Finite Group Databasebeta

← #28510652

Subgroups — #28510652

Order 15104.

2222 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273685C211yesmaximal, characteristic
2C2×#16837264C211yesmaximal, characteristic
3C2×#16837259C212yesmaximal
4C2×#95×D59C211yesmaximal, characteristic
5#21422361C211yesmaximal, characteristic, complemented
6#21422355C211yesmaximal, characteristic, complemented
7#21422352C211yesmaximal, characteristic, complemented
8#21422330C212yesmaximal, complemented
9#21422328C212yesmaximal, complemented
10(C2×D59)⋊#95C212yesmaximal
11C59×#1444C211yesmaximal, characteristic, complemented
12C23×C4×D59C2212yes—
13C23×C4×D59C2211yescharacteristic
14C2×#14273685—24no—
15C2×#14273685—28no—
16C22×C4⋊Dic59C2211yescharacteristic
17C22×C4⋊Dic59C2212yes—
18C22×(C2×D59)⋊C4C2211yescharacteristic
19C22×(C2×D59)⋊C4C2211yescharacteristic
20C22×(C2×D59)⋊C4C2212yes—
21C22×C4×Dic59C2212yes—
22#16837264—24no—
23#16837259—28no—
24C2×C59⋊#95C2212yes—
25C2×C59⋊#95C2211yescharacteristic
26C2×C59⋊#95C2212yes—
27C2×C59⋊#95C2212yes—
28C2×C59⋊#95C2211yescharacteristic
29C2×C59⋊#95C2211yescharacteristic
30C2×C59⋊#95C2212yes—
31C2×C59⋊#95C2212yes—
32C2×C59⋊#95C2211yescharacteristic
33C2×C59⋊#95C2212yes—
34#95×D59C418yescomplemented
35C22×C4⋊C4×C59C2211yescharacteristic
36C22×C4⋊C4×C59C2212yes—
37C2×#95×C59C2211yescharacteristic
38C2×#95×C59C2212yes—
39C2×#95×C59C2211yescharacteristic
40C24×D59C2311yescharacteristic
41C23×Dic59C2311yescharacteristic
42C23×Dic59C2311yescharacteristic
43C23×Dic59C2312yes—
44C23×Dic59C2312yes—
45C23×Dic59C2311yescharacteristic
46C22×C4×D59—24no—
47C22×C4×D59—24no—
48C22×C4×D59—24no—
49C22×C4×D59—24no—
50C22×C4×D59—24no—
51C22×C4×D59—24no—
52C22×C4×D59—42no—
53C22×C4×D59—42no—
54C22×C4×D59C2×C418yes—
55C22×C4×D59C2×C414yes—
56C22×C4×D59D414yes—
57C22×C4×D59Q814yes—
58#14273685—232no—
59C2×C4⋊Dic59—22no—
60C2×C4⋊Dic59—24no—
61C2×C4⋊Dic59—24no—
62C2×C4⋊Dic59—24no—
63C2×C4⋊Dic59—42no—
64C2×(C2×D59)⋊C4—24no—
65C2×(C2×D59)⋊C4—24no—
66C2×(C2×D59)⋊C4—24no—
67C2×(C2×D59)⋊C4—22no—
68C2×(C2×D59)⋊C4—22no—
69C2×(C2×D59)⋊C4—24no—
70C2×(C2×D59)⋊C4D418yes—
71C2×C4×Dic59—24no—
72C2×C4×Dic59—24no—
73C2×C4×Dic59—24no—
74C59⋊#95—24no—
75C59⋊#95—22no—
76C59⋊#95—24no—
77C59⋊#95—24no—
78C59⋊#95—22no—
79C59⋊#95—22no—
80C59⋊#95—42no—
81C59⋊#95—42no—
82C59⋊#95C2×C414yes—
83C59⋊#95C2×C418yes—
84C23×C4×C59C2311yescharacteristic
85C23×C4×C59C2311yescharacteristic
86C23×C4×C59C2312yes—
87C23×C4×C59C2311yescharacteristic
88C23×C4×C59C2312yes—
89C2×C4⋊C4×C59—24no—
90C2×C4⋊C4×C59—22no—
91C2×C4⋊C4×C59—24no—
92C2×C4⋊C4×C59—24no—
93C2×C4⋊C4×C59—42no—
94#95×C59—22no—
95#95×C59—24no—
96#95×C59C2×C414yes—
97C24×C59C2411yescharacteristic
98C23×D59—22no—
