Finite Group Databasebeta

← D59×#2101 #28618587

Subgroups — D59×#2101 #28618587

Order 15104.

910 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#389×D59C211yesmaximal, characteristic
2#382×D59C211yesmaximal, characteristic
3#21425669C211yesmaximal, characteristic, complemented
4#21425668C211yesmaximal, characteristic, complemented
5#21425667C211yesmaximal, characteristic, complemented
6#21425666C211yesmaximal, characteristic, complemented
7#21425665C211yesmaximal, characteristic, complemented
8#21425664C211yesmaximal, characteristic, complemented
9#21425663C211yesmaximal, characteristic, complemented
10#340×D59C211yesmaximal, characteristic
11#340×D59C211yesmaximal, characteristic
12#340×D59C211yesmaximal, characteristic
13#340×D59C211yesmaximal, characteristic
14#336×D59C211yesmaximal, characteristic
15C59×#2101C211yesmaximal, characteristic, complemented
16C22×C8×D59C2211yescharacteristic
17C22×C8×D59C2211yescharacteristic
18C2×C4⋊C4×D59—21no—
19C2×C4⋊C4×D59C2211yescharacteristic
20C2×C4⋊C4×D59C2211yescharacteristic
21C2×C4⋊C4×D59C2211yescharacteristic
22C2×C4⋊C4×D59C2211yescharacteristic
23C2×C42×D59C2211yescharacteristic
24#16837301C2211yescharacteristic
25#16837300C2211yescharacteristic
26#16837299—21no—
27#16837298C2211yescharacteristic
28#16837297—21no—
29#16837268C2211yescharacteristic
30#16837267C2211yescharacteristic
31#16837267C2211yescharacteristic
32#16837266—21no—
33#16837266—21no—
34#95×D59—21no—
35#95×D59—21no—
36#95×D59—21no—
37#16837165C2211yescharacteristic
38#16837165C2211yescharacteristic
39#16837165C2211yescharacteristic
40#16837165C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837157C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C59×#389C2211yescharacteristic
53C59×#382C2211yescharacteristic
54C59×#340C2211yescharacteristic
55C59×#340C2211yescharacteristic
56C59×#340C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C22×C4×D59—21no—
60C22×C4×D59—21no—
61C22×C4×D59—21no—
62C22×C4×D59—21no—
63C22×C4×D59D411yescharacteristic
64C22×C4×D59D411yescharacteristic
65C22×C4×D59C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C2×C8×D59—22no—
69C2×C8×D59—22no—
70C2×C8×D59—22no—
71C2×C8×D59—22no—
72C2×C8×D59—22no—
73C2×C8×D59—22no—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C4⋊C4×D59—22no—
86C4⋊C4×D59—22no—
87C4⋊C4×D59—22no—
88C4⋊C4×D59—22no—
89C4⋊C4×D59—22no—
90C4⋊C4×D59—22no—
91C4⋊C4×D59—22no—
92C4⋊C4×D59—44no—
93C4⋊C4×D59Q814yes—
94C2×C4×Dic59D411yescharacteristic
95C2×C4×Dic59D411yescharacteristic
96C2×C4×Dic59C2311yescharacteristic
97C42×D59—22no—
98C42×D59—22no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C22×C8×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110C2×C4⋊C4×C59—21no—
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C42×C59C2311yescharacteristic
116#95×C59—21no—
117#95×C59—21no—
118#95×C59—21no—
119C23×D59C2×D411yescharacteristic
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2411yescharacteristic
127C22×Dic59—21no—
128C22×Dic59—21no—
129C22×Dic59—21no—
130C22×Dic59—21no—
131C22×Dic59C2×D411yescharacteristic
132C22×Dic59C2×D411yescharacteristic
133C22×Dic59C2×D411yescharacteristic
134C2×C4×D59—24no—
135C2×C4×D59—22no—
136C2×C4×D59—22no—
137C2×C4×D59—22no—
138C2×C4×D59—22no—
139C2×C4×D59—22no—
140C2×C4×D59—22no—
141C2×C4×D59—42no—
142C2×C4×D59—42no—
143C2×C4×D59—42no—
144C2×C4×D59—42no—
145C2×C4×D59—42no—
146C2×C4×D59—42no—
147C2×C4×D59—42no—
148C2×C4×D59—42no—
149C2×C4×D59—42no—
150C2×C4×D59—42no—
151C2×C4×D59—42no—
152C2×C4×D59C2×Q812yes—
153C2×C4×D59#4112yes—
154C2×C4×D59#4112yes—
155C2×C4×D59#4112yes—
156C2×C4×D59#4112yes—
157C2×C4×D59#4112yes—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—42no—
172C4⋊C4×C59C2×Q812yes—
173C42×C59—21no—
174C42×C59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—42no—
183C4⋊Dic59C2×Q812yes—
184Dic59⋊C4—24no—
185Dic59⋊C4—42no—
