Finite Group Databasebeta

← #28637463

Subgroups — #28637463

Order 15104.

850 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837740C211yesmaximal, characteristic, complemented
2C2×#16837474C211yesmaximal, characteristic
3#21428393C211yesmaximal, characteristic
4#21425587C211yesmaximal, characteristic, complemented
5#21425533C211yesmaximal, characteristic, complemented
6#21425491C211yesmaximal, characteristic, complemented
7#21425296C211yesmaximal, characteristic, complemented
8#21424831C211yesmaximal, characteristic, complemented
9#21424789C211yesmaximal, characteristic, complemented
10#21424530C211yesmaximal, characteristic, complemented
11#21424520C211yesmaximal, characteristic, complemented
12#21424284C211yesmaximal, characteristic, complemented
13#21424123C211yesmaximal, characteristic, complemented
14#21423815C211yesmaximal, characteristic, complemented
15C59×#2417C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C23×C59⋊C8C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19#16837740—24no—
20#16837474—24no—
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×#14273651C2211yescharacteristic
30C2×C22⋊(C59⋊C8)C2211yescharacteristic
31C2×C22⋊(C59⋊C8)C2211yescharacteristic
32C2×#14273638C2211yescharacteristic
33#16837330—21no—
34#16837316—21no—
35#16837300C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837290—21no—
38#16837283—21no—
39#16837262C2211yescharacteristic
40#16837262C2211yescharacteristic
41#16837262C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50C2×C22⋊Q8×C59C2211yescharacteristic
51C2×C22⋊Q8×C59C2211yescharacteristic
52C2×C4×D4×C59C2211yescharacteristic
53C22×C4⋊C4×C59C2211yescharacteristic
54C59×#398—21no—
55C59×#396—21no—
56C59×#391C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#386—21no—
59C59×#380—21no—
60C59×#380C2211yescharacteristic
61C2×#11745664—22no—
62C2×#11745664—22no—
63C2×#11745664—41no—
64C22×C59⋊C8—22no—
65C22×C59⋊C8—22no—
66C22×C59⋊C8—41no—
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71#14273686—22no—
72#14273686—22no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C22⋊Dic59—21no—
82C2×C22⋊Dic59—21no—
83C2×C4×Dic59—21no—
84#14273660—22no—
85#14273660—22no—
86#14273660—41no—
87#14273658—22no—
88C4⋊(C59⋊C8)—22no—
89#14273651—22no—
90#14273651—22no—
91#14273651—22no—
92#14273651—41no—
93C22⋊(C59⋊C8)—22no—
94C22⋊(C59⋊C8)D414yes—
95#14273638—22no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C22×Q8×C59C2311yescharacteristic
102C22×D4×C59D411yescharacteristic
103C23×C4×C59—21no—
104C23×C4×C59C2311yescharacteristic
105C23×C4×C59C2311yescharacteristic
106C22⋊Q8×C59—22no—
107C22⋊Q8×C59—22no—
108C22⋊Q8×C59—22no—
109C22⋊Q8×C59—41no—
110C4×D4×C59—22no—
111C4×D4×C59—22no—
112C2×C4⋊C4×C59—22no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59—21no—
116C2×C4⋊C4×C59—21no—
117C2×C4⋊C4×C59—41no—
118C2×C4⋊C4×C59D411yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C4⋊C4×C59C2311yescharacteristic
121C2×C4⋊C4×C59C2311yescharacteristic
122C2×C22⋊C4×C59—21no—
123C2×C22⋊C4×C59—21no—
124C2×C22⋊C4×C59—21no—
125C2×C22⋊C4×C59D411yescharacteristic
126C2×C22⋊C4×C59D411yescharacteristic
127C2×C42×C59C2311yescharacteristic