99C23×D59—22no—
100C23×D59—21no—
101C23×D59—21no—
102C23×D59—21no—
103C23×D59—21no—
104C23×D59C22×C412yes—
105C23×D59C2×D412yes—
106C23×D59C2×D414yes—
107C23×D59C2×Q812yes—
108C23×D59#4114yes—
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—22no—
119C22×C4×C59—22no—
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—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59C22×C412yes—
132C22×C4×C59C22×C414yes—
133C22×C4×C59C2×D414yes—
134C22×C4×C59C2×D412yes—
135C22×C4×C59C2×Q812yes—
136C22×C4×C59#4114yes—
137C22×C4×C59#4114yes—
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—21no—
153C22×Dic59—21no—
154C22×Dic59—22no—
155C22×Dic59—22no—
156C22×Dic59—22no—
157C22×Dic59—22no—
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C22×Dic59C22×C414yes—
163C22×Dic59C22×C412yes—
164C22×Dic59C2×D414yes—
165C22×Dic59C2×D412yes—
166C22×Dic59C2×Q812yes—
167C2×C4×D59—216no—
168C2×C4×D59—28no—
169C2×C4×D59—216no—
170C2×C4×D59—28no—
171C2×C4×D59—216no—
172C2×C4×D59—216no—
173C2×C4×D59—48no—
174C2×C4×D59—48no—
175C2×C4×D59—48no—
176C2×C4×D59—48no—
177C2×C4×D59—48no—
178C2×C4×D59C4⋊C4116yescomplemented
179C4⋊C4×C59—28no—
180C4⋊C4×C59—48no—
181(C2×D59)⋊C4—216no—
182(C2×D59)⋊C4—44no—
183(C2×D59)⋊C4—44no—
184C4⋊Dic59—28no—
185C4⋊Dic59—48no—
186C4×Dic59—216no—
187C23×C59C23×C411yescharacteristic
188C23×C59C22×D411yescharacteristic
189C23×C59C22×D412yes—
190C23×C59C22×Q811yescharacteristic
191C23×C59C2×#4112yes—
192C23×C59C2×#4112yes—
193C23×C59C2×#4112yes—
194C23×C59H(5,𝔽2)11yescharacteristic
195C23×C59H(5,𝔽2)11yescharacteristic
196C23×C592−1+411yescharacteristic
197C23×C592−1+411yescharacteristic
198C22×D59—24no—
199C22×D59—22no—
200C22×D59—22no—
201C22×D59—22no—
202C22×D59—22no—
203C22×D59—24no—
204C22×D59—22no—
205C22×D59—22no—
206C22×D59—22no—
207C22×D59—22no—
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×D418yes—
218C22×D59#12014yes—
219C22×D59C22⋊Q818yes—
220C22×D59#12314yes—
221C2×C4×C59—24no—
222C2×C4×C59—24no—
223C2×C4×C59—24no—
224C2×C4×C59—22no—
225C2×C4×C59—24no—
226C2×C4×C59—22no—
227C2×C4×C59—24no—
228C2×C4×C59—24no—
229C2×C4×C59—24no—
230C2×C4×C59—24no—
231C2×C4×C59—24no—
232C2×C4×C59—24no—
233C2×C4×C59—24no—
234C2×C4×C59—42no—
235C2×C4×C59—42no—
236C2×C4×C59—42no—
237C2×C4×C59—42no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—42no—
245C2×C4×C59—42no—
246C2×C4×C59—42no—
247C2×C4×C59—42no—
248C2×C4×C59C2×C4⋊C414yes—
249C2×C4×C59#11718yescomplemented
250C2×Dic59—24no—
251C2×Dic59—22no—
252C2×Dic59—24no—
253C2×Dic59—22no—
254C2×Dic59—24no—
255C2×Dic59—24no—
256C2×Dic59—24no—
257C2×Dic59—24no—
258C2×Dic59—24no—
259C2×Dic59—24no—
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—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—42no—
