Finite Group Databasebeta

← #28623034

Subgroups — #28623034

Order 15104.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837684C211yesmaximal, characteristic, complemented
2C2×#16837599C211yesmaximal, characteristic, complemented
3C4×#14273706C211yesmaximal, characteristic, complemented
4#21425282C212yesmaximal, complemented
5#21425099C212yesmaximal, complemented
6#21425049C211yesmaximal, characteristic, complemented
7#21425024C211yesmaximal, characteristic, complemented
8#21424430C212yesmaximal, complemented
9#21424357C212yesmaximal, complemented
10#21424076C211yesmaximal, characteristic, complemented
11C2×C59×#532C211yesmaximal, characteristic
12C22×#11745670C2212yes—
13C2×#14273706C2211yescharacteristic
14C2×C4×Dic118C2211yescharacteristic
15#16837684—24no—
16#16837599—24no—
17C2×#14273658C2212yes—
18C2×C4⋊(C59⋊C8)C2211yescharacteristic
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×#14273650C2212yes—
21C2×#14273648C2212yes—
22C2×#14273648C2212yes—
23C2×#14273648C2212yes—
24C2×C22⋊(C59⋊C8)C2212yes—
25#16837308C2211yescharacteristic
26#16837300C2211yescharacteristic
27#16837286—21no—
28#16837275—21no—
29#16837262C2211yescharacteristic
30C4×Dic59⋊C4—21no—
31C4×C4⋊Dic59C2211yescharacteristic
32C4×C22⋊Dic59—21no—
33#16837164C2212yes—
34#16837164C2212yes—
35#16837157C2212yes—
36C22×#41×C59—21no—
37C59×#532—24no—
38C59×#532—24no—
39C2×C4⋊Q8×C59C2211yescharacteristic
40C2×#127×C59C2211yescharacteristic
41C2×#124×C59—21no—
42C2×#121×C59—21no—
43C2×#121×C59C2212yes—
44C2×C4×D4×C59—21no—
45C2×C4×D4×C59C2212yes—
46C22×C42×C59C2211yescharacteristic
47C22×Dic118C2311yescharacteristic
48C2×#11745670—24no—
49C2×#11745670—24no—
50C2×#11745670—42no—
51C22×C59⋊C8C2312yes—
52C22×C59⋊C8C2312yes—
53#14273706—22no—
54#14273706—22no—
55C4×Dic118—22no—
56C4×Dic118—22no—
57C2×C4⋊Dic59C2311yescharacteristic
58C2×C4⋊Dic59C2311yescharacteristic
59C2×C4⋊Dic59C2311yescharacteristic
60C2×Dic59⋊C4—21no—
61C2×Dic59⋊C4—21no—
62C2×Dic59⋊C4—21no—
63C2×C22⋊Dic59—21no—
64C2×C22⋊Dic59—21no—
65C2×C4×Dic59—21no—
66C2×C4×Dic59—21no—
67#14273658—24no—
68C4⋊(C59⋊C8)—22no—
69C4⋊(C59⋊C8)—22no—
70#14273650—24no—
71#14273648—24no—
72#14273648—24no—
73#14273648—42no—
74C22⋊(C59⋊C8)—42no—
75C59⋊#95—21no—
76C59⋊#95—21no—
77C59⋊#95—21no—
78C59⋊#95—21no—
79C59⋊#95—21no—
80C2×#41×C59—28no—
81C2×#41×C59—44no—
82C2×#41×C59—44no—
83C22×Q8×C59—21no—
84C22×D4×C59—22no—
85C22×D4×C59—21no—
86C22×D4×C59—21no—
87C22×D4×C59C2312yes—
88C23×C4×C59—22no—
89C23×C4×C59D411yescharacteristic
90C23×C4×C59D411yescharacteristic
91C23×C4×C59C2311yescharacteristic
92C4⋊Q8×C59—22no—
93C4⋊Q8×C59D414yes—
94#127×C59—22no—
95#127×C59D414yes—
96#124×C59—28no—
97#121×C59—28no—
98#121×C59—24no—
99#121×C59—24no—
100C4×D4×C59—28no—
101C4×D4×C59—24no—
102C4×D4×C59—24no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59C2312yes—
105C2×C22⋊C4×C59—22no—
106C2×C22⋊C4×C59—22no—
107C2×C42×C59—22no—
108C2×C42×C59—22no—
109C2×C42×C59D411yescharacteristic
110C2×C42×C59D411yescharacteristic
111C2×C42×C59C2311yescharacteristic
112C2×C42×C59C2311yescharacteristic
113C24×C59—22no—
114C24×C59C2×D411yescharacteristic
115#41×C59—432no—
116C2×Q8×C59—24no—
117C2×Q8×C59—42no—
118C2×Q8×C59—42no—
