Finite Group Databasebeta

← #28617563

Subgroups — #28617563

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×#16837442C211yesmaximal, characteristic, complemented
2C2×#16837366C211yesmaximal, characteristic
3#21426479C211yesmaximal, characteristic
4#21425533C212yesmaximal, complemented
5#21425532C212yesmaximal, complemented
6#21425527C212yesmaximal, complemented
7#21425522C212yesmaximal, complemented
8#21423815C211yesmaximal, characteristic, complemented
9#21423815C211yesmaximal, characteristic, complemented
10#21423815C211yesmaximal, characteristic, complemented
11C2×C59×#380C211yesmaximal, characteristic
12C22×#11745664C2211yescharacteristic
13C22×#11745664C2211yescharacteristic
14C2×#14273686C2211yescharacteristic
15C2×#14273686C2211yescharacteristic
16#16837442—24no—
17C2×#14273660C2212yescomplemented
18C2×#14273658C2212yescomplemented
19#16837366—24no—
20C2×C22⋊(C59⋊C8)C2212yes—
21C2×C22⋊(C59⋊C8)C2212yes—
22#16837265—21no—
23#16837262C2211yescharacteristic
24#16837262C2211yescharacteristic
25#16837262C2211yescharacteristic
26#16837262C2211yescharacteristic
27#16837258—21no—
28C4×C22⋊Dic59—21no—
29C4×C22⋊Dic59—21no—
30#16837164C2212yescomplemented
31#16837164C2212yescomplemented
32#16837164C2212yescomplemented
33#16837164C2212yescomplemented
34#16837164C2212yescomplemented
35#16837164C2212yescomplemented
36C24×C4×C59C2211yescharacteristic
37C22×C4⋊C4×C59C2211yescharacteristic
38C22×C4⋊C4×C59C2211yescharacteristic
39C22×C22⋊C4×C59—21no—
40C59×#380—21no—
41C59×#380—21no—
42C59×#380C2212yes—
43C59×#380C2212yes—
44C2×#95×C59—21no—
45C2×#11745664—22no—
46C2×#11745664—22no—
47C2×#11745664—22no—
48C2×#11745664—22no—
49C2×#11745664—22no—
50C2×#11745664D414yes—
51C22×C59⋊C8C2312yes—
52C22×C59⋊C8C2312yes—
53#14273686—22no—
54#14273686—22no—
55#14273686—22no—
56#14273686—22no—
57C2×C4⋊Dic59C2311yescharacteristic
58C2×C4⋊Dic59C2311yescharacteristic
59C2×C4⋊Dic59C2311yescharacteristic
60C2×C4⋊Dic59C2311yescharacteristic
61C2×C22⋊Dic59—21no—
62C2×C22⋊Dic59—21no—
63C2×C22⋊Dic59—21no—
64C2×C22⋊Dic59—21no—
65C2×C22⋊Dic59—21no—
66C2×C22⋊Dic59—21no—
67C2×C4×Dic59—21no—
68C2×C4×Dic59—21no—
69#14273660—24no—
70#14273658—24no—
71C22⋊(C59⋊C8)—24no—
72C22⋊(C59⋊C8)—24no—
73C59⋊#95—21no—
74C59⋊#95—21no—
75C59⋊#95—21no—
76C59⋊#95—21no—
77C25×C59D411yescharacteristic
78C23×C4×C59—22no—
79C23×C4×C59—22no—
80C23×C4×C59—22no—
81C23×C4×C59—22no—
82C23×C4×C59—22no—
83C23×C4×C59—22no—
84C23×C4×C59—21no—
85C23×C4×C59—21no—
86C23×C4×C59D411yescharacteristic
87C23×C4×C59D411yescharacteristic
88C23×C4×C59D411yescharacteristic
89C23×C4×C59C2311yescharacteristic
90C23×C4×C59C2311yescharacteristic
91C23×C4×C59C2311yescharacteristic
92C2×C4⋊C4×C59—22no—
93C2×C4⋊C4×C59—22no—
94C2×C4⋊C4×C59—22no—
95C2×C4⋊C4×C59—22no—
96C2×C4⋊C4×C59—21no—
97C2×C4⋊C4×C59—21no—
98C2×C4⋊C4×C59C2312yes—
99C2×C4⋊C4×C59C2312yes—
100C2×C22⋊C4×C59—22no—
101C2×C22⋊C4×C59—22no—
102C2×C22⋊C4×C59—44no—
103#95×C59—22no—
104#95×C59—22no—
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—22no—
137C22×C4×C59—22no—
138C22×C4×C59—44no—
139C22×C4×C59—44no—
140C22×C4×C59—44no—
141C22×C4×C59—44no—
142C22×C4×C59—42no—
143C22×C4×C59—42no—
144C22×C4×C59—42no—
145C22×C4×C59—42no—
146C22×C4×C59—42no—
147C22×C4×C59—42no—
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D412yes—
152C22×C4×C59#4112yes—
153C22×C4×C59#4112yes—
154C22×C4×C59#4112yes—
155C22×C4×C59#4112yes—
156C22×C4×C59#4112yes—
157C22×C4×C59C2411yescharacteristic
