Finite Group Databasebeta

← #28622620

Subgroups — #28622620

Order 15104.

1350 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837684C211yesmaximal, characteristic, complemented
2C2×C4×#11745670C211yesmaximal, characteristic, complemented
3#21429018C211yesmaximal, characteristic, complemented
4#21425282C212yesmaximal, complemented
5#21425099C212yesmaximal, complemented
6#21425058C211yesmaximal, characteristic, complemented
7#21424891C211yesmaximal, characteristic, complemented
8#21424403C212yesmaximal, complemented
9#21424289C212yesmaximal, complemented
10#21424096C211yesmaximal, characteristic, complemented
11C2×C59×#547C211yesmaximal, characteristic
12C22×#11745670C2211yescharacteristic
13C2×#14273706C2212yes—
14C2×C4×Dic118C2211yescharacteristic
15#16837684—24no—
16C4×#11745670—24no—
17C2×#14273658C2212yes—
18C2×#14273658C2211yescharacteristic
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×#14273650C2212yes—
21C2×#14273650C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×#14273648C2212yes—
24C2×C22⋊(C59⋊C8)C2212yes—
25C2×C22⋊(C59⋊C8)C2212yes—
26C2×C4×C59⋊C8C2211yescharacteristic
27#16837325—22no—
28#16837305—21no—
29#16837298—21no—
30#16837272C2211yescharacteristic
31#16837262C2212yes—
32#16837164C2212yes—
33#16837164C2212yes—
34#16837164C2212yes—
35C22×#41×C59—21no—
36C59×#547—24no—
37C59×#547—24no—
38C2×C4⋊Q8×C59C2211yescharacteristic
39C2×#123×C59—22no—
40C2×C22⋊Q8×C59—21no—
41C2×#121×C59C2212yes—
42C2×C4×D4×C59C2211yescharacteristic
43C2×C4×D4×C59C2211yescharacteristic
44C22×C4⋊C4×C59C2212yes—
45C22×Dic118C2311yescharacteristic
46C2×#11745670—22no—
47C2×#11745670—22no—
48C2×#11745670—22no—
49C22×C59⋊C8C2311yescharacteristic
50C22×C59⋊C8C2312yes—
51C22×C59⋊C8C2311yescharacteristic
52#14273706—24no—
53#14273706—42no—
54C4×Dic118—22no—
55C4×Dic118—22no—
56C2×C4⋊Dic59—21no—
57C2×C4⋊Dic59—21no—
58C2×C4⋊Dic59C2312yes—
59C2×C4⋊Dic59C2311yescharacteristic
60C2×Dic59⋊C4—22no—
61C2×Dic59⋊C4—21no—
62C2×C22⋊Dic59—22no—
63C2×C22⋊Dic59—22no—
64C2×C4×Dic59—21no—
65#14273658—24no—
66#14273658—22no—
67C4⋊(C59⋊C8)—22no—
68#14273650—24no—
69#14273650—22no—
70#14273648—22no—
71#14273648—42no—
72C22⋊(C59⋊C8)—24no—
73C22⋊(C59⋊C8)—42no—
74C4×C59⋊C8—22no—
75C59⋊#95—22no—
76C2×#41×C59—28no—
77C2×#41×C59—44no—
78C2×#41×C59—44no—
79C22×Q8×C59—21no—
80C22×D4×C59—22no—
81C22×D4×C59D411yescharacteristic
82C22×D4×C59C2311yescharacteristic
83C23×C4×C59—22no—
84C23×C4×C59—21no—
85C23×C4×C59C2312yes—
86C4⋊Q8×C59—22no—
87C4⋊Q8×C59D414yes—
88#123×C59—216no—
89C22⋊Q8×C59—28no—
90#121×C59—24no—
91#121×C59—24no—
92C4×D4×C59—22no—
93C4×D4×C59—22no—
94C4×D4×C59—22no—
95C4×D4×C59D414yes—
96C2×C4⋊C4×C59—22no—
97C2×C4⋊C4×C59—21no—
98C2×C4⋊C4×C59—24no—
99C2×C4⋊C4×C59—24no—
100C2×C4⋊C4×C59D411yescharacteristic
101C2×C4⋊C4×C59C2311yescharacteristic
102C2×C4⋊C4×C59C2312yes—
103C2×C22⋊C4×C59—21no—
104C2×C22⋊C4×C59—22no—
105C2×C22⋊C4×C59D412yes—
106C2×C42×C59C2311yescharacteristic
107C24×C59—21no—
108C24×C59C2×D412yes—
109#41×C59—432no—
110C2×Q8×C59—24no—
111C2×Q8×C59—42no—
112C2×Q8×C59—42no—
113C2×D4×C59—28no—
114C2×D4×C59—22no—
115C2×D4×C59—22no—
116C2×D4×C59—21no—
117C2×D4×C59—21no—
118C2×D4×C59—21no—
