Finite Group Databasebeta

← #28596259

Subgroups — #28596259

Order 15104.

1000 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837690C211yesmaximal, characteristic
2#21426667C211yesmaximal, characteristic, complemented
3C4×D4×Dic59C211yesmaximal, characteristic
4#21425042C211yesmaximal, characteristic, complemented
5#21424856C211yesmaximal, characteristic, complemented
6#21424682C211yesmaximal, characteristic, complemented
7#21424678C211yesmaximal, characteristic, complemented
8#21424357C211yesmaximal, characteristic
9#21424094C211yesmaximal, characteristic
10#106×Dic59C211yesmaximal, characteristic, complemented
11#21423984C211yesmaximal, characteristic, complemented
12#21423944C211yesmaximal, characteristic, complemented
13#102×Dic59C211yesmaximal, characteristic
14#102×Dic59C211yesmaximal, characteristic
15C59×#1864C211yesmaximal, characteristic
16C2×#14273726C2211yescharacteristic
17C2×#14273692—21no—
18C2×D4×Dic59C2211yescharacteristic
19C2×D4×Dic59C2211yescharacteristic
20C22×C4×Dic59—21no—
21#16837690C418yescomplemented
22C2×#14273659C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×Dic59⋊C8C2211yescharacteristic
26C2×Dic59⋊C8C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273648C2211yescharacteristic
29C2×C8×Dic59C2211yescharacteristic
30C2×C8×Dic59C2211yescharacteristic
31#16837306—21no—
32#16837300C2211yescharacteristic
33#16837297—21no—
34#16837271C2211yescharacteristic
35C4⋊C4×Dic59C2211yescharacteristic
36C4⋊C4×Dic59C2211yescharacteristic
37C4⋊C4×Dic59C2211yescharacteristic
38C4⋊C4×Dic59C2211yescharacteristic
39C4×C4⋊Dic59C2211yescharacteristic
40#16837253—21no—
41C22⋊C4×Dic59—21no—
42C22⋊C4×Dic59—21no—
43C4×C22⋊Dic59—21no—
44C42×Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837157C2211yescharacteristic
53C2×C4×D4×C59C2211yescharacteristic
54C2×#106×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C2×#102×C59C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×Dic118—21no—
61C23×Dic59—21no—
62C23×Dic59—21no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273726—22no—
66#14273726C2×C414yescomplemented
67#14273692—44no—
68D4×Dic59—22no—
69D4×Dic59—22no—
70D4×Dic59C2×C414yes—
71D4×Dic59Q814yescomplemented
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×C22⋊Dic59—21no—
81C2×C22⋊Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—44no—
88C2×C4×Dic59D411yescharacteristic
89C2×C4×Dic59D411yescharacteristic
90C2×C4×Dic59D411yescharacteristic
91C2×C4×Dic59D411yescharacteristic
92C2×C4×Dic59C2311yescharacteristic
93C2×C4×Dic59C2311yescharacteristic
94#14273659C2×C414yescomplemented
95#14273658—22no—
96#14273658C2×C414yescomplemented
97Dic59⋊C8—22no—
98Dic59⋊C8C2×C414yescomplemented
99#14273648—22no—
100#14273648C2×C414yes—
101C8×Dic59—22no—
102C8×Dic59—22no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C22×D4×C59C2311yescharacteristic
110C23×C4×C59—21no—
111C22×C8×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C4×D4×C59—22no—
114C4×D4×C59—22no—
115C2×C4⋊C4×C59—21no—
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C22⋊C4×C59—21no—
120C2×C42×C59C2311yescharacteristic
121#106×C59—22no—
122#102×C59—22no—
123#102×C59C2×C414yes—
124#95×C59—21no—
125C24×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59C22×C412yes—
