Finite Group Databasebeta

← #28624079

Subgroups — #28624079

Order 15104.

1438 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837609C211yesmaximal, characteristic, complemented
2C2×#16837564C212yesmaximal, complemented
3C2×C4×#11745670C211yesmaximal, characteristic, complemented
4#21429018C211yesmaximal, characteristic, complemented
5#21425289C212yesmaximal, complemented
6#21425104C212yesmaximal, complemented
7#21424899C211yesmaximal, characteristic, complemented
8#21424891C211yesmaximal, characteristic, complemented
9#21424096C211yesmaximal, characteristic, complemented
10#21424036C212yesmaximal, complemented
11C2×C59×#547C211yesmaximal, characteristic
12C22×#11745672C2212yes—
13C22×#11745670C2211yescharacteristic
14C2×#14273706C2212yes—
15C2×C4×Dic118C2211yescharacteristic
16#16837609—24no—
17#16837564—28no—
18C4×#11745670—24no—
19C2×#14273658C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273651C2212yes—
22C2×#14273651C2212yes—
23C2×#14273651C2212yes—
24C2×#14273650C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2212yes—
27C2×C22⋊(C59⋊C8)C2212yes—
28C2×C22⋊(C59⋊C8)C2212yes—
29C2×C4×C59⋊C8C2211yescharacteristic
30#16837325—22no—
31#16837305—21no—
32#16837298—21no—
33#16837272C2211yescharacteristic
34#16837262C2212yes—
35#16837164C2212yes—
36C22×#41×C59C2212yes—
37C59×#547—24no—
38C59×#547—24no—
39C2×C4⋊Q8×C59C2211yescharacteristic
40C2×#123×C59—22no—
41C2×C22⋊Q8×C59C2212yes—
42C2×#121×C59—21no—
43C2×C4×D4×C59C2211yescharacteristic
44C2×C4×D4×C59C2211yescharacteristic
45C22×C4⋊C4×C59—21no—
46C22×Dic118C2311yescharacteristic
47C2×#11745672—24no—
48C2×#11745672—24no—
49C2×#11745672—42no—
50C2×#11745670—22no—
51C2×#11745670—22no—
52C2×#11745670D414yescomplemented
53C22×C59⋊C8C2311yescharacteristic
54C22×C59⋊C8C2312yes—
55C22×C59⋊C8C2311yescharacteristic
56#14273706—24no—
57#14273706—42no—
58C4×Dic118—22no—
59C4×Dic118—22no—
60C2×C4⋊Dic59—21no—
61C2×C4⋊Dic59—21no—
62C2×C4⋊Dic59C2312yes—
63C2×C4⋊Dic59C2311yescharacteristic
64C2×Dic59⋊C4—22no—
65C2×Dic59⋊C4—21no—
66C2×C22⋊Dic59—22no—
67C2×C22⋊Dic59—22no—
68C2×C4×Dic59—21no—
69#14273658—22no—
70C4⋊(C59⋊C8)—22no—
71#14273651—24no—
72#14273651—24no—
73#14273651—42no—
74#14273650—22no—
75#14273648—22no—
76#14273648—24no—
77C22⋊(C59⋊C8)—24no—
78C22⋊(C59⋊C8)—42no—
79C4×C59⋊C8—22no—
80C59⋊#95—22no—
81C2×#41×C59—24no—
82C2×#41×C59—24no—
83C2×#41×C59—24no—
84C2×#41×C59—24no—
85C2×#41×C59—42no—
86C2×#41×C59—42no—
87C22×Q8×C59C2312yes—
88C22×D4×C59—22no—
89C22×D4×C59D411yescharacteristic
90C22×D4×C59C2311yescharacteristic
91C23×C4×C59—21no—
92C23×C4×C59—22no—
93C23×C4×C59C2312yes—
94C4⋊Q8×C59—22no—
95C4⋊Q8×C59D414yes—
96#123×C59—216no—
97C22⋊Q8×C59—24no—
98C22⋊Q8×C59—24no—
99#121×C59—28no—
100C4×D4×C59—22no—
101C4×D4×C59—22no—
102C4×D4×C59—22no—
103C4×D4×C59D414yes—
104C2×C4⋊C4×C59—28no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—22no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C22⋊C4×C59—22no—
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59D412yes—
113C2×C42×C59C2311yescharacteristic
114C24×C59—21no—
115C24×C59C2×D412yes—
116#41×C59—48no—
117#41×C59—48no—
118#41×C59—48no—
119#41×C59—48no—
120C2×Q8×C59—22no—
121C2×Q8×C59—22no—
122C2×Q8×C59—22no—
123C2×Q8×C59—22no—
124C2×Q8×C59—22no—
125C2×Q8×C59C2×D414yes—
126C2×D4×C59—28no—
127C2×D4×C59—22no—
