Finite Group Databasebeta

← #28646205

Subgroups — #28646205

Order 15104.

920 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837711C211yesmaximal, characteristic, complemented
2#21427803C211yesmaximal, characteristic
3Q8×C22⋊Dic59C211yesmaximal, characteristic
4#21425632C211yesmaximal, characteristic, complemented
5#21425475C211yesmaximal, characteristic, complemented
6#21425473C211yesmaximal, characteristic, complemented
7#21425245C211yesmaximal, characteristic, complemented
8#21424366C211yesmaximal, characteristic, complemented
9#21424348C211yesmaximal, characteristic, complemented
10#21424295C211yesmaximal, characteristic
11#21424279C211yesmaximal, characteristic
12#21424129C211yesmaximal, characteristic
13#21424099C211yesmaximal, characteristic, complemented
14#21424031C211yesmaximal, characteristic
15C2×C59×#419C211yesmaximal, characteristic
16C2×#14273718C2211yescharacteristic
17C2×Q8×Dic59C2211yescharacteristic
18C2×Q8×Dic59C2211yescharacteristic
19#16837711—24no—
20C2×#14273661C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273659C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×Dic59⋊C8C2211yescharacteristic
25C2×Dic59⋊C8C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×#14273651C2211yescharacteristic
30C2×C22⋊(C59⋊C8)C2211yescharacteristic
31#16837338—21no—
32#16837325—21no—
33#16837309—21no—
34#16837309—21no—
35#16837309C2211yescharacteristic
36#16837309C2211yescharacteristic
37#16837301C2211yescharacteristic
38#16837300—21no—
39#16837298C2211yescharacteristic
40#16837286—21no—
41#16837271C2211yescharacteristic
42#16837262—21no—
43#16837262C2211yescharacteristic
44#16837262C2211yescharacteristic
45C4×C22⋊Dic59—21no—
46C4×C22⋊Dic59C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C23×Q8×C59C2211yescharacteristic
54C22×M16×C59C2211yescharacteristic
55C2×#117×C59C2211yescharacteristic
56C59×#419—24no—
57C2×#103×C59C2211yescharacteristic
58C2×#103×C59C2211yescharacteristic
59C2×#103×C59C2211yescharacteristic
60C2×#103×C59C2211yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63#14273718—22no—
64#14273718—41no—
65Q8×Dic59—22no—
66Q8×Dic59—22no—
67Q8×Dic59—41no—
68Q8×Dic59—41no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—21no—
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⋊Dic59D411yescharacteristic
84C2×C22⋊Dic59D411yescharacteristic
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59C2311yescharacteristic
88C2×C4×Dic59C2311yescharacteristic
89#14273661—22no—
90#14273660—41no—
91#14273659—41no—
92#14273658—22no—
93Dic59⋊C8—22no—
94Dic59⋊C8—41no—
95#14273651—22no—
96#14273651—22no—
97#14273651—41no—
98#14273651—41no—
99C22⋊(C59⋊C8)—22no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95D411yescharacteristic
109C59⋊#95D411yescharacteristic
110C22×Q8×C59—21no—
111C22×Q8×C59—22no—
112C22×Q8×C59—22no—
113C22×Q8×C59—22no—
114C22×Q8×C59—41no—
115C22×Q8×C59—41no—
116C22×Q8×C59Q814yes—
117C22×Q8×C59C2311yescharacteristic
118C22×Q8×C59C2311yescharacteristic
119C23×C4×C59—21no—
120C23×C4×C59C2311yescharacteristic
121C2×M16×C59—22no—
122C2×M16×C59—22no—
123C2×M16×C59—41no—
124C22×C8×C59C2311yescharacteristic
125C22×C8×C59C2311yescharacteristic
126#117×C59—22no—
127#117×C59—41no—
128C2×C4⋊C4×C59C2311yescharacteristic
129C2×C4⋊C4×C59C2311yescharacteristic
130C2×C22⋊C4×C59—21no—
131C2×C42×C59—21no—
132#103×C59—22no—
133#103×C59—22no—
134#103×C59—22no—
135#103×C59D414yes—
136C24×C59C2×D411yescharacteristic
137C2×Q8×C59—24no—
