Finite Group Databasebeta

← #28646350

Subgroups — #28646350

Order 15104.

774 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427798C211yesmaximal, characteristic
2#21426971C211yesmaximal, characteristic
3#21425655C211yesmaximal, characteristic, complemented
4#21425594C211yesmaximal, characteristic, complemented
5#21425594C211yesmaximal, characteristic, complemented
6#21425430C211yesmaximal, characteristic, complemented
7#21425047C211yesmaximal, characteristic, complemented
8#21424860C211yesmaximal, characteristic, complemented
9#21424675C211yesmaximal, characteristic, complemented
10#21424675C211yesmaximal, characteristic, complemented
11#21424291C211yesmaximal, characteristic
12#21424187C211yesmaximal, characteristic
13#21424121C211yesmaximal, characteristic
14C2×#16837165C211yesmaximal, characteristic
15C59×#1850C211yesmaximal, characteristic
16C2×#14273727C2211yescharacteristic
17C2×#14273704—21no—
18C22×C4⋊Dic59C2211yescharacteristic
19C22×C4⋊Dic59C2211yescharacteristic
20C2×#14273659C2211yescharacteristic
21C2×#14273659C2211yescharacteristic
22C2×#14273659C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24#16837334—21no—
25#16837323—21no—
26#16837300C2211yescharacteristic
27#16837300C2211yescharacteristic
28#16837300C2211yescharacteristic
29#16837300C2211yescharacteristic
30#16837285—21no—
31#16837285—21no—
32#16837277—21no—
33#16837272C2211yescharacteristic
34#16837262C2211yescharacteristic
35#16837262C2211yescharacteristic
36#16837260—21no—
37C4×C4⋊Dic59C2211yescharacteristic
38C4×C4⋊Dic59C2211yescharacteristic
39C22⋊C4×Dic59—21no—
40#16837165C2211yescharacteristic
41#16837165C2211yescharacteristic
42#16837165C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52C23×C8×C59C2211yescharacteristic
53C2×C4⋊C8×C59C2211yescharacteristic
54C2×C4⋊C8×C59C2211yescharacteristic
55C2×C22⋊C8×C59C2211yescharacteristic
56C4×C22⋊C4×C59C2211yescharacteristic
57C59×#336C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C22×Dic118—21no—
60C23×Dic59—21no—
61C23×Dic59—21no—
62#14273727—22no—
63#14273727—41no—
64#14273704—44no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59—22no—
67C2×C4⋊Dic59—22no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—41no—
73C2×C4⋊Dic59Q814yescomplemented
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C2×Dic59⋊C4—21no—
86C2×C22⋊Dic59—21no—
87C2×C22⋊Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90#14273659—22no—
91#14273659—22no—
92#14273659—41no—
93#14273659—41no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C23×C4×C59C2311yescharacteristic
103C22×C8×C59—22no—
104C22×C8×C59—22no—
105C22×C8×C59—41no—
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110C2×C22⋊C4×C59D411yescharacteristic
111C2×C22⋊C4×C59D411yescharacteristic
112C2×C42×C59C2311yescharacteristic
113C2×C42×C59C2311yescharacteristic
114C4⋊C8×C59—22no—
115C4⋊C8×C59—41no—
116C22⋊C8×C59—22no—
117#95×C59D411yescharacteristic
118#95×C59D411yescharacteristic
119C24×C59C2×D411yescharacteristic
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×Q812yes—
130C22×C4×C59#4112yes—
131C22×C4×C59C2411yescharacteristic
132C22×Dic59—21no—
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59—24no—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—42no—
142C22×Dic59—42no—
143C22×Dic59—81no—
144C22×Dic59—81no—
145C22×Dic59—81no—
146C2×Dic118—42no—
147C2×Dic118—42no—
148C2×Dic118—81no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—44no—
161C2×C8×C59#4112yes—
162C22⋊C4×C59—22no—
163C22⋊C4×C59SD1614yes—
164C42×C59—21no—
165C42×C59—21no—
166C42×C59—21no—
167C42×C59—21no—
168C22⋊Dic59—42no—
169C22⋊Dic59—42no—
170C4⋊Dic59—24no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—42no—
185C4⋊Dic59—42no—
186C4⋊Dic59—42no—
187C4⋊Dic59—81no—
188C4⋊Dic59C2×Q812yes—
189C4⋊Dic59C2×Q812yes—
190C4⋊Dic59#4112yes—
