Finite Group Databasebeta

← #28630041

Subgroups — #28630041

Order 15104.

794 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837702C211yesmaximal, characteristic, complemented
2C2×#16837463C211yesmaximal, characteristic, complemented
3#21429926C211yesmaximal, characteristic, complemented
4#21425532C211yesmaximal, characteristic, complemented
5#21425527C211yesmaximal, characteristic, complemented
6#21425275C211yesmaximal, characteristic
7#21425275C211yesmaximal, characteristic
8#21425053C211yesmaximal, characteristic, complemented
9#21424854C211yesmaximal, characteristic, complemented
10#21424831C211yesmaximal, characteristic, complemented
11#21424826C211yesmaximal, characteristic, complemented
12#21424796C211yesmaximal, characteristic, complemented
13#21424782C211yesmaximal, characteristic, complemented
14#21424096C211yesmaximal, characteristic, complemented
15C59×#2324C211yesmaximal, characteristic
16C2×#14273719C2211yescharacteristic
17#16837702—24no—
18#16837463—24no—
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273648C2211yescharacteristic
29C2×C22⋊(C59⋊C8)C2211yescharacteristic
30C2×C22⋊(C59⋊C8)C2211yescharacteristic
31C2×#14273638C2211yescharacteristic
32C2×C4×C59⋊C8C2211yescharacteristic
33#16837335—21no—
34#16837335—21no—
35#16837325—21no—
36#16837325—21no—
37#16837300C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837272C2211yescharacteristic
40#16837262C2211yescharacteristic
41#16837262C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50C2×#121×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C2×#117×C59C2211yescharacteristic
53C59×#390—21no—
54C59×#390C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C59×#388—21no—
57C59×#386—21no—
58C59×#385—21no—
59C59×#380C2211yescharacteristic
60C4×C4⋊C4×C59C2211yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273719—22no—
66#14273719—22no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59—21no—
79#14273660—22no—
80#14273660—22no—
81#14273660—22no—
82#14273658—22no—
83C4⋊(C59⋊C8)—22no—
84C4⋊(C59⋊C8)D414yescomplemented
85#14273648—22no—
86#14273648—22no—
87#14273648—22no—
88#14273648—22no—
89C22⋊(C59⋊C8)—22no—
90C22⋊(C59⋊C8)—22no—
91#14273638—22no—
92C4×C59⋊C8—22no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C22×D4×C59—21no—
98C22×D4×C59C2311yescharacteristic
99C23×C4×C59—21no—
100C23×C4×C59C2311yescharacteristic
101#121×C59—22no—
102#121×C59—22no—
103C4×D4×C59—22no—
104C4×D4×C59—22no—
105#117×C59—22no—
106#117×C59—22no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59D411yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59—21no—
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59D411yescharacteristic
119C2×C22⋊C4×C59D411yescharacteristic
120C2×C42×C59—21no—
121C2×C42×C59C2311yescharacteristic
122C2×C42×C59C2311yescharacteristic
123#95×C59—21no—
124#95×C59—21no—
125#95×C59—21no—
126C24×C59—21no—
127C24×C59C2×D411yescharacteristic
128C2×D4×C59—24no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59—81no—
135C2×D4×C59—81no—
136C2×D4×C59#4112yes—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—42no—
147C22×C4×C59—81no—
148C22×C4×C59—81no—
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2411yescharacteristic
155C22×Dic59—21no—
156C22×Dic59—21no—
157C22×Dic59—21no—
158C22×Dic59—21no—
159C4⋊C4×C59—22no—
160C4⋊C4×C59—22no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—81no—
167C4⋊C4×C59#4112yes—
168C22⋊C4×C59—24no—
169C22⋊C4×C59—22no—
170C22⋊C4×C59—22no—
171C22⋊C4×C59—42no—
172C22⋊C4×C59—81no—
173C22⋊C4×C59—81no—
174C22⋊C4×C59—81no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—81no—
178C42×C59C2×D412yes—
179C42×C59#4112yes—
180C22⋊Dic59—81no—
181C22⋊Dic59—81no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—81no—
191C4⋊Dic59—81no—
192Dic59⋊C4—42no—
193Dic59⋊C4—42no—
194Dic59⋊C4—81no—
195Dic59⋊C4—81no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8C2×D412yes—
207C2×C59⋊C8C2×D412yes—
208C23×C59—21no—