128C2×D4×C59—22no—
129C2×D4×C59—22no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—44no—
134C2×D4×C59—44no—
135C2×D4×C59C2×D412yes—
136C2×D4×C59C2×D412yes—
137C2×D4×C59#4112yes—
138C22×C4×C59—28no—
139C22×C4×C59—21no—
140C22×C4×C59—22no—
141C22×C4×C59—24no—
142C22×C4×C59—22no—
143C22×C4×C59—22no—
144C22×C4×C59—22no—
145C22×C4×C59—22no—
146C22×C4×C59—22no—
147C22×C4×C59—22no—
148C22×C4×C59—44no—
149C22×C4×C59—44no—
150C22×C4×C59—42no—
151C22×C4×C59—42no—
152C22×C4×C59C2×D412yes—
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2411yescharacteristic
156C22×Dic59—21no—
157C22×Dic59—21no—
158C22×Dic59—22no—
159C2×Dic118—21no—
160C2×Dic118—21no—
161C2×Dic118—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118C2×D412yes—
165C4⋊C4×C59—28no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—24no—
168C4⋊C4×C59—24no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59C2×D412yes—
171C22⋊C4×C59—28no—
172C22⋊C4×C59—24no—
173C22⋊C4×C59—24no—
174C42×C59C2×D412yes—
175C42×C59C2×D412yes—
176C22⋊Dic59—44no—
177C22⋊Dic59—82no—
178#11745672—48no—
179#11745670—28no—
180C4⋊Dic59—22no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—22no—
184C4⋊Dic59—42no—
185C4⋊Dic59—81no—
186Dic59⋊C4—44no—
187Dic59⋊C4—42no—
188C4×Dic59—42no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—22no—
193C2×C59⋊C8—22no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—22no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8C2×D412yes—
198C23×C59—22no—
199C23×C59—22no—
200C23×C59—22no—
201C23×C59—22no—
202C23×C59—22no—
203C23×C59—22no—
204C23×C59—22no—
205C23×C59—22no—
206C23×C59—22no—
207C23×C59—22no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59C22×D411yescharacteristic
213Q8×C59—24no—
214Q8×C59—24no—
215Q8×C59—42no—
216Q8×C59—42no—
217D4×C59—22no—
218D4×C59—22no—
219D4×C59—22no—
220D4×C59—416no—
221D4×C59—44no—
222D4×C59#12114yes—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—22no—
226C2×C4×C59—22no—
227C2×C4×C59—24no—
228C2×C4×C59—22no—
229C2×C4×C59—22no—
230C2×C4×C59—416no—
231C2×C4×C59—42no—
232C2×C4×C59—42no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—42no—
236C2×C4×C59—42no—
237C2×C4×C59—42no—
238C2×C4×C59—42no—
239C2×C4×C59—48no—
240C2×C4×C59#12012yes—
241C2×C4×C59#12012yes—
242C2×C4×C59#13512yes—
243C2×C4×C59#13712yes—
244C2×C4×C59C22×D411yescharacteristic
245C2×C4×C59C22×D411yescharacteristic
246C2×C4×C59C22×D411yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59H(5,𝔽2)11yescharacteristic
249C2×C4×C59H(5,𝔽2)11yescharacteristic
250C2×Dic59—22no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—42no—
255C2×Dic59—42no—
256C2×Dic59—42no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259Dic118—22no—
260Dic118—22no—
261Dic118—41no—
262Dic118—41no—
263C59⋊C8—22no—
264C59⋊C8—41no—
265C59⋊C8—42no—
266#24241—591nomaximal, Sylow
267C22×C59—24no—
268C22×C59—42no—
269C22×C59—42no—
270C22×C59—42no—
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—42no—
274C22×C59—42no—
275C22×C59—42no—
276C22×C59—42no—
277C22×C59—42no—
278C22×C59—42no—
279C22×C59—42no—
280C22×C59—42no—
281C22×C59C2×#12011yescharacteristic
282C22×C59D4211yescharacteristic
283C22×C59#54611yescharacteristic
284C22×C59C2×#13511yescharacteristic
285C22×C59C2×#13711yescharacteristic
286C22×C59#57711yescharacteristic
287C22×C59#57811yescharacteristic
