Finite Group Databasebeta

← #28595949

Subgroups — #28595949

Order 15104.

900 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×#16837718C211yesmaximal, characteristic
2#21426791C211yesmaximal, characteristic
3#21426452C211yesmaximal, characteristic, complemented
4#21424726C211yesmaximal, characteristic, complemented
5#21424719C211yesmaximal, characteristic, complemented
6#21424684C211yesmaximal, characteristic, complemented
7#21424677C211yesmaximal, characteristic, complemented
8#21424295C211yesmaximal, characteristic, complemented
9#21424183C211yesmaximal, characteristic, complemented
10#21423992C211yesmaximal, characteristic, complemented
11#21423984C211yesmaximal, characteristic, complemented
12#21423923C211yesmaximal, characteristic, complemented
13#102×Dic59C211yesmaximal, characteristic, complemented
14#21423864C211yesmaximal, characteristic, complemented
15C59×#1875C211yesmaximal, characteristic
16C2×D4×Dic59C2211yescharacteristic
17C2×#14273686C2211yescharacteristic
18C22×C4⋊Dic59C2211yescharacteristic
19#16837718C418yescomplemented
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×C22⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273637C2211yescharacteristic
28C2×C8×Dic59C2211yescharacteristic
29#16837305—21no—
30#16837305C2211yescharacteristic
31#16837301C2211yescharacteristic
32#16837295—21no—
33#16837282—21no—
34#16837277—21no—
35#16837262C2211yescharacteristic
36C4⋊C4×Dic59C2211yescharacteristic
37C4⋊C4×Dic59C2211yescharacteristic
38C4⋊C4×Dic59C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C4⋊C4×Dic59C2211yescharacteristic
41C4×C4⋊Dic59—21no—
42C4×C4⋊Dic59—21no—
43C22⋊C4×Dic59—21no—
44C22⋊C4×Dic59—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×#121×C59C2211yescharacteristic
54C2×#117×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C2×#102×C59C2211yescharacteristic
57C2×C22⋊C8×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C23×Dic59—21no—
61C23×Dic59—21no—
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64D4×Dic59—22no—
65D4×Dic59—22no—
66#14273686—22no—
67#14273686—22no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—22no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59C2×C414yescomplemented
72C2×C4⋊Dic59D411yescharacteristic
73C2×C4⋊Dic59D411yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C22⋊Dic59—21no—
82C2×C22⋊Dic59D411yescharacteristic
83C2×C22⋊Dic59D411yescharacteristic
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59C2311yescharacteristic
90C2×C4×Dic59C2311yescharacteristic
91#14273660—22no—
92#14273658—22no—
93#14273658C2×C414yescomplemented
94#14273658C2×C414yescomplemented
95#14273648C2×C414yes—
96#14273648C2×C414yes—
97C22⋊(C59⋊C8)C2×C414yescomplemented
98#14273637—22no—
99C8×Dic59—22no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C22×D4×C59—21no—
108C22×D4×C59C2311yescharacteristic
109C23×C4×C59C2311yescharacteristic
110C22×C8×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112#121×C59—22no—
113#121×C59C2×C414yes—
114#117×C59—22no—
115#117×C59—22no—
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59—21no—
121C2×C42×C59—21no—
122#102×C59—22no—
123#102×C59—22no—
124C22⋊C8×C59—22no—
125C24×C59—21no—
126C24×C59C2×D411yescharacteristic
127C2×D4×C59—24no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—81no—