209C23×C59—22no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—21no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59—41no—
222C23×C59C22×D411yescharacteristic
223D4×C59—22no—
224D4×C59—22no—
225D4×C59—41no—
226D4×C59—41no—
227D4×C59—84no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—22no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—84no—
255C2×C4×C59#12112yes—
256C2×C4×C59#12112yes—
257C2×C4×C59C22×D411yescharacteristic
258C2×C4×C59C2×#4111yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C592−1+411yescharacteristic
262C2×C4×C592−1+411yescharacteristic
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—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C59⋊C8—22no—
276C59⋊C8—22no—
277C59⋊C8—41no—
278C59⋊C8—41no—
279#24713—591nomaximal, Sylow
280C22×C59—41no—
281C22×C59—42no—
282C22×C59—41no—
283C22×C59—42no—
284C22×C59—41no—
285C22×C59—42no—
286C22×C59—41no—
287C22×C59—41no—
288C22×C59—41no—
289C22×C59—41no—
290C22×C59—81no—
291C22×C59—81no—
292C22×C59—81no—
293C22×C59—81no—
294C22×C59C2×#12111yescharacteristic
295C22×C59#52511yescharacteristic
296C22×C59#53611yescharacteristic
297C22×C59#57611yescharacteristic
298C22×C59#57611yescharacteristic
299C22×C59#57711yescharacteristic
300C22×C59#57811yescharacteristic
301C4×C59—22no—
302C4×C59—22no—
303C4×C59—42no—
304C4×C59—41no—
305C4×C59—41no—
306C4×C59—41no—
307C4×C59—81no—
308C4×C59—81no—
309C4×C59#54711yescharacteristic
310C4×C59#54811yescharacteristic
311C4×C59#55211yescharacteristic
312C4×C59#55611yescharacteristic
313Dic59—81no—
314Dic59—81no—
315Dic59—81no—
316Dic59—81no—
317C2×#478—591no—
318C2×#433—591no—
319#2629—591no—
320#2324D5911yescharacteristic
321#2077—591no—
322#2076—591no—
323#2033—591no—
324#2033—591no—
325#1989—591no—
326#1947—591no—
327#1943—591no—
328#1942—591no—
329#1936—591no—
330#1934—591no—
331#1802—591no—
332C2×C59—81no—
333C2×C59—82no—
334C2×C59#241511yescharacteristic
335C2×C59#306111yescharacteristic
336C2×C59#306211yescharacteristic
337C2×C59#312011yescharacteristic
338C2×C59#312211yescharacteristic
339C2×C59#324111yescharacteristic
340C2×C59#324311yescharacteristic
341C2×#125—591no—
342C2×#121C2×D5911yescharacteristic
343C2×C4×D4C2×D5911yescharacteristic
344C2×#117C2×D5911yescharacteristic
345#478—1184no—
346#433—1184no—
347C2×#107—591no—
348C2×#107—591no—
349C2×#107—591no—
350C2×#106—591no—
351C2×C4⋊C8—591no—
352C2×C4⋊C8—591no—
353C2×#102—591no—
354C2×#102—591no—
355C2×#102—591no—
356C2×#102—591no—
357C2×C22⋊C8—591no—
358C2×C22⋊C8—591no—
359C2×#97—591no—
360C2×C4×C8—591no—
361#398—1181no—
362#398—1181no—
363#396—1181no—
364#396—1181no—
365#390—21no—
366#390C2×D5911yescharacteristic
367#389—591no—
368#389—591no—
369#389C2×D5911yescharacteristic
370#388—21no—
371#386—21no—
372#385—21no—
373#384—591no—
374#380—591no—
375#380—591no—
376#380C2×D5911yescharacteristic
377C4×C4⋊C4C2×D5911yescharacteristic
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#340—591no—
386C59#2471311yescharacteristic, complemented, Sylow
387C22×D4—21no—
388C22×D4C22×D5911yescharacteristic
389C23×C4—21no—
390C23×C4C22×D5911yescharacteristic
391C22×C8—591no—
392C22×C8—591no—
393C22×C8—591no—
394C22×C8—591no—
395#125—1182no—
396#125—1182no—
397#121—22no—
398#121—22no—
399C4×D4—22no—
400C4×D4—22no—
401#117—22no—
402#117—22no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—591no—
405C2×C4⋊C4—591no—
406C2×C4⋊C4—591no—
407C2×C4⋊C4—591no—
408C2×C4⋊C4—1181no—
409C2×C4⋊C4—1181no—
410C2×C4⋊C4—1181no—
411C2×C4⋊C4—1181no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4C22⋊D5911yescharacteristic
415C2×C4⋊C4C22⋊D5911yescharacteristic
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C4⋊C4C22×D5911yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1181no—
425C2×C22⋊C4—1181no—
426C2×C22⋊C4C22⋊D5911yescharacteristic
427C2×C22⋊C4C22⋊D5911yescharacteristic
428C2×C42—21no—
429C2×C42C22×D5911yescharacteristic
430C2×C42C22×D5911yescharacteristic
431#107—1182no—
432#107—1182no—
433#107—1182no—
