Finite Group Databasebeta

← #28656806

Subgroups — #28656806

Order 15104.

764 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427099C211yesmaximal, characteristic
2#21425418C211yesmaximal, characteristic, complemented
3#21425299C211yesmaximal, characteristic
4#21425234C211yesmaximal, characteristic, complemented
5#21425104C211yesmaximal, characteristic
6#21425045C211yesmaximal, characteristic, complemented
7#21424775C211yesmaximal, characteristic, complemented
8#21424734C211yesmaximal, characteristic, complemented
9#21424704C211yesmaximal, characteristic
10#21424688C211yesmaximal, characteristic, complemented
11#21424440C211yesmaximal, characteristic, complemented
12#21424404C211yesmaximal, characteristic
13#21424184C211yesmaximal, characteristic
14C4×#14273650C211yesmaximal, characteristic, complemented
15C59×#2492C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C2×#14273706C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19C22×C4⋊Dic59C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×C22⋊(C59⋊C8)C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837308C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837299—21no—
34#16837289—21no—
35#16837274—21no—
36C4×C4⋊Dic59C2211yescharacteristic
37C22⋊C4×Dic59—21no—
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×C4⋊Q8×C59C2211yescharacteristic
49C2×C22⋊Q8×C59C2211yescharacteristic
50C59×#400—21no—
51C59×#393—21no—
52C59×#391—21no—
53C59×#391C2211yescharacteristic
54C59×#391C2211yescharacteristic
55C59×#388—21no—
56C59×#380C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59C2211yescharacteristic
59C22×Dic118C2311yescharacteristic
60C23×Dic59—21no—
61C2×#11745672—22no—
62C2×#11745672—41no—
63C2×#11745672—41no—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273706—22no—
69#14273706—22no—
70C4×Dic118—22no—
71C4×Dic118—41no—
72C2×C4⋊Dic59—22no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59—41no—
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×C22⋊Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84#14273660—22no—
85C4⋊(C59⋊C8)—41no—
86#14273651—22no—
87#14273651—22no—
88#14273651—41no—
89#14273650—22no—
90#14273650—22no—
91#14273650—41no—
92C22⋊(C59⋊C8)—41no—
93C22⋊(C59⋊C8)—41no—
94C4×C59⋊C8—41no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C22×Q8×C59—21no—
100C22×Q8×C59C2311yescharacteristic
101C23×C4×C59C2311yescharacteristic
102C4⋊Q8×C59—22no—
103C4⋊Q8×C59—41no—
104C22⋊Q8×C59—22no—
105C22⋊Q8×C59—41no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59D411yescharacteristic
114C2×C22⋊C4×C59D411yescharacteristic
115C2×C42×C59—21no—
116C2×C42×C59C2311yescharacteristic
117C2×C42×C59C2311yescharacteristic
118#95×C59—21no—
119#95×C59—21no—
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124#95×C59D411yescharacteristic
125#95×C59D411yescharacteristic
126C24×C59C2×D411yescharacteristic
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59—21no—
132C2×Q8×C59—21no—
133C2×Q8×C59—42no—
134C2×Q8×C59—42no—
135C2×Q8×C59—81no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
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×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2411yescharacteristic
152C22×Dic59—21no—
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—42no—
157C22×Dic59—42no—
158C22×Dic59—81no—
159C2×Dic118—21no—
160C2×Dic118—21no—
161C2×Dic118—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118#4112yes—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—42no—
171C4⋊C4×C59—42no—
172C4⋊C4×C59#4112yes—
173C22⋊C4×C59—41no—
174C22⋊C4×C59—42no—
175C22⋊C4×C59—81no—
176C22⋊C4×C59Q1614yescomplemented
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—21no—
180C42×C59—21no—
181C42×C59—81no—
182C22⋊Dic59—24no—
183#11745672—44no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—42no—
189C4⋊Dic59#4112yes—
190C4⋊Dic59#4112yes—
191Dic59⋊C4—24no—
192Dic59⋊C4—42no—
193Dic59⋊C4—81no—
194C4×Dic59—42no—
195C4×Dic59—81no—