275C2×Dic59—42no—
276C2×Dic59—42no—
277C2×Dic59—42no—
278C2×Dic59C2×C4⋊C414yes—
279C4×D59—232no—
280C4×D59—432no—
281#13878—591nomaximal, Sylow
282C22×C59C22×C4⋊C411yescharacteristic
283C22×C59C2×#11712yes—
284C22×C59C2×C4×D412yes—
285C22×C59#51811yescharacteristic
286C22×C59#51911yescharacteristic
287C22×C59C2×#12011yescharacteristic
288C22×C59C2×C22⋊Q812yes—
289C22×C59C2×#12311yescharacteristic
290C22×C59#53211yescharacteristic
291C22×C59#53211yescharacteristic
292C22×C59#53312yes—
293C22×C59#53711yescharacteristic
294C22×C59#53711yescharacteristic
295C22×C59#54411yescharacteristic
296C22×C59#54411yescharacteristic
297C22×C59#54612yes—
298C22×C59#54612yes—
299C22×C59#54712yes—
300C22×C59#54712yes—
301C22×C59#55112yes—
302C22×C59#55112yes—
303C22×C59#55212yes—
304C22×C59#55212yes—
305C2×D59—24no—
306C2×D59—28no—
307C2×D59—28no—
308C2×D59—24no—
309C2×D59—24no—
310C2×D59—28no—
311C2×D59—28no—
312C2×D59—24no—
313C2×D59—24no—
314C2×D59—24no—
315C2×D59#38518yes—
316C4×C59—28no—
317C4×C59—24no—
318C4×C59—44no—
319C4×C59—81no—
320C4×C59—81no—
321Dic59—24no—
322Dic59—44no—
323Dic59—44no—
324Dic59—81no—
325Dic59—81no—
326C22×#117—591no—
327C2×#381—591no—
328C2×#380—592no—
329C22×#95—591no—
330#1444—591no—
331#1444D5911yescharacteristic, complemented
332#1442—591no—
333#1442—591no—
334#1436—592no—
335#1436—592no—
336C22⋊#95—592no—
337C2×C59#228511yescharacteristic
338C2×C59C2×#38511yescharacteristic
339C2×C59#231611yescharacteristic
340C2×C59#234212yes—
341C2×C59#234412yes—
342C2×C59#243011yescharacteristic
343C2×C59#243011yescharacteristic
344C2×C59#243812yes—
345C2×C59#243812yes—
346C2×C59#249811yescharacteristic
347C2×C59#249811yescharacteristic
348D59—28no—
349C24×C4—591no—
350C24×C4—592no—
351C2×#117—1188no—
352C2×#117—1184no—
353C22×C4⋊C4—591no—
354C22×C4⋊C4—592no—
355C22×C4⋊C4C2×D5911yescharacteristic
356C22×C4⋊C4C2×D5912yes—
357C22×C22⋊C4—592no—
358C22×C22⋊C4—591no—
359C22×C22⋊C4—591no—
360C22×C42—592no—
361#381—1184no—
362#380—1188no—
363C2×#95—592no—
364C2×#95—591no—
365C2×#95—598no—
366C2×#95—592no—
367C2×#95—591no—
368C2×#95—591no—
369C2×#95—592no—
370C2×#95—592no—
371C2×#95—591no—
372C2×#95—592no—
373C2×#95—592no—
374C2×#95C2×D5912yes—
375C2×#95C2×D5911yescharacteristic
376C2×#95C2×D5911yescharacteristic
377C59#1387811yescharacteristic, complemented, Sylow
378C25—591no—
379C23×C4—591no—
380C23×C4—594no—
381C23×C4—594no—
382C23×C4—594no—
383C23×C4—592no—
384C23×C4—592no—
385C23×C4—598no—
386C23×C4—591no—
387C23×C4—591no—
388C23×C4—1184no—
389C23×C4—1184no—
390C23×C4—1184no—
391C23×C4—1184no—
392C23×C4—1184no—
393C23×C4—1184no—
394C23×C4—2362no—
395C23×C4—2362no—
396C23×C4C22×D5911yescharacteristic
