Finite Group Databasebeta

← #28519724

Subgroups — #28519724

Order 15104.

1702 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273705C211yesmaximal, characteristic
2C2×#16837264C211yesmaximal, characteristic
3C2×#16837259C211yesmaximal, characteristic
4#21422366C212yesmaximal, complemented
5#21422362C212yesmaximal, complemented
6#21422350C211yesmaximal, characteristic, complemented
7#21422349C211yesmaximal, characteristic, complemented
8#21422346C211yesmaximal, characteristic, complemented
9#21422344C212yesmaximal
10#21422344C212yesmaximal
11C59×#1441C211yesmaximal, characteristic, complemented
12C23×Dic118C2211yescharacteristic
13C23×C4×D59C2211yescharacteristic
14C2×#14273705—24no—
15C2×#14273705—24no—
16C2×#14273705C418yescomplemented
17C22×C4⋊Dic59C2211yescharacteristic
18C22×Dic59⋊C4C2211yescharacteristic
19C22×Dic59⋊C4C2211yescharacteristic
20C22×(C2×D59)⋊C4C2211yescharacteristic
21C22×(C2×D59)⋊C4C2211yescharacteristic
22C22×(C2×D59)⋊C4C2212yes—
23C22×(C2×D59)⋊C4C2212yes—
24#16837264—24no—
25#16837259—24no—
26C2×C59⋊#95C2211yescharacteristic
27C2×C59⋊#95C2212yes—
28C2×C59⋊#95C2212yes—
29C2×C59⋊#95C2212yes—
30C2×C59⋊#95C2212yes—
31C2×C59⋊#95C2212yes—
32C2×C59⋊#95C2212yes—
33C2×C59⋊#95C2212yes—
34C2×C59⋊#95C2212yes—
35C22×C4⋊C4×C59C2211yescharacteristic
36C22×C4⋊C4×C59C2211yescharacteristic
37C2×#95×C59C2211yescharacteristic
38C2×#95×C59C2212yes—
39C2×#95×C59C2212yes—
40C24×D59C2311yescharacteristic
41C22×Dic118—22no—
42C22×Dic118—22no—
43C22×Dic118—42no—
44C22×Dic118—42no—
45C22×Dic118C2×C414yes—
46C23×Dic59C2312yes—
47C23×Dic59C2312yes—
48C23×Dic59C2311yescharacteristic
49C23×Dic59C2311yescharacteristic
50C23×Dic59C2311yescharacteristic
51C22×C4×D59—42no—
52C22×C4×D59—42no—
53C22×C4×D59C2×C414yes—
54C22×C4×D59D414yes—
55C22×C4×D59D414yes—
56#14273705—232no—
57C2×C4⋊Dic59—22no—
58C2×C4⋊Dic59—42no—
59C2×Dic59⋊C4—22no—
60C2×Dic59⋊C4—22no—
61C2×Dic59⋊C4—22no—
62C2×Dic59⋊C4—22no—
63C2×Dic59⋊C4C2×C414yes—
64C2×Dic59⋊C4C2×C414yes—
65C2×(C2×D59)⋊C4—22no—
66C2×(C2×D59)⋊C4—22no—
67C2×(C2×D59)⋊C4—22no—
68C2×(C2×D59)⋊C4—22no—
69C2×(C2×D59)⋊C4—24no—
70C2×(C2×D59)⋊C4—24no—
71C2×(C2×D59)⋊C4—42no—
72C2×(C2×D59)⋊C4—42no—
73C2×(C2×D59)⋊C4—42no—
74C2×(C2×D59)⋊C4—42no—
75C2×(C2×D59)⋊C4C2×C414yes—
76C2×(C2×D59)⋊C4C2×C414yes—
77C59⋊#95—22no—
78C59⋊#95—24no—
79C59⋊#95—24no—
80C59⋊#95—42no—
81C59⋊#95—42no—
82C59⋊#95—42no—
83C59⋊#95—42no—
84C59⋊#95—42no—
85C59⋊#95—42no—
86C23×C4×C59C2312yes—
87C23×C4×C59C2312yes—
88C23×C4×C59C2311yescharacteristic
89C23×C4×C59C2311yescharacteristic
90C23×C4×C59C2311yescharacteristic
91C2×C4⋊C4×C59—22no—
92C2×C4⋊C4×C59—22no—
93C2×C4⋊C4×C59—22no—
94C2×C4⋊C4×C59—42no—
95C2×C4⋊C4×C59C2×C414yes—
96#95×C59—22no—
97#95×C59—42no—
98#95×C59—42no—
99C24×C59C2411yescharacteristic
100C23×D59—21no—
101C23×D59—21no—
102C23×D59—22no—
103C23×D59—22no—