130C2×D4×C59C2×Q812yes—
131C2×D4×C59#4112yes—
132C22×C4×C59—24no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—42no—
137C22×C4×C59—42no—
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2411yescharacteristic
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—24no—
145C22×Dic59—24no—
146C22×Dic59—21no—
147C22×Dic59—42no—
148C22×Dic59—42no—
149C22×Dic59—42no—
150C22×Dic59—42no—
151C22×Dic59C2×D411yescharacteristic
152C22×Dic59C2×D411yescharacteristic
153C22×Dic59C2×D411yescharacteristic
154C22×Dic59C2×D411yescharacteristic
155C2×Dic118—24no—
156C2×Dic118—42no—
157C2×Dic118—42no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59C22×C412yes—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—42no—
169C4⋊C4×C59C22×C412yes—
170C4⋊C4×C59#4112yes—
171C22⋊C4×C59—24no—
172C42×C59—21no—
173C42×C59—21no—
174C22⋊Dic59—24no—
175C22⋊Dic59—24no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59C22×C412yes—
182C4⋊Dic59C22×C412yes—
183C4⋊Dic59C2×Q812yes—
184Dic59⋊C4—24no—
185Dic59⋊C4—42no—
186Dic59⋊C4—42no—
187Dic59⋊C4—42no—
188C4×Dic59—22no—
189C4×Dic59—22no—
190C4×Dic59—41no—
191C4×Dic59—42no—
192C4×Dic59—42no—
193C4×Dic59—42no—
194C4×Dic59—42no—
195C4×Dic59—42no—
196C4×Dic59SD1614yes—
197C4×Dic59C22×C412yes—
198C4×Dic59C2×Q812yes—
199C4×Dic59#4112yes—
200C4×Dic59#4112yes—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8C22×C412yes—
207C23×C59—22no—
208C23×C59—22no—
209C23×C59—22no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59—41no—
213C23×C59—41no—
214C23×C59C22×D411yescharacteristic
215D4×C59—22no—
216D4×C59—22no—
217D4×C59—22no—
218D4×C59C4×Q814yescomplemented
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—22no—
222C2×C4×C59—21no—
223C2×C4×C59—22no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—48no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59C2×SD1612yes—
235C2×C4×C59C23×C411yescharacteristic
236C2×C4×C59C22×Q811yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—22no—
246C2×Dic59—22no—
247C2×Dic59—22no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—48no—
255C2×Dic59—48no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59C4×D412yes—
260C2×Dic59C4×D412yes—
261C2×Dic59C22⋊Q812yes—
262C2×Dic59C22⋊Q812yes—
263C2×Dic59C2×SD1612yes—
264C2×Dic59C2×SD1612yes—
265C2×Dic59#13512yes—
266C2×Dic59AGL(1,ℤ/8)12yes—
267Dic118—48no—
268C8×C59—22no—
269C8×C59—41no—
270C59⋊C8—22no—
271C59⋊C8—41no—
272C4×#476—591nomaximal, Sylow
273C22×C59—24no—
274C22×C59—42no—
275C22×C59—42no—
276C22×C59—42no—
277C22×C59—42no—
278C22×C59—42no—
279C22×C59—42no—
280C22×C59C2×C4×D411yescharacteristic
281C22×C59C2×C22⋊Q811yescharacteristic
282C22×C59#52511yescharacteristic
283C22×C59C22×SD1611yescharacteristic
284C22×C59C2×#13511yescharacteristic
285C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C59#57511yescharacteristic
287C4×C59—22no—
288C4×C59—42no—
289C4×C59—41no—
290C4×C59—42no—
291C4×C59—81no—
292C4×C59C2×C4×Q811yescharacteristic
293C4×C59C4×#4111yescharacteristic
294C4×C59#52911yescharacteristic
295C4×C59#53311yescharacteristic
296Dic59—22no—
297Dic59—22no—
298Dic59—42no—
299Dic59—42no—
300Dic59—42no—
301Dic59—81no—
302Dic59C4×SD1614yes—
303Dic59#47614yescomplemented
304C2×#476—591no—
