Finite Group Databasebeta

← #28656808

Subgroups — #28656808

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#21428470C211yesmaximal, characteristic
2#21427099C211yesmaximal, characteristic
3#21425419C211yesmaximal, characteristic, complemented
4#21425300C211yesmaximal, characteristic
5#21425238C211yesmaximal, characteristic, complemented
6#21425106C211yesmaximal, characteristic
7#21425043C211yesmaximal, characteristic, complemented
8#21424772C211yesmaximal, characteristic, complemented
9#21424738C211yesmaximal, characteristic, complemented
10#21424689C211yesmaximal, characteristic, complemented
11#21424436C211yesmaximal, characteristic, complemented
12#21424401C211yesmaximal, characteristic
13#21424187C211yesmaximal, characteristic
14C4×#14273652C211yesmaximal, characteristic, complemented
15C59×#1923C211yesmaximal, characteristic
16C22×Dic236C2211yescharacteristic
17C2×#14273727C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C2×#14273706C2211yescharacteristic
20C2×C4×Dic118C2211yescharacteristic
21C22×C4⋊Dic59C2211yescharacteristic
22C2×#14273661C2211yescharacteristic
23C2×#14273652C2211yescharacteristic
24C2×#14273652C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273652C2211yescharacteristic
27C2×#14273652C2211yescharacteristic
28C2×#14273652C2211yescharacteristic
29#16837344—21no—
30#16837314—21no—
31#16837308C2211yescharacteristic
32#16837308C2211yescharacteristic
33#16837308C2211yescharacteristic
34#16837306—21no—
35#16837300C2211yescharacteristic
36#16837299—21no—
37#16837295—21no—
38#16837289—21no—
39#16837274—21no—
40#16837262C2211yescharacteristic
41C4×C4⋊Dic59C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43C22⋊C4×Dic59—21no—
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52C2×C4⋊C8×C59C2211yescharacteristic
53C2×C22⋊C8×C59C2211yescharacteristic
54C2×C22⋊C8×C59C2211yescharacteristic
55C2×C4×C8×C59C2211yescharacteristic
56C4×C22⋊C4×C59C2211yescharacteristic
57C59×#336C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C22×Dic118—21no—
60C22×Dic118C2311yescharacteristic
61C22×Dic118C2311yescharacteristic
62C23×Dic59—21no—
63C2×Dic236—22no—
64C2×Dic236—41no—
65C2×Dic236—41no—
66#14273727—22no—
67#14273727—41no—
68#14273706—22no—
69#14273706—22no—
70#14273706—22no—
71#14273706—41no—
72C4×Dic118—22no—
73C4×Dic118—41no—
74C2×C4⋊Dic59—22no—
75C2×C4⋊Dic59—21no—
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59—41no—
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×C4⋊Dic59C2311yescharacteristic
84C2×Dic59⋊C4—21no—
85C2×Dic59⋊C4—21no—
86C2×Dic59⋊C4—21no—
87C2×Dic59⋊C4—21no—
88C2×C22⋊Dic59—21no—
89C2×C22⋊Dic59—21no—
90C2×C22⋊Dic59—21no—
91C2×C4×Dic59—21no—
92C2×C4×Dic59—21no—
93C2×C4×Dic59—21no—
94#14273661—22no—
95#14273652—22no—
96#14273652—22no—
97#14273652—22no—
98#14273652—22no—
99#14273652—41no—
100#14273652—41no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C23×C4×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C22×C8×C59C2311yescharacteristic
115C22×C8×C59C2311yescharacteristic
116C2×C22⋊C4×C59D411yescharacteristic
117C2×C22⋊C4×C59D411yescharacteristic
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120C4⋊C8×C59—41no—
121C22⋊C8×C59—41no—
122C22⋊C8×C59—41no—
123C4×C8×C59—41no—
124#95×C59D411yescharacteristic
125#95×C59D411yescharacteristic
126C24×C59C2×D411yescharacteristic
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2411yescharacteristic
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—42no—
148C22×Dic59—42no—
149C22×Dic59—81no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118—21no—
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118—21no—
159C2×Dic118—21no—
160C2×Dic118—21no—
161C2×Dic118—42no—
162C2×Dic118—42no—
163C2×Dic118—81no—
164C2×Dic118#4112yes—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59—21no—
171C2×C8×C59—21no—
172C2×C8×C59—21no—
