Finite Group Databasebeta

← #28510787

Subgroups — #28510787

Order 15104.

1746 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837294C212yesmaximal
2C2×#16837261C211yesmaximal, characteristic
3C2×#16837253C211yesmaximal, characteristic, complemented
4C2×C4×(C2×D59)⋊C4C211yesmaximal, characteristic
5#21422360C212yesmaximal, complemented
6#21422340C212yesmaximal
7#21422331C212yesmaximal, complemented
8#21422326C211yesmaximal, characteristic, complemented
9#21422320C211yesmaximal, characteristic, complemented
10(C2×D59)⋊#95C211yesmaximal, characteristic
11C59×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D59C2211yescharacteristic
13C22×C4⋊Dic59C2212yes—
14C22×Dic59⋊C4C2212yes—
15C22×Dic59⋊C4C2212yes—
16C22×(C2×D59)⋊C4C2212yes—
17C22×(C2×D59)⋊C4C2212yes—
18C22×(C2×D59)⋊C4C2211yescharacteristic
19C22×(C2×D59)⋊C4C2211yescharacteristic
20C22×C4×Dic59C2212yes—
21C22×C4×Dic59C2211yescharacteristic
22#16837294—28no—
23#16837261—24no—
24#16837253—24no—
25C4×(C2×D59)⋊C4C418yescomplemented
26C2×C59⋊#95C2212yes—
27C2×C59⋊#95C2212yes—
28C2×C59⋊#95C2211yescharacteristic
29C2×C59⋊#95C2212yes—
30C2×C59⋊#95C2211yescharacteristic
31C2×C59⋊#95C2212yes—
32C2×C59⋊#95C2211yescharacteristic
33C2×C59⋊#95C2211yescharacteristic
34C22×C42×C59C2211yescharacteristic
35C2×#95×C59C2212yes—
36C2×#95×C59C2211yescharacteristic
37C2×#95×C59C2212yes—
38C2×#95×C59C2211yescharacteristic
39C24×D59C2311yescharacteristic
40C23×Dic59C2312yes—
41C23×Dic59C2312yes—
42C23×Dic59C2311yescharacteristic
43C23×Dic59C2311yescharacteristic
44C23×Dic59C2311yescharacteristic
45C22×C4×D59—22no—
46C22×C4×D59—22no—
47C22×C4×D59—24no—
48C22×C4×D59—42no—
49C22×C4×D59C2×C414yes—
50C2×C4⋊Dic59—24no—
51C2×C4⋊Dic59—24no—
52C2×C4⋊Dic59—42no—
53C2×Dic59⋊C4—24no—
54C2×Dic59⋊C4—24no—
55C2×Dic59⋊C4—24no—
56C2×Dic59⋊C4—24no—
57C2×Dic59⋊C4—24no—
58C2×Dic59⋊C4—42no—
59C2×(C2×D59)⋊C4—24no—
60C2×(C2×D59)⋊C4—24no—
61C2×(C2×D59)⋊C4—24no—
62C2×(C2×D59)⋊C4—24no—
63C2×(C2×D59)⋊C4—24no—
64C2×(C2×D59)⋊C4—42no—
65C2×(C2×D59)⋊C4—42no—
66C2×(C2×D59)⋊C4C2×C414yes—
67C2×(C2×D59)⋊C4C2×C414yes—
68C2×(C2×D59)⋊C4D414yes—
69C2×(C2×D59)⋊C4Q814yes—
70C2×C4×Dic59—22no—
71C2×C4×Dic59—22no—
72C2×C4×Dic59—24no—
73C2×C4×Dic59—24no—
74C2×C4×Dic59—42no—
75C2×C4×Dic59C2×C414yes—
76C59⋊#95—22no—
77C59⋊#95—22no—
78C59⋊#95—24no—
79C59⋊#95—24no—
80C59⋊#95—24no—
81C59⋊#95—42no—
82C59⋊#95C2×C414yes—
83C59⋊#95C2×C414yes—
84C23×C4×C59C2312yes—
85C23×C4×C59C2311yescharacteristic
86C23×C4×C59C2312yes—
87C23×C4×C59C2311yescharacteristic
88C23×C4×C59C2311yescharacteristic
89C2×C42×C59—42no—
90C2×C42×C59C2×C414yes—
91#95×C59—24no—
92#95×C59—22no—
93#95×C59—42no—
94#95×C59D414yes—
95C24×C59C2411yescharacteristic
96C23×D59—21no—
97C23×D59—22no—
98C23×D59—22no—
99C23×D59—22no—
100C23×D59—21no—
