Finite Group Databasebeta

← #28622006

Subgroups — #28622006

Order 15104.

1176 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837730C211yesmaximal, characteristic, complemented
2C2×#16837713C211yesmaximal, characteristic
3C2×#16837300C211yesmaximal, characteristic, complemented
4#21425431C212yesmaximal, complemented
5#21425332C212yesmaximal, complemented
6#21424414C211yesmaximal, characteristic
7#21424404C211yesmaximal, characteristic, complemented
8#21424357C212yesmaximal, complemented
9#21424350C212yesmaximal, complemented
10#21424164C211yesmaximal, characteristic, complemented
11C2×C59×#529C211yesmaximal, characteristic
12C22×C4⋊Dic59C2211yescharacteristic
13C22×C4⋊Dic59C2211yescharacteristic
14C22×Dic59⋊C4—21no—
15#16837730—24no—
16#16837713—24no—
17C2×#14273660C2212yes—
18C2×#14273658C2212yes—
19C2×#14273651C2212yes—
20C2×#14273651C2212yes—
21C2×#14273648C2212yes—
22C2×#14273648C2212yes—
23C2×C22⋊(C59⋊C8)C2211yescharacteristic
24C2×C22⋊(C59⋊C8)C2211yescharacteristic
25#16837300—21no—
26#16837300—21no—
27#16837300C2212yes—
28#16837300C2212yes—
29C2×C59⋊#95—21no—
30#16837164C2212yes—
31#16837164C2212yes—
32#16837157C2212yes—
33#16837157C2212yes—
34C59×#529—24no—
35C59×#529—24no—
36C2×#126×C59—21no—
37C2×#125×C59C2211yescharacteristic
38C2×#124×C59C2211yescharacteristic
39C2×#123×C59—21no—
40C2×C22⋊Q8×C59C2211yescharacteristic
41C2×#121×C59C2211yescharacteristic
42C2×C4×Q8×C59C2211yescharacteristic
43C2×C4×D4×C59—21no—
44C2×C4×D4×C59C2211yescharacteristic
45C2×#117×C59—21no—
46C22×C42×C59C2211yescharacteristic
47C23×Dic59—21no—
48C23×Dic59—21no—
49C22×C59⋊C8C2312yes—
50C22×C59⋊C8C2312yes—
51C2×C4⋊Dic59—21no—
52C2×C4⋊Dic59—24no—
53C2×C4⋊Dic59—22no—
54C2×C4⋊Dic59—22no—
55C2×C4⋊Dic59—21no—
56C2×C4⋊Dic59C2312yes—
57C2×C4⋊Dic59C2312yes—
58C2×Dic59⋊C4—22no—
59C2×Dic59⋊C4—22no—
60C2×Dic59⋊C4—44no—
61#14273660—24no—
62#14273658—24no—
63#14273651—24no—
64#14273651—24no—
65#14273648—24no—
66#14273648—24no—
67C22⋊(C59⋊C8)—22no—
68C22⋊(C59⋊C8)—22no—
69C59⋊#95—22no—
70C59⋊#95—22no—
71C22×Q8×C59C2311yescharacteristic
72C22×D4×C59—21no—
73C22×D4×C59C2311yescharacteristic
74C23×C4×C59—21no—
75C23×C4×C59D411yescharacteristic
76C23×C4×C59D411yescharacteristic
77C23×C4×C59C2311yescharacteristic
78#126×C59—28no—
79#125×C59—22no—
80#125×C59—22no—
81#124×C59—22no—
82#124×C59—22no—
83#123×C59—28no—
84C22⋊Q8×C59—22no—
85C22⋊Q8×C59D414yes—
86#121×C59—22no—
87#121×C59D414yes—
88C4×Q8×C59—22no—
89C4×Q8×C59—22no—
90C4×D4×C59—28no—
91C4×D4×C59—22no—
92C4×D4×C59—22no—
93#117×C59—28no—
94C2×C4⋊C4×C59—21no—
95C2×C4⋊C4×C59—21no—
96C2×C4⋊C4×C59—21no—
97C2×C4⋊C4×C59—21no—
98C2×C4⋊C4×C59C2311yescharacteristic
99C2×C4⋊C4×C59C2311yescharacteristic
100C2×C22⋊C4×C59—21no—
101C2×C22⋊C4×C59—21no—
102C2×C22⋊C4×C59—21no—
103C2×C22⋊C4×C59—21no—
104C2×C22⋊C4×C59—21no—
105C2×C42×C59—22no—
106C2×C42×C59—22no—
107C2×C42×C59—21no—
108C2×C42×C59D411yescharacteristic
109C2×C42×C59D411yescharacteristic
110C2×C42×C59C2311yescharacteristic
