Finite Group Databasebeta

← #28610112

Subgroups — #28610112

Order 15104.

930 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837720C211yesmaximal, characteristic
2#21427112C211yesmaximal, characteristic
3#21426973C211yesmaximal, characteristic, complemented
4#21424838C211yesmaximal, characteristic, complemented
5#21424794C211yesmaximal, characteristic, complemented
6#21424714C211yesmaximal, characteristic, complemented
7#21424679C211yesmaximal, characteristic, complemented
8#21424677C211yesmaximal, characteristic, complemented
9#21424524C211yesmaximal, characteristic
10#21424344C211yesmaximal, characteristic, complemented
11#21424284C211yesmaximal, characteristic, complemented
12#21423992C211yesmaximal, characteristic, complemented
13#21423920C211yesmaximal, characteristic, complemented
14#21423872C211yesmaximal, characteristic, complemented
15C2×C59×#340C211yesmaximal, characteristic, complemented
16C2×#14273701C2211yescharacteristic
17C2×#14273689C2211yescharacteristic
18C2×C4×C22⋊D59—21no—
19#16837720C418yescomplemented
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273649C2211yescharacteristic
25C2×#14273649C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×#14273637C2211yescharacteristic
30C2×#14273637C2211yescharacteristic
31#16837303C2211yescharacteristic
32#16837301C2211yescharacteristic
33#16837301C2211yescharacteristic
34#16837298C2211yescharacteristic
35#16837292—21no—
36#16837288—21no—
37#16837288—21no—
38#16837281—21no—
39#16837273—21no—
40#16837262—21no—
41#16837262C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C4⋊C4×Dic59C2211yescharacteristic
44C4⋊C4×Dic59C2211yescharacteristic
45C4×C22⋊Dic59—21no—
46C4×C22⋊Dic59C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C23×C8×C59C2211yescharacteristic
54C22×C4⋊C4×C59C2211yescharacteristic
55C22×C4⋊C4×C59C2211yescharacteristic
56C59×#340C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×D236C2311yescharacteristic
61C22×C22⋊D59—21no—
62C22×C4×D59—21no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273701—22no—
66#14273701C2×C414yes—
67#14273689—22no—
68#14273689—41no—
69C4×C22⋊D59—44no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×C22⋊Dic59—21no—
77C2×C22⋊Dic59D411yescharacteristic
78C2×C22⋊Dic59D411yescharacteristic
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59C2311yescharacteristic
87C2×C4×Dic59C2311yescharacteristic
88#14273660—22no—
89#14273660—41no—
90#14273658C2×C414yescomplemented
91#14273658C2×C414yescomplemented
92#14273649—22no—
93#14273649—41no—
94#14273647C2×C414yescomplemented
95#14273647C2×C414yescomplemented
96C22⋊(C59⋊C8)C2×C414yescomplemented
97#14273637—22no—
98#14273637—41no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95D411yescharacteristic
107C59⋊#95D411yescharacteristic
108C23×C4×C59—21no—
109C23×C4×C59—21no—
110C23×C4×C59C2311yescharacteristic
111C22×C8×C59—22no—
112C22×C8×C59—22no—
113C22×C8×C59—41no—
114C22×C8×C59C2311yescharacteristic
115C22×C8×C59C2311yescharacteristic
116C2×C4⋊C4×C59—21no—
117C2×C4⋊C4×C59—22no—
118C2×C4⋊C4×C59—22no—
119C2×C4⋊C4×C59—21no—
120C2×C4⋊C4×C59—41no—
121C2×C4⋊C4×C59C2×C414yescomplemented
122C2×C4⋊C4×C59C2311yescharacteristic
123C2×C4⋊C4×C59C2311yescharacteristic
124C2×C4⋊C4×C59C2311yescharacteristic
125C2×C4⋊C4×C59C2311yescharacteristic
126C24×C59C2×D411yescharacteristic
127C23×D59—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—24no—
136C22×C4×C59—21no—
137C22×C4×C59—42no—
138C22×C4×C59—42no—
139C22×C4×C59—81no—
140C22×C4×C59—81no—
