Finite Group Databasebeta

← #28588172

Subgroups — #28588172

Order 15104.

1450 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D59C211yesmaximal, characteristic
2C2×#16837424C211yesmaximal, characteristic
3C2×#16837409C212yesmaximal
4#21426501C211yesmaximal, characteristic
5#21424110C212yesmaximal, complemented
6#21424109C212yesmaximal, complemented
7#21424102C211yesmaximal, characteristic, complemented
8#21424101C211yesmaximal, characteristic, complemented
9#21424098C211yesmaximal, characteristic, complemented
10#21423814C212yesmaximal
11C59×#1803C211yesmaximal, characteristic, complemented
12C22×Q8×D59C2211yescharacteristic
13C22×#11745659C2212yes—
14C2×Q8×Dic59C2211yescharacteristic
15C2×C4×Dic118—21no—
16C2×C4×Dic118C2211yescharacteristic
17C2×#14273692—21no—
18C2×#14273692C2211yescharacteristic
19C4×Q8×D59—24no—
20C4×Q8×D59C418yescomplemented
21C2×#14273685C2212yes—
22C2×C4⋊C4×D59—21no—
23C2×C4⋊C4×D59C2211yescharacteristic
24C2×C42×D59—21no—
25C2×C42×D59C2211yescharacteristic
26#16837424—24no—
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×Dic59⋊C8C2212yes—
29#16837409—28no—
30C2×#14273652C2212yes—
31C2×#14273651C2212yes—
32C2×#14273650C2212yes—
33#16837272C2211yescharacteristic
34#16837267C2212yes—
35#16837259—21no—
36#16837259—21no—
37C4⋊C4×Dic59—21no—
38C4⋊C4×Dic59—21no—
39#16837165C2212yes—
40#16837164C2212yes—
41#16837164C2212yes—
42C2×C4×Q8×C59C2211yescharacteristic
43C2×C4⋊C8×C59C2211yescharacteristic
44C2×#103×C59C2212yes—
45C59×#384C2211yescharacteristic
46C59×#340C2212yes—
47C22×Dic118—21no—
48C22×Dic118C2311yescharacteristic
49C2×Q8×D59—22no—
50C2×Q8×D59—22no—
51C2×Q8×D59—22no—
52C2×Q8×D59C2×C414yes—
53C2×Q8×D59D414yes—
54C2×Q8×D59Q814yes—
55C22×C4×D59—21no—
56C22×C4×D59—21no—
57C22×C4×D59D411yescharacteristic
58C22×C4×D59D411yescharacteristic
59C22×C4×D59C2311yescharacteristic
60C2×#11745659—24no—
61C2×#11745659—24no—
62C2×#11745659—24no—
63C22×C59⋊C8C2312yes—
64Q8×Dic59—22no—
65Q8×Dic59C2×C414yes—
66C4×Dic118—22no—
67C4×Dic118—28no—
68C4×Dic118C2×C414yes—
69#14273692—28no—
70#14273692—22no—
71#14273692C2×C414yes—
72#14273685—24no—
73#14273685—24no—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59—21no—
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59C2312yes—
78C2×C4⋊Dic59C2311yescharacteristic
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C4⋊C4×D59—28no—
83C4⋊C4×D59—22no—
84C4⋊C4×D59C2×C414yes—
85C2×(C2×D59)⋊C4—22no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59—22no—
90C2×C4×Dic59D411yescharacteristic
91C2×C4×Dic59D411yescharacteristic
92C2×C4×Dic59C2311yescharacteristic
93C42×D59—28no—
94C42×D59C2×C414yes—
95C42×D59D414yes—
96C4⋊(C59⋊C8)—22no—
97Dic59⋊C8—24no—
98#14273652—24no—
