Finite Group Databasebeta

← #28515702

Subgroups — #28515702

Order 15104.

1590 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837338C211yesmaximal, characteristic
2C2×#16837300C211yesmaximal, characteristic, complemented
3C2×#16837297C212yesmaximal, complemented
4C2×#16837296C212yesmaximal
5#21422365C211yesmaximal, characteristic, complemented
6#21422362C212yesmaximal, complemented
7#21422360C211yesmaximal, characteristic, complemented
8#21422341C211yesmaximal, characteristic
9#21422339C212yesmaximal
10C4×C59⋊#95C211yesmaximal, characteristic, complemented
11C2×C59×#388C211yesmaximal, characteristic
12C22×C4⋊Dic59C2211yescharacteristic
13C22×C4⋊Dic59C2211yescharacteristic
14C22×Dic59⋊C4C2212yes—
15C22×Dic59⋊C4C2212yes—
16C22×Dic59⋊C4C2212yes—
17C22×C22⋊Dic59C2212yes—
18C22×C22⋊Dic59C2212yes—
19C22×C4×Dic59C2212yes—
20#16837338C418yescomplemented
21#16837300—24no—
22#16837297—28no—
23#16837296—28no—
24C2×C59⋊#95C2212yes—
25C2×C59⋊#95C2212yes—
26C2×C59⋊#95C2212yes—
27C2×C59⋊#95C2212yes—
28C2×C59⋊#95C2211yescharacteristic
29C2×C59⋊#95C2212yes—
30C2×C59⋊#95C2212yes—
31C2×C59⋊#95C2211yescharacteristic
32C22×C4⋊C4×C59C2211yescharacteristic
33C22×C22⋊C4×C59C2211yescharacteristic
34C22×C22⋊C4×C59C2211yescharacteristic
35C22×C22⋊C4×C59C2211yescharacteristic
36C22×C42×C59C2211yescharacteristic
37C59×#388—24no—
38C2×#95×C59C2211yescharacteristic
39C2×#95×C59C2211yescharacteristic
40C23×Dic59C2312yes—
41C23×Dic59C2312yes—
42C23×Dic59C2312yes—
43C23×Dic59C2312yes—
44C2×C4⋊Dic59—22no—
45C2×C4⋊Dic59—22no—
46C2×C4⋊Dic59—22no—
47C2×C4⋊Dic59—42no—
48C2×C4⋊Dic59C2×C414yes—
49C2×Dic59⋊C4—24no—
50C2×Dic59⋊C4—24no—
51C2×Dic59⋊C4—24no—
52C2×Dic59⋊C4—24no—
53C2×Dic59⋊C4—24no—
54C2×Dic59⋊C4—24no—
55C2×Dic59⋊C4—42no—
56C2×Dic59⋊C4—42no—
57C2×Dic59⋊C4C2×C418yes—
58C2×C22⋊Dic59—24no—
59C2×C22⋊Dic59—24no—
60C2×C22⋊Dic59—24no—
61C2×C22⋊Dic59—42no—
62C2×C22⋊Dic59—42no—
63C2×C22⋊Dic59C2×C418yes—
64C2×C4×Dic59—42no—
65C2×C4×Dic59—42no—
66C2×C4×Dic59D418yes—
67C59⋊#95—24no—
68C59⋊#95—22no—
69C59⋊#95—24no—
70C59⋊#95—24no—
71C59⋊#95—24no—
72C59⋊#95—24no—
73C59⋊#95—24no—
74C59⋊#95C2×C414yes—
75C25×C59C2311yescharacteristic
76C23×C4×C59C2311yescharacteristic
77C23×C4×C59C2311yescharacteristic
78C23×C4×C59C2311yescharacteristic
79C23×C4×C59C2311yescharacteristic
80C23×C4×C59C2311yescharacteristic
81C23×C4×C59C2311yescharacteristic
82C2×C4⋊C4×C59—22no—
83C2×C4⋊C4×C59—24no—
84C2×C22⋊C4×C59—22no—
85C2×C22⋊C4×C59—22no—
86C2×C22⋊C4×C59—22no—
87C2×C22⋊C4×C59—22no—
88C2×C22⋊C4×C59—24no—
89C2×C22⋊C4×C59—42no—
90C2×C22⋊C4×C59C2×C414yes—
91C2×C42×C59—42no—
92C2×C42×C59D414yes—
93#95×C59—22no—
94#95×C59—22no—