134C2×D4×C59—81no—
135C2×D4×C59C22×C412yes—
136C22×C4×C59—21no—
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×C59C22×C412yes—
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59#4112yes—
146C22×C4×C59C2411yescharacteristic
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—24no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—42no—
153C22×Dic59—81no—
154C22×Dic59—81no—
155C22×Dic59—81no—
156C22×Dic59—81no—
157C22×Dic59C2×D411yescharacteristic
158C22×Dic59C2×D411yescharacteristic
159C22×Dic59C2×D411yescharacteristic
160C22×Dic59C2×D411yescharacteristic
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59C22×C412yes—
171C4⋊C4×C59#4112yes—
172C4⋊C4×C59#4112yes—
173C22⋊C4×C59—24no—
174C22⋊C4×C59—24no—
175C42×C59—24no—
176C22⋊Dic59—22no—
177C22⋊Dic59—22no—
178C22⋊Dic59—42no—
179C22⋊Dic59—81no—
180C22⋊Dic59—81no—
181C22⋊Dic59—81no—
182C4⋊Dic59—24no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—22no—
186C4⋊Dic59—22no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—81no—
192C4⋊Dic59C22×C412yes—
193C4⋊Dic59C22×C412yes—
194C4×Dic59—21no—
195C4×Dic59—21no—
196C4×Dic59—81no—
197C4×Dic59—81no—
198C4×Dic59—81no—
199C4×Dic59—81no—
200C4×Dic59—81no—
201C4×Dic59#4112yes—
202C4×Dic59#4112yes—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8C22×C412yes—
208C2×C59⋊C8C22×C412yes—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—22no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59—41no—
222C23×C59C4×D412yes—
223C23×C59C22×D411yescharacteristic
224D4×C59—22no—
225D4×C59—22no—
226D4×C59—41no—
227D4×C59—41no—
228D4×C59—84no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—22no—
232C2×C4×C59—22no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—22no—
236C2×C4×C59—22no—
237C2×C4×C59—22no—
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×C59C4×D412yes—
245C2×C4×C59C23×C411yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C2×C4×C59C2×#4111yescharacteristic
251C2×Dic59—21no—
252C2×Dic59—21no—
253C2×Dic59—21no—
254C2×Dic59—21no—
255C2×Dic59—22no—
256C2×Dic59—22no—
257C2×Dic59—21no—
258C2×Dic59—21no—
259C2×Dic59—21no—
260C2×Dic59—21no—
261C2×Dic59—21no—
262C2×Dic59—21no—
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—84no—
274C2×Dic59—84no—
275C2×Dic59#12312yes—
276C2×Dic59#12312yes—
277C8×C59—41no—
278C8×C59—41no—
279C59⋊C8—22no—
280C59⋊C8—22no—
281#21584—591nomaximal, Sylow
282C22×C59—22no—
283C22×C59—22no—
284C22×C59—22no—
285C22×C59—41no—
286C22×C59—41no—
287C22×C59—41no—
288C22×C59—41no—
289C22×C59—42no—
290C22×C59—42no—
291C22×C59—42no—
292C22×C59—81no—
293C22×C59—81no—
294C22×C59—81no—
295C22×C59—81no—
296C22×C59C2×C4×D411yescharacteristic
297C22×C59C2×#12311yescharacteristic
298C22×C59#52511yescharacteristic
299C22×C59#57611yescharacteristic
300C22×C59#57711yescharacteristic
301C22×C59#57711yescharacteristic
302C22×C59#57711yescharacteristic
303C4×C59—22no—
304C4×C59—42no—
305C4×C59—42no—
306C4×C59—42no—
307C4×C59C4×#4111yescharacteristic
308C4×C59C4×#4111yescharacteristic
309C4×C59#52911yescharacteristic
310C4×C59#52911yescharacteristic
311Dic59—22no—
312Dic59—22no—
313Dic59—41no—
314Dic59—41no—
315Dic59—42no—
316Dic59—42no—
317Dic59—81no—
318Dic59—81no—
319C2×#481—591no—
320#2324—591no—
321#2274—591no—
322#1926—591no—
