Finite Group Databasebeta

← #28651805

Subgroups — #28651805

Order 15104.

828 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×#16837709C211yesmaximal, characteristic, complemented
2C2×C4×#11745671C211yesmaximal, characteristic
3#21427968C211yesmaximal, characteristic
4#21425625C211yesmaximal, characteristic
5#21425573C211yesmaximal, characteristic, complemented
6#21425481C211yesmaximal, characteristic
7#21425434C211yesmaximal, characteristic, complemented
8#21424826C211yesmaximal, characteristic, complemented
9#21424655C211yesmaximal, characteristic, complemented
10#21424451C211yesmaximal, characteristic
11#21424451C211yesmaximal, characteristic
12#21424356C211yesmaximal, characteristic
13#21424350C211yesmaximal, characteristic, complemented
14#21424350C211yesmaximal, characteristic, complemented
15Q8×C4⋊C4×C59C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C2×#14273719C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19#16837709—24no—
20C4×#11745671—24no—
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×#14273651C2211yescharacteristic
30C2×#14273647C2211yescharacteristic
31C2×#14273647C2211yescharacteristic
32C2×#14273647C2211yescharacteristic
33C2×C4×C59⋊C8C2211yescharacteristic
34#16837337—21no—
35#16837326—21no—
36#16837304C2211yescharacteristic
37#16837304C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837273—21no—
41#16837261—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C4×Q8×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C2×C4×Q8×C59C2211yescharacteristic
53C59×#389—21no—
54C59×#389—21no—
55C59×#389C2211yescharacteristic
56C59×#389C2211yescharacteristic
57C59×#384—21no—
58C59×#384C2211yescharacteristic
59C4×C4⋊C4×C59—21no—
60C4×C4⋊C4×C59C2211yescharacteristic
61C22×D236C2311yescharacteristic
62C22×C4×D59—21no—
63C2×#11745671—22no—
64C2×#11745671—22no—
65C2×#11745671Q814yes—
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70#14273719—22no—
71#14273719—41no—
72C4×D236—22no—
73C4×D236—41no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86#14273660—22no—
87#14273660—22no—
88#14273660—41no—
89#14273658—22no—
90#14273658—22no—
91C4⋊(C59⋊C8)—22no—
92#14273651—22no—
93#14273651—22no—
94#14273651D414yes—
95#14273647—22no—
96#14273647—22no—
97#14273647—41no—
98C4×C59⋊C8—22no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C22×Q8×C59C2311yescharacteristic
102C4×Q8×C59—22no—
103C4×Q8×C59—22no—
104C4×Q8×C59—22no—
105C4×Q8×C59—22no—
106C4×Q8×C59—41no—
107C4×Q8×C59—41no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59—21no—
110C2×C4⋊C4×C59—21no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59D411yescharacteristic
116C2×C4⋊C4×C59D411yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C42×C59—21no—
121C2×C42×C59—21no—
122C2×C42×C59—21no—
123C2×C42×C59C2311yescharacteristic
124C2×C42×C59C2311yescharacteristic
125C2×C42×C59C2311yescharacteristic
126#95×C59—21no—
127#95×C59—21no—
128#95×C59D411yescharacteristic
129#95×C59D411yescharacteristic
130C23×D59—21no—
131C2×Q8×C59—21no—
132C2×Q8×C59—21no—
133C2×Q8×C59—21no—
134C2×Q8×C59C2×D412yes—
135C2×Q8×C59C2×Q812yes—
136C2×Q8×C59#4112yes—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2411yescharacteristic
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C2×D236—21no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236—21no—
155C2×D236—21no—
156C2×D236C2×Q812yes—
157C2×C4×D59—42no—
158C2×C4×D59—42no—
159C2×C4×D59—81no—
160C4⋊C4×C59—24no—
161C4⋊C4×C59—22no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—42no—
169C4⋊C4×C59—42no—
170C4⋊C4×C59—42no—
171C4⋊C4×C59—81no—
172C4⋊C4×C59—81no—
173C4⋊C4×C59—81no—
174C4⋊C4×C59#4112yes—
175C42×C59—24no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—21no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—42no—
183C42×C59—42no—
184#11745671—28no—
185(C2×D59)⋊C4—42no—
186(C2×D59)⋊C4—42no—
187(C2×D59)⋊C4—81no—
188(C2×D59)⋊C4—81no—
189(C2×D59)⋊C4—81no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59C2×D412yes—
196Dic59⋊C4—42no—
197Dic59⋊C4—42no—
198Dic59⋊C4—81no—
199Dic59⋊C4—81no—
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—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8C2×D412yes—