95C24×C59—21no—
96C24×C59—22no—
97C24×C59—21no—
98C24×C59—22no—
99C24×C59—22no—
100C24×C59—21no—
101C24×C59—21no—
102C24×C59—21no—
103C24×C59—22no—
104C24×C59—21no—
105C24×C59C22×C412yes—
106C24×C59C2411yescharacteristic
107C22×C4×C59—22no—
108C22×C4×C59—22no—
109C22×C4×C59—21no—
110C22×C4×C59—22no—
111C22×C4×C59—21no—
112C22×C4×C59—21no—
113C22×C4×C59—22no—
114C22×C4×C59—21no—
115C22×C4×C59—22no—
116C22×C4×C59—22no—
117C22×C4×C59—22no—
118C22×C4×C59—21no—
119C22×C4×C59—21no—
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—22no—
129C22×C4×C59—21no—
130C22×C4×C59C22×C412yes—
131C22×C4×C59C22×C412yes—
132C22×C4×C59C2×D412yes—
133C22×C4×C59C2×D412yes—
134C22×C4×C59C2×D414yes—
135C22×C4×C59C2×D412yes—
136C22×C4×C59#4112yes—
137C22×Dic59—22no—
138C22×Dic59—22no—
139C22×Dic59—22no—
140C22×Dic59—22no—
141C22×Dic59—22no—
142C22×Dic59—22no—
143C22×Dic59—22no—
144C22×Dic59—22no—
145C22×Dic59—22no—
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×Dic59C22×C414yes—
161C22×Dic59C22×C414yes—
162C22×Dic59C2×D414yes—
163C22×Dic59C2×D414yes—
164C22×Dic59#4114yes—
165C4⋊C4×C59—44no—
166C22⋊C4×C59—28no—
167C22⋊C4×C59—44no—
168C22⋊C4×C59—44no—
169C42×C59—44no—
170C22⋊Dic59—216no—
171C22⋊Dic59—48no—
172C4⋊Dic59—28no—
173C4⋊Dic59—44no—
174Dic59⋊C4—216no—
175Dic59⋊C4—48no—
176Dic59⋊C4—48no—
177C4×Dic59—48no—
178C23×C59—24no—
179C23×C59—22no—
180C23×C59—22no—
181C23×C59—24no—
182C23×C59—22no—
183C23×C59—41no—
184C23×C59—41no—
185C23×C59—42no—
186C23×C59—42no—
187C23×C59—41no—
188C23×C59—42no—
189C23×C59—42no—
190C23×C59—41no—
191C23×C59—41no—
192C23×C59—42no—
193C23×C59—41no—
194C23×C59—42no—
195C23×C59—42no—
196C23×C59—42no—
197C23×C59—42no—
198C23×C59—41no—
199C23×C59—42no—
200C23×C59—41no—
201C23×C59C23×C411yescharacteristic
202C23×C59C22×D412yes—
203C23×C59C22×D411yescharacteristic
204C23×C59C2×#4112yes—
205C23×C59C2×#4111yescharacteristic
206C23×C59H(5,𝔽2)12yes—
207C23×C59H(5,𝔽2)11yescharacteristic
208C23×C592−1+411yescharacteristic
209C23×C592−1+411yescharacteristic
210C23×C592−1+412yes—
211C23×C592−1+411yescharacteristic
212C2×C4×C59—24no—
213C2×C4×C59—24no—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—24no—
218C2×C4×C59—22no—
219C2×C4×C59—24no—
220C2×C4×C59—24no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—42no—
228C2×C4×C59—42no—
229C2×C4×C59—42no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—42no—
233C2×C4×C59—41no—
234C2×C4×C59—42no—
235C2×C4×C59—42no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—42no—
239C2×C4×C59—41no—
240C2×C4×C59C4×D414yes—
241C2×C4×C59#12714yes—
242C2×Dic59—24no—
243C2×Dic59—24no—
244C2×Dic59—24no—
245C2×Dic59—24no—