141C22×C4×C59—81no—
142C22×C4×C59C22×C412yes—
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59#4112yes—
145C22×C4×C59C2411yescharacteristic
146C2×C22⋊D59—24no—
147C2×C22⋊D59—42no—
148C2×C22⋊D59—42no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59C2×D411yescharacteristic
153C22×Dic59C2×D411yescharacteristic
154C22×Dic59C2×D411yescharacteristic
155C22×Dic59C2×D411yescharacteristic
156C2×D236—21no—
157C2×D236—21no—
158C2×D236—21no—
159C2×D236—21no—
160C2×D236—21no—
161C2×D236C22×C412yes—
162C2×C4×D59—42no—
163C2×C4×D59—42no—
164C2×C4×D59—81no—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59—44no—
171C2×C8×C59#4112yes—
172C4⋊C4×C59—24no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—42no—
179C4⋊C4×C59C22×C412yes—
180C4⋊C4×C59C22×C412yes—
181C4⋊C4×C59#4112yes—
182C22⋊Dic59—22no—
183C22⋊Dic59—22no—
184C22⋊Dic59—42no—
185(C2×D59)⋊C4—24no—
186(C2×D59)⋊C4—42no—
187(C2×D59)⋊C4—42no—
188(C2×D59)⋊C4—81no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—81no—
195C4⋊Dic59—81no—
196C4⋊Dic59C22×C412yes—
197Dic59⋊C4—42no—
198C4×Dic59—21no—
199C4×Dic59—21no—
200C4×Dic59—21no—
201C4×Dic59—21no—
202C4×Dic59—42no—
203C4×Dic59—81no—
204C4×Dic59—81no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8C22×C412yes—
210C2×C59⋊C8C22×C412yes—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59C4×D412yes—
216C23×C59AGL(1,ℤ/8)12yes—
217C23×C59#13712yes—
218C23×C59C22×D411yescharacteristic
219C22×D59—22no—
220C22×D59—41no—
221C22×D59—41no—
222C22×D59—41no—
223C22×D59—41no—
224C22×D59—41no—
225C22×D59—41no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—22no—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
232C2×C4×C59—22no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
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×C59—84no—
245C2×C4×C59—84no—
246C2×C4×C59C4×D412yes—
247C2×C4×C59C23×C411yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C2×C4×C59C2×#4111yescharacteristic
251C2×C4×C59H(5,𝔽2)11yescharacteristic
252C2×C4×C592−1+411yescharacteristic
253C22⋊D59—48no—
254C2×Dic59—22no—
255C2×Dic59—21no—
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—21no—
259C2×Dic59—21no—
260C2×Dic59—21no—
261C2×Dic59—21no—
262C2×Dic59—21no—
263C2×Dic59—21no—
264C2×Dic59—21no—
265C2×Dic59—21no—
266C2×Dic59—21no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275D236—22no—
276D236—22no—
277D236—41no—
278D236—41no—
279C4×D59—84no—
280C8×C59—22no—
281C8×C59—41no—
282C59⋊C8—22no—
283C59⋊C8—22no—
284#22936—591nomaximal, Sylow
285C22×C59—22no—
286C22×C59—22no—
287C22×C59—22no—
288C22×C59—22no—
289C22×C59—22no—
290C22×C59—22no—
291C22×C59#44014yes—
292C22×C59C2×C4×D411yescharacteristic
293C22×C59#54611yescharacteristic
294C22×C59#54711yescharacteristic
295C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
296C22×C59C2×#13711yescharacteristic
297C22×C59#57611yescharacteristic
298C22×C59#57711yescharacteristic
299C2×D59—42no—
300C2×D59—42no—
301C2×D59—42no—
302C2×D59—81no—
303C2×D59—81no—
304C2×D59—81no—
305C2×D59—81no—
306C4×C59—22no—
307C4×C59—42no—
308C4×C59—42no—
309C4×C59—42no—
310C4×C59—81no—
311C4×C59C4×#4111yescharacteristic
312C4×C59#52011yescharacteristic
313C4×C59#55111yescharacteristic
314C4×C59#55211yescharacteristic
315Dic59—41no—
316Dic59—41no—
317Dic59—42no—
318Dic59—41no—
319Dic59—41no—
320Dic59—81no—
321Dic59—81no—
322C2×#481—591no—
323#2361—591no—
324#2345—591no—
325#1944—591no—
326#1936—591no—
327#1925—591no—
328#1920—591no—
329#1919—591no—
330#1893—591no—