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—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59C2×Q1612yes—
215C23×C59C22×D411yescharacteristic
216Q8×C59—41no—
217Q8×C59—41no—
218Q8×C59—41no—
219Q8×C59—41no—
220Q8×C59—84no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—22no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—41no—
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×C59C2×Q1612yes—
251C2×C4×C59C2×Q1612yes—
252C2×C4×C59C2×#4111yescharacteristic
253C2×C4×C59C2×#4111yescharacteristic
254C2×C4×C59C2×#4111yescharacteristic
255C2×C4×C59C2×#4111yescharacteristic
256C2×C4×C59H(5,𝔽2)11yescharacteristic
257C2×C4×C59H(5,𝔽2)11yescharacteristic
258C2×Dic59—22no—
259C2×Dic59—22no—
260C2×Dic59—22no—
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—
270C2×Dic59—84no—
271Dic118—22no—
272Dic118—22no—
273Dic118—41no—
274Dic118—41no—
275C59⋊C8—41no—
276C59⋊C8—41no—
277C59⋊C8—41no—
278C59⋊C8—41no—
279#26726—591nomaximal, Sylow
280C22×C59—22no—
281C22×C59—22no—
282C22×C59—22no—
283C22×C59—41no—
284C22×C59—41no—
285C22×C59—41no—
286C22×C59—41no—
287C22×C59#52511yescharacteristic
288C22×C59#54611yescharacteristic
289C22×C59#54611yescharacteristic
290C22×C59C22×Q1611yescharacteristic
291C22×C59#57511yescharacteristic
292C22×C59#57711yescharacteristic
293C22×C59#57711yescharacteristic
294C4×C59—22no—
295C4×C59—22no—
296C4×C59—41no—
297C4×C59—41no—
298C4×C59—41no—
299C4×C59—42no—
300C4×C59—81no—
301C4×C59—81no—
302C4×C59—81no—
303C4×C59#52911yescharacteristic
304C4×C59#53811yescharacteristic
305C4×C59#55511yescharacteristic
306C4×C59#55911yescharacteristic
307Dic59—42no—
308Dic59—42no—
309Dic59—42no—
310Dic59—81no—
311#2492D5911yescharacteristic
312#2360—591no—
313#2056—591no—
314#2036—591no—
315#2024—591no—
316#2001—591no—
317#1987—591no—
318#1933—591no—
319#1928—591no—
320#1923—591no—
321#1921—591no—
322#1878—591no—
323#1872—591no—
324#1824—591no—
325C4×#103—591no—
326C2×C59—42no—
327C2×C59#245911yescharacteristic
328C2×C59#310411yescharacteristic
329C2×C59#312511yescharacteristic
330C2×C59#329911yescharacteristic
331C2×C59#329911yescharacteristic
332C2×C59#331111yescharacteristic
333C2×C59#332311yescharacteristic
334C22×Q16—591no—
335C2×C4⋊Q8C2×D5911yescharacteristic
336C2×C22⋊Q8—591no—
337C2×C22⋊Q8C2×D5911yescharacteristic
338C2×C4×Q8—591no—
339C22×C4⋊C4—591no—
340C2×#107—591no—
341C2×C4⋊C8—591no—
342C2×#103—591no—
343C2×#103—591no—
344C2×#103—591no—
345C2×#103—591no—
346C2×#103—591no—
347C2×#103—591no—
348C2×C22⋊C8—591no—
349C2×C22⋊C8—591no—
350C2×C4×C8—591no—
351#400—21no—
352#393—21no—
353#391—21no—
354#391—591no—
355#391C2×D5911yescharacteristic
356#391C2×D5911yescharacteristic
357#389—591no—
358#389—1181no—
359#388—21no—
360#387—1181no—
361#385—1181no—
362#380C2×D5911yescharacteristic
363C4×C4⋊C4—591no—
364C4×C4⋊C4C2×D5911yescharacteristic
365C4×C22⋊C4—1181no—
366C4×C22⋊C4C2×D5911yescharacteristic
367#340—591no—
368#340—591no—
369#340—591no—
370#340—591no—
371#340—591no—
372#340—591no—
373#340—591no—
374#340—591no—
375#336—591no—
376#336—591no—
377C59#2672611yescharacteristic, complemented, Sylow
378C22×Q8—21no—
379C22×Q8—591no—
380C22×Q8C22×D5911yescharacteristic
381C23×C4—1181no—
382C23×C4C22×D5911yescharacteristic
383C2×Q16—1182no—
384C2×Q16—2361no—
385C2×Q16—2361no—
386C22×C8—591no—
387C22×C8—591no—
388C22×C8—591no—
389C22×C8—591no—
390C4⋊Q8—22no—
391C4⋊Q8—41no—
392C22⋊Q8—22no—
393C22⋊Q8—41no—
394C22⋊Q8—1182no—
395C22⋊Q8—1182no—
396C4×Q8—1182no—
397C4×Q8—2361no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—591no—
401C2×C4⋊C4—591no—
402C2×C4⋊C4—591no—
403C2×C4⋊C4—1182no—
404C2×C4⋊C4—1181no—
405C2×C4⋊C4—1181no—
406C2×C4⋊C4—1181no—
407C2×C4⋊C4—1181no—
408C2×C4⋊C4—2361no—
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⋊C4C22⋊D5911yescharacteristic
416C2×C22⋊C4C22⋊D5911yescharacteristic
417C2×C42—21no—
418C2×C42—1181no—
419C2×C42—1181no—
420C2×C42C22×D5911yescharacteristic
421C2×C42C22×D5911yescharacteristic
422#107—1182no—
423C4⋊C8—2361no—
424#103—1182no—
425#103—1182no—
426#103—1182no—
