Finite Group Databasebeta

← #28617557

Subgroups — #28617557

Order 15104.

1166 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837444C211yesmaximal, characteristic, complemented
2#21426479C211yesmaximal, characteristic
3C2×#16837262C211yesmaximal, characteristic
4#21425529C212yesmaximal, complemented
5#21425529C212yesmaximal, complemented
6#21425524C212yesmaximal, complemented
7#21425519C212yesmaximal, complemented
8#21423816C211yesmaximal, characteristic, complemented
9#21423816C211yesmaximal, characteristic, complemented
10#21423816C211yesmaximal, characteristic, complemented
11C2×C59×#407C211yesmaximal, characteristic
12C2×#14273686C2211yescharacteristic
13C2×#14273686C2211yescharacteristic
14C22×C4⋊Dic59C2211yescharacteristic
15C22×C4⋊Dic59C2211yescharacteristic
16C22×C22⋊Dic59—21no—
17#16837444—24no—
18C2×#14273661C2212yescomplemented
19C2×#14273659C2212yescomplemented
20#16837265—21no—
21#16837262—21no—
22#16837262—21no—
23#16837262C2212yes—
24#16837262C2212yes—
25#16837262C2211yescharacteristic
26#16837262C2211yescharacteristic
27#16837262C2211yescharacteristic
28#16837262C2211yescharacteristic
29#16837258—21no—
30C4×C22⋊Dic59—21no—
31C4×C22⋊Dic59—21no—
32C2×C59⋊#95—21no—
33#16837165C2212yescomplemented
34#16837165C2212yescomplemented
35#16837165C2212yescomplemented
36#16837165C2212yescomplemented
37#16837165C2212yescomplemented
38#16837165C2212yescomplemented
39C24×C4×C59C2211yescharacteristic
40C22×M16×C59C2211yescharacteristic
41C22×M16×C59C2211yescharacteristic
42C59×#407—24no—
43C2×C22⋊C8×C59C2212yes—
44C2×C22⋊C8×C59C2212yes—
45C23×Dic59—21no—
46C23×Dic59—21no—
47#14273686—22no—
48#14273686—22no—
49#14273686—22no—
50#14273686—22no—
51C2×C4⋊Dic59—22no—
52C2×C4⋊Dic59—22no—
53C2×C4⋊Dic59—22no—
54C2×C4⋊Dic59—22no—
55C2×C4⋊Dic59—21no—
56C2×C4⋊Dic59—21no—
57C2×C4⋊Dic59C2312yes—
58C2×C4⋊Dic59C2311yescharacteristic
59C2×C4⋊Dic59C2311yescharacteristic
60C2×C4⋊Dic59C2311yescharacteristic
61C2×C4⋊Dic59C2312yes—
62C2×C4⋊Dic59C2311yescharacteristic
63C2×C22⋊Dic59—21no—
64C2×C22⋊Dic59—21no—
65C2×C22⋊Dic59—21no—
66C2×C22⋊Dic59—21no—
67C2×C22⋊Dic59—21no—
68C2×C22⋊Dic59—21no—
69C2×C22⋊Dic59—22no—
70C2×C22⋊Dic59—22no—
71C2×C22⋊Dic59—44no—
72C2×C4×Dic59—21no—
73C2×C4×Dic59—21no—
74#14273661—24no—
75#14273659—24no—
76C59⋊#95—22no—
77C59⋊#95—22no—
78C59⋊#95—21no—
79C59⋊#95—21no—
80C59⋊#95—21no—
81C59⋊#95—21no—
82C25×C59D411yescharacteristic
83C23×C4×C59—22no—
84C23×C4×C59—22no—
85C23×C4×C59—22no—
86C23×C4×C59—22no—
87C23×C4×C59—22no—
88C23×C4×C59—22no—
89C23×C4×C59D411yescharacteristic
90C23×C4×C59D411yescharacteristic
91C23×C4×C59D411yescharacteristic
92C23×C4×C59C2311yescharacteristic
93C23×C4×C59C2311yescharacteristic
94C23×C4×C59C2311yescharacteristic
95C2×M16×C59—22no—
96C2×M16×C59—22no—
97C2×M16×C59—22no—
98C2×M16×C59—22no—
99C2×M16×C59—22no—
100C2×M16×C59D414yes—
101C22×C8×C59C2312yes—
102C22×C8×C59C2312yes—
103C22⋊C8×C59—24no—
104C22⋊C8×C59—24no—
105C24×C59—22no—
106C24×C59—22no—
107C24×C59—22no—
108C24×C59—22no—
109C24×C59—22no—
110C24×C59—41no—
111C24×C59—41no—
112C24×C59C2×D411yescharacteristic
113C24×C59C2×D411yescharacteristic
114C24×C59C2×D411yescharacteristic
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—22no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—44no—
