Finite Group Databasebeta

← #28634410

Subgroups — #28634410

Order 15104.

794 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837641C211yesmaximal, characteristic
2C2×C4×#11745672C211yesmaximal, characteristic, complemented
3#21427599C211yesmaximal, characteristic
4#21425515C211yesmaximal, characteristic, complemented
5#21425514C211yesmaximal, characteristic, complemented
6#21425292C211yesmaximal, characteristic
7#21425289C211yesmaximal, characteristic
8#21424956C211yesmaximal, characteristic, complemented
9#21424826C211yesmaximal, characteristic, complemented
10#21424364C211yesmaximal, characteristic, complemented
11#21424363C211yesmaximal, characteristic, complemented
12#21424258C211yesmaximal, characteristic, complemented
13#21424123C211yesmaximal, characteristic
14C22⋊C4×C59⋊C8C211yesmaximal, characteristic
15C59×#2421C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C23×C59⋊C8C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20#16837641—24no—
21C4×#11745672—24no—
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273660C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×#14273651C2211yescharacteristic
30C2×#14273650C2211yescharacteristic
31C2×#14273650C2211yescharacteristic
32C2×#14273650C2211yescharacteristic
33C2×C22⋊(C59⋊C8)C2211yescharacteristic
34C2×C4×C59⋊C8C2211yescharacteristic
35C2×C4×C59⋊C8C2211yescharacteristic
36#16837343—21no—
37#16837330—21no—
38#16837308C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837296—21no—
41#16837284—21no—
42#16837268C2211yescharacteristic
43#16837262C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C22⋊Q8×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C59×#400—21no—
53C59×#397—21no—
54C59×#391C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C59×#388—21no—
57C59×#387—21no—
58C59×#382C2211yescharacteristic
59C59×#380C2211yescharacteristic
60C4×C22⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C2×#11745672—22no—
63C2×#11745672—22no—
64C2×#11745672D414yes—
65C22×C59⋊C8—22no—
66C22×C59⋊C8—22no—
67C22×C59⋊C8—22no—
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72#14273706—22no—
73#14273706—41no—
74C4×Dic118—22no—
75C4×Dic118—41no—
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×C22⋊Dic59—21no—
83C2×C22⋊Dic59—21no—
84C2×C4×Dic59—21no—
85#14273660—22no—
86#14273660—22no—
87#14273660—22no—
88#14273658—41no—
89#14273658—41no—
90#14273651—22no—
91#14273651—22no—
92#14273651—22no—
93#14273650—22no—
94#14273650—22no—
95#14273650—22no—
96C22⋊(C59⋊C8)—22no—
97C4×C59⋊C8—22no—
98C4×C59⋊C8—22no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C22×Q8×C59C2311yescharacteristic
106C23×C4×C59C2311yescharacteristic
107C22⋊Q8×C59—22no—
108C22⋊Q8×C59—41no—
109C4×Q8×C59—22no—
110C4×Q8×C59—41no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59D411yescharacteristic
120C2×C22⋊C4×C59D411yescharacteristic
121C2×C42×C59—21no—
122C2×C42×C59C2311yescharacteristic
123C2×C42×C59C2311yescharacteristic
124#95×C59—21no—
125#95×C59—21no—
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—21no—
148C22×C4×C59—21no—
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2411yescharacteristic
155C22×Dic59—21no—
156C22×Dic59—21no—
157C22×Dic59—21no—
158C22×Dic59—21no—
159C2×Dic118—21no—
160C2×Dic118—21no—
161C2×Dic118—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118C2×D412yes—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—42no—
172C4⋊C4×C59—42no—
173C4⋊C4×C59—42no—
174C22⋊C4×C59—22no—
175C22⋊C4×C59—22no—
176C22⋊C4×C59—42no—
177C22⋊C4×C59—42no—
178C42×C59—21no—
179C42×C59—21no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—42no—
183C22⋊Dic59—42no—
184C22⋊Dic59—42no—
185#11745672—28no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192Dic59⋊C4—42no—
193Dic59⋊C4—42no—
194Dic59⋊C4—42no—
195C4×Dic59—42no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—44no—
205C2×C59⋊C8C2×D412yes—
206C2×C59⋊C8#4112yes—
207C2×C59⋊C8#4112yes—
