Finite Group Databasebeta

← #28623007

Subgroups — #28623007

Order 15104.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837959C211yesmaximal, characteristic, complemented
2C2×#16837692C211yesmaximal, characteristic, complemented
3C2×#16837607C211yesmaximal, characteristic, complemented
4#21425292C212yesmaximal, complemented
5#21425099C212yesmaximal
6#21425044C211yesmaximal, characteristic, complemented
7#21425019C211yesmaximal, characteristic, complemented
8#21424454C212yesmaximal, complemented
9#21424363C212yesmaximal, complemented
10#21424076C211yesmaximal, characteristic, complemented
11C4×#121×C59C211yesmaximal, characteristic
12C22×#11745670C2212yes—
13#16837959—24no—
14#16837959—24no—
15C2×#14273727C2211yescharacteristic
16C2×#14273727C2211yescharacteristic
17C2×#14273719—21no—
18C2×#14273706—21no—
19C2×#14273706C2212yes—
20C2×C4×Dic118—21no—
21C2×C4×Dic118C2212yes—
22C2×#14273686—21no—
23#16837692—24no—
24#16837607—24no—
25C2×#14273658C2212yes—
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273650C2212yes—
29C2×#14273650C2212yes—
30C2×#14273650C2212yes—
31C2×#14273648C2212yes—
32C2×C22⋊(C59⋊C8)C2212yes—
33#16837164C2212yes—
34#16837164C2212yes—
35#16837157C2212yes—
36C2×#121×C59C2211yescharacteristic
37C2×C4×D4×C59—21no—
38C2×C4×D4×C59C2211yescharacteristic
39C22×C42×C59C2211yescharacteristic
40C59×#390C2211yescharacteristic
41C59×#389C2211yescharacteristic
42C59×#388—21no—
43C59×#386—21no—
44C59×#380C2211yescharacteristic
45C4×C4⋊C4×C59C2211yescharacteristic
46C4×C22⋊C4×C59—21no—
47C22×Dic118—21no—
48C22×Dic118C2312yes—
49C2×#11745670—24no—
50C2×#11745670—24no—
51C2×#11745670—42no—
52C22×C59⋊C8C2312yes—
53C22×C59⋊C8C2312yes—
54#14273727—22no—
55#14273727—22no—
56#14273727D414yescomplemented
57#14273727D414yescomplemented
58#14273719—28no—
59#14273706—28no—
60#14273706—24no—
61#14273706—24no—
62C4×Dic118—28no—
63C4×Dic118—24no—
64C4×Dic118—24no—
65#14273686—28no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59C2312yes—
70C2×Dic59⋊C4—22no—
71C2×Dic59⋊C4—22no—
72C2×C22⋊Dic59—22no—
73C2×C4×Dic59—22no—
74#14273658—24no—
75C4⋊(C59⋊C8)—22no—
76C4⋊(C59⋊C8)—22no—
77#14273650—24no—
78#14273650—24no—
79#14273650—42no—
80#14273648—24no—
81C22⋊(C59⋊C8)—42no—
82C22×D4×C59—21no—
83C22×D4×C59C2311yescharacteristic
84C23×C4×C59—21no—
85C23×C4×C59D411yescharacteristic
86C23×C4×C59D411yescharacteristic
87C23×C4×C59C2311yescharacteristic
88#121×C59—22no—
89#121×C59—22no—
90C4×D4×C59—22no—
91C4×D4×C59—22no—
92C4×D4×C59—44no—
93C2×C4⋊C4×C59—21no—
94C2×C4⋊C4×C59C2311yescharacteristic
95C2×C4⋊C4×C59C2311yescharacteristic
96C2×C4⋊C4×C59C2311yescharacteristic
97C2×C22⋊C4×C59—21no—
98C2×C22⋊C4×C59—21no—
99C2×C22⋊C4×C59—21no—
100C2×C22⋊C4×C59—21no—
101C2×C22⋊C4×C59—21no—
102C2×C42×C59—22no—
103C2×C42×C59—22no—
104C2×C42×C59—21no—
105C2×C42×C59D411yescharacteristic
106C2×C42×C59D411yescharacteristic
107C2×C42×C59C2311yescharacteristic
108C2×C42×C59C2311yescharacteristic
109#95×C59—21no—
110#95×C59—21no—
111#95×C59—21no—
112C24×C59—21no—
113C24×C59C2×D411yescharacteristic
114C2×D4×C59—24no—
115C2×D4×C59—21no—
116C2×D4×C59—21no—