331#1862—591no—
332#1852—591no—
333#1776—591no—
334#1762—591no—
335#1752—591no—
336C2×#340D5911yescharacteristic, complemented
337C2×C59—24no—
338C2×C59#235911yescharacteristic
339C2×C59C2×#44011yescharacteristic
340C2×C59#294611yescharacteristic
341C2×C59#331611yescharacteristic
342C2×C59#331811yescharacteristic
343C2×C59#334511yescharacteristic
344C2×C59#334611yescharacteristic
345D59—82no—
346C23×C8C2×D5911yescharacteristic
347C2×#121—591no—
348C2×C4×D4—591no—
349C2×C4×D4—1181no—
350C22×C4⋊C4C2×D5911yescharacteristic
351C22×C4⋊C4C2×D5911yescharacteristic
352#481—598no—
353C2×#107—591no—
354C2×#107—591no—
355C2×#106—591no—
356C2×#106—591no—
357C2×#102—591no—
358C2×#102—591no—
359C2×#102—591no—
360C2×#102—591no—
361C2×C22⋊C8—591no—
362C2×#97—591no—
363C2×#97—591no—
364#390—591no—
365#389—591no—
366#389—591no—
367#389—591no—
368#388—1181no—
369#387—1181no—
370#387—1181no—
371#386—1181no—
372#385—1181no—
373#380—591no—
374#380—1181no—
375C4×C4⋊C4—591no—
376C4×C4⋊C4—591no—
377C4×C4⋊C4—591no—
378C4×C22⋊C4—591no—
379C4×C22⋊C4—1181no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340C2×D5911yescharacteristic
385#340C2×D5911yescharacteristic
386#340C2×D5911yescharacteristic
387#340C2×D5911yescharacteristic
388#336—591no—
389#336—591no—
390C59#2293611yescharacteristic, complemented, Sylow
391C22×D4—591no—
392C22×D4—1181no—
393C23×C4—21no—
394C23×C4—21no—
395C23×C4—1181no—
396C23×C4C22×D5911yescharacteristic
397C22×C8—22no—
398C22×C8—22no—
399C22×C8—41no—
400C22×C8—591no—
401C22×C8—591no—
402C22×C8C22×D5911yescharacteristic
403C22×C8C22×D5911yescharacteristic
404#121—594no—
405#121—1182no—
406C4×D4—1182no—
407C4×D4—2361no—
408C4×D4—2364no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—41no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—591no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—1181no—
418C2×C4⋊C4—1181no—
419C2×C4⋊C4—1181no—
420C2×C4⋊C4C4×D5914yescomplemented
421C2×C4⋊C4C22×D5911yescharacteristic
422C2×C4⋊C4C22×D5911yescharacteristic
423C2×C4⋊C4C22×D5911yescharacteristic
424C2×C4⋊C4C22×D5911yescharacteristic
425C2×C22⋊C4—591no—
426C2×C22⋊C4—591no—
427C2×C22⋊C4—1181no—
428C2×C22⋊C4—1181no—
429C2×C22⋊C4—1181no—
430C2×C22⋊C4—1181no—
431C2×C22⋊C4—1181no—
432C2×C42—591no—
433C2×C42—591no—
434C2×C42—1181no—
435C2×C42—1181no—
436C2×C42—1181no—
437#107—1182no—
438#107—2361no—
439#106—594no—
440#106—594no—
441#102—594no—
442#102—594no—
443#102—1182no—
444#102—2361no—
445C22⋊C8—594no—
446#97—1182no—
447#97—2361no—
448#95—591no—
449#95—591no—
450#95—1181no—
451#95—1181no—
452#95—1181no—
453#95—1181no—
454#95—1181no—
455#95—1181no—
456#95—1181no—
457C24—1181no—
458C24D4×D5911yescharacteristic
459C2×D4—592no—
460C2×D4—1184no—
461C2×D4—1181no—
462C2×D4—1181no—
463C2×D4—1181no—
464C2×D4—1181no—
465C2×D4—1181no—
466C2×D4—2362no—
467C2×D4—2362no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—81no—
482C22×C4—591no—
483C22×C4—591no—
484C22×C4—591no—
485C22×C4—591no—
486C22×C4—1181no—
487C22×C4—1181no—
488C22×C4—1181no—
489C22×C4—4721no—
490C22×C4—2362no—
491C22×C4—2362no—
492C22×C4C2×C4×D5912yes—
493C22×C4D4×D5911yescharacteristic
494C22×C4#1174568712yes—
495C22×C4C23×D5911yescharacteristic
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—44no—
502C2×C8—592no—
503C2×C8—592no—
504C2×C8—1181no—
505C2×C8—1181no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8#1174568512yes—
509C4⋊C4—24no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—42no—
516C4⋊C4—592no—
517C4⋊C4—1181no—
518C4⋊C4—1181no—
