Finite Group Databasebeta

← #28610148

Subgroups — #28610148

Order 15104.

958 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837716C211yesmaximal, characteristic
2#21427110C211yesmaximal, characteristic
3#21426963C211yesmaximal, characteristic, complemented
4#21424834C211yesmaximal, characteristic, complemented
5#21424793C211yesmaximal, characteristic, complemented
6#21424724C211yesmaximal, characteristic, complemented
7#21424689C211yesmaximal, characteristic, complemented
8#21424689C211yesmaximal, characteristic, complemented
9#21424524C211yesmaximal, characteristic
10#21424360C211yesmaximal, characteristic, complemented
11#21424291C211yesmaximal, characteristic, complemented
12#21423993C211yesmaximal, characteristic, complemented
13#21423922C211yesmaximal, characteristic, complemented
14C2×#16837165C211yesmaximal, characteristic, complemented
15C59×#1752C211yesmaximal, characteristic, complemented
16C2×#14273701C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×(C22⋊D59)⋊C4—21no—
19C2×#14273683—21no—
20C22×C4⋊Dic59C2211yescharacteristic
21C22×C4⋊Dic59C2211yescharacteristic
22#16837716C418yescomplemented
23C2×#14273661C2211yescharacteristic
24C2×#14273661C2211yescharacteristic
25C2×#14273659C2211yescharacteristic
26C2×#14273659C2211yescharacteristic
27C2×#14273649C2211yescharacteristic
28C2×#14273649C2211yescharacteristic
29C2×#14273649C2211yescharacteristic
30C2×#14273649C2211yescharacteristic
31#16837304C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837300C2211yescharacteristic
35#16837294—21no—
36#16837287—21no—
37#16837287—21no—
38#16837279—21no—
39#16837276—21no—
40#16837263—21no—
41#16837262C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43C4×C4⋊Dic59C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53C23×C8×C59C2211yescharacteristic
54C2×C22⋊C8×C59C2211yescharacteristic
55C2×#97×C59C2211yescharacteristic
56C2×#97×C59C2211yescharacteristic
57C4×C22⋊C4×C59C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×D236C2311yescharacteristic
61C22×C22⋊D59—21no—
62C23×Dic59—21no—
63C23×Dic59—21no—
64C22×C4×D59—21no—
65#14273701—22no—
66#14273701C2×C414yes—
67C4×D236—22no—
68C4×D236—41no—
69(C22⋊D59)⋊C4—44no—
70#14273683—44no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—22no—
73C2×C4⋊Dic59—22no—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59—41no—
76C2×C4⋊Dic59C2×C414yescomplemented
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×C4⋊Dic59C2311yescharacteristic
84C2×Dic59⋊C4—21no—
85C2×Dic59⋊C4—21no—
86C2×Dic59⋊C4—21no—
87C2×C22⋊Dic59—21no—
88C2×(C2×D59)⋊C4—21no—
89C2×(C2×D59)⋊C4—21no—
90C2×(C2×D59)⋊C4—21no—
91C2×(C2×D59)⋊C4—21no—
92C2×C4×Dic59—21no—
93C2×C4×Dic59—21no—
94C2×C4×Dic59—21no—
95#14273661C2×C414yescomplemented
96#14273661C2×C414yescomplemented
97#14273659—22no—
98#14273659—41no—
99#14273649—22no—
100#14273649—41no—
101#14273649C2×C414yescomplemented
102#14273649C2×C414yescomplemented
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C23×C4×C59C2311yescharacteristic
111C22×C8×C59—22no—
112C22×C8×C59—22no—
113C22×C8×C59—41no—
114C22×C8×C59C2311yescharacteristic
115C22×C8×C59C2311yescharacteristic
116C22×C8×C59C2311yescharacteristic
117C22×C8×C59C2311yescharacteristic
118C2×C22⋊C4×C59D411yescharacteristic
119C2×C22⋊C4×C59D411yescharacteristic
120C2×C42×C59C2311yescharacteristic
121C2×C42×C59C2311yescharacteristic
122C22⋊C8×C59C2×C414yescomplemented
123#97×C59—22no—
124#97×C59—41no—
125#95×C59D411yescharacteristic
