Finite Group Databasebeta

← #28515229

Subgroups — #28515229

Order 15104.

1926 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837276C212yesmaximal
2C2×#16837261C211yesmaximal, characteristic
3C2×#16837259C211yesmaximal, characteristic
4C2×#95×D59C211yesmaximal, characteristic
5#21422361C211yesmaximal, characteristic, complemented
6#21422360C211yesmaximal, characteristic, complemented
7#21422355C211yesmaximal, characteristic, complemented
8#21422350C212yesmaximal, complemented
9#21422347C212yesmaximal, complemented
10(C2×D59)⋊#95C212yesmaximal
11C59×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D59C2211yescharacteristic
13C23×C4×D59C2211yescharacteristic
14C23×C4×D59C2211yescharacteristic
15C22×C4⋊Dic59C2212yes—
16C22×C4⋊Dic59C2211yescharacteristic
17C22×Dic59⋊C4C2212yes—
18C22×Dic59⋊C4C2212yes—
19C22×(C2×D59)⋊C4C2212yes—
20C22×(C2×D59)⋊C4C2212yes—
21#16837276—28no—
22#16837261C418yescomplemented
23#16837259—24no—
24C2×C59⋊#95C2211yescharacteristic
25C2×C59⋊#95C2212yes—
26C2×C59⋊#95C2211yescharacteristic
27C2×C59⋊#95C2212yes—
28C2×C59⋊#95C2211yescharacteristic
29C2×C59⋊#95C2212yes—
30C2×C59⋊#95C2212yes—
31C2×C59⋊#95C2212yes—
32C2×C59⋊#95C2211yescharacteristic
33#95×D59—24no—
34C22×C4⋊C4×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—22no—
48C22×C4×D59—22no—
49C22×C4×D59—42no—
50C22×C4×D59—42no—
51C22×C4×D59—42no—
52C22×C4×D59—42no—
53C22×C4×D59—42no—
54C22×C4×D59—42no—
55C22×C4×D59C2×C414yes—
56C22×C4×D59D414yes—
57C22×C4×D59D414yes—
58C22×C4×D59Q814yes—
59C22×C4×D59Q814yes—
60C2×C4⋊Dic59—22no—
61C2×C4⋊Dic59—24no—
62C2×C4⋊Dic59—42no—
63C2×C4⋊Dic59—42no—
64C2×C4⋊Dic59—42no—
65C2×Dic59⋊C4—24no—
66C2×Dic59⋊C4—42no—
67C2×Dic59⋊C4—42no—
68C2×Dic59⋊C4—42no—
69C2×Dic59⋊C4—42no—
70C2×Dic59⋊C4C2×C418yescomplemented
71C2×(C2×D59)⋊C4—24no—
72C2×(C2×D59)⋊C4—24no—
73C2×(C2×D59)⋊C4—24no—
74C2×(C2×D59)⋊C4—24no—
75C2×(C2×D59)⋊C4—24no—
76C2×(C2×D59)⋊C4C2×C418yes—
77C59⋊#95—22no—
78C59⋊#95—22no—
79C59⋊#95—22no—
80C59⋊#95—24no—
81C59⋊#95—24no—
82C59⋊#95—42no—
83C59⋊#95—42no—
84C59⋊#95—42no—
85C59⋊#95C2×C414yescomplemented
86C23×C4×C59C2312yes—
87C23×C4×C59C2312yes—
88C23×C4×C59C2311yescharacteristic
89C23×C4×C59C2311yescharacteristic
90C23×C4×C59C2311yescharacteristic
91C2×C4⋊C4×C59—22no—
92C2×C4⋊C4×C59—42no—
93#95×C59—22no—
94#95×C59—24no—
95#95×C59—42no—
96#95×C59C2×C414yescomplemented
97C24×C59C2411yescharacteristic
98C23×D59—22no—
99C23×D59—22no—
100C23×D59—22no—
101C23×D59—22no—
102C23×D59C22×C412yes—
103C23×D59C2×D412yes—
104C23×D59C2×D412yes—
105C23×D59C2×Q812yes—
106C23×D59C2×Q812yes—
107C23×D59#4112yes—
108C23×D59#4112yes—
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—21no—
116C22×C4×C59—21no—