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—22no—
164C4⋊C4×C59—22no—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—42no—
167C4⋊C4×C59—42no—
168C22⋊C4×C59—44no—
169C22⋊C4×C59—44no—
170C22⋊Dic59—42no—
171C22⋊Dic59—42no—
172C22⋊Dic59—42no—
173C22⋊Dic59—42no—
174C22⋊Dic59—42no—
175C22⋊Dic59—42no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4×Dic59—42no—
185C4×Dic59—42no—
186#11745664—28no—
187#11745664—44no—
188C2×C59⋊C8—22no—
189C2×C59⋊C8—22no—
190C2×C59⋊C8—22no—
191C2×C59⋊C8—22no—
192C2×C59⋊C8—22no—
193C2×C59⋊C8C2×D414yes—
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—42no—
241C2×C4×C59—42no—
242C2×C4×C59—42no—
243C2×C4×C59—42no—
244C2×C4×C59—42no—
245C2×C4×C59—42no—
246C2×C4×C59—84no—
247C2×C4×C59—84no—
248C2×C4×C59#12112yes—
249C2×C4×C59#12312yes—
250C2×C4×C59#12314yes—
251C2×C4×C59#12312yes—
252C2×C4×C59#12414yes—
253C2×C4×C59#12614yes—
254C2×C4×C59#12614yes—
255C2×C4×C59C22×D411yescharacteristic
256C2×C4×C59C2×#4111yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×C4×C59C2×#4111yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
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—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C59⋊C8—24no—
274C59⋊C8—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—82no—
308C4×C59—82no—
309C4×C59#39714yes—
310C4×C59C2×#12311yescharacteristic
311C4×C59C2×#12411yescharacteristic
312C4×C59C2×#12611yescharacteristic
313C4×C59C2×#12611yescharacteristic
314Dic59—81no—
315Dic59—81no—
316Dic59—81no—
317Dic59—81no—
318C2×#428—591no—
319C2×#407—591no—
320#2280—591no—
321C2×#380D5911yescharacteristic
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×M16—591no—
341C22×M16—591no—
342C2×#117—591no—
343C2×#117—591no—
344C22×C4⋊C4C2×D5911yescharacteristic
345C22×C4⋊C4C2×D5911yescharacteristic
346C22×C22⋊C4—21no—
347#428—1184no—
348C2×#107—592no—
349C2×#106—592no—
350#407—1184no—
351C2×C22⋊C8—592no—
352C2×C22⋊C8—592no—
353#381—1181no—
354#380—21no—
355#380—21no—
356#380—591no—
357#380—591no—
358#380—591no—
359#380—591no—
360#380C2×D5912yes—
361#380C2×D5912yes—
362#379—1181no—
363C4×C22⋊C4—1181no—
364C4×C22⋊C4—1181no—
365C2×#95—21no—
366#340—592no—
367#340—592no—
368#340—592no—
369#340—592no—
370#340—592no—
371#340—592no—
372C59#2364211yescharacteristic, complemented, Sylow
373C25C22⋊D5911yescharacteristic
374C23×C4—22no—
375C23×C4—22no—
376C23×C4—22no—
377C23×C4—22no—
378C23×C4—22no—
379C23×C4—22no—
380C23×C4—21no—
381C23×C4—21no—
382C23×C4C22⋊D5911yescharacteristic
383C23×C4C22⋊D5911yescharacteristic
384C23×C4C22⋊D5911yescharacteristic
385C23×C4C22×D5911yescharacteristic
386C23×C4C22×D5911yescharacteristic
387C23×C4C22×D5911yescharacteristic
388C2×M16—594no—
389C2×M16—1182no—
390C2×M16—1182no—
391C2×M16—1182no—
392C2×M16—1182no—
393C2×M16—1182no—
394C22×C8—592no—
395C22×C8—592no—
396#117—1182no—
397#117—1182no—
398#117—1182no—
399#117—1182no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—591no—
407C2×C4⋊C4—591no—
408C2×C4⋊C4—591no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4C22×D5912yes—
411C2×C4⋊C4C22×D5912yes—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—44no—
415C2×C22⋊C4—1181no—
416C2×C22⋊C4—1181no—
417C2×C22⋊C4—1181no—
418C2×C22⋊C4—1181no—
419C2×C22⋊C4—1181no—
420C2×C22⋊C4—1181no—
421C2×C42—1181no—
422C2×C42—1181no—
423#107—1184no—
424#106—1184no—
425C22⋊C8—1184no—
426C22⋊C8—1184no—
427#95—22no—
428#95—22no—
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×C22⋊D5911yescharacteristic
441C24C2×C22⋊D5911yescharacteristic
