Finite Group Databasebeta

← #28622622

Subgroups — #28622622

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×#16838056C211yesmaximal, characteristic
2C2×#16837687C211yesmaximal, characteristic, complemented
3C2×C4×#11745671C211yesmaximal, characteristic, complemented
4#21425285C212yesmaximal, complemented
5#21425097C212yesmaximal, complemented
6#21425056C211yesmaximal, characteristic, complemented
7#21424891C211yesmaximal, characteristic, complemented
8#21424404C212yesmaximal, complemented
9#21424288C212yesmaximal, complemented
10#21424093C211yesmaximal, characteristic, complemented
11C59×#2538C211yesmaximal, characteristic, complemented
12C22×#11745687—21no—
13C22×#11745671C2211yescharacteristic
14#16838056—24no—
15#16838056—24no—
16C2×#14273727C2211yescharacteristic
17C2×#14273711—22no—
18C2×#14273707—21no—
19C2×#14273701C2212yes—
20C2×C4×D236C2211yescharacteristic
21C22×C4⋊Dic59C2212yes—
22#16837687—24no—
23C4×#11745671—24no—
24C2×#14273658C2212yes—
25C2×#14273658C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273651C2212yes—
28C2×#14273651C2211yescharacteristic
29C2×#14273647C2211yescharacteristic
30C2×#14273647C2212yes—
31C2×C22⋊(C59⋊C8)C2212yes—
32C2×C22⋊(C59⋊C8)C2212yes—
33C2×C4×C59⋊C8C2211yescharacteristic
34#16837164C2212yes—
35#16837164C2212yes—
36#16837164C2212yes—
37C2×C22⋊Q8×C59C2212yes—
38C2×C4×Q8×C59C2211yescharacteristic
39C2×C4×D4×C59C2211yescharacteristic
40C59×#396—22no—
41C59×#390—21no—
42C59×#389—21no—
43C59×#384C2211yescharacteristic
44C59×#380C2212yes—
45C2×#11745687—28no—
46C2×#11745687—44no—
47C2×#11745687—44no—
48C22×Dic118—21no—
49C22×D236C2311yescharacteristic
50C22×C22⋊D59—22no—
51C23×Dic59—22no—
52C22×C4×D59—21no—
53C2×#11745671—22no—
54C2×#11745671—22no—
55C2×#11745671—22no—
56C22×C59⋊C8C2311yescharacteristic
57C22×C59⋊C8C2312yes—
58C22×C59⋊C8C2311yescharacteristic
59#14273727—22no—
60#14273727D414yes—
61#14273711—216no—
62#14273707—28no—
63#14273701—24no—
64#14273701—24no—
65C4×D236—22no—
66C4×D236D414yes—
67C2×C4⋊Dic59—22no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—24no—
70C2×C4⋊Dic59—24no—
71C2×C4⋊Dic59C2312yes—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×(C2×D59)⋊C4—22no—
74C2×(C2×D59)⋊C4—21no—
75#14273658—24no—
76#14273658—22no—
77C4⋊(C59⋊C8)—22no—
78#14273651—24no—
79#14273651—22no—
80#14273647—22no—
81#14273647—42no—
82C22⋊(C59⋊C8)—24no—
83C22⋊(C59⋊C8)—42no—
84C4×C59⋊C8—22no—
85C22×Q8×C59C2311yescharacteristic
86C22×D4×C59D411yescharacteristic
87C23×C4×C59C2312yes—
88C22⋊Q8×C59—24no—
89C22⋊Q8×C59—42no—
90C4×Q8×C59—22no—
91C4×Q8×C59—22no—
92C4×D4×C59—22no—
93C4×D4×C59—22no—
94C2×C4⋊C4×C59—21no—
95C2×C4⋊C4×C59—22no—
96C2×C4⋊C4×C59—21no—
97C2×C4⋊C4×C59—21no—
98C2×C4⋊C4×C59D411yescharacteristic
99C2×C4⋊C4×C59C2312yes—
100C2×C4⋊C4×C59C2311yescharacteristic
101C2×C22⋊C4×C59—22no—
102C2×C22⋊C4×C59—22no—
103C2×C22⋊C4×C59D412yes—
104C2×C42×C59—21no—
105C2×C42×C59C2311yescharacteristic
106#95×C59—22no—
107C24×C59C2×D412yes—