119C2×D4×C59—21no—
120C2×D4×C59—21no—
121C2×D4×C59—44no—
122C2×D4×C59—44no—
123C2×D4×C59SD1614yescomplemented
124C2×D4×C59C2×D412yes—
125C22×C4×C59—28no—
126C22×C4×C59—21no—
127C22×C4×C59—22no—
128C22×C4×C59—24no—
129C22×C4×C59—22no—
130C22×C4×C59—22no—
131C22×C4×C59—22no—
132C22×C4×C59—22no—
133C22×C4×C59—22no—
134C22×C4×C59—22no—
135C22×C4×C59—44no—
136C22×C4×C59—44no—
137C22×C4×C59—42no—
138C22×C4×C59—42no—
139C22×C4×C59C2×D412yes—
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2411yescharacteristic
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—22no—
146C2×Dic118—21no—
147C2×Dic118—21no—
148C2×Dic118—21no—
149C2×Dic118—21no—
150C2×Dic118—21no—
151C2×Dic118#4112yescomplemented
152C4⋊C4×C59—28no—
153C4⋊C4×C59—24no—
154C4⋊C4×C59—22no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—22no—
157C4⋊C4×C59C2×D412yes—
158C4⋊C4×C59C2×D414yes—
159C22⋊C4×C59—28no—
160C22⋊C4×C59—24no—
161C22⋊C4×C59—24no—
162C42×C59C2×D412yes—
163C42×C59C2×D412yes—
164C22⋊Dic59—44no—
165C22⋊Dic59—82no—
166#11745670—28no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—22no—
170C4⋊Dic59—22no—
171C4⋊Dic59—42no—
172C4⋊Dic59—81no—
173Dic59⋊C4—44no—
174Dic59⋊C4—42no—
175C4×Dic59—42no—
176C2×C59⋊C8—21no—
177C2×C59⋊C8—21no—
178C2×C59⋊C8—21no—
179C2×C59⋊C8—22no—
180C2×C59⋊C8—22no—
181C2×C59⋊C8—21no—
182C2×C59⋊C8—22no—
183C2×C59⋊C8—21no—
184C2×C59⋊C8#4112yescomplemented
185C23×C59—22no—
186C23×C59—22no—
187C23×C59—22no—
188C23×C59—22no—
189C23×C59—22no—
190C23×C59—22no—
191C23×C59—22no—
192C23×C59—41no—
193C23×C59—41no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59C2×SD1614yes—
199C23×C59C22×D411yescharacteristic
200Q8×C59—48no—
201D4×C59—22no—
202D4×C59—22no—
203D4×C59—28no—
204D4×C59—416no—
205D4×C59—41no—
206D4×C59—41no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—22no—
210C2×C4×C59—22no—
211C2×C4×C59—24no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—416no—
215C2×C4×C59—41no—
216C2×C4×C59—42no—
217C2×C4×C59—41no—
218C2×C4×C59—42no—
219C2×C4×C59—42no—
220C2×C4×C59—42no—
221C2×C4×C59—42no—
222C2×C4×C59—42no—
223C2×C4×C59—48no—
224C2×C4×C59#12012yes—
225C2×C4×C59#12012yes—
226C2×C4×C59C2×SD1612yes—
227C2×C4×C59AGL(1,ℤ/8)12yes—
228C2×C4×C59C22×D411yescharacteristic
229C2×C4×C59C22×D411yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59H(5,𝔽2)11yescharacteristic
233C2×C4×C592−1+411yescharacteristic
234C2×Dic59—22no—
235C2×Dic59—41no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—42no—
239C2×Dic59—42no—
240C2×Dic59—42no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243Dic118—22no—
244Dic118—22no—
245Dic118—41no—
246Dic118—41no—
247C59⋊C8—22no—
248C59⋊C8—41no—
249C59⋊C8—42no—
250#24120—591nomaximal, Sylow
251C22×C59—24no—
252C22×C59—24no—
253C22×C59—24no—
254C22×C59—42no—
255C22×C59—42no—
256C22×C59—42no—
257C22×C59—42no—
258C22×C59—42no—
259C22×C59—42no—
260C22×C59—42no—
261C22×C59—81no—
262C22×C59—81no—
263C22×C59—81no—
264C22×C59—81no—
265C22×C59C2×#12011yescharacteristic
266C22×C59#54611yescharacteristic
267C22×C59#54711yescharacteristic
268C22×C59C22×SD1611yescharacteristic
269C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C59#57711yescharacteristic