99#14273651—24no—
100#14273650—24no—
101C59⋊#95—22no—
102C59⋊#95—22no—
103C22×Q8×C59C2311yescharacteristic
104C22×C8×C59C2312yes—
105C4×Q8×C59—22no—
106C4×Q8×C59C2×C414yes—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59—21no—
110C2×C4⋊C4×C59C2312yes—
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C42×C59—21no—
113C2×C42×C59C2311yescharacteristic
114C4⋊C8×C59—22no—
115#103×C59—24no—
116C23×D59C2×D411yescharacteristic
117C2×Q8×C59—21no—
118C2×Q8×C59—21no—
119C2×Q8×C59—21no—
120C2×Q8×C59C22×C412yes—
121C2×Q8×C59C2×D412yes—
122C2×Q8×C59C2×Q812yes—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—22no—
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2411yescharacteristic
129C22×Dic59—21no—
130C22×Dic59—21no—
131C22×Dic59—22no—
132C22×Dic59C2×D411yescharacteristic
133C22×Dic59C2×D411yescharacteristic
134C22×Dic59C2×D411yescharacteristic
135Q8×D59—28no—
136Q8×D59—28no—
137Q8×D59—28no—
138Q8×D59C4⋊C4116yescomplemented
139C2×C4×D59—24no—
140C2×C4×D59—24no—
141C2×C4×D59—24no—
142C2×C4×D59—22no—
143C2×C4×D59—22no—
144C2×C4×D59—24no—
145C2×C4×D59—41no—
146C2×C4×D59—41no—
147C2×C4×D59—42no—
148C2×C4×D59—42no—
149C2×C4×D59C22⋊C414yes—
150C2×C4×D59C22⋊C414yes—
151C2×C4×D59C22×C412yes—
152C2×C4×D59C2×D412yes—
153C2×C4×D59C2×D412yes—
154C2×C4×D59C2×Q812yes—
155C2×C4×D59#4112yes—
156C2×C4×D59#4112yes—
157C2×Dic118—21no—
158C2×Dic118—21no—
159C2×Dic118—21no—
160C2×Dic118—24no—
161C2×Dic118—24no—
162C2×Dic118—24no—
163C2×Dic118C22×C412yes—
164C2×Dic118C2×D412yes—
165C2×Dic118C2×Q812yes—
166C2×C8×C59—22no—
167C2×C8×C59—22no—
168C2×C8×C59—22no—
169C4⋊C4×C59—24no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—22no—
172C4⋊C4×C59—22no—
173C4⋊C4×C59—42no—
174C4⋊C4×C59—42no—
175C4⋊C4×C59C22×C412yes—
176C42×C59—24no—
177C42×C59C22×C412yes—
178C42×C59C2×D412yes—
179(C2×D59)⋊C4—44no—
180C4⋊Dic59—21no—
181C4⋊Dic59—24no—
182C4⋊Dic59—22no—
183C4⋊Dic59—22no—
184C4⋊Dic59—42no—
185C4⋊Dic59—42no—
186C4⋊Dic59C22×C412yes—
187Dic59⋊C4—24no—
188Dic59⋊C4—24no—
189Dic59⋊C4—24no—
190C4×Dic59—24no—
191C4×Dic59—24no—
192C4×Dic59—24no—
193C4×Dic59—44no—
194C4×Dic59C22⋊C414yes—
195C4×Dic59C22⋊C414yes—
196C4×Dic59C22×C412yes—
197C4×Dic59C2×D412yes—
198C2×C59⋊C8—22no—
199C2×C59⋊C8—22no—
200C2×C59⋊C8—22no—
201#11745659—48no—
202C23×C59C22×D411yescharacteristic
203C22×D59—21no—
204C22×D59—21no—
205C22×D59—21no—
206C22×D59—21no—
207C22×D59C2×C22⋊C412yes—
208C22×D59#12112yes—
209C22×D59C22⋊Q812yes—
210Q8×C59—22no—
211Q8×C59—22no—
212Q8×C59—22no—
213Q8×C59C2×C4⋊C414yes—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—22no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—42no—
223C2×C4×C59—42no—
224C2×C4×C59—42no—
225C2×C4×C59C2×C22⋊C412yes—