186Dic59⋊C4—42no—
187Dic59⋊C4—42no—
188Dic59⋊C4—42no—
189Dic59⋊C4—42no—
190C4×Dic59—21no—
191C4×Dic59—21no—
192C4×Dic59—22no—
193C4×Dic59—22no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C8×D59—44no—
201C8×D59—44no—
202C23×C59C22×D411yescharacteristic
203C22×D59C22⋊Q812yes—
204C22×D59#12312yes—
205C22×D59#12412yes—
206C22×D59#13512yes—
207C22×D59#13512yes—
208C22×D59AGL(1,ℤ/8)12yes—
209C22×D59#13712yes—
210C2×C4×C59—22no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59C22×Q811yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×Dic59—22no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59C22⋊Q812yes—
253C2×Dic59#12312yes—
254C2×Dic59#12412yes—
255C4×D59—48no—
256C4×D59—44no—
257C4×D59—44no—
258C4×D59—84no—
259C4×D59—84no—
260C4×D59—84no—
261C4×D59#12514yes—
262C4×D59#12514yes—
263C4×D59#12614yes—
264C4×D59#12614yes—
265C8×C59—41no—
266C8×C59—41no—
267C59⋊C8—41no—
268C59⋊C8—41no—
269C2×#2101—591nomaximal, Sylow
270C22×C59C2×C22⋊Q811yescharacteristic
271C22×C59C2×#12311yescharacteristic
272C22×C59C2×#12411yescharacteristic
273C22×C59C2×#13511yescharacteristic
274C22×C59C2×#13511yescharacteristic
275C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C59C2×#13711yescharacteristic
277C2×D59#40014yes—
278C2×D59#47814yes—
279C2×D59#47914yes—
280C2×D59#48214yes—
281C2×D59#48514yes—
282C2×D59#48814yes—
283C2×D59#48914yes—
284C4×C59—42no—
285C4×C59—41no—
286C4×C59—41no—
287C4×C59—81no—
288C4×C59—81no—
289C4×C59—81no—
290C4×C59C2×#12511yescharacteristic
291C4×C59C2×#12511yescharacteristic
292C4×C59C2×#12611yescharacteristic
293C4×C59C2×#12611yescharacteristic
294Dic59—42no—
295Dic59—41no—
296Dic59—41no—
297Dic59—81no—
298Dic59—81no—
299Dic59—81no—
300Dic59#40014yes—
301C2×#389—591no—
302C2×#382—591no—
303#2101—591no—
304#2101—591no—
305#2101—591no—
306#2101—591no—
307#2101—591no—
308#2101—591no—
309#2101—591no—
310#2101D5911yescharacteristic, direct product factor
311C2×#340—591no—
312C2×#340—591no—
313C2×#340—591no—
314C2×#340—591no—
315C2×#336—591no—
316C2×C59C2×#40011yescharacteristic
317C2×C59C2×#47811yescharacteristic
318C2×C59C2×#47911yescharacteristic
319C2×C59C2×#48211yescharacteristic
320C2×C59C2×#48511yescharacteristic
321C2×C59C2×#48811yescharacteristic
322C2×C59C2×#48911yescharacteristic
323D59#210118yesdirect product factor
324C23×C8—591no—
325C23×C8—591no—
326C22×C4⋊C4—591no—
327C22×C4⋊C4—591no—
328C22×C4⋊C4—591no—
329C22×C4⋊C4—591no—
330C22×C4⋊C4—1181no—
331C22×C42—591no—
332#389—591no—
333#389—591no—
334#389—591no—
335#389—1181no—
336#389—1181no—
337#389C2×D5911yescharacteristic
338#382—591no—
339#382—591no—
340#382—591no—
341#382—1181no—
342#382—1181no—
343#382C2×D5911yescharacteristic
344C2×#95—1181no—
345C2×#95—1181no—
346C2×#95—1181no—
347#340—591no—
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#340C2×D5911yescharacteristic
360#340C2×D5911yescharacteristic
361#340C2×D5911yescharacteristic
362#340C2×D5911yescharacteristic
363#336—591no—
364#336—591no—
365#336—591no—
366#336C2×D5911yescharacteristic
367C59C2×#210111yescharacteristic, complemented, Sylow
368C23×C4—591no—
369C23×C4—591no—
370C23×C4—591no—
371C23×C4—1181no—
372C23×C4—1181no—
373C23×C4—1181no—
374C23×C4—1181no—
375C22×C8—591no—
376C22×C8—591no—
377C22×C8—1182no—
378C22×C8—1182no—
379C22×C8—1182no—
380C22×C8—1182no—
381C22×C8—1182no—
382C22×C8—1182no—
383C22×C8C22×D5911yescharacteristic
384C22×C8C22×D5911yescharacteristic
385C2×C4⋊C4—21no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—594no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—1182no—
392C2×C4⋊C4—1182no—
393C2×C4⋊C4—1182no—
394C2×C4⋊C4—1182no—
395C2×C4⋊C4—1182no—
396C2×C4⋊C4—1182no—
397C2×C4⋊C4—1182no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4—1181no—
405C2×C4⋊C4—2364no—
406C2×C4⋊C4C22×D5911yescharacteristic
407C2×C4⋊C4C22×D5911yescharacteristic
408C2×C4⋊C4C22×D5911yescharacteristic
