Finite Group Databasebeta

← #28651546

Subgroups — #28651546

Order 15104.

812 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837706C211yesmaximal, characteristic, complemented
2C2×#16837445C211yesmaximal, characteristic
3#21427099C211yesmaximal, characteristic
4#21425635C211yesmaximal, characteristic, complemented
5#21425587C211yesmaximal, characteristic, complemented
6#21425474C211yesmaximal, characteristic, complemented
7#21425431C211yesmaximal, characteristic, complemented
8#21424852C211yesmaximal, characteristic, complemented
9#21424671C211yesmaximal, characteristic, complemented
10#21424527C211yesmaximal, characteristic
11#21424408C211yesmaximal, characteristic
12#21424349C211yesmaximal, characteristic, complemented
13#21424283C211yesmaximal, characteristic
14#21424281C211yesmaximal, characteristic
15C59×#2441C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C2×#14273706C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19C22×C4⋊Dic59C2211yescharacteristic
20#16837706—24no—
21#16837445—24no—
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273660C2211yescharacteristic
25C2×#14273660C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×#14273658C2211yescharacteristic
28C2×C4⋊(C59⋊C8)C2211yescharacteristic
29C2×C4⋊(C59⋊C8)C2211yescharacteristic
30C2×#14273650C2211yescharacteristic
31C2×#14273650C2211yescharacteristic
32C2×#14273650C2211yescharacteristic
33C2×#14273650C2211yescharacteristic
34C2×C22⋊(C59⋊C8)C2211yescharacteristic
35#16837308C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837299—21no—
38#16837289—21no—
39#16837274—21no—
40C4×C4⋊Dic59C2211yescharacteristic
41C22⋊C4×Dic59—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#125×C59C2211yescharacteristic
51C2×#117×C59C2211yescharacteristic
52C59×#399—21no—
53C59×#396—21no—
54C59×#389—21no—
55C59×#389C2211yescharacteristic
56C59×#389C2211yescharacteristic
57C59×#387—21no—
58C59×#384C2211yescharacteristic
59C59×#380C2211yescharacteristic
60C4×C22⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C23×Dic59—21no—
63C2×#11745664—22no—
64C2×#11745664—22no—
65C2×#11745664—22no—
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70#14273706—22no—
71#14273706Q814yes—
72C4×Dic118—22no—
73C4×Dic118—41no—
74C2×C4⋊Dic59—22no—
75C2×C4⋊Dic59—21no—
76C2×C4⋊Dic59—41no—
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×C22⋊Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86#14273660—22no—
87#14273660—22no—
88#14273660—22no—
89#14273660—41no—
90#14273658—22no—
91#14273658D414yes—
92C4⋊(C59⋊C8)—22no—
93C4⋊(C59⋊C8)—22no—
94#14273650—22no—
95#14273650—22no—
96#14273650—22no—
97#14273650—41no—
98C22⋊(C59⋊C8)—22no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C23×C4×C59C2311yescharacteristic
104#125×C59—22no—
105#125×C59—41no—
106#117×C59—22no—
107#117×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×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59D411yescharacteristic
121C2×C22⋊C4×C59D411yescharacteristic
122C2×C42×C59—21no—
123C2×C42×C59C2311yescharacteristic
124C2×C42×C59C2311yescharacteristic
125#95×C59—21no—
126#95×C59—21no—
127#95×C59—21no—
128#95×C59D411yescharacteristic
129#95×C59D411yescharacteristic
130C24×C59C2×D411yescharacteristic
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×Q812yes—
145C22×C4×C59#4112yes—
146C22×C4×C59C2411yescharacteristic
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—42no—
152C22×Dic59—42no—
153C22×Dic59—81no—
154C2×Dic118—21no—
155C2×Dic118—21no—
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118—21no—
159C2×Dic118C2×Q812yes—
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—21no—
167C4⋊C4×C59—42no—
168C4⋊C4×C59—42no—
169C4⋊C4×C59—42no—
170C4⋊C4×C59—42no—
171C4⋊C4×C59—81no—
172C4⋊C4×C59—81no—
173C4⋊C4×C59C2×D412yes—
174C22⋊C4×C59—22no—
175C22⋊C4×C59—41no—
176C22⋊C4×C59—42no—