226C2×C4×C59C2×C22⋊C412yes—
227C2×C4×C59#13712yes—
228C2×C4×C59#13712yes—
229C2×C4×C59C23×C411yescharacteristic
230C2×C4×C59C22×D411yescharacteristic
231C2×C4×C59C22×D411yescharacteristic
232C2×C4×C59C22×Q811yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×Dic59—22no—
236C2×Dic59—22no—
237C2×Dic59—22no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—22no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—42no—
246C2×Dic59—42no—
247C2×Dic59—42no—
248C2×Dic59C2×C22⋊C412yes—
249C2×Dic59C2×C22⋊C412yes—
250C2×Dic59C2×C22⋊C412yes—
251C2×Dic59#12112yes—
252C2×Dic59C22⋊Q812yes—
253C4×D59—216no—
254C4×D59—28no—
255C4×D59—44no—
256C4×D59—48no—
257C4×D59C2×C4⋊C414yes—
258C4×D59#11714yes—
259C4×D59#12114yes—
260C4×D59C22⋊Q814yes—
261Dic118—22no—
262Dic118—22no—
263Dic118—22no—
264Dic118—216no—
265Dic118C2×C4⋊C414yes—
266C8×C59—42no—
267C59⋊C8—42no—
268#20647—591nomaximal, Sylow
269C22×C59C22×C22⋊C411yescharacteristic
270C22×C59C2×#12111yescharacteristic
271C22×C59C2×C22⋊Q811yescharacteristic
272C22×C59C2×#13711yescharacteristic
273C22×C59C2×#13711yescharacteristic
274C22×C59#57511yescharacteristic
275C22×C59#57511yescharacteristic
276C2×D59—22no—
277C2×D59—22no—
278C2×D59—22no—
279C2×D59—22no—
280C2×D59—22no—
281C2×D59—22no—
282C2×D59#38014yes—
283C4×C59—22no—
284C4×C59—24no—
285C4×C59—42no—
286C4×C59—82no—
287C4×C59#41914yescomplemented
288C4×C59C22×C4⋊C411yescharacteristic
289C4×C59C2×#11711yescharacteristic
290C4×C59C2×#12111yescharacteristic
291C4×C59C2×C22⋊Q811yescharacteristic
292Dic59—22no—
293Dic59—22no—
294Dic59—24no—
295Dic59—42no—
296Dic59—82no—
297Dic59#38014yes—
298C22×C4×Q8—591no—
299C2×#423—591no—
300C2×#419—592no—
301#2285—591no—
302#1805—592no—
303#1805—592no—
304#1803—591no—
305#1803—591no—
306#1803—591no—
307#1803D5911yescharacteristic, complemented
308#1736—592no—
309C2×C59C2×#38011yescharacteristic
310C2×C59C2×#41911yescharacteristic
311C2×C59#289611yescharacteristic
312C2×C59#303711yescharacteristic
313C2×C59#303711yescharacteristic
314C2×C59#307811yescharacteristic
315C2×C59#307811yescharacteristic
316D59—24no—
317C23×Q8—591no—
318C22×M16—592no—
319C2×C4×Q8—591no—
320C2×C4×Q8—598no—
321C2×C4×Q8—591no—
322C2×C4×Q8—591no—
323C2×C4×Q8—1181no—
324C2×C4×Q8—1181no—
325C2×C4×Q8—1184no—
326C2×C4×Q8C2×D5911yescharacteristic
327C2×#117—592no—
328C22×C4⋊C4—591no—
329C22×C4⋊C4—1181no—
330C22×C42—591no—
331C22×C42—1181no—
332#423—1184no—
333C2×C4⋊C8—592no—
334C2×C4⋊C8—591no—
335C2×C4⋊C8C2×D5911yescharacteristic
336#419—1188no—
337C2×#103—592no—
338C2×#103—592no—
339C2×#103—592no—
340C2×#103C2×D5912yes—
341#384—591no—
342#384C2×D5911yescharacteristic
343#382—592no—
344#380—1181no—
345#380—1181no—
346C4×C4⋊C4—1181no—
347C4×C4⋊C4—1181no—