104C23×D59—22no—
105C23×D59—22no—
106C23×D59—21no—
107C23×D59—21no—
108C23×D59C22×C412yes—
109C23×D59C2×D412yes—
110C23×D59C2×D412yes—
111C22×C4×C59—22no—
112C22×C4×C59—22no—
113C22×C4×C59—21no—
114C22×C4×C59—21no—
115C22×C4×C59—21no—
116C22×C4×C59—21no—
117C22×C4×C59—21no—
118C22×C4×C59—22no—
119C22×C4×C59—21no—
120C22×C4×C59—22no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—22no—
127C22×C4×C59—22no—
128C22×C4×C59—22no—
129C22×C4×C59—22no—
130C22×C4×C59—22no—
131C22×C4×C59—22no—
132C22×C4×C59—22no—
133C22×C4×C59—22no—
134C22×C4×C59—22no—
135C22×C4×C59—22no—
136C22×C4×C59—22no—
137C22×C4×C59C22×C412yes—
138C22×C4×C59C22×C412yes—
139C22×C4×C59C22×C412yes—
140C22×C4×C59C2×D412yes—
141C22×C4×C59C2×D412yes—
142C22×Dic59—22no—
143C22×Dic59—22no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—22no—
151C22×Dic59—22no—
152C22×Dic59—21no—
153C22×Dic59—21no—
154C22×Dic59—22no—
155C22×Dic59—22no—
156C22×Dic59—22no—
157C22×Dic59—22no—
158C22×Dic59—22no—
159C22×Dic59—22no—
160C22×Dic59—22no—
161C22×Dic59—22no—
162C22×Dic59—22no—
163C22×Dic59—22no—
164C22×Dic59—22no—
165C22×Dic59—22no—
166C22×Dic59—22no—
167C22×Dic59—22no—
168C22×Dic59C22×C412yes—
169C22×Dic59C22×C412yes—
170C22×Dic59C22×C412yes—
171C22×Dic59C2×D412yes—
172C22×Dic59C2×D412yes—
173C2×C4×D59—48no—
174C2×C4×D59—44no—
175C2×C4×D59—44no—
176C2×C4×D59—48no—
177C2×C4×D59C22⋊C4116yes—
178C2×Dic118—28no—
179C2×Dic118—48no—
180C2×Dic118—48no—
181C2×Dic118—44no—
182C2×Dic118—44no—
183C4⋊C4×C59—28no—
184C4⋊C4×C59—44no—
185(C2×D59)⋊C4—28no—
186(C2×D59)⋊C4—28no—
187(C2×D59)⋊C4—48no—
188(C2×D59)⋊C4—48no—
189C4⋊Dic59—44no—
190Dic59⋊C4—28no—
191Dic59⋊C4—28no—
192C23×C59C23×C411yescharacteristic
193C23×C59C22×D411yescharacteristic
194C23×C59C22×D411yescharacteristic
195C23×C59H(5,𝔽2)12yes—
196C23×C59H(5,𝔽2)11yescharacteristic
197C23×C59H(5,𝔽2)12yes—
198C23×C59H(5,𝔽2)11yescharacteristic
199C23×C592−1+412yes—
200C23×C592−1+411yescharacteristic
201C23×C592−1+412yes—
202C23×C592−1+411yescharacteristic
203C22×D59—24no—
204C22×D59—24no—
205C22×D59—22no—
206C22×D59—22no—
207C22×D59—24no—
208C22×D59—22no—
209C22×D59—22no—
210C22×D59—22no—
211C22×D59—22no—
212C22×D59—24no—
213C22×D59—22no—
214C22×D59—22no—
215C22×D59—22no—
216C22×D59—22no—
217C22×D59—42no—
218C22×D59—42no—
219C22×D59—42no—
220C22×D59—42no—
221C22×D59—42no—
222C22×D59—42no—
223C22×D59—42no—
224C22×D59—42no—
225C22×D59C2×C22⋊C414yes—
226C2×C4×C59—24no—
227C2×C4×C59—22no—
228C2×C4×C59—22no—
229C2×C4×C59—24no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
232C2×C4×C59—24no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—42no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—42no—
243C2×C4×C59—42no—
244C2×C4×C59—42no—
245C2×C4×C59—42no—
246C2×C4×C59—42no—
247C2×C4×C59—42no—
248C2×C4×C59—42no—
249C2×C4×C59—42no—
250C2×C4×C59—42no—
251C2×C4×C59—42no—