137C22×C4×C59—44no—
138C22×C4×C59—44no—
139C22×C4×C59—44no—
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D412yes—
144C22×C4×C59#4112yes—
145C22×C4×C59#4112yes—
146C22×C4×C59#4112yes—
147C22×C4×C59#4112yes—
148C22×C4×C59#4112yes—
149C22×C4×C59C2411yescharacteristic
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153C22×Dic59—21no—
154C22×Dic59—22no—
155C22×Dic59—22no—
156C22×Dic59—42no—
157C22×Dic59—42no—
158C22×Dic59—42no—
159C22×Dic59—42no—
160C22×Dic59—42no—
161C22×Dic59—42no—
162M16×C59—28no—
163M16×C59—44no—
164C2×C8×C59—22no—
165C2×C8×C59—22no—
166C2×C8×C59—22no—
167C2×C8×C59—22no—
168C2×C8×C59—22no—
169C2×C8×C59C2×D414yes—
170C22⋊Dic59—42no—
171C22⋊Dic59—42no—
172C22⋊Dic59—44no—
173C22⋊Dic59—42no—
174C22⋊Dic59—42no—
175C22⋊Dic59—42no—
176C22⋊Dic59—44no—
177C22⋊Dic59—42no—
178C4⋊Dic59—21no—
179C4⋊Dic59—22no—
180C4⋊Dic59—21no—
181C4⋊Dic59—22no—
182C4⋊Dic59—22no—
183C4⋊Dic59—22no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—42no—
191C4⋊Dic59—42no—
192C4×Dic59—42no—
193C4×Dic59—42no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—28no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—44no—
212C23×C59—44no—
213C23×C59—44no—
214C23×C59—44no—
215C23×C59—44no—
216C23×C59—44no—
217C23×C59#12112yes—
218C23×C59#12312yes—
219C23×C59#12312yes—
220C23×C59AGL(1,ℤ/8)12yes—
221C23×C59#13712yes—
222C23×C59C22×D411yescharacteristic
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—44no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—44no—
239C2×C4×C59—44no—
240C2×C4×C59#12112yes—
241C2×C4×C59#12314yes—
242C2×C4×C59#12312yes—
243C2×C4×C59#12312yes—
244C2×C4×C59#12414yes—
245C2×C4×C59#12614yes—
246C2×C4×C59#12614yes—
247C2×C4×C59C22×D411yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C2×C4×C59C2×#4111yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59C2×#4111yescharacteristic
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—42no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—42no—
259C2×Dic59—42no—
260C2×Dic59—42no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—42no—
270C2×Dic59—42no—
271C2×Dic59—84no—
272C2×Dic59—84no—
273C8×C59—24no—
274C8×C59—42no—
275#23642—591nomaximal, Sylow
276C22×C59—22no—
277C22×C59—22no—
278C22×C59—22no—
279C22×C59—22no—
280C22×C59—22no—
281C22×C59—22no—
282C22×C59—22no—
283C22×C59—22no—
284C22×C59—22no—
285C22×C59—22no—
286C22×C59—22no—
287C22×C59—22no—
288C22×C59—416no—
289C22×C59—41no—
290C22×C59—41no—
291C22×C59—41no—
292C22×C59—41no—
293C22×C59—41no—
294C22×C59—41no—
295C22×C59—41no—
296C22×C59—41no—
297C22×C59#39714yes—
298C22×C59C2×#12111yescharacteristic
299C22×C59C2×#12311yescharacteristic
300C22×C59C2×#12311yescharacteristic
301C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
302C22×C59C2×#13711yescharacteristic
303C22×C59#57511yescharacteristic
304C22×C59#57511yescharacteristic
305C4×C59—41no—
306C4×C59—41no—
307C4×C59#39714yes—
308C4×C59C2×#12311yescharacteristic
309C4×C59C2×#12411yescharacteristic
310C4×C59C2×#12611yescharacteristic
311C4×C59C2×#12611yescharacteristic
312Dic59—81no—
313Dic59—81no—
314Dic59—82no—
315Dic59—81no—
316Dic59—82no—
317Dic59—81no—
318C2×#428—591no—
319C2×#407D5911yescharacteristic