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×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×C59C22×C414yes—
133C22×C4×C59C22×C412yes—
134C22×C4×C59C2×D412yes—
135C22×C4×C59C2×D412yes—
136C22×C4×C59C2×Q812yes—
137C22×C4×C59C2×Q812yes—
138C22×Dic59—22no—
139C22×Dic59—22no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—22no—
143C22×Dic59—22no—
144C22×Dic59—21no—
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×Dic59—22no—
160C22×Dic59—22no—
161C22×Dic59C22×C414yes—
162C22×Dic59C22×C412yes—
163C22×Dic59C2×D412yes—
164C22×Dic59C2×D412yes—
165C22×Dic59C2×Q812yes—
166C22×Dic59C2×Q812yes—
167C2×C4×D59—28no—
168C2×C4×D59—28no—
169C2×C4×D59—48no—
170C2×C4×D59—44no—
171C2×C4×D59—44no—
172C2×C4×D59—48no—
173C2×C4×D59—48no—
174C2×C4×D59—48no—
175C2×C4×D59—44no—
176C2×C4×D59—44no—
177C2×C4×D59—44no—
178C2×C4×D59—44no—
179C2×C4×D59—48no—
180C2×C4×D59—48no—
181C2×C4×D59C4⋊C4116yescomplemented
182C4⋊C4×C59—44no—
183(C2×D59)⋊C4—216no—
184(C2×D59)⋊C4—48no—
185C4⋊Dic59—48no—
186C4⋊Dic59—44no—
187Dic59⋊C4—48no—
188Dic59⋊C4—48no—
189C23×C59C23×C411yescharacteristic
190C23×C59C22×D411yescharacteristic
191C23×C59C22×D411yescharacteristic
192C23×C59C22×Q811yescharacteristic
193C23×C59C22×Q811yescharacteristic
194C23×C59C2×#4111yescharacteristic
195C23×C59C2×#4111yescharacteristic
196C23×C59H(5,𝔽2)12yes—
197C23×C59H(5,𝔽2)12yes—
198C23×C59H(5,𝔽2)12yes—
199C23×C592−1+412yes—
200C22×D59—24no—
201C22×D59—24no—
202C22×D59—22no—
203C22×D59—22no—
204C22×D59—24no—
205C22×D59—22no—
206C22×D59—22no—
207C22×D59—24no—
208C22×D59—24no—
209C22×D59—24no—
210C22×D59—24no—
211C22×D59—24no—
212C22×D59—24no—
213C22×D59—24no—
214C22×D59—24no—
215C22×D59—24no—
216C22×D59C2×C4⋊C414yes—
217C22×D59C4×D414yes—
218C22×D59C4×Q814yes—
219C22×D59#12114yes—
220C22×D59C22⋊Q814yes—
221C22×D59#12514yes—
222C22×D59C4⋊Q814yes—
223C2×C4×C59—24no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—24no—
227C2×C4×C59—24no—
228C2×C4×C59—24no—
229C2×C4×C59—24no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
232C2×C4×C59—24no—
233C2×C4×C59—24no—
234C2×C4×C59—42no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—42no—
240C2×C4×C59—42no—
241C2×C4×C59—42no—
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×C59C2×C4⋊C414yes—
252C2×Dic59—24no—
253C2×Dic59—22no—
254C2×Dic59—22no—
255C2×Dic59—24no—
256C2×Dic59—24no—
257C2×Dic59—24no—
258C2×Dic59—24no—
259C2×Dic59—22no—
260C2×Dic59—22no—
261C2×Dic59—24no—
262C2×Dic59—24no—
263C2×Dic59—42no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—42no—
269C2×Dic59—42no—
270C2×Dic59—42no—
271C2×Dic59—42no—
272C2×Dic59—42no—
273C2×Dic59—42no—
274C2×Dic59—42no—
275C2×Dic59—42no—
276C2×Dic59—42no—
277C2×Dic59—42no—