211C2×C59⋊C8C2×Q812yes—
212C23×C59C22×D411yescharacteristic
213C22×D59—22no—
214C22×D59—41no—
215C22×D59—41no—
216C22×D59—41no—
217C22×D59—41no—
218C22×D59—41no—
219C22×D59—41no—
220Q8×C59—22no—
221Q8×C59—22no—
222Q8×C59—22no—
223Q8×C59C22⋊Q814yes—
224C2×C4×C59—21no—
225C2×C4×C59—22no—
226C2×C4×C59—21no—
227C2×C4×C59—22no—
228C2×C4×C59—22no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—22no—
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—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
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—41no—
252C2×C4×C59#13512yes—
253C2×C4×C59#13712yes—
254C2×C4×C59C22×D411yescharacteristic
255C2×C4×C59C22×Q811yescharacteristic
256C2×C4×C59C2×#4111yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×C4×C59H(5,𝔽2)11yescharacteristic
259C2×C4×C592−1+411yescharacteristic
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—
269D236—22no—
270D236—22no—
271D236—41no—
272D236—41no—
273C4×D59—84no—
274C59⋊C8—22no—
275C59⋊C8—22no—
276C59⋊C8—41no—
277C59⋊C8—41no—
278#26379—591nomaximal, Sylow
279C22×C59#52511yescharacteristic
280C22×C59D4211yescharacteristic
281C22×C59D4×Q811yescharacteristic
282C22×C59C2×#13511yescharacteristic
283C22×C59C2×#13711yescharacteristic
284C22×C59#57611yescharacteristic
285C22×C59#57711yescharacteristic
286C2×D59—42no—
287C2×D59—42no—
288C2×D59—42no—
289C2×D59—81no—
290C2×D59—81no—
291C2×D59—81no—
292C2×D59—81no—
293C4×C59—22no—
294C4×C59—24no—
295C4×C59—22no—
296C4×C59—41no—
297C4×C59—42no—
298C4×C59—41no—
299C4×C59—41no—
300C4×C59—41no—
301C4×C59—81no—
302C4×C59—81no—
303C4×C59C2×C22⋊Q811yescharacteristic
304C4×C59#53911yescharacteristic
305C4×C59#54711yescharacteristic
306C4×C59#55411yescharacteristic
307Dic59—42no—
308Dic59—81no—
309Dic59—81no—
310C2×#479—591no—
311C2×C4×SD16—591no—
312#2454—591no—
313Q8×C4⋊C4D5911yescharacteristic, complemented
314#2093—591no—
315#2084—591no—
316#2067—591no—
317#2059—591no—
318#1942—591no—
319#1916—591no—
320#1880—591no—
321#1880—591no—
322#1864—591no—
323#1863—591no—
324#1863—591no—
325C2×C59#244911yescharacteristic
326C2×C59#310811yescharacteristic
327C2×C59#311911yescharacteristic
328C2×C59#326611yescharacteristic
329C2×C59#328711yescharacteristic
330C2×C59#338111yescharacteristic
331C2×C59#338511yescharacteristic
332D59—82no—
333C22×SD16—591no—
334C2×#125—591no—
335C2×C4×Q8C2×D5911yescharacteristic
336C2×C4×Q8C2×D5911yescharacteristic
337C2×C4×Q8C2×D5911yescharacteristic
338C2×C4×D4—591no—
339#479—1184no—
340C4×SD16—1184no—
341C2×#107—591no—
342C2×#107—591no—
343C2×#107—591no—
344C2×#106—591no—
345C2×#106—591no—
346C2×C4⋊C8—591no—
347C2×#103—591no—
348C2×#103—591no—
349C2×#103—591no—
350C2×#102—591no—
351C2×#102—591no—
352C2×#102—591no—
353C2×C4×C8—591no—
354#399—1181no—
355#396—1181no—
356#390—591no—
357#390—591no—
358#389—21no—
359#389—21no—
360#389—591no—
361#389—591no—
362#389C2×D5911yescharacteristic
363#389C2×D5911yescharacteristic
364#385—1181no—
365#384—21no—
366#384C2×D5911yescharacteristic
367#380—1181no—
368C4×C4⋊C4—21no—
369C4×C4⋊C4C2×D5911yescharacteristic
370#340—591no—
371#340—591no—
372#340—591no—
373#340—591no—
374#336—591no—
375#336—591no—
376#336—591no—
377#336—591no—
378C59#2637911yescharacteristic, complemented, Sylow
379C22×Q8C22×D5911yescharacteristic
380C22×D4—591no—
381C23×C4—1181no—
382C2×SD16—594no—
383C2×SD16—1182no—
384C2×SD16—1182no—
385C22×C8—591no—
386C22×C8—591no—
387C22×C8—591no—
388C22×C8—591no—
389#125—1182no—
390#125—2361no—
391C4×Q8—22no—
392C4×Q8—22no—
393C4×Q8—22no—
394C4×Q8—22no—
395C4×Q8—41no—
396C4×Q8—41no—
397C4×D4—1182no—
398C4×D4—2361no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—591no—
407C2×C4⋊C4—591no—
408C2×C4⋊C4—591no—
409C2×C4⋊C4—1181no—
410C2×C4⋊C4—1181no—
411C2×C4⋊C4—1181no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4C22⋊D5911yescharacteristic
414C2×C4⋊C4C22⋊D5911yescharacteristic
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C22⋊C4—1181no—
419C2×C22⋊C4—1181no—
420C2×C22⋊C4—1181no—
421C2×C22⋊C4—1181no—
422C2×C22⋊C4—1181no—
423C2×C42—21no—
424C2×C42—21no—
