Finite Group Databasebeta

← #28661342

Subgroups — #28661342

Order 15104.

812 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427110C211yesmaximal, characteristic
2#21425500C211yesmaximal, characteristic, complemented
3#21425481C211yesmaximal, characteristic, complemented
4#21425278C211yesmaximal, characteristic
5#21425278C211yesmaximal, characteristic
6#21424852C211yesmaximal, characteristic, complemented
7#21424826C211yesmaximal, characteristic, complemented
8#21424704C211yesmaximal, characteristic, complemented
9#21424667C211yesmaximal, characteristic, complemented
10#21424664C211yesmaximal, characteristic, complemented
11#21424342C211yesmaximal, characteristic, complemented
12#21424283C211yesmaximal, characteristic, complemented
13#21424180C211yesmaximal, characteristic, complemented
14C4×#14273647C211yesmaximal, characteristic, complemented
15C59×#2496C211yesmaximal, characteristic, complemented
16C2×#14273701C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×(C22⋊D59)⋊C4—21no—
19C22×C4⋊Dic59C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×C22⋊(C59⋊C8)C2211yescharacteristic
30C2×C22⋊(C59⋊C8)C2211yescharacteristic
31C2×C4×C59⋊C8C2211yescharacteristic
32#16837304C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837294—21no—
35#16837279—21no—
36#16837276—21no—
37C4×C4⋊Dic59C2211yescharacteristic
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×#126×C59—21no—
49C59×#400—21no—
50C59×#397—21no—
51C59×#389C2211yescharacteristic
52C59×#389C2211yescharacteristic
53C59×#387—21no—
54C59×#384C2211yescharacteristic
55C59×#380C2211yescharacteristic
56C59×#380C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59C2211yescharacteristic
59C22×D236C2311yescharacteristic
60C22×C22⋊D59—21no—
61C23×Dic59—21no—
62C22×C4×D59—21no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273701—22no—
68#14273701—22no—
69C4×D236—22no—
70C4×D236—41no—
71(C22⋊D59)⋊C4—44no—
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×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×C4×Dic59—21no—
84#14273660—22no—
85#14273660—22no—
86#14273660—22no—
87#14273660—41no—
88C4⋊(C59⋊C8)—22no—
89#14273647—22no—
90#14273647—22no—
91#14273647—22no—
92#14273647—41no—
93C22⋊(C59⋊C8)—22no—
94C22⋊(C59⋊C8)—41no—
95C4×C59⋊C8—41no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C23×C4×C59C2311yescharacteristic
99#126×C59—44no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59C2311yescharacteristic
105C2×C4⋊C4×C59C2311yescharacteristic
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C22⋊C4×C59—21no—
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59D411yescharacteristic
111C2×C22⋊C4×C59D411yescharacteristic
112C2×C42×C59—21no—
113C2×C42×C59C2311yescharacteristic
114C2×C42×C59C2311yescharacteristic
115#95×C59—21no—
116#95×C59—21no—
117#95×C59—21no—
118#95×C59—21no—
119#95×C59—21no—
120#95×C59D411yescharacteristic
121#95×C59D411yescharacteristic
122C24×C59C2×D411yescharacteristic
123C23×D59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
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×C59—21no—
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2411yescharacteristic
140C2×C22⋊D59—24no—
141C2×C22⋊D59—42no—
142C2×C22⋊D59—42no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—42no—
147C22×Dic59—42no—
148C22×Dic59—81no—
149C2×D236—21no—
150C2×D236—21no—
151C2×D236—21no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236#4112yes—
155C2×C4×D59—42no—
156C2×C4×D59—42no—
157C2×C4×D59—81no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—42no—
167C4⋊C4×C59—42no—
168C4⋊C4×C59—42no—
169C4⋊C4×C59—42no—
170C22⋊C4×C59—41no—
171C22⋊C4×C59—42no—
172C22⋊C4×C59—42no—
173C22⋊C4×C59D814yes—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—42no—
179(C2×D59)⋊C4—24no—
180(C2×D59)⋊C4—42no—
181(C2×D59)⋊C4—42no—
182(C2×D59)⋊C4—81no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—42no—
188C4⋊Dic59#4112yes—
189C4⋊Dic59#4112yes—
190Dic59⋊C4—42no—
191C4×Dic59—42no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