348#340—592no—
349#340—592no—
350#340—592no—
351#340C2×D5912yes—
352C59#2064711yescharacteristic, complemented, Sylow
353C22×Q8—594no—
354C22×Q8—594no—
355C22×Q8—594no—
356C22×Q8—591no—
357C22×Q8—1181no—
358C22×Q8—1182no—
359C22×Q8—1182no—
360C22×Q8—1182no—
361C22×Q8C22×D5911yescharacteristic
362C23×C4—591no—
363C23×C4—591no—
364C23×C4—591no—
365C23×C4—1181no—
366C23×C4—1181no—
367C2×M16—1184no—
368C2×M16—1184no—
369C2×M16—1184no—
370C22×C8—592no—
371C22×C8C22×D5912yes—
372C4×Q8—22no—
373C4×Q8—594no—
374C4×Q8—594no—
375C4×Q8—594no—
376C4×Q8—1188no—
377C4×Q8—1188no—
378C4×Q8—1182no—
379C4×Q8—1182no—
380C4×Q8—1182no—
381C4×Q8C4×D5914yes—
382#117—1184no—
383#117—1184no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—594no—
389C2×C4⋊C4—592no—
390C2×C4⋊C4—1182no—
391C2×C4⋊C4—1188no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4C22×D5912yes—
399C2×C4⋊C4C22×D5911yescharacteristic
400C2×C22⋊C4—1182no—
401C2×C42—21no—
402C2×C42—591no—
403C2×C42—591no—
404C2×C42—591no—
405C2×C42—594no—
406C2×C42—594no—
407C2×C42—1188no—
408C2×C42—1181no—
409C2×C42—1181no—
410C2×C42—1181no—
411C2×C42—1182no—
412C2×C42C22×D5911yescharacteristic
413C4⋊C8—22no—
414C4⋊C8—1184no—
415C4⋊C8—1182no—
416#103—24no—
417#103—1184no—
418#103—1184no—
419#103—1184no—
420#95—1182no—
421#95—1182no—
422C24—591no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—5916no—
427C2×Q8—592no—
428C2×Q8—592no—
429C2×Q8—592no—
430C2×Q8—1184no—
431C2×Q8—1184no—
432C2×Q8—1184no—
433C2×Q8—1188no—
434C2×Q8—1188no—
435C2×Q8—1188no—
436C2×Q8—1181no—
437C2×Q8—1181no—
438C2×Q8—1181no—
439C2×Q8C2×C4×D5912yes—
440C2×Q8D4×D5912yes—
441C2×Q8Q8×D5912yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—591no—
446C22×C4—592no—
447C22×C4—592no—
448C22×C4—592no—
449C22×C4—592no—
450C22×C4—592no—
451C22×C4—592no—
452C22×C4—591no—
453C22×C4—591no—
454C22×C4—594no—
455C22×C4—594no—
456C22×C4—1184no—
457C22×C4—1184no—
458C22×C4—1184no—
459C22×C4—1184no—
460C22×C4—1181no—
461C22×C4—1182no—
462C22×C4—1182no—
463C22×C4—1181no—
464C22×C4—1182no—
465C22×C4—2361no—
466C22×C4—2361no—
467C22×C4—2362no—
468C22×C4—2362no—
469C22×C4D4×D5911yescharacteristic
470C22×C4D4×D5911yescharacteristic
471C22×C4C23×D5911yescharacteristic
472M16—2368no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—1182no—
477C2×C8—1182no—
478C2×C8—1182no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—592no—
486C4⋊C4—1184no—
487C4⋊C4—1184no—
488C4⋊C4—1184no—
489C4⋊C4—1184no—
490C4⋊C4—1181no—
491C4⋊C4—1182no—
492C4⋊C4—1182no—
493C4⋊C4—2362no—
494C4⋊C4—2362no—
495C4⋊C4C2×C4×D5912yes—
496C22⋊C4—2364no—
497C42—24no—
498C42—594no—
499C42—594no—
500C42—592no—
501C42—592no—
502C42—1184no—
503C42—1184no—
504C42—1184no—
505C42—2364no—