252C2×C4×C59—42no—
253C2×C4×C59—42no—
254C2×C4×C59—42no—
255C2×C4×C59—42no—
256C2×C4×C59C2×C22⋊C414yes—
257C2×Dic59—24no—
258C2×Dic59—24no—
259C2×Dic59—22no—
260C2×Dic59—22no—
261C2×Dic59—22no—
262C2×Dic59—22no—
263C2×Dic59—24no—
264C2×Dic59—22no—
265C2×Dic59—22no—
266C2×Dic59—42no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—42no—
274C2×Dic59—42no—
275C2×Dic59—42no—
276C2×Dic59—42no—
277C2×Dic59—42no—
278C2×Dic59—42no—
279C2×Dic59—42no—
280C2×Dic59—42no—
281C2×Dic59—42no—
282C2×Dic59—42no—
283C2×Dic59—42no—
284C2×Dic59—42no—
285C2×Dic59—42no—
286C2×Dic59—42no—
287C2×Dic59C2×C22⋊C414yes—
288C4×D59—416no—
289Dic118—416no—
290#14660—591nomaximal, Sylow
291C22×C59C22×C22⋊C411yescharacteristic
292C22×C59#51812yes—
293C22×C59#51912yes—
294C22×C59#52011yescharacteristic
295C22×C59#52011yescharacteristic
296C22×C59#53412yes—
297C22×C59#53411yescharacteristic
298C22×C59#53411yescharacteristic
299C22×C59#53511yescharacteristic
300C22×C59#53611yescharacteristic
301C22×C59#53611yescharacteristic
302C22×C59#53612yes—
303C22×C59#53611yescharacteristic
304C22×C59#53711yescharacteristic
305C22×C59#53711yescharacteristic
306C22×C59#56212yes—
307C22×C59#56212yes—
308C22×C59#56212yes—
309C22×C59#56212yes—
310C22×C59#56312yes—
311C22×C59#56312yes—
312C22×C59#56312yes—
313C22×C59#56312yes—
314C2×D59—24no—
315C2×D59—24no—
316C2×D59—24no—
317C2×D59—42no—
318C2×D59—42no—
319C2×D59—42no—
320C2×D59—42no—
321C2×D59—44no—
322C2×D59—44no—
323C2×D59—44no—
324C2×D59—44no—
325C4×C59—24no—
326C4×C59—42no—
327C4×C59—42no—
328C4×C59—82no—
329C4×C59—82no—
330Dic59—24no—
331Dic59—42no—
332Dic59—42no—
333Dic59—82no—
334Dic59—82no—
335C22×C22⋊Q8—591no—
336C2×#381—591no—
337C2×#380—591no—
338#1446—592no—
339#1445—592no—
340#1441—591no—
341#1441—591no—
342#1441—591no—
343#1441D5911yescharacteristic, complemented
344#1440—592no—
345#1440—592no—
346C2×C59#230811yescharacteristic
347C2×C59#231011yescharacteristic
348C2×C59#231111yescharacteristic
349C2×C59#237812yes—
350C2×C59#237812yes—
351C2×C59#281412yes—
352C2×C59#281412yes—
353C2×C59#281611yescharacteristic
354C2×C59#281611yescharacteristic
355C2×C59#281711yescharacteristic
356C2×C59#281711yescharacteristic
357D59—44no—
358C23×Q8—591no—
359C24×C4—591no—
360C2×C22⋊Q8—598no—
361C2×C22⋊Q8—1184no—
362C2×C22⋊Q8—1184no—
363C22×C4⋊C4—591no—
364C22×C4⋊C4—591no—
365C22×C4⋊C4—591no—
366C22×C4⋊C4C2×D5911yescharacteristic
367C22×C4⋊C4C2×D5911yescharacteristic
368C22×C22⋊C4—591no—
369C22×C22⋊C4—591no—
370C22×C22⋊C4—592no—
371C22×C22⋊C4—592no—
372#381—1184no—
373#380—1184no—
374C2×#95—591no—
375C2×#95—592no—
376C2×#95—592no—
377C2×#95—592no—
378C2×#95—592no—
379C2×#95—592no—
380C2×#95—592no—
381C2×#95—592no—
382C2×#95—592no—
383C2×#95C2×D5911yescharacteristic
384C2×#95C2×D5912yes—
385C2×#95C2×D5912yes—
386C59#1466011yescharacteristic, complemented, Sylow
