Finite Group Databasebeta

← #28636988

Subgroups — #28636988

Order 15104.

884 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837737C211yesmaximal, characteristic, complemented
2C2×#16837629C211yesmaximal, characteristic
3#21427099C211yesmaximal, characteristic, complemented
4#21425578C211yesmaximal, characteristic, complemented
5#21425515C211yesmaximal, characteristic, complemented
6#21425484C211yesmaximal, characteristic, complemented
7#21425413C211yesmaximal, characteristic, complemented
8#21424862C211yesmaximal, characteristic, complemented
9#21424713C211yesmaximal, characteristic, complemented
10#21424454C211yesmaximal, characteristic, complemented
11#21424414C211yesmaximal, characteristic
12#21424276C211yesmaximal, characteristic, complemented
13#21424108C211yesmaximal, characteristic, complemented
14C2×#16837164C211yesmaximal, characteristic, complemented
15C59×#2504C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C23×C59⋊C8C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20C22×C4⋊Dic59C2211yescharacteristic
21#16837737—24no—
22#16837629—24no—
23C2×#14273660C2211yescharacteristic
24C2×#14273660C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273650C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×#14273648C2211yescharacteristic
33C2×#14273648C2211yescharacteristic
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×#121×C59C2211yescharacteristic
51C22×C4⋊C4×C59C2211yescharacteristic
52C59×#397—21no—
53C59×#397—21no—
54C59×#390C2211yescharacteristic
55C59×#390C2211yescharacteristic
56C59×#386—21no—
57C59×#386—21no—
58C59×#382C2211yescharacteristic
59C59×#382C2211yescharacteristic
60C4×C22⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C23×Dic59—21no—
63C2×#11745670—22no—
64C2×#11745670—22no—
65C2×#11745670—41no—
66C22×C59⋊C8—22no—
67C22×C59⋊C8—22no—
68C22×C59⋊C8—41no—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73#14273706—22no—
74#14273706D414yes—
75C4×Dic118—22no—
76C4×Dic118—41no—
77C2×C4⋊Dic59—22no—
78C2×C4⋊Dic59—21no—
79C2×C4⋊Dic59—41no—
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C2×Dic59⋊C4—21no—
86C2×C22⋊Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89#14273660—22no—
90#14273660—22no—
91#14273658—22no—
92C4⋊(C59⋊C8)—22no—
93C4⋊(C59⋊C8)—41no—
94#14273650—22no—
95#14273650—22no—
96#14273650—22no—
97#14273648—22no—
98#14273648—22no—
99#14273648—41no—
100C22⋊(C59⋊C8)—22no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C22×D4×C59—21no—
106C22×D4×C59C2311yescharacteristic
107C23×C4×C59—21no—
108C23×C4×C59C2311yescharacteristic
109#121×C59—22no—
110#121×C59—22no—
111C2×C4⋊C4×C59—22no—
112C2×C4⋊C4×C59—22no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59—21no—
121C2×C22⋊C4×C59—21no—
122C2×C22⋊C4×C59D411yescharacteristic
123C2×C22⋊C4×C59D411yescharacteristic
124C2×C42×C59C2311yescharacteristic
125C2×C42×C59C2311yescharacteristic
126#95×C59—21no—
127#95×C59—21no—
128#95×C59—21no—
129#95×C59—21no—
130#95×C59D411yescharacteristic
131#95×C59D411yescharacteristic
132C24×C59—21no—
133C24×C59C2×D411yescharacteristic
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59—21no—
139C2×D4×C59—21no—
140C2×D4×C59—42no—
141C2×D4×C59—42no—
142C2×D4×C59—42no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—42no—
151C22×C4×C59—81no—
152C22×C4×C59—81no—
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D412yes—
