Finite Group Databasebeta

← #28554623

Subgroups — #28554623

Order 15104.

1174 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837882C211yesmaximal, characteristic
2#21433180C212yesmaximal, complemented
3#21422436C212yesmaximal, complemented
4#21422435C212yesmaximal, complemented
5#21422430C211yesmaximal, characteristic, complemented
6#21422429C212yesmaximal, complemented
7#21422415C212yesmaximal, complemented
8#21422409C212yesmaximal, complemented
9C59×#1464C211yesmaximal, characteristic, complemented
10C22×#11745685C411yescharacteristic
11#16837882—24no—
12#16837882—24no—
13C2×C4×Dic118C411yescharacteristic
14C2×C4×Dic118C2211yescharacteristic
15C2×C4×D236C411yescharacteristic
16C2×C4×D236C2211yescharacteristic
17C2×(C22⋊D59)⋊C4C411yescharacteristic
18C2×(C22⋊D59)⋊C4C411yescharacteristic
19C2×(C22⋊D59)⋊C4C2211yescharacteristic
20C2×(C22⋊D59)⋊C4C2211yescharacteristic
21C2×#14273685C2211yescharacteristic
22C2×#14273685C2211yescharacteristic
23C22×C4⋊Dic59C411yescharacteristic
24C2×C4⋊C4×D59C411yescharacteristic
25C2×C4⋊C4×D59C411yescharacteristic
26#16837531—24no—
27#16837530—24no—
28#16837459C2212yes—
29#16837456C2214yes—
30#16837452C2214yes—
31#16837128C2212yes—
32#16837126C2212yes—
33#16837125C2214yes—
34#16837125C2214yes—
35C2×#117×C59C2211yescharacteristic
36C59×#432C2212yes—
37C59×#324C2212yes—
38C59×#323C2212yes—
39C2×#11745685—24no—
40C2×#11745685—24no—
41C2×#11745685—22no—
42C2×#11745685—22no—
43C22×Dic118C2×C411yescharacteristic
44C22×D236C2×C411yescharacteristic
45C22×C22⋊D59C2×C411yescharacteristic
46C22×C22⋊D59C2×C411yescharacteristic
47C23×Dic59C2×C411yescharacteristic
48C23×Dic59C2×C411yescharacteristic
49C22×C4×D59C2×C411yescharacteristic
50C22×C4×D59C2×C411yescharacteristic
51C22×C4×D59C2×C411yescharacteristic
52C22×C4×D59C2×C411yescharacteristic
53C22×C4×D59C2×C411yescharacteristic
54C22×C4×D59C2×C411yescharacteristic
55C4×Dic118—22no—
56C4×Dic118—22no—
57C4×Dic118D414yes—
58C4×Dic118D414yes—
59C4×D236—22no—
60C4×D236—22no—
61C4×D236—22no—
62C4×D236—22no—
63(C22⋊D59)⋊C4—24no—
64(C22⋊D59)⋊C4—24no—
65(C22⋊D59)⋊C4—24no—
66(C22⋊D59)⋊C4—24no—
67#14273685—24no—
68#14273685—24no—
69C2×C4⋊Dic59—22no—
70C2×C4⋊Dic59—22no—
71C2×C4⋊Dic59C2×C411yescharacteristic
72C2×C4⋊Dic59C2×C411yescharacteristic
73C2×C4⋊Dic59C2×C411yescharacteristic
74C2×C4⋊Dic59C2×C411yescharacteristic
75C2×Dic59⋊C4C2×C411yescharacteristic
76C2×Dic59⋊C4C2×C411yescharacteristic
77C2×Dic59⋊C4C2×C411yescharacteristic
78C2×Dic59⋊C4C2×C411yescharacteristic
79C4⋊C4×D59—24no—
80C4⋊C4×D59—24no—
81C2×(C2×D59)⋊C4C2×C411yescharacteristic
82C2×(C2×D59)⋊C4C2×C411yescharacteristic
83C2×(C2×D59)⋊C4C2311yescharacteristic
84C2×(C2×D59)⋊C4C2311yescharacteristic
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—
96C23×C4×C59C2×C411yescharacteristic
97#117×C59—22no—
98#117×C59—22no—
99C2×C4⋊C4×C59C2×C411yescharacteristic
100C2×C4⋊C4×C59C2311yescharacteristic
101C2×C22⋊C4×C59C2×C411yescharacteristic
102C2×C22⋊C4×C59C2311yescharacteristic
103C2×C42×C59C2×C411yescharacteristic
104C2×C42×C59C2311yescharacteristic
105C4⋊C8×C59—22no—
106C22⋊C8×C59C2314yes—
107#97×C59—22no—
108C24×C59C22×C411yescharacteristic
109C23×D59C22×C411yescharacteristic