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#4112yes—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59C2×D812yes—
211C23×C59C22×D411yescharacteristic
212C22×D59—22no—
213C22×D59—41no—
214C22×D59—41no—
215C22×D59—41no—
216C22×D59—41no—
217C22×D59—41no—
218C22×D59—41no—
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—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
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×C59C2×D812yes—
249C2×C4×C59C2×D812yes—
250C2×C4×C59C2×#4111yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59C2×#4111yescharacteristic
253C2×C4×C59C2×#4111yescharacteristic
254C2×C4×C59H(5,𝔽2)11yescharacteristic
255C2×C4×C592−1+411yescharacteristic
256C22⋊D59—48no—
257C2×Dic59—22no—
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—84no—
267D236—22no—
268D236—22no—
269D236—41no—
270D236—41no—
271C4×D59—84no—
272C59⋊C8—22no—
273C59⋊C8—41no—
274C59⋊C8—41no—
275C59⋊C8—41no—
276#27030—591nomaximal, Sylow
277C22×C59—22no—
278C22×C59—22no—
279C22×C59—22no—
280C22×C59—41no—
281C22×C59—41no—
282C22×C59—41no—
283C22×C59—41no—
284C22×C59#52511yescharacteristic
285C22×C59#54611yescharacteristic
286C22×C59#54711yescharacteristic
287C22×C59C22×D811yescharacteristic
288C22×C59#57511yescharacteristic
289C22×C59#57711yescharacteristic
290C22×C59#57811yescharacteristic
291C2×D59—42no—
292C2×D59—42no—
293C2×D59—42no—
294C2×D59—81no—
295C2×D59—81no—
296C2×D59—81no—
297C2×D59—81no—
298C4×C59—22no—
299C4×C59—22no—
300C4×C59—41no—
301C4×C59—41no—
302C4×C59—41no—
303C4×C59—81no—
304C4×C59—81no—
305C4×C59—81no—
306C4×C59—81no—
307C4×C59#52911yescharacteristic
308C4×C59#53911yescharacteristic
309C4×C59#55311yescharacteristic
310C4×C59#55611yescharacteristic
311Dic59—42no—
312Dic59—42no—
313Dic59—81no—
314#2496D5911yescharacteristic, complemented
315#2361—591no—
316#2071—591no—
317#2067—591no—
318#2033—591no—
319#2033—591no—
320#1947—591no—
321#1942—591no—
322#1923—591no—
323#1918—591no—
324#1918—591no—
325#1862—591no—
326#1852—591no—
327#1823—591no—
328C4×#102—591no—
329C2×C59—42no—
330C2×C59#246611yescharacteristic
331C2×C59#310011yescharacteristic
332C2×C59#312011yescharacteristic
333C2×C59#329411yescharacteristic
334C2×C59#331311yescharacteristic
335C2×C59#333411yescharacteristic
336C2×C59#335111yescharacteristic
337D59—82no—
338C2×#126—21no—
339C2×#121—591no—
340C2×C4×D4—591no—
341C2×C4×D4—1181no—
342C22×C4⋊C4—591no—
343C2×#107—591no—
344C2×#107—591no—
345C2×#107—591no—
346C2×#107—591no—
347C2×C4⋊C8—591no—
348C2×#102—591no—
349C2×#102—591no—
350C2×#102—591no—
351C2×#102—591no—
352C2×C22⋊C8—591no—
353C2×C22⋊C8—591no—
354C2×C4×C8—591no—
355#400—21no—
356#397—21no—
357#390—591no—
358#389—591no—
359#389C2×D5911yescharacteristic
360#389C2×D5911yescharacteristic
361#388—1181no—
362#387—21no—
363#386—1181no—
364#385—1181no—
365#384C2×D5911yescharacteristic
366#380C2×D5911yescharacteristic
367#380C2×D5911yescharacteristic
368C4×C4⋊C4—591no—
369C4×C4⋊C4C2×D5911yescharacteristic
370C4×C22⋊C4C2×D5911yescharacteristic
371#340—591no—
372#340—591no—
373#340—591no—
374#340—591no—
375#340—591no—
376#340—591no—
377#340—591no—
378#340—591no—
379#336—591no—
380#336—591no—
381C59#2703011yescharacteristic, complemented, Sylow
382C22×D4—591no—
383C22×D4—1181no—
384C23×C4—1181no—
385C23×C4—1181no—
386C23×C4C22×D5911yescharacteristic
387C22×C8—591no—
388C22×C8—591no—
389C22×C8—591no—
390C22×C8—591no—
391#126—44no—
392#121—1182no—
393#121—1182no—
394C4×D4—1182no—
395C4×D4—2361no—
396C4×D4—2364no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
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—2361no—
408C2×C4⋊C4C22×D5911yescharacteristic
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⋊C4—1181no—
417C2×C22⋊C4—1181no—
418C2×C22⋊C4C22⋊D5911yescharacteristic
419C2×C22⋊C4C22⋊D5911yescharacteristic
420C2×C42—21no—
421C2×C42—1181no—
422C2×C42C22×D5911yescharacteristic
423C2×C42C22×D5911yescharacteristic
424#107—1182no—
425#107—1182no—
426#107—1182no—
427#107—2361no—
428C4⋊C8—1182no—
429#102—1182no—
430#102—1182no—
431#102—1182no—
432#102—2361no—
433C22⋊C8—1182no—
434C22⋊C8—2361no—
