Finite Group Databasebeta

← #28554626

Subgroups — #28554626

Order 15104.

1174 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837881C211yesmaximal, characteristic
2#21433180C212yesmaximal, complemented
3#21422437C211yesmaximal, characteristic, complemented
4#21422435C212yesmaximal, complemented
5#21422428C212yesmaximal, complemented
6#21422427C212yesmaximal, complemented
7#21422415C212yesmaximal, complemented
8#21422407C212yesmaximal, complemented
9C59×#1466C211yesmaximal, characteristic, complemented
10C22×#11745687C411yescharacteristic
11#16837881—24no—
12#16837881—24no—
13C2×C4×Dic118C2211yescharacteristic
14C2×#14273692C411yescharacteristic
15C2×D4×Dic59C411yescharacteristic
16C2×C4×C22⋊D59C411yescharacteristic
17C2×C4×C22⋊D59C411yescharacteristic
18C2×(C22⋊D59)⋊C4C2211yescharacteristic
19C2×(C22⋊D59)⋊C4C2211yescharacteristic
20C2×#14273684C2211yescharacteristic
21C2×#14273683C2211yescharacteristic
22C2×#14273683C2211yescharacteristic
23C22×Dic59⋊C4C411yescharacteristic
24C22×Dic59⋊C4C411yescharacteristic
25C2×C4⋊C4×D59C411yescharacteristic
26#16837531—24no—
27#16837530—24no—
28#16837459C2212yes—
29#16837456C2214yes—
30#16837452C2214yes—
31#16837127C2214yes—
32#16837126C2212yes—
33#16837126C2212yes—
34#16837125C2214yes—
35C2×C4×D4×C59C2211yescharacteristic
36C59×#432C2212yes—
37C59×#323C2212yes—
38C59×#323C2212yes—
39C2×#11745687—24no—
40C2×#11745687—24no—
41C2×#11745687—24no—
42C22×Dic118C2×C411yescharacteristic
43C22×C22⋊D59C2×C411yescharacteristic
44C22×C22⋊D59C2×C411yescharacteristic
45C23×Dic59C2×C411yescharacteristic
46C23×Dic59C2×C411yescharacteristic
47C23×Dic59C2×C411yescharacteristic
48C23×Dic59C2×C411yescharacteristic
49C22×C4×D59C2×C411yescharacteristic
50C22×C4×D59C2×C411yescharacteristic
51C22×C4×D59C2×C411yescharacteristic
52C4×Dic118D414yes—
53C4×Dic118D414yes—
54#14273692—24no—
55D4×Dic59—24no—
56C4×C22⋊D59—22no—
57C4×C22⋊D59—22no—
58C4×C22⋊D59—22no—
59C4×C22⋊D59—22no—
60(C22⋊D59)⋊C4—24no—
61(C22⋊D59)⋊C4—24no—
62#14273684—22no—
63#14273684—22no—
64#14273683—24no—
65#14273683—24no—
66C2×C4⋊Dic59C2×C411yescharacteristic
67C2×Dic59⋊C4—22no—
68C2×Dic59⋊C4—22no—
69C2×Dic59⋊C4—22no—
70C2×Dic59⋊C4—22no—
71C2×Dic59⋊C4C2×C411yescharacteristic
72C2×Dic59⋊C4C2×C411yescharacteristic
73C2×Dic59⋊C4C2×C411yescharacteristic
74C2×Dic59⋊C4C2×C411yescharacteristic
75C2×Dic59⋊C4C2×C411yescharacteristic
76C2×Dic59⋊C4C2×C411yescharacteristic
77C2×Dic59⋊C4C2×C411yescharacteristic
78C2×Dic59⋊C4C2311yescharacteristic
79C4⋊C4×D59—24no—
80C2×C22⋊Dic59C2×C411yescharacteristic
81C2×C22⋊Dic59C2×C411yescharacteristic
