Finite Group Databasebeta

← #28610110

Subgroups — #28610110

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×#16837718C211yesmaximal, characteristic
2#21426971C211yesmaximal, characteristic, complemented
3#21424836C211yesmaximal, characteristic, complemented
4#21424796C211yesmaximal, characteristic, complemented
5#21424713C211yesmaximal, characteristic, complemented
6#21424678C211yesmaximal, characteristic, complemented
7#21424674C211yesmaximal, characteristic, complemented
8#21424525C211yesmaximal, characteristic
9#21424343C211yesmaximal, characteristic, complemented
10#21424284C211yesmaximal, characteristic, complemented
11#21423994C211yesmaximal, characteristic, complemented
12#21423921C211yesmaximal, characteristic, complemented
13#21423874C211yesmaximal, characteristic, complemented
14C2×#16837164C211yesmaximal, characteristic, complemented
15C59×#2361C211yesmaximal, characteristic
16C23×C59⋊C8C2211yescharacteristic
17C22×C4⋊Dic59C2211yescharacteristic
18#16837718C418yescomplemented
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837285—21no—
33#16837285—21no—
34#16837262C2211yescharacteristic
35#16837260—21no—
36C4×C4⋊Dic59C2211yescharacteristic
37C4×C4⋊Dic59C2211yescharacteristic
38C22⋊C4×Dic59—21no—
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×#121×C59C2211yescharacteristic
50C2×C4×D4×C59—21no—
51C2×C4×D4×C59C2211yescharacteristic
52C22×C4⋊C4×C59C2211yescharacteristic
53C59×#390C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#388—21no—
56C59×#386—21no—
57C59×#385—21no—
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C22⋊C4×C59C2211yescharacteristic
60C23×Dic59—21no—
61C22×C59⋊C8—22no—
62C22×C59⋊C8—22no—
63C22×C59⋊C8—41no—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—22no—
70C2×C4⋊Dic59C2×C414yescomplemented
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×C22⋊Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80#14273660—22no—
81#14273660—41no—
82#14273658C2×C414yescomplemented
83#14273658C2×C414yescomplemented
84#14273648—22no—
85#14273648—41no—
86#14273648C2×C414yescomplemented
87#14273648C2×C414yescomplemented
88C22⋊(C59⋊C8)C2×C414yescomplemented
89#14273638—22no—
90#14273638—41no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C22×D4×C59—21no—
97C22×D4×C59C2311yescharacteristic
98C23×C4×C59—21no—
99C23×C4×C59—21no—
100C23×C4×C59C2311yescharacteristic
101#121×C59—22no—
102#121×C59C2×C414yes—
103C4×D4×C59—22no—
104C4×D4×C59—41no—
105C4×D4×C59—44no—
106C2×C4⋊C4×C59—22no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59—41no—
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×C59D411yescharacteristic
118C2×C22⋊C4×C59D411yescharacteristic
119C2×C42×C59—21no—
120C2×C42×C59C2311yescharacteristic
121C2×C42×C59C2311yescharacteristic
122#95×C59—21no—
123#95×C59—21no—
124#95×C59D411yescharacteristic
125#95×C59D411yescharacteristic
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—42no—
135C2×D4×C59—42no—
136C2×D4×C59C22×C412yes—
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—42no—
145C22×C4×C59—42no—
146C22×C4×C59—42no—
147C22×C4×C59—42no—
148C22×C4×C59—81no—
149C22×C4×C59—81no—
150C22×C4×C59C22×C412yes—
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59#4112yes—
157C22×C4×C59C2411yescharacteristic
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—24no—
161C22×Dic59—21no—
162C22×Dic59—21no—
163C22×Dic59—81no—
164C22×Dic59—81no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—42no—
170C4⋊C4×C59—42no—
171C4⋊C4×C59C22×C412yes—
172C4⋊C4×C59#4112yes—
173C22⋊C4×C59—22no—
174C22⋊C4×C59—22no—
175C22⋊C4×C59—24no—
176C22⋊C4×C59—42no—
177C22⋊C4×C59—42no—
178C22⋊C4×C59—81no—
179C42×C59—21no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59—42no—
184C22⋊Dic59—42no—
185C4⋊Dic59—24no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59C22×C412yes—
193C4⋊Dic59C22×C412yes—
194Dic59⋊C4—81no—
195Dic59⋊C4—81no—
196C4×Dic59—81no—
197C4×Dic59—81no—
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⋊C8—21no—
207C2×C59⋊C8—44no—
208C2×C59⋊C8C22×C412yes—