246C2×Dic59—24no—
247C2×Dic59—24no—
248C2×Dic59—24no—
249C2×Dic59—24no—
250C2×Dic59—24no—
251C2×Dic59—24no—
252C2×Dic59—24no—
253C2×Dic59—24no—
254C2×Dic59—42no—
255C2×Dic59—42no—
256C2×Dic59—42no—
257C2×Dic59—42no—
258C2×Dic59—42no—
259C2×Dic59—42no—
260C2×Dic59—42no—
261C2×Dic59—42no—
262C2×Dic59—42no—
263C2×Dic59—42no—
264C2×Dic59—42no—
265C2×Dic59—42no—
266C2×Dic59—42no—
267C2×Dic59—42no—
268C2×Dic59—42no—
269C2×Dic59C4×D418yes—
270#14322—591nomaximal, Sylow
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—44no—
274C22×C59—44no—
275C22×C59—42no—
276C22×C59—81no—
277C22×C59—81no—
278C22×C59—81no—
279C22×C59—82no—
280C22×C59—81no—
281C22×C59—82no—
282C22×C59C2×C4×D412yes—
283C22×C59C2×C4×D411yescharacteristic
284C22×C59#51911yescharacteristic
285C22×C59#52012yes—
286C22×C59#52011yescharacteristic
287C22×C59C2×#12711yescharacteristic
288C22×C59#53211yescharacteristic
289C22×C59#53212yes—
290C22×C59#53412yes—
291C22×C59#53411yescharacteristic
292C22×C59#53612yes—
293C22×C59#53612yes—
294C22×C59#53611yescharacteristic
295C22×C59#53611yescharacteristic
296C22×C59#53712yes—
297C22×C59#53711yescharacteristic
298C22×C59#53912yes—
299C22×C59#53912yes—
300C22×C59#53912yes—
301C22×C59#54112yes—
302C22×C59#54111yescharacteristic
303C22×C59#54212yes—
304C22×C59#54211yescharacteristic
305C4×C59—24no—
306C4×C59—24no—
307C4×C59—42no—
308C4×C59—42no—
309C4×C59—81no—
310C4×C59—81no—
311Dic59—28no—
312Dic59—44no—
313Dic59—44no—
314Dic59—82no—
315C2×#399—591no—
316C2×#389—591no—
317C2×#389—592no—
318C2×#388—592no—
319C2×#388D5911yescharacteristic
320#1445—592no—
321#1445—591no—
322#1444—591no—
323#1439—592no—
324#1439—591no—
325C4×#95—591no—
326C2×C59—82no—
327C2×C59C4×#12711yescharacteristic
328C2×C59#232211yescharacteristic
329C2×C59#232212yes—
330C2×C59#232312yes—
331C2×C59#232311yescharacteristic
332C2×C59#265611yescharacteristic
333C2×C59#265811yescharacteristic
334C2×C59#266311yescharacteristic
335C2×C59#266312yes—
336C2×C59#266411yescharacteristic
337C2×C59#266412yes—
338C22×C4⋊C4—592no—
339C22×C4⋊C4—591no—
340C22×C4⋊C4—592no—
341C22×C4⋊C4—592no—
342C22×C4⋊C4—591no—
343C22×C4⋊C4C2×D5911yescharacteristic
344C22×C22⋊C4—592no—
345C22×C22⋊C4—592no—
346C22×C22⋊C4C2×D5911yescharacteristic
347C22×C22⋊C4C2×D5911yescharacteristic
348C22×C22⋊C4C2×D5911yescharacteristic
349C22×C42—592no—
350C22×C42C2×D5911yescharacteristic
351#399—598no—
352#389—1188no—
353#389—1184no—
354#388—24no—
355#388—1188no—
356C2×#95—591no—
357C2×#95—592no—
358C2×#95—591no—
359C2×#95—592no—
360C2×#95—592no—
361C2×#95—592no—
362C2×#95—592no—
363C2×#95—592no—
364C2×#95C2×D5911yescharacteristic