108C23×D59—21no—
109C2×Q8×C59—21no—
110C2×Q8×C59—21no—
111C2×Q8×C59—21no—
112C2×Q8×C59—21no—
113C2×Q8×C59—21no—
114C2×Q8×C59#4112yescomplemented
115C2×D4×C59—22no—
116C2×D4×C59—22no—
117C2×D4×C59SD1614yescomplemented
118C22×C4×C59—21no—
119C22×C4×C59—22no—
120C22×C4×C59—22no—
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—21no—
126C22×C4×C59—22no—
127C22×C4×C59C2×D412yes—
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2411yescharacteristic
131C2×C22⋊D59—28no—
132C2×C22⋊D59—44no—
133C2×C22⋊D59—44no—
134C22×Dic59—28no—
135C22×Dic59—21no—
136C22×Dic59—22no—
137C22×Dic59—44no—
138C22×Dic59—44no—
139#11745687—432no—
140C2×D236—21no—
141C2×D236—21no—
142C2×D236—21no—
143C2×D236—21no—
144C2×D236—21no—
145C2×D236C2×D412yes—
146C2×C4×D59—24no—
147C2×C4×D59—42no—
148C2×C4×D59—42no—
149C2×Dic118—24no—
150C2×Dic118—42no—
151C2×Dic118—42no—
152C4⋊C4×C59—22no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—22no—
156C4⋊C4×C59—22no—
157C4⋊C4×C59—44no—
158C4⋊C4×C59—42no—
159C4⋊C4×C59—42no—
160C4⋊C4×C59—81no—
161C22⋊C4×C59—24no—
162C22⋊C4×C59—44no—
163C22⋊C4×C59—82no—
164C42×C59—42no—
165C42×C59C2×D412yes—
166C42×C59C2×D412yes—
167#11745671—28no—
168(C2×D59)⋊C4—28no—
169(C2×D59)⋊C4—24no—
170C4⋊Dic59—28no—
171C4⋊Dic59—24no—
172C4⋊Dic59—21no—
173C4⋊Dic59—22no—
174C4⋊Dic59C2×D412yes—
175C4⋊Dic59C2×D414yes—
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×C59C2×SD1614yes—
192C23×C59C22×D411yescharacteristic
193C22×D59—22no—
194C22×D59—41no—
195C22×D59—41no—
196C22×D59—41no—
197C22×D59—41no—
198C22×D59—41no—
199C22×D59—41no—
200Q8×C59—22no—
201Q8×C59—22no—
202Q8×C59—41no—
203Q8×C59—41no—
204D4×C59—28no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—22no—
208C2×C4×C59—22no—
209C2×C4×C59—22no—
210C2×C4×C59—22no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
213C2×C4×C59—42no—
214C2×C4×C59—42no—
215C2×C4×C59—42no—
216C2×C4×C59—42no—
217C2×C4×C59—41no—
218C2×C4×C59—42no—
219C2×C4×C59—42no—
220C2×C4×C59—41no—
221C2×C4×C59—42no—
222C2×C4×C59—41no—
223C2×C4×C59#12012yes—
224C2×C4×C59#12012yes—
225C2×C4×C59C2×SD1612yes—
226C2×C4×C59AGL(1,ℤ/8)12yes—
227C2×C4×C59C22×D411yescharacteristic
228C2×C4×C59C22×D411yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59H(5,𝔽2)11yescharacteristic
232C2×C4×C592−1+411yescharacteristic
233C22⋊D59—416no—
234C2×Dic59—24no—
235C2×Dic59—22no—
236C2×Dic59—416no—
237C2×Dic59—42no—
238C2×Dic59—41no—
239C2×Dic59—42no—
240C2×Dic59—41no—
241D236—22no—
242D236—22no—
243D236—41no—
244D236—41no—
245C4×D59—48no—
246Dic118—48no—
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×C59C2×#12011yescharacteristic
259C22×C59#54611yescharacteristic
260C22×C59#54711yescharacteristic
261C22×C59C22×SD1611yescharacteristic
262C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C59#57711yescharacteristic
264C22×C59#57811yescharacteristic
265C2×D59—42no—
266C2×D59—42no—
267C2×D59—42no—
268C2×D59—81no—
269C2×D59—81no—
270C2×D59—81no—
271C2×D59—81no—
272C4×C59—22no—