288C4×C59—22no—
289C4×C59—42no—
290C4×C59—44no—
291C4×C59—42no—
292C4×C59#45214yescomplemented
293C4×C59C2×#12111yescharacteristic
294C4×C59#53511yescharacteristic
295C4×C59D4211yescharacteristic
296C4×C59#54611yescharacteristic
297Dic59—42no—
298Dic59—81no—
299Dic59—82no—
300C2×#547D5911yescharacteristic
301C2×#462—591no—
302C2×#452—592no—
303C2×C4×SD16—591no—
304#2538—591no—
305#2035—592no—
306#2001—592no—
307#1956—591no—
308#1954—591no—
309#1802—591no—
310#1788—592no—
311C2×C59—44no—
312C2×C59—82no—
313C2×C59#240811yescharacteristic
314C2×C59C2×#45211yescharacteristic
315C2×C59#301311yescharacteristic
316C2×C59#327611yescharacteristic
317C2×C59#328411yescharacteristic
318C2×C59#330211yescharacteristic
319C2×C59#333111yescharacteristic
320C22×#41C2×D5912yes—
321C22×Q16—592no—
322C22×SD16—591no—
323#547—24no—
324#547—24no—
325C2×C4⋊Q8C2×D5911yescharacteristic
326C2×#123—22no—
327C2×C22⋊Q8—592no—
328C2×C22⋊Q8C2×D5912yes—
329C2×#121—21no—
330C2×C4×Q8—591no—
331C2×C4×D4C2×D5911yescharacteristic
332C2×C4×D4C2×D5911yescharacteristic
333C22×C4⋊C4—21no—
334#462—1184no—
335#452—1188no—
336C4×SD16—1184no—
337C2×#106—591no—
338C2×C4⋊C8—591no—
339C2×#103—592no—
340C2×#103—592no—
341C2×#103—591no—
342C2×#103—592no—
343C2×#102—592no—
344C2×#102—591no—
345C2×C22⋊C8—592no—
346C2×C22⋊C8—592no—
347C2×C4×C8—591no—
348#396—1182no—
349#390—1181no—
350#389—1181no—
351#384—591no—
352#380—592no—
353#340—592no—
354C59#2424111yescharacteristic, complemented, Sylow
355C2×#41—24no—
356C2×#41—24no—
357C2×#41—24no—
358C2×#41—24no—
359C2×#41—42no—
360C2×#41—42no—
361C22×Q8—591no—
362C22×Q8C22×D5912yes—
363C22×D4—22no—
364C22×D4C22⋊D5911yescharacteristic
365C22×D4C22×D5911yescharacteristic
366C23×C4—21no—
367C23×C4—22no—
368C23×C4C22×D5912yes—
369C2×Q16—1184no—
370C2×Q16—1184no—
371C2×Q16—2362no—
372C2×SD16—594no—
373C2×SD16—1182no—
374C2×SD16—1182no—
375C22×C8—591no—
376C22×C8—592no—
377C22×C8—591no—
378C4⋊Q8—22no—
379C4⋊Q8C22⋊D5914yes—
380#123—216no—
381C22⋊Q8—24no—
382C22⋊Q8—24no—
383C22⋊Q8—1184no—
384C22⋊Q8—2362no—
385#121—28no—
386C4×Q8—1182no—
387C4×Q8—1182no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4C22⋊D5914yes—
392C2×C4⋊C4—28no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—592no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1182no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4C22⋊D5911yescharacteristic
403C2×C4⋊C4C22×D5911yescharacteristic
404C2×C22⋊C4—22no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1182no—
407C2×C22⋊C4—1182no—
408C2×C22⋊C4C22⋊D5912yes—
409C2×C42—1181no—
410C2×C42C22×D5911yescharacteristic
411#106—1182no—
412C4⋊C8—1182no—
413#103—1184no—
414#103—1184no—
415#103—1182no—
416#103—2362no—
417#102—1184no—
418#102—1182no—
419C22⋊C8—1184no—
420C22⋊C8—2362no—
421C4×C8—1182no—
422#95—1182no—
423C24—21no—
424C24C2×C22⋊D5912yes—
425#41—48no—
426#41—48no—
427#41—48no—
428#41—48no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—592no—
435C2×Q8—1181no—
436C2×Q8—1181no—
437C2×Q8—1181no—
438C2×Q8—1181no—
439C2×Q8—1181no—
440C2×Q8C2×C22⋊D5914yes—
441C2×D4—28no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—44no—
449C2×D4—44no—
450C2×D4D4×D5912yes—
451C2×D4#1174568712yes—
452C2×D4C2×C22⋊D5912yes—
453C22×C4—28no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—1181no—
468C22×C4—1181no—