271C22×C59#57811yescharacteristic
272C4×C59—22no—
273C4×C59—42no—
274C4×C59—42no—
275C4×C59—44no—
276C4×C59#45014yes—
277C4×C59#52511yescharacteristic
278C4×C59#53711yescharacteristic
279C4×C59#54811yescharacteristic
280C4×C59#55011yescharacteristic
281Dic59—42no—
282Dic59—82no—
283Dic59—81no—
284C2×#547D5911yescharacteristic
285C2×#475—591no—
286C2×C4×SD16—591no—
287#2538—591no—
288#2034—592no—
289#2000—592no—
290#1990—591no—
291#1954—591no—
292#1872—592no—
293#1853—592no—
294#1802—591no—
295C2×C59—44no—
296C2×C59—82no—
297C2×C59#240711yescharacteristic
298C2×C59C2×#45011yescharacteristic
299C2×C59#301311yescharacteristic
300C2×C59#329811yescharacteristic
301C2×C59#333211yescharacteristic
302C2×C59#333511yescharacteristic
303C2×C59#335311yescharacteristic
304C22×#41—21no—
305C22×SD16—591no—
306#547—24no—
307#547—24no—
308C2×C4⋊Q8C2×D5911yescharacteristic
309C2×#123—22no—
310C2×C22⋊Q8—21no—
311C2×C22⋊Q8—592no—
312C2×#121C2×D5912yes—
313C2×C4×Q8—591no—
314C2×C4×D4C2×D5911yescharacteristic
315C2×C4×D4C2×D5911yescharacteristic
316C22×C4⋊C4C2×D5912yes—
317#475—1184no—
318C4×SD16—1184no—
319C2×#106—592no—
320C2×#106—591no—
321C2×C4⋊C8—591no—
322C2×#103—592no—
323C2×#103—591no—
324C2×#102—591no—
325C2×#102—592no—
326C2×C22⋊C8—592no—
327C2×C22⋊C8—592no—
328C2×C4×C8—591no—
329#396—1182no—
330#390—1181no—
331#389—1181no—
332#384—591no—
333#380—592no—
334#340—592no—
335#340—592no—
336#340—592no—
337C59#2412011yescharacteristic, complemented, Sylow
338C2×#41—28no—
339C2×#41—44no—
340C2×#41—44no—
341C22×Q8—21no—
342C22×Q8—591no—
343C22×D4—22no—
344C22×D4C22⋊D5911yescharacteristic
345C22×D4C22×D5911yescharacteristic
346C23×C4—22no—
347C23×C4—21no—
348C23×C4C22×D5912yes—
349C2×SD16—1182no—
350C2×SD16—1182no—
351C2×SD16—1182no—
352C22×C8—591no—
353C22×C8—591no—
354C22×C8—592no—
355C4⋊Q8—22no—
356C4⋊Q8C22⋊D5914yes—
357#123—216no—
358C22⋊Q8—28no—
359C22⋊Q8—1184no—
360C22⋊Q8—2362no—
361#121—24no—
362#121—24no—
363C4×Q8—1182no—
364C4×Q8—1182no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4C22⋊D5914yes—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—592no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4—1182no—
378C2×C4⋊C4—1181no—
379C2×C4⋊C4C22⋊D5911yescharacteristic
380C2×C4⋊C4C22×D5911yescharacteristic
381C2×C4⋊C4C22×D5912yes—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—1182no—
385C2×C22⋊C4—1182no—
386C2×C22⋊C4C22⋊D5912yes—
387C2×C42—1181no—
388C2×C42C22×D5911yescharacteristic
389#106—1182no—
390#106—1184no—
391C4⋊C8—1182no—
392#103—1182no—
393#103—1184no—
394#102—1182no—
395#102—2362no—
396C22⋊C8—1184no—
397C22⋊C8—2362no—
398C4×C8—1182no—
399#95—1182no—
400C24—21no—
401C24C2×C22⋊D5912yes—
402#41—432no—
403C2×Q8—24no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—592no—
407C2×Q8—1181no—
408C2×Q8—1181no—
409C2×Q8—1181no—
410C2×Q8—1181no—
411C2×Q8—1181no—
412C2×D4—28no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4#1174567014yescomplemented
423C2×D4C2×C22⋊D5912yes—
424C22×C4—28no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—1181no—
439C22×C4—1181no—
440C22×C4—1182no—
441C22×C4C2×C22⋊D5912yes—
442C22×C4C2×C22⋊D5911yescharacteristic
