Finite Group Databasebeta

← #28657194

Subgroups — #28657194

Order 15104.

832 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21428612C211yesmaximal, characteristic, complemented
2#21425499C211yesmaximal, characteristic, complemented
3#21425484C211yesmaximal, characteristic, complemented
4#21425327C211yesmaximal, characteristic
5#21425327C211yesmaximal, characteristic
6#21424782C211yesmaximal, characteristic, complemented
7#21424713C211yesmaximal, characteristic, complemented
8#21424667C211yesmaximal, characteristic, complemented
9#21424667C211yesmaximal, characteristic, complemented
10#21424655C211yesmaximal, characteristic, complemented
11#21424343C211yesmaximal, characteristic, complemented
12#21424343C211yesmaximal, characteristic, complemented
13#21424096C211yesmaximal, characteristic, complemented
14C4×#14273648C211yesmaximal, characteristic, complemented
15C59×#2366C211yesmaximal, characteristic
16C2×#14273717—21no—
17C2×#14273660C2211yescharacteristic
18C2×#14273660C2211yescharacteristic
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×#14273648C2211yescharacteristic
21C2×#14273648C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×C4×C59⋊C8C2211yescharacteristic
29#16837342—21no—
30#16837332—21no—
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837300C2211yescharacteristic
35#16837287—21no—
36#16837272C2211yescharacteristic
37C4×C4⋊Dic59C2211yescharacteristic
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837157C2211yescharacteristic
45#16837157C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×#127×C59C2211yescharacteristic
49C2×C4×D4×C59—21no—
50C2×C4×D4×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C59×#390—21no—
53C59×#390C2211yescharacteristic
54C59×#390C2211yescharacteristic
55C59×#386—21no—
56C59×#386—21no—
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C22×C59⋊C8C2311yescharacteristic
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63#14273717—44no—
64C2×C4⋊Dic59—21no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59C2311yescharacteristic
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×Dic59⋊C4—21no—
71C2×Dic59⋊C4—21no—
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×C4×Dic59—21no—
75#14273660—22no—
76#14273660—22no—
77C4⋊(C59⋊C8)—41no—
78#14273648—22no—
79#14273648—22no—
80#14273648—22no—
81#14273648—22no—
82#14273648—22no—
83#14273648—22no—
84#14273648—41no—
85#14273648—41no—
86C4×C59⋊C8—41no—
87C59⋊#95—21no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C22×D4×C59—21no—
93C22×D4×C59—21no—
94C22×D4×C59C2311yescharacteristic
95C22×D4×C59C2311yescharacteristic
96C23×C4×C59—21no—
97C23×C4×C59—21no—
98#127×C59—22no—
99#127×C59—22no—
100C4×D4×C59—22no—
101C4×D4×C59—22no—
102C4×D4×C59—41no—
103C4×D4×C59—41no—
104C4×D4×C59—44no—
105C2×C4⋊C4×C59D411yescharacteristic
106C2×C4⋊C4×C59D411yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C42×C59—21no—
116C2×C42×C59C2311yescharacteristic
117C2×C42×C59C2311yescharacteristic
118C2×C42×C59C2311yescharacteristic
119#95×C59—21no—
120#95×C59D411yescharacteristic
121#95×C59D411yescharacteristic
122C24×C59—21no—
123C24×C59—21no—
124C2×D4×C59—24no—
125C2×D4×C59—24no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—42no—
137C2×D4×C59—42no—
138C2×D4×C59—81no—
139C2×D4×C59—81no—
140C2×D4×C59#4112yes—
141C2×D4×C59#4112yes—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—42no—
145C22×C4×C59—42no—
146C22×C4×C59—42no—
147C22×C4×C59—42no—
148C22×C4×C59—81no—
149C22×C4×C59—81no—
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2411yescharacteristic
157C22×Dic59—21no—
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—41no—
165C4⋊C4×C59D814yes—
166C4⋊C4×C59#4112yes—
167C22⋊C4×C59—42no—
168C22⋊C4×C59—42no—
169C22⋊C4×C59—42no—
170C22⋊C4×C59—42no—
171C22⋊C4×C59—81no—
172C22⋊C4×C59—81no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—42no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—42no—
188C4⋊Dic59—42no—