442C24C2×C22⋊D5911yescharacteristic
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—22no—
465C22×C4—22no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—1181no—
477C22×C4—1181no—
478C22×C4—1181no—
479C22×C4—1181no—
480C22×C4#1174568512yes—
481C22×C4#1174568512yes—
482C22×C4D4×D5912yes—
483C22×C4#1174568712yes—
484C22×C4#1174568712yes—
485C22×C4#1174568712yes—
486C22×C4C2×C22⋊D5911yescharacteristic
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C2×C22⋊D5911yescharacteristic
489C22×C4C23×D5911yescharacteristic
490M16—1188no—
491M16—2364no—
492C2×C8—594no—
493C2×C8—1182no—
494C2×C8—1182no—
495C2×C8—1182no—
496C2×C8—1182no—
497C2×C8—1182no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—1181no—
509C4⋊C4—1181no—
510C4⋊C4—1181no—
511C4⋊C4—1181no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—2362no—
515C22⋊C4—2362no—
516C22⋊C4—2362no—
517C22⋊C4—2362no—
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#1427369912yes—
546C23#1427370812yes—
547C23#1427371012yes—
548C23#1427375312yes—
549C23#1427375812yes—
550C23C22×C22⋊D5911yescharacteristic
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—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—2361no—
577C2×C4—2361no—
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—2361no—
587C2×C4—2361no—
588C2×C4#1427369912yes—
589C2×C4#1427370812yes—
590C2×C4#1427370914yes—
591C2×C4#1427371012yes—
592C2×C4#1427371314yes—
593C2×C4#1427372014yes—
594C2×C4#1427372014yes—
595C2×C4C2×D4×D5911yescharacteristic
596C2×C4C2×#1174568711yescharacteristic
597C2×C4C2×#1174568711yescharacteristic
598C2×C4C2×#1174568711yescharacteristic
599C2×C4C2×#1174568511yescharacteristic
600C2×C4C2×#1174568511yescharacteristic
601C8—1184no—
602C8—2362no—
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#1683733014yes—
625C22C2×#1427369911yescharacteristic
626C22C2×#1427370811yescharacteristic
627C22C2×#1427371011yescharacteristic
628C22C2×#1427375311yescharacteristic
629C22C2×#1427375811yescharacteristic
630C22#1683820011yescharacteristic
631C22#1683820011yescharacteristic
632C4—41no—
633C4—41no—
634C4—82no—
635C4—82no—
636C4—4721no—
637C4—4721no—
638C4—4721no—
639C4—4721no—
640C4#1683733014yes—
641C4C2×#1427370911yescharacteristic
642C4C2×#1427371311yescharacteristic
643C4C2×#1427372011yescharacteristic
644C4C2×#1427372011yescharacteristic
645C2—24no—
646C2—42no—
647C2—42no—
648C2C2×#1683733011yescharacteristic
649C2#2143371111yescharacteristic
650C2#2143372211yescharacteristic
651C2#2143401911yescharacteristic
652C2#2143402711yescharacteristic
653C2#2143403211yescharacteristic
654C2#2143404011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#380C2
socleC23×C59C22×D4
prime radicalC25×C59D4
hypercenterC2×#380D59
hypocenterC59#23642
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C2×#16837442C2semidirect
4. #21425533C2semidirect
5. #21425532C2semidirect
6. #21425527C2semidirect
7. #21425522C2semidirect
8. #21423815C2semidirect
9. #21423815C2semidirect
10. #21423815C2semidirect
17. C2×#14273660C22semidirect
18. C2×#14273658C22semidirect
30. #16837164C22semidirect
31. #16837164C22semidirect
32. #16837164C22semidirect
33. #16837164C22semidirect
34. #16837164C22semidirect
35. #16837164C22semidirect
372. C59#23642semidirect

Maximal subgroups

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

subgroupindex
C2×#168374422
C2×#168373662
#214264792
#214255332
#214255322
#214255272
#214255222
#214238152
C2×C59×#3802
#2364259