278C2×Dic59—42no—
279C2×Dic59—42no—
280C2×Dic59C2×C4⋊C414yes—
281C4×D59—416no—
282C4×D59—416no—
283C4×D59—416no—
284#14275—591nomaximal, Sylow
285C22×C59C22×C4⋊C411yescharacteristic
286C22×C59C2×C4×D411yescharacteristic
287C22×C59C2×C4×Q811yescharacteristic
288C22×C59#51812yes—
289C22×C59#52012yes—
290C22×C59C2×#12111yescharacteristic
291C22×C59C2×C22⋊Q811yescharacteristic
292C22×C59C2×#12511yescharacteristic
293C22×C59C2×C4⋊Q811yescharacteristic
294C22×C59#53411yescharacteristic
295C22×C59#53411yescharacteristic
296C22×C59#53412yes—
297C22×C59#53511yescharacteristic
298C22×C59#53611yescharacteristic
299C22×C59#53712yes—
300C22×C59#53812yes—
301C22×C59#53812yes—
302C22×C59#53912yes—
303C22×C59#53912yes—
304C22×C59#54312yes—
305C22×C59#54312yes—
306C22×C59#54412yes—
307C22×C59#54412yes—
308C2×D59—24no—
309C2×D59—24no—
310C2×D59—28no—
311C2×D59—28no—
312C2×D59—24no—
313C2×D59—24no—
314C2×D59—24no—
315C2×D59—24no—
316C2×D59—28no—
317C2×D59—28no—
318C2×D59#38918yes—
319C4×C59—24no—
320C4×C59—44no—
321C4×C59—42no—
322C4×C59—42no—
323C4×C59—82no—
324Dic59—24no—
325Dic59—44no—
326Dic59—42no—
327Dic59—42no—
328Dic59—82no—
329C2×#385—592no—
330C2×#380—591no—
331C2×#380—591no—
332C22×#95—591no—
333#1444—591no—
334#1444—591no—
335#1442—591no—
336#1442D5911yescharacteristic, complemented
337#1441—592no—
338#1441—592no—
339C22⋊#95—592no—
340C2×C59C2×#38911yescharacteristic
341C2×C59#231511yescharacteristic
342C2×C59#231711yescharacteristic
343C2×C59#232512yes—
344C2×C59#232912yes—
345C2×C59#261711yescharacteristic
346C2×C59#261811yescharacteristic
347C2×C59#262611yescharacteristic
348C2×C59#262711yescharacteristic
349C2×C59#264612yes—
350C2×C59#265912yes—
351D59—28no—
352C24×C4—591no—
353C24×C4—591no—
354C24×C4—591no—
355C22×C4⋊C4—592no—
356C22×C4⋊C4—592no—
357C22×C4⋊C4—592no—
358C22×C4⋊C4—591no—
359C22×C4⋊C4C2×D5911yescharacteristic
360C22×C22⋊C4—592no—
361C22×C22⋊C4—592no—
362#385—1188no—
363#380—598no—
364#380—1184no—
365C2×#95—591no—
366C2×#95—591no—
367C2×#95—591no—
368C2×#95—592no—
369C2×#95—592no—
370C2×#95—591no—
371C2×#95—592no—
372C2×#95—592no—
373C2×#95—592no—
374C2×#95—1184no—
375C2×#95C2×D5912yes—
376C2×#95C2×D5911yescharacteristic
377C2×#95C2×D5912yes—
378C2×#95C2×D5911yescharacteristic
379C59#1427511yescharacteristic, complemented, Sylow
380C25—591no—
381C23×C4—591no—
382C23×C4—594no—
383C23×C4—594no—
384C23×C4—594no—
385C23×C4—592no—
386C23×C4—591no—
387C23×C4—591no—
388C23×C4—594no—
389C23×C4—594no—
390C23×C4—592no—
391C23×C4—1182no—
392C23×C4—1182no—
393C23×C4—1182no—
394C23×C4—1182no—
395C23×C4—2362no—
396C23×C4—2362no—
397C23×C4—2362no—
398C23×C4—2362no—
399C23×C4—2362no—
400C23×C4—2362no—
401C23×C4C22×D5912yes—