82C2×(C2×D59)⋊C4C2×C411yescharacteristic
83C2×(C2×D59)⋊C4C2311yescharacteristic
84C2×C4×Dic59C2×C411yescharacteristic
85C2×C4×Dic59C2×C411yescharacteristic
86C2×C4×Dic59C2×C411yescharacteristic
87C2×C4×Dic59C2311yescharacteristic
88C2×C4×Dic59C2311yescharacteristic
89C4⋊(C59⋊C8)—22no—
90Dic59⋊C8—24no—
91Dic59⋊C8—24no—
92C22⋊(C59⋊C8)C2314yes—
93#14273638—22no—
94#14273637—24no—
95C8×Dic59—24no—
96C22×D4×C59C2×C411yescharacteristic
97C23×C4×C59C2×C411yescharacteristic
98C23×C4×C59C2×C411yescharacteristic
99C4×D4×C59—22no—
100C4×D4×C59—22no—
101C2×C4⋊C4×C59C2×C411yescharacteristic
102C2×C22⋊C4×C59C2311yescharacteristic
103C2×C22⋊C4×C59C2311yescharacteristic
104C2×C42×C59C2311yescharacteristic
105C4⋊C8×C59—22no—
106C22⋊C8×C59C2314yes—
107#97×C59—22no—
108C24×C59C22×C411yescharacteristic
109C24×C59C22×C411yescharacteristic
110C23×D59C22×C411yescharacteristic
111C2×D4×C59—22no—
112C2×D4×C59—22no—
113C2×D4×C59—22no—
114C22×C4×C59—21no—
115C22×C4×C59—21no—
116C22×C4×C59—21no—
117C22×C4×C59—21no—
118C22×C4×C59—81no—
119C22×C4×C59—81no—
120C22×C4×C59C22×C411yescharacteristic
121C22×C4×C59C22×C411yescharacteristic
122C22×C4×C59C22×C411yescharacteristic
123C22×C4×C59C22×C411yescharacteristic
124C22×C4×C59C2411yescharacteristic
125C2×C22⋊D59—22no—
126C2×C22⋊D59—22no—
127C2×C22⋊D59—22no—
128C2×C22⋊D59—22no—
129C2×C22⋊D59—22no—
130C2×C22⋊D59M1614yes—
131C22×Dic59—22no—
132C22×Dic59—22no—
133C22×Dic59—22no—
134C22×Dic59—22no—
135C22×Dic59—22no—
136C22×Dic59—22no—
137C22×Dic59—22no—
138C22×Dic59—81no—
139C22×Dic59—81no—
140C22×Dic59M1614yes—
141C22×Dic59C22×C411yescharacteristic
142C22×Dic59C22×C411yescharacteristic
143C22×Dic59C22×C411yescharacteristic
144C22×Dic59C22×C411yescharacteristic
145C22×Dic59C22×C411yescharacteristic
146C22×Dic59C22×C411yescharacteristic
147C22×Dic59C22×C411yescharacteristic
148#11745687—232no—
149C2×C4×D59—22no—
150C2×C4×D59—22no—
151C2×C4×D59—22no—
152C2×C4×D59—22no—
153C2×C4×D59—22no—
154C2×C4×D59—81no—
155C2×C4×D59—81no—
156C2×Dic118—22no—
157C2×Dic118—22no—
158C2×Dic118C22⋊C414yes—
159C2×C8×C59—24no—
160C4⋊C4×C59—22no—
161C22⋊C4×C59—22no—
162C22⋊C4×C59—22no—
163C42×C59C2×D412yes—
164C42×C59C2×D412yes—
165C22⋊Dic59—22no—
166C22⋊Dic59—22no—
167(C2×D59)⋊C4—21no—
168(C2×D59)⋊C4—21no—
169(C2×D59)⋊C4—22no—
170C4⋊Dic59C22⋊C414yes—
171Dic59⋊C4—21no—
172Dic59⋊C4—21no—
173Dic59⋊C4—22no—
174Dic59⋊C4—22no—
175Dic59⋊C4—22no—
176Dic59⋊C4—22no—
177Dic59⋊C4—22no—
178Dic59⋊C4C22⋊C412yes—
179Dic59⋊C4C22⋊C412yes—
180Dic59⋊C4C22⋊C412yes—
181Dic59⋊C4C22⋊C412yes—
182C4×Dic59—22no—
183C4×Dic59—22no—
184C4×Dic59—22no—
185C4×Dic59C2×D414yes—
186C4×Dic59C2×D414yes—
