Finite Group Databasebeta

← #28636926

Subgroups — #28636926

Order 15104.

858 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837735C211yesmaximal, characteristic
2C2×#16837641C211yesmaximal, characteristic
3#21428574C211yesmaximal, characteristic
4#21425567C211yesmaximal, characteristic, complemented
5#21425504C211yesmaximal, characteristic, complemented
6#21425488C211yesmaximal, characteristic, complemented
7#21425418C211yesmaximal, characteristic, complemented
8#21424854C211yesmaximal, characteristic, complemented
9#21424737C211yesmaximal, characteristic, complemented
10#21424437C211yesmaximal, characteristic, complemented
11#21424403C211yesmaximal, characteristic
12#21424276C211yesmaximal, characteristic
13#21424101C211yesmaximal, characteristic, complemented
14C2×#16837164C211yesmaximal, characteristic
15C59×#2360C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C23×C59⋊C8C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C22×C4⋊Dic59C2211yescharacteristic
20#16837735—24no—
21#16837641—24no—
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273660C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×#14273651C2211yescharacteristic
30C2×#14273650C2211yescharacteristic
31C2×#14273650C2211yescharacteristic
32C2×#14273650C2211yescharacteristic
33C2×C22⋊(C59⋊C8)C2211yescharacteristic
34#16837334—21no—
35#16837334—21no—
36#16837308C2211yescharacteristic
37#16837308C2211yescharacteristic
38#16837277—21no—
39#16837277—21no—
40#16837272C2211yescharacteristic
41#16837272C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C22⋊Q8×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C22×C4⋊C4×C59C2211yescharacteristic
53C59×#391C2211yescharacteristic
54C59×#389—21no—
55C59×#389C2211yescharacteristic
56C59×#387—21no—
57C59×#385—21no—
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C22⋊C4×C59—21no—
60C4×C22⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C23×Dic59—21no—
63C2×#11745672—22no—
64C2×#11745672—22no—
65C2×#11745672—41no—
66C22×C59⋊C8—22no—
67C22×C59⋊C8—22no—
68C22×C59⋊C8—41no—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73#14273706—22no—
74#14273706—22no—
75C2×C4⋊Dic59—22no—
76C2×C4⋊Dic59—22no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59—21no—
79C2×C4⋊Dic59—21no—
80C2×C4⋊Dic59—21no—
81C2×C4⋊Dic59—21no—
82C2×C4⋊Dic59C2311yescharacteristic
83C2×C4⋊Dic59C2311yescharacteristic
84C2×C4⋊Dic59C2311yescharacteristic
85C2×Dic59⋊C4—21no—
86C2×C22⋊Dic59—21no—
87#14273660—22no—
88#14273660—22no—
89#14273660—22no—
90C4⋊(C59⋊C8)—22no—
91C4⋊(C59⋊C8)—41no—
92#14273651—22no—
93#14273651—22no—
94#14273651—22no—
95#14273650—22no—
96#14273650—22no—
97#14273650—41no—
98C22⋊(C59⋊C8)—22no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×Q8×C59C2311yescharacteristic
104C23×C4×C59—21no—
105C23×C4×C59C2311yescharacteristic
106C22⋊Q8×C59—22no—
107C22⋊Q8×C59D414yes—
108C4×Q8×C59—22no—
109C4×Q8×C59—41no—
110C2×C4⋊C4×C59—21no—
111C2×C4⋊C4×C59—22no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59—41no—
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59D411yescharacteristic
121C2×C22⋊C4×C59D411yescharacteristic
122C2×C42×C59—21no—
123C2×C42×C59—21no—
124C2×C42×C59C2311yescharacteristic