128#95×C59—21no—
129#95×C59—21no—
130#95×C59—21no—
131#95×C59—21no—
132C24×C59C2×D411yescharacteristic
133C24×C59C2×D411yescharacteristic
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×D4×C59—22no—
141C2×D4×C59—22no—
142C2×D4×C59—41no—
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—21no—
151C22×C4×C59—21no—
152C22×C4×C59—21no—
153C22×C4×C59—21no—
154C22×C4×C59—21no—
155C22×C4×C59—21no—
156C22×C4×C59—21no—
157C22×C4×C59—42no—
158C22×C4×C59—42no—
159C22×C4×C59—81no—
160C22×C4×C59C2×D411yescharacteristic
161C22×C4×C59C2×D411yescharacteristic
162C22×C4×C59C2×D411yescharacteristic
163C22×C4×C59C2×D411yescharacteristic
164C22×C4×C59C2×D412yes—
165C22×C4×C59#4112yes—
166C22×C4×C59C2411yescharacteristic
167C22×Dic59—21no—
168C22×Dic59—21no—
169C22×Dic59—21no—
170C22×Dic59—21no—
171C4⋊C4×C59—22no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—42no—
178C4⋊C4×C59—42no—
179C4⋊C4×C59—42no—
180C4⋊C4×C59—81no—
181C4⋊C4×C59#4112yes—
182C22⋊C4×C59—22no—
183C22⋊C4×C59—22no—
184C22⋊C4×C59—42no—
185C22⋊C4×C59—42no—
186C22⋊C4×C59—81no—
187C42×C59—21no—
188C42×C59—21no—
189C22⋊Dic59—42no—
190C22⋊Dic59—42no—
191C22⋊Dic59—81no—
192C22⋊Dic59—81no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201Dic59⋊C4—42no—
202Dic59⋊C4—81no—
203C4×Dic59—42no—
204#11745664—44no—
205C2×C59⋊C8—21no—
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—44no—
215C2×C59⋊C8C2×D412yes—
216C2×C59⋊C8C2×D412yes—
217C2×C59⋊C8#4112yes—
218C23×C59—21no—
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#12112yes—
231C23×C59#13712yes—
232C23×C59C22×D411yescharacteristic
233Q8×C59—41no—
234Q8×C59—41no—
235Q8×C59—41no—
236Q8×C59—41no—
237D4×C59—44no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—22no—
241C2×C4×C59—22no—
242C2×C4×C59—22no—
243C2×C4×C59—22no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—22no—
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—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
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—41no—
267C2×C4×C59—41no—
268C2×C4×C59—41no—
269C2×C4×C59—84no—
270C2×C4×C59#12112yes—
271C2×C4×C59C22×D411yescharacteristic
272C2×C4×C59C2×#4111yescharacteristic
273C2×C4×C59C2×#4111yescharacteristic
274C2×C4×C59C2×#4111yescharacteristic
275C2×C4×C59H(5,𝔽2)11yescharacteristic
276C2×C4×C592−1+411yescharacteristic
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—41no—
287C2×Dic59—41no—
288C2×Dic59—41no—
289C59⋊C8—22no—
290C59⋊C8—22no—
291C59⋊C8—22no—
292C59⋊C8—41no—
293#25241—591nomaximal, Sylow
294C22×C59—22no—
295C22×C59—22no—
296C22×C59—22no—
297C22×C59—22no—
298C22×C59—22no—
299C22×C59—22no—
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×C59C2×#12111yescharacteristic
309C22×C59#54611yescharacteristic
310C22×C59#54711yescharacteristic
311C22×C59C2×#13711yescharacteristic
312C22×C59#57511yescharacteristic
313C22×C59#57611yescharacteristic
314C22×C59#57811yescharacteristic
315C4×C59—22no—
316C4×C59—41no—
317C4×C59—41no—
318C4×C59—41no—
319C4×C59—42no—
320C4×C59—81no—
321C4×C59—81no—
322C4×C59—81no—
323C4×C59#52911yescharacteristic
324C4×C59#53911yescharacteristic
325C4×C59#54611yescharacteristic
326C4×C59#54711yescharacteristic