191Dic59⋊C4—42no—
192Dic59⋊C4—42no—
193Dic59⋊C4—81no—
194Dic59⋊C4—81no—
195C4×Dic59—81no—
196C4×Dic59—81no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59C22⋊Q812yes—
202C23×C59C2×SD1612yes—
203C23×C59#13712yes—
204C23×C59C22×D411yescharacteristic
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—22no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59C22⋊Q812yes—
222C2×C4×C59C2×SD1612yes—
223C2×C4×C59C2×SD1612yes—
224C2×C4×C59C22×Q811yescharacteristic
225C2×C4×C59C2×#4111yescharacteristic
226C2×C4×C59C2×#4111yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C2×C4×C59H(5,𝔽2)11yescharacteristic
229C2×C4×C592−1+411yescharacteristic
230C2×Dic59—22no—
231C2×Dic59—22no—
232C2×Dic59—22no—
233C2×Dic59—41no—
234C2×Dic59—41no—
235C2×Dic59—41no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—84no—
255C2×Dic59—84no—
256Dic118—84no—
257C8×C59—22no—
258C8×C59—41no—
259C8×C59—41no—
260C8×C59—41no—
261#25992—591nomaximal, Sylow
262C22×C59—22no—
263C22×C59—22no—
264C22×C59—22no—
265C22×C59—22no—
266C22×C59—22no—
267C22×C59—22no—
268C22×C59#47514yes—
269C22×C59C2×C22⋊Q811yescharacteristic
270C22×C59#54611yescharacteristic
271C22×C59#54711yescharacteristic
272C22×C59C22×SD1611yescharacteristic
273C22×C59C2×#13711yescharacteristic
274C22×C59#57611yescharacteristic
275C22×C59#57711yescharacteristic
276C4×C59—22no—
277C4×C59—22no—
278C4×C59—22no—
279C4×C59—41no—
280C4×C59—41no—
281C4×C59#53311yescharacteristic
282C4×C59#54411yescharacteristic
283C4×C59#55111yescharacteristic
284C4×C59#55211yescharacteristic
285Dic59—42no—
286Dic59—42no—
287Dic59—42no—
288Dic59—81no—
289Dic59—81no—
290Dic59—81no—
291Dic59—81no—
292Dic59—81no—
293#2440—591no—
294#2345—591no—
295#2099—591no—
296#2088—591no—
297#2088—591no—
298#2058—591no—
299#1988—591no—
300#1948—591no—
301#1919—591no—
302#1919—591no—
303#1853—591no—
304#1850D5911yescharacteristic
305#1824—591no—
306#1807—591no—
307C2×#340—591no—
308C2×C59—24no—
309C2×C59#243811yescharacteristic
310C2×C59C2×#47511yescharacteristic
311C2×C59#308111yescharacteristic
312C2×C59#329511yescharacteristic
313C2×C59#332011yescharacteristic
314C2×C59#333511yescharacteristic
315C2×C59#334611yescharacteristic
316C23×C8C2×D5911yescharacteristic
317C2×C4⋊Q8—591no—
318C2×C22⋊Q8—1181no—
319C22×C4⋊C4—591no—
320C22×C4⋊C4—591no—
321C2×#106—591no—
322C2×#106—591no—
323C2×#106—591no—
324C2×#106—591no—
325C2×C4⋊C8C2×D5911yescharacteristic
326C2×C4⋊C8C2×D5911yescharacteristic
327C2×C22⋊C8C2×D5911yescharacteristic
328#398—1181no—
329#395—1181no—
330#389—591no—
331#389—591no—
332#389—591no—
333#389—591no—
334#387—1181no—
335#387—1181no—
336#385—1181no—
337#384—591no—
338#380—591no—
339#380—591no—
340#380—1181no—
341C4×C4⋊C4—591no—
342C4×C4⋊C4—591no—
343C4×C22⋊C4—1181no—
344C4×C22⋊C4C2×D5911yescharacteristic
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357#336C2×D5911yescharacteristic
358#336C2×D5911yescharacteristic
359C59#2599211yescharacteristic, complemented, Sylow
360C22×Q8—1181no—
361C23×C4—1181no—
362C23×C4—1181no—
363C23×C4C22×D5911yescharacteristic
364C22×C8—22no—
365C22×C8—22no—
366C22×C8—41no—
367C22×C8C22×D5911yescharacteristic
368C22×C8C22×D5911yescharacteristic
369C22×C8C22×D5911yescharacteristic
370C22×C8C22×D5911yescharacteristic
371C4⋊Q8—1182no—
372C4⋊Q8—2361no—
373C22⋊Q8—2364no—
374C2×C4⋊C4—594no—
375C2×C4⋊C4—591no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—591no—
378C2×C4⋊C4—591no—
379C2×C4⋊C4—591no—
380C2×C4⋊C4—591no—
381C2×C4⋊C4—591no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—1182no—
384C2×C4⋊C4—1181no—
385C2×C4⋊C4—1182no—
386C2×C4⋊C4—1181no—
387C2×C4⋊C4—1181no—
388C2×C4⋊C4—1181no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—2361no—
395C2×C22⋊C4—1181no—
396C2×C22⋊C4—1181no—
397C2×C22⋊C4D23611yescharacteristic
398C2×C22⋊C4D23611yescharacteristic
399C2×C42—1181no—