125C2×C42×C59C2311yescharacteristic
126#95×C59—21no—
127#95×C59—21no—
128#95×C59—21no—
129#95×C59—21no—
130#95×C59D411yescharacteristic
131#95×C59D411yescharacteristic
132C24×C59C2×D411yescharacteristic
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59C2×D412yes—
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—21no—
146C22×C4×C59—21no—
147C22×C4×C59—42no—
148C22×C4×C59—42no—
149C22×C4×C59—81no—
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D412yes—
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59#4112yes—
157C22×C4×C59C2411yescharacteristic
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C22×Dic59—42no—
163C22×Dic59—81no—
164C22×Dic59—81no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118—21no—
171C4⋊C4×C59—24no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—42no—
177C4⋊C4×C59—42no—
178C4⋊C4×C59—81no—
179C4⋊C4×C59C2×D412yes—
180C4⋊C4×C59#4112yes—
181C22⋊C4×C59—24no—
182C22⋊C4×C59—22no—
183C22⋊C4×C59Q1614yescomplemented
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—21no—
188C42×C59—42no—
189C42×C59—81no—
190C22⋊Dic59—42no—
191#11745672—44no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—42no—
199C4⋊Dic59—81no—
200C4⋊Dic59—81no—
201C4⋊Dic59—81no—
202C4⋊Dic59—81no—
203Dic59⋊C4—42no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—44no—
216C2×C59⋊C8#4112yes—
217C23×C59—21no—
218C23×C59—21no—
219C23×C59—21no—
220C23×C59—21no—
221C23×C59#12112yes—
222C23×C59C2×Q1612yes—
223C23×C59#13712yes—
224C23×C59C22×D411yescharacteristic
225Q8×C59—22no—
226Q8×C59—22no—
227Q8×C59—41no—
228Q8×C59—41no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—22no—
236C2×C4×C59—22no—
237C2×C4×C59—22no—
238C2×C4×C59—22no—
239C2×C4×C59—22no—
240C2×C4×C59—22no—
241C2×C4×C59—21no—
242C2×C4×C59—22no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—84no—
258C2×C4×C59#12112yes—
259C2×C4×C59C2×Q1612yes—
260C2×C4×C59C2×Q1612yes—
261C2×C4×C59C22×D411yescharacteristic
262C2×C4×C59C2×#4111yescharacteristic
263C2×C4×C59C2×#4111yescharacteristic
264C2×C4×C59C2×#4111yescharacteristic
265C2×C4×C59H(5,𝔽2)11yescharacteristic
266C2×C4×C592−1+411yescharacteristic
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—84no—
280Dic118—41no—
281Dic118—41no—
282Dic118—41no—
283Dic118—41no—
284C59⋊C8—22no—
285C59⋊C8—41no—
286C59⋊C8—41no—
287C59⋊C8—41no—
288#25206—591nomaximal, Sylow
289C22×C59—22no—
290C22×C59—22no—
291C22×C59—22no—
292C22×C59—22no—
293C22×C59—22no—
294C22×C59—22no—
295C22×C59#46214yes—
296C22×C59C2×#12111yescharacteristic
297C22×C59#54611yescharacteristic
298C22×C59#54711yescharacteristic
299C22×C59C22×Q1611yescharacteristic
300C22×C59C2×#13711yescharacteristic
301C22×C59#57611yescharacteristic
302C22×C59#57711yescharacteristic
303C4×C59—22no—
304C4×C59—22no—
305C4×C59—22no—
306C4×C59—41no—
307C4×C59—42no—
308C4×C59—42no—
309C4×C59—41no—
310C4×C59—42no—
311C4×C59—81no—
312C4×C59#52911yescharacteristic
313C4×C59#53911yescharacteristic
314C4×C59#54611yescharacteristic
315C4×C59#54711yescharacteristic
316Dic59—81no—
317Dic59—81no—
318Dic59—81no—
319Dic59—81no—
320C2×#484—591no—
321C2×#467—591no—
322#2502—591no—
323#2360D5911yescharacteristic
324#2083—591no—
325#2072—591no—
326#2068—591no—
327#2056—591no—