126#95×C59D411yescharacteristic
127C24×C59C2×D411yescharacteristic
128C23×D59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59C22×C412yes—
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59#4112yes—
140C22×C4×C59C2411yescharacteristic
141C2×C22⋊D59—24no—
142C2×C22⋊D59—42no—
143C2×C22⋊D59—42no—
144C22×Dic59—24no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—42no—
153C22×Dic59—42no—
154C22×Dic59—42no—
155C22×Dic59—42no—
156C22×Dic59—81no—
157C2×D236—21no—
158C2×D236—21no—
159C2×D236—21no—
160C2×D236—21no—
161C2×D236—21no—
162C2×D236C22×C412yes—
163C2×C4×D59—42no—
164C2×C4×D59—42no—
165C2×C4×D59—81no—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59—21no—
171C2×C8×C59—21no—
172C2×C8×C59—21no—
173C2×C8×C59—21no—
174C2×C8×C59—21no—
175C2×C8×C59—44no—
176C2×C8×C59C22×C412yes—
177C2×C8×C59C22×C412yes—
178C2×C8×C59#4112yes—
179C22⋊C4×C59—22no—
180C22⋊C4×C59—22no—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—21no—
185C22⋊Dic59—42no—
186(C2×D59)⋊C4—24no—
187(C2×D59)⋊C4—42no—
188(C2×D59)⋊C4—42no—
189(C2×D59)⋊C4—81no—
190C4⋊Dic59—24no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—42no—
202C4⋊Dic59C22×C412yes—
203C4⋊Dic59C22×C412yes—
204C4⋊Dic59C22×C412yes—
205C4⋊Dic59#4112yes—
206Dic59⋊C4—42no—
207Dic59⋊C4—42no—
208Dic59⋊C4—42no—
209C4×Dic59—42no—
210C4×Dic59—42no—
211C4×Dic59—42no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—21no—
216C23×C59C4×D412yes—
217C23×C59AGL(1,ℤ/8)12yes—
218C23×C59AGL(1,ℤ/8)12yes—
219C23×C59C22×D411yescharacteristic
220C22×D59—22no—
221C22×D59—41no—
222C22×D59—41no—
223C22×D59—41no—
224C22×D59—41no—
225C22×D59—41no—
226C22×D59—41no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
232C2×C4×C59—22no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59C4×D412yes—
246C2×C4×C59C23×C411yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C2×C4×C59H(5,𝔽2)11yescharacteristic
251C2×C4×C592−1+411yescharacteristic
252C22⋊D59—48no—
253C2×Dic59—22no—
254C2×Dic59—22no—
255C2×Dic59—22no—
256C2×Dic59—22no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—48no—
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—
275C2×Dic59—84no—
276D236—22no—
277D236—22no—
278D236—41no—
279D236—41no—
280C4×D59—84no—
281C8×C59—22no—
282C8×C59—22no—
283C8×C59—22no—
284C8×C59—41no—
285#22938—591nomaximal, Sylow
286C22×C59—22no—
287C22×C59—22no—
288C22×C59—22no—
289C22×C59—22no—
290C22×C59—22no—
291C22×C59—22no—
292C22×C59#44214yes—
293C22×C59C2×C4×D411yescharacteristic
294C22×C59#54611yescharacteristic
295C22×C59#54711yescharacteristic
296C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
298C22×C59#57711yescharacteristic
299C22×C59#57811yescharacteristic
300C2×D59—42no—
301C2×D59—42no—
302C2×D59—42no—
303C2×D59—81no—
304C2×D59—81no—
305C2×D59—81no—
306C2×D59—81no—
307C4×C59—22no—
308C4×C59—41no—
309C4×C59—41no—
310C4×C59—41no—
311C4×C59—41no—
312C4×C59C4×#4111yescharacteristic
313C4×C59#52011yescharacteristic
314C4×C59#55111yescharacteristic
315C4×C59#55211yescharacteristic
316Dic59—42no—
317Dic59—42no—
318Dic59—42no—
319Dic59—42no—
320Dic59—81no—
321Dic59—81no—
322Dic59—81no—
323C2×#480—591no—
324#2361—591no—
325#2344—591no—
326#1943—591no—
327#1936—591no—
328#1926—591no—
329#1921—591no—
330#1921—591no—
331#1893—591no—
332#1864—591no—
333#1853—591no—
334#1776—591no—
335#1762—591no—
336#1752D5911yescharacteristic, complemented
337C2×#340—591no—
338C2×C59—24no—
339C2×C59#235911yescharacteristic
340C2×C59C2×#44211yescharacteristic
341C2×C59#294711yescharacteristic