117C2×D4×C59—21no—
118C2×D4×C59—21no—
119C2×D4×C59—21no—
120C2×D4×C59—42no—
121C2×D4×C59—42no—
122C2×D4×C59#4112yes—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—21no—
126C22×C4×C59—22no—
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×C59—21no—
135C22×C4×C59—41no—
136C22×C4×C59—41no—
137C22×C4×C59—42no—
138C22×C4×C59—42no—
139C22×C4×C59—42no—
140C22×C4×C59SD1614yescomplemented
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2411yescharacteristic
147C22×Dic59—22no—
148C22×Dic59—22no—
149C2×Dic118—22no—
150C2×Dic118—22no—
151C2×Dic118—22no—
152C2×Dic118—22no—
153C2×Dic118—24no—
154C2×Dic118—22no—
155C2×Dic118—42no—
156C2×Dic118—42no—
157C2×Dic118C2×D414yes—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—42no—
164C4⋊C4×C59#4112yes—
165C22⋊C4×C59—24no—
166C22⋊C4×C59—42no—
167C22⋊C4×C59—42no—
168C22⋊C4×C59—42no—
169C22⋊C4×C59—42no—
170C42×C59—22no—
171C42×C59—22no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—42no—
175C42×C59C2×D412yes—
176C42×C59C2×D412yes—
177C22⋊Dic59—28no—
178#11745670—48no—
179C4⋊Dic59—24no—
180C4⋊Dic59—24no—
181C4⋊Dic59—24no—
182C4⋊Dic59—22no—
183C4⋊Dic59C2×D414yes—
184Dic59⋊C4—28no—
185Dic59⋊C4—28no—
186C4×Dic59—28no—
187C2×C59⋊C8—22no—
188C2×C59⋊C8—22no—
189C2×C59⋊C8—22no—
190C2×C59⋊C8—22no—
191C2×C59⋊C8—22no—
192C2×C59⋊C8—22no—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—22no—
200C23×C59—41no—
201C23×C59—41no—
202C23×C59—41no—
203C23×C59—41no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59C2×SD1612yes—
207C23×C59C22×D411yescharacteristic
208D4×C59—22no—
209D4×C59—22no—
210D4×C59—48no—
211D4×C59—41no—
212D4×C59—41no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—28no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—22no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—44no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
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—84no—
239C2×C4×C59#12012yes—
240C2×C4×C59#12012yes—
241C2×C4×C59C2×SD1612yes—
242C2×C4×C59C2×SD1612yes—
243C2×C4×C59#13712yes—
244C2×C4×C59C22×D411yescharacteristic
245C2×C4×C59C22×D411yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59H(5,𝔽2)11yescharacteristic
249C2×C4×C592−1+411yescharacteristic
250C2×Dic59—24no—
251C2×Dic59—24no—
252C2×Dic59—42no—
253C2×Dic59—42no—
254C2×Dic59—42no—
255C2×Dic59—42no—
256Dic118—24no—
257Dic118—24no—
258Dic118—48no—
259Dic118—42no—
260Dic118—42no—
261C59⋊C8—42no—
262C59⋊C8—42no—
263#24151—591nomaximal, Sylow
264C22×C59—22no—
265C22×C59—22no—
266C22×C59—22no—
267C22×C59—41no—
268C22×C59—41no—
269C22×C59—41no—
270C22×C59—41no—
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—42no—
274C22×C59—81no—
275C22×C59—81no—
276C22×C59—81no—
277C22×C59—81no—
278C22×C59C2×#12011yescharacteristic
279C22×C59#52511yescharacteristic
280C22×C59#53711yescharacteristic
281C22×C59C22×SD1611yescharacteristic
282C22×C59C2×#13711yescharacteristic
283C22×C59#57511yescharacteristic
284C22×C59#57511yescharacteristic
285C4×C59—22no—
286C4×C59—22no—
287C4×C59—42no—
288C4×C59—41no—
289C4×C59—41no—
290C4×C59—81no—
291C4×C59—81no—
292C4×C59#44814yes—