365C2×#95C2×D5911yescharacteristic
366C59#1432211yescharacteristic, complemented, Sylow
367C25C22×D5911yescharacteristic
368C23×C4—592no—
369C23×C4—592no—
370C23×C4—592no—
371C23×C4—592no—
372C23×C4C22×D5911yescharacteristic
373C23×C4C22×D5911yescharacteristic
374C23×C4C22×D5911yescharacteristic
375C23×C4C22×D5911yescharacteristic
376C23×C4C22×D5911yescharacteristic
377C23×C4C22×D5911yescharacteristic
378C2×C4⋊C4—24no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—598no—
381C2×C4⋊C4—594no—
382C2×C4⋊C4—1184no—
383C2×C4⋊C4—1184no—
384C2×C4⋊C4—1182no—
385C2×C4⋊C4—1182no—
386C2×C4⋊C4—1184no—
387C2×C4⋊C4—1184no—
388C2×C4⋊C4—1184no—
389C2×C4⋊C4—1184no—
390C2×C4⋊C4—1182no—
391C2×C4⋊C4—2362no—
392C2×C4⋊C4—2362no—
393C2×C4⋊C4—2362no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—42no—
400C2×C22⋊C4—598no—
401C2×C22⋊C4—1184no—
402C2×C22⋊C4—1184no—
403C2×C22⋊C4—1184no—
404C2×C22⋊C4—2362no—
405C2×C22⋊C4—2362no—
406C2×C22⋊C4C4×D5914yes—
407C2×C42—42no—
408C2×C42—598no—
409C2×C42—2362no—
410C2×C42—2362no—
411C2×C42C22⋊D5914yes—
412#95—22no—
413#95—22no—
414#95—594no—
415#95—1184no—
416#95—1184no—
417#95—1184no—
418#95—1184no—
419#95—1184no—
420#95—1182no—
421#95—1184no—
422C24—22no—
423C24—21no—
424C24—22no—
425C24—21no—
426C24—22no—
427C24—21no—
428C24—21no—
429C24—21no—
430C24—22no—
431C24—21no—
432C24C2×C4×D5912yes—
433C24C23×D5911yescharacteristic
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—594no—
458C22×C4—594no—
459C22×C4—594no—
460C22×C4—594no—
461C22×C4—594no—
462C22×C4—1182no—
463C22×C4—1182no—
464C22×C4—1182no—
465C22×C4—1182no—
466C22×C4—1182no—
467C22×C4—1182no—
468C22×C4—1182no—
469C22×C4—1182no—
470C22×C4—1182no—
471C22×C4—1182no—
472C22×C4—1182no—
473C22×C4—1182no—
474C22×C4—1182no—
475C22×C4—1182no—
476C22×C4—1182no—
477C22×C4—1182no—
478C22×C4—1182no—
479C22×C4—1182no—
480C22×C4—1182no—
481C22×C4—1182no—
482C22×C4—1182no—
483C22×C4—1182no—
484C22×C4—1182no—
485C22×C4C2×C4×D5912yes—
486C22×C4C2×C4×D5912yes—
487C22×C4#1174568512yes—
488C22×C4D4×D5914yes—
489C22×C4C2×C22⋊D5912yes—
490C22×C4C2×C22⋊D5912yes—
491C22×C4C2×C22⋊D5912yes—
492C4⋊C4—44no—
493C4⋊C4—11816no—
494C4⋊C4—1188no—
495C4⋊C4—2364no—
496C4⋊C4—2368no—
497C4⋊C4—2368no—
498C22⋊C4—28no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—11816no—
502C22⋊C4—2368no—
503C42—44no—
504C42—2368no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—41no—
528C23C22×C4×D5911yescharacteristic
529C23C2×D4×D5912yes—
530C23C22×C22⋊D5911yescharacteristic
531C23#41×D5912yes—
532C23C2×#1174568511yescharacteristic
533C23#1427377212yes—
534C23#1427377311yescharacteristic
535C23#1427377412yes—
536C23#1427377411yescharacteristic