342C2×C59#330811yescharacteristic
343C2×C59#331011yescharacteristic
344C2×C59#334311yescharacteristic
345C2×C59#334511yescharacteristic
346D59—82no—
347C23×C8C2×D5911yescharacteristic
348C2×#121—591no—
349C2×C4×D4—591no—
350C2×C4×D4—1181no—
351C2×#117—1181no—
352C22×C4⋊C4—591no—
353C22×C4⋊C4—591no—
354#480—598no—
355C2×#107—591no—
356C2×#107—591no—
357C2×#106—591no—
358C2×#106—591no—
359C2×#102—591no—
360C2×#102—591no—
361C2×#102—591no—
362C2×#102—591no—
363C2×C22⋊C8C2×D5911yescharacteristic
364C2×#97C2×D5911yescharacteristic
365C2×#97C2×D5911yescharacteristic
366#390—591no—
367#389—591no—
368#389—591no—
369#389—591no—
370#388—1181no—
371#387—1181no—
372#387—1181no—
373#386—1181no—
374#385—1181no—
375#381—1181no—
376#380—591no—
377C4×C4⋊C4—591no—
378C4×C4⋊C4—591no—
379C4×C4⋊C4—591no—
380C4×C22⋊C4C2×D5911yescharacteristic
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#340—591no—
386#340—591no—
387#340—591no—
388#340—591no—
389#336C2×D5911yescharacteristic
390#336C2×D5911yescharacteristic
391C59#2293811yescharacteristic, complemented, Sylow
392C22×D4—591no—
393C22×D4—1181no—
394C23×C4—1181no—
395C23×C4—1181no—
396C23×C4—1181no—
397C23×C4C22×D5911yescharacteristic
398C22×C8—22no—
399C22×C8—22no—
400C22×C8—41no—
401C22×C8C22×D5911yescharacteristic
402C22×C8C22×D5911yescharacteristic
403C22×C8C22×D5911yescharacteristic
404C22×C8C22×D5911yescharacteristic
405#121—594no—
406#121—1182no—
407C4×D4—1182no—
408C4×D4—2361no—
409C4×D4—2364no—
410#117—2364no—
411C2×C4⋊C4—594no—
412C2×C4⋊C4—591no—
413C2×C4⋊C4—591no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—591no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—591no—
419C2×C4⋊C4—1182no—
420C2×C4⋊C4—1181no—
421C2×C4⋊C4—1182no—
422C2×C4⋊C4—1181no—
423C2×C4⋊C4—1181no—
424C2×C4⋊C4—1181no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—2361no—
427C2×C22⋊C4—1181no—
428C2×C22⋊C4—1181no—
429C2×C22⋊C4—1181no—
430C2×C22⋊C4—1181no—
431C2×C22⋊C4—1181no—
432C2×C22⋊C4D23611yescharacteristic
433C2×C22⋊C4D23611yescharacteristic
434C2×C42—1181no—
435C2×C42—1181no—
436C2×C42—1181no—
437C2×C42C22×D5911yescharacteristic
438C2×C42C22×D5911yescharacteristic
439#107—594no—
440#107—594no—
441#106—1182no—
442#106—2361no—
443#102—594no—
444#102—594no—
445#102—1182no—
446#102—2361no—
447C22⋊C8C4×D5914yescomplemented
448#97—22no—
449#97—41no—
450#95—1181no—
451#95—1181no—
452#95—1181no—
453#95—1181no—
454#95—1181no—
455#95—1181no—
456#95—1181no—
457#95D23611yescharacteristic
458#95D23611yescharacteristic
459C24—1181no—
460C24C2×D23611yescharacteristic
461C2×D4—592no—
462C2×D4—1184no—
463C2×D4—1181no—
464C2×D4—1181no—
465C2×D4—1181no—
466C2×D4—1181no—
467C2×D4—1181no—
468C2×D4—2362no—
469C2×D4—2362no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—1184no—
475C22×C4—1181no—
476C22×C4—1181no—
477C22×C4—1181no—
478C22×C4—1181no—
479C22×C4—1181no—
480C22×C4—1181no—
481C22×C4—1181no—
482C22×C4—4721no—
483C22×C4—4721no—
484C22×C4—2362no—
485C22×C4—2362no—
486C22×C4—2362no—
487C22×C4—2362no—
488C22×C4—2362no—
489C22×C4—2362no—
490C22×C4C2×C4×D5912yes—
491C22×C4C2×D23611yescharacteristic
492C22×C4C2×D23611yescharacteristic
493C22×C4C2×D23611yescharacteristic
494C22×C4C2×D23611yescharacteristic
495C22×C4C2×D23611yescharacteristic
496C22×C4#1174568512yes—
497C22×C4C23×D5911yescharacteristic
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—44no—
508C2×C8C2×C4×D5912yes—
509C2×C8C2×C4×D5912yes—
510C2×C8#1174568712yes—
511C4⋊C4—592no—