101C23×D59—21no—
102C23×D59—22no—
103C23×D59—21no—
104C23×D59C22×C412yes—
105C23×D59C2×D412yes—
106C23×D59C2×Q812yes—
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—22no—
116C22×C4×C59—22no—
117C22×C4×C59—22no—
118C22×C4×C59—21no—
119C22×C4×C59—22no—
120C22×C4×C59—21no—
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—22no—
127C22×C4×C59—22no—
128C22×C4×C59C22×C412yes—
129C22×C4×C59C22×C412yes—
130C22×C4×C59C22×C412yes—
131C22×C4×C59C2×D414yes—
132C22×C4×C59C2×D412yes—
133C22×C4×C59C2×D412yes—
134C22×C4×C59C2×Q812yes—
135C22×C4×C59#4112yes—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—22no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—22no—
145C22×Dic59—21no—
146C22×Dic59—22no—
147C22×Dic59—22no—
148C22×Dic59—22no—
149C22×Dic59—22no—
150C22×Dic59—22no—
151C22×Dic59—22no—
152C22×Dic59—22no—
153C22×Dic59—22no—
154C22×Dic59—22no—
155C22×Dic59—22no—
156C22×Dic59—22no—
157C22×Dic59—22no—
158C22×Dic59—22no—
159C22×Dic59C22×C412yes—
160C22×Dic59C22×C412yes—
161C22×Dic59C22×C412yes—
162C22×Dic59C2×D412yes—
163C22×Dic59C2×Q812yes—
164C22×Dic59#4114yes—
165C22×Dic59#4114yes—
166C22×Dic59#4114yes—
167C2×C4×D59—28no—
168C2×C4×D59—28no—
169C2×C4×D59—44no—
170C2×C4×D59—48no—
171C2×C4×D59—48no—
172C42×C59—44no—
173(C2×D59)⋊C4—28no—
174(C2×D59)⋊C4—48no—
175(C2×D59)⋊C4—48no—
176(C2×D59)⋊C4C4⋊C4116yescomplemented
177C4⋊Dic59—48no—
178Dic59⋊C4—216no—
179Dic59⋊C4—48no—
180C4×Dic59—28no—
181C4×Dic59—48no—
182C23×C59C23×C411yescharacteristic
183C23×C59C22×D411yescharacteristic
184C23×C59C22×D411yescharacteristic
185C23×C59C22×Q811yescharacteristic
186C23×C59C2×#4112yes—
187C23×C59C2×#4112yes—
188C23×C59C2×#4112yes—
189C23×C59C2×#4111yescharacteristic
190C23×C59H(5,𝔽2)12yes—
191C23×C59H(5,𝔽2)11yescharacteristic
192C23×C592−1+411yescharacteristic
193C22×D59—22no—
194C22×D59—22no—
195C22×D59—22no—
196C22×D59—22no—
197C22×D59—22no—
198C22×D59—22no—
199C22×D59—24no—
200C22×D59—24no—
201C22×D59—22no—
202C22×D59—24no—
203C22×D59—22no—
204C22×D59—22no—
205C22×D59—24no—
206C22×D59—22no—
207C22×D59—42no—
208C22×D59—42no—
209C22×D59—42no—
210C22×D59—42no—
211C22×D59—42no—
212C22×D59—42no—
213C22×D59—42no—
214C22×D59—42no—
215C22×D59C2×C4⋊C414yes—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
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—22no—
225C2×C4×C59—22no—
226C2×C4×C59—22no—
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—42no—
239C2×C4×C59—42no—
240C2×C4×C59—42no—
241C2×C4×C59—42no—
242C2×C4×C59—42no—
243C2×C4×C59C2×C4⋊C414yes—
244C2×C4×C59C4×D414yes—
245C2×Dic59—22no—
246C2×Dic59—22no—
247C2×Dic59—22no—
248C2×Dic59—22no—
249C2×Dic59—22no—
250C2×Dic59—22no—
251C2×Dic59—24no—
252C2×Dic59—24no—
253C2×Dic59—22no—
254C2×Dic59—24no—
255C2×Dic59—22no—
256C2×Dic59—22no—
257C2×Dic59—24no—
