Finite Group Databasebeta

← #28640409

Subgroups — #28640409

Order 15104.

792 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837812C211yesmaximal, characteristic, complemented
2C2×#16837692C211yesmaximal, characteristic
3#21427154C211yesmaximal, characteristic, complemented
4#21425578C211yesmaximal, characteristic, complemented
5#21425573C211yesmaximal, characteristic, complemented
6#21425484C211yesmaximal, characteristic
7#21425470C211yesmaximal, characteristic
8#21425054C211yesmaximal, characteristic, complemented
9#21424862C211yesmaximal, characteristic, complemented
10#21424857C211yesmaximal, characteristic, complemented
11#21424820C211yesmaximal, characteristic, complemented
12#21424720C211yesmaximal, characteristic
13#21424713C211yesmaximal, characteristic
14#21423921C211yesmaximal, characteristic
15C59×#2566C211yesmaximal, characteristic
16C2×#14273727C2211yescharacteristic
17C2×#14273692—21no—
18#16837812—24no—
19#16837692—24no—
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×C4⋊(C59⋊C8)C2211yescharacteristic
29C2×#14273648C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×#14273648C2211yescharacteristic
33C2×#14273638C2211yescharacteristic
34#16837307—21no—
35#16837300C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837287—21no—
38#16837287—21no—
39C4×C4⋊Dic59C2211yescharacteristic
40C4×C4⋊Dic59C2211yescharacteristic
41C4×C4⋊Dic59C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#125×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C59×#396—21no—
53C59×#396—21no—
54C59×#390C2211yescharacteristic
55C59×#390C2211yescharacteristic
56C59×#385—21no—
57C59×#385—21no—
58C59×#384C2211yescharacteristic
59C59×#382C2211yescharacteristic
60C4×C4⋊C4×C59C2211yescharacteristic
61C22×Dic118—21no—
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66#14273727—22no—
67#14273727D414yes—
68#14273692—44no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×C4×Dic59—21no—
77C2×C4×Dic59—21no—
78C2×C4×Dic59—21no—
79#14273660—22no—
80#14273660—22no—
81#14273658—22no—
82#14273658—22no—
83#14273658—22no—
84#14273658—22no—
85C4⋊(C59⋊C8)—22no—
86C4⋊(C59⋊C8)—22no—
87C4⋊(C59⋊C8)—22no—
88#14273648—22no—
89#14273648—22no—
90#14273648—22no—
91#14273648Q814yescomplemented
92#14273638—22no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C22×D4×C59C2311yescharacteristic
99C23×C4×C59—21no—
100#125×C59—22no—
101#125×C59—22no—
102C4×D4×C59—22no—
103C4×D4×C59—22no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59D411yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59—21no—
117C2×C22⋊C4×C59—21no—
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120C2×C42×C59C2311yescharacteristic
121#95×C59—21no—
122#95×C59—21no—
123#95×C59D411yescharacteristic
124#95×C59D411yescharacteristic
125C24×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59C2×Q812yes—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—42no—
136C22×C4×C59—81no—
137C22×C4×C59—81no—
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2411yescharacteristic
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C2×Dic118—24no—
150C2×Dic118—42no—
151C2×Dic118—42no—
152C4⋊C4×C59—22no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—41no—
160C4⋊C4×C59—42no—
161C4⋊C4×C59—42no—
162C4⋊C4×C59—81no—
163C4⋊C4×C59—81no—
164C22⋊C4×C59—42no—
165C22⋊C4×C59—42no—
166C22⋊C4×C59—42no—
167C22⋊C4×C59—81no—
168C22⋊C4×C59—81no—
169C42×C59—21no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59C2×D412yes—
174C42×C59#4112yes—
175C4⋊Dic59—24no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59C2×D412yes—
182C4⋊Dic59C2×D412yes—
183C4⋊Dic59C2×Q812yes—
184Dic59⋊C4—42no—
185Dic59⋊C4—42no—
186C4×Dic59—42no—
187C4×Dic59—42no—
188C4×Dic59—42no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