320#2280—591no—
321C2×#380—591no—
322#2077—592no—
323#2077—592no—
324#2076—592no—
325#2075—592no—
326#1736—591no—
327#1736—591no—
328#1736—591no—
329C2×C59—24no—
330C2×C59—42no—
331C2×C59—42no—
332C2×C59C2×#39711yescharacteristic
333C2×C59#305511yescharacteristic
334C2×C59#305611yescharacteristic
335C2×C59#309211yescharacteristic
336C2×C59#309311yescharacteristic
337C2×C59#309411yescharacteristic
338C2×C59#309511yescharacteristic
339C24×C4C2×D5911yescharacteristic
340C22×M16C2×D5911yescharacteristic
341C22×M16C2×D5911yescharacteristic
342C2×#117—591no—
343C2×#117—591no—
344C22×C4⋊C4—591no—
345C22×C4⋊C4—591no—
346C22×C22⋊C4—1181no—
347#428—1184no—
348C2×#107—592no—
349C2×#106—592no—
350#407—24no—
351C2×C22⋊C8C2×D5912yes—
352C2×C22⋊C8C2×D5912yes—
353#381—1181no—
354#380—591no—
355#380—592no—
356#380—591no—
357#380—592no—
358#380—591no—
359#380—591no—
360#380—1181no—
361#380—1181no—
362#379—1181no—
363C4×C22⋊C4—1181no—
364C4×C22⋊C4—1181no—
365C2×#95—1181no—
366#340—592no—
367#340—592no—
368#340—592no—
369#340—592no—
370#340—592no—
371#340—592no—
372C59#2364211yescharacteristic, complemented, Sylow
373C25D23611yescharacteristic
374C23×C4—22no—
375C23×C4—22no—
376C23×C4—22no—
377C23×C4—22no—
378C23×C4—22no—
379C23×C4—22no—
380C23×C4—1181no—
381C23×C4—1181no—
382C23×C4D23611yescharacteristic
383C23×C4D23611yescharacteristic
384C23×C4D23611yescharacteristic
385C23×C4C22×D5911yescharacteristic
386C23×C4C22×D5911yescharacteristic
387C23×C4C22×D5911yescharacteristic
388C2×M16—22no—
389C2×M16—22no—
390C2×M16—22no—
391C2×M16—22no—
392C2×M16—22no—
393C2×M16C22⋊D5914yes—
394C22×C8C22×D5912yes—
395C22×C8C22×D5912yes—
396#117—1182no—
397#117—1182no—
398#117—1182no—
399#117—1182no—
400C2×C4⋊C4—592no—
401C2×C4⋊C4—592no—
402C2×C4⋊C4—591no—
403C2×C4⋊C4—591no—
404C2×C4⋊C4—591no—
405C2×C4⋊C4—591no—
406C2×C4⋊C4—1182no—
407C2×C4⋊C4—1182no—
408C2×C4⋊C4—1182no—
409C2×C4⋊C4—1182no—
410C2×C4⋊C4—1181no—
411C2×C4⋊C4—1181no—
412C2×C22⋊C4—1182no—
413C2×C22⋊C4—1182no—
414C2×C22⋊C4—1181no—
415C2×C22⋊C4—1181no—
416C2×C22⋊C4—1181no—
417C2×C22⋊C4—1181no—
418C2×C22⋊C4—1181no—
419C2×C22⋊C4—1181no—
420C2×C22⋊C4—2364no—
421C2×C42—1181no—
422C2×C42—1181no—
423#107—1184no—
424#106—1184no—
425C22⋊C8—24no—
426C22⋊C8—24no—
427#95—1182no—
428#95—1182no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—41no—
439C24—41no—
440C24C2×D23611yescharacteristic
441C24C2×D23611yescharacteristic
442C24C2×D23611yescharacteristic
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—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—1181no—
469C22×C4—1181no—
470C22×C4—1181no—
471C22×C4—1181no—
472C22×C4—1182no—
473C22×C4—1182no—
474C22×C4—2362no—
475C22×C4—2362no—
476C22×C4—2362no—
477C22×C4—2362no—
478C22×C4—2362no—
479C22×C4—2362no—
480C22×C4C2×D23611yescharacteristic
481C22×C4C2×D23611yescharacteristic
482C22×C4C2×D23611yescharacteristic
483C22×C4#1174568512yes—
484C22×C4#1174568712yes—
485C22×C4#1174568712yes—
486C22×C4#1174568712yes—
487C22×C4#1174568712yes—
488C22×C4C2×C22⋊D5912yes—
489C22×C4C23×D5911yescharacteristic
490M16—28no—
491M16—44no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8C2×C22⋊D5914yes—
498C4⋊C4—1181no—
499C4⋊C4—1182no—
500C4⋊C4—1181no—
501C4⋊C4—1182no—