397C23×C4C22×D5911yescharacteristic
398C23×C4C22×D5912yes—
399C23×C4C22×D5911yescharacteristic
400C23×C4C22×D5912yes—
401#117—11832no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—24no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4—42no—
407C2×C4⋊C4—1184no—
408C2×C4⋊C4—1182no—
409C2×C4⋊C4—1184no—
410C2×C4⋊C4—1184no—
411C2×C4⋊C4—2362no—
412C2×C22⋊C4—598no—
413C2×C22⋊C4—1184no—
414C2×C22⋊C4—1184no—
415C2×C22⋊C4—1184no—
416C2×C22⋊C4—1182no—
417C2×C22⋊C4—1182no—
418C2×C22⋊C4—1184no—
419C2×C42—1184no—
420C2×C42—1184no—
421C2×C42—1184no—
422#95—22no—
423#95—24no—
424#95—598no—
425#95—594no—
426#95—1184no—
427#95—1182no—
428#95—1182no—
429#95—1184no—
430#95—1182no—
431#95—1184no—
432#95—2362no—
433#95—2362no—
434#95C4×D5914yes—
435C24—592no—
436C24—594no—
437C24—592no—
438C24—592no—
439C24—594no—
440C24—1182no—
441C24—1182no—
442C24—1181no—
443C24—1181no—
444C24—1181no—
445C24—1181no—
446C24C23×D5911yescharacteristic
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—592no—
470C22×C4—592no—
471C22×C4—592no—
472C22×C4—5916no—
473C22×C4—594no—
474C22×C4—594no—
475C22×C4—1188no—
476C22×C4—11816no—
477C22×C4—1188no—
478C22×C4—11816no—
479C22×C4—11816no—
480C22×C4—1182no—
481C22×C4—1182no—
482C22×C4—1182no—
483C22×C4—1182no—
484C22×C4—1182no—
485C22×C4—1182no—
486C22×C4—1182no—
487C22×C4—1182no—
488C22×C4—1182no—
489C22×C4—1182no—
490C22×C4—1182no—
491C22×C4—1182no—
492C22×C4—1182no—
493C22×C4—1182no—
494C22×C4—11816no—
495C22×C4—1181no—
496C22×C4—1181no—
497C22×C4—1182no—
498C22×C4—1182no—
499C22×C4—1182no—
500C22×C4—1182no—
501C22×C4—1181no—
502C22×C4—1181no—
503C22×C4—1181no—
504C22×C4—1181no—
505C22×C4—2368no—
506C22×C4—2368no—
507C22×C4—2368no—
508C22×C4—2368no—
509C22×C4—2368no—
510C22×C4C2×C4×D5912yes—
511C22×C4C2×C4×D5914yes—
512C22×C4D4×D5914yes—
513C22×C4D4×D5912yes—
514C22×C4#1174568714yes—
515C22×C4Q8×D5912yes—
516C22×C4#1174568914yes—
517C4⋊C4—28no—
518C4⋊C4—48no—
519C4⋊C4—1188no—
520C4⋊C4—2368no—
521C22⋊C4—11816no—
522C22⋊C4—2364no—
523C22⋊C4—2364no—
524C42—11816no—
525C23—598no—
526C23—594no—
527C23—594no—
528C23—598no—
529C23—594no—
530C23—1182no—
531C23—1182no—
532C23—1184no—
533C23—1182no—
534C23—1182no—
535C23—1182no—
536C23—1182no—
537C23—1184no—
538C23—1184no—
539C23—1184no—
540C23—1184no—
541C23—1184no—
542C23—1184no—
543C23—1184no—
544C23—1184no—
545C23—1184no—
546C23—1182no—
547C23—1182no—
548C23C22×C4×D5911yescharacteristic
549C23C2×D4×D5912yes—
550C23C2×D4×D5911yescharacteristic
551C23C2×Q8×D5911yescharacteristic
552C23#41×D5912yes—
553C23C2×#1174568712yes—
554C23C2×#1174568912yes—
555C23#1427377211yescharacteristic
556C23#1427377311yescharacteristic
557C23#1427377411yescharacteristic