293C4×C59#54611yescharacteristic
294C4×C59#54711yescharacteristic
295C4×C59#54811yescharacteristic
296C4×C59#55011yescharacteristic
297Dic59—44no—
298Dic59—44no—
299C2×#533—591no—
300C2×#476—591no—
301C2×#461—591no—
302C4×#121D5911yescharacteristic
303#2035—592no—
304#2000—592no—
305#1987—591no—
306#1981—591no—
307#1880—592no—
308#1865—592no—
309#1797—591no—
310C2×C59—42no—
311C2×C59—82no—
312C2×C59#240711yescharacteristic
313C2×C59C2×#44811yescharacteristic
314C2×C59#300511yescharacteristic
315C2×C59#310111yescharacteristic
316C2×C59#311611yescharacteristic
317C2×C59#324611yescharacteristic
318C2×C59#325211yescharacteristic
319C22×SD16—592no—
320#533—1184no—
321#533—1184no—
322C2×C4⋊Q8—591no—
323C2×C4⋊Q8—591no—
324C2×#125—1181no—
325C2×C22⋊Q8—592no—
326C2×C22⋊Q8—1181no—
327C2×#121C2×D5911yescharacteristic
328C2×C4×Q8—592no—
329C2×C4×Q8—1181no—
330C2×C4×D4—21no—
331C2×C4×D4C2×D5911yescharacteristic
332C2×#117—1181no—
333C22×C42C2×D5911yescharacteristic
334#476—1184no—
335#461—1184no—
336C2×#106—592no—
337C2×C4⋊C8—591no—
338C2×C4⋊C8—591no—
339C2×#103—592no—
340C2×#103—592no—
341C2×#103—592no—
342C2×#102—592no—
343C2×C22⋊C8—592no—
344#390C2×D5911yescharacteristic
345#389C2×D5911yescharacteristic
346#388—21no—
347#386—21no—
348#380C2×D5911yescharacteristic
349C4×C4⋊C4C2×D5911yescharacteristic
350C4×C22⋊C4—21no—
351#340—592no—
352#340—592no—
353#336—592no—
354C59#2415111yescharacteristic, complemented, Sylow
355C22×Q8—592no—
356C22×Q8—1181no—
357C22×D4—21no—
358C22×D4C22×D5911yescharacteristic
359C23×C4—21no—
360C23×C4C22⋊D5911yescharacteristic
361C23×C4C22⋊D5911yescharacteristic
362C23×C4C22×D5911yescharacteristic
363C2×SD16—1184no—
364C2×SD16—1184no—
365C2×SD16—2362no—
366C22×C8—592no—
367C22×C8—592no—
368C4⋊Q8—594no—
369C4⋊Q8—594no—
370C4⋊Q8—1182no—
371C4⋊Q8—1182no—
372#125—1188no—
373C22⋊Q8—1184no—
374C22⋊Q8—1184no—
375C22⋊Q8—1188no—
376#121—22no—
377#121—22no—
378C4×Q8—1184no—
379C4×Q8—1184no—
380C4×Q8—1188no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—44no—
384#117—1188no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—592no—
387C2×C4⋊C4—1181no—
388C2×C4⋊C4—1181no—
389C2×C4⋊C4—1182no—
390C2×C4⋊C4—1182no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4C22×D5911yescharacteristic
393C2×C4⋊C4C22×D5911yescharacteristic
394C2×C4⋊C4C22×D5911yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1182no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—21no—
404C2×C42—1182no—
405C2×C42C22⋊D5911yescharacteristic
406C2×C42C22⋊D5911yescharacteristic
407C2×C42C22×D5911yescharacteristic
408C2×C42C22×D5911yescharacteristic
409#106—1184no—
410C4⋊C8—1182no—
411C4⋊C8—1182no—
412#103—1184no—
413#103—1184no—
414#103—2362no—
415#102—1184no—
416C22⋊C8—2362no—
417#95—21no—
418#95—21no—
419#95—21no—
420C24—21no—
421C24C2×C22⋊D5911yescharacteristic
422C2×Q8—594no—
423C2×Q8—1184no—
424C2×Q8—1182no—
425C2×Q8—1182no—
426C2×Q8—1182no—
427C2×Q8—1182no—
428C2×Q8—1182no—
429C2×Q8—2362no—
430C2×Q8—2362no—
431C2×D4—24no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4#1174568512yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—1182no—
