Finite Group Databasebeta

← #28638972

Subgroups — #28638972

Order 15104.

860 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837698C211yesmaximal, characteristic, complemented
2#21429088C211yesmaximal, characteristic
3#21429005C211yesmaximal, characteristic
4#21425524C211yesmaximal, characteristic, complemented
5#21425519C211yesmaximal, characteristic, complemented
6#21425300C211yesmaximal, characteristic, complemented
7#21425293C211yesmaximal, characteristic, complemented
8#21425069C211yesmaximal, characteristic, complemented
9#21424866C211yesmaximal, characteristic, complemented
10#21424529C211yesmaximal, characteristic, complemented
11#21424519C211yesmaximal, characteristic, complemented
12#21424129C211yesmaximal, characteristic, complemented
13#21424125C211yesmaximal, characteristic, complemented
14#21423927C211yesmaximal, characteristic, complemented
15C2×C59×#435C211yesmaximal, characteristic
16C2×#14273727C2211yescharacteristic
17C2×#14273714—21no—
18C2×#14273706C2211yescharacteristic
19C2×#14273706C2211yescharacteristic
20C2×C4×Dic118C2211yescharacteristic
21#16837698—24no—
22C2×#14273661C2211yescharacteristic
23C2×#14273661C2211yescharacteristic
24C2×#14273661C2211yescharacteristic
25C2×#14273659C2211yescharacteristic
26C2×#14273652C2211yescharacteristic
27C2×#14273652C2211yescharacteristic
28C2×#14273652C2211yescharacteristic
29C2×#14273652C2211yescharacteristic
30#16837333—21no—
31#16837327—21no—
32#16837327—21no—
33#16837316—21no—
34#16837307—21no—
35#16837290—21no—
36#16837290—21no—
37#16837268C2211yescharacteristic
38#16837262C2211yescharacteristic
39#16837262C2211yescharacteristic
40#16837262C2211yescharacteristic
41#16837262C2211yescharacteristic
42#16837262C2211yescharacteristic
43#16837262C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53C22×M16×C59C2211yescharacteristic
54C23×C8×C59C2211yescharacteristic
55C2×C4×D4×C59C2211yescharacteristic
56C59×#435—24no—
57C2×C4⋊C8×C59C2211yescharacteristic
58C2×C22⋊C8×C59C2211yescharacteristic
59C2×C22⋊C8×C59C2211yescharacteristic
60C2×#97×C59C2211yescharacteristic
61C22×Dic118—21no—
62C22×Dic118C2311yescharacteristic
63#14273727—22no—
64#14273727—22no—
65#14273714—44no—
66#14273706—22no—
67#14273706—22no—
68#14273706—41no—
69#14273706—41no—
70C4×Dic118—22no—
71C4×Dic118—22no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
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×C22⋊Dic59—21no—
85C2×C22⋊Dic59—21no—
86C2×C22⋊Dic59—21no—
87C2×C22⋊Dic59—21no—
88C2×C22⋊Dic59—21no—
89C2×C22⋊Dic59—21no—
90C2×C22⋊Dic59—21no—
91C2×C4×Dic59—21no—
92C2×C4×Dic59—21no—
93#14273661—22no—
94#14273661—22no—
95#14273661—41no—
96#14273659—41no—
97#14273652—22no—
98#14273652—22no—
99#14273652—22no—
100#14273652—22no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C22×D4×C59D411yescharacteristic
111C23×C4×C59C2311yescharacteristic
112C23×C4×C59C2311yescharacteristic
113C2×M16×C59—22no—
114C2×M16×C59—22no—
115C2×M16×C59—41no—
116C22×C8×C59—22no—
117C22×C8×C59—22no—
118C22×C8×C59—41no—
119C22×C8×C59C2311yescharacteristic
120C22×C8×C59C2311yescharacteristic
121C22×C8×C59C2311yescharacteristic
122C22×C8×C59C2311yescharacteristic
123C4×D4×C59—22no—
124C4×D4×C59—22no—
125C2×C4⋊C4×C59D411yescharacteristic
126C2×C22⋊C4×C59D411yescharacteristic
127C2×C22⋊C4×C59D411yescharacteristic
128C2×C42×C59C2311yescharacteristic
129C4⋊C8×C59D414yes—
130C22⋊C8×C59—22no—