537C23#1427377411yescharacteristic
538C23#1427377411yescharacteristic
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—598no—
568C2×C4—1184no—
569C2×C4—1184no—
570C2×C4—1184no—
571C2×C4—1184no—
572C2×C4—1184no—
573C2×C4—1184no—
574C2×C4—1184no—
575C2×C4—1184no—
576C2×C4—1184no—
577C2×C4—1184no—
578C2×C4—1184no—
579C2×C4—1184no—
580C2×C4—2362no—
581C2×C4—2362no—
582C2×C4—2362no—
583C2×C4—2362no—
584C2×C4—2362no—
585C2×C4—2362no—
586C2×C4—2362no—
587C2×C4—2362no—
588C2×C4—2362no—
589C2×C4—2362no—
590C2×C4—2362no—
591C2×C4—2362no—
592C2×C4—2362no—
593C2×C4—2362no—
594C2×C4—2362no—
595C2×C4C4×C22⋊D5914yes—
596C2×C4#1427372314yes—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—82no—
606C22—81no—
607C22—82no—
608C22C4×D4×D5912yes—
609C22C2×C4×C22⋊D5911yescharacteristic
610C22#1683787911yescharacteristic
611C22#1683788112yes—
612C22#1683788211yescharacteristic
613C22C2×#1427372311yescharacteristic
614C22#1683794811yescharacteristic
615C22#1683794912yes—
616C22#1683796012yes—
617C22#1683796111yescharacteristic
618C22#1683797212yes—
619C22#1683797212yes—
620C22#1683797311yescharacteristic
621C22#1683797311yescharacteristic
622C22#1683797912yes—
623C22#1683798011yescharacteristic
624C22#1683799312yes—
625C22#1683799412yes—
626C22#1683799512yes—
627C22#1683800912yes—
628C22#1683801411yescharacteristic
629C22#1683802012yes—
630C22#1683802311yescharacteristic
631C4—24no—
632C4—24no—
633C4—42no—
634C4—42no—
635C4—81no—
636C4—81no—
637C4—1188no—
638C4—4722no—
639C4—2364no—
640C4—2364no—
641C2—82no—
642C2C4×#1427372311yescharacteristic
643C2#2142676111yescharacteristic
644C2#2142676612yes—
645C2#2142677112yes—
646C2#2142677811yescharacteristic
647C2#2143020011yescharacteristic
648C2#2143020711yescharacteristic
649C2#2143024611yescharacteristic
650C2#2143024812yes—
651C2#2143025811yescharacteristic
652C2#2143026112yes—
subgroupG/H
center, FrattiniC24C23×D59
commutatorC23×C59C23×C4
FittingC2×C59×#388C2
socle, cosocleC24×C59C24
prime radicalC25×C59C23
hypercenterC2×#388D59
hypocenterC59#14322
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ C2×#388

Complements

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

NH ≅ G/Ntype
2. C2×#16837300C2semidirect
3. C2×#16837297C2semidirect
5. #21422365C2semidirect
6. #21422362C2semidirect
7. #21422360C2semidirect
10. C4×C59⋊#95C2semidirect
20. #16837338C4semidirect
366. C59#14322semidirect

Maximal subgroups

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

subgroupindex
C2×#168373382
C2×#168373002
C2×#168372972
C2×#168372962
#214223652
#214223622
#214223602
#214223412
#214223392
C4×C59⋊#952
C2×C59×#3882
#1432259