208C2×C59⋊C8#4112yes—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59#13512yes—
216C23×C59C22×D411yescharacteristic
217Q8×C59—22no—
218Q8×C59—22no—
219Q8×C59—41no—
220Q8×C59—41no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—22no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
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—21no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59#13512yes—
252C2×C4×C59C22×D411yescharacteristic
253C2×C4×C59C2×#4111yescharacteristic
254C2×C4×C59C2×#4111yescharacteristic
255C2×C4×C59C2×#4111yescharacteristic
256C2×C4×C59H(5,𝔽2)11yescharacteristic
257C2×C4×C592−1+411yescharacteristic
258C2×Dic59—22no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270Dic118—22no—
271Dic118—22no—
272Dic118—41no—
273Dic118—41no—
274C59⋊C8—22no—
275C59⋊C8—41no—
276C59⋊C8—41no—
277C59⋊C8#12314yes—
278#25033—591nomaximal, Sylow
279C22×C59—22no—
280C22×C59—22no—
281C22×C59—22no—
282C22×C59—41no—
283C22×C59—41no—
284C22×C59—41no—
285C22×C59—41no—
286C22×C59#53211yescharacteristic
287C22×C59D4211yescharacteristic
288C22×C59#54711yescharacteristic
289C22×C59C2×#13511yescharacteristic
290C22×C59C2×#13511yescharacteristic
291C22×C59#57811yescharacteristic
292C22×C59#57811yescharacteristic
293C4×C59—22no—
294C4×C59—41no—
295C4×C59—41no—
296C4×C59—41no—
297C4×C59—42no—
298C4×C59—41no—
299C4×C59—81no—
300C4×C59—81no—
301C4×C59—81no—
302C4×C59C2×#12311yescharacteristic
303C4×C59#54211yescharacteristic
304C4×C59#54711yescharacteristic
305C4×C59#55311yescharacteristic
306Dic59—42no—
307Dic59—81no—
308Dic59—81no—
309Dic59—81no—
310C2×#467—591no—
311C2×C4×Q16—591no—
312#2421—591no—
313#2421D5911yescharacteristic
314#2074—591no—
315#2074—591no—
316#2035—591no—
317#2035—591no—
318#1969—591no—
319#1942—591no—
320#1865—591no—
321#1865—591no—
322#1846—591no—
323#1808—591no—
324C8×C22⋊C4—591no—
325C2×C59—42no—
326C2×C59#242211yescharacteristic
327C2×C59#316811yescharacteristic
328C2×C59#317711yescharacteristic
329C2×C59#328511yescharacteristic
330C2×C59#328511yescharacteristic
331C2×C59#333611yescharacteristic
332C2×C59#334211yescharacteristic
333C22×Q16—591no—
334C23×C8—591no—
335C2×C22⋊Q8—591no—
336C2×C22⋊Q8C2×D5911yescharacteristic
337C2×C4×Q8—591no—
338C2×C4×Q8C2×D5911yescharacteristic
339#467—1184no—
340C4×Q16—1184no—
341C2×#107—591no—
342C2×#107—591no—
343C2×#107—591no—
344C2×#106—591no—
345C2×#106—591no—
346C2×#103—591no—
347C2×#103—591no—
348C2×#103—591no—
349C2×#103—591no—
350C2×#103—591no—
351C2×#103—591no—
352C2×C22⋊C8—591no—
353C2×C4×C8—591no—
354C2×C4×C8—591no—
355#400—21no—
356#400—1181no—
357#397—21no—
358#397—1181no—
359#391—591no—
360#391C2×D5911yescharacteristic
361#389—591no—
362#389C2×D5911yescharacteristic
363#388—21no—
364#388—1181no—
365#387—21no—
366#387—1181no—
367#382—591no—
368#382C2×D5911yescharacteristic
369#380—591no—
370#380C2×D5911yescharacteristic
371C4×C22⋊C4C2×D5911yescharacteristic
372#340—591no—
373#340—591no—
374#340—591no—
375#340—591no—
376#336—591no—
377#336—591no—
378C59#2503311yescharacteristic, complemented, Sylow
379C22×Q8—591no—
380C22×Q8C22×D5911yescharacteristic
381C23×C4C22×D5911yescharacteristic
382C2×Q16—594no—
383C2×Q16—1182no—
384C2×Q16—1182no—
385C22×C8—591no—
386C22×C8—591no—
387C22×C8—591no—
388C22×C8—591no—
389C22×C8—1182no—
390C22×C8—1182no—
391C22×C8—1182no—
392C22⋊Q8—22no—
393C22⋊Q8—41no—
394C22⋊Q8—1182no—
395C22⋊Q8—2361no—
396C4×Q8—22no—
397C4×Q8—41no—
398C4×Q8—1182no—
399C4×Q8—2361no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—591no—
404C2×C4⋊C4—591no—
405C2×C4⋊C4—591no—
406C2×C4⋊C4—1181no—
407C2×C4⋊C4—1181no—
408C2×C4⋊C4—1181no—
409C2×C4⋊C4C22×D5911yescharacteristic
410C2×C4⋊C4C22×D5911yescharacteristic
411C2×C4⋊C4C22×D5911yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—1181no—
415C2×C22⋊C4—1181no—
416C2×C22⋊C4C22⋊D5911yescharacteristic
417C2×C22⋊C4C22⋊D5911yescharacteristic
418C2×C42—21no—
419C2×C42—1181no—