435C4×C8—2361no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—1181no—
442#95—1181no—
443#95C22⋊D5911yescharacteristic
444#95C22⋊D5911yescharacteristic
445C24—1181no—
446C24C2×C22⋊D5911yescharacteristic
447C2×D4—592no—
448C2×D4—1184no—
449C2×D4—1181no—
450C2×D4—1181no—
451C2×D4—1181no—
452C2×D4—1181no—
453C2×D4—1181no—
454C2×D4—2362no—
455C2×D4—2362no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—1181no—
467C22×C4—1181no—
468C22×C4—1181no—
469C22×C4—4721no—
470C22×C4—4721no—
471C22×C4—2362no—
472C22×C4—2362no—
473C22×C4—2362no—
474C22×C4—2362no—
475C22×C4C2×C22⋊D5911yescharacteristic
476C22×C4C2×C22⋊D5911yescharacteristic
477C22×C4C2×C22⋊D5911yescharacteristic
478C22×C4C2×C22⋊D5911yescharacteristic
479C22×C4C2×C22⋊D5911yescharacteristic
480C22×C4C23×D5911yescharacteristic
481C2×C8—592no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—1181no—
485C2×C8—1181no—
486C2×C8—1181no—
487C2×C8—1181no—
488C2×C8—1181no—
489C2×C8—1181no—
490C2×C8—1181no—
491C2×C8—1181no—
492C2×C8—1181no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—592no—
506C4⋊C4—592no—
507C4⋊C4—1181no—
508C4⋊C4—1181no—
509C4⋊C4—1181no—
510C4⋊C4—1181no—
511C4⋊C4—2362no—
512C4⋊C4—2362no—
513C22⋊C4—41no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—1184no—
517C22⋊C4—4721no—
518C22⋊C4—2362no—
519C22⋊C4—2362no—
520C22⋊C4#1174566914yes—
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—1182no—
534C23—2361no—
535C23—2361no—
536C23—2361no—
537C23—2361no—
538C23—2361no—
539C23—2361no—
540C23C2×#1174566912yes—
541C23C22×C22⋊D5911yescharacteristic
542D4—1182no—
543D4—1182no—
544D4—2368no—
545D4—2361no—
546D4—2361no—
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—21no—
558C2×C4—21no—
559C2×C4—21no—
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—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—1182no—
577C2×C4—1182no—
578C2×C4—4724no—
579C2×C4—4724no—
580C2×C4—2361no—
581C2×C4—2361no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4C2×#1174566912yes—
588C2×C4C2×#1174566912yes—
589C2×C4#41×D5911yescharacteristic
590C2×C4#41×D5911yescharacteristic
591C2×C4#41×D5911yescharacteristic
592C2×C4#41×D5911yescharacteristic
593C2×C4#1427377311yescharacteristic
594C2×C4#1427377411yescharacteristic
595C8—1182no—
596C8—2361no—
597C8—2361no—
598C8—2361no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—4721no—
607C22—4721no—
608C22—4721no—
609C22—4721no—
610C22—2362no—
611C22—2362no—
612C22—2362no—
613C22#1683790911yescharacteristic
614C22#1683804011yescharacteristic
615C22#1683805111yescharacteristic
616C22C22×#1174566911yescharacteristic
617C22#1683820011yescharacteristic
618C22#1683821011yescharacteristic
619C22#1683821711yescharacteristic
620C4—22no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—4721no—
630C4—2362no—
631C4—2362no—
632C4#1683793511yescharacteristic
633C4#1683799311yescharacteristic
634C4#1683810311yescharacteristic
635C4#1683811911yescharacteristic
636C2—42no—
637C2—4722no—
638C2#2142812811yescharacteristic
639C2#2143409211yescharacteristic
640C2#2143430411yescharacteristic
641C2#2143590311yescharacteristic
642C2#2143615611yescharacteristic
643C2#2143643911yescharacteristic
644C2#2143665611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2496C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
hypercenter#2496D59
hypocenterC59#27030
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2496

Complements

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

NH ≅ G/Ntype
2. #21425500C2semidirect
3. #21425481C2semidirect
6. #21424852C2semidirect
7. #21424826C2semidirect
8. #21424704C2semidirect
9. #21424667C2semidirect
10. #21424664C2semidirect
11. #21424342C2semidirect
12. #21424283C2semidirect
13. #21424180C2semidirect
14. C4×#14273647C2semidirect
15. C59×#2496C2semidirect
314. #2496D59semidirect
381. C59#27030semidirect

Maximal subgroups

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

subgroupindex
#214271102
#214255002
#214254812
#214252782
#214248522
#214248262
#214247042
#214246672
#214246642
#214243422
#214242832
#214241802
C4×#142736472
C59×#24962
#2703059