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59—21no—
141C2×Q8×C59—21no—
142C2×Q8×C59—28no—
143C2×Q8×C59—21no—
144C2×Q8×C59—21no—
145C2×Q8×C59—21no—
146C2×Q8×C59—21no—
147C2×Q8×C59—42no—
148C2×Q8×C59—42no—
149C2×Q8×C59—44no—
150C2×Q8×C59—44no—
151C2×Q8×C59—44no—
152C2×Q8×C59C2×D412yes—
153C2×Q8×C59C2×Q812yes—
154C2×Q8×C59C2×Q812yes—
155C2×Q8×C59#4112yes—
156C22×C4×C59—21no—
157C22×C4×C59—21no—
158C22×C4×C59—21no—
159C22×C4×C59—21no—
160C22×C4×C59—21no—
161C22×C4×C59—24no—
162C22×C4×C59—21no—
163C22×C4×C59—21no—
164C22×C4×C59—21no—
165C22×C4×C59—42no—
166C22×C4×C59—42no—
167C22×C4×C59C2×D411yescharacteristic
168C22×C4×C59C2×Q812yes—
169C22×C4×C59#4112yes—
170C22×C4×C59C2411yescharacteristic
171C22×Dic59—21no—
172C22×Dic59—21no—
173C22×Dic59—21no—
174C22×Dic59—21no—
175C22×Dic59C2×D411yescharacteristic
176C22×Dic59C2×D411yescharacteristic
177C22×Dic59C2×D411yescharacteristic
178C22×Dic59C2×D411yescharacteristic
179M16×C59—44no—
180C2×C8×C59—21no—
181C2×C8×C59—21no—
182C2×C8×C59—21no—
183C2×C8×C59—21no—
184C2×C8×C59—21no—
185C2×C8×C59C2×D412yes—
186C4⋊C4×C59—21no—
187C4⋊C4×C59—21no—
188C4⋊C4×C59—21no—
189C4⋊C4×C59C2×D412yes—
190C22⋊C4×C59—42no—
191C42×C59—42no—
192C22⋊Dic59—22no—
193C22⋊Dic59—41no—
194C22⋊Dic59—42no—
195C22⋊Dic59—81no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—21no—
203C4⋊Dic59—21no—
204C4⋊Dic59—42no—
205C4⋊Dic59—42no—
206C4⋊Dic59—42no—
207C4⋊Dic59—42no—
208Dic59⋊C4—42no—
209Dic59⋊C4—42no—
210Dic59⋊C4—81no—
211Dic59⋊C4—81no—
212C4×Dic59—21no—
213C4×Dic59—21no—
214C4×Dic59—21no—
215C4×Dic59—21no—
216C4×Dic59—42no—
217C4×Dic59—42no—
218C2×C59⋊C8—21no—
219C2×C59⋊C8—21no—
220C2×C59⋊C8—21no—
221C2×C59⋊C8—21no—
222C2×C59⋊C8—21no—
223C2×C59⋊C8—21no—
224C23×C59—21no—
225C23×C59—21no—
226C23×C59—21no—
227C23×C59—21no—
228C23×C59—21no—
229C23×C59—21no—
230C23×C59C22⋊Q812yes—
231C23×C59C22×D411yescharacteristic
232Q8×C59—22no—
233Q8×C59—22no—
234Q8×C59—22no—
235Q8×C59—22no—
236Q8×C59—22no—
237Q8×C59—48no—
238Q8×C59—41no—
239Q8×C59—41no—
240Q8×C59C22⋊Q814yes—
241C2×C4×C59—22no—
242C2×C4×C59—22no—
243C2×C4×C59—21no—
244C2×C4×C59—22no—
245C2×C4×C59—21no—
246C2×C4×C59—22no—
247C2×C4×C59—22no—
248C2×C4×C59—22no—
249C2×C4×C59—22no—
250C2×C4×C59—22no—
251C2×C4×C59—22no—
252C2×C4×C59—48no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59C22⋊Q812yes—
261C2×C4×C59C22×D411yescharacteristic
262C2×C4×C59C22×Q811yescharacteristic
263C2×C4×C59C2×#4111yescharacteristic
264C2×C4×C59C2×#4111yescharacteristic
265C2×C4×C59H(5,𝔽2)11yescharacteristic
266C2×C4×C59H(5,𝔽2)11yescharacteristic
267C2×Dic59—21no—
268C2×Dic59—21no—
269C2×Dic59—21no—
270C2×Dic59—21no—
271C2×Dic59—21no—
272C2×Dic59—21no—
273C2×Dic59—21no—
274C2×Dic59—21no—
275C2×Dic59—21no—
276C2×Dic59—21no—
277C2×Dic59—21no—
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—41no—
287C2×Dic59—41no—
288C2×Dic59—41no—
289C2×Dic59—41no—
290C2×Dic59#13712yes—
291C8×C59—22no—
292C8×C59—41no—
293C59⋊C8—41no—
294C59⋊C8—41no—
295#25981—591nomaximal, Sylow
296C22×C59—22no—
297C22×C59—22no—
298C22×C59—22no—
299C22×C59—41no—
300C22×C59—41no—
301C22×C59—41no—
302C22×C59—41no—
303C22×C59C2×C22⋊Q811yescharacteristic
304C22×C59D4211yescharacteristic
305C22×C59#54611yescharacteristic
306C22×C59C2×#13711yescharacteristic
307C22×C59#57511yescharacteristic
308C22×C59#57711yescharacteristic
309C22×C59#57811yescharacteristic
310C4×C59—22no—
311C4×C59—24no—
312C4×C59—42no—
313C4×C59—42no—
314C4×C59—81no—