119C2×D4×C59—28no—
120C2×D4×C59—24no—
121C2×D4×C59—24no—
122C2×D4×C59—22no—
123C2×D4×C59—22no—
124C2×D4×C59—22no—
125C2×D4×C59—22no—
126C2×D4×C59—22no—
127C2×D4×C59—42no—
128C2×D4×C59—42no—
129C2×D4×C59—44no—
130C2×D4×C59—44no—
131C2×D4×C59—42no—
132C2×D4×C59—42no—
133C2×D4×C59C2×D414yes—
134C22×C4×C59—22no—
135C22×C4×C59—22no—
136C22×C4×C59—28no—
137C22×C4×C59—22no—
138C22×C4×C59—22no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—41no—
146C22×C4×C59—41no—
147C22×C4×C59—44no—
148C22×C4×C59—44no—
149C22×C4×C59SD1614yescomplemented
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2411yescharacteristic
156C22×Dic59—21no—
157C22×Dic59—21no—
158C22×Dic59—21no—
159C22×Dic59—21no—
160C2×Dic118—21no—
161C2×Dic118—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118#4112yescomplemented
166C4⋊C4×C59—24no—
167C4⋊C4×C59—22no—
168C4⋊C4×C59C2×D414yes—
169C22⋊C4×C59—28no—
170C22⋊C4×C59—28no—
171C42×C59—22no—
172C42×C59—22no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59C2×D412yes—
176C42×C59C2×D412yes—
177C22⋊Dic59—24no—
178C22⋊Dic59—42no—
179#11745670—48no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59#4112yescomplemented
186Dic59⋊C4—24no—
187Dic59⋊C4—42no—
188Dic59⋊C4—42no—
189C4×Dic59—42no—
190C4×Dic59—42no—
191C2×C59⋊C8—22no—
192C2×C59⋊C8—22no—
193C2×C59⋊C8—22no—
194C2×C59⋊C8—22no—
195C2×C59⋊C8—22no—
196C2×C59⋊C8—22no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59—21no—
203C23×C59—24no—
204C23×C59—42no—
205C23×C59—42no—
206C23×C59—42no—
207C23×C59—42no—
208C23×C59—42no—
209C23×C59—42no—
210C23×C59C2×SD1612yes—
211C23×C59C22×D411yescharacteristic
212Q8×C59—48no—
213D4×C59—24no—
214D4×C59—24no—
215D4×C59—48no—
216D4×C59—416no—
217D4×C59—42no—
218D4×C59—42no—
219D4×C59—48no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—28no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—24no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—44no—
233C2×C4×C59—416no—
234C2×C4×C59—42no—
235C2×C4×C59—42no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59#12012yes—
241C2×C4×C59#12012yes—
242C2×C4×C59C2×SD1612yes—
243C2×C4×C59C2×SD1612yes—
244C2×C4×C59AGL(1,ℤ/8)12yes—
245C2×C4×C59C22×D411yescharacteristic
246C2×C4×C59C22×D411yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59H(5,𝔽2)11yescharacteristic
250C2×C4×C592−1+411yescharacteristic
251C2×Dic59—22no—
252C2×Dic59—22no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263Dic118—22no—
264Dic118—22no—
265Dic118—41no—
266Dic118—41no—
267C59⋊C8—42no—
268C59⋊C8—42no—
269#24154—591nomaximal, Sylow
270C22×C59—22no—
271C22×C59—22no—
272C22×C59—22no—
273C22×C59—41no—
274C22×C59—41no—
275C22×C59—41no—
276C22×C59—41no—
277C22×C59—44no—
278C22×C59—44no—
279C22×C59—44no—
280C22×C59—82no—
281C22×C59—82no—
282C22×C59—82no—
283C22×C59—82no—
284C22×C59C2×#12011yescharacteristic
285C22×C59#52511yescharacteristic
286C22×C59#53711yescharacteristic
287C22×C59C22×SD1611yescharacteristic
288C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
289C22×C59#57511yescharacteristic
290C22×C59#57511yescharacteristic
291C4×C59—22no—
292C4×C59—22no—
293C4×C59—41no—
294C4×C59—44no—
295C4×C59—41no—
296C4×C59#45014yes—
297C4×C59#54611yescharacteristic