328#1947—591no—
329#1928—591no—
330#1878—591no—
331#1872—591no—
332#1850—591no—
333#1803—591no—
334C2×#340—591no—
335C2×C59—24no—
336C2×C59#242511yescharacteristic
337C2×C59C2×#46211yescharacteristic
338C2×C59#304511yescharacteristic
339C2×C59#329911yescharacteristic
340C2×C59#331411yescharacteristic
341C2×C59#333811yescharacteristic
342C2×C59#334411yescharacteristic
343C22×Q16—591no—
344C23×C8—591no—
345C2×C22⋊Q8—591no—
346C2×C22⋊Q8C2×D5911yescharacteristic
347C2×C4×Q8C2×D5911yescharacteristic
348C22×C4⋊C4—591no—
349C22×C4⋊C4C2×D5911yescharacteristic
350#484—1184no—
351#467—1184no—
352C2×#107—591no—
353C2×#107—591no—
354C2×#107—591no—
355C2×C4⋊C8—591no—
356C2×C4⋊C8—591no—
357C2×#103—591no—
358C2×#103—591no—
359C2×#103—591no—
360C2×#103—591no—
361C2×#103—591no—
362C2×#103—591no—
363C2×C22⋊C8—591no—
364#398—1181no—
365#398—1181no—
366#391—591no—
367#391—591no—
368#391C2×D5911yescharacteristic
369#389—21no—
370#389C2×D5911yescharacteristic
371#387—21no—
372#385—21no—
373#385—1181no—
374#385—1181no—
375#384—591no—
376#384—591no—
377C4×C4⋊C4C2×D5911yescharacteristic
378C4×C22⋊C4—21no—
379C4×C22⋊C4C2×D5911yescharacteristic
380#340—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#340—591no—
386#336—591no—
387#336—591no—
388C59#2520611yescharacteristic, complemented, Sylow
389C22×Q8—591no—
390C22×Q8C22×D5911yescharacteristic
391C23×C4—21no—
392C23×C4—1181no—
393C23×C4C22×D5911yescharacteristic
394C2×Q16—1182no—
395C2×Q16—1182no—
396C2×Q16—2361no—
397C22×C8—591no—
398C22×C8—591no—
399C22×C8—591no—
400C22×C8—591no—
401C22×C8—1182no—
402C22×C8—1182no—
403C22×C8—2361no—
404C22⋊Q8—22no—
405C22⋊Q8—1182no—
406C22⋊Q8—1182no—
407C22⋊Q8D23614yes—
408C4×Q8—22no—
409C4×Q8—41no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—41no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—591no—
419C2×C4⋊C4—1182no—
420C2×C4⋊C4—1182no—
421C2×C4⋊C4—1181no—
422C2×C4⋊C4—1181no—
423C2×C4⋊C4—1181no—
424C2×C4⋊C4—1181no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4C22×D5911yescharacteristic
428C2×C4⋊C4C22×D5911yescharacteristic
429C2×C4⋊C4C22×D5911yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1181no—
432C2×C22⋊C4C22⋊D5911yescharacteristic
433C2×C22⋊C4C22⋊D5911yescharacteristic
434C2×C42—21no—
435C2×C42—21no—
436C2×C42C22×D5911yescharacteristic
437C2×C42C22×D5911yescharacteristic
438#107—1182no—
439#107—1182no—
440#107—1182no—
441C4⋊C8—1182no—
442C4⋊C8—2361no—
443#103—1182no—
444#103—1182no—
445#103—1182no—
446#103—1182no—
447#103—1182no—
448#103—2361no—
449C22⋊C8—1182no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—1181no—
455#95—1181no—
456#95—1181no—
457#95—1181no—
458#95C22⋊D5911yescharacteristic
459#95C22⋊D5911yescharacteristic
460C24C2×C22⋊D5911yescharacteristic
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—1181no—
467C2×Q8—1181no—
468C2×Q8—1181no—
469C2×Q8—1181no—
470C2×Q8—1181no—
471C2×Q8—1181no—
472C2×Q8C2×D23612yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—81no—
484C22×C4—1181no—
485C22×C4—1181no—
486C22×C4—1181no—
487C22×C4—1181no—
488C22×C4—4721no—
489C22×C4—4721no—
490C22×C4—2362no—
491C22×C4C2×D23612yes—
492C22×C4#1174568512yes—
493C22×C4C2×C22⋊D5911yescharacteristic
494C22×C4C2×C22⋊D5911yescharacteristic