209C2×C59⋊C8C22×C412yes—
210C2×C59⋊C8#4112yes—
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×C59AGL(1,ℤ/8)12yes—
224C23×C59#13712yes—
225C23×C59C22×D411yescharacteristic
226D4×C59—22no—
227D4×C59—22no—
228D4×C59—48no—
229D4×C59—41no—
230D4×C59—41no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—22no—
236C2×C4×C59—22no—
237C2×C4×C59—22no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—22no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—21no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—84no—
259C2×C4×C59—84no—
260C2×C4×C59C4×D412yes—
261C2×C4×C59C23×C411yescharacteristic
262C2×C4×C59C2×#4111yescharacteristic
263C2×C4×C59C2×#4111yescharacteristic
264C2×C4×C59C2×#4111yescharacteristic
265C2×C4×C59H(5,𝔽2)11yescharacteristic
266C2×C4×C592−1+411yescharacteristic
267C2×Dic59—22no—
268C2×Dic59—22no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—84no—
280C59⋊C8—22no—
281C59⋊C8—22no—
282C59⋊C8—22no—
283C59⋊C8—41no—
284#22936—591nomaximal, Sylow
285C22×C59—22no—
286C22×C59—22no—
287C22×C59—22no—
288C22×C59—22no—
289C22×C59—22no—
290C22×C59—22no—
291C22×C59—42no—
292C22×C59—42no—
293C22×C59—42no—
294C22×C59—81no—
295C22×C59—81no—
296C22×C59—81no—
297C22×C59—81no—
298C22×C59#44014yes—
299C22×C59C2×C4×D411yescharacteristic
300C22×C59#54611yescharacteristic
301C22×C59#54711yescharacteristic
302C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
303C22×C59C2×#13711yescharacteristic
304C22×C59#57611yescharacteristic
305C22×C59#57711yescharacteristic
306C4×C59—22no—
307C4×C59—41no—
308C4×C59—41no—
309C4×C59—42no—
310C4×C59—41no—
311C4×C59—41no—
312C4×C59—42no—
313C4×C59—81no—
314C4×C59C4×#4111yescharacteristic
315C4×C59#52011yescharacteristic
316C4×C59#55111yescharacteristic
317C4×C59#55211yescharacteristic
318Dic59—42no—
319Dic59—42no—
320Dic59—81no—
321Dic59—81no—
322C2×#481—591no—
323#2361D5911yescharacteristic
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×#340—591no—
337C2×C59—24no—
338C2×C59—82no—
339C2×C59#235911yescharacteristic
340C2×C59C2×#44011yescharacteristic
341C2×C59#294611yescharacteristic
342C2×C59#331611yescharacteristic
343C2×C59#331811yescharacteristic
344C2×C59#334511yescharacteristic
345C2×C59#334611yescharacteristic
346C23×C8—591no—
347C2×#121C2×D5911yescharacteristic
348C2×C4×D4—21no—
349C2×C4×D4C2×D5911yescharacteristic
350C22×C4⋊C4—591no—
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#390C2×D5911yescharacteristic
365#389—591no—
366#389—591no—
367#389C2×D5911yescharacteristic
368#388—21no—
369#387—1181no—
370#387—1181no—
371#386—21no—
372#385—21no—
373#380—591no—
374#380—1181no—
375C4×C4⋊C4—591no—
376C4×C4⋊C4—591no—
377C4×C4⋊C4C2×D5911yescharacteristic
378C4×C22⋊C4—1181no—
379C4×C22⋊C4C2×D5911yescharacteristic
380#340—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#340—591no—
386#340—591no—
387#340—591no—
388#336—591no—
389#336—591no—
390C59#2293611yescharacteristic, complemented, Sylow
391C22×D4—21no—
392C22×D4C22×D5911yescharacteristic
393C23×C4—21no—
394C23×C4—21no—
395C23×C4—1181no—
396C23×C4C22×D5911yescharacteristic
397C22×C8—591no—
398C22×C8—591no—
399C22×C8—591no—
400C22×C8—591no—
401C22×C8—1182no—
402C22×C8—1182no—
403C22×C8—2361no—
404#121—22no—
405#121C4×D5914yes—
406C4×D4—22no—
407C4×D4—41no—
408C4×D4—44no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—41no—
413C2×C4⋊C4—594no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—591no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—1182no—
419C2×C4⋊C4—1181no—
420C2×C4⋊C4—1181no—
421C2×C4⋊C4—1181no—
422C2×C4⋊C4C22×D5911yescharacteristic
423C2×C4⋊C4C22×D5911yescharacteristic
424C2×C4⋊C4C22×D5911yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—1181no—
430C2×C22⋊C4C22⋊D5911yescharacteristic
431C2×C22⋊C4C22⋊D5911yescharacteristic
432C2×C42—21no—
433C2×C42—1181no—
434C2×C42—1181no—
435C2×C42C22×D5911yescharacteristic
436C2×C42C22×D5911yescharacteristic
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—21no—
449#95—21no—
450#95—1181no—
451#95—1181no—
452#95—1181no—
453#95—1181no—