111C2×C42×C59C2311yescharacteristic
112C24×C59—21no—
113C24×C59C2×D411yescharacteristic
114C2×Q8×C59—21no—
115C2×Q8×C59—22no—
116C2×Q8×C59—22no—
117C2×Q8×C59C2×D412yes—
118C2×D4×C59—24no—
119C2×D4×C59—21no—
120C2×D4×C59—22no—
121C2×D4×C59—22no—
122C2×D4×C59—42no—
123C2×D4×C59—42no—
124C2×D4×C59C2×D412yes—
125C22×C4×C59—22no—
126C22×C4×C59—22no—
127C22×C4×C59—24no—
128C22×C4×C59—22no—
129C22×C4×C59—22no—
130C22×C4×C59—22no—
131C22×C4×C59—22no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—22no—
136C22×C4×C59—41no—
137C22×C4×C59—42no—
138C22×C4×C59—42no—
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D412yes—
144C22×C4×C59C2×D412yes—
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2411yescharacteristic
147C22×Dic59—24no—
148C22×Dic59—22no—
149C22×Dic59—22no—
150C22×Dic59—42no—
151C22×Dic59—42no—
152C22×Dic59—42no—
153C22×Dic59—42no—
154C22×Dic59—42no—
155C4⋊C4×C59—24no—
156C4⋊C4×C59—24no—
157C4⋊C4×C59—24no—
158C4⋊C4×C59—24no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59C2×D412yes—
162C4⋊C4×C59C2×D412yes—
163C22⋊C4×C59—24no—
164C22⋊C4×C59—24no—
165C22⋊C4×C59—24no—
166C22⋊C4×C59—24no—
167C22⋊C4×C59—24no—
168C42×C59—24no—
169C42×C59—22no—
170C42×C59—22no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—21no—
175C4⋊Dic59—24no—
176C4⋊Dic59—22no—
177C4⋊Dic59—22no—
178C4⋊Dic59—22no—
179C4⋊Dic59—42no—
180C4⋊Dic59#4114yes—
181Dic59⋊C4—44no—
182Dic59⋊C4—44no—
183C2×C59⋊C8—22no—
184C2×C59⋊C8—22no—
185C2×C59⋊C8—22no—
186C2×C59⋊C8—22no—
187C2×C59⋊C8—22no—
188C2×C59⋊C8—22no—
189C23×C59—21no—
190C23×C59—21no—
191C23×C59—21no—
192C23×C59—21no—
193C23×C59—22no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59—41no—
199C23×C59—41no—
200C23×C59#12012yes—
201C23×C59#13512yes—
202C23×C59AGL(1,ℤ/8)12yes—
203C23×C59C22×D411yescharacteristic
204Q8×C59—24no—
205Q8×C59—42no—
206D4×C59—24no—
207D4×C59—48no—
208D4×C59—42no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—24no—
214C2×C4×C59—24no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—44no—
228C2×C4×C59—48no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—44no—
236C2×C4×C59#12012yes—
237C2×C4×C59#13712yes—
238C2×C4×C59C22×D411yescharacteristic
239C2×C4×C59C22×D411yescharacteristic
240C2×C4×C59C2×#4112yes—
241C2×C4×C592−1+412yes—
242C2×Dic59—24no—
243C2×Dic59—48no—
244C2×Dic59—42no—
245C2×Dic59—42no—
246C2×Dic59—42no—
247C2×Dic59—42no—
248C2×Dic59—42no—
249C2×Dic59—84no—
250C59⋊C8—42no—
251C59⋊C8—42no—
252#24066—591nomaximal, Sylow
253C22×C59—22no—
254C22×C59—22no—
255C22×C59—22no—
256C22×C59—22no—
257C22×C59—22no—
258C22×C59—22no—
259C22×C59—42no—
260C22×C59—42no—
261C22×C59—42no—
262C22×C59—81no—
263C22×C59—81no—
264C22×C59—81no—
265C22×C59—81no—
266C22×C59#44914yes—
267C22×C59C2×#12011yescharacteristic
268C22×C59#52511yescharacteristic
269C22×C59#53611yescharacteristic
270C22×C59C2×#13511yescharacteristic
271C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C59C2×#13711yescharacteristic
273C22×C59#57511yescharacteristic