315C4×C59C2×C22⋊Q811yescharacteristic
316C4×C59#54311yescharacteristic
317C4×C59#54611yescharacteristic
318C4×C59#55111yescharacteristic
319Dic59—22no—
320Dic59—41no—
321Dic59—41no—
322Dic59—41no—
323Dic59—81no—
324Dic59—81no—
325Dic59—81no—
326Dic59—81no—
327C2×#479—591no—
328C2×#419D5911yescharacteristic
329#2441—591no—
330Q8×C22⋊C4—591no—
331#2095—591no—
332#2066—591no—
333#2066—591no—
334#2026—591no—
335#1865—591no—
336#1863—591no—
337#1854—591no—
338#1851—591no—
339#1809—591no—
340#1803—591no—
341#1787—591no—
342C2×C59—42no—
343C2×C59#243711yescharacteristic
344C2×C59#307811yescharacteristic
345C2×C59#308211yescharacteristic
346C2×C59#327111yescharacteristic
347C2×C59#327611yescharacteristic
348C2×C59#332011yescharacteristic
349C2×C59#332711yescharacteristic
350C23×Q8C2×D5911yescharacteristic
351C22×M16C2×D5911yescharacteristic
352C2×#125—591no—
353C2×C4×Q8—591no—
354C2×C4×Q8—591no—
355C2×#117C2×D5911yescharacteristic
356#479—1184no—
357C2×#107—591no—
358C2×#107—591no—
359C2×#106—591no—
360C2×#106—591no—
361C2×C4⋊C8—591no—
362C2×C4⋊C8—591no—
363#419—24no—
364C2×#103—591no—
365C2×#103—591no—
366C2×#103—591no—
367C2×#103—591no—
368C2×#103C2×D5911yescharacteristic
369C2×#103C2×D5911yescharacteristic
370C2×#103C2×D5911yescharacteristic
371C2×#103C2×D5911yescharacteristic
372C2×C22⋊C8—591no—
373#399—1181no—
374#396—1181no—
375#391—591no—
376#391—591no—
377#391—1181no—
378#391—1181no—
379#389—591no—
380#389—591no—
381#389—1181no—
382#387—1181no—
383#384—591no—
384#380—591no—
385#380—591no—
386#380—1181no—
387C4×C22⋊C4—591no—
388C4×C22⋊C4—1181no—
389#340—591no—
390#340—591no—
391#340—591no—
392#340—591no—
393#336—591no—
394#336—591no—
395C59#2598111yescharacteristic, complemented, Sylow
396C22×Q8—21no—
397C22×Q8—22no—
398C22×Q8—22no—
399C22×Q8—22no—
400C22×Q8—41no—
401C22×Q8—41no—
402C22×Q8Dic11814yes—
403C22×Q8C22×D5911yescharacteristic
404C22×Q8C22×D5911yescharacteristic
405C23×C4—21no—
406C23×C4C22×D5911yescharacteristic
407C2×M16—22no—
408C2×M16—22no—
409C2×M16—41no—
410C22×C8—591no—
411C22×C8—591no—
412C22×C8C22×D5911yescharacteristic
413C22×C8C22×D5911yescharacteristic
414#125—1182no—
415#125—2361no—
416C4×Q8—1182no—
417C4×Q8—1182no—
418C4×Q8—2361no—
419C4×Q8—2361no—
420#117—22no—
421#117—41no—
422C2×C4⋊C4—591no—
423C2×C4⋊C4—591no—
424C2×C4⋊C4—591no—
425C2×C4⋊C4—591no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4—1181no—
428C2×C4⋊C4—1181no—
429C2×C4⋊C4—1181no—
430C2×C4⋊C4—1181no—
431C2×C4⋊C4—1181no—
432C2×C4⋊C4—1181no—
433C2×C4⋊C4—1181no—
434C2×C4⋊C4C22×D5911yescharacteristic
435C2×C4⋊C4C22×D5911yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—591no—
438C2×C22⋊C4—591no—
439C2×C22⋊C4—1181no—
440C2×C22⋊C4—1181no—
441C2×C42—21no—
442C2×C42—591no—
443C2×C42—591no—
444C2×C42—1181no—
445C2×C42—1181no—
446#107—1182no—
447#107—2361no—
448#106—1182no—
449#106—2361no—
450C4⋊C8—1182no—
451C4⋊C8—2361no—
452#103—22no—
453#103—22no—
454#103—22no—
455#103—1182no—
456#103—1182no—
457#103—2361no—
458#103—2361no—
459#103C22⋊D5914yes—
460C22⋊C8—1182no—
461#95—591no—
462#95—591no—
463#95—1181no—
464#95—1181no—
465#95—1181no—
466#95—1181no—
467#95—1181no—
468#95—1181no—
469#95—1181no—
470#95—1181no—
471C24D4×D5911yescharacteristic
472C2×Q8—24no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—28no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—42no—
483C2×Q8—42no—
484C2×Q8—44no—
485C2×Q8—44no—
486C2×Q8—44no—
487C2×Q8C2×Dic11812yes—
488C2×Q8C2×Dic11812yes—
489C2×Q8#1174568512yes—