187C2×C59⋊C8—24no—
188C23×C59—21no—
189C23×C59—21no—
190C23×C59—21no—
191C23×C59—21no—
192C23×C59—41no—
193C23×C59—41no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59C2×M1612yes—
197C23×C59C2×M1612yes—
198C23×C59C23×C411yescharacteristic
199C22×D59—21no—
200C22×D59—21no—
201C22×D59—41no—
202C22×D59—41no—
203C22×D59C2×M1612yes—
204D4×C59—28no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—41no—
210C2×C4×C59—41no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
213C2×C4×C59—81no—
214C2×C4×C59—81no—
215C2×C4×C59—81no—
216C2×C4×C59—81no—
217C2×C4×C59C2×C22⋊C412yes—
218C2×C4×C59C2×C22⋊C411yescharacteristic
219C2×C4×C59C2×C22⋊C411yescharacteristic
220C2×C4×C59C22×D412yes—
221C22⋊D59—28no—
222C22⋊D59—28no—
223C2×Dic59—28no—
224C2×Dic59—21no—
225C2×Dic59—21no—
226C2×Dic59—21no—
227C2×Dic59—21no—
228C2×Dic59—28no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—41no—
232C2×Dic59—41no—
233C2×Dic59—41no—
234C2×Dic59—41no—
235C2×Dic59—82no—
236C2×Dic59—82no—
237C2×Dic59C2×C22⋊C412yes—
238C2×Dic59C2×C22⋊C412yes—
239C2×Dic59C2×C22⋊C412yes—
240C2×Dic59C2×C22⋊C412yes—
241C2×Dic59C2×M1612yes—
242C2×Dic59C2×M1612yes—
243C2×Dic59C2×M1612yes—
244C4×D59—28no—
245C4×D59—82no—
246C4×D59—82no—
247Dic118—28no—
248C8×C59—44no—
249C59⋊C8—44no—
250#17372—591nomaximal, Sylow
251C22×C59—21no—
252C22×C59—21no—
253C22×C59—22no—
254C22×C59—22no—
255C22×C59—42no—
256C22×C59—42no—
257C22×C59—42no—
258C22×C59—42no—
259C22×C59—41no—
260C22×C59—41no—
261C22×C59—41no—
262C22×C59—41no—
263C22×C59C22×C22⋊C411yescharacteristic
264C22×C59C22×M1611yescharacteristic
265C22×C59C22×M1611yescharacteristic
266C2×D59—22no—
267C2×D59—42no—
268C2×D59—41no—
269C2×D59—41no—
270C2×D59—42no—
271C4×C59—22no—
272C4×C59—22no—
273C4×C59—21no—
274C4×C59—21no—
275C4×C59—41no—
276C4×C59—41no—
277Dic59—22no—
278Dic59—22no—
279Dic59—22no—
280Dic59—22no—
281Dic59—41no—
282Dic59—41no—
283Dic59#40714yes—
284C2×#520—591no—
285#2983—592no—
286#1466—592no—
287#1466—591no—
288#1466D5911yescharacteristic, complemented
289#1464—592no—
290#1464—592no—
291#1460—592no—
292#1458—592no—
293C2×C59—21no—
294C2×C59—21no—
295C2×C59—42no—
296C2×C59—42no—
297C2×C59C2×#40711yescharacteristic
298C2×C59#288411yescharacteristic
299C2×C59#288811yescharacteristic
300D59—42no—
301C22×#41—591no—
302#520—1184no—
303#520—1184no—
304C2×C4×Q8—591no—
305C2×C4×Q8—591no—
306C2×C4×D4—591no—
307C2×C4×D4—591no—
308C2×C4×D4—591no—
309C2×C4×D4—591no—
310C2×C4×D4—591no—
311C2×C4×D4C2×D5911yescharacteristic
312C2×#117—591no—
313C2×#117—591no—
314C2×#117—591no—
315C22×C4⋊C4—591no—
316C22×C4⋊C4—591no—
317C22×C4⋊C4—591no—