173C2×C8×C59—21no—
174C2×C8×C59—21no—
175C2×C8×C59—21no—
176C2×C8×C59—21no—
177C22⋊C4×C59—41no—
178C22⋊C4×C59Q1614yescomplemented
179C42×C59—21no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—21no—
183C22⋊Dic59—24no—
184C22⋊Dic59—42no—
185C22⋊Dic59—81no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—42no—
196C4⋊Dic59—42no—
197C4⋊Dic59#4112yes—
198C4⋊Dic59#4112yes—
199C4⋊Dic59#4112yes—
200Dic59⋊C4—24no—
201Dic59⋊C4—42no—
202Dic59⋊C4—42no—
203Dic59⋊C4—81no—
204C4×Dic59—42no—
205C4×Dic59—81no—
206C4×Dic59—81no—
207Dic236—44no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59C2×Q1612yes—
215C23×C59C22×D411yescharacteristic
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
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—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59C2×Q1612yes—
234C2×C4×C59C2×Q1612yes—
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59H(5,𝔽2)11yescharacteristic
240C2×C4×C59H(5,𝔽2)11yescharacteristic
241C2×Dic59—22no—
242C2×Dic59—22no—
243C2×Dic59—22no—
244C2×Dic59—22no—
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—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—84no—
266Dic118—22no—
267Dic118—22no—
268Dic118—41no—
269Dic118—41no—
270Dic118—41no—
271Dic118—41no—
272Dic118—41no—
273Dic118—41no—
274Dic118—84no—
275C8×C59—41no—
276C8×C59—41no—
277C8×C59—41no—
278C8×C59—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#52911yescharacteristic
300C4×C59#53811yescharacteristic
301C4×C59#55511yescharacteristic
302C4×C59#55911yescharacteristic
303Dic59—42no—
304Dic59—42no—
305Dic59—42no—
306Dic59—42no—
307Dic59—81no—
308Dic59—81no—
309Dic59—81no—
310Dic59—81no—
311#2492—591no—
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#1923D5911yescharacteristic
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⋊Q8—591no—
336C2×C22⋊Q8—591no—
337C2×C22⋊Q8—591no—
338C2×C4×Q8—591no—
339C22×C4⋊C4—591no—
340C2×#107—591no—
341C2×C4⋊C8C2×D5911yescharacteristic
342C2×#103—591no—
343C2×#103—591no—
344C2×#103—591no—
345C2×#103—591no—
346C2×#103—591no—
347C2×#103—591no—
348C2×C22⋊C8C2×D5911yescharacteristic
349C2×C22⋊C8C2×D5911yescharacteristic
350C2×C4×C8C2×D5911yescharacteristic
351#400—1181no—
352#393—1181no—
353#391—591no—
354#391—591no—
355#391—591no—
356#391—1181no—
357#389—591no—
358#389—1181no—
359#388—1181no—
360#387—1181no—
361#385—1181no—
362#380—591no—
363C4×C4⋊C4—591no—
364C4×C4⋊C4—591no—
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#336C2×D5911yescharacteristic
376#336C2×D5911yescharacteristic
377C59#2672611yescharacteristic, complemented, Sylow
378C22×Q8—591no—
379C22×Q8—591no—
380C22×Q8—1181no—
381C23×C4—1181no—
382C23×C4C22×D5911yescharacteristic
383C2×Q16—1182no—
384C2×Q16—2361no—
385C2×Q16—2361no—
386C22×C8C22×D5911yescharacteristic
387C22×C8C22×D5911yescharacteristic
388C22×C8C22×D5911yescharacteristic
389C22×C8C22×D5911yescharacteristic
390C4⋊Q8—1182no—
391C4⋊Q8—2361no—
392C22⋊Q8—1182no—
393C22⋊Q8—1182no—
394C22⋊Q8—1182no—
395C22⋊Q8—2361no—
396C4×Q8—1182no—
397C4×Q8—2361no—
398C2×C4⋊C4—591no—
399C2×C4⋊C4—591no—
400C2×C4⋊C4—591no—
401C2×C4⋊C4—591no—
402C2×C4⋊C4—591no—
403C2×C4⋊C4—591no—
404C2×C4⋊C4—1182no—
405C2×C4⋊C4—1181no—
406C2×C4⋊C4—1181no—
407C2×C4⋊C4—1181no—
408C2×C4⋊C4—1181no—
409C2×C4⋊C4—1181no—
410C2×C4⋊C4—1181no—
411C2×C4⋊C4—2361no—
412C2×C22⋊C4—1181no—
413C2×C22⋊C4—1181no—
414C2×C22⋊C4—1181no—
415C2×C22⋊C4D23611yescharacteristic
416C2×C22⋊C4D23611yescharacteristic
417C2×C42—1181no—
418C2×C42—1181no—
419C2×C42—1181no—
420C2×C42C22×D5911yescharacteristic
421C2×C42C22×D5911yescharacteristic
422#107—1182no—
423C4⋊C8—41no—
424#103—1182no—
425#103—1182no—
426#103—1182no—
427#103—1182no—
428#103—2361no—
429#103—2361no—
430C22⋊C8—41no—
431C22⋊C8—41no—
432C4×C8—41no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438#95—1181no—
439#95—1181no—
440#95—1181no—