427#103—1182no—
428#103—2361no—
429#103—2361no—
430C22⋊C8—2361no—
431C22⋊C8—2361no—
432C4×C8—2361no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—1181no—
440#95—1181no—
441#95—1181no—
442#95—1181no—
443#95C22⋊D5911yescharacteristic
444#95C22⋊D5911yescharacteristic
445C24C2×C22⋊D5911yescharacteristic
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—81no—
455C2×Q8—592no—
456C2×Q8—1181no—
457C2×Q8—1181no—
458C2×Q8—1181no—
459C2×Q8—1181no—
460C2×Q8—1181no—
461C22×C4—21no—
462C22×C4—21no—
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—1181no—
472C22×C4—1181no—
473C22×C4—1181no—
474C22×C4—1181no—
475C22×C4—4721no—
476C22×C4—2362no—
477C22×C4—2362no—
478C22×C4C2×C22⋊D5911yescharacteristic
479C22×C4C2×C22⋊D5911yescharacteristic
480C22×C4C2×C22⋊D5911yescharacteristic
481C22×C4C2×C22⋊D5911yescharacteristic
482C22×C4C2×C22⋊D5911yescharacteristic
483C22×C4C23×D5911yescharacteristic
484Q16—2364no—
485C2×C8—1181no—
486C2×C8—1181no—
487C2×C8—1181no—
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—1181no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—592no—
505C4⋊C4—592no—
506C4⋊C4—1184no—
507C4⋊C4—1181no—
508C4⋊C4—1181no—
509C4⋊C4—1181no—
510C4⋊C4—1181no—
511C4⋊C4—4721no—
512C4⋊C4—2362no—
513C4⋊C4—2362no—
514C4⋊C4#1174568712yes—
515C22⋊C4—41no—
516C22⋊C4—42no—
517C22⋊C4—81no—
518C22⋊C4—1184no—
519C22⋊C4#1174567214yescomplemented
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—81no—
525C42—4721no—
526C42—2362no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23C2×#1174567212yes—
534C23C22×C22⋊D5911yescharacteristic
535Q8—41no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
539Q8—84no—
540Q8—1182no—
541Q8—1182no—
542Q8—2361no—
543Q8—2361no—
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—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
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—1182no—
575C2×C4—1182no—
576C2×C4—4724no—
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—2361no—
586C2×C4C2×#1174567212yes—
587C2×C4C2×#1174567212yes—
588C2×C4#41×D5911yescharacteristic
589C2×C4#41×D5911yescharacteristic
590C2×C4#41×D5911yescharacteristic
591C2×C4C2×#1174568711yescharacteristic
592C2×C4#1427377211yescharacteristic
593C2×C4#1427377311yescharacteristic
594C8—2361no—
595C8—2361no—
596C8—2361no—
597C8—2361no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22#1683790911yescharacteristic
606C22#1683804011yescharacteristic
607C22#1683804611yescharacteristic
608C22C22×#1174567211yescharacteristic
609C22#1683820011yescharacteristic
610C22#1683821011yescharacteristic
611C22#1683821311yescharacteristic
612C4—22no—
613C4—22no—
614C4—41no—
615C4—41no—
616C4—42no—
617C4—41no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—4721no—
622C4—2362no—
623C4—2362no—
624C4—2362no—
625C4#1683793111yescharacteristic
626C4#1683798611yescharacteristic
627C4#1683811411yescharacteristic
628C4#1683813411yescharacteristic
629C2—42no—
630C2#2142802911yescharacteristic
631C2#2143413111yescharacteristic
632C2#2143436711yescharacteristic
633C2#2143596611yescharacteristic
634C2#2143597211yescharacteristic
635C2#2143612611yescharacteristic
636C2#2143631211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2492C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2492D59
hypocenterC59#26726
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2492

Complements

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

NH ≅ G/Ntype
2. #21425418C2semidirect
4. #21425234C2semidirect
6. #21425045C2semidirect
7. #21424775C2semidirect
8. #21424734C2semidirect
10. #21424688C2semidirect
11. #21424440C2semidirect
14. C4×#14273650C2semidirect
176. C22⋊C4×C59Q16semidirect
377. C59#26726semidirect
519. C22⋊C4#11745672semidirect

Maximal subgroups

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

subgroupindex
#214270992
#214254182
#214252992
#214252342
#214251042
#214250452
#214247752
#214247342
#214247042
#214246882
#214244402
#214244042
#214241842
C4×#142736502
C59×#24922
#2672659