318#446—1184no—
319#446—1184no—
320#432—592no—
321#432—594no—
322#432C2×D5912yes—
323#431—594no—
324#324—594no—
325#323—592no—
326#323—592no—
327#323—594no—
328#323C2×D5912yes—
329#323C2×D5912yes—
330C59#1737211yescharacteristic, complemented, Sylow
331C2×#41—1184no—
332C2×#41—1184no—
333C2×#41—1184no—
334C22×Q8—591no—
335C22×D4—591no—
336C22×D4—591no—
337C22×D4C4×D5911yescharacteristic
338C23×C4—591no—
339C23×C4—591no—
340C23×C4—591no—
341C23×C4—591no—
342C23×C4—591no—
343C23×C4—591no—
344C23×C4—591no—
345C23×C4C4×D5911yescharacteristic
346C23×C4C4×D5911yescharacteristic
347C4×Q8—594no—
348C4×Q8—594no—
349C4×Q8—1184no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—1184no—
353C4×D4—1184no—
354C4×D4—1182no—
355C4×D4—1182no—
356C4×D4—1182no—
357C4×D4—1182no—
358C4×D4—1184no—
359#117—1184no—
360#117—1184no—
361#117—1182no—
362#117—1182no—
363C2×C4⋊C4—591no—
364C2×C4⋊C4—591no—
365C2×C4⋊C4—591no—
366C2×C4⋊C4—591no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—1184no—
373C2×C4⋊C4—1182no—
374C2×C4⋊C4—1182no—
375C2×C4⋊C4—1182no—
376C2×C4⋊C4—1182no—
377C2×C4⋊C4C4×D5911yescharacteristic
378C2×C22⋊C4—591no—
379C2×C22⋊C4—591no—
380C2×C22⋊C4—591no—
381C2×C22⋊C4—591no—
382C2×C22⋊C4C22×D5911yescharacteristic
383C2×C22⋊C4C22×D5911yescharacteristic
384C2×C42—591no—
385C2×C42—591no—
386C2×C42—591no—
387C2×C42—591no—
388C2×C42—591no—
389C2×C42C22×D5911yescharacteristic
390C4⋊C8—22no—
391C4⋊C8—1184no—
392C4⋊C8—1184no—
393C4⋊C8—1182no—
394C22⋊C8—594no—
395C22⋊C8C22×D5914yes—
396#97—22no—
397#97—1184no—
398#97—1182no—
399C4×C8—1184no—
400C24—591no—
401C24C2×C4×D5911yescharacteristic
402C24C2×C4×D5911yescharacteristic
403#41—11832no—
404C2×Q8—594no—
405C2×Q8—1182no—
406C2×Q8—1182no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—594no—
411C2×D4—1182no—
412C2×D4—1182no—
413C2×D4—1182no—
414C2×D4—1182no—
415C2×D4—1182no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—591no—
423C22×C4—594no—
424C22×C4—591no—
425C22×C4—591no—
426C22×C4—591no—
427C22×C4—591no—
428C22×C4—591no—
429C22×C4—591no—
430C22×C4—1182no—
431C22×C4—1182no—
432C22×C4—1182no—
433C22×C4—1182no—
434C22×C4—1182no—
435C22×C4—1182no—
436C22×C4—1182no—
437C22×C4—1182no—
438C22×C4—1182no—
439C22×C4—1182no—
440C22×C4—1182no—
441C22×C4—1182no—
442C22×C4—4721no—
443C22×C4—4721no—
444C22×C4—4721no—
445C22×C4—4721no—
446C22×C4C2×C4×D5911yescharacteristic
447C22×C4C2×C4×D5911yescharacteristic
448C22×C4C2×C4×D5911yescharacteristic
449C22×C4C2×C4×D5911yescharacteristic
450C22×C4C23×D5911yescharacteristic
451C2×C8—24no—
452C2×C8—1184no—
453C4⋊C4—22no—
454C4⋊C4—592no—
455C4⋊C4—592no—