159C22×C4×C59#4112yes—
160C22×C4×C59C2411yescharacteristic
161C22×Dic59—21no—
162C22×Dic59—21no—
163C22×Dic59—21no—
164C22×Dic59—21no—
165C22×Dic59—42no—
166C22×Dic59—42no—
167C22×Dic59—81no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118—21no—
171C2×Dic118—21no—
172C2×Dic118—21no—
173C2×Dic118C2×D412yes—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—21no—
179C4⋊C4×C59—21no—
180C4⋊C4×C59—42no—
181C22⋊C4×C59—22no—
182C22⋊C4×C59—22no—
183C22⋊C4×C59—42no—
184C22⋊C4×C59—81no—
185C22⋊C4×C59—81no—
186C22⋊C4×C59—81no—
187C22⋊C4×C59—81no—
188C42×C59—21no—
189C42×C59—21no—
190C42×C59—21no—
191C42×C59—21no—
192C22⋊Dic59—24no—
193#11745670—44no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—42no—
199C4⋊Dic59C2×D412yes—
200C4⋊Dic59#4112yes—
201Dic59⋊C4—24no—
202Dic59⋊C4—42no—
203Dic59⋊C4—81no—
204C4×Dic59—42no—
205C4×Dic59—81no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8—21no—
217C2×C59⋊C8—44no—
218C2×C59⋊C8#4112yes—
219C23×C59—21no—
220C23×C59—21no—
221C23×C59—21no—
222C23×C59—21no—
223C23×C59—41no—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59—41no—
227C23×C59—41no—
228C23×C59—41no—
229C23×C59—41no—
230C23×C59#12112yes—
231C23×C59#13512yes—
232C23×C59#13712yes—
233C23×C59C22×D411yescharacteristic
234D4×C59—41no—
235D4×C59—41no—
236D4×C59—41no—
237D4×C59—41no—
238D4×C59—84no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—22no—
242C2×C4×C59—22no—
243C2×C4×C59—22no—
244C2×C4×C59—22no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—21no—
251C2×C4×C59—21no—
252C2×C4×C59—21no—
253C2×C4×C59—21no—
254C2×C4×C59—21no—
255C2×C4×C59—21no—
256C2×C4×C59—21no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59—41no—
263C2×C4×C59—41no—
264C2×C4×C59—41no—
265C2×C4×C59—41no—
266C2×C4×C59—84no—
267C2×C4×C59#12112yes—
268C2×C4×C59C22×D411yescharacteristic
269C2×C4×C59C2×#4111yescharacteristic
270C2×C4×C59C2×#4111yescharacteristic
271C2×C4×C59C2×#4111yescharacteristic
272C2×C4×C59H(5,𝔽2)11yescharacteristic
273C2×C4×C592−1+411yescharacteristic
274C2×Dic59—22no—
275C2×Dic59—22no—
276C2×Dic59—22no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283C2×Dic59—41no—
284C2×Dic59—41no—
285C2×Dic59—41no—
286C2×Dic59—84no—
287Dic118—22no—
288Dic118—22no—
289Dic118—41no—
290Dic118—41no—
291C59⋊C8—22no—
292C59⋊C8—41no—
293C59⋊C8—41no—
294C59⋊C8—41no—
295#25210—591nomaximal, Sylow
296C22×C59—22no—
297C22×C59—22no—
298C22×C59—22no—
299C22×C59—22no—
300C22×C59—22no—
301C22×C59—22no—
302C22×C59—81no—
303C22×C59—81no—
304C22×C59—81no—
305C22×C59—81no—
306C22×C59—81no—
307C22×C59—81no—
308C22×C59—81no—
309C22×C59#46414yes—
310C22×C59C2×#12111yescharacteristic
311C22×C59#54611yescharacteristic
312C22×C59#54711yescharacteristic
313C22×C59C2×#13511yescharacteristic
314C22×C59C2×#13711yescharacteristic
315C22×C59#57611yescharacteristic
316C22×C59#57711yescharacteristic
317C4×C59—22no—
318C4×C59—41no—
319C4×C59—41no—
320C4×C59—41no—
321C4×C59—41no—
322C4×C59—81no—
323C4×C59—81no—
324C4×C59—81no—
325C4×C59#52911yescharacteristic
326C4×C59#53911yescharacteristic
327C4×C59#54611yescharacteristic
328C4×C59#54711yescharacteristic
329Dic59—42no—
330Dic59—42no—
331Dic59—42no—
332Dic59—81no—
333C2×#484—591no—
334C2×#465—591no—
335#2504D5911yescharacteristic
336#2360—591no—
337#2085—591no—
338#2074—591no—
339#2067—591no—
340#2055—591no—
341#1949—591no—
342#1925—591no—
343#1880—591no—