434#106—1182no—
435C4⋊C8—594no—
436C4⋊C8—1182no—
437#102—1182no—
438#102—1182no—
439#102—1182no—
440#102—1182no—
441C22⋊C8—1182no—
442C22⋊C8—1182no—
443#97—1182no—
444C4×C8—1182no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—1181no—
449#95—1181no—
450#95—1181no—
451#95—1181no—
452C24—21no—
453C24C2×C22⋊D5911yescharacteristic
454C2×D4—24no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—81no—
461C2×D4—81no—
462C2×D4#1174568512yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—1181no—
476C22×C4—1181no—
477C22×C4—1181no—
478C22×C4—1181no—
479C22×C4C2×C22⋊D5911yescharacteristic
480C22×C4C2×C22⋊D5911yescharacteristic
481C22×C4C2×C22⋊D5911yescharacteristic
482C22×C4C2×C22⋊D5911yescharacteristic
483C22×C4C2×C22⋊D5911yescharacteristic
484C22×C4C23×D5911yescharacteristic
485C2×C8—592no—
486C2×C8—592no—
487C2×C8—1181no—
488C2×C8—1181no—
489C2×C8—1181no—
490C2×C8—1181no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—81no—
505C4⋊C4—1181no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—1181no—
509C4⋊C4—1181no—
510C4⋊C4—1181no—
511C4⋊C4—1181no—
512C4⋊C4—1181no—
513C4⋊C4—4721no—
514C4⋊C4—4721no—
515C4⋊C4—4721no—
516C4⋊C4—4721no—
517C4⋊C4—2362no—
518C4⋊C4—2362no—
519C4⋊C4#1174568512yes—
520C22⋊C4—24no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—42no—
524C22⋊C4—81no—
525C22⋊C4—81no—
526C22⋊C4—81no—
527C22⋊C4—4721no—
528C22⋊C4—4721no—
529C42—21no—
530C42—21no—
531C42—81no—
532C42D4×D5912yes—
533C42#1174568712yes—
534C23—21no—
535C23—22no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C22×C22⋊D5911yescharacteristic
549D4—22no—
550D4—22no—
551D4—41no—
552D4—41no—
553D4—84no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—84no—
581C2×C4—2361no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4#1427369912yes—
594C2×C4#1427369912yes—
595C2×C4C2×D4×D5911yescharacteristic
596C2×C4C2×#1174568711yescharacteristic
597C2×C4C2×#1174568511yescharacteristic
598C2×C4C2×#1174568511yescharacteristic
599C2×C4#1427377411yescharacteristic
600C2×C4#1427377411yescharacteristic
601C8—1182no—
602C8—1182no—
603C8—2361no—
604C8—2361no—
605C22—41no—
606C22—42no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—42no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—41no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22C2×#1427369911yescharacteristic
620C22#1683791311yescharacteristic
621C22#1683797311yescharacteristic
622C22#1683820511yescharacteristic
623C22#1683820511yescharacteristic
624C22#1683821311yescharacteristic
625C22#1683821711yescharacteristic
626C4—22no—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—81no—
633C4—81no—
634C4—4721no—
635C4—4721no—
636C4—4721no—
637C4—4721no—
638C4#1683805711yescharacteristic
639C4#1683806111yescharacteristic
640C4#1683809211yescharacteristic
641C4#1683812211yescharacteristic
642C2—81no—
643C2—82no—
644C2#2142752511yescharacteristic
645C2#2143376311yescharacteristic
646C2#2143377211yescharacteristic
647C2#2143431011yescharacteristic
648C2#2143433911yescharacteristic
649C2#2143534311yescharacteristic
650C2#2143537311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2324C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2324D59
hypocenterC59#24713
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2324

Complements

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

NH ≅ G/Ntype
1. C2×#16837702C2semidirect
2. C2×#16837463C2semidirect
3. #21429926C2semidirect
4. #21425532C2semidirect
5. #21425527C2semidirect
8. #21425053C2semidirect
9. #21424854C2semidirect
10. #21424831C2semidirect
11. #21424826C2semidirect
12. #21424796C2semidirect
13. #21424782C2semidirect
14. #21424096C2semidirect
84. C4⋊(C59⋊C8)D4semidirect
386. C59#24713semidirect

Maximal subgroups

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

subgroupindex
C2×#168377022
C2×#168374632
#214299262
#214255322
#214255272
#214252752
#214250532
#214248542
#214248312
#214248262
#214247962
#214247822
#214240962
C59×#23242
#2471359