456C4⋊C4—592no—
457C4⋊C4—592no—
458C4⋊C4—594no—
459C4⋊C4—1182no—
460C4⋊C4—1182no—
461C4⋊C4—1182no—
462C4⋊C4—1182no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1182no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—1182no—
469C22⋊C4—1182no—
470C22⋊C4—1181no—
471C22⋊C4—1181no—
472C22⋊C4—1182no—
473C42—594no—
474C42—594no—
475C42—1182no—
476C42—1182no—
477C42—1182no—
478C42D4×D5912yes—
479C42D4×D5912yes—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—592no—
489C23—1181no—
490C23—1181no—
491C23—2361no—
492C23—2361no—
493C23M16×D5912yes—
494C23M16×D5912yes—
495C23C22×C4×D5911yescharacteristic
496Q8—1188no—
497D4—28no—
498D4—1188no—
499D4—1188no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—592no—
513C2×C4—592no—
514C2×C4—592no—
515C2×C4—592no—
516C2×C4—592no—
517C2×C4—592no—
518C2×C4—592no—
519C2×C4—1188no—
520C2×C4—1188no—
521C2×C4—1188no—
522C2×C4—1181no—
523C2×C4—1181no—
524C2×C4—1181no—
525C2×C4—1181no—
526C2×C4—1181no—
527C2×C4—1181no—
528C2×C4—4722no—
529C2×C4—4722no—
530C2×C4—4722no—
531C2×C4—4722no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4C22⋊C4×D5912yes—
537C2×C4C22⋊C4×D5911yescharacteristic
538C2×C4C22⋊C4×D5911yescharacteristic
539C2×C4C2×D4×D5912yes—
540C8—44no—
541C8—2364no—
542C22—21no—
543C22—21no—
544C22—22no—
545C22—22no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—1182no—
555C22—2362no—
556C22—2362no—
557C22—2361no—
558C22—2361no—
559C22C2×C22⋊C4×D5911yescharacteristic
560C22C2×M16×D5911yescharacteristic
561C22C2×M16×D5911yescharacteristic
562C4—22no—
563C4—22no—
564C4—21no—
565C4—21no—
566C4—41no—
567C4—41no—
568C4—594no—
569C4—1182no—
570C4—1182no—
571C4—1182no—
572C4—1182no—
573C4—2361no—
574C4—2361no—
575C2—21no—
576C2—21no—
577C2—42no—
578C2—42no—
579C2—2362no—
580C2#407×D5911yescharacteristic
581C2#2143255811yescharacteristic
582C2#2143257211yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D59
commutatorC23×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1466C2
socleC22×C59C22×C22⋊C4
prime radicalC22×#11745687C4
cosocleC22×C4×C59C24
hypercenter#1466D59
hypocenterC59#17372
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
2. #21433180C2semidirect
3. #21422437C2semidirect
4. #21422435C2semidirect
5. #21422428C2semidirect
6. #21422427C2semidirect
7. #21422415C2semidirect
8. #21422407C2semidirect
9. C59×#1466C2semidirect
288. #1466D59semidirect
330. C59#17372semidirect

Maximal subgroups

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

subgroupindex
C2×#168378812
#214331802
#214224372
#214224352
#214224282
#214224272
#214224152
#214224072
C59×#14662
#1737259