, Sylow
263C22×C59C2×C4×D411yescharacteristic
264C22×C59C4×#4112yes—
265C22×C59#51812yes—
266C22×C59#51911yescharacteristic
267C22×C59#52011yescharacteristic
268C22×C59C2×#12311yescharacteristic
269C22×C59#52511yescharacteristic
270C22×C59#52912yes—
271C22×C59#53511yescharacteristic
272C22×C59#53511yescharacteristic
273C22×C59#53611yescharacteristic
274C22×C59#53711yescharacteristic
275C22×C59#53812yes—
276C22×C59#53912yes—
277C22×C59#53912yes—
278C22×C59#53912yes—
279C22×C59#54012yes—
280C22×C59#54012yes—
281C22×C59#54111yescharacteristic
282C22×C59#54212yes—
283C22×C59#54211yescharacteristic
284C22×C59#54212yes—
285C22×C59#54212yes—
286C2×D59—42no—
287C2×D59—42no—
288C2×D59—44no—
289C2×D59—44no—
290C2×D59—42no—
291C2×D59—81no—
292C2×D59—81no—
293C2×D59—81no—
294C2×D59—81no—
295C2×D59—82no—
296C2×D59—82no—
297C4×C59—24no—
298C4×C59—24no—
299C4×C59—44no—
300C4×C59—42no—
301C4×C59—44no—
302Dic59—42no—
303Dic59—44no—
304Dic59—81no—
305Dic59—81no—
306Dic59—82no—
307C2×#399—591no—
308C2×#388—591no—
309C2×C4×C4⋊C4D5911yescharacteristic, complemented
310#1445—591no—
311#1442—591no—
312#1441—592no—
313#1440—592no—
314#1439—591no—
315#1437—592no—
316#1436—592no—
317#1434—591no—
318C2×C59C4×#12311yescharacteristic
319C2×C59#232311yescharacteristic
320C2×C59#232811yescharacteristic
321C2×C59#233212yes—
322C2×C59#233512yes—
323C2×C59#262311yescharacteristic
324C2×C59#262411yescharacteristic
325C2×C59#263211yescharacteristic
326C2×C59#263611yescharacteristic
327C2×C59#265112yes—
328C2×C59#265412yes—
329D59—82no—
330C22×C4⋊C4—592no—
331C22×C4⋊C4—591no—
332C22×C4⋊C4C2×D5911yescharacteristic
333C22×C4⋊C4C2×D5911yescharacteristic
334C22×C22⋊C4—591no—
335C22×C22⋊C4—592no—
336C22×C22⋊C4—591no—
337C22×C22⋊C4—592no—
338C22×C22⋊C4—591no—
339C22×C42C2×D5911yescharacteristic
340C22×C42C2×D5912yes—
341#399—598no—
342#388—1184no—
343C4×C4⋊C4—24no—
344C2×#95—591no—
345C2×#95—592no—
346C2×#95—591no—
347C2×#95—591no—
348C2×#95—592no—
349C2×#95—591no—
350C2×#95—592no—
351C2×#95—592no—
352C2×#95—592no—
353C2×#95—592no—
354C2×#95—592no—
355C2×#95C2×D5912yes—
356C59#1430511yescharacteristic, complemented, Sylow
357C25—591no—
358C23×C4—591no—
359C23×C4—592no—
360C23×C4—591no—
361C23×C4—591no—
362C23×C4—592no—
363C23×C4C22×D5912yes—
364C23×C4C22×D5912yes—
365C23×C4C22×D5911yescharacteristic
366C23×C4C22×D5911yescharacteristic
367C23×C4C22×D5911yescharacteristic
368C2×C4⋊C4—24no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—598no—
373C2×C4⋊C4—1182no—
374C2×C4⋊C4—2362no—
375C2×C4⋊C4—2362no—
376C2×C4⋊C4—2362no—
377C2×C4⋊C4C4×D5914yes—
378C2×C22⋊C4—594no—
379C2×C22⋊C4—598no—
380C2×C22⋊C4—1182no—
381C2×C22⋊C4—1182no—
382C2×C22⋊C4—1184no—
383C2×C22⋊C4—1184no—
384C2×C22⋊C4—1182no—
385C2×C22⋊C4—1182no—
386C2×C22⋊C4—2362no—
387C2×C22⋊C4—2362no—
388C2×C22⋊C4—2362no—
389C2×C22⋊C4—2362no—
390C2×C22⋊C4—2362no—
391C2×C42—24no—
392C2×C42—24no—
393C2×C42—42no—
394C2×C42—42no—
395C2×C42C22⋊D5914yes—
396#95—24no—
397#95—594no—
398#95—1184no—
399#95—1184no—
400#95—1182no—
401#95—1182no—
402#95—1182no—
403#95—2362no—
404#95—2362no—
405#95—2362no—
406#95—2362no—
407#95—2362no—
408C24—592no—
409C24—1181no—
410C24—1181no—
411C24—1182no—
412C24—1182no—
413C24—1182no—
414C24—1182no—
415C24—1181no—
416C24—1181no—
417C24—1181no—
418C24—1181no—
419C24C23×D5911yescharacteristic
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—594no—
442C22×C4—592no—
443C22×C4—1181no—
444C22×C4—1182no—
445C22×C4—1182no—
446C22×C4—1181no—
447C22×C4—1182no—
448C22×C4—1182no—
449C22×C4—1182no—
450C22×C4—1182no—
451C22×C4—1182no—
452C22×C4—1182no—
453C22×C4—1181no—
454C22×C4—1181no—
455C22×C4—1182no—
456C22×C4—1182no—
457C22×C4—1182no—
458C22×C4—1182no—
459C22×C4—1182no—
460C22×C4—1182no—
461C22×C4—1181no—
462C22×C4—1182no—
463C22×C4—1182no—
464C22×C4—1182no—