258C2×Dic59—22no—
259C2×Dic59—24no—
260C2×Dic59—24no—
261C2×Dic59—24no—
262C2×Dic59—42no—
263C2×Dic59—42no—
264C2×Dic59—42no—
265C2×Dic59—42no—
266C2×Dic59—42no—
267C2×Dic59—42no—
268C2×Dic59—42no—
269C2×Dic59—42no—
270C2×Dic59—42no—
271C2×Dic59—42no—
272C2×Dic59—42no—
273C2×Dic59C2×C4⋊C414yes—
274C2×Dic59#11714yes—
275C2×Dic59#11714yes—
276C4×D59—416no—
277#13891—591nomaximal, Sylow
278C22×C59C22×C4⋊C411yescharacteristic
279C22×C59C2×#11711yescharacteristic
280C22×C59C2×#11711yescharacteristic
281C22×C59C2×C4×D411yescharacteristic
282C22×C59C4×#4111yescharacteristic
283C22×C59#51811yescharacteristic
284C22×C59#52011yescharacteristic
285C22×C59#52512yes—
286C22×C59#52912yes—
287C22×C59#53212yes—
288C22×C59#53312yes—
289C22×C59#54012yes—
290C22×C59#54212yes—
291C22×C59D4211yescharacteristic
292C22×C59#54612yes—
293C22×C59#54612yes—
294C22×C59#54711yescharacteristic
295C22×C59D4×Q811yescharacteristic
296C22×C59#55112yes—
297C22×C59#55312yes—
298C22×C59#55412yes—
299C22×C59#55411yescharacteristic
300C22×C59#55612yes—
301C2×D59—24no—
302C2×D59—24no—
303C2×D59—24no—
304C2×D59—42no—
305C2×D59—42no—
306C2×D59—42no—
307C2×D59—44no—
308C2×D59—44no—
309C2×D59—44no—
310C2×D59—44no—
311C2×D59—42no—
312C4×C59—24no—
313C4×C59—24no—
314C4×C59—42no—
315C4×C59—44no—
316C4×C59—82no—
317Dic59—24no—
318Dic59—24no—
319Dic59—24no—
320Dic59—44no—
321Dic59—82no—
322C2×#388—592no—
323C2×#380—591no—
324C2×C4×C4⋊C4—591no—
325C2×C4×C22⋊C4—591no—
326#1444—592no—
327#1439—592no—
328#1436—592no—
329C4⋊#95—591no—
330#1434—591no—
331#1434D5911yescharacteristic, complemented
332C22⋊#95—591no—
333C2×C59#228511yescharacteristic
334C2×C59#229011yescharacteristic
335C2×C59#231911yescharacteristic
336C2×C59#234211yescharacteristic
337C2×C59#234711yescharacteristic
338C2×C59D4×C4⋊C411yescharacteristic
339C2×C59#234911yescharacteristic
340C2×C59#246412yes—
341C2×C59#246512yes—
342C2×C59#252112yes—
343C2×C59#252212yes—
344D59—44no—
345C24×C4—591no—
346C22×C4⋊C4—592no—
347C22×C4⋊C4—592no—
348C22×C4⋊C4—592no—
349C22×C22⋊C4—591no—
350C22×C22⋊C4—591no—
351C22×C22⋊C4—592no—
352C22×C22⋊C4—592no—
353C22×C42—591no—
354C22×C42—592no—
355C22×C42C2×D5911yescharacteristic
356#388—1188no—
357#380—1184no—
358C4×C4⋊C4—1184no—
359C4×C22⋊C4—598no—
360C2×#95—591no—
361C2×#95—591no—
362C2×#95—591no—
363C2×#95—592no—
364C2×#95—591no—
365C2×#95—592no—
366C2×#95—592no—
367C2×#95—592no—
368C2×#95C2×D5911yescharacteristic
369C2×#95C2×D5912yes—
370C2×#95C2×D5912yes—
371C2×#95C2×D5911yescharacteristic
372C59#1389111yescharacteristic, complemented, Sylow
373C25—591no—
374C23×C4—591no—
375C23×C4—591no—
376C23×C4—591no—
377C23×C4—592no—
378C23×C4—594no—
379C23×C4—592no—
380C23×C4—1184no—
381C23×C4—1182no—
382C23×C4—1182no—
383C23×C4—2362no—
384C23×C4C22×D5912yes—