177C22⋊C4×C59—81no—
178C42×C59—21no—
179C42×C59—21no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—42no—
183C22⋊Dic59—24no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—42no—
189C4⋊Dic59C2×D412yes—
190C4⋊Dic59C2×Q812yes—
191Dic59⋊C4—24no—
192Dic59⋊C4—42no—
193Dic59⋊C4—81no—
194C4×Dic59—42no—
195C4×Dic59—81no—
196#11745664—28no—
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⋊C8C2×D412yes—
208C2×C59⋊C8#4112yes—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59C22⋊Q812yes—
216C23×C59C22×D411yescharacteristic
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—22no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
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×C59C22⋊Q812yes—
247C2×C4×C59#13712yes—
248C2×C4×C59C22×D411yescharacteristic
249C2×C4×C59C22×Q811yescharacteristic
250C2×C4×C59C2×#4111yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59H(5,𝔽2)11yescharacteristic
253C2×C4×C592−1+411yescharacteristic
254C2×Dic59—22no—
255C2×Dic59—22no—
256C2×Dic59—22no—
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—41no—
266C2×Dic59—84no—
267Dic118—22no—
268Dic118—22no—
269Dic118—41no—
270Dic118—41no—
271C59⋊C8—22no—
272C59⋊C8—22no—
273C59⋊C8—41no—
274C59⋊C8—41no—
275#26362—591nomaximal, Sylow
276C22×C59—22no—
277C22×C59—22no—
278C22×C59—22no—
279C22×C59—41no—
280C22×C59—41no—
281C22×C59—41no—
282C22×C59—41no—
283C22×C59C2×C22⋊Q811yescharacteristic
284C22×C59D4211yescharacteristic
285C22×C59#54711yescharacteristic
286C22×C59C2×#13711yescharacteristic
287C22×C59#57511yescharacteristic
288C22×C59#57611yescharacteristic
289C22×C59#57811yescharacteristic
290C4×C59—22no—
291C4×C59—22no—
292C4×C59—41no—
293C4×C59—41no—
294C4×C59—41no—
295C4×C59—42no—
296C4×C59—81no—
297C4×C59—81no—
298C4×C59—81no—
299C4×C59#52511yescharacteristic
300C4×C59#53911yescharacteristic
301C4×C59D4×Q811yescharacteristic
302C4×C59#55411yescharacteristic
303Dic59—42no—
304Dic59—42no—
305Dic59—42no—
306Dic59—81no—
307C2×#479—591no—
308C2×#428—591no—
309#2441D5911yescharacteristic
310#2360—591no—
311#2095—591no—
312#2086—591no—
313#2066—591no—
314#2059—591no—
315#1947—591no—
316#1919—591no—
317#1894—591no—
318#1873—591no—
319#1863—591no—
320#1852—591no—
321#1851—591no—
322C2×C59—42no—
323C2×C59#244911yescharacteristic
324C2×C59#307811yescharacteristic
325C2×C59#308311yescharacteristic
326C2×C59#326811yescharacteristic
327C2×C59#327611yescharacteristic
328C2×C59#334611yescharacteristic
329C2×C59#335211yescharacteristic
330C22×M16—591no—
331C2×#125C2×D5911yescharacteristic
332C2×C22⋊Q8—591no—
333C2×C4×Q8—591no—
334C2×#117C2×D5911yescharacteristic
335C22×C4⋊C4—591no—
336#479—1184no—
337#428—1184no—
338C2×#107—591no—
339C2×#107—591no—
340C2×#107—591no—
341C2×#107—591no—
342C2×#106—591no—
343C2×#106—591no—
344C2×C4⋊C8—591no—
345C2×C4⋊C8—591no—
346C2×#103—591no—
347C2×#103—591no—
348C2×#103—591no—
349C2×#103—591no—
350C2×C22⋊C8—591no—
351#399—21no—
352#396—21no—
353#391—591no—
354#389—21no—
355#389—591no—
356#389—1181no—
357#389C2×D5911yescharacteristic
358#389C2×D5911yescharacteristic
359#387—21no—
360#387—1181no—
361#385—1181no—
362#384C2×D5911yescharacteristic
363#380C2×D5911yescharacteristic
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#336—591no—
374#336—591no—
375C59#2636211yescharacteristic, complemented, Sylow
376C22×Q8—591no—
377C23×C4—1181no—
378C23×C4C22×D5911yescharacteristic
379C2×M16—1182no—
380C2×M16—1182no—
381C2×M16—1182no—
382C22×C8—591no—
383C22×C8—591no—
384C22×C8—591no—
385C22×C8—591no—
386#125—22no—
387#125—41no—
388C22⋊Q8—594no—
389C22⋊Q8—1182no—
390C4×Q8—1182no—
391C4×Q8—2361no—
392#117—22no—
393#117—41no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
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×C4⋊C4C22×D5911yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—1181no—