305C4×C4⋊Q8—591no—
306C42×D4—591no—
307#1987—591no—
308#1948—591no—
309#1920—591no—
310#1920—591no—
311#1864—591no—
312#1864D5911yescharacteristic
313#1802—591no—
314C4×#106—591no—
315C4×#106—591no—
316C4×C4⋊C8—591no—
317C4×#102—591no—
318C4×#102—591no—
319C2×C59—44no—
320C2×C59C4×C22⋊Q811yescharacteristic
321C2×C59C2×C4×SD1611yescharacteristic
322C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
323C2×C59C2×#47611yescharacteristic
324C2×C59#307611yescharacteristic
325C2×C59#310111yescharacteristic
326C2×C59#310611yescharacteristic
327C2×C4⋊Q8—591no—
328C2×C4×Q8—1181no—
329C2×C4×D4—591no—
330C2×C4×D4—591no—
331C2×C4×D4C2×D5911yescharacteristic
332C22×C42—1181no—
333#476—598no—
334C2×#106—591no—
335C2×#106—591no—
336C2×#106—591no—
337C2×#106C2×D5911yescharacteristic
338C2×C4⋊C8—591no—
339C2×C4⋊C8—591no—
340C2×#102—591no—
341C2×#102—591no—
342C2×#102C2×D5911yescharacteristic
343C2×#102C2×D5911yescharacteristic
344C2×C4×C8—591no—
345C2×C4×C8—591no—
346#391—1181no—
347#389—591no—
348#389—1181no—
349#389C2×D5911yescharacteristic
350#384—591no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—591no—
353C4×C4⋊C4—591no—
354C4×C4⋊C4—591no—
355C4×C4⋊C4—591no—
356C4×C4⋊C4—1181no—
357C4×C22⋊C4—1181no—
358C4×C22⋊C4—1181no—
359C4×C22⋊C4—1181no—
360C43—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340C2×D5911yescharacteristic
369#336—591no—
370#336C2×D5911yescharacteristic
371C59C4×#47611yescharacteristic, complemented, Sylow
372C22×Q8—1181no—
373C22×D4C22×D5911yescharacteristic
374C23×C4—21no—
375C23×C4—1181no—
376C23×C4—1181no—
377C22×C8—591no—
378C22×C8—591no—
379C22×C8C22×D5911yescharacteristic
380C22×C8C22×D5911yescharacteristic
381C4⋊Q8—594no—
382C4⋊Q8—1182no—
383C4×Q8—2364no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—594no—
387C4×D4—594no—
388C4×D4—1182no—
389C4×D4—1182no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4C22×D5911yescharacteristic
400C2×C4⋊C4C22×D5911yescharacteristic
401C2×C4⋊C4C22×D5911yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—1181no—
404C2×C22⋊C4—1181no—
405C2×C42—591no—
406C2×C42—591no—
407C2×C42—591no—
408C2×C42—591no—
409C2×C42—591no—
410C2×C42—591no—
411C2×C42—1181no—
412C2×C42—1181no—
413C2×C42—1181no—
414C2×C42—1181no—
415C2×C42—1181no—
416C2×C42—2364no—
417C2×C42C22×D5911yescharacteristic
418#106—22no—
419#106—594no—
420#106—594no—
421#106—1182no—
422C4⋊C8—594no—
423C4⋊C8—1182no—
424#102—22no—
425#102—594no—
426#102—1182no—
427#102C4×D5914yes—
428C4×C8—1182no—
429C4×C8—1182no—
430#95—21no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437C24—21no—
438C2×Q8—1184no—
439C2×Q8—2362no—
440C2×Q8—2362no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4C2×C4×D5912yes—
445C2×D4#1174568712yes—
446C2×D4Q8×D5912yes—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—591no—
454C22×C4—591no—
455C22×C4—591no—
456C22×C4—591no—
457C22×C4—1184no—
458C22×C4—1184no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4—1181no—
463C22×C4—2362no—
464C22×C4—2362no—
465C22×C4—2362no—
466C22×C4—2362no—
467C22×C4D4×D5911yescharacteristic
468C22×C4D4×D5911yescharacteristic
469C22×C4C23×D5911yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—592no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8C2×C4×D5912yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—592no—
