Finite Group Databasebeta

← #28640477

Subgroups — #28640477

Order 15104.

812 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837812C211yesmaximal, characteristic, complemented
2#21429049C211yesmaximal, characteristic
3#21427196C211yesmaximal, characteristic, complemented
4#21425575C211yesmaximal, characteristic, complemented
5#21425570C211yesmaximal, characteristic, complemented
6#21425480C211yesmaximal, characteristic
7#21425466C211yesmaximal, characteristic
8#21425043C211yesmaximal, characteristic, complemented
9#21424856C211yesmaximal, characteristic, complemented
10#21424851C211yesmaximal, characteristic, complemented
11#21424817C211yesmaximal, characteristic, complemented
12#21424726C211yesmaximal, characteristic
13#21424719C211yesmaximal, characteristic
14#21423921C211yesmaximal, characteristic
15C2×C59×#478C211yesmaximal, characteristic
16C2×#14273727C2211yescharacteristic
17C2×#14273706—21no—
18C2×#14273706—21no—
19#16837812—24no—
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×Dic59⋊C8C2211yescharacteristic
25C2×Dic59⋊C8C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29#16837338—21no—
30#16837338—21no—
31#16837305C2211yescharacteristic
32#16837305C2211yescharacteristic
33#16837301—21no—
34#16837301C2211yescharacteristic
35#16837301C2211yescharacteristic
36#16837288—21no—
37#16837288—21no—
38#16837271C2211yescharacteristic
39#16837271C2211yescharacteristic
40C4⋊C4×Dic59—21no—
41C4⋊C4×Dic59C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C4×C4⋊Dic59C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×#125×C59C2211yescharacteristic
54C2×C4×D4×C59C2211yescharacteristic
55C59×#478—24no—
56C2×#107×C59C2211yescharacteristic
57C2×#106×C59C2211yescharacteristic
58C2×C4⋊C8×C59C2211yescharacteristic
59C2×#102×C59C2211yescharacteristic
60C2×#102×C59C2211yescharacteristic
61C22×Dic118—21no—
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273727—22no—
65#14273727—22no—
66#14273706—44no—
67#14273706—44no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59D411yescharacteristic
72C2×C4⋊Dic59D411yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
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×C22⋊Dic59—21no—
82C2×C22⋊Dic59—21no—
83C2×C22⋊Dic59—21no—
84C2×C22⋊Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59C2311yescharacteristic
89C2×C4×Dic59C2311yescharacteristic
90#14273660—22no—
91#14273660—22no—
92#14273658—22no—
93#14273658—22no—
94Dic59⋊C8—22no—
95Dic59⋊C8—22no—
96#14273648—22no—
97#14273648Q814yescomplemented
98#14273638—22no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95D411yescharacteristic
105C59⋊#95D411yescharacteristic
106C22×D4×C59C2311yescharacteristic
107C23×C4×C59—21no—
108C22×C8×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110#125×C59—22no—
111#125×C59D414yes—
112C4×D4×C59—22no—
113C4×D4×C59—22no—
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59—21no—
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C22⋊C4×C59—21no—
120C2×C42×C59C2311yescharacteristic
121#107×C59—22no—
122#106×C59—22no—
123C4⋊C8×C59—22no—
124#102×C59—22no—
125#102×C59—22no—
126C24×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59C2×Q812yes—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—42no—
137C22×C4×C59—42no—
138C22×C4×C59—42no—
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2411yescharacteristic
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59C2×D411yescharacteristic
147C22×Dic59C2×D411yescharacteristic
148C22×Dic59C2×D411yescharacteristic
149C22×Dic59C2×D411yescharacteristic
150C2×Dic118—42no—
151C2×Dic118—42no—
152C2×Dic118—42no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C4⋊C4×C59—24no—
160C4⋊C4×C59—24no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59C2×D412yes—
166C4⋊C4×C59C2×D412yes—
167C22⋊C4×C59—42no—
168C42×C59C2×D412yes—
169C42×C59#4112yes—