416C2×C22⋊C4C22⋊D5911yescharacteristic
417C2×C22⋊C4C22⋊D5911yescharacteristic
418C2×C42—21no—
419C2×C42—1181no—
420C2×C42—1181no—
421C2×C42C22×D5911yescharacteristic
422C2×C42C22×D5911yescharacteristic
423#107—1182no—
424#107—1182no—
425#107—1182no—
426#107—2361no—
427#106—594no—
428#106—1182no—
429C4⋊C8—1182no—
430C4⋊C8—1182no—
431#103—1182no—
432#103—1182no—
433#103—1182no—
434#103—2361no—
435C22⋊C8—1182no—
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—592no—
447C2×Q8—1181no—
448C2×Q8—1181no—
449C2×Q8—1181no—
450C2×Q8—1181no—
451C2×Q8—1181no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4—1181no—
463C22×C4—1181no—
464C22×C4—4721no—
465C22×C4—2362no—
466C22×C4—2362no—
467C22×C4Q8×D5912yes—
468C22×C4#1174568912yes—
469C22×C4C2×C22⋊D5911yescharacteristic
470C22×C4C2×C22⋊D5911yescharacteristic
471C22×C4C2×C22⋊D5911yescharacteristic
472C22×C4C2×C22⋊D5911yescharacteristic
473C22×C4C2×C22⋊D5911yescharacteristic
474C22×C4C23×D5911yescharacteristic
475M16—1188no—
476C2×C8—592no—
477C2×C8—592no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—1181no—
485C2×C8—1181no—
486C2×C8—1181no—
487C2×C8—1181no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C4⋊C4—592no—
502C4⋊C4—592no—
503C4⋊C4—1184no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—4721no—
509C4⋊C4—2362no—
510C4⋊C4—2362no—
511C4⋊C4D4×D5912yes—
512C22⋊C4—22no—
513C22⋊C4—41no—
514C22⋊C4—42no—
515C22⋊C4—81no—
516C22⋊C4—1184no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—42no—
522C42—4721no—
523C42—2362no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#1427370312yes—
531C23C22×C22⋊D5911yescharacteristic
532Q8—1182no—
533Q8—1182no—
534Q8—2361no—
535Q8—2361no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
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—22no—
553C2×C4—21no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—1182no—
566C2×C4—1182no—
567C2×C4—1182no—
568C2×C4—4724no—
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#1427370312yes—
579C2×C4#1427375712yes—
580C2×C4C2×D4×D5911yescharacteristic
581C2×C4C2×Q8×D5911yescharacteristic
582C2×C4#41×D5911yescharacteristic
583C2×C4C2×#1174568911yescharacteristic
584C2×C4#1427377211yescharacteristic
585C2×C4#1427377411yescharacteristic
586C8—1182no—
587C8—1182no—
588C8—2361no—
589C8—2361no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22C2×#1427370311yescharacteristic
598C22D4×C22⋊D5911yescharacteristic
599C22#1683805111yescharacteristic
600C22C2×#1427375711yescharacteristic
601C22#1683820011yescharacteristic
602C22#1683820511yescharacteristic
603C22#1683821711yescharacteristic
604C4—22no—
605C4—22no—
606C4—41no—
607C4—42no—
608C4—41no—
609C4—41no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—4721no—
614C4—2362no—
615C4—2362no—
616C4—2362no—
617C4#1683791511yescharacteristic
618C4#1683800111yescharacteristic
619C4#1683806511yescharacteristic
620C4#1683810611yescharacteristic
621C2—42no—
622C2#2142790111yescharacteristic
623C2#2143389711yescharacteristic
624C2#2143394311yescharacteristic
625C2#2143563711yescharacteristic
626C2#2143573811yescharacteristic
627C2#2143658111yescharacteristic
628C2#2143666511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2441C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2441D59
hypocenterC59#26362
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
1. C2×#16837706C2semidirect
4. #21425635C2semidirect
5. #21425587C2semidirect
6. #21425474C2semidirect
7. #21425431C2semidirect
8. #21424852C2semidirect
9. #21424671C2semidirect
12. #21424349C2semidirect
375. C59#26362semidirect

Maximal subgroups

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

subgroupindex
C2×#168377062
C2×#168374452
#214270992
#214256352
#214255872
#214254742
#214254312
#214248522
#214246712
#214245272
#214244082
#214243492
#214242832
#214242812
C59×#24412
#2636259