387C25—591no—
388C22×Q8—594no—
389C22×Q8—1182no—
390C22×Q8—1182no—
391C22×Q8—2362no—
392C22×Q8—2362no—
393C23×C4—591no—
394C23×C4—594no—
395C23×C4—594no—
396C23×C4—594no—
397C23×C4—591no—
398C23×C4—591no—
399C23×C4—592no—
400C23×C4—592no—
401C23×C4—2362no—
402C23×C4—2362no—
403C23×C4C22×D5912yes—
404C23×C4C22×D5912yes—
405C23×C4C22×D5911yescharacteristic
406C23×C4C22×D5911yescharacteristic
407C23×C4C22×D5911yescharacteristic
408C22⋊Q8—11832no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—42no—
413C2×C4⋊C4—594no—
414C2×C4⋊C4—594no—
415C2×C4⋊C4—1182no—
416C2×C4⋊C4—1182no—
417C2×C4⋊C4—1182no—
418C2×C4⋊C4—1182no—
419C2×C4⋊C4—1182no—
420C2×C4⋊C4—2362no—
421C2×C4⋊C4C4×D5914yes—
422C2×C22⋊C4—594no—
423C2×C22⋊C4—594no—
424C2×C22⋊C4—1182no—
425C2×C22⋊C4—1182no—
426C2×C22⋊C4—1182no—
427C2×C22⋊C4—1182no—
428C2×C22⋊C4—1184no—
429C2×C22⋊C4—1184no—
430C2×C22⋊C4—2362no—
431C2×C22⋊C4—2362no—
432C2×C22⋊C4—2362no—
433C2×C22⋊C4—2362no—
434#95—22no—
435#95—42no—
436#95—42no—
437#95—1184no—
438#95—1184no—
439#95—1182no—
440#95—2362no—
441#95—2362no—
442#95—2362no—
443#95—2362no—
444#95—2362no—
445#95—2362no—
446C24—592no—
447C24—592no—
448C24—592no—
449C24—1181no—
450C24—1182no—
451C24—1181no—
452C24—1182no—
453C24—1182no—
454C24—1181no—
455C24—1182no—
456C24—1181no—
457C24C23×D5911yescharacteristic
458C2×Q8—1188no—
459C2×Q8—2368no—
460C2×Q8—2368no—
461C2×Q8—2364no—
462C2×Q8—2364no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—592no—
490C22×C4—592no—
491C22×C4—592no—
492C22×C4—5916no—
493C22×C4—592no—
494C22×C4—592no—
495C22×C4—1182no—
496C22×C4—1182no—
497C22×C4—1181no—
498C22×C4—1181no—
499C22×C4—1181no—
500C22×C4—1181no—
501C22×C4—1181no—
502C22×C4—1182no—
503C22×C4—1181no—
504C22×C4—1182no—
505C22×C4—1181no—
506C22×C4—1181no—
507C22×C4—1182no—
508C22×C4—1182no—
509C22×C4—1182no—
510C22×C4—1182no—
511C22×C4—1182no—
512C22×C4—1182no—
513C22×C4—1182no—
514C22×C4—1182no—
515C22×C4—1182no—
516C22×C4—1182no—
517C22×C4—1182no—
518C22×C4—1182no—
519C22×C4—1182no—
520C22×C4—1182no—
521C22×C4—2368no—
522C22×C4—2364no—
523C22×C4—2364no—
524C22×C4—2368no—
525C22×C4C2×C4×D5912yes—
526C22×C4C2×C4×D5912yes—
527C22×C4C2×C4×D5912yes—
528C22×C4D4×D5912yes—
529C22×C4D4×D5912yes—
530C4⋊C4—28no—
531C4⋊C4—44no—
532C4⋊C4—1188no—
533C4⋊C4—1188no—
534C4⋊C4—2364no—
535C22⋊C4—1188no—
536C22⋊C4—1188no—
537C22⋊C4—2368no—
538C22⋊C4—2368no—
539C23—594no—
540C23—1184no—
541C23—1182no—
542C23—1182no—
543C23—1182no—
544C23—1182no—
545C23—1184no—
546C23—1182no—
547C23—1182no—
548C23—1182no—
549C23—1182no—
550C23—1184no—
551C23—1184no—
552C23—1182no—
553C23—1182no—
554C23—2362no—
555C23—2362no—
556C23—2362no—
557C23—2362no—
558C23—2362no—
559C23—2362no—
560C23—2362no—
561C23—2362no—
562C23C22×C4×D5911yescharacteristic
563C23C2×D4×D5911yescharacteristic
564C23C2×D4×D5911yescharacteristic