385C23×C4C22×D5911yescharacteristic
386C23×C4C22×D5912yes—
387C23×C4C22×D5911yescharacteristic
388C23×C4C22×D5911yescharacteristic
389C2×C4⋊C4—1184no—
390C2×C4⋊C4—1184no—
391C2×C4⋊C4—1184no—
392C2×C4⋊C4—1184no—
393C2×C4⋊C4—1184no—
394C2×C4⋊C4—1184no—
395C2×C4⋊C4—1184no—
396C2×C4⋊C4—2362no—
397C2×C4⋊C4—2362no—
398C2×C22⋊C4—594no—
399C2×C22⋊C4—594no—
400C2×C22⋊C4—594no—
401C2×C22⋊C4—594no—
402C2×C22⋊C4—1184no—
403C2×C22⋊C4—1184no—
404C2×C22⋊C4—1184no—
405C2×C22⋊C4—1184no—
406C2×C22⋊C4—1184no—
407C2×C22⋊C4—2362no—
408C2×C22⋊C4—2362no—
409C2×C42—42no—
410C2×C42—594no—
411C2×C42—1182no—
412C2×C42—1182no—
413C2×C42—1184no—
414C2×C42—1184no—
415C2×C42—2362no—
416C2×C42C4×D5914yes—
417#95—22no—
418#95—24no—
419#95—42no—
420#95—594no—
421#95—594no—
422#95—1184no—
423#95—1184no—
424#95—1184no—
425#95—1182no—
426#95—1182no—
427#95—2362no—
428#95D23614yes—
429C24—592no—
430C24—592no—
431C24—592no—
432C24—1181no—
433C24—1182no—
434C24—1182no—
435C24—1182no—
436C24—1181no—
437C24—1182no—
438C24—1181no—
439C24—1181no—
440C24C23×D5911yescharacteristic
441C22×C4—22no—
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—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—592no—
463C22×C4—592no—
464C22×C4—592no—
465C22×C4—592no—
466C22×C4—592no—
467C22×C4—594no—
468C22×C4—594no—
469C22×C4—594no—
470C22×C4—1188no—
471C22×C4—1181no—
472C22×C4—1181no—
473C22×C4—1181no—
474C22×C4—1182no—
475C22×C4—1181no—
476C22×C4—1181no—
477C22×C4—1181no—
478C22×C4—1182no—
479C22×C4—1181no—
480C22×C4—1181no—
481C22×C4—1182no—
482C22×C4—1182no—
483C22×C4—1182no—
484C22×C4—1182no—
485C22×C4—1182no—
486C22×C4—1182no—
487C22×C4—1188no—
488C22×C4—1182no—
489C22×C4—1182no—
490C22×C4—1182no—
491C22×C4—1182no—
492C22×C4—1182no—
493C22×C4—1182no—
494C22×C4—1182no—
495C22×C4—2364no—
496C22×C4—2368no—
497C22×C4—2368no—
498C22×C4C2×C4×D5912yes—
499C22×C4C2×C4×D5912yes—
500C22×C4C2×C4×D5912yes—
501C22×C4C2×D23614yes—
502C22×C4C2×D23612yes—
503C22×C4#1174568512yes—
504C22×C4D4×D5912yes—
505C22×C4Q8×D5912yes—
506C4⋊C4—11816no—
507C4⋊C4—2368no—
508C4⋊C4—2368no—
509C22⋊C4—5916no—
510C22⋊C4—1188no—
511C22⋊C4—2368no—
512C22⋊C4—2368no—
513C42—44no—
514C42—1188no—
515C42—2368no—
516C23—594no—
517C23—1184no—
518C23—1184no—
519C23—1182no—
520C23—1184no—
521C23—1182no—
522C23—1182no—
523C23—1184no—
524C23—1182no—
525C23—1182no—
526C23—1182no—
527C23—1182no—
528C23—1182no—
529C23—1182no—
530C23—1182no—
531C23—2362no—
532C23—2362no—
533C23—2362no—
534C23—2362no—
535C23—2362no—
536C23—2362no—
537C23—2362no—
538C23—2362no—
539C23C22×C4×D5911yescharacteristic
540C23C2×D4×D5911yescharacteristic
541C23C22×D23611yescharacteristic
542C23C2×Q8×D5911yescharacteristic
543C23#41×D5912yes—
544C23#41×D5912yes—
545C23#41×D5912yes—
546C23C2×#1174568511yescharacteristic