298C4×C59#54711yescharacteristic
299C4×C59#54811yescharacteristic
300C4×C59#55011yescharacteristic
301Dic59—42no—
302Dic59—42no—
303Dic59—81no—
304Dic59—81no—
305C2×#532D5911yescharacteristic
306C2×#475—591no—
307C2×#460—591no—
308C4×C22⋊Q8—591no—
309#2034—592no—
310#2000—592no—
311#1988—591no—
312#1982—591no—
313#1877—592no—
314#1864—592no—
315#1797—591no—
316C2×C59—42no—
317C2×C59—84no—
318C2×C59#240711yescharacteristic
319C2×C59C2×#45011yescharacteristic
320C2×C59#300511yescharacteristic
321C2×C59#310311yescharacteristic
322C2×C59#311511yescharacteristic
323C2×C59#324611yescharacteristic
324C2×C59#325111yescharacteristic
325C22×#41—21no—
326C22×SD16—592no—
327#532—24no—
328#532—24no—
329C2×C4⋊Q8C2×D5911yescharacteristic
330C2×#127C2×D5911yescharacteristic
331C2×#124—21no—
332C2×C22⋊Q8—591no—
333C2×#121—21no—
334C2×#121C2×D5912yes—
335C2×C4×Q8—591no—
336C2×C4×D4—21no—
337C2×C4×D4C2×D5912yes—
338C22×C42C2×D5911yescharacteristic
339#475—1184no—
340#460—1184no—
341C2×#106—592no—
342C2×C4⋊C8—591no—
343C2×C4⋊C8—591no—
344C2×#103—592no—
345C2×#102—592no—
346C2×#102—592no—
347C2×#102—592no—
348C2×C22⋊C8—592no—
349#391—591no—
350#389—591no—
351#387—1181no—
352#385—1181no—
353#380—591no—
354C4×C4⋊C4—591no—
355C4×C4⋊C4—1181no—
356C4×C22⋊C4—1181no—
357#340—592no—
358#340—592no—
359#336—592no—
360C59#2415411yescharacteristic, complemented, Sylow
361C2×#41—28no—
362C2×#41—44no—
363C2×#41—44no—
364C22×Q8—21no—
365C22×Q8—591no—
366C22×D4—22no—
367C22×D4—21no—
368C22×D4—21no—
369C22×D4C22×D5912yes—
370C23×C4—22no—
371C23×C4C22⋊D5911yescharacteristic
372C23×C4C22⋊D5911yescharacteristic
373C23×C4C22×D5911yescharacteristic
374C2×SD16—1184no—
375C2×SD16—1184no—
376C2×SD16—2362no—
377C22×C8—592no—
378C22×C8—592no—
379C4⋊Q8—22no—
380C4⋊Q8C22⋊D5914yes—
381#127—22no—
382#127C22⋊D5914yes—
383#124—28no—
384C22⋊Q8—1182no—
385C22⋊Q8—1182no—
386#121—28no—
387#121—24no—
388#121—24no—
389C4×Q8—1182no—
390C4×Q8—1182no—
391C4×D4—28no—
392C4×D4—24no—
393C4×D4—24no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4C22×D5912yes—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—1181no—
405C2×C22⋊C4—1181no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—1181no—
409C2×C42—1181no—
410C2×C42C22⋊D5911yescharacteristic
411C2×C42C22⋊D5911yescharacteristic
412C2×C42C22×D5911yescharacteristic
413C2×C42C22×D5911yescharacteristic
414#106—1184no—
415C4⋊C8—1182no—
416C4⋊C8—1182no—
417#103—1184no—
418#102—1184no—
419#102—1184no—
420#102—2362no—
421C22⋊C8—2362no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427C24—22no—
428C24C2×C22⋊D5911yescharacteristic
429#41—432no—
430C2×Q8—24no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—592no—
434C2×Q8—1181no—
435C2×Q8—1181no—
436C2×Q8—1181no—
437C2×Q8—1181no—
438C2×Q8—1181no—
439C2×D4—28no—
440C2×D4—24no—
441C2×D4—24no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4C2×C22⋊D5914yes—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—28no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—1181no—
470C22×C4—1181no—
471C22×C4—1181no—
472C22×C4—1181no—
473C22×C4#1174567014yescomplemented
474C22×C4C2×C22⋊D5911yescharacteristic
475C22×C4C2×C22⋊D5911yescharacteristic
476C22×C4C2×C22⋊D5911yescharacteristic