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⋊C8C2×Q812yes—
200C2×C59⋊C8#4112yes—
201C23×C59—22no—
202C23×C59—41no—
203C23×C59—41no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59C22×D411yescharacteristic
209D4×C59—22no—
210D4×C59—22no—
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—21no—
218C2×C4×C59—21no—
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—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—84no—
238C2×C4×C59#12112yes—
239C2×C4×C59#12112yes—
240C2×C4×C59AGL(1,ℤ/8)12yes—
241C2×C4×C59C22×D411yescharacteristic
242C2×C4×C59C22×Q811yescharacteristic
243C2×C4×C59C2×#4111yescharacteristic
244C2×C4×C59C2×#4111yescharacteristic
245C2×C4×C592−1+411yescharacteristic
246C2×C4×C592−1+411yescharacteristic
247C2×Dic59—22no—
248C2×Dic59—22no—
249C2×Dic59—22no—
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—
259Dic118—48no—
260C59⋊C8—22no—
261C59⋊C8—22no—
262C59⋊C8—41no—
263C59⋊C8—41no—
264#25461—591nomaximal, Sylow
265C22×C59—42no—
266C22×C59—42no—
267C22×C59—42no—
268C22×C59—81no—
269C22×C59—81no—
270C22×C59—81no—
271C22×C59—81no—
272C22×C59C2×#12111yescharacteristic
273C22×C59#54711yescharacteristic
274C22×C59D4×Q811yescharacteristic
275C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C59#57511yescharacteristic
277C22×C59#57711yescharacteristic
278C22×C59#57711yescharacteristic
279C4×C59—22no—
280C4×C59—22no—
281C4×C59—22no—
282C4×C59—41no—
283C4×C59—41no—
284C4×C59—41no—
285C4×C59—81no—
286C4×C59—81no—
287C4×C59—81no—
288C4×C59#53311yescharacteristic
289C4×C59#54111yescharacteristic
290C4×C59#54811yescharacteristic
291C4×C59D4×Q811yescharacteristic
292Dic59—42no—
293Dic59—42no—
294Dic59—42no—
295Dic59—81no—
296C2×#501—591no—
297C2×#476—591no—
298#2566D5911yescharacteristic
299#2365—591no—
300#2085—591no—
301#2084—591no—
302#2067—591no—
303#2065—591no—
304#1989—591no—
305#1949—591no—
306#1948—591no—
307#1940—591no—
308#1926—591no—
309#1925—591no—
310#1762—591no—
311C2×C59—82no—
312C2×C59#242911yescharacteristic
313C2×C59#303611yescharacteristic
314C2×C59#304311yescharacteristic
315C2×C59#334511yescharacteristic
316C2×C59#335011yescharacteristic
317C2×C59#338411yescharacteristic
318C2×C59#338811yescharacteristic
319C2×C4⋊Q8—591no—
320C2×#125C2×D5911yescharacteristic
321C2×C4×Q8—1181no—
322C2×C4×D4C2×D5911yescharacteristic
323#501—1184no—
324#476—1184no—
325C2×#107—591no—
326C2×#107—591no—
327C2×#106—591no—
328C2×#106—591no—
329C2×#106—591no—
330C2×#106—591no—
331C2×C4⋊C8—591no—
332C2×C4⋊C8—591no—
333C2×C4⋊C8—591no—
334C2×#102—591no—
335C2×#102—591no—
336C2×#102—591no—
337C2×#102—591no—
338C2×#97—591no—
339#396—21no—
340#396—21no—
341#391—1181no—
342#390C2×D5911yescharacteristic
343#390C2×D5911yescharacteristic
344#389—591no—
345#389—591no—
346#387—1181no—
347#387—1181no—
348#385—21no—
349#385—21no—
350#384C2×D5911yescharacteristic
351#382C2×D5911yescharacteristic
352C4×C4⋊C4—591no—
353C4×C4⋊C4—591no—
354C4×C4⋊C4—591no—
355C4×C4⋊C4C2×D5911yescharacteristic
356#340—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#336—591no—
363#336—591no—
364C59#2546111yescharacteristic, complemented, Sylow
365C22×Q8—1181no—
366C22×D4C22×D5911yescharacteristic
367C23×C4—21no—
368C22×C8—591no—
369C22×C8—591no—
370C22×C8—591no—
371C22×C8—591no—
372C4⋊Q8—594no—
373C4⋊Q8—1182no—
374#125—22no—
375#125—22no—
376C4×Q8—2364no—
377C4×D4—22no—
378C4×D4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—1181no—
388C2×C4⋊C4—1181no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4C22⋊D5911yescharacteristic
391C2×C4⋊C4C22⋊D5911yescharacteristic
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×C42—1181no—
401C2×C42—1181no—
402C2×C42—1181no—
403C2×C42C22×D5911yescharacteristic
404C2×C42C22×D5911yescharacteristic
405C2×C42C22×D5911yescharacteristic