512C4⋊C4—592no—
513C4⋊C4—592no—
514C4⋊C4—592no—
515C4⋊C4—1184no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4—1181no—
519C4⋊C4—1181no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—1181no—
524C4⋊C4—1181no—
525C4⋊C4—1181no—
526C4⋊C4—2362no—
527C4⋊C4—2362no—
528C4⋊C4—2362no—
529C4⋊C4—2362no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—1184no—
533C22⋊C4—4721no—
534C22⋊C4—2362no—
535C22⋊C4—2362no—
536C22⋊C4—2362no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—2362no—
542C42—2362no—
543C42—2362no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—1182no—
549C23—2361no—
550C23—2361no—
551C23—2361no—
552C23—2361no—
553C23—2361no—
554C23—2361no—
555C23C4×D23612yes—
556C23#1427375512yes—
557C23#1427375512yes—
558C23C22×D23611yescharacteristic
559D4—1182no—
560D4—1182no—
561D4—2368no—
562D4—2361no—
563D4—2361no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—1182no—
583C2×C4—1182no—
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—2361no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4—2361no—
605C2×C4—2368no—
606C2×C4C4×D23612yes—
607C2×C4C22×C4×D5911yescharacteristic
608C2×C4#41×D5911yescharacteristic
609C2×C4C2×#1174568711yescharacteristic
610C2×C4C2×#1174568511yescharacteristic
611C2×C4#1427377211yescharacteristic
612C2×C4#1427377411yescharacteristic
613C8—22no—
614C8—22no—
615C8—22no—
616C8—41no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—4721no—
624C22—4721no—
625C22—4721no—
626C22—4721no—
627C22—2362no—
628C22—2362no—
629C22—2362no—
630C22#1683751514yes—
631C22C2×C4×D23611yescharacteristic
632C22#1683804711yescharacteristic
633C22#1683805611yescharacteristic
634C22C2×#1427375511yescharacteristic
635C22C2×#1427375511yescharacteristic
636C22#1683821411yescharacteristic
637C22#1683821911yescharacteristic
638C4—22no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—4721no—
644C4—4721no—
645C4—4721no—
646C4—2362no—
647C4—2362no—
648C4—2362no—
649C4—2362no—
650C4C4×#1174568711yescharacteristic
651C4#1683788111yescharacteristic
652C4#1683807811yescharacteristic
653C4#1683808511yescharacteristic
654C2—24no—
655C2—4722no—
656C2#2142709011yescharacteristic
657C2C2×#1683751511yescharacteristic
658C2#2143294111yescharacteristic
659C2#2143608911yescharacteristic
660C2#2143611911yescharacteristic
661C2#2143653811yescharacteristic
662C2#2143656811yescharacteristic
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1752C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
cosocleC22×C4×C59C24
hypercenter#1752D59
hypocenterC59#22938
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1752

Complements

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

NH ≅ G/Ntype
3. #21426963C2semidirect
4. #21424834C2semidirect
5. #21424793C2semidirect
6. #21424724C2semidirect
7. #21424689C2semidirect
8. #21424689C2semidirect
10. #21424360C2semidirect
11. #21424291C2semidirect
12. #21423993C2semidirect
13. #21423922C2semidirect
14. C2×#16837165C2semidirect
15. C59×#1752C2semidirect
22. #16837716C4semidirect
76. C2×C4⋊Dic59C2×C4semidirect
95. #14273661C2×C4semidirect
96. #14273661C2×C4semidirect
101. #14273649C2×C4semidirect
102. #14273649C2×C4semidirect
122. C22⋊C8×C59C2×C4semidirect
336. #1752D59semidirect
391. C59#22938semidirect
447. C22⋊C8C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377162
#214271102
#214269632
#214248342
#214247932
#214247242
#214246892
#214245242
#214243602
#214242912
#214239932
#214239222
C2×#168371652
C59×#17522
#2293859