274C4×C59—22no—
275C4×C59—22no—
276C4×C59—42no—
277C4×C59—41no—
278C4×C59—41no—
279C4×C59—42no—
280C4×C59—41no—
281C4×C59—42no—
282C4×C59#54712yes—
283C4×C59#54712yes—
284Dic59—44no—
285Dic59—82no—
286C2×#529D5911yescharacteristic
287C2×#483—591no—
288C2×#480—591no—
289C2×#389—591no—
290#2059—592no—
291#2042—592no—
292#1874—591no—
293#1872—591no—
294#1864—592no—
295#1863—592no—
296#1817—591no—
297C2×C59—24no—
298C2×C59—82no—
299C2×C59#240611yescharacteristic
300C2×C59C2×#44911yescharacteristic
301C2×C59#300711yescharacteristic
302C2×C59#310811yescharacteristic
303C2×C59#311411yescharacteristic
304C2×C59#323211yescharacteristic
305C2×C59#323811yescharacteristic
306#529—24no—
307#529—24no—
308C2×#126—21no—
309C2×#125C2×D5911yescharacteristic
310C2×#124C2×D5911yescharacteristic
311C2×#123—21no—
312C2×C22⋊Q8C2×D5911yescharacteristic
313C2×#121C2×D5911yescharacteristic
314C2×C4×Q8C2×D5911yescharacteristic
315C2×C4×D4—21no—
316C2×C4×D4C2×D5911yescharacteristic
317C2×#117—21no—
318C22×C4⋊C4—591no—
319C22×C4⋊C4—591no—
320C22×C4⋊C4—1181no—
321C22×C42C2×D5911yescharacteristic
322#483—1184no—
323#480—1184no—
324C2×#107—592no—
325C2×#106—592no—
326C2×#103—592no—
327C2×#103—592no—
328C2×#102—592no—
329C2×#102—592no—
330C2×C22⋊C8—591no—
331C2×C22⋊C8—591no—
332#389—592no—
333#389—592no—
334#389—1181no—
335#389—1181no—
336C2×#95—1181no—
337#340—592no—
338#340—592no—
339#336—592no—
340#336—592no—
341C59#2406611yescharacteristic, complemented, Sylow
342C22×Q8C22×D5911yescharacteristic
343C22×D4—21no—
344C22×D4C22×D5911yescharacteristic
345C23×C4—21no—
346C23×C4—1181no—
347C23×C4—1181no—
348C23×C4C22⋊D5911yescharacteristic
349C23×C4C22⋊D5911yescharacteristic
350C23×C4C22×D5911yescharacteristic
351C22×C8—592no—
352C22×C8—592no—
353#126—28no—
354#125—22no—
355#125—22no—
356#124—22no—
357#124—22no—
358#123—28no—
359C22⋊Q8—22no—
360C22⋊Q8C22⋊D5914yes—
361#121—22no—
362#121C22⋊D5914yes—
363C4×Q8—22no—
364C4×Q8—22no—
365C4×D4—28no—
366C4×D4—22no—
367C4×D4—22no—
368#117—28no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—592no—
374C2×C4⋊C4—592no—
375C2×C4⋊C4—1184no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4—1182no—
378C2×C4⋊C4—1182no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1182no—
381C2×C4⋊C4—1182no—
382C2×C4⋊C4—2364no—
383C2×C4⋊C4C22×D5911yescharacteristic
384C2×C4⋊C4C22×D5911yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C42—21no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42C22⋊D5911yescharacteristic
394C2×C42C22⋊D5911yescharacteristic
395C2×C42C22×D5911yescharacteristic
396C2×C42C22×D5911yescharacteristic
397#107—1184no—
398#106—1184no—
399#103—1184no—
400#103—1184no—
401#102—1184no—
402#102—1184no—
403C22⋊C8—1182no—
404C22⋊C8—1182no—
405#95—1182no—
406#95—1182no—
407C24—21no—
408C24C2×C22⋊D5911yescharacteristic
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8C2×C22⋊D5912yes—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4C2×C22⋊D5912yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—1184no—