131C22⋊C8×C59—22no—
132#97×C59—22no—
133C24×C59C2×D411yescharacteristic
134C24×C59C2×D411yescharacteristic
135C2×D4×C59—22no—
136C2×D4×C59—22no—
137C2×D4×C59—41no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
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×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2411yescharacteristic
155C22×Dic59—21no—
156C22×Dic59—21no—
157C22×Dic59—21no—
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C22×Dic59—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C2×Dic118—42no—
169C2×Dic118—42no—
170C2×Dic118—42no—
171C2×Dic118#4112yes—
172M16×C59—44no—
173C2×C8×C59—21no—
174C2×C8×C59—21no—
175C2×C8×C59—21no—
176C2×C8×C59—21no—
177C2×C8×C59—21no—
178C2×C8×C59—21no—
179C2×C8×C59—21no—
180C2×C8×C59—21no—
181C2×C8×C59—21no—
182C2×C8×C59—44no—
183C2×C8×C59C2×D412yes—
184C2×C8×C59C2×D412yes—
185C2×C8×C59#4112yes—
186C4⋊C4×C59—22no—
187C22⋊C4×C59—22no—
188C22⋊C4×C59—22no—
189C42×C59C2×D412yes—
190C42×C59#4112yes—
191C22⋊Dic59—42no—
192C22⋊Dic59—42no—
193C22⋊Dic59—42no—
194C22⋊Dic59—42no—
195C22⋊Dic59—42no—
196C22⋊Dic59—42no—
197C22⋊Dic59—42no—
198C22⋊Dic59—42no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—21no—
203C4⋊Dic59—21no—
204C4⋊Dic59—21no—
205C4⋊Dic59—21no—
206C4⋊Dic59—21no—
207C4⋊Dic59—21no—
208C4⋊Dic59—21no—
209C4⋊Dic59—21no—
210C4⋊Dic59—21no—
211C4⋊Dic59—21no—
212C4⋊Dic59—21no—
213C4⋊Dic59—42no—
214Dic59⋊C4—42no—
215Dic59⋊C4—42no—
216Dic59⋊C4—42no—
217C4×Dic59—42no—
218C4×Dic59—42no—
219C23×C59—21no—
220C23×C59—21no—
221C23×C59—21no—
222C23×C59—21no—
223C23×C59—21no—
224C23×C59—21no—
225C23×C59—21no—
226C23×C59—21no—
227C23×C59—21no—
228C23×C59—21no—
229C23×C59—21no—
230C23×C59—21no—
231C23×C59—21no—
232C23×C59AGL(1,ℤ/8)12yes—
233C23×C59C22×D411yescharacteristic
234D4×C59—44no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59#12112yes—
254C2×C4×C59#12112yes—
255C2×C4×C59C22×D411yescharacteristic
256C2×C4×C59C2×#4111yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×C4×C59C2×#4111yescharacteristic
259C2×C4×C59H(5,𝔽2)11yescharacteristic
260C2×C4×C592−1+411yescharacteristic
261C2×Dic59—22no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283C2×Dic59—41no—
284C2×Dic59—41no—
285Dic118—22no—
286Dic118—22no—
287Dic118—41no—
288Dic118—41no—
289Dic118—84no—
290C8×C59—22no—
291C8×C59—22no—
292C8×C59—22no—
293C8×C59—41no—
294#25352—591nomaximal, Sylow
295C22×C59—22no—
296C22×C59—22no—
297C22×C59—22no—
298C22×C59—41no—
299C22×C59—41no—
300C22×C59—41no—
301C22×C59—41no—
302C22×C59—41no—
303C22×C59—41no—
304C22×C59—41no—
305C22×C59—41no—
306C22×C59—41no—
307C22×C59—41no—
308C22×C59—41no—
309C22×C59C2×#12111yescharacteristic
310C22×C59#54611yescharacteristic
311C22×C59#54711yescharacteristic
312C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
313C22×C59#57511yescharacteristic
314C22×C59#57811yescharacteristic
315C22×C59#57811yescharacteristic
316C4×C59—22no—
317C4×C59—22no—
318C4×C59—41no—
319C4×C59—41no—
320C4×C59#52911yescharacteristic
321C4×C59#54211yescharacteristic
322C4×C59#54611yescharacteristic
323C4×C59#54811yescharacteristic
324Dic59—42no—
325Dic59—81no—
326Dic59—81no—
327Dic59—81no—
328Dic59—81no—
329Dic59—81no—
330Dic59—81no—
331Dic59—81no—
332C2×#477—591no—
333C2×#435D5911yescharacteristic
334#2547—591no—
335#2536—591no—
336#2076—591no—
337#2075—591no—
338#2036—591no—
339#2035—591no—
340#1993—591no—
341#1949—591no—
342#1894—591no—