402C23×C4C22×D5912yes—
403C23×C4C22×D5911yescharacteristic
404C23×C4C22×D5911yescharacteristic
405C23×C4C22×D5911yescharacteristic
406C2×C4⋊C4—22no—
407C2×C4⋊C4—42no—
408C2×C4⋊C4—598no—
409C2×C4⋊C4—1184no—
410C2×C4⋊C4—1182no—
411C2×C4⋊C4—1184no—
412C2×C4⋊C4—2362no—
413C2×C4⋊C4—2362no—
414C2×C4⋊C4—2362no—
415C2×C4⋊C4—2362no—
416C2×C4⋊C4—2362no—
417C2×C4⋊C4—2362no—
418C2×C4⋊C4—2362no—
419C2×C22⋊C4—598no—
420C2×C22⋊C4—1184no—
421C2×C22⋊C4—1184no—
422C2×C22⋊C4—1184no—
423C2×C22⋊C4—1184no—
424C2×C22⋊C4—1184no—
425#95—22no—
426#95—24no—
427#95—42no—
428#95—594no—
429#95—1182no—
430#95—1182no—
431#95—1184no—
432#95—1184no—
433#95—1182no—
434#95—2362no—
435#95—2362no—
436#95—2362no—
437#95C4×D5914yescomplemented
438C24—592no—
439C24—592no—
440C24—592no—
441C24—592no—
442C24—592no—
443C24—592no—
444C24—592no—
445C24—1182no—
446C24—1182no—
447C24—1182no—
448C24—1182no—
449C24C23×D5911yescharacteristic
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—592no—
474C22×C4—592no—
475C22×C4—592no—
476C22×C4—5916no—
477C22×C4—594no—
478C22×C4—592no—
479C22×C4—592no—
480C22×C4—1188no—
481C22×C4—1188no—
482C22×C4—1182no—
483C22×C4—1182no—
484C22×C4—1181no—
485C22×C4—1181no—
486C22×C4—1182no—
487C22×C4—1182no—
488C22×C4—1181no—
489C22×C4—1181no—
490C22×C4—1182no—
491C22×C4—1182no—
492C22×C4—1182no—
493C22×C4—1182no—
494C22×C4—1182no—
495C22×C4—1182no—
496C22×C4—1182no—
497C22×C4—1182no—
498C22×C4—1182no—
499C22×C4—1182no—
500C22×C4—1182no—
501C22×C4—1182no—
502C22×C4—1182no—
503C22×C4—1182no—
504C22×C4—1182no—
505C22×C4—2364no—
506C22×C4—2364no—
507C22×C4—2368no—
508C22×C4—2368no—
509C22×C4—2368no—
510C22×C4—2368no—
511C22×C4—2364no—
512C22×C4—2364no—
513C22×C4—2364no—
514C22×C4—2364no—
515C22×C4—2368no—
516C22×C4—2368no—
517C22×C4C2×C4×D5914yes—
518C22×C4C2×C4×D5912yes—
519C22×C4D4×D5912yes—
520C22×C4D4×D5912yes—
521C22×C4Q8×D5912yes—
522C22×C4Q8×D5912yes—
523C4⋊C4—44no—
524C4⋊C4—2368no—
525C4⋊C4—2368no—
526C4⋊C4—2364no—
527C4⋊C4—2368no—
528C22⋊C4—11816no—
529C22⋊C4—2368no—
530C23—594no—
531C23—594no—
532C23—594no—
533C23—594no—
534C23—594no—
535C23—594no—
536C23—594no—
537C23—1182no—
538C23—1182no—
539C23—1184no—
540C23—1182no—
541C23—1182no—
542C23—1184no—
543C23—1184no—
544C23—1184no—
545C23—1184no—
546C23—1184no—
547C23—1184no—
548C23—1184no—
549C23—1184no—
550C23—1184no—
551C23—1184no—
552C23—1184no—
553C23C22×C4×D5911yescharacteristic
554C23C2×D4×D5911yescharacteristic
555C23C2×D4×D5911yescharacteristic
556C23C2×Q8×D5911yescharacteristic
557C23C2×Q8×D5911yescharacteristic
558C23#41×D5911yescharacteristic
559C23#41×D5911yescharacteristic
560C23#1427377212yes—
561C23#1427377312yes—
562C23#1427377312yes—