469C22×C4—1182no—
470C22×C4C2×C22⋊D5912yes—
471C22×C4C2×C22⋊D5911yescharacteristic
472C22×C4C2×C22⋊D5911yescharacteristic
473C22×C4C23×D5911yescharacteristic
474Q16—2368no—
475SD16—1188no—
476C2×C8—592no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1182no—
480C2×C8—1181no—
481C2×C8—1182no—
482C2×C8—1181no—
483C2×C8—1182no—
484C2×C8—1181no—
485C4⋊C4—28no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—21no—
490C4⋊C4—1182no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1182no—
494C4⋊C4—4721no—
495C4⋊C4—2364no—
496C4⋊C4—2362no—
497C4⋊C4—2362no—
498C4⋊C4C2×C22⋊D5912yes—
499C22⋊C4—28no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—4722no—
503C22⋊C4—2364no—
504C42—2362no—
505C42C2×C22⋊D5912yes—
506C42C2×C22⋊D5912yes—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C22×C22⋊D5911yescharacteristic
522Q8—24no—
523Q8—24no—
524Q8—42no—
525Q8—42no—
526Q8—1182no—
527Q8—1182no—
528Q8—2361no—
529Q8—2361no—
530D4—22no—
531D4—22no—
532D4—22no—
533D4—416no—
534D4—44no—
535D4#1427369814yes—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—416no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—48no—
553C2×C4—1182no—
554C2×C4—2361no—
555C2×C4—2362no—
556C2×C4—2361no—
557C2×C4—2362no—
558C2×C4—2361no—
559C2×C4—2362no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4#1427369612yes—
563C2×C4#1427369612yes—
564C2×C4#1427374712yes—
565C2×C4#1427375712yes—
566C2×C4C2×D4×D5911yescharacteristic
567C2×C4C22×C22⋊D5911yescharacteristic
568C2×C4C22×C22⋊D5911yescharacteristic
569C2×C4C2×#1174568711yescharacteristic
570C2×C4#1427377211yescharacteristic
571C2×C4#1427377211yescharacteristic
572C8—1182no—
573C8—2361no—
574C8—2362no—
575C22—24no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22C2×#1427369611yescharacteristic
590C22D4×C22⋊D5911yescharacteristic
591C22#1683804611yescharacteristic
592C22C2×#1427374711yescharacteristic
593C22C2×#1427375711yescharacteristic
594C22#1683821011yescharacteristic
595C22#1683821711yescharacteristic
596C4—22no—
597C4—42no—
598C4—42no—
599C4—44no—
600C4—4721no—
601C4—4722no—
602C4—2362no—
603C4#1683756414yescomplemented
604C4C2×#1427369811yescharacteristic
605C4#1683796911yescharacteristic
606C4D4×C22⋊D5911yescharacteristic
607C4#1683804611yescharacteristic
608C2—44no—
609C2—82no—
610C2#2142744611yescharacteristic
611C2C2×#1683756411yescharacteristic
612C2#2143341111yescharacteristic
613C2#2143573811yescharacteristic
614C2#2143582211yescharacteristic
615C2#2143601311yescharacteristic
616C2#2143641211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
Fitting, prime radicalC2×C59×#547C2
socleC23×C59C22×D4
hypercenterC2×#547D59
hypocenterC59#24241
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×#16837609C2semidirect
2. C2×#16837564C2semidirect
3. C2×C4×#11745670C2semidirect
4. #21429018C2semidirect
5. #21425289C2semidirect
6. #21425104C2semidirect
7. #21424899C2semidirect
8. #21424891C2semidirect
9. #21424096C2semidirect
10. #21424036C2semidirect
52. C2×#11745670D4semidirect
292. C4×C59#452semidirect
354. C59#24241semidirect
603. C4#16837564semidirect

Maximal subgroups

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

subgroupindex
C2×#168376092
C2×#168375642
C2×C4×#117456702
#214290182
#214252892
#214251042
#214248992
#214248912
#214240962
#214240362
C2×C59×#5472
#2424159