Finite Group Databasebeta

← #28628722

Subgroups — #28628722

Order 15104.

850 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837723C211yesmaximal, characteristic
2C2×#16837459C211yesmaximal, characteristic, complemented
3#21426579C211yesmaximal, characteristic, complemented
4#21425597C211yesmaximal, characteristic, complemented
5#21425587C211yesmaximal, characteristic, complemented
6#21425484C211yesmaximal, characteristic, complemented
7#21425470C211yesmaximal, characteristic, complemented
8#21424836C211yesmaximal, characteristic, complemented
9#21424831C211yesmaximal, characteristic, complemented
10#21424796C211yesmaximal, characteristic, complemented
11#21424796C211yesmaximal, characteristic, complemented
12#21424419C211yesmaximal, characteristic
13#21424293C211yesmaximal, characteristic, complemented
14#21424188C211yesmaximal, characteristic, complemented
15C59×#2854C211yesmaximal, characteristic
16C2×#14273686C2211yescharacteristic
17C2×#14273686C2211yescharacteristic
18#16837723—24no—
19#16837459—24no—
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273648C2211yescharacteristic
29C2×#14273648C2211yescharacteristic
30C2×C22⋊(C59⋊C8)C2211yescharacteristic
31C2×C22⋊(C59⋊C8)C2211yescharacteristic
32C2×#14273638C2211yescharacteristic
33C2×#14273638C2211yescharacteristic
34#16837300C2211yescharacteristic
35#16837300C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837275—21no—
39#16837275—21no—
40C4×Dic59⋊C4—21no—
41C4×Dic59⋊C4—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50C2×#123×C59—21no—
51C2×#121×C59C2211yescharacteristic
52C22×C42×C59C2211yescharacteristic
53C59×#390C2211yescharacteristic
54C59×#390C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C59×#389C2211yescharacteristic
57C59×#388—21no—
58C59×#388—21no—
59C59×#386—21no—
60C59×#380C2211yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273686—22no—
66#14273686—22no—
67#14273686—22no—
68#14273686—22no—
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—
79C2×C4×Dic59—21no—
80C2×C4×Dic59—21no—
81#14273660—22no—
82#14273660—22no—
83#14273658—22no—
84#14273658—22no—
85C4⋊(C59⋊C8)—22no—
86C4⋊(C59⋊C8)—22no—
87#14273648—22no—
88#14273648—22no—
89#14273648—22no—
90#14273648—22no—
91C22⋊(C59⋊C8)—22no—
92C22⋊(C59⋊C8)D414yes—
93#14273638—22no—
94#14273638—22no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C22×D4×C59—21no—
100C22×D4×C59C2311yescharacteristic
101C23×C4×C59D411yescharacteristic
102C23×C4×C59D411yescharacteristic
103C23×C4×C59C2311yescharacteristic
104#123×C59—44no—
105#121×C59—22no—
106#121×C59—22no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C22⋊C4×C59—21no—
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×C42×C59—22no—
119C2×C42×C59—22no—
120C2×C42×C59D411yescharacteristic
121C2×C42×C59D411yescharacteristic
122C2×C42×C59C2311yescharacteristic
123C2×C42×C59C2311yescharacteristic
124#95×C59—21no—
125#95×C59—21no—
126#95×C59—21no—
127C24×C59—21no—
128C24×C59C2×D411yescharacteristic
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59—42no—
136C2×D4×C59—42no—
137C2×D4×C59—42no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—22no—
143C22×C4×C59—22no—
144C22×C4×C59—22no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—41no—
149C22×C4×C59—41no—
150C22×C4×C59—41no—
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D412yes—
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59#4112yes—
158C22×C4×C59C2411yescharacteristic
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C22×Dic59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—42no—
170C4⋊C4×C59—42no—
171C22⋊C4×C59—42no—
172C22⋊C4×C59—42no—
173C22⋊C4×C59—42no—
174C22⋊C4×C59—42no—
175C22⋊C4×C59—42no—
176C22⋊C4×C59—42no—
177C42×C59—22no—
178C42×C59—22no—
179C42×C59—21no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—21no—
183C22⋊Dic59—24no—
184C22⋊Dic59—42no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59#4112yes—
192C4⋊Dic59#4112yes—
193Dic59⋊C4—42no—
194Dic59⋊C4—42no—
195Dic59⋊C4—42no—
196Dic59⋊C4—42no—