490C2×Q8C2×C22⋊D5912yes—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—24no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—591no—
503C22×C4—591no—
504C22×C4—591no—
505C22×C4—591no—
506C22×C4—1181no—
507C22×C4—1181no—
508C22×C4—1181no—
509C22×C4—1181no—
510C22×C4C2×Dic11812yes—
511C22×C4D4×D5911yescharacteristic
512C22×C4#1174568712yes—
513C22×C4C23×D5911yescharacteristic
514M16—44no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—1181no—
521C2×C8—1181no—
522C2×C8—1181no—
523C2×C8—1181no—
524C2×C8—1181no—
525C2×C8—1181no—
526C2×C8C2×C22⋊D5912yes—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
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—4721no—
539C4⋊C4—4721no—
540C4⋊C4—2362no—
541C4⋊C4—2362no—
542C4⋊C4—2362no—
543C4⋊C4—2362no—
544C4⋊C4—2362no—
545C4⋊C4—2362no—
546C4⋊C4C2×C22⋊D5912yes—
547C22⋊C4—42no—
548C22⋊C4—1182no—
549C22⋊C4—4721no—
550C22⋊C4—2361no—
551C22⋊C4—2362no—
552C42—42no—
553C42—1181no—
554C42—1181no—
555C42—1181no—
556C42—1181no—
557C42—2362no—
558C42—2362no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23#1427370412yes—
566C23C2×D4×D5911yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—22no—
572Q8—48no—
573Q8—41no—
574Q8—41no—
575Q8#1427370614yes—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—48no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—592no—
596C2×C4—1181no—
597C2×C4—1181no—
598C2×C4—1181no—
599C2×C4—1181no—
600C2×C4—1181no—
601C2×C4—1181no—
602C2×C4—1181no—
603C2×C4—1181no—
604C2×C4—1181no—
605C2×C4—1181no—
606C2×C4—1181no—
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#1427370412yes—
620C2×C4C22×C22⋊D5911yescharacteristic
621C2×C4C22×Dic11811yescharacteristic
622C2×C4C2×#1174568711yescharacteristic
623C2×C4C2×#1174568511yescharacteristic
624C2×C4#1427377211yescharacteristic
625C2×C4#1427377211yescharacteristic
626C8—22no—
627C8—41no—
628C8—2361no—
629C8—2361no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22C2×#1427370411yescharacteristic
638C22D4×C22⋊D5911yescharacteristic
639C22#1683804211yescharacteristic
640C22#137×D5911yescharacteristic
641C22#1683820111yescharacteristic
642C22#1683821111yescharacteristic
643C22#1683821811yescharacteristic
644C4—22no—
645C4—24no—
646C4—42no—
647C4—42no—
648C4—81no—
649C4—1182no—
650C4—4721no—
651C4—4721no—
652C4—4721no—
653C4—4721no—
654C4—2361no—
655C4—2361no—
656C4—2361no—
657C4C2×#1427370611yescharacteristic
658C4#1683802811yescharacteristic
659C4#1683804611yescharacteristic
660C4#1683807811yescharacteristic
661C2—42no—
662C2#2142775711yescharacteristic
663C2#2143389811yescharacteristic
664C2#2143394011yescharacteristic
665C2#2143567511yescharacteristic
666C2#2143573611yescharacteristic
667C2#2143626811yescharacteristic
668C2#2143636011yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#419C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenterC2×#419D59
hypocenterC59#25981
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
1. C2×#16837711C2semidirect
4. #21425632C2semidirect
5. #21425475C2semidirect
6. #21425473C2semidirect
7. #21425245C2semidirect
8. #21424366C2semidirect
9. #21424348C2semidirect
13. #21424099C2semidirect
395. C59#25981semidirect

Maximal subgroups

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

subgroupindex
C2×#168377112
#214278032
Q8×C22⋊Dic592
#214256322
#214254752
#214254732
#214252452
#214243662
#214243482
#214242952
#214242792
#214241292
#214240992
#214240312
C2×C59×#4192
#2598159