420C2×C42C22×D5911yescharacteristic
421C2×C42C22×D5911yescharacteristic
422#107—1182no—
423#107—1182no—
424#107—1182no—
425#106—2361no—
426#106—2361no—
427#103—1182no—
428#103—1182no—
429#103—1182no—
430#103—1182no—
431#103—1182no—
432#103—1182no—
433C22⋊C8—1182no—
434C4×C8—1182no—
435C4×C8—1182no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—1181no—
443#95—1181no—
444#95—1181no—
445#95—1181no—
446#95—1181no—
447#95—1181no—
448#95C22⋊D5911yescharacteristic
449#95C22⋊D5911yescharacteristic
450C24C2×C22⋊D5911yescharacteristic
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—592no—
457C2×Q8—1181no—
458C2×Q8—1181no—
459C2×Q8—1181no—
460C2×Q8—1181no—
461C2×Q8—1181no—
462C2×Q8D4×D5912yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—1181no—
474C22×C4—1181no—
475C22×C4—1181no—
476C22×C4—1181no—
477C22×C4C2×C22⋊D5911yescharacteristic
478C22×C4C2×C22⋊D5911yescharacteristic
479C22×C4C2×C22⋊D5911yescharacteristic
480C22×C4C2×C22⋊D5911yescharacteristic
481C22×C4C2×C22⋊D5911yescharacteristic
482C22×C4C23×D5911yescharacteristic
483Q16—1188no—
484C2×C8—592no—
485C2×C8—592no—
486C2×C8—592no—
487C2×C8—592no—
488C2×C8—1181no—
489C2×C8—1181no—
490C2×C8—1181no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—2364no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—1181no—
509C4⋊C4—1181no—
510C4⋊C4—1181no—
511C4⋊C4—1181no—
512C4⋊C4—2362no—
513C4⋊C4—2362no—
514C4⋊C4—2362no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—2362no—
520C22⋊C4—2362no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—42no—
526C42—2362no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23#1427374712yes—
534C23C22×C22⋊D5911yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—41no—
538Q8—41no—
539Q8—1182no—
540Q8—1182no—
541Q8—2361no—
542Q8—2361no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—1182no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4—2361no—
578C2×C4—2361no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4—2361no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4#1427374712yes—
586C2×C4C2×D4×D5911yescharacteristic
587C2×C4#41×D5911yescharacteristic
588C2×C4#41×D5911yescharacteristic
589C2×C4#41×D5911yescharacteristic
590C2×C4#1427377211yescharacteristic
591C2×C4#1427377411yescharacteristic
592C8—594no—
593C8—1182no—
594C8—2361no—
595C8—2361no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22#1683794811yescharacteristic
604C22D4×C22⋊D5911yescharacteristic
605C22#1683805111yescharacteristic
606C22C2×#1427374711yescharacteristic
607C22C2×#1427374711yescharacteristic
608C22#1683821711yescharacteristic
609C22#1683821711yescharacteristic
610C4—22no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—42no—
615C4—41no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—4721no—
620C4—4721no—
621C4—4721no—
622C4—2362no—
623C4#123×D5911yescharacteristic
624C4#1683802011yescharacteristic
625C4#1683805011yescharacteristic
626C4#1683809711yescharacteristic
627C2—42no—
628C2#2142760711yescharacteristic
629C2#2143466011yescharacteristic
630C2#2143474911yescharacteristic
631C2#2143583311yescharacteristic
632C2#2143583311yescharacteristic
633C2#2143645811yescharacteristic
634C2#2143652211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2421C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2421D59
hypocenterC59#25033
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2421

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11745672C2semidirect
4. #21425515C2semidirect
5. #21425514C2semidirect
8. #21424956C2semidirect
9. #21424826C2semidirect
10. #21424364C2semidirect
11. #21424363C2semidirect
12. #21424258C2semidirect
378. C59#25033semidirect

Maximal subgroups

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

subgroupindex
C2×#168376412
C2×C4×#117456722
#214275992
#214255152
#214255142
#214252922
#214252892
#214249562
#214248262
#214243642
#214243632
#214242582
#214241232
C22⋊C4×C59⋊C82
C59×#24212
#2503359