563C23#1427377412yes—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—594no—
593C2×C4—1182no—
594C2×C4—1182no—
595C2×C4—1184no—
596C2×C4—1184no—
597C2×C4—1182no—
598C2×C4—1182no—
599C2×C4—1184no—
600C2×C4—1184no—
601C2×C4—1184no—
602C2×C4—1184no—
603C2×C4—1184no—
604C2×C4—23616no—
605C2×C4—23616no—
606C2×C4—23616no—
607C2×C4—2362no—
608C2×C4—2362no—
609C2×C4—2362no—
610C2×C4—2362no—
611C2×C4—2362no—
612C2×C4—2362no—
613C2×C4—2362no—
614C2×C4—2361no—
615C2×C4—2362no—
616C2×C4—2361no—
617C2×C4—2361no—
618C2×C4—2361no—
619C2×C4—2362no—
620C2×C4—2362no—
621C2×C4—2362no—
622C2×C4—2362no—
623C2×C4—2362no—
624C2×C4C4⋊C4×D5914yes—
625C22—598no—
626C22—1184no—
627C22—1184no—
628C22—1188no—
629C22—1184no—
630C22—1184no—
631C22—1188no—
632C22—1188no—
633C22—1184no—
634C22—1184no—
635C22—1188no—
636C22C2×C4⋊C4×D5911yescharacteristic
637C22C4×D4×D5911yescharacteristic
638C22C4×Q8×D5911yescharacteristic
639C22#1683787412yes—
640C22#1683788212yes—
641C22#121×D5911yescharacteristic
642C22C22⋊Q8×D5911yescharacteristic
643C22#125×D5911yescharacteristic
644C22C4⋊Q8×D5911yescharacteristic
645C22#1683796211yescharacteristic
646C22#1683796311yescharacteristic
647C22#1683796312yes—
648C22#1683797011yescharacteristic
649C22#1683797211yescharacteristic
650C22#1683798112yes—
651C22#1683798612yes—
652C22#1683799112yes—
653C22#1683799312yes—
654C22#1683799512yes—
655C22#1683802712yes—
656C22#1683802712yes—
657C22#1683803212yes—
658C22#1683803212yes—
659C4—24no—
660C4—42no—
661C4—44no—
662C4—42no—
663C4—82no—
664C4—1184no—
665C4—4722no—
666C4—2362no—
667C4—2364no—
668C4—2362no—
669C2—1188no—
670C2#389×D5911yescharacteristic
671C2#2142670711yescharacteristic
672C2#2142672111yescharacteristic
673C2#2142680012yes—
674C2#2142682312yes—
675C2#2142983111yescharacteristic
676C2#2142983711yescharacteristic
677C2#2142990011yescharacteristic
678C2#2142991011yescharacteristic
679C2#2143006712yes—
680C2#2143021112yes—
subgroupG/H
center, FrattiniC24C23×D59
commutatorC23×C59C23×C4
FittingC59×#1442C2
socle, cosocleC24×C59C24
prime radicalC24×D59C23
hypercenter#1442D59
hypocenterC59#14275
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
5. #21422361C2semidirect
6. #21422360C2semidirect
7. #21422355C2semidirect
8. #21422350C2semidirect
9. #21422347C2semidirect
11. C59×#1442C2semidirect
22. #16837261C4semidirect
70. C2×Dic59⋊C4C2×C4semidirect
85. C59⋊#95C2×C4semidirect
96. #95×C59C2×C4semidirect
181. C2×C4×D59C4⋊C4semidirect
336. #1442D59semidirect
379. C59#14275semidirect
437. #95C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168372762
C2×#168372612
C2×#168372592
C2×#95×D592
#214223612
#214223602
#214223552
#214223502
#214223472
(C2×D59)⋊#952
C59×#14422
#1427559