323#1925—591no—
324#1920—591no—
325#1919—591no—
326#1875D5911yescharacteristic
327#1854—591no—
328#1823—591no—
329#1776—591no—
330C4×#106—591no—
331#1762—591no—
332C4×#102—591no—
333#1750—591no—
334C2×C59—42no—
335C2×C59—82no—
336C2×C59C4×#12311yescharacteristic
337C2×C59#294611yescharacteristic
338C2×C59#294911yescharacteristic
339C2×C59#309611yescharacteristic
340C2×C59#309911yescharacteristic
341C2×C59#311911yescharacteristic
342C2×C59#312511yescharacteristic
343C2×#121C2×D5911yescharacteristic
344C2×C4×D4—591no—
345C2×#117—591no—
346C2×#117C2×D5911yescharacteristic
347C22×C4⋊C4—591no—
348#481—598no—
349C2×#107—591no—
350C2×#106—591no—
351C2×#106—591no—
352C2×#106—591no—
353C2×#102—591no—
354C2×#102—591no—
355C2×#102C2×D5911yescharacteristic
356C2×#102C2×D5911yescharacteristic
357C2×C22⋊C8—591no—
358C2×C22⋊C8C2×D5911yescharacteristic
359C2×#97—591no—
360C2×C4×C8—591no—
361#390—591no—
362#390—1181no—
363#389—591no—
364#388—1181no—
365#386—1181no—
366#385—1181no—
367#380—591no—
368C4×C4⋊C4—591no—
369C4×C4⋊C4—591no—
370C4×C4⋊C4—591no—
371C4×C4⋊C4—591no—
372C4×C4⋊C4—591no—
373C4×C4⋊C4—1181no—
374C4×C4⋊C4—1181no—
375C4×C22⋊C4—1181no—
376C4×C22⋊C4—1181no—
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340C2×D5911yescharacteristic
384#340C2×D5911yescharacteristic
385#336—591no—
386#336—591no—
387C59#2158411yescharacteristic, complemented, Sylow
388C22×D4—21no—
389C22×D4C22×D5911yescharacteristic
390C23×C4—1181no—
391C23×C4—1181no—
392C23×C4C22×D5911yescharacteristic
393C22×C8—591no—
394C22×C8—591no—
395C22×C8C22×D5911yescharacteristic
396C22×C8C22×D5911yescharacteristic
397#121—22no—
398#121C4×D5914yes—
399C4×D4—1182no—
400C4×D4—1182no—
401#117—22no—
402#117—22no—
403#117—1182no—
404#117—1182no—
405C2×C4⋊C4—594no—
406C2×C4⋊C4—591no—
407C2×C4⋊C4—591no—
408C2×C4⋊C4—591no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—591no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4—1182no—
414C2×C4⋊C4—1181no—
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—591no—
421C2×C22⋊C4—591no—
422C2×C22⋊C4—1181no—
423C2×C22⋊C4—1181no—
424C2×C22⋊C4—1181no—
425C2×C22⋊C4—1181no—
426C2×C42—21no—
427C2×C42—591no—
428C2×C42—591no—
429C2×C42—1181no—
430C2×C42—1181no—
431C2×C42—1181no—
432C2×C42—1181no—
433C2×C42—1181no—
434#107—1182no—
435#106—594no—
436#106—594no—
437#106—1182no—
438#102—22no—
439#102—22no—
440#102—594no—
441#102—594no—
442C22⋊C8—22no—
443C22⋊C8—594no—
444#97—1182no—
445C4×C8—1182no—
446#95—1181no—
447#95—1181no—
448#95—1181no—
449#95—1181no—
450#95—1181no—
451#95—1181no—
452#95—1181no—
453C24—21no—
454C24D4×D5911yescharacteristic
455C2×D4—24no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—81no—
462C2×D4—81no—
463C2×D4C2×C4×D5912yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—591no—
472C22×C4—591no—
473C22×C4—591no—
474C22×C4—591no—
475C22×C4—1184no—
476C22×C4—1181no—
477C22×C4—1181no—
478C22×C4—1181no—
479C22×C4—1181no—
480C22×C4—4721no—
481C22×C4—4721no—
482C22×C4—4721no—
483C22×C4—4721no—
484C22×C4—2362no—
485C22×C4C2×C4×D5912yes—
486C22×C4D4×D5911yescharacteristic
487C22×C4#1174568712yes—
488C22×C4C23×D5911yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—592no—
496C2×C8—592no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—592no—
505C4⋊C4—592no—
506C4⋊C4—1182no—