273C4×C59—42no—
274C4×C59—42no—
275C4×C59—82no—
276C4×C59—81no—
277C4×C59#45014yes—
278C4×C59#52511yescharacteristic
279C4×C59#53711yescharacteristic
280C4×C59#54811yescharacteristic
281C4×C59#55011yescharacteristic
282Dic59—44no—
283Dic59—42no—
284C2×#547—591no—
285C2×#475—591no—
286C2×C4×SD16—591no—
287#2538D5911yescharacteristic, complemented
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#240711yescharacteristic
297C2×C59C2×#45011yescharacteristic
298C2×C59#301311yescharacteristic
299C2×C59#329811yescharacteristic
300C2×C59#333211yescharacteristic
301C2×C59#333511yescharacteristic
302C2×C59#335311yescharacteristic
303D59—82no—
304C22×#41—1181no—
305C22×SD16—591no—
306#547—1184no—
307#547—1184no—
308C2×C4⋊Q8—591no—
309C2×#123—1182no—
310C2×C22⋊Q8—1181no—
311C2×C22⋊Q8C2×D5912yes—
312C2×#121—592no—
313C2×C4×Q8C2×D5911yescharacteristic
314C2×C4×D4—591no—
315C2×C4×D4C2×D5911yescharacteristic
316C22×C4⋊C4—592no—
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—22no—
330#390—21no—
331#389—21no—
332#384C2×D5911yescharacteristic
333#380C2×D5912yes—
334#340—592no—
335#340—592no—
336#340—592no—
337C59#2412011yescharacteristic, complemented, Sylow
338C2×#41—1188no—
339C2×#41—2364no—
340C2×#41—2364no—
341C22×Q8—1181no—
342C22×Q8C22×D5911yescharacteristic
343C22×D4—591no—
344C22×D4—1182no—
345C22×D4C22⋊D5911yescharacteristic
346C23×C4—1182no—
347C23×C4—1181no—
348C23×C4C22×D5912yes—
349C2×SD16—1182no—
350C2×SD16—1182no—
351C2×SD16—1182no—
352C22×C8—591no—
353C22×C8—591no—
354C22×C8—592no—
355C4⋊Q8—594no—
356C4⋊Q8—1182no—
357#123—11816no—
358C22⋊Q8—24no—
359C22⋊Q8—42no—
360C22⋊Q8—1188no—
361#121—1184no—
362#121—1184no—
363C4×Q8—22no—
364C4×Q8—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—594no—
368C4×D4—1182no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—592no—
375C2×C4⋊C4—1184no—
376C2×C4⋊C4—1184no—
377C2×C4⋊C4—1182no—
378C2×C4⋊C4—1181no—
379C2×C4⋊C4C22⋊D5911yescharacteristic
380C2×C4⋊C4C22×D5911yescharacteristic
381C2×C4⋊C4C22×D5912yes—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—1181no—
385C2×C22⋊C4—1182no—
386C2×C22⋊C4C22⋊D5912yes—
387C2×C42—21no—
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—22no—
400C24—1181no—
401C24C2×C22⋊D5912yes—
402#41—23632no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—1184no—
409C2×Q8—2362no—
410C2×Q8—2362no—
411C2×Q8#1174568712yescomplemented
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—592no—
415C2×D4—1188no—
416C2×D4—1181no—
417C2×D4—1181no—
418C2×D4—1181no—
419C2×D4—1181no—
420C2×D4—1181no—
421C2×D4—2364no—
422C2×D4—2364no—
423C2×D4#1174567114yescomplemented
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—1188no—
434C22×C4—1181no—
435C22×C4—1182no—
436C22×C4—1184no—
437C22×C4—2364no—
438C22×C4—2364no—
439C22×C4—2362no—
440C22×C4—2362no—
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—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—81no—
464C4⋊C4—592no—
465C4⋊C4—594no—
466C4⋊C4—1188no—