197C4×Dic59—24no—
198C4×Dic59—42no—
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—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8C2×D412yes—
210C2×C59⋊C8C2×D412yes—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—21no—
216C23×C59—21no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59—41no—
222C23×C59—41no—
223C23×C59—41no—
224C23×C59#12112yes—
225C23×C59C22×D411yescharacteristic
226D4×C59—41no—
227D4×C59—41no—
228D4×C59—41no—
229D4×C59—41no—
230D4×C59—84no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—22no—
236C2×C4×C59—22no—
237C2×C4×C59—22no—
238C2×C4×C59—22no—
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×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—44no—
249C2×C4×C59—44no—
250C2×C4×C59—41no—
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—41no—
259C2×C4×C59#12112yes—
260C2×C4×C59AGL(1,ℤ/8)12yes—
261C2×C4×C59C22×D411yescharacteristic
262C2×C4×C59C2×#4111yescharacteristic
263C2×C4×C59C2×#4111yescharacteristic
264C2×C4×C59C2×#4111yescharacteristic
265C2×C4×C592−1+411yescharacteristic
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—
279C59⋊C8—22no—
280C59⋊C8—22no—
281C59⋊C8—41no—
282C59⋊C8—41no—
283#24610—591nomaximal, Sylow
284C22×C59—22no—
285C22×C59—22no—
286C22×C59—22no—
287C22×C59—41no—
288C22×C59—41no—
289C22×C59—41no—
290C22×C59—41no—
291C22×C59—81no—
292C22×C59—81no—
293C22×C59—81no—
294C22×C59—81no—
295C22×C59—81no—
296C22×C59—81no—
297C22×C59—81no—
298C22×C59C2×#12111yescharacteristic
299C22×C59#52511yescharacteristic
300C22×C59#53611yescharacteristic
301C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
302C22×C59#57511yescharacteristic
303C22×C59#57511yescharacteristic
304C22×C59#57511yescharacteristic
305C4×C59—22no—
306C4×C59—22no—
307C4×C59—41no—
308C4×C59—41no—
309C4×C59—41no—
310C4×C59—81no—
311C4×C59—81no—
312C4×C59—81no—
313C4×C59#54711yescharacteristic
314C4×C59#54711yescharacteristic
315C4×C59#55211yescharacteristic
316C4×C59#55211yescharacteristic
317Dic59—42no—
318Dic59—42no—
319Dic59—81no—
320Dic59—81no—
321C2×#482—591no—
322C2×#432—591no—
323#2854D5911yescharacteristic
324#2296—591no—
325#2088—591no—
326#2086—591no—
327#2067—591no—
328#2065—591no—
329#1944—591no—
330#1943—591no—
331#1936—591no—
332#1936—591no—
333#1875—591no—
334#1854—591no—
335#1825—591no—
336C2×C59—42no—
337C2×C59—161no—
338C2×C59#241311yescharacteristic
339C2×C59#305411yescharacteristic
340C2×C59#305511yescharacteristic
341C2×C59#311111yescharacteristic
342C2×C59#311511yescharacteristic
343C2×C59#323511yescharacteristic
344C2×C59#324011yescharacteristic
345C2×#123—21no—
346C2×#121C2×D5911yescharacteristic
347C2×#117—591no—
348C2×#117—591no—
349C22×C42C2×D5911yescharacteristic
350#482—1184no—
351#432—1184no—
352C2×#107—591no—
353C2×#107—591no—
354C2×#106—591no—
355C2×#106—591no—
356C2×C4⋊C8—591no—
357C2×C4⋊C8—591no—
358C2×#102—591no—
359C2×#102—591no—
360C2×#102—591no—
361C2×#102—591no—
362C2×C22⋊C8—591no—
363C2×C22⋊C8—591no—
364C2×#97—591no—
365C2×#97—591no—
366#390C2×D5911yescharacteristic
367#390C2×D5911yescharacteristic
368#389—591no—
369#389—591no—
370#389—591no—
371#389—591no—
372#389C2×D5911yescharacteristic
373#389C2×D5911yescharacteristic
374#388—21no—
375#388—21no—
376#386—21no—
377#385—1181no—
378#385—1181no—
379#380C2×D5911yescharacteristic
380C4×C4⋊C4—1181no—
381C4×C4⋊C4—1181no—
382#340—591no—
383#340—591no—
384#340—591no—
385#340—591no—
386#340—591no—
387#340—591no—
388#340—591no—
389#340—591no—
390C59#2461011yescharacteristic, complemented, Sylow
391C22×D4—21no—
392C22×D4C22×D5911yescharacteristic
393C23×C4C22⋊D5911yescharacteristic
394C23×C4C22⋊D5911yescharacteristic
395C23×C4C22×D5911yescharacteristic
396C22×C8—591no—
397C22×C8—591no—
398C22×C8—591no—
399C22×C8—591no—
400#123—44no—
401#121—22no—
402#121—22no—
403#117—1182no—
404#117—1182no—
405#117—1182no—
406#117—1182no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—591no—
412C2×C4⋊C4—591no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4—1181no—
415C2×C4⋊C4—1181no—