488C4⋊C4—592no—
489C4⋊C4—592no—
490C4⋊C4—1184no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—2362no—
497C4⋊C4—2362no—
498C4⋊C4—2362no—
499C4⋊C4C2×C4×D5912yes—
500C4⋊C4#1174568712yes—
501C22⋊C4—24no—
502C22⋊C4—1184no—
503C22⋊C4—1184no—
504C42—21no—
505C42—21no—
506C42—594no—
507C42—592no—
508C42—592no—
509C42—592no—
510C42—592no—
511C42—1182no—
512C42—1182no—
513C42—2361no—
514C42—2362no—
515C42—2362no—
516C42—2362no—
517C42—2362no—
518C42—2362no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C2×D4×D5911yescharacteristic
527Q8—2368no—
528D4—22no—
529D4—22no—
530D4—22no—
531D4#1427369214yescomplemented
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—48no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—592no—
548C2×C4—592no—
549C2×C4—592no—
550C2×C4—592no—
551C2×C4—592no—
552C2×C4—592no—
553C2×C4—592no—
554C2×C4—592no—
555C2×C4—1181no—
556C2×C4—1181no—
557C2×C4—1181no—
558C2×C4—1181no—
559C2×C4—1182no—
560C2×C4—1182no—
561C2×C4—1182no—
562C2×C4—2368no—
563C2×C4—2368no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2361no—
573C2×C4SD16×D5912yes—
574C2×C4C22×C4×D5911yescharacteristic
575C2×C4C2×Q8×D5911yescharacteristic
576C2×C4#41×D5911yescharacteristic
577C2×C4#41×D5911yescharacteristic
578C2×C4C2×#1174568711yescharacteristic
579C2×C4C2×#1174568711yescharacteristic
580C8—22no—
581C8—41no—
582C8—1182no—
583C8—2361no—
584C22—24no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22C4×D4×D5911yescharacteristic
592C22C22⋊Q8×D5911yescharacteristic
593C22#1683790811yescharacteristic
594C22C2×SD16×D5911yescharacteristic
595C22#135×D5911yescharacteristic
596C22AGL(1,ℤ/8)×D5911yescharacteristic
597C22#1683819811yescharacteristic
598C4—22no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—81no—
603C4—594no—
604C4—594no—
605C4—1182no—
606C4—1182no—
607C4—4721no—
608C4—2362no—
609C4—2362no—
610C4—2362no—
611C4C2×#1427369211yescharacteristic
612C4C4×#1174568711yescharacteristic
613C4#1683792911yescharacteristic
614C4#1683795411yescharacteristic
615C2—44no—
616C2#2142662811yescharacteristic
617C2C4×SD16×D5911yescharacteristic
618C2#2143290711yescharacteristic
619C2#476×D5911yescharacteristic
620C2#2143388111yescharacteristic
621C2#2143409911yescharacteristic
622C2#2143414911yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1864C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1864D59
hypocenterC59C4×#476
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #21426667C2semidirect
4. #21425042C2semidirect
5. #21424856C2semidirect
6. #21424682C2semidirect
7. #21424678C2semidirect
10. #106×Dic59C2semidirect
11. #21423984C2semidirect
12. #21423944C2semidirect
21. #16837690C4semidirect
66. #14273726C2×C4semidirect
71. D4×Dic59Q8semidirect
94. #14273659C2×C4semidirect
96. #14273658C2×C4semidirect
98. Dic59⋊C8C2×C4semidirect
218. D4×C59C4×Q8semidirect
303. Dic59#476semidirect
371. C59C4×#476semidirect
531. D4#14273692semidirect

Maximal subgroups

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

subgroupindex
C2×#168376902
#214266672
C4×D4×Dic592
#214250422
#214248562
#214246822
#214246782
#214243572
#214240942
#106×Dic592
#214239842
#214239442
#102×Dic592
C59×#18642
C4×#47659