467C4⋊C4—1184no—
468C4⋊C4—1181no—
469C4⋊C4—1182no—
470C22⋊C4—24no—
471C22⋊C4—44no—
472C22⋊C4—82no—
473C22⋊C4—1188no—
474C22⋊C4—1184no—
475C42—42no—
476C42D4×D5912yes—
477C42D4×D5912yes—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—1182no—
485C23—2361no—
486C23—2361no—
487C23—2361no—
488C23—2361no—
489C23—2361no—
490C23—2361no—
491C23C2×#1174567114yes—
492C23C22×C22⋊D5911yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497Q8—2368no—
498D4—28no—
499D4—1182no—
500D4—1182no—
501D4—23616no—
502D4—2361no—
503D4—2361no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—1182no—
523C2×C4—1184no—
524C2×C4—23616no—
525C2×C4—2361no—
526C2×C4—2362no—
527C2×C4—2361no—
528C2×C4—2362no—
529C2×C4—2368no—
530C2×C4#1427369512yes—
531C2×C4#1427369512yes—
532C2×C4C2×#1174567112yes—
533C2×C4#1427375212yes—
534C2×C4C2×D4×D5911yescharacteristic
535C2×C4C2×D4×D5911yescharacteristic
536C2×C4#41×D5911yescharacteristic
537C2×C4C2×#1174568711yescharacteristic
538C2×C4#1427377211yescharacteristic
539C2×C4#1427377411yescharacteristic
540C8—1182no—
541C8—2361no—
542C8—2362no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—4721no—
551C22—4721no—
552C22—4721no—
553C22—4721no—
554C22—2362no—
555C22—2362no—
556C22—2362no—
557C22C2×#1427369511yescharacteristic
558C22#1683804611yescharacteristic
559C22#1683805111yescharacteristic
560C22C22×#1174567111yescharacteristic
561C22C2×#1427375211yescharacteristic
562C22#1683821311yescharacteristic
563C22#1683821711yescharacteristic
564C4—22no—
565C4—42no—
566C4—42no—
567C4—82no—
568C4—81no—
569C4—2362no—
570C4—2364no—
571C4#1683755214yes—
572C4#1683791211yescharacteristic
573C4#1683798111yescharacteristic
574C4#1683805811yescharacteristic
575C4#1683807011yescharacteristic
576C2—44no—
577C2—4722no—
578C2#2142743711yescharacteristic
579C2C2×#1683755211yescharacteristic
580C2#2143340811yescharacteristic
581C2#2143596111yescharacteristic
582C2#2143642311yescharacteristic
583C2#2143644711yescharacteristic
584C2#2143667411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2538C2
socleC23×C59C22×D4
prime radicalC2×#16838056C2
hypercenter#2538D59
hypocenterC59#24120
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2538

Complements

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

NH ≅ G/Ntype
2. C2×#16837687C2semidirect
3. C2×C4×#11745671C2semidirect
4. #21425285C2semidirect
5. #21425097C2semidirect
6. #21425056C2semidirect
7. #21424891C2semidirect
8. #21424404C2semidirect
9. #21424288C2semidirect
10. #21424093C2semidirect
11. C59×#2538C2semidirect
114. C2×Q8×C59#41semidirect
117. C2×D4×C59SD16semidirect
184. C2×C59⋊C8#41semidirect
287. #2538D59semidirect
337. C59#24120semidirect
411. C2×Q8#11745687semidirect
423. C2×D4#11745671semidirect

Maximal subgroups

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

subgroupindex
C2×#168380562
C2×#168376872
C2×C4×#117456712
#214252852
#214250972
#214250562
#214248912
#214244042
#214242882
#214240932
C59×#25382
#2412059