558C23#1427377511yescharacteristic
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—594no—
587C2×C4—11832no—
588C2×C4—1184no—
589C2×C4—1182no—
590C2×C4—1184no—
591C2×C4—1182no—
592C2×C4—1184no—
593C2×C4—1184no—
594C2×C4—1184no—
595C2×C4—1184no—
596C2×C4—1184no—
597C2×C4—1184no—
598C2×C4—23632no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4—2362no—
605C2×C4—2362no—
606C2×C4—2362no—
607C2×C4—2362no—
608C2×C4—2361no—
609C2×C4—2362no—
610C2×C4—2362no—
611C2×C4—2362no—
612C2×C4—2362no—
613C2×C4—2362no—
614C2×C4—2362no—
615C2×C4—2362no—
616C2×C4—2362no—
617C2×C4C4⋊C4×D5914yes—
618C2×C4#1427368518yescomplemented
619C22—598no—
620C22—1188no—
621C22—1184no—
622C22—1188no—
623C22—1184no—
624C22—1188no—
625C22—1184no—
626C22—1184no—
627C22—1184no—
628C22—1184no—
629C22—1188no—
630C22C2×C4⋊C4×D5911yescharacteristic
631C22C2×#1427368512yes—
632C22C4×D4×D5912yes—
633C22#1683787411yescharacteristic
634C22#1683787811yescharacteristic
635C22#120×D5911yescharacteristic
636C22C22⋊Q8×D5912yes—
637C22#123×D5911yescharacteristic
638C22#1683795011yescharacteristic
639C22#1683795311yescharacteristic
640C22#1683795712yes—
641C22#1683798111yescharacteristic
642C22#1683798211yescharacteristic
643C22#1683802911yescharacteristic
644C22#1683803211yescharacteristic
645C22#1683803912yes—
646C22#1683804312yes—
647C22#1683805212yes—
648C22#1683805512yes—
649C22#1683807512yes—
650C22#1683807912yes—
651C22#1683808612yes—
652C22#1683809012yes—
653C4—24no—
654C4—28no—
655C4—44no—
656C4—81no—
657C4—81no—
658C4—1184no—
659C4—4721no—
660C4—4721no—
661C4—2364no—
662C4—2364no—
663C2—1188no—
664C2#2142650111yescharacteristic
665C2#385×D5911yescharacteristic
666C2#2142671111yescharacteristic
667C2#2142694312yes—
668C2#2142696412yes—
669C2#2142769911yescharacteristic
670C2#2142770711yescharacteristic
671C2#2142776512yes—
672C2#2142777312yes—
673C2#2142853311yescharacteristic
674C2#2142854111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D59
commutatorC23×C59C23×C4
FittingC59×#1444C2
socle, cosocleC24×C59C24
prime radicalC24×D59C23
hypercenter#1444D59
hypocenterC59#13878
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
5. #21422361C2semidirect
6. #21422355C2semidirect
7. #21422352C2semidirect
8. #21422330C2semidirect
9. #21422328C2semidirect
11. C59×#1444C2semidirect
34. #95×D59C4semidirect
178. C2×C4×D59C4⋊C4semidirect
249. C2×C4×C59#117semidirect
331. #1444D59semidirect
377. C59#13878semidirect
618. C2×C4#14273685semidirect

Maximal subgroups

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

subgroupindex
C22×#142736852
C2×#168372642
C2×#168372592
C2×#95×D592
#214223612
#214223552
#214223522
#214223302
#214223282
(C2×D59)⋊#952
C59×#14442
#1387859