443C22×C4C2×C22⋊D5911yescharacteristic
444C22×C4C23×D5911yescharacteristic
445SD16—1188no—
446C2×C8—592no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8—1182no—
451C2×C8—1182no—
452C2×C8—1181no—
453C2×C8—1182no—
454C2×C8—1181no—
455C4⋊C4—28no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—1182no—
463C4⋊C4—1182no—
464C4⋊C4—4721no—
465C4⋊C4—2364no—
466C4⋊C4—2362no—
467C4⋊C4—2362no—
468C4⋊C4C2×C22⋊D5912yes—
469C4⋊C4C2×C22⋊D5914yes—
470C22⋊C4—28no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—4722no—
474C22⋊C4—2364no—
475C42—2362no—
476C42C2×C22⋊D5912yes—
477C42C2×C22⋊D5912yes—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C2×#1174567014yes—
492C23C22×C22⋊D5911yescharacteristic
493Q8—48no—
494Q8—1182no—
495Q8—1182no—
496Q8—2361no—
497Q8—2361no—
498D4—22no—
499D4—22no—
500D4—28no—
501D4—416no—
502D4—41no—
503D4—41no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—416no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—48no—
520C2×C4—42no—
521C2×C4—1182no—
522C2×C4—2361no—
523C2×C4—2362no—
524C2×C4—2361no—
525C2×C4—2362no—
526C2×C4—2361no—
527C2×C4—2362no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4#1427369612yes—
531C2×C4#1427369612yes—
532C2×C4C2×#1174567012yes—
533C2×C4#1427375312yes—
534C2×C4C22×C22⋊D5911yescharacteristic
535C2×C4C22×C22⋊D5911yescharacteristic
536C2×C4#41×D5911yescharacteristic
537C2×C4#41×D5911yescharacteristic
538C2×C4#1427377311yescharacteristic
539C2×C4#1427377411yescharacteristic
540C8—1182no—
541C8—2361no—
542C8—2362no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#1427369611yescharacteristic
558C22#1683804011yescharacteristic
559C22#1683805111yescharacteristic
560C22C22×#1174567011yescharacteristic
561C22C2×#1427375311yescharacteristic
562C22#1683821011yescharacteristic
563C22#1683821711yescharacteristic
564C4—22no—
565C4—42no—
566C4—44no—
567C4—42no—
568C4—4722no—
569C4—4721no—
570C4—2362no—
571C4#1683755314yes—
572C4#1683790911yescharacteristic
573C4#1683798011yescharacteristic
574C4#1683805911yescharacteristic
575C4#1683807111yescharacteristic
576C2—44no—
577C2—82no—
578C2#2142743511yescharacteristic
579C2C2×#1683755311yescharacteristic
580C2#2143341111yescharacteristic
581C2#2143595511yescharacteristic
582C2#2143641711yescharacteristic
583C2#2143645311yescharacteristic
584C2#2143668011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
Fitting, prime radicalC2×C59×#547C2
socleC23×C59C22×D4
hypercenterC2×#547D59
hypocenterC59#24120
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#16837684C2semidirect
2. C2×C4×#11745670C2semidirect
3. #21429018C2semidirect
4. #21425282C2semidirect
5. #21425099C2semidirect
6. #21425058C2semidirect
7. #21424891C2semidirect
8. #21424403C2semidirect
9. #21424289C2semidirect
10. #21424096C2semidirect
123. C2×D4×C59SD16semidirect
151. C2×Dic118#41semidirect
184. C2×C59⋊C8#41semidirect
337. C59#24120semidirect
422. C2×D4#11745670semidirect

Maximal subgroups

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

subgroupindex
C2×#168376842
C2×C4×#117456702
#214290182
#214252822
#214250992
#214250582
#214248912
#214244032
#214242892
#214240962
C2×C59×#5472
#2412059