519C4⋊C4—1181no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—4721no—
523C4⋊C4—4721no—
524C4⋊C4—2362no—
525C4⋊C4C2×C4×D5912yes—
526C4⋊C4C2×C4×D5912yes—
527C4⋊C4#1174568512yes—
528C22⋊C4—1182no—
529C22⋊C4—1182no—
530C22⋊C4—1184no—
531C22⋊C4—4721no—
532C22⋊C4—2362no—
533C22⋊C4—2362no—
534C22⋊C4—2362no—
535C42—1181no—
536C42—1181no—
537C42—1181no—
538C42—1181no—
539C42—4721no—
540C42—4721no—
541C42—2362no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—1182no—
547C23—2361no—
548C23—2361no—
549C23—2361no—
550C23—2361no—
551C23—2361no—
552C23—2361no—
553C23(C22⋊D59)⋊C412yes—
554C23#1427375412yes—
555C23#1427375912yes—
556C23C2×D4×D5911yescharacteristic
557D4—1182no—
558D4—1182no—
559D4—2368no—
560D4—2361no—
561D4—2361no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
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—84no—
582C2×C4—1181no—
583C2×C4—1181no—
584C2×C4—1181no—
585C2×C4—1181no—
586C2×C4—1181no—
587C2×C4—1181no—
588C2×C4—1181no—
589C2×C4—1181no—
590C2×C4—1181no—
591C2×C4—1181no—
592C2×C4—1181no—
593C2×C4—1181no—
594C2×C4—1182no—
595C2×C4—4724no—
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(C22⋊D59)⋊C412yes—
605C2×C4C22×C4×D5911yescharacteristic
606C2×C4C2×#1174568711yescharacteristic
607C2×C4C2×#1174568511yescharacteristic
608C2×C4C2×#1174568511yescharacteristic
609C2×C4#1427377211yescharacteristic
610C2×C4#1427377511yescharacteristic
611C8—22no—
612C8—41no—
613C8—1182no—
614C8—1182no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—4721no—
622C22—4721no—
623C22—4721no—
624C22—4721no—
625C22—2362no—
626C22—2362no—
627C22—2362no—
628C22#1683750514yes—
629C22C2×(C22⋊D59)⋊C411yescharacteristic
630C22#1683804211yescharacteristic
631C22#1683805311yescharacteristic
632C22C2×#1427375411yescharacteristic
633C22C2×#1427375911yescharacteristic
634C22#1683820311yescharacteristic
635C22#1683820811yescharacteristic
636C4—22no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—81no—
641C4—4721no—
642C4—4721no—
643C4—2361no—
644C4—2362no—
645C4—2361no—
646C4—2361no—
647C4—2361no—
648C4C4×#1174568511yescharacteristic
649C4#1683788311yescharacteristic
650C4#1683807811yescharacteristic
651C4#1683808711yescharacteristic
652C2—24no—
653C2—4722no—
654C2#2142708411yescharacteristic
655C2C2×#1683750511yescharacteristic
656C2#2143292211yescharacteristic
657C2#2143620311yescharacteristic
658C2#2143623311yescharacteristic
659C2#2143656211yescharacteristic
660C2#2143657711yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#340C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
cosocleC22×C4×C59C24
hypercenterC2×#340D59
hypocenterC59#22936
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. #21426973C2semidirect
4. #21424838C2semidirect
5. #21424794C2semidirect
6. #21424714C2semidirect
7. #21424679C2semidirect
8. #21424677C2semidirect
10. #21424344C2semidirect
11. #21424284C2semidirect
12. #21423992C2semidirect
13. #21423920C2semidirect
14. #21423872C2semidirect
15. C2×C59×#340C2semidirect
19. #16837720C4semidirect
90. #14273658C2×C4semidirect
91. #14273658C2×C4semidirect
94. #14273647C2×C4semidirect
95. #14273647C2×C4semidirect
96. C22⋊(C59⋊C8)C2×C4semidirect
121. C2×C4⋊C4×C59C2×C4semidirect
336. C2×#340D59semidirect
390. C59#22936semidirect
420. C2×C4⋊C4C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377202
#214271122
#214269732
#214248382
#214247942
#214247142
#214246792
#214246772
#214245242
#214243442
#214242842
#214239922
#214239202
#214238722
C2×C59×#3402
#2293659