344#1874—591no—
345#1850—591no—
346#1805—591no—
347C2×#340—591no—
348C2×C59—24no—
349C2×C59—161no—
350C2×C59#242511yescharacteristic
351C2×C59C2×#46411yescharacteristic
352C2×C59#304511yescharacteristic
353C2×C59#329711yescharacteristic
354C2×C59#332011yescharacteristic
355C2×C59#333711yescharacteristic
356C2×C59#334611yescharacteristic
357C22×SD16—591no—
358C23×C8—591no—
359C2×C22⋊Q8—591no—
360C2×#121C2×D5911yescharacteristic
361C2×C4×Q8—591no—
362C22×C4⋊C4—591no—
363C22×C4⋊C4C2×D5911yescharacteristic
364#484—1184no—
365#465—1184no—
366C2×#107—591no—
367C2×#107—591no—
368C2×#106—591no—
369C2×C4⋊C8—591no—
370C2×C4⋊C8—591no—
371C2×#103—591no—
372C2×#103—591no—
373C2×#103—591no—
374C2×#102—591no—
375C2×#102—591no—
376C2×#102—591no—
377C2×C22⋊C8—591no—
378#397—21no—
379#397—21no—
380#391—591no—
381#390C2×D5911yescharacteristic
382#390C2×D5911yescharacteristic
383#389—591no—
384#389—1181no—
385#387—1181no—
386#386—21no—
387#386—21no—
388#385—1181no—
389#382C2×D5911yescharacteristic
390#382C2×D5911yescharacteristic
391C4×C4⋊C4—591no—
392C4×C22⋊C4—1181no—
393C4×C22⋊C4C2×D5911yescharacteristic
394#340—591no—
395#340—591no—
396#340—591no—
397#340—591no—
398#340—591no—
399#340—591no—
400#336—591no—
401#336—591no—
402C59#2521011yescharacteristic, complemented, Sylow
403C22×Q8—591no—
404C22×D4—21no—
405C22×D4C22×D5911yescharacteristic
406C23×C4—21no—
407C23×C4—1181no—
408C23×C4C22×D5911yescharacteristic
409C2×SD16—1182no—
410C2×SD16—1182no—
411C2×SD16—2361no—
412C22×C8—591no—
413C22×C8—591no—
414C22×C8—591no—
415C22×C8—591no—
416C22×C8—1182no—
417C22×C8—1182no—
418C22×C8—2361no—
419C22⋊Q8—594no—
420C22⋊Q8—1182no—
421#121—22no—
422#121—22no—
423C4×Q8—1182no—
424C4×Q8—2361no—
425C2×C4⋊C4—22no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—591no—
429C2×C4⋊C4—591no—
430C2×C4⋊C4—591no—
431C2×C4⋊C4—1182no—
432C2×C4⋊C4—1181no—
433C2×C4⋊C4—1181no—
434C2×C4⋊C4—1181no—
435C2×C4⋊C4—1181no—
436C2×C4⋊C4—2361no—
437C2×C4⋊C4C22×D5911yescharacteristic
438C2×C4⋊C4C22×D5911yescharacteristic
439C2×C4⋊C4C22×D5911yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—1181no—
446C2×C22⋊C4C22⋊D5911yescharacteristic
447C2×C22⋊C4C22⋊D5911yescharacteristic
448C2×C42—1181no—
449C2×C42—1181no—
450C2×C42C22×D5911yescharacteristic
451C2×C42C22×D5911yescharacteristic
452#107—1182no—
453#107—1182no—
454#106—1182no—
455C4⋊C8—1182no—
456C4⋊C8—2361no—
457#103—1182no—
458#103—1182no—
459#103—1182no—
460#102—1182no—
461#102—1182no—
462#102—2361no—
463C22⋊C8—1182no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95—1181no—
469#95—1181no—
470#95—1181no—
471#95—1181no—
472#95C22⋊D5911yescharacteristic
473#95C22⋊D5911yescharacteristic
474C24—21no—
475C24C2×C22⋊D5911yescharacteristic
476C2×Q8—592no—
477C2×Q8—1181no—
478C2×Q8—1181no—
479C2×Q8—1181no—
480C2×Q8—1181no—
481C2×Q8—1181no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—42no—
489C2×D4—42no—
490C2×D4—42no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—42no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—1181no—
502C22×C4—1181no—
503C22×C4—1181no—
504C22×C4—1181no—
505C22×C4—4721no—
506C22×C4—2362no—
507C22×C4—2362no—
508C22×C4D4×D5912yes—
509C22×C4#1174568712yes—
510C22×C4C2×C22⋊D5911yescharacteristic
511C22×C4C2×C22⋊D5911yescharacteristic
512C22×C4C2×C22⋊D5911yescharacteristic
513C22×C4C2×C22⋊D5911yescharacteristic
514C22×C4C2×C22⋊D5911yescharacteristic