454#95—1181no—
455#95C22⋊D5911yescharacteristic
456#95C22⋊D5911yescharacteristic
457C24—21no—
458C24C2×C22⋊D5911yescharacteristic
459C2×D4—24no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4C2×C4×D5912yes—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—1184no—
482C22×C4—1181no—
483C22×C4—1181no—
484C22×C4—1181no—
485C22×C4—1181no—
486C22×C4—4721no—
487C22×C4—4721no—
488C22×C4C2×C4×D5912yes—
489C22×C4#1174568512yes—
490C22×C4C2×C22⋊D5911yescharacteristic
491C22×C4C2×C22⋊D5911yescharacteristic
492C22×C4C2×C22⋊D5911yescharacteristic
493C22×C4C2×C22⋊D5911yescharacteristic
494C22×C4C2×C22⋊D5911yescharacteristic
495C22×C4C23×D5911yescharacteristic
496C2×C8—592no—
497C2×C8—592no—
498C2×C8—592no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8—2364no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—592no—
516C4⋊C4—592no—
517C4⋊C4—1184no—
518C4⋊C4—1181no—
519C4⋊C4—1181no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—1181no—
524C4⋊C4—4721no—
525C4⋊C4—4721no—
526C4⋊C4C2×C4×D5912yes—
527C4⋊C4#1174568712yes—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—24no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—81no—
534C22⋊C4—2362no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—42no—
540C42—4721no—
541C42—4721no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—22no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23C4×C22⋊D5912yes—
554C23#1427375312yes—
555C23#1427375712yes—
556C23C22×C22⋊D5911yescharacteristic
557D4—22no—
558D4—22no—
559D4—48no—
560D4—41no—
561D4—41no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
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—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—1182no—
592C2×C4—1182no—
593C2×C4—4724no—
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×C4C4×C22⋊D5912yes—
605C2×C4C22×C4×D5911yescharacteristic
606C2×C4#41×D5911yescharacteristic
607C2×C4C2×#1174568711yescharacteristic
608C2×C4C2×#1174568511yescharacteristic
609C2×C4#1427377211yescharacteristic
610C2×C4#1427377411yescharacteristic
611C8—1182no—
612C8—1182no—
613C8—1182no—
614C8—2361no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22#1683750214yes—
629C22C2×C4×C22⋊D5911yescharacteristic
630C22#1683804611yescharacteristic
631C22#1683805111yescharacteristic
632C22C2×#1427375311yescharacteristic
633C22C2×#1427375711yescharacteristic
634C22#1683820511yescharacteristic
635C22#1683821011yescharacteristic
636C4—22no—
637C4—42no—
638C4—41no—
639C4—41no—
640C4—42no—
641C4—41no—
642C4—41no—
643C4—81no—
644C4—4721no—
645C4—4721no—
646C4—2362no—
647C4—2362no—
648C4C4×#1174568711yescharacteristic
649C4#1683788111yescharacteristic
650C4#1683808111yescharacteristic
651C4#1683809211yescharacteristic
652C2—24no—
653C2—82no—
654C2#2142708211yescharacteristic
655C2C2×#1683750211yescharacteristic
656C2#2143292411yescharacteristic
657C2#2143620711yescharacteristic
658C2#2143623711yescharacteristic
659C2#2143656611yescharacteristic
660C2#2143658111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2361C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
cosocleC22×C4×C59C24
hypercenter#2361D59
hypocenterC59#22936
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
2. #21426971C2semidirect
3. #21424836C2semidirect
4. #21424796C2semidirect
5. #21424713C2semidirect
6. #21424678C2semidirect
7. #21424674C2semidirect
9. #21424343C2semidirect
10. #21424284C2semidirect
11. #21423994C2semidirect
12. #21423921C2semidirect
13. #21423874C2semidirect
14. C2×#16837164C2semidirect
18. #16837718C4semidirect
70. C2×C4⋊Dic59C2×C4semidirect
82. #14273658C2×C4semidirect
83. #14273658C2×C4semidirect
86. #14273648C2×C4semidirect
87. #14273648C2×C4semidirect
88. C22⋊(C59⋊C8)C2×C4semidirect
390. C59#22936semidirect

Maximal subgroups

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

subgroupindex
C2×#168377182
#214269712
#214248362
#214247962
#214247132
#214246782
#214246742
#214245252
#214243432
#214242842
#214239942
#214239212
#214238742
C2×#168371642
C59×#23612
#2293659