441#95—1181no—
442#95—1181no—
443#95D23611yescharacteristic
444#95D23611yescharacteristic
445C24C2×D23611yescharacteristic
446C2×Q8—592no—
447C2×Q8—1181no—
448C2×Q8—1181no—
449C2×Q8—1181no—
450C2×Q8—1181no—
451C2×Q8—1181no—
452C2×Q8—1181no—
453C2×Q8—1181no—
454C2×Q8—1181no—
455C2×Q8—1181no—
456C2×Q8—1181no—
457C2×Q8—1181no—
458C2×Q8—4721no—
459C2×Q8—2362no—
460C2×Q8—2362no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—1181no—
468C22×C4—1181no—
469C22×C4—1181no—
470C22×C4—1181no—
471C22×C4—1181no—
472C22×C4—1181no—
473C22×C4—1181no—
474C22×C4—1181no—
475C22×C4—4721no—
476C22×C4—2362no—
477C22×C4—2362no—
478C22×C4C2×D23611yescharacteristic
479C22×C4C2×D23611yescharacteristic
480C22×C4C2×D23611yescharacteristic
481C22×C4C2×D23611yescharacteristic
482C22×C4C2×D23611yescharacteristic
483C22×C4C23×D5911yescharacteristic
484Q16—2364no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C4⋊C4—592no—
498C4⋊C4—592no—
499C4⋊C4—592no—
500C4⋊C4—1184no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
503C4⋊C4—1181no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—1181no—
509C4⋊C4—1181no—
510C4⋊C4—4721no—
511C4⋊C4—2362no—
512C4⋊C4—2362no—
513C4⋊C4—2362no—
514C4⋊C4—2362no—
515C22⋊C4—41no—
516C22⋊C4—1184no—
517C22⋊C4—4721no—
518C22⋊C4—2362no—
519C22⋊C4Dic23614yescomplemented
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—4721no—
525C42—4721no—
526C42—2362no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23C2×Dic23612yes—
534C23C22×D23611yescharacteristic
535Q8—1182no—
536Q8—1182no—
537Q8—4724no—
538Q8—2361no—
539Q8—2361no—
540Q8—2361no—
541Q8—2361no—
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—21no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—1182no—
562C2×C4—1182no—
563C2×C4—1182no—
564C2×C4—1182no—
565C2×C4—4724no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2361no—
573C2×C4—2361no—
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—2361no—
586C2×C4C2×Dic23612yes—
587C2×C4C2×Dic23612yes—
588C2×C4#41×D5911yescharacteristic
589C2×C4#41×D5911yescharacteristic
590C2×C4#41×D5911yescharacteristic
591C2×C4#41×D5911yescharacteristic
592C2×C4#1427377211yescharacteristic
593C2×C4#1427377211yescharacteristic
594C8—41no—
595C8—41no—
596C8—41no—
597C8—41no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22#1683791411yescharacteristic
606C22#1683804711yescharacteristic
607C22#1683804711yescharacteristic
608C22C22×Dic23611yescharacteristic
609C22#1683820211yescharacteristic
610C22#1683821411yescharacteristic
611C22#1683821411yescharacteristic
612C4—22no—
613C4—22no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—4721no—
618C4—4721no—
619C4—4721no—
620C4—4721no—
621C4—2362no—
622C4—2362no—
623C4—2362no—
624C4—2362no—
625C4#1683793511yescharacteristic
626C4#1683798411yescharacteristic
627C4#1683811411yescharacteristic
628C4#1683813611yescharacteristic
629C2—42no—
630C2#2142803111yescharacteristic
631C2#2143413611yescharacteristic
632C2#2143437511yescharacteristic
633C2#2143597311yescharacteristic
634C2#2143597311yescharacteristic
635C2#2143613411yescharacteristic
636C2#2143631411yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1923C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#1923D59
hypocenterC59#26726
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
3. #21425419C2semidirect
5. #21425238C2semidirect
7. #21425043C2semidirect
8. #21424772C2semidirect
9. #21424738C2semidirect
10. #21424689C2semidirect
11. #21424436C2semidirect
14. C4×#14273652C2semidirect
178. C22⋊C4×C59Q16semidirect
377. C59#26726semidirect
519. C22⋊C4Dic236semidirect

Maximal subgroups

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

subgroupindex
#214284702
#214270992
#214254192
#214253002
#214252382
#214251062
#214250432
#214247722
#214247382
#214246892
#214244362
#214244012
#214241872
C4×#142736522
C59×#19232
#2672659