425C2×C42—21no—
426C2×C42C22×D5911yescharacteristic
427C2×C42C22×D5911yescharacteristic
428C2×C42C22×D5911yescharacteristic
429#107—1182no—
430#107—1182no—
431#107—2361no—
432#106—1182no—
433#106—1182no—
434C4⋊C8—1182no—
435#103—594no—
436#103—1182no—
437#103—1182no—
438#102—1182no—
439#102—1182no—
440#102—2361no—
441C4×C8—1182no—
442#95—21no—
443#95—21no—
444#95—1181no—
445#95—1181no—
446#95C22⋊D5911yescharacteristic
447#95C22⋊D5911yescharacteristic
448C24—1181no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8#1174568512yes—
453C2×Q8D4×D5912yes—
454C2×Q8Q8×D5912yes—
455C2×D4—592no—
456C2×D4—1181no—
457C2×D4—1181no—
458C2×D4—1181no—
459C2×D4—1181no—
460C2×D4—1181no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—1181no—
466C22×C4—1181no—
467C22×C4—1181no—
468C22×C4—4721no—
469C22×C4—2362no—
470C22×C4—2362no—
471C22×C4C2×C22⋊D5911yescharacteristic
472C22×C4C2×C22⋊D5911yescharacteristic
473C22×C4C2×C22⋊D5911yescharacteristic
474C22×C4C2×C22⋊D5911yescharacteristic
475C22×C4C2×C22⋊D5911yescharacteristic
476C22×C4C2×C22⋊D5911yescharacteristic
477C22×C4C23×D5911yescharacteristic
478SD16—1188no—
479C2×C8—592no—
480C2×C8—592no—
481C2×C8—1181no—
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—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—81no—
503C4⋊C4—81no—
504C4⋊C4—81no—
505C4⋊C4—592no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—1181no—
509C4⋊C4—1181no—
510C4⋊C4—1181no—
511C4⋊C4—4721no—
512C4⋊C4—4721no—
513C4⋊C4—2362no—
514C4⋊C4—2362no—
515C4⋊C4#1174568512yes—
516C22⋊C4—4721no—
517C22⋊C4—4721no—
518C22⋊C4—4721no—
519C22⋊C4—2362no—
520C22⋊C4—2362no—
521C42—24no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—42no—
529C42—42no—
530C23—1182no—
531C23—2361no—
532C23—2361no—
533C23—2361no—
534C23—2361no—
535C23—2361no—
536C23—2361no—
537C23C22×C22⋊D5911yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—22no—
541Q8#1427370514yes—
542D4—1182no—
543D4—1182no—
544D4—2361no—
545D4—2361no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
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—1182no—
575C2×C4—4724no—
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#1427374712yes—
585C2×C4#1427375812yes—
586C2×C4C2×D4×D5911yescharacteristic
587C2×C4C2×Q8×D5911yescharacteristic
588C2×C4C2×#1174568511yescharacteristic
589C2×C4C2×#1174568511yescharacteristic
590C2×C4#1427377211yescharacteristic
591C2×C4#1427377511yescharacteristic
592C8—1182no—
593C8—1182no—
594C8—2361no—
595C8—2361no—
596C22—4721no—
597C22—4721no—
598C22—4721no—
599C22—4721no—
600C22—2362no—
601C22—2362no—
602C22—2362no—
603C22#1683791311yescharacteristic
604C22D4×C22⋊D5911yescharacteristic
605C22Q8×C22⋊D5911yescharacteristic
606C22C2×#1427374711yescharacteristic
607C22C2×#1427375811yescharacteristic
608C22#1683820511yescharacteristic
609C22#1683821011yescharacteristic
610C4—22no—
611C4—24no—
612C4—22no—
613C4—41no—
614C4—42no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—81no—
619C4—81no—
620C4—4721no—
621C4—4721no—
622C4—2362no—
623C4C2×#1427370511yescharacteristic
624C4#1683799911yescharacteristic
625C4#1683805311yescharacteristic
626C4#1683810711yescharacteristic
627C2—4722no—
628C2#2142790511yescharacteristic
629C2#2143417611yescharacteristic
630C2#2143429511yescharacteristic
631C2#2143560711yescharacteristic
632C2#2143585211yescharacteristic
633C2#2143696711yescharacteristic
634C2#2143700211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingQ8×C4⋊C4×C59C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenterQ8×C4⋊C4D59
hypocenterC59#26379
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16837709C2semidirect
5. #21425573C2semidirect
7. #21425434C2semidirect
8. #21424826C2semidirect
9. #21424655C2semidirect
13. #21424350C2semidirect
14. #21424350C2semidirect
15. Q8×C4⋊C4×C59C2semidirect
313. Q8×C4⋊C4D59semidirect
378. C59#26379semidirect

Maximal subgroups

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

subgroupindex
C2×#168377092
C2×C4×#117456712
#214279682
#214256252
#214255732
#214254812
#214254342
#214248262
#214246552
#214244512
#214243562
#214243502
Q8×C4⋊C4×C592
#2637959