110C23×D59C22×C411yescharacteristic
111C22×C4×C59—22no—
112C22×C4×C59—21no—
113C22×C4×C59—21no—
114C22×C4×C59—22no—
115C22×C4×C59C22×C411yescharacteristic
116C22×C4×C59C22×C411yescharacteristic
117C22×C4×C59C22×C411yescharacteristic
118C22×C4×C59C22×C411yescharacteristic
119C22×C4×C59C22×C411yescharacteristic
120C22×C4×C59C2411yescharacteristic
121C2×C22⋊D59—22no—
122C2×C22⋊D59—22no—
123C2×C22⋊D59—22no—
124C2×C22⋊D59—22no—
125C2×C22⋊D59—22no—
126C2×C22⋊D59M1614yes—
127C22×Dic59—22no—
128C22×Dic59—22no—
129C22×Dic59—81no—
130C22×Dic59—81no—
131C22×Dic59C22×C411yescharacteristic
132C22×Dic59C22×C411yescharacteristic
133C22×Dic59C22×C411yescharacteristic
134C22×Dic59C22×C411yescharacteristic
135C22×Dic59C22×C411yescharacteristic
136C22×Dic59C22×C411yescharacteristic
137#11745685—216no—
138#11745685—216no—
139C2×D236—22no—
140C2×D236—22no—
141C2×D236—21no—
142C2×D236—21no—
143C2×C4×D59—22no—
144C2×C4×D59—22no—
145C2×C4×D59—22no—
146C2×C4×D59—22no—
147C2×C4×D59—22no—
148C2×C4×D59—22no—
149C2×C4×D59—22no—
150C2×C4×D59—22no—
151C2×C4×D59—22no—
152C2×C4×D59—81no—
153C2×C4×D59—81no—
154C2×C4×D59—81no—
155C2×C4×D59—81no—
156C2×C4×D59M1614yes—
157C2×Dic118—22no—
158C2×Dic118—22no—
159C2×Dic118C22⋊C412yes—
160C2×Dic118C22⋊C412yes—
161C2×C8×C59—24no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—22no—
164C22⋊C4×C59—22no—
165C22⋊C4×C59—22no—
166C42×C59—21no—
167C42×C59—21no—
168C42×C59C2×D412yes—
169C42×C59C2×D412yes—
170(C2×D59)⋊C4—22no—
171(C2×D59)⋊C4—22no—
172(C2×D59)⋊C4—22no—
173(C2×D59)⋊C4—22no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—22no—
177C4⋊Dic59—22no—
178C4⋊Dic59C22⋊C412yes—
179C4⋊Dic59C22⋊C412yes—
180Dic59⋊C4—22no—
181Dic59⋊C4—22no—
182Dic59⋊C4C22⋊C414yes—
183Dic59⋊C4C22⋊C414yes—
184C4×Dic59—22no—
185C4×Dic59—22no—
186C4×Dic59C2×D414yes—
187C4×Dic59C2×D414yes—
188C2×C59⋊C8—24no—
189C23×C59—21no—
190C23×C59—21no—
191C23×C59—41no—
192C23×C59—41no—
193C23×C59C2×M1612yes—
194C23×C59C23×C411yescharacteristic
195C22×D59—21no—
196C22×D59—21no—
197C22×D59—21no—
198C22×D59—21no—
199C22×D59—41no—
200C22×D59—41no—
201C22×D59—41no—
202C22×D59—41no—
203C22×D59C2×M1612yes—
204C22×D59C2×M1612yes—
205C2×C4×C59—24no—
206C2×C4×C59—24no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—41no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59C2×C22⋊C412yes—
218C2×C4×C59C2×C22⋊C411yescharacteristic
219C2×C4×C59C2×C22⋊C411yescharacteristic
220C2×C4×C59C2×M1612yes—
221C2×C4×C59C22×D412yes—
222C22⋊D59—28no—
223C22⋊D59—28no—
224C2×Dic59—21no—
225C2×Dic59—21no—
226C2×Dic59—21no—
227C2×Dic59—21no—
228C2×Dic59—41no—
229C2×Dic59—41no—
230C2×Dic59—41no—
231C2×Dic59—41no—
232C2×Dic59—82no—
233C2×Dic59—82no—
234C2×Dic59C2×C22⋊C412yes—
235C2×Dic59C2×C22⋊C412yes—
236C2×Dic59C2×C22⋊C412yes—
237C2×Dic59C2×C22⋊C412yes—
238C2×Dic59C2×M1612yes—
239C2×Dic59C2×M1612yes—
240D236—24no—
241D236—24no—
242C4×D59—28no—
243C4×D59—28no—
244C4×D59—82no—
245C4×D59—82no—
246C4×D59—82no—
247C4×D59—82no—
248Dic118—24no—
249Dic118—24no—
250C8×C59—44no—
251C59⋊C8—44no—
252#17372—591nomaximal, Sylow
253C22×C59—21no—
254C22×C59—21no—
255C22×C59—22no—
256C22×C59—42no—
257C22×C59—42no—
258C22×C59—41no—
259C22×C59—41no—