406#107—1182no—
407#107—1182no—
408#106—1182no—
409#106—1182no—
410#106—1182no—
411#106—1182no—
412C4⋊C8—1182no—
413C4⋊C8—1182no—
414C4⋊C8—1182no—
415#102—594no—
416#102—1182no—
417#102—1182no—
418#102—1182no—
419#97—1182no—
420#95—21no—
421#95—21no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95C22⋊D5911yescharacteristic
428#95C22⋊D5911yescharacteristic
429C24—21no—
430C2×Q8—1184no—
431C2×Q8—2362no—
432C2×Q8—2362no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4Q8×D5912yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—1181no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4C2×C22⋊D5911yescharacteristic
450C22×C4C2×C22⋊D5911yescharacteristic
451C22×C4C2×C22⋊D5911yescharacteristic
452C22×C4C2×C22⋊D5911yescharacteristic
453C22×C4C2×C22⋊D5911yescharacteristic
454C22×C4C2×C22⋊D5911yescharacteristic
455C22×C4C23×D5911yescharacteristic
456C2×C8—592no—
457C2×C8—592no—
458C2×C8—1181no—
459C2×C8—1181no—
460C2×C8—1181no—
461C2×C8—1181no—
462C2×C8—1181no—
463C2×C8—1181no—
464C2×C8—1181no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8—1181no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—41no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—592no—
481C4⋊C4—592no—
482C4⋊C4—592no—
483C4⋊C4—1184no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—2362no—
490C4⋊C4—2362no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—2362no—
501C42—2362no—
502C42—2362no—
503C42D4×D5912yes—
504C42#1174568712yes—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C22×C22⋊D5911yescharacteristic
513Q8—2368no—
514D4—22no—
515D4—22no—
516D4—41no—
517D4—41no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—84no—
543C2×C4—1182no—
544C2×C4—1182no—
545C2×C4—1182no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4#1427369812yes—
556C2×C4#1427369812yes—
557C2×C4#1427375212yes—
558C2×C4C2×D4×D5911yescharacteristic
559C2×C4C2×Q8×D5911yescharacteristic
560C2×C4#41×D5911yescharacteristic
561C2×C4C2×#1174568711yescharacteristic
562C2×C4#1427377411yescharacteristic
563C2×C4#1427377511yescharacteristic
564C8—1182no—
565C8—1182no—
566C8—2361no—
567C8—2361no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22C2×#1427369811yescharacteristic
576C22#1683805111yescharacteristic
577C22Q8×C22⋊D5911yescharacteristic
578C22C2×#1427375211yescharacteristic
579C22#1683820011yescharacteristic
580C22#1683821311yescharacteristic
581C22#1683821311yescharacteristic
582C4—22no—
583C4—22no—
584C4—22no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—4721no—
592C4—2362no—
593C4—2362no—
594C4—2362no—
595C4#1683795411yescharacteristic
596C4#1683801211yescharacteristic
597C4#1683806211yescharacteristic
598C4#1683806511yescharacteristic
599C2—82no—
600C2#2142769111yescharacteristic
601C2#2143355611yescharacteristic
602C2#2143361311yescharacteristic
603C2#2143656011yescharacteristic
604C2#2143663511yescharacteristic
605C2#2143699511yescharacteristic
606C2#2143704111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2566C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2566D59
hypocenterC59#25461
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2566

Complements

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

NH ≅ G/Ntype
1. C2×#16837812C2semidirect
3. #21427154C2semidirect
4. #21425578C2semidirect
5. #21425573C2semidirect
8. #21425054C2semidirect
9. #21424862C2semidirect
10. #21424857C2semidirect
11. #21424820C2semidirect
91. #14273648Q8semidirect
364. C59#25461semidirect

Maximal subgroups

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

subgroupindex
C2×#168378122
C2×#168376922
#214271542
#214255782
#214255732
#214254842
#214254702
#214250542
#214248622
#214248572
#214248202
#214247202
#214247132
#214239212
C59×#25662
#2546159