565C23#1427377211yescharacteristic
566C23#1427377212yes—
567C23#1427377312yes—
568C23#1427377311yescharacteristic
569C23#1427377412yes—
570C23#1427377411yescharacteristic
571C23#1427377512yes—
572C23#1427377511yescharacteristic
573Q8—23616no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—594no—
605C2×C4—1184no—
606C2×C4—1182no—
607C2×C4—1182no—
608C2×C4—1184no—
609C2×C4—1182no—
610C2×C4—1182no—
611C2×C4—1184no—
612C2×C4—1182no—
613C2×C4—1182no—
614C2×C4—23616no—
615C2×C4—2362no—
616C2×C4—2362no—
617C2×C4—2362no—
618C2×C4—2362no—
619C2×C4—2362no—
620C2×C4—2362no—
621C2×C4—2362no—
622C2×C4—2362no—
623C2×C4—2362no—
624C2×C4—2362no—
625C2×C4—2362no—
626C2×C4—2362no—
627C2×C4—2362no—
628C2×C4—2362no—
629C2×C4—2362no—
630C2×C4—2361no—
631C2×C4—2361no—
632C2×C4—2361no—
633C2×C4—2361no—
634C2×C4—2361no—
635C2×C4—2361no—
636C2×C4C22⋊C4×D5914yes—
637C22—1184no—
638C22—1184no—
639C22—1184no—
640C22—2364no—
641C22—2364no—
642C22—2364no—
643C22—2364no—
644C22—2362no—
645C22—2362no—
646C22—2362no—
647C22—2362no—
648C22C2×C22⋊C4×D5911yescharacteristic
649C22#1683787412yes—
650C22#1683787812yes—
651C22#1683788211yescharacteristic
652C22#1683788311yescharacteristic
653C22#1683796012yes—
654C22#1683796211yescharacteristic
655C22#1683796311yescharacteristic
656C22#1683796811yescharacteristic
657C22#1683797211yescharacteristic
658C22#1683797512yes—
659C22#1683797511yescharacteristic
660C22#1683797711yescharacteristic
661C22#1683798111yescharacteristic
662C22#1683798211yescharacteristic
663C22#1683814712yes—
664C22#1683814812yes—
665C22#1683814912yes—
666C22#1683815012yes—
667C22#1683815112yes—
668C22#1683815312yes—
669C22#1683815412yes—
670C22#1683815612yes—
671C4—24no—
672C4—42no—
673C4—42no—
674C4—82no—
675C4—82no—
676C4—1184no—
677C4—4722no—
678C4—4722no—
679C4—2362no—
680C4—2362no—
681C2—2364no—
682C2#2142667011yescharacteristic
683C2#2142667911yescharacteristic
684C2#2142668511yescharacteristic
685C2#2142725812yes—
686C2#2142726112yes—
687C2#2143211412yes—
688C2#2143211612yes—
689C2#2143212711yescharacteristic
690C2#2143212911yescharacteristic
691C2#2143213011yescharacteristic
692C2#2143213211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D59
commutatorC23×C59C23×C4
FittingC59×#1441C2
socle, cosocleC24×C59C24
prime radicalC24×D59C23
hypercenter#1441D59
hypocenterC59#14660
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
4. #21422366C2semidirect
5. #21422362C2semidirect
6. #21422350C2semidirect
7. #21422349C2semidirect
8. #21422346C2semidirect
11. C59×#1441C2semidirect
16. C2×#14273705C4semidirect
343. #1441D59semidirect
386. C59#14660semidirect

Maximal subgroups

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

subgroupindex
C22×#142737052
C2×#168372642
C2×#168372592
#214223662
#214223622
#214223502
#214223492
#214223462
#214223442
C59×#14412
#1466059