400C2×C42—1181no—
401C2×C42C22×D5911yescharacteristic
402C2×C42C22×D5911yescharacteristic
403#106—1182no—
404#106—1182no—
405#106—2361no—
406#106—2361no—
407C4⋊C8—22no—
408C4⋊C8—41no—
409C22⋊C8—22no—
410#95—1181no—
411#95—1181no—
412#95—1181no—
413#95—1181no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417#95—1181no—
418#95D23611yescharacteristic
419#95D23611yescharacteristic
420C24C2×D23611yescharacteristic
421C2×Q8—4721no—
422C2×Q8—2362no—
423C2×Q8—2362no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—1184no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4—1181no—
432C22×C4—1181no—
433C22×C4—1181no—
434C22×C4—1181no—
435C22×C4—1181no—
436C22×C4—1181no—
437C22×C4—4721no—
438C22×C4—4721no—
439C22×C4—4721no—
440C22×C4—2362no—
441C22×C4—2362no—
442C22×C4C2×D23611yescharacteristic
443C22×C4C2×D23611yescharacteristic
444C22×C4C2×D23611yescharacteristic
445C22×C4C2×D23611yescharacteristic
446C22×C4C2×D23611yescharacteristic
447C22×C4#1174568712yes—
448C22×C4Q8×D5912yes—
449C22×C4C23×D5911yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—44no—
462C2×C8#1174568712yes—
463C4⋊C4—592no—
464C4⋊C4—592no—
465C4⋊C4—592no—
466C4⋊C4—1184no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—1181no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—4721no—
481C4⋊C4—4721no—
482C4⋊C4—4721no—
483C4⋊C4—2362no—
484C4⋊C4—2362no—
485C4⋊C4—2362no—
486C4⋊C4—2362no—
487C4⋊C4—2362no—
488C22⋊C4—22no—
489C22⋊C4—2362no—
490C22⋊C4—2362no—
491C22⋊C4#1174566014yes—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—4721no—
497C42—4721no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23#1427370712yes—
503C23C2×#1174566012yes—
504C23#1427376012yes—
505C23C22×D23611yescharacteristic
506Q8—4724no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—1182no—
524C2×C4—1182no—
525C2×C4—1182no—
526C2×C4—4724no—
527C2×C4—4724no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4#1427370712yes—
550C2×C4C2×#1174566012yes—
551C2×C4C2×#1174566012yes—
552C2×C4C2×Q8×D5911yescharacteristic
553C2×C4#41×D5911yescharacteristic
554C2×C4C2×#1174568711yescharacteristic
555C2×C4C2×#1174568711yescharacteristic
556C2×C4#1427377211yescharacteristic
557C2×C4#1427377411yescharacteristic
558C8—22no—
559C8—41no—
560C8—41no—
561C8—41no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22#1683768814yes—
569C22C2×#1427370711yescharacteristic
570C22#1683804711yescharacteristic
571C22#1683805611yescharacteristic
572C22C22×#1174566011yescharacteristic
573C22C2×#1427376011yescharacteristic
574C22#1683820711yescharacteristic
575C22#1683821411yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—41no—
580C4—41no—
581C4—4721no—
582C4—4721no—
583C4—4721no—
584C4—4721no—
585C4—4721no—
586C4—2362no—
587C4—2362no—
588C4—2362no—
589C4#1683795411yescharacteristic
590C4#1683803111yescharacteristic
591C4#1683808311yescharacteristic
592C4#1683808911yescharacteristic
593C2—24no—
594C2#2142777011yescharacteristic
595C2C2×#1683768811yescharacteristic
596C2#2143392811yescharacteristic
597C2#2143592111yescharacteristic
598C2#2143626911yescharacteristic
599C2#2143645411yescharacteristic
600C2#2143658311yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1850C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#1850D59
hypocenterC59#25992
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
3. #21425655C2semidirect
4. #21425594C2semidirect
5. #21425594C2semidirect
6. #21425430C2semidirect
7. #21425047C2semidirect
8. #21424860C2semidirect
9. #21424675C2semidirect
10. #21424675C2semidirect
73. C2×C4⋊Dic59Q8semidirect
359. C59#25992semidirect

Maximal subgroups

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

subgroupindex
#214277982
#214269712
#214256552
#214255942
#214254302
#214250472
#214248602
#214246752
#214242912
#214241872
#214241212
C2×#168371652
C59×#18502
#2599259