189C4⋊Dic59#4112yes—
190Dic59⋊C4—42no—
191Dic59⋊C4—42no—
192Dic59⋊C4—42no—
193Dic59⋊C4—42no—
194C4×Dic59—42no—
195C2×C59⋊C8—21no—
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—
207C23×C59—22no—
208C23×C59—22no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59—41no—
213C23×C59—41no—
214C23×C59—41no—
215C23×C59—41no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59C22×D411yescharacteristic
222D4×C59—22no—
223D4×C59—22no—
224D4×C59—22no—
225D4×C59—22no—
226D4×C59—48no—
227D4×C59—41no—
228D4×C59—41no—
229D4×C59—41no—
230D4×C59—41no—
231D4×C59—84no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—22no—
242C2×C4×C59—21no—
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—84no—
254C2×C4×C59—84no—
255C2×C4×C59C2×D812yes—
256C2×C4×C59C2×D812yes—
257C2×C4×C59C2×D812yes—
258C2×C4×C59C2×#4111yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59C2×#4111yescharacteristic
262C2×C4×C59H(5,𝔽2)11yescharacteristic
263C2×C4×C592−1+411yescharacteristic
264C2×Dic59—22no—
265C2×Dic59—41no—
266C2×Dic59—41no—
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—
276C59⋊C8—41no—
277C59⋊C8—41no—
278C59⋊C8—41no—
279C59⋊C8—41no—
280#26752—591nomaximal, Sylow
281C22×C59—42no—
282C22×C59—42no—
283C22×C59—42no—
284C22×C59—42no—
285C22×C59—42no—
286C22×C59—42no—
287C22×C59—81no—
288C22×C59—81no—
289C22×C59—81no—
290C22×C59—81no—
291C22×C59—81no—
292C22×C59—81no—
293C22×C59—81no—
294C22×C59—81no—
295C22×C59#52511yescharacteristic
296C22×C59#54611yescharacteristic
297C22×C59#54811yescharacteristic
298C22×C59C22×D811yescharacteristic
299C22×C59#57511yescharacteristic
300C22×C59#57711yescharacteristic
301C22×C59#57711yescharacteristic
302C4×C59—22no—
303C4×C59—22no—
304C4×C59—22no—
305C4×C59—41no—
306C4×C59—42no—
307C4×C59—41no—
308C4×C59—41no—
309C4×C59—81no—
310C4×C59#52911yescharacteristic
311C4×C59#53911yescharacteristic
312C4×C59#55211yescharacteristic
313C4×C59#55311yescharacteristic
314Dic59—42no—
315Dic59—81no—
316Dic59—81no—
317Dic59—81no—
318#2506—591no—
319#2366D5911yescharacteristic
320#2071—591no—
321#2067—591no—
322#2041—591no—
323#2041—591no—
324#1934—591no—
325#1925—591no—
326#1918—591no—
327#1918—591no—
328#1916—591no—
329#1862—591no—
330#1862—591no—
331#1802—591no—
332C4×#102—591no—
333C2×C59—82no—
334C2×C59—82no—
335C2×C59#246011yescharacteristic
336C2×C59#310011yescharacteristic
337C2×C59#312311yescharacteristic
338C2×C59#329411yescharacteristic
339C2×C59#331711yescharacteristic
340C2×C59#335811yescharacteristic
341C2×C59#337111yescharacteristic
342C2×#127C2×D5911yescharacteristic
343C2×#125—1181no—
344C2×C4×D4—21no—
345C2×C4×D4C2×D5911yescharacteristic
346C2×C4×D4C2×D5911yescharacteristic
347C2×#107—591no—
348C2×#107—591no—
349C2×C4⋊C8—591no—
350C2×#102—591no—
351C2×#102—591no—
352C2×#102—591no—
353C2×#102—591no—
354C2×#102—591no—
355C2×#102—591no—
356C2×#102—591no—
357C2×#102—591no—
358C2×C4×C8—591no—
359#400—1181no—
360#398—1181no—
361#390—21no—
362#390C2×D5911yescharacteristic
363#390C2×D5911yescharacteristic
364#389—591no—
365#389—591no—
366#389—591no—
367#389—591no—
368#387—1181no—
369#386—21no—
370#386—21no—
371#384—591no—
372C4×C4⋊C4—591no—
373C4×C4⋊C4C2×D5911yescharacteristic
374C4×C4⋊C4C2×D5911yescharacteristic
375#340—591no—
376#340—591no—
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#336—591no—
382#336—591no—
383#336—591no—
384#336—591no—
385C59#2675211yescharacteristic, complemented, Sylow
386C22×D4—21no—
387C22×D4—21no—
388C22×D4C22×D5911yescharacteristic
389C22×D4C22×D5911yescharacteristic
390C23×C4—21no—
391C23×C4—21no—
392C22×C8—591no—
393C22×C8—591no—
394C22×C8—591no—
395C22×C8—591no—
396#127—22no—
397#127—22no—
398#125—2364no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—41no—
402C4×D4—41no—
403C4×D4—44no—
404C2×C4⋊C4—591no—
405C2×C4⋊C4—591no—
406C2×C4⋊C4—591no—
407C2×C4⋊C4—591no—
408C2×C4⋊C4—1181no—
409C2×C4⋊C4—1181no—
410C2×C4⋊C4—1181no—