327Dic59—81no—
328Dic59—81no—
329Dic59—81no—
330Dic59—81no—
331C2×#485—591no—
332C2×#435—591no—
333#2485—591no—
334#2417D5911yescharacteristic
335#2086—591no—
336#2077—591no—
337#2068—591no—
338#2036—591no—
339#1943—591no—
340#1935—591no—
341#1894—591no—
342#1892—591no—
343#1852—591no—
344#1808—591no—
345#1736—591no—
346C2×C59—42no—
347C2×C59—42no—
348C2×C59#242511yescharacteristic
349C2×C59#305611yescharacteristic
350C2×C59#306111yescharacteristic
351C2×C59#331211yescharacteristic
352C2×C59#331911yescharacteristic
353C2×C59#334611yescharacteristic
354C2×C59#334911yescharacteristic
355C22×M16—591no—
356C23×C8—591no—
357C2×C22⋊Q8C2×D5911yescharacteristic
358C2×C22⋊Q8C2×D5911yescharacteristic
359C2×C4×D4C2×D5911yescharacteristic
360C2×#117—591no—
361C22×C4⋊C4C2×D5911yescharacteristic
362#485—1184no—
363#435—1184no—
364C2×#107—591no—
365C2×#107—591no—
366C2×#107—591no—
367C2×#106—591no—
368C2×C4⋊C8—591no—
369C2×#103—591no—
370C2×#103—591no—
371C2×#103—591no—
372C2×#103—591no—
373C2×C22⋊C8—591no—
374C2×C22⋊C8—591no—
375C2×#97—591no—
376#398—21no—
377#397—1181no—
378#396—21no—
379#394—1181no—
380#391C2×D5911yescharacteristic
381#389—591no—
382#389—591no—
383#389C2×D5911yescharacteristic
384#388—1181no—
385#387—1181no—
386#386—21no—
387#380—21no—
388#380—591no—
389#380—591no—
390#380—591no—
391#380C2×D5911yescharacteristic
392#340—591no—
393#340—591no—
394#340—591no—
395#340—591no—
396#340—591no—
397#340—591no—
398#340—591no—
399#340—591no—
400C59#2524111yescharacteristic, complemented, Sylow
401C22×Q8C22×D5911yescharacteristic
402C22×D4C22⋊D5911yescharacteristic
403C23×C4—21no—
404C23×C4C22×D5911yescharacteristic
405C23×C4C22×D5911yescharacteristic
406C2×M16—1182no—
407C2×M16—1182no—
408C2×M16—2361no—
409C22×C8—591no—
410C22×C8—591no—
411C22×C8—591no—
412C22×C8—591no—
413C22×C8—1182no—
414C22×C8—1182no—
415C22×C8—2361no—
416C22⋊Q8—22no—
417C22⋊Q8—22no—
418C22⋊Q8—22no—
419C22⋊Q8—41no—
420C4×D4—22no—
421C4×D4—22no—
422#117—1182no—
423#117—1182no—
424C2×C4⋊C4—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—41no—
430C2×C4⋊C4—591no—
431C2×C4⋊C4—591no—
432C2×C4⋊C4—591no—
433C2×C4⋊C4—591no—
434C2×C4⋊C4—1181no—
435C2×C4⋊C4—1181no—
436C2×C4⋊C4C22⋊D5911yescharacteristic
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—1181no—
444C2×C22⋊C4—1181no—
445C2×C22⋊C4—1181no—
446C2×C22⋊C4—1181no—
447C2×C22⋊C4C22⋊D5911yescharacteristic
448C2×C22⋊C4C22⋊D5911yescharacteristic
449C2×C42—1181no—
450C2×C42C22×D5911yescharacteristic
451#107—1182no—
452#107—1182no—
453#107—2361no—
454#106—1182no—
455C4⋊C8—1182no—
456#103—1182no—
457#103—1182no—
458#103—1182no—
459#103—2361no—
460C22⋊C8—594no—
461C22⋊C8—1182no—
462#97—1182no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—1181no—
468#95—1181no—
469#95—1181no—
470#95—1181no—
471#95—1181no—
472C24C2×C22⋊D5911yescharacteristic
473C24C2×C22⋊D5911yescharacteristic
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—41no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
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—42no—
498C22×C4—42no—
499C22×C4—81no—
500C22×C4—1181no—
501C22×C4—1181no—
502C22×C4—1181no—
503C22×C4—1181no—
504C22×C4D4×D5912yes—
505C22×C4#1174568712yes—