458C22×C4—1182no—
459C22×C4#1174567014yescomplemented
460C22×C4C2×C22⋊D5911yescharacteristic
461C22×C4C2×C22⋊D5911yescharacteristic
462C22×C4C2×C22⋊D5911yescharacteristic
463C22×C4C2×C22⋊D5911yescharacteristic
464C22×C4C2×C22⋊D5911yescharacteristic
465C22×C4C23×D5911yescharacteristic
466SD16—2368no—
467C2×C8—1182no—
468C2×C8—1182no—
469C2×C8—1182no—
470C2×C8—1182no—
471C2×C8—1182no—
472C2×C8—1182no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—42no—
479C4⋊C4—594no—
480C4⋊C4—1188no—
481C4⋊C4—1188no—
482C4⋊C4—1184no—
483C4⋊C4—1184no—
484C4⋊C4—1184no—
485C4⋊C4—1182no—
486C4⋊C4#1174568512yes—
487C22⋊C4—24no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—1188no—
493C42—22no—
494C42—22no—
495C42—21no—
496C42—21no—
497C42—42no—
498C42—1188no—
499C42D4×D5912yes—
500C42D4×D5912yes—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C2×#1174567012yes—
515C23C22×C22⋊D5911yescharacteristic
516Q8—1184no—
517Q8—1184no—
518Q8—2368no—
519Q8—2362no—
520Q8—2362no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—84no—
552C2×C4—1184no—
553C2×C4—1184no—
554C2×C4—2362no—
555C2×C4—2362no—
556C2×C4—2362no—
557C2×C4—2362no—
558C2×C4#1427369512yes—
559C2×C4#1427369512yes—
560C2×C4C2×#1174567012yes—
561C2×C4C2×#1174567012yes—
562C2×C4#1427375712yes—
563C2×C4C2×D4×D5911yescharacteristic
564C2×C4C2×D4×D5911yescharacteristic
565C2×C4C2×#1174568511yescharacteristic
566C2×C4C2×#1174568511yescharacteristic
567C2×C4#1427377311yescharacteristic
568C2×C4#1427377411yescharacteristic
569C8—2362no—
570C8—2362no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×#1427369511yescharacteristic
586C22#1683791311yescharacteristic
587C22#1683798011yescharacteristic
588C22C22×#1174567011yescharacteristic
589C22C2×#1427375711yescharacteristic
590C22#1683820011yescharacteristic
591C22#1683820011yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—2364no—
600C4—2364no—
601C4#1683754014yes—
602C4#1683804811yescharacteristic
603C4#1683805711yescharacteristic
604C4#1683806111yescharacteristic
605C4#1683807311yescharacteristic
606C2—42no—
607C2—82no—
608C2#2142744111yescharacteristic
609C2C2×#1683754011yescharacteristic
610C2#2143333611yescharacteristic
611C2#2143410411yescharacteristic
612C2#2143425811yescharacteristic
613C2#2143541211yescharacteristic
614C2#2143546911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC4×#121×C59C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenterC4×#121D59
hypocenterC59#24151
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16837959C2semidirect
2. C2×#16837692C2semidirect
3. C2×#16837607C2semidirect
4. #21425292C2semidirect
6. #21425044C2semidirect
7. #21425019C2semidirect
8. #21424454C2semidirect
9. #21424363C2semidirect
10. #21424076C2semidirect
56. #14273727D4semidirect
57. #14273727D4semidirect
140. C22×C4×C59SD16semidirect
354. C59#24151semidirect
459. C22×C4#11745670semidirect

Maximal subgroups

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

subgroupindex
C2×#168379592
C2×#168376922
C2×#168376072
#214252922
#214250992
#214250442
#214250192
#214244542
#214243632
#214240762
C4×#121×C592
#2415159