411C2×C4⋊C4—1181no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4C22⋊D5911yescharacteristic
415C2×C4⋊C4C22⋊D5911yescharacteristic
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C42—21no—
425C2×C42—1181no—
426C2×C42C22×D5911yescharacteristic
427C2×C42C22×D5911yescharacteristic
428C2×C42C22×D5911yescharacteristic
429#107—1182no—
430#107—1182no—
431C4⋊C8—2361no—
432#102—1182no—
433#102—1182no—
434#102—1182no—
435#102—1182no—
436#102—1182no—
437#102—1182no—
438#102—2361no—
439#102—2361no—
440C4×C8—2361no—
441#95—21no—
442#95—1181no—
443#95—1181no—
444#95—1181no—
445#95—1181no—
446#95—1181no—
447#95C22⋊D5911yescharacteristic
448#95C22⋊D5911yescharacteristic
449C24—21no—
450C24—21no—
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—81no—
466C2×D4—81no—
467C2×D4#1174568512yes—
468C2×D4#1174568512yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4—1181no—
478C22×C4—1181no—
479C22×C4—1181no—
480C22×C4—1181no—
481C22×C4C2×C22⋊D5911yescharacteristic
482C22×C4C2×C22⋊D5911yescharacteristic
483C22×C4C2×C22⋊D5911yescharacteristic
484C22×C4C2×C22⋊D5911yescharacteristic
485C22×C4C2×C22⋊D5911yescharacteristic
486C22×C4C2×C22⋊D5911yescharacteristic
487C22×C4C23×D5911yescharacteristic
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—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—41no—
504C4⋊C4—592no—
505C4⋊C4—1181no—
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—
515C4⋊C4—2362no—
516C4⋊C4—2362no—
517C4⋊C4—2362no—
518C4⋊C4#1174566914yes—
519C4⋊C4#1174568712yes—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—81no—
525C22⋊C4—81no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—42no—
533C42—2362no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C22×C22⋊D5911yescharacteristic
549D4—22no—
550D4—22no—
551D4—22no—
552D4—22no—
553D4—48no—
554D4—41no—
555D4—41no—
556D4—41no—
557D4—41no—
558D4—84no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—1182no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4C2×#1174566912yes—
595C2×C4C2×#1174566912yes—
596C2×C4C2×#1174566912yes—
597C2×C4#41×D5911yescharacteristic
598C2×C4C2×#1174568711yescharacteristic
599C2×C4C2×#1174568511yescharacteristic
600C2×C4C2×#1174568511yescharacteristic
601C2×C4#1427377211yescharacteristic
602C2×C4#1427377411yescharacteristic
603C8—2361no—
604C8—2361no—
605C8—2361no—
606C8—2361no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22#1683791311yescharacteristic
622C22#1683804611yescharacteristic
623C22#1683805911yescharacteristic
624C22C22×#1174566911yescharacteristic
625C22#1683820011yescharacteristic
626C22#1683821311yescharacteristic
627C22#1683821311yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—81no—
636C4—4721no—
637C4—4721no—
638C4—4721no—
639C4—2362no—
640C4#1683793211yescharacteristic
641C4#1683799411yescharacteristic
642C4#1683809211yescharacteristic
643C4#1683810111yescharacteristic
644C2—82no—
645C2—82no—
646C2#2142804211yescharacteristic
647C2#2143409611yescharacteristic
648C2#2143434811yescharacteristic
649C2#2143590911yescharacteristic
650C2#2143622211yescharacteristic
651C2#2143672711yescharacteristic
652C2#2143685811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2366C2
socleC23×C59C22×D4
prime radicalC2×#127×C59C22
hypercenter#2366D59
hypocenterC59#26752
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2366

Complements

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

NH ≅ G/Ntype
1. #21428612C2semidirect
2. #21425499C2semidirect
3. #21425484C2semidirect
6. #21424782C2semidirect
7. #21424713C2semidirect
8. #21424667C2semidirect
9. #21424667C2semidirect
10. #21424655C2semidirect
11. #21424343C2semidirect
12. #21424343C2semidirect
13. #21424096C2semidirect
14. C4×#14273648C2semidirect
385. C59#26752semidirect

Maximal subgroups

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

subgroupindex
#214286122
#214254992
#214254842
#214253272
#214247822
#214247132
#214246672
#214246552
#214243432
#214240962
C4×#142736482
C59×#23662
#2675259