495C22×C4C2×C22⋊D5911yescharacteristic
496C22×C4C2×C22⋊D5911yescharacteristic
497C22×C4C2×C22⋊D5911yescharacteristic
498C22×C4C23×D5911yescharacteristic
499Q16—2364no—
500C2×C8—592no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8—1181no—
510C2×C8—1181no—
511C2×C8—1181no—
512C2×C8—2364no—
513C4⋊C4—24no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—81no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—1181no—
524C4⋊C4—1181no—
525C4⋊C4—1181no—
526C4⋊C4—1181no—
527C4⋊C4—4721no—
528C4⋊C4—4721no—
529C4⋊C4—4721no—
530C4⋊C4—4721no—
531C4⋊C4—2362no—
532C4⋊C4—2362no—
533C4⋊C4C2×D23612yes—
534C4⋊C4#1174568712yes—
535C22⋊C4—24no—
536C22⋊C4—22no—
537C22⋊C4—2362no—
538C22⋊C4#1174567214yescomplemented
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—42no—
544C42—81no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23#1427370112yes—
550C23C2×#1174567212yes—
551C23#1427375712yes—
552C23C22×C22⋊D5911yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—41no—
556Q8—41no—
557Q8—2361no—
558Q8—2361no—
559Q8—2361no—
560Q8—2361no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—4724no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4—2361no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4#1427370112yes—
604C2×C4C2×#1174567212yes—
605C2×C4C2×#1174567212yes—
606C2×C4C22×D23611yescharacteristic
607C2×C4#41×D5911yescharacteristic
608C2×C4C2×#1174568711yescharacteristic
609C2×C4C2×#1174568511yescharacteristic
610C2×C4#1427377211yescharacteristic
611C2×C4#1427377411yescharacteristic
612C8—1182no—
613C8—2361no—
614C8—2361no—
615C8—2361no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22#1683761214yes—
623C22C2×#1427370111yescharacteristic
624C22#1683804611yescharacteristic
625C22#1683805111yescharacteristic
626C22C22×#1174567211yescharacteristic
627C22C2×#1427375711yescharacteristic
628C22#1683820511yescharacteristic
629C22#1683821011yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—41no—
634C4—42no—
635C4—42no—
636C4—41no—
637C4—42no—
638C4—81no—
639C4—4721no—
640C4—4721no—
641C4—4721no—
642C4—4721no—
643C4#1683793211yescharacteristic
644C4#1683799411yescharacteristic
645C4#1683804711yescharacteristic
646C4#1683805611yescharacteristic
647C2—24no—
648C2#2142764111yescharacteristic
649C2C2×#1683761211yescharacteristic
650C2#2143363511yescharacteristic
651C2#2143597211yescharacteristic
652C2#2143617711yescharacteristic
653C2#2143648411yescharacteristic
654C2#2143655111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2360C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2360D59
hypocenterC59#25206
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
4. #21425567C2semidirect
5. #21425504C2semidirect
6. #21425488C2semidirect
7. #21425418C2semidirect
8. #21424854C2semidirect
9. #21424737C2semidirect
10. #21424437C2semidirect
13. #21424101C2semidirect
183. C22⋊C4×C59Q16semidirect
388. C59#25206semidirect
538. C22⋊C4#11745672semidirect

Maximal subgroups

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

subgroupindex
C2×#168377352
C2×#168376412
#214285742
#214255672
#214255042
#214254882
#214254182
#214248542
#214247372
#214244372
#214244032
#214242762
#214241012
C2×#168371642
C59×#23602
#2520659