435C22×C4—1182no—
436C22×C4—1182no—
437C22×C4—2362no—
438C22×C4—2362no—
439C22×C4—2362no—
440C22×C4—2362no—
441C22×C4—2362no—
442C22×C4C2×C22⋊D5911yescharacteristic
443C22×C4C2×C22⋊D5911yescharacteristic
444C22×C4C2×C22⋊D5911yescharacteristic
445C22×C4C2×C22⋊D5911yescharacteristic
446C22×C4C2×C22⋊D5912yes—
447C22×C4C2×C22⋊D5912yes—
448C22×C4C2×C22⋊D5911yescharacteristic
449C22×C4C23×D5911yescharacteristic
450C2×C8—1182no—
451C2×C8—1182no—
452C2×C8—1182no—
453C2×C8—1182no—
454C2×C8—1182no—
455C2×C8—1182no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—594no—
463C4⋊C4—1184no—
464C4⋊C4—1182no—
465C4⋊C4—1182no—
466C4⋊C4—1182no—
467C4⋊C4—2364no—
468C4⋊C4—2364no—
469C4⋊C4—2362no—
470C4⋊C4C2×C22⋊D5912yes—
471C4⋊C4C2×C22⋊D5912yes—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C42—24no—
478C42—22no—
479C42—22no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23#1427369612yes—
496C23#1427374712yes—
497C23#1427375212yes—
498C23C22×C22⋊D5911yescharacteristic
499Q8—24no—
500Q8—42no—
501D4—24no—
502D4—48no—
503D4—42no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—44no—
523C2×C4—48no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—1184no—
532C2×C4—4724no—
533C2×C4—2362no—
534C2×C4—2362no—
535C2×C4—2362no—
536C2×C4—2368no—
537C2×C4—2362no—
538C2×C4—2362no—
539C2×C4#1427369612yes—
540C2×C4#1427375712yes—
541C2×C4C22×C22⋊D5911yescharacteristic
542C2×C4C22×C22⋊D5911yescharacteristic
543C2×C4#41×D5912yes—
544C2×C4#1427377412yes—
545C8—2362no—
546C8—2362no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22#1683754714yes—
561C22C2×#1427369611yescharacteristic
562C22#1683790911yescharacteristic
563C22#1683797311yescharacteristic
564C22C2×#1427374711yescharacteristic
565C22C2×#1427375211yescharacteristic
566C22C2×#1427375711yescharacteristic
567C22#1683820011yescharacteristic
568C4—22no—
569C4—22no—
570C4—41no—
571C4—42no—
572C4—41no—
573C4—41no—
574C4—42no—
575C4—42no—
576C4—4722no—
577C4—2364no—
578C4#1683805112yes—
579C4#1683805112yes—
580C2—24no—
581C2—82no—
582C2#2142742711yescharacteristic
583C2C2×#1683754711yescharacteristic
584C2#2143335611yescharacteristic
585C2#2143417211yescharacteristic
586C2#2143423011yescharacteristic
587C2#2143526011yescharacteristic
588C2#2143531411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#529C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenterC2×#529D59
hypocenterC59#24066
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
1. C2×#16837730C2semidirect
3. C2×#16837300C2semidirect
4. #21425431C2semidirect
5. #21425332C2semidirect
7. #21424404C2semidirect
8. #21424357C2semidirect
9. #21424350C2semidirect
10. #21424164C2semidirect
341. C59#24066semidirect

Maximal subgroups

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

subgroupindex
C2×#168377302
C2×#168377132
C2×#168373002
#214254312
#214253322
#214244142
#214244042
#214243572
#214243502
#214241642
C2×C59×#5292
#2406659