260C22×C59C22×C22⋊C411yescharacteristic
261C22×C59C22×M1611yescharacteristic
262C22×C59C22×M1611yescharacteristic
263C2×D59—22no—
264C2×D59—22no—
265C2×D59—42no—
266C2×D59—42no—
267C2×D59—41no—
268C2×D59—41no—
269C2×D59—41no—
270C2×D59—41no—
271C2×D59—42no—
272C2×D59—42no—
273C4×C59—21no—
274C4×C59—21no—
275C4×C59—22no—
276C4×C59—22no—
277C4×C59—22no—
278C4×C59—41no—
279C4×C59—41no—
280Dic59—22no—
281Dic59—22no—
282Dic59—22no—
283Dic59—41no—
284Dic59—41no—
285Dic59#40714yes—
286C2×#520—591no—
287#2983—592no—
288#1466—592no—
289#1466—592no—
290#1464—592no—
291#1464—591no—
292#1464D5911yescharacteristic, complemented
293#1460—592no—
294#1458—592no—
295C2×C59—21no—
296C2×C59—21no—
297C2×C59—42no—
298C2×C59C2×#40711yescharacteristic
299C2×C59#288411yescharacteristic
300C2×C59#288811yescharacteristic
301D59—42no—
302D59—42no—
303C22×#41—591no—
304#520—1184no—
305#520—1184no—
306C2×C4×Q8—591no—
307C2×C4×Q8—591no—
308C2×C4×D4—591no—
309C2×C4×D4—591no—
310C2×C4×D4—591no—
311C2×C4×D4—591no—
312C2×C4×D4—591no—
313C2×C4×D4—591no—
314C2×#117—591no—
315C2×#117—591no—
316C2×#117C2×D5911yescharacteristic
317C22×C4⋊C4—591no—
318C22×C4⋊C4—591no—
319C22×C4⋊C4—591no—
320#446—1184no—
321#446—1184no—
322#432—592no—
323#432—594no—
324#432C2×D5912yes—
325#431—594no—
326#324—592no—
327#324C2×D5912yes—
328#323—592no—
329#323—594no—
330#323—594no—
331#323C2×D5912yes—
332C59#1737211yescharacteristic, complemented, Sylow
333C2×#41—1184no—
334C2×#41—1184no—
335C2×#41—1182no—
336C2×#41—1182no—
337C22×Q8—591no—
338C22×D4—591no—
339C22×D4—591no—
340C22×D4—591no—
341C23×C4—591no—
342C23×C4—591no—
343C23×C4—591no—
344C23×C4—591no—
345C23×C4—591no—
346C23×C4—591no—
347C23×C4—591no—
348C23×C4—591no—
349C23×C4C4×D5911yescharacteristic
350C4×Q8—594no—
351C4×Q8—594no—
352C4×Q8—1182no—
353C4×Q8—1182no—
354C4×D4—1182no—
355C4×D4—1182no—
356C4×D4—1184no—
357C4×D4—1184no—
358C4×D4—1184no—
359C4×D4—1184no—
360C4×D4—1182no—
361C4×D4—1182no—
362#117—22no—
363#117—22no—
364#117—1184no—
365#117—1184no—
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—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—1184no—
375C2×C4⋊C4—1184no—
376C2×C4⋊C4—1182no—
377C2×C4⋊C4—1182no—
378C2×C4⋊C4C4×D5911yescharacteristic
379C2×C4⋊C4C22×D5911yescharacteristic
380C2×C22⋊C4—591no—
381C2×C22⋊C4—591no—
382C2×C22⋊C4—591no—
383C2×C22⋊C4—591no—
384C2×C22⋊C4C4×D5911yescharacteristic
385C2×C22⋊C4C22×D5911yescharacteristic
386C2×C42—591no—
387C2×C42—591no—
388C2×C42—591no—
389C2×C42—591no—
390C2×C42C4×D5911yescharacteristic
391C2×C42C22×D5911yescharacteristic
392C4⋊C8—22no—
393C4⋊C8—1184no—
394C4⋊C8—1184no—
395C4⋊C8—1182no—
396C22⋊C8—594no—
397C22⋊C8C22×D5914yes—
398#97—22no—
399#97—1184no—
400#97—1182no—
401C4×C8—1184no—
402C24—591no—
403C24—591no—
404C24C2×C4×D5911yescharacteristic
405#41—11816no—
406#41—11816no—
407C2×Q8—592no—
408C2×Q8—592no—
409C2×Q8—1182no—
410C2×Q8—1182no—
411C2×D4—594no—
412C2×D4—1182no—
413C2×D4—1182no—
414C2×D4—1182no—
415C2×D4—1182no—
416C2×D4—1182no—
417C2×D4—1182no—
418C2×D4—1182no—
419C2×D4—1181no—
420C2×D4—1181no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—591no—
426C22×C4—591no—
427C22×C4—591no—
428C22×C4—591no—
429C22×C4—594no—
430C22×C4—591no—