547C23#1427377212yes—
548C23#1427377311yescharacteristic
549C23#1427377411yescharacteristic
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—594no—
578C2×C4—594no—
579C2×C4—594no—
580C2×C4—1184no—
581C2×C4—1184no—
582C2×C4—1182no—
583C2×C4—1184no—
584C2×C4—1182no—
585C2×C4—1182no—
586C2×C4—1184no—
587C2×C4—1182no—
588C2×C4—1182no—
589C2×C4—1182no—
590C2×C4—1182no—
591C2×C4—1182no—
592C2×C4—1182no—
593C2×C4—1182no—
594C2×C4—1184no—
595C2×C4—1184no—
596C2×C4—1184no—
597C2×C4—23616no—
598C2×C4—2362no—
599C2×C4—2362no—
600C2×C4—2362no—
601C2×C4—2362no—
602C2×C4—2362no—
603C2×C4—2362no—
604C2×C4—2362no—
605C2×C4—2362no—
606C2×C4—2362no—
607C2×C4—2362no—
608C2×C4—2362no—
609C2×C4C4⋊C4×D5914yes—
610C2×C4C4×D23614yes—
611C22—1184no—
612C22—1184no—
613C22—1184no—
614C22—2362no—
615C22—2362no—
616C22—2362no—
617C22—2364no—
618C22—2364no—
619C22—2362no—
620C22—2364no—
621C22—2364no—
622C22C2×C4⋊C4×D5911yescharacteristic
623C22#117×D5911yescharacteristic
624C22#117×D5911yescharacteristic
625C22C2×C4×D23611yescharacteristic
626C22#1683787011yescharacteristic
627C22#1683787511yescharacteristic
628C22#1683788211yescharacteristic
629C22#1683791412yes—
630C22#1683793012yes—
631C22#1683795112yes—
632C22#1683795812yes—
633C22#1683800412yes—
634C22#1683802012yes—
635C22D4×D23611yescharacteristic
636C22#1683804112yes—
637C22#1683804712yes—
638C22#1683805711yescharacteristic
639C22Q8×D23611yescharacteristic
640C22#1683808112yes—
641C22#1683810312yes—
642C22#1683810612yes—
643C22#1683811211yescharacteristic
644C22#1683811912yes—
645C4—24no—
646C4—24no—
647C4—42no—
648C4—44no—
649C4—82no—
650C4—1184no—
651C4—1184no—
652C4—1184no—
653C4—4722no—
654C4—2364no—
655C2—2364no—
656C2#2142650211yescharacteristic
657C2#2142654011yescharacteristic
658C2#2142673611yescharacteristic
659C2#2142694811yescharacteristic
660C2#2142698411yescharacteristic
661C2C4⋊C4×D23611yescharacteristic
662C2#2142700011yescharacteristic
663C2#2142810012yes—
664C2#2142811512yes—
665C2#2142879712yes—
666C2#2142881212yes—
subgroupG/H
center, FrattiniC24C23×D59
commutatorC23×C59C23×C4
FittingC59×#1434C2
socle, cosocleC24×C59C24
prime radicalC24×D59C23
hypercenter#1434D59
hypocenterC59#13891
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
3. C2×#16837253C2semidirect
5. #21422360C2semidirect
7. #21422331C2semidirect
8. #21422326C2semidirect
9. #21422320C2semidirect
11. C59×#1434C2semidirect
25. C4×(C2×D59)⋊C4C4semidirect
176. (C2×D59)⋊C4C4⋊C4semidirect
331. #1434D59semidirect
372. C59#13891semidirect

Maximal subgroups

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

subgroupindex
C2×#168372942
C2×#168372612
C2×#168372532
C2×C4×(C2×D59)⋊C42
#214223602
#214223402
#214223312
#214223262
#214223202
(C2×D59)⋊#952
C59×#14342
#1389159