506C22×C4C2×C22⋊D5911yescharacteristic
507C22×C4C2×C22⋊D5911yescharacteristic
508C22×C4C2×C22⋊D5911yescharacteristic
509C22×C4C2×C22⋊D5911yescharacteristic
510C22×C4C23×D5911yescharacteristic
511M16—2364no—
512C2×C8—592no—
513C2×C8—592no—
514C2×C8—592no—
515C2×C8—1181no—
516C2×C8—1181no—
517C2×C8—1181no—
518C2×C8—1181no—
519C2×C8—1181no—
520C2×C8—1181no—
521C2×C8—1181no—
522C2×C8—1181no—
523C2×C8—1181no—
524C2×C8—2364no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—81no—
535C4⋊C4—1181no—
536C4⋊C4—1181no—
537C4⋊C4—1181no—
538C4⋊C4—1181no—
539C4⋊C4—1181no—
540C4⋊C4—1181no—
541C4⋊C4—1181no—
542C4⋊C4—1181no—
543C4⋊C4—4721no—
544C4⋊C4—2362no—
545C4⋊C4#1174568712yes—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—81no—
551C22⋊C4—4721no—
552C22⋊C4—4721no—
553C22⋊C4—2362no—
554C22⋊C4—2362no—
555C42—21no—
556C42—21no—
557C42—2362no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23#1427369912yes—
571C23#1427375812yes—
572C23C22×C22⋊D5911yescharacteristic
573Q8—41no—
574Q8—41no—
575Q8—41no—
576Q8—41no—
577D4—44no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—84no—
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#1427369912yes—
623C2×C4C2×D4×D5911yescharacteristic
624C2×C4#41×D5911yescharacteristic
625C2×C4C2×#1174568711yescharacteristic
626C2×C4C2×#1174568711yescharacteristic
627C2×C4#1427377211yescharacteristic
628C2×C4#1427377411yescharacteristic
629C8—1182no—
630C8—1182no—
631C8—1182no—
632C8—2361no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22C2×#1427369911yescharacteristic
648C22#1683804611yescharacteristic
649C22#1683805111yescharacteristic
650C22C2×#1427375811yescharacteristic
651C22#1683820011yescharacteristic
652C22#1683820511yescharacteristic
653C22#1683821711yescharacteristic
654C4—22no—
655C4—42no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4—4721no—
663C4—4721no—
664C4—4721no—
665C4—4721no—
666C4#1683793311yescharacteristic
667C4#1683799711yescharacteristic
668C4#1683804111yescharacteristic
669C4#1683805211yescharacteristic
670C2—42no—
671C2—42no—
672C2#2142765311yescharacteristic
673C2#2143372211yescharacteristic
674C2#2143376311yescharacteristic
675C2#2143614711yescharacteristic
676C2#2143625211yescharacteristic
677C2#2143657511yescharacteristic
678C2#2143662611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2417C2
socleC23×C59C22×D4
prime radicalC22×D4×C59D4
hypercenter#2417D59
hypocenterC59#25241
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2417

Complements

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

NH ≅ G/Ntype
1. C2×#16837740C2semidirect
4. #21425587C2semidirect
5. #21425533C2semidirect
6. #21425491C2semidirect
7. #21425296C2semidirect
8. #21424831C2semidirect
9. #21424789C2semidirect
10. #21424530C2semidirect
11. #21424520C2semidirect
12. #21424284C2semidirect
13. #21424123C2semidirect
14. #21423815C2semidirect
400. C59#25241semidirect

Maximal subgroups

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

subgroupindex
C2×#168377402
C2×#168374742
#214283932
#214255872
#214255332
#214254912
#214252962
#214248312
#214247892
#214245302
#214245202
#214242842
#214241232
#214238152
C59×#24172
#2524159