465C22×C4—1182no—
466C22×C4—1181no—
467C22×C4—1181no—
468C22×C4—1181no—
469C22×C4—1182no—
470C22×C4C2×C4×D5914yes—
471C22×C4C2×C4×D5912yes—
472C22×C4#1174568514yes—
473C22×C4#1174568512yes—
474C22×C4#1174568514yes—
475C22×C4C2×C22⋊D5912yes—
476C22×C4C2×C22⋊D5912yes—
477C22×C4C2×C22⋊D5912yes—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—2368no—
481C4⋊C4—2364no—
482C22⋊C4—1188no—
483C22⋊C4—2368no—
484C22⋊C4—2364no—
485C22⋊C4—2368no—
486C22⋊C4—2364no—
487C42—28no—
488C42—48no—
489C23—1182no—
490C23—1184no—
491C23—1184no—
492C23—1182no—
493C23—1182no—
494C23—2361no—
495C23—2361no—
496C23—2362no—
497C23—2361no—
498C23—2361no—
499C23—2361no—
500C23—2361no—
501C23—2362no—
502C23—2362no—
503C23—2362no—
504C23—2362no—
505C23—2362no—
506C23—2362no—
507C23—2361no—
508C23—2362no—
509C23—2361no—
510C23—2362no—
511C23—2362no—
512C23C22×C4×D5911yescharacteristic
513C23C22×C22⋊D5911yescharacteristic
514C23C2×#1174568512yes—
515C23C2×#1174568512yes—
516C23C2×#1174568511yescharacteristic
517C23#1427377212yes—
518C23#1427377312yes—
519C23#1427377311yescharacteristic
520C23#1427377411yescharacteristic
521C23#1427377511yescharacteristic
522C23#1427377511yescharacteristic
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—1182no—
551C2×C4—1184no—
552C2×C4—1184no—
553C2×C4—1184no—
554C2×C4—1184no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2362no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2362no—
563C2×C4—2362no—
564C2×C4—2362no—
565C2×C4—2362no—
566C2×C4—2362no—
567C2×C4—2362no—
568C2×C4—2362no—
569C2×C4—2362no—
570C2×C4—2362no—
571C2×C4—2362no—
572C2×C4—2361no—
573C2×C4—2361no—
574C2×C4—2362no—
575C2×C4—2362no—
576C2×C4—2362no—
577C2×C4—2362no—
578C2×C4—2362no—
579C2×C4C4×C22⋊D5914yes—
580C2×C4#1427371014yes—
581C22—4722no—
582C22—4721no—
583C22—4721no—
584C22—4722no—
585C22—4721no—
586C22—4721no—
587C22—2362no—
588C22—2362no—
589C22—2364no—
590C22—2362no—
591C22—2364no—
592C22C2×C4×C22⋊D5911yescharacteristic
593C22C4×#1174568512yes—
594C22#1683787412yes—
595C22#1683787811yescharacteristic
596C22#1683788411yescharacteristic
597C22C2×#1427371011yescharacteristic
598C22#1683791311yescharacteristic
599C22#1683793612yes—
600C22#1683796611yescharacteristic
601C22#1683796911yescharacteristic
602C22#1683797611yescharacteristic
603C22#1683798011yescharacteristic
604C22#1683798512yes—
605C22#1683799412yes—
606C22#1683799912yes—
607C22#1683799912yes—
608C22#1683800312yes—
609C22#1683800712yes—
610C22#1683801011yescharacteristic
611C22#1683802312yes—
612C22#1683802411yescharacteristic
613C22#1683802412yes—
614C22#1683802412yes—
615C4—24no—
616C4—24no—
617C4—42no—
618C4—44no—
619C4—44no—
620C4—4722no—
621C4—4721no—
622C4—4721no—
623C4—2362no—
624C4—2364no—
625C2—4722no—
626C2C4×#1427371011yescharacteristic
627C2#2142677211yescharacteristic
628C2#2142681811yescharacteristic
629C2#2142684712yes—
630C2#2142688012yes—
631C2#2142987611yescharacteristic
632C2#2142988511yescharacteristic
633C2#2142995611yescharacteristic
634C2#2142999111yescharacteristic
635C2#2143012312yes—
636C2#2143016812yes—
subgroupG/H
center, FrattiniC24C23×D59
commutatorC23×C59C23×C4
FittingC2×C4×C4⋊C4×C59C2
socle, cosocleC24×C59C24
prime radicalC24×D59C23
hypercenterC2×C4×C4⋊C4D59
hypocenterC59#14305
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #21422364C2semidirect
4. #21422355C2semidirect
5. #21422347C2semidirect
9. #21422329C2semidirect
10. #21422323C2semidirect
11. C2×C4×C4⋊C4×C59C2semidirect
19. #16837337C4semidirect
309. C2×C4×C4⋊C4D59semidirect
356. C59#14305semidirect

Maximal subgroups

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

subgroupindex
C2×#168373372
C2×#168372932
#214223642
#214223552
#214223472
#214223442
#214223382
#214223342
#214223292
#214223232
C2×C4×C4⋊C4×C592
#1430559