409C2×C4⋊C4C22×D5911yescharacteristic
410C2×C42—591no—
411C2×C42—591no—
412C2×C42—591no—
413C2×C42—1182no—
414C2×C42—1182no—
415C2×C42C22×D5911yescharacteristic
416#95—21no—
417#95—21no—
418#95—21no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428C24—591no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—591no—
434C22×C4—592no—
435C22×C4—592no—
436C22×C4—592no—
437C22×C4—592no—
438C22×C4—592no—
439C22×C4—592no—
440C22×C4—591no—
441C22×C4—591no—
442C22×C4—1182no—
443C22×C4—1182no—
444C22×C4—1184no—
445C22×C4—1182no—
446C22×C4—1182no—
447C22×C4—1182no—
448C22×C4—1182no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—1181no—
452C22×C4—1181no—
453C22×C4—2362no—
454C22×C4—2362no—
455C22×C4—2362no—
456C22×C4—2362no—
457C22×C4—2362no—
458C22×C4—2362no—
459C22×C4—2362no—
460C22×C4—2362no—
461C22×C4—2362no—
462C22×C4—2362no—
463C22×C4—2362no—
464C22×C4D4×D5911yescharacteristic
465C22×C4D4×D5911yescharacteristic
466C22×C4C23×D5911yescharacteristic
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—2364no—
480C2×C8—2364no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—592no—
490C4⋊C4—1184no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—2362no—
499C4⋊C4—2362no—
500C4⋊C4—2362no—
501C4⋊C4—2362no—
502C4⋊C4—2362no—
503C4⋊C4—2362no—
504C4⋊C4Q8×D5912yes—
505C42—21no—
506C42—21no—
507C42—1182no—
508C42—1182no—
509C42—1181no—
510C42—1181no—
511C23—592no—
512C23—592no—
513C23—592no—
514C23—592no—
515C23—592no—
516C23—592no—
517C23—592no—
518C23C2×D4×D5911yescharacteristic
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—592no—
538C2×C4—592no—
539C2×C4—592no—
540C2×C4—594no—
541C2×C4—594no—
542C2×C4—594no—
543C2×C4—594no—
544C2×C4—1182no—
545C2×C4—1181no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—4724no—
552C2×C4—4724no—
553C2×C4—4724no—
554C2×C4—2364no—
555C2×C4—2364no—
556C2×C4—2368no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4C2×Q8×D5911yescharacteristic
569C2×C4#41×D5911yescharacteristic
570C2×C4#41×D5911yescharacteristic
571C2×C4#41×D5911yescharacteristic
572C2×C4#41×D5911yescharacteristic
573C2×C4#41×D5911yescharacteristic
574C8—41no—
575C8—41no—
576C8—2361no—
577C8—2361no—
578C22—594no—
579C22—594no—
580C22—594no—
581C22—594no—
582C22—594no—
583C22—594no—
584C22—594no—
585C22C22⋊Q8×D5911yescharacteristic
586C22#123×D5911yescharacteristic
587C22#124×D5911yescharacteristic
588C22#135×D5911yescharacteristic
589C22#135×D5911yescharacteristic
590C22AGL(1,ℤ/8)×D5911yescharacteristic
591C22#137×D5911yescharacteristic
592C4—41no—
593C4—41no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4—594no—
599C4—4721no—
600C4—4721no—
601C4—4721no—
602C4—2361no—
603C4—2361no—
604C4—2362no—
605C4#125×D5911yescharacteristic
606C4#125×D5911yescharacteristic
607C4#126×D5911yescharacteristic
608C4#126×D5911yescharacteristic
609C2—598no—
610C2#400×D5911yescharacteristic
611C2#478×D5911yescharacteristic
612C2#479×D5911yescharacteristic
613C2#482×D5911yescharacteristic
614C2#485×D5911yescharacteristic
615C2#488×D5911yescharacteristic
616C2#489×D5911yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2101C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
hypercenter#2101D59
hypocenterC59C2×#2101
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2101

Complements

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

NH ≅ G/Ntype
3. #21425669C2semidirect
4. #21425668C2semidirect
5. #21425667C2semidirect
6. #21425666C2semidirect
7. #21425665C2semidirect
8. #21425664C2semidirect
9. #21425663C2semidirect
15. C59×#2101C2semidirect
310. #2101D59direct (N × H)
323. D59#2101direct (N × H)
367. C59C2×#2101semidirect

Maximal subgroups

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

subgroupindex
#389×D592
#382×D592
#214256692
#214256682
#214256672
#214256662
#214256652
#214256642
#214256632
#340×D592
#336×D592
C59×#21012
C2×#210159