502C4⋊C4—1182no—
503C4⋊C4—1181no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C4⋊C4—1181no—
507C4⋊C4—1182no—
508C4⋊C4—1181no—
509C4⋊C4—1181no—
510C4⋊C4—2362no—
511C4⋊C4—2362no—
512C22⋊C4—2364no—
513C22⋊C4—2362no—
514C22⋊C4—2362no—
515C22⋊C4—2362no—
516C22⋊C4—2362no—
517C22⋊C4—2364no—
518C22⋊C4—2362no—
519C22⋊C4—2362no—
520C42—2362no—
521C42—2362no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—28no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23#1427370112yes—
546C23#1427371112yes—
547C23#1427371112yes—
548C23#1427375512yes—
549C23#1427376012yes—
550C23C22×D23611yescharacteristic
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—4724no—
569C2×C4—4724no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2362no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2362no—
576C2×C4—2362no—
577C2×C4—2362no—
578C2×C4—2361no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4—2361no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2362no—
587C2×C4—2362no—
588C2×C4#1427370112yes—
589C2×C4#1427370814yes—
590C2×C4#1427371112yes—
591C2×C4#1427371112yes—
592C2×C4#1427371414yes—
593C2×C4#1427372014yes—
594C2×C4#1427372014yes—
595C2×C4C22×C22⋊D5911yescharacteristic
596C2×C4C2×#1174568711yescharacteristic
597C2×C4C2×#1174568711yescharacteristic
598C2×C4C2×#1174568711yescharacteristic
599C2×C4C2×#1174568711yescharacteristic
600C2×C4C2×#1174568511yescharacteristic
601C8—24no—
602C8—42no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—416no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22#1683732714yes—
625C22C2×#1427370111yescharacteristic
626C22C2×#1427371111yescharacteristic
627C22C2×#1427371111yescharacteristic
628C22C2×#1427375511yescharacteristic
629C22C2×#1427376011yescharacteristic
630C22#1683820211yescharacteristic
631C22#1683820211yescharacteristic
632C4—41no—
633C4—41no—
634C4—4722no—
635C4—4721no—
636C4—4721no—
637C4—4721no—
638C4—4721no—
639C4—4722no—
640C4#1683732714yes—
641C4C2×#1427370811yescharacteristic
642C4C2×#1427371411yescharacteristic
643C4C2×#1427372011yescharacteristic
644C4C2×#1427372011yescharacteristic
645C2—24no—
646C2—42no—
647C2—42no—
648C2C2×#1683732711yescharacteristic
649C2#2143371511yescharacteristic
650C2#2143372611yescharacteristic
651C2#2143402111yescharacteristic
652C2#2143402911yescharacteristic
653C2#2143403711yescharacteristic
654C2#2143404511yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#407C2
socleC23×C59C22×D4
prime radicalC25×C59D4
hypercenterC2×#407D59
hypocenterC59#23642
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
1. C2×#16837444C2semidirect
4. #21425529C2semidirect
5. #21425529C2semidirect
6. #21425524C2semidirect
7. #21425519C2semidirect
8. #21423816C2semidirect
9. #21423816C2semidirect
10. #21423816C2semidirect
18. C2×#14273661C22semidirect
19. C2×#14273659C22semidirect
33. #16837165C22semidirect
34. #16837165C22semidirect
35. #16837165C22semidirect
36. #16837165C22semidirect
37. #16837165C22semidirect
38. #16837165C22semidirect
372. C59#23642semidirect

Maximal subgroups

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

subgroupindex
C2×#168374442
#214264792
C2×#168372622
#214255292
#214255242
#214255192
#214238162
C2×C59×#4072
#2364259