170C22⋊Dic59—42no—
171C22⋊Dic59—42no—
172C22⋊Dic59—42no—
173C22⋊Dic59—42no—
174C4⋊Dic59—21no—
175C4⋊Dic59—22no—
176C4⋊Dic59—22no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—42no—
184C4⋊Dic59—42no—
185C4⋊Dic59—81no—
186C4⋊Dic59C2×Q812yes—
187Dic59⋊C4—42no—
188Dic59⋊C4—42no—
189Dic59⋊C4—42no—
190Dic59⋊C4—42no—
191C4×Dic59—21no—
192C4×Dic59—21no—
193C4×Dic59—21no—
194C4×Dic59—21no—
195C4×Dic59—42no—
196C4×Dic59—81no—
197C4×Dic59—81no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8C2×Q812yes—
203C2×C59⋊C8#4112yes—
204C23×C59—22no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59C22×D411yescharacteristic
212D4×C59—22no—
213D4×C59—22no—
214D4×C59—41no—
215D4×C59—41no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—84no—
230C2×C4×C59#12112yes—
231C2×C4×C59#12112yes—
232C2×C4×C59C22×D411yescharacteristic
233C2×C4×C59C22×Q811yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C592−1+411yescharacteristic
237C2×C4×C592−1+411yescharacteristic
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—22no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59AGL(1,ℤ/8)12yes—
261C2×Dic59AGL(1,ℤ/8)12yes—
262Dic118—84no—
263C8×C59—41no—
264C8×C59—41no—
265C59⋊C8—22no—
266C59⋊C8—22no—
267#25465—591nomaximal, Sylow
268C22×C59—42no—
269C22×C59—42no—
270C22×C59—42no—
271C22×C59—81no—
272C22×C59—81no—
273C22×C59—81no—
274C22×C59—81no—
275C22×C59C2×#12111yescharacteristic
276C22×C59#54711yescharacteristic
277C22×C59D4×Q811yescharacteristic
278C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C59#57711yescharacteristic
281C22×C59#57811yescharacteristic
282C4×C59—22no—
283C4×C59—22no—
284C4×C59—42no—
285C4×C59—42no—
286C4×C59—81no—
287C4×C59#53311yescharacteristic
288C4×C59#54111yescharacteristic
289C4×C59#54811yescharacteristic
290C4×C59D4×Q811yescharacteristic
291Dic59—22no—
292Dic59—22no—
293Dic59—41no—
294Dic59—41no—
295Dic59—42no—
296Dic59—81no—
297Dic59—81no—
298Dic59—81no—
299C2×#501—591no—
300C2×#478D5911yescharacteristic
301#2542—591no—
302#2370—591no—
303#2084—591no—
304#2083—591no—
305#2067—591no—
306#2065—591no—
307#1987—591no—
308#1948—591no—
309#1947—591no—
310#1940—591no—
311#1926—591no—
312#1925—591no—
313#1762—591no—
314C2×C59—82no—
315C2×C59#242911yescharacteristic
316C2×C59#303811yescharacteristic
317C2×C59#304611yescharacteristic
318C2×C59#334311yescharacteristic
319C2×C59#334511yescharacteristic
320C2×C59#338411yescharacteristic
321C2×C59#338611yescharacteristic
322C2×C4⋊Q8—591no—
323C2×#125C2×D5911yescharacteristic
324C2×C22⋊Q8—1181no—
325C2×C22⋊Q8—1181no—
326C2×C4×D4C2×D5911yescharacteristic
327#501—1184no—
328#478—24no—
329C2×#107—591no—
330C2×#107—591no—
331C2×#107C2×D5911yescharacteristic
332C2×#106—591no—
333C2×#106—591no—
334C2×#106C2×D5911yescharacteristic
335C2×C4⋊C8—591no—
336C2×C4⋊C8—591no—
337C2×C4⋊C8C2×D5911yescharacteristic
338C2×#102—591no—
339C2×#102—591no—
340C2×#102C2×D5911yescharacteristic
341C2×#102C2×D5911yescharacteristic
342C2×#97—591no—
343#399—1181no—
344#399—1181no—
345#390—591no—
346#390—591no—
347#389—591no—
348#389—591no—
349#389—1181no—
350#387—1181no—
351#387—1181no—
352#384—591no—
353#384—591no—
354C4×C4⋊C4—591no—
355C4×C4⋊C4—591no—
356C4×C4⋊C4—591no—
357C4×C4⋊C4—591no—
358C4×C4⋊C4—1181no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#336—591no—
366#336—591no—
367C59#2546511yescharacteristic, complemented, Sylow
368C22×Q8—1181no—
369C22×D4C22×D5911yescharacteristic
370C23×C4—21no—
371C22×C8—591no—
372C22×C8—591no—
373C22×C8C22×D5911yescharacteristic
374C22×C8C22×D5911yescharacteristic
375C4⋊Q8—1182no—
376C4⋊Q8—1182no—
377#125—22no—
378#125C22⋊D5914yes—
379C22⋊Q8—2364no—
380C22⋊Q8—2364no—
381C4×D4—22no—
382C4×D4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4C22×D5911yescharacteristic
399C2×C4⋊C4C22×D5911yescharacteristic
400C2×C4⋊C4C22×D5911yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1181no—