431C22×C4—591no—
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—1182no—
443C22×C4—4721no—
444C22×C4—4721no—
445C22×C4—4721no—
446C22×C4—4721no—
447C22×C4—4721no—
448C22×C4—4721no—
449C22×C4C2×C4×D5911yescharacteristic
450C22×C4C2×C4×D5911yescharacteristic
451C22×C4C2×C4×D5911yescharacteristic
452C22×C4C2×C4×D5911yescharacteristic
453C22×C4C2×C4×D5911yescharacteristic
454C22×C4C23×D5911yescharacteristic
455C2×C8—24no—
456C2×C8—1184no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—594no—
460C4⋊C4—594no—
461C4⋊C4—592no—
462C4⋊C4—592no—
463C4⋊C4—1182no—
464C4⋊C4—1182no—
465C4⋊C4—1182no—
466C4⋊C4—1182no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—1182no—
472C22⋊C4—1182no—
473C22⋊C4—1182no—
474C22⋊C4—1182no—
475C42—21no—
476C42—21no—
477C42—594no—
478C42—594no—
479C42—1182no—
480C42—1182no—
481C42D4×D5912yes—
482C42D4×D5912yes—
483C23—21no—
484C23—21no—
485C23—41no—
486C23—41no—
487C23—592no—
488C23—592no—
489C23—1181no—
490C23—1181no—
491C23—1181no—
492C23—1181no—
493C23—2361no—
494C23—2361no—
495C23—2361no—
496C23—2361no—
497C23M16×D5912yes—
498C23C22×C4×D5911yescharacteristic
499Q8—1184no—
500Q8—1184no—
501D4—1184no—
502D4—1184no—
503D4—1188no—
504D4—1188no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—592no—
518C2×C4—592no—
519C2×C4—592no—
520C2×C4—592no—
521C2×C4—592no—
522C2×C4—592no—
523C2×C4—1188no—
524C2×C4—1188no—
525C2×C4—1181no—
526C2×C4—1181no—
527C2×C4—1181no—
528C2×C4—1181no—
529C2×C4—4722no—
530C2×C4—4722no—
531C2×C4—4722no—
532C2×C4—4722no—
533C2×C4—4722no—
534C2×C4—4722no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4C22⋊C4×D5911yescharacteristic
540C2×C4C22⋊C4×D5911yescharacteristic
541C2×C4C22⋊C4×D5912yes—
542C2×C4M16×D5912yes—
543C2×C4C2×D4×D5912yes—
544C8—44no—
545C8—2364no—
546C22—21no—
547C22—21no—
548C22—22no—
549C22—42no—
550C22—42no—
551C22—41no—
552C22—41no—
553C22—1182no—
554C22—1182no—
555C22—2362no—
556C22—2362no—
557C22—2362no—
558C22—2362no—
559C22—2361no—
560C22—2361no—
561C22—2361no—
562C22—2361no—
563C22C2×C22⋊C4×D5911yescharacteristic
564C22C2×M16×D5911yescharacteristic
565C22C2×M16×D5911yescharacteristic
566C4—22no—
567C4—21no—
568C4—21no—
569C4—22no—
570C4—22no—
571C4—41no—
572C4—41no—
573C4—594no—
574C4—1182no—
575C4—1182no—
576C4—1182no—
577C4—2361no—
578C4—2361no—
579C2—21no—
580C2—21no—
581C2—42no—
582C2—2362no—
583C2—2362no—
584C2#407×D5911yescharacteristic
585C2#2143255811yescharacteristic
586C2#2143257311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D59
commutatorC23×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1464C2
socleC22×C59C22×C22⋊C4
prime radicalC22×#11745685C4
cosocleC22×C4×C59C24
hypercenter#1464D59
hypocenterC59#17372
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #21433180C2semidirect
3. #21422436C2semidirect
4. #21422435C2semidirect
5. #21422430C2semidirect
6. #21422429C2semidirect
7. #21422415C2semidirect
8. #21422409C2semidirect
9. C59×#1464C2semidirect
292. #1464D59semidirect
332. C59#17372semidirect

Maximal subgroups

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

subgroupindex
C2×#168378822
#214331802
#214224362
#214224352
#214224302
#214224292
#214224152
#214224092
C59×#14642
#1737259