343#1892—591no—
344#1809—591no—
345#1808—591no—
346#1763—591no—
347C2×C59—42no—
348C2×C59—81no—
349C2×C59#242711yescharacteristic
350C2×C59#305511yescharacteristic
351C2×C59#305911yescharacteristic
352C2×C59#330811yescharacteristic
353C2×C59#331311yescharacteristic
354C2×C59#334311yescharacteristic
355C2×C59#334911yescharacteristic
356C22×M16C2×D5911yescharacteristic
357C23×C8C2×D5911yescharacteristic
358C2×C4⋊Q8—591no—
359C2×#124—1181no—
360C2×C22⋊Q8—591no—
361C2×C22⋊Q8—591no—
362C2×C4×Q8—591no—
363C2×C4×D4C2×D5911yescharacteristic
364#477—1184no—
365#435—24no—
366C2×#107—591no—
367C2×#107—591no—
368C2×#107—591no—
369C2×#106—591no—
370C2×C4⋊C8C2×D5911yescharacteristic
371C2×#103—591no—
372C2×#103—591no—
373C2×#103—591no—
374C2×#103—591no—
375C2×C22⋊C8C2×D5911yescharacteristic
376C2×C22⋊C8C2×D5911yescharacteristic
377C2×#97C2×D5911yescharacteristic
378#398—1181no—
379#397—1181no—
380#397—1181no—
381#394—1181no—
382#391—1181no—
383#388—1181no—
384#388—1181no—
385#382—591no—
386#380—591no—
387#380—591no—
388#380—591no—
389#380—591no—
390#380—591no—
391#380—591no—
392C4×C4⋊C4—591no—
393#340—591no—
394#340—591no—
395#340—591no—
396#340—591no—
397#340—591no—
398#340—591no—
399#340—591no—
400#340—591no—
401C59#2535211yescharacteristic, complemented, Sylow
402C22×Q8—591no—
403C22×Q8—1181no—
404C22×D4D23611yescharacteristic
405C23×C4C22×D5911yescharacteristic
406C23×C4C22×D5911yescharacteristic
407C2×M16—22no—
408C2×M16—22no—
409C2×M16—41no—
410C22×C8—22no—
411C22×C8—22no—
412C22×C8—41no—
413C22×C8C22×D5911yescharacteristic
414C22×C8C22×D5911yescharacteristic
415C22×C8C22×D5911yescharacteristic
416C22×C8C22×D5911yescharacteristic
417C4⋊Q8—1182no—
418C4⋊Q8—1182no—
419#124—2364no—
420C22⋊Q8—1182no—
421C22⋊Q8—1182no—
422C22⋊Q8—2361no—
423C22⋊Q8—2361no—
424C4×Q8—1182no—
425C4×Q8—1182no—
426C4×D4—22no—
427C4×D4—22no—
428C2×C4⋊C4—591no—
429C2×C4⋊C4—591no—
430C2×C4⋊C4—591no—
431C2×C4⋊C4—591no—
432C2×C4⋊C4—591no—
433C2×C4⋊C4—591no—
434C2×C4⋊C4—591no—
435C2×C4⋊C4—1181no—
436C2×C4⋊C4—1181no—
437C2×C4⋊C4—1181no—
438C2×C4⋊C4—1181no—
439C2×C4⋊C4D23611yescharacteristic
440C2×C22⋊C4—1181no—
441C2×C22⋊C4—1181no—
442C2×C22⋊C4—1181no—
443C2×C22⋊C4—1181no—
444C2×C22⋊C4—1181no—
445C2×C22⋊C4—1181no—
446C2×C22⋊C4—1181no—
447C2×C22⋊C4—1181no—
448C2×C22⋊C4D23611yescharacteristic
449C2×C22⋊C4D23611yescharacteristic
450C2×C42—1181no—
451C2×C42—1181no—
452C2×C42C22×D5911yescharacteristic
453#107—1182no—
454#107—1182no—
455#107—2361no—
456#106—2361no—
457C4⋊C8C22⋊D5914yes—
458#103—1182no—
459#103—1182no—
460#103—1182no—
461#103—1182no—
462C22⋊C8—22no—
463C22⋊C8—22no—
464#97—22no—
465#95—1181no—
466#95—1181no—
467#95—1181no—
468#95—1181no—
469#95—1181no—
470#95—1181no—
471#95—1181no—
472#95—1181no—
473#95—1181no—
474C24C2×D23611yescharacteristic
475C24C2×D23611yescharacteristic
476C2×Q8—592no—
477C2×Q8—1181no—
478C2×Q8—1181no—
479C2×Q8—1181no—
480C2×Q8—1181no—
481C2×Q8—1181no—
482C2×Q8—2362no—
483C2×Q8—2362no—
484C2×Q8—2362no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—41no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—1181no—
501C22×C4—1181no—
502C22×C4—1181no—
503C22×C4—1181no—
504C22×C4—1181no—
505C22×C4—1181no—
506C22×C4—1181no—
507C22×C4—1181no—
508C22×C4C2×D23611yescharacteristic
509C22×C4C2×D23611yescharacteristic
510C22×C4C2×D23611yescharacteristic
511C22×C4C2×D23611yescharacteristic
512C22×C4C23×D5911yescharacteristic