506C42C2×C4×D5912yes—
507C42D4×D5912yes—
508C23—592no—
509C23—592no—
510C23—592no—
511C23—1181no—
512C23—1181no—
513C23—1181no—
514C23—1181no—
515C23C2×D4×D5911yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—22no—
519Q8—594no—
520Q8—11816no—
521Q8—1182no—
522Q8—1182no—
523Q8—1182no—
524Q8C4⋊C4×D5914yes—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—592no—
537C2×C4—592no—
538C2×C4—592no—
539C2×C4—594no—
540C2×C4—594no—
541C2×C4—594no—
542C2×C4—594no—
543C2×C4—592no—
544C2×C4—592no—
545C2×C4—11816no—
546C2×C4—1182no—
547C2×C4—1182no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1182no—
553C2×C4—1188no—
554C2×C4—1182no—
555C2×C4—2364no—
556C2×C4—2368no—
557C2×C4—2362no—
558C2×C4—2362no—
559C2×C4—2362no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4C22⋊C4×D5912yes—
563C2×C4C22⋊C4×D5912yes—
564C2×C4#1427375612yes—
565C2×C4#1427375912yes—
566C2×C4C22×C4×D5911yescharacteristic
567C2×C4C2×D4×D5911yescharacteristic
568C2×C4C2×D4×D5911yescharacteristic
569C2×C4C2×Q8×D5911yescharacteristic
570C2×C4#41×D5911yescharacteristic
571C2×C4#41×D5911yescharacteristic
572C8—42no—
573C8—2362no—
574C22—594no—
575C22—1182no—
576C22—1182no—
577C22—1182no—
578C22—1182no—
579C22—1182no—
580C22—1182no—
581C22C2×C22⋊C4×D5911yescharacteristic
582C22#121×D5911yescharacteristic
583C22C22⋊Q8×D5911yescharacteristic
584C22C2×#1427375611yescharacteristic
585C22C2×#1427375911yescharacteristic
586C22#1683819811yescharacteristic
587C22#1683820111yescharacteristic
588C4—24no—
589C4—22no—
590C4—42no—
591C4—82no—
592C4—594no—
593C4—1182no—
594C4—1184no—
595C4—1182no—
596C4—4722no—
597C4—2362no—
598C4#1683740914yescomplemented
599C4C2×C4⋊C4×D5911yescharacteristic
600C4#117×D5911yescharacteristic
601C4#121×D5911yescharacteristic
602C4C22⋊Q8×D5911yescharacteristic
603C2—1184no—
604C2#380×D5911yescharacteristic
605C2C2×#1683740911yescharacteristic
606C2#2143261211yescharacteristic
607C2#2143356311yescharacteristic
608C2#2143356911yescharacteristic
609C2#2143389611yescharacteristic
610C2#2143390211yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1803C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenter#1803D59
hypocenterC59#20647
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
5. #21424110C2semidirect
6. #21424109C2semidirect
7. #21424102C2semidirect
8. #21424101C2semidirect
9. #21424098C2semidirect
11. C59×#1803C2semidirect
20. C4×Q8×D59C4semidirect
138. Q8×D59C4⋊C4semidirect
287. C4×C59#419semidirect
307. #1803D59semidirect
352. C59#20647semidirect
598. C4#16837409semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D592
C2×#168374242
C2×#168374092
#214265012
#214241102
#214241092
#214241022
#214241012
#214240982
#214238142
C59×#18032
#2064759