515C22×C4C23×D5911yescharacteristic
516SD16—2364no—
517C2×C8—592no—
518C2×C8—1181no—
519C2×C8—1181no—
520C2×C8—1181no—
521C2×C8—1181no—
522C2×C8—1181no—
523C2×C8—1181no—
524C2×C8—1181no—
525C2×C8—1181no—
526C2×C8—1181no—
527C2×C8—1181no—
528C2×C8—1181no—
529C2×C8—2364no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—42no—
537C4⋊C4—592no—
538C4⋊C4—592no—
539C4⋊C4—1184no—
540C4⋊C4—1181no—
541C4⋊C4—1181no—
542C4⋊C4—1181no—
543C4⋊C4—1181no—
544C4⋊C4—4721no—
545C4⋊C4—2362no—
546C4⋊C4—2362no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—42no—
550C22⋊C4—81no—
551C22⋊C4—81no—
552C22⋊C4—81no—
553C22⋊C4—81no—
554C22⋊C4—1184no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—4721no—
560C42—2362no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23#1427369812yes—
573C23#1427374712yes—
574C23#1427375712yes—
575C23C22×C22⋊D5911yescharacteristic
576Q8—1182no—
577Q8—1182no—
578Q8—2361no—
579Q8—2361no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—41no—
584D4—84no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—84no—
613C2×C4—1182no—
614C2×C4—1182no—
615C2×C4—1182no—
616C2×C4—4724no—
617C2×C4—2361no—
618C2×C4—2361no—
619C2×C4—2361no—
620C2×C4—2361no—
621C2×C4—2361no—
622C2×C4—2361no—
623C2×C4—2361no—
624C2×C4—2361no—
625C2×C4—2361no—
626C2×C4#1427369812yes—
627C2×C4C2×D4×D5911yescharacteristic
628C2×C4#41×D5911yescharacteristic
629C2×C4#41×D5911yescharacteristic
630C2×C4C2×#1174568711yescharacteristic
631C2×C4#1427377311yescharacteristic
632C2×C4#1427377411yescharacteristic
633C8—1182no—
634C8—2361no—
635C8—2361no—
636C8—2361no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22#1683762114yes—
651C22C2×#1427369811yescharacteristic
652C22#1683804011yescharacteristic
653C22#1683805111yescharacteristic
654C22C2×#1427374711yescharacteristic
655C22C2×#1427375711yescharacteristic
656C22#1683820511yescharacteristic
657C22#1683821311yescharacteristic
658C4—22no—
659C4—41no—
660C4—41no—
661C4—41no—
662C4—41no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4—4721no—
667C4—2362no—
668C4—2362no—
669C4—2362no—
670C4#1683792911yescharacteristic
671C4#1683799611yescharacteristic
672C4#1683804411yescharacteristic
673C4#1683805011yescharacteristic
674C2—24no—
675C2—161no—
676C2#2142764311yescharacteristic
677C2C2×#1683762111yescharacteristic
678C2#2143363111yescharacteristic
679C2#2143594011yescharacteristic
680C2#2143626111yescharacteristic
681C2#2143646911yescharacteristic
682C2#2143658111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2504C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2504D59
hypocenterC59#25210
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2504

Complements

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

NH ≅ G/Ntype
1. C2×#16837737C2semidirect
3. #21427099C2semidirect
4. #21425578C2semidirect
5. #21425515C2semidirect
6. #21425484C2semidirect
7. #21425413C2semidirect
8. #21424862C2semidirect
9. #21424713C2semidirect
10. #21424454C2semidirect
12. #21424276C2semidirect
13. #21424108C2semidirect
14. C2×#16837164C2semidirect
402. C59#25210semidirect

Maximal subgroups

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

subgroupindex
C2×#168377372
C2×#168376292
#214270992
#214255782
#214255152
#214254842
#214254132
#214248622
#214247132
#214244542
#214244142
#214242762
#214241082
C2×#168371642
C59×#25042
#2521059