477C22×C4C2×C22⋊D5911yescharacteristic
478C22×C4C2×C22⋊D5911yescharacteristic
479C22×C4C23×D5911yescharacteristic
480SD16—2368no—
481C2×C8—1182no—
482C2×C8—1182no—
483C2×C8—1182no—
484C2×C8—1182no—
485C2×C8—1182no—
486C2×C8—1182no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—592no—
490C4⋊C4—1184no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—2362no—
497C4⋊C4—2362no—
498C4⋊C4C2×C22⋊D5914yes—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—1184no—
502C22⋊C4—2362no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—21no—
507C42—2362no—
508C42—2362no—
509C42C2×C22⋊D5912yes—
510C42C2×C22⋊D5912yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—24no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23C2×#1174567012yes—
525C23C22×C22⋊D5911yescharacteristic
526Q8—48no—
527Q8—1182no—
528Q8—1182no—
529Q8—2361no—
530Q8—2361no—
531D4—24no—
532D4—24no—
533D4—48no—
534D4—416no—
535D4—42no—
536D4—48no—
537D4—42no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—44no—
551C2×C4—416no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—1182no—
559C2×C4—1182no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4#1427369612yes—
571C2×C4#1427369612yes—
572C2×C4C2×#1174567012yes—
573C2×C4C2×#1174567012yes—
574C2×C4#1427375312yes—
575C2×C4C22×C22⋊D5911yescharacteristic
576C2×C4C22×C22⋊D5911yescharacteristic
577C2×C4#41×D5911yescharacteristic
578C2×C4#41×D5911yescharacteristic
579C2×C4#1427377311yescharacteristic
580C2×C4#1427377411yescharacteristic
581C8—2362no—
582C8—2362no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22C2×#1427369611yescharacteristic
598C22#1683790911yescharacteristic
599C22#1683798011yescharacteristic
600C22C22×#1174567011yescharacteristic
601C22C2×#1427375311yescharacteristic
602C22#1683820011yescharacteristic
603C22#1683820011yescharacteristic
604C4—22no—
605C4—22no—
606C4—44no—
607C4—41no—
608C4—41no—
609C4—4721no—
610C4—4721no—
611C4—2362no—
612C4—2362no—
613C4#1683755314yes—
614C4#1683804011yescharacteristic
615C4#1683805111yescharacteristic
616C4#1683805911yescharacteristic
617C4#1683807111yescharacteristic
618C2—42no—
619C2—84no—
620C2#2142743511yescharacteristic
621C2C2×#1683755311yescharacteristic
622C2#2143333811yescharacteristic
623C2#2143412311yescharacteristic
624C2#2143424111yescharacteristic
625C2#2143541211yescharacteristic
626C2#2143545211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
Fitting, prime radicalC2×C59×#532C2
socleC23×C59C22×D4
hypercenterC2×#532D59
hypocenterC59#24154
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×#16837684C2semidirect
2. C2×#16837599C2semidirect
3. C4×#14273706C2semidirect
4. #21425282C2semidirect
5. #21425099C2semidirect
6. #21425049C2semidirect
7. #21425024C2semidirect
8. #21424430C2semidirect
9. #21424357C2semidirect
10. #21424076C2semidirect
149. C22×C4×C59SD16semidirect
165. C2×Dic118#41semidirect
185. C4⋊Dic59#41semidirect
360. C59#24154semidirect
473. C22×C4#11745670semidirect

Maximal subgroups

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

subgroupindex
C2×#168376842
C2×#168375992
C4×#142737062
#214252822
#214250992
#214250492
#214250242
#214244302
#214243572
#214240762
C2×C59×#5322
#2415459