507C4⋊C4—1182no—
508C4⋊C4—1184no—
509C4⋊C4—1181no—
510C4⋊C4—1181no—
511C4⋊C4—1181no—
512C4⋊C4—1181no—
513C4⋊C4—1181no—
514C4⋊C4—1181no—
515C4⋊C4—4721no—
516C4⋊C4C2×C4×D5912yes—
517C4⋊C4#1174568712yes—
518C4⋊C4#1174568712yes—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—1182no—
522C22⋊C4—1182no—
523C22⋊C4—4721no—
524C22⋊C4—4721no—
525C22⋊C4—4721no—
526C22⋊C4—2362no—
527C42—24no—
528C42—592no—
529C42—592no—
530C42—1181no—
531C42—1181no—
532C42—4721no—
533C42—4721no—
534C42—4721no—
535C42—4721no—
536C42—4721no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23(C22⋊D59)⋊C412yes—
551C23C2×D4×D5911yescharacteristic
552D4—22no—
553D4—22no—
554D4—41no—
555D4—41no—
556D4—84no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—592no—
573C2×C4—592no—
574C2×C4—1181no—
575C2×C4—1181no—
576C2×C4—1181no—
577C2×C4—1181no—
578C2×C4—1181no—
579C2×C4—1181no—
580C2×C4—1181no—
581C2×C4—1181no—
582C2×C4—1181no—
583C2×C4—1181no—
584C2×C4—1182no—
585C2×C4—1182no—
586C2×C4—4724no—
587C2×C4—4724no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4(C22⋊D59)⋊C412yes—
599C2×C4C22×C4×D5911yescharacteristic
600C2×C4#41×D5911yescharacteristic
601C2×C4#41×D5911yescharacteristic
602C2×C4C2×#1174568711yescharacteristic
603C2×C4C2×#1174568711yescharacteristic
604C2×C4C2×#1174568711yescharacteristic
605C8—41no—
606C8—41no—
607C8—1182no—
608C8—1182no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22C2×(C22⋊D59)⋊C411yescharacteristic
624C22#123×D5911yescharacteristic
625C22#1683790811yescharacteristic
626C22#1683820411yescharacteristic
627C22#1683820911yescharacteristic
628C22#1683821211yescharacteristic
629C22#1683821211yescharacteristic
630C4—22no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—1182no—
635C4—1182no—
636C4—4721no—
637C4—4721no—
638C4—2362no—
639C4—2362no—
640C4—2361no—
641C4—2361no—
642C4C4×#1174568711yescharacteristic
643C4C4×#1174568711yescharacteristic
644C4#1683793311yescharacteristic
645C4#1683793311yescharacteristic
646C2—42no—
647C2—82no—
648C2#2142662111yescharacteristic
649C2#2143293111yescharacteristic
650C2#2143295811yescharacteristic
651C2#2143405111yescharacteristic
652C2#2143408811yescharacteristic
653C2#2143428811yescharacteristic
654C2#2143437411yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1875C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
cosocleC22×C4×C59C24
hypercenter#1875D59
hypocenterC59#21584
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

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

NH ≅ G/Ntype
3. #21426452C2semidirect
4. #21424726C2semidirect
5. #21424719C2semidirect
6. #21424684C2semidirect
7. #21424677C2semidirect
8. #21424295C2semidirect
9. #21424183C2semidirect
10. #21423992C2semidirect
11. #21423984C2semidirect
12. #21423923C2semidirect
13. #102×Dic59C2semidirect
14. #21423864C2semidirect
19. #16837718C4semidirect
71. C2×C4⋊Dic59C2×C4semidirect
93. #14273658C2×C4semidirect
94. #14273658C2×C4semidirect
97. C22⋊(C59⋊C8)C2×C4semidirect
387. C59#21584semidirect

Maximal subgroups

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

subgroupindex
C2×#168377182
#214267912
#214264522
#214247262
#214247192
#214246842
#214246772
#214242952
#214241832
#214239922
#214239842
#214239232
#102×Dic592
#214238642
C59×#18752
#2158459