403C2×C22⋊C4—1181no—
404C2×C22⋊C4—1181no—
405C2×C22⋊C4—1181no—
406C2×C42—591no—
407C2×C42—591no—
408C2×C42—1181no—
409C2×C42—1181no—
410C2×C42—1181no—
411C2×C42C22×D5911yescharacteristic
412#107—22no—
413#107—1182no—
414#107—1182no—
415#106—22no—
416#106—1182no—
417#106—1182no—
418C4⋊C8—22no—
419C4⋊C8—1182no—
420C4⋊C8—1182no—
421#102—22no—
422#102—22no—
423#102—594no—
424#102—1182no—
425#97—1182no—
426#95—591no—
427#95—591no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433C24—21no—
434C2×Q8—2362no—
435C2×Q8—2362no—
436C2×Q8—2362no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4Q8×D5912yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—591no—
450C22×C4—591no—
451C22×C4—591no—
452C22×C4—591no—
453C22×C4—1181no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4D4×D5911yescharacteristic
458C22×C4D4×D5911yescharacteristic
459C22×C4C23×D5911yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—592no—
467C2×C8—592no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—592no—
479C4⋊C4—1182no—
480C4⋊C4—1182no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—4721no—
489C4⋊C4—2362no—
490C4⋊C4—2362no—
491C4⋊C4—2362no—
492C4⋊C4—2362no—
493C4⋊C4—2362no—
494C4⋊C4—2362no—
495C4⋊C4C2×C22⋊D5912yes—
496C4⋊C4C2×C22⋊D5912yes—
497C22⋊C4—42no—
498C22⋊C4—2362no—
499C22⋊C4—2362no—
500C22⋊C4—2362no—
501C22⋊C4—2362no—
502C42—1181no—
503C42—1181no—
504C42—1181no—
505C42—1181no—
506C42—4721no—
507C42—4721no—
508C42—2362no—
509C42#1174568712yes—
510C42C2×C22⋊D5912yes—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23C2×D4×D5911yescharacteristic
519Q8—4724no—
520D4—22no—
521D4—22no—
522D4—41no—
523D4—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—592no—
539C2×C4—592no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—1181no—
545C2×C4—1181no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1182no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4#1427369912yes—
563C2×C4#1427369912yes—
564C2×C4C22×C22⋊D5911yescharacteristic
565C2×C4C2×Q8×D5911yescharacteristic
566C2×C4#41×D5911yescharacteristic
567C2×C4C2×#1174568711yescharacteristic
568C2×C4#1427377411yescharacteristic
569C2×C4#1427377511yescharacteristic
570C8—41no—
571C8—41no—
572C8—1182no—
573C8—1182no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×#1427369911yescharacteristic
582C22#1683805411yescharacteristic
583C22#1683806511yescharacteristic
584C22AGL(1,ℤ/8)×D5911yescharacteristic
585C22AGL(1,ℤ/8)×D5911yescharacteristic
586C22#1683820911yescharacteristic
587C22#1683821611yescharacteristic
588C4—22no—
589C4—22no—
590C4—42no—
591C4—42no—
592C4—81no—
593C4—1182no—
594C4—1182no—
595C4—4721no—
596C4—4721no—
597C4—4721no—
598C4—2362no—
599C4—2361no—
600C4—2361no—
601C4#1683795411yescharacteristic
602C4#1683801211yescharacteristic
603C4#1683805911yescharacteristic
604C4Q8×C22⋊D5911yescharacteristic
605C2—82no—
606C2#2142769711yescharacteristic
607C2#2143357411yescharacteristic
608C2#2143364411yescharacteristic
609C2#2143654311yescharacteristic
610C2#2143656311yescharacteristic
611C2#2143699811yescharacteristic
612C2#2143701211yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#478C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenterC2×#478D59
hypocenterC59#25465
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. C2×#16837812C2semidirect
3. #21427196C2semidirect
4. #21425575C2semidirect
5. #21425570C2semidirect
8. #21425043C2semidirect
9. #21424856C2semidirect
10. #21424851C2semidirect
11. #21424817C2semidirect
97. #14273648Q8semidirect
367. C59#25465semidirect

Maximal subgroups

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

subgroupindex
C2×#168378122
#214290492
#214271962
#214255752
#214255702
#214254802
#214254662
#214250432
#214248562
#214248512
#214248172
#214247262
#214247192
#214239212
C2×C59×#4782
#2546559