416C2×C4⋊C4—1181no—
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C4⋊C4C22×D5911yescharacteristic
419C2×C4⋊C4C22×D5911yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—1181no—
427C2×C22⋊C4—1181no—
428C2×C42—22no—
429C2×C42—22no—
430C2×C42—1181no—
431C2×C42—1181no—
432C2×C42C22⋊D5911yescharacteristic
433C2×C42C22⋊D5911yescharacteristic
434C2×C42C22×D5911yescharacteristic
435C2×C42C22×D5911yescharacteristic
436#107—1182no—
437#107—1182no—
438#106—1182no—
439#106—1182no—
440C4⋊C8—1182no—
441C4⋊C8—1182no—
442#102—1182no—
443#102—1182no—
444#102—1182no—
445#102—1182no—
446C22⋊C8—594no—
447C22⋊C8—1182no—
448#97—1182no—
449#97—1182no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—1181no—
454#95—1181no—
455#95—1181no—
456#95—1181no—
457C24—21no—
458C24C2×C22⋊D5911yescharacteristic
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—1181no—
482C22×C4—1181no—
483C22×C4—1181no—
484C22×C4—1181no—
485C22×C4D4×D5912yes—
486C22×C4#1174568712yes—
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C2×C22⋊D5911yescharacteristic
489C22×C4C2×C22⋊D5911yescharacteristic
490C22×C4C2×C22⋊D5911yescharacteristic
491C22×C4C2×C22⋊D5911yescharacteristic
492C22×C4C23×D5911yescharacteristic
493C2×C8—592no—
494C2×C8—592no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—592no—
514C4⋊C4—592no—
515C4⋊C4—1181no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4—1181no—
519C4⋊C4—1181no—
520C4⋊C4—1181no—
521C4⋊C4—2362no—
522C4⋊C4—2362no—
523C4⋊C4—2362no—
524C4⋊C4—2362no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—1184no—
532C22⋊C4—2362no—
533C42—22no—
534C42—22no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—1184no—
540C42—2362no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23#1427369912yes—
555C23C22×C22⋊D5911yescharacteristic
556D4—41no—
557D4—41no—
558D4—41no—
559D4—41no—
560D4—84no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
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—44no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—1182no—
590C2×C4—1182no—
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#1427369912yes—
602C2×C4#1427375312yes—
603C2×C4C2×D4×D5911yescharacteristic
604C2×C4#41×D5911yescharacteristic
605C2×C4#41×D5911yescharacteristic
606C2×C4C2×#1174568711yescharacteristic
607C2×C4#1427377411yescharacteristic
608C2×C4#1427377411yescharacteristic
609C8—1182no—
610C8—1182no—
611C8—2361no—
612C8—2361no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22C2×#1427369911yescharacteristic
628C22#1683790911yescharacteristic
629C22#1683797311yescharacteristic
630C22C2×#1427375311yescharacteristic
631C22#1683820011yescharacteristic
632C22#1683820011yescharacteristic
633C22#1683820011yescharacteristic
634C4—22no—
635C4—22no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—4721no—
643C4—4721no—
644C4—2362no—
645C4—2362no—
646C4#1683805011yescharacteristic
647C4#1683805011yescharacteristic
648C4#1683808911yescharacteristic
649C4#1683808911yescharacteristic
650C2—42no—
651C2—161no—
652C2#2142750411yescharacteristic
653C2#2143370211yescharacteristic
654C2#2143371111yescharacteristic
655C2#2143419911yescharacteristic
656C2#2143424111yescharacteristic
657C2#2143529011yescharacteristic
658C2#2143533211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2854C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2854D59
hypocenterC59#24610
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
2. C2×#16837459C2semidirect
3. #21426579C2semidirect
4. #21425597C2semidirect
5. #21425587C2semidirect
6. #21425484C2semidirect
7. #21425470C2semidirect
8. #21424836C2semidirect
9. #21424831C2semidirect
10. #21424796C2semidirect
11. #21424796C2semidirect
13. #21424293C2semidirect
14. #21424188C2semidirect
390. C59#24610semidirect

Maximal subgroups

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

subgroupindex
C2×#168377232
C2×#168374592
#214265792
#214255972
#214255872
#214254842
#214254702
#214248362
#214248312
#214247962
#214244192
#214242932
#214241882
C59×#28542
#2461059