513M16—44no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—44no—
524C2×C8#1174568712yes—
525C2×C8C2×C22⋊D5912yes—
526C2×C8C2×C22⋊D5912yes—
527C4⋊C4—22no—
528C4⋊C4—1181no—
529C4⋊C4—1181no—
530C4⋊C4—1181no—
531C4⋊C4—1181no—
532C4⋊C4—1181no—
533C4⋊C4—1181no—
534C4⋊C4—1181no—
535C4⋊C4—1181no—
536C4⋊C4—1181no—
537C4⋊C4—1181no—
538C4⋊C4—1181no—
539C4⋊C4—1181no—
540C4⋊C4—1181no—
541C4⋊C4—1181no—
542C4⋊C4—2362no—
543C4⋊C4—2362no—
544C4⋊C4—2362no—
545C4⋊C4—2362no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—2362no—
549C22⋊C4—2362no—
550C22⋊C4—2362no—
551C22⋊C4—2362no—
552C22⋊C4—2362no—
553C22⋊C4—2362no—
554C22⋊C4—2362no—
555C22⋊C4—2362no—
556C42—2362no—
557C42—2362no—
558C42#1174568512yes—
559C42C2×C22⋊D5912yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23#1427375512yes—
574C23C22×D23611yescharacteristic
575Q8—1182no—
576Q8—1182no—
577Q8—4724no—
578Q8—2361no—
579Q8—2361no—
580D4—44no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—1182no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4—2361no—
605C2×C4—2361no—
606C2×C4—2361no—
607C2×C4—2361no—
608C2×C4—2361no—
609C2×C4—2361no—
610C2×C4—2361no—
611C2×C4—2361no—
612C2×C4—2361no—
613C2×C4—2361no—
614C2×C4—2361no—
615C2×C4—2361no—
616C2×C4—2361no—
617C2×C4—2361no—
618C2×C4—2361no—
619C2×C4—2361no—
620C2×C4—2361no—
621C2×C4—2361no—
622C2×C4—2361no—
623C2×C4#1427370112yes—
624C2×C4#1427370112yes—
625C2×C4C22×C22⋊D5911yescharacteristic
626C2×C4#41×D5911yescharacteristic
627C2×C4C2×#1174568711yescharacteristic
628C2×C4C2×#1174568511yescharacteristic
629C2×C4#1427377211yescharacteristic
630C2×C4#1427377411yescharacteristic
631C8—22no—
632C8—22no—
633C8—22no—
634C8—41no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22C2×#1427370111yescharacteristic
650C22#1683804711yescharacteristic
651C22#1683805611yescharacteristic
652C22C2×#1427375511yescharacteristic
653C22#1683820211yescharacteristic
654C22#1683821911yescharacteristic
655C22#1683821911yescharacteristic
656C4—22no—
657C4—22no—
658C4—41no—
659C4—41no—
660C4—4721no—
661C4—4721no—
662C4—4721no—
663C4—4721no—
664C4—4721no—
665C4—4721no—
666C4—4721no—
667C4—2362no—
668C4#1683793411yescharacteristic
669C4#1683801711yescharacteristic
670C4#1683804611yescharacteristic
671C4#1683805911yescharacteristic
672C2—42no—
673C2—81no—
674C2#2142766711yescharacteristic
675C2#2143371511yescharacteristic
676C2#2143375111yescharacteristic
677C2#2143608911yescharacteristic
678C2#2143616411yescharacteristic
679C2#2143653811yescharacteristic
680C2#2143662811yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#435C2
socleC23×C59C22×D4
prime radicalC22×D4×C59D4
hypercenterC2×#435D59
hypocenterC59#25352
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#435

Complements

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

NH ≅ G/Ntype
1. C2×#16837698C2semidirect
4. #21425524C2semidirect
5. #21425519C2semidirect
6. #21425300C2semidirect
7. #21425293C2semidirect
8. #21425069C2semidirect
9. #21424866C2semidirect
10. #21424529C2semidirect
11. #21424519C2semidirect
12. #21424129C2semidirect
13. #21424125C2semidirect
14. #21423927C2semidirect
401. C59#25352semidirect

Maximal subgroups

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

subgroupindex
C2×#168376982
#214290882
#214290052
#214255242
#214255192
#214253002
#214252932
#214250692
#214248662
#214245292
#214245192
#214241292
#214241252
#214239272
C2×C59×#4352
#2535259