Finite Group Databasebeta

← #28656776

Subgroups — #28656776

Order 15104.

826 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427099C211yesmaximal, characteristic, complemented
2#21425413C211yesmaximal, characteristic, complemented
3#21425292C211yesmaximal, characteristic, complemented
4#21425234C211yesmaximal, characteristic, complemented
5#21425099C211yesmaximal, characteristic
6#21425049C211yesmaximal, characteristic, complemented
7#21424782C211yesmaximal, characteristic, complemented
8#21424720C211yesmaximal, characteristic, complemented
9#21424704C211yesmaximal, characteristic, complemented
10#21424681C211yesmaximal, characteristic, complemented
11#21424454C211yesmaximal, characteristic, complemented
12#21424414C211yesmaximal, characteristic
13#21424184C211yesmaximal, characteristic, complemented
14C4×#14273650C211yesmaximal, characteristic, complemented
15C59×#2491C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C2×#14273706C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19C22×C4⋊Dic59C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×#14273650C2211yescharacteristic
23C2×#14273650C2211yescharacteristic
24C2×#14273650C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×C22⋊(C59⋊C8)C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837308C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837299—21no—
34#16837289—21no—
35#16837274—21no—
36C4×C4⋊Dic59C2211yescharacteristic
37C22⋊C4×Dic59—21no—
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×C4⋊Q8×C59C2211yescharacteristic
49C2×#124×C59—21no—
50C2×#121×C59C2211yescharacteristic
51C59×#394—21no—
52C59×#393—21no—
53C59×#390C2211yescharacteristic
54C59×#390C2211yescharacteristic
55C59×#388—21no—
56C59×#380C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59C2211yescharacteristic
59C22×Dic118C2311yescharacteristic
60C23×Dic59—21no—
61C2×#11745670—22no—
62C2×#11745670—41no—
63C2×#11745670—41no—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273706—22no—
69#14273706—22no—
70C4×Dic118—22no—
71C4×Dic118—41no—
72C2×C4⋊Dic59—22no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59—41no—
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×C22⋊Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84#14273658—22no—
85C4⋊(C59⋊C8)—41no—
86#14273650—22no—
87#14273650—22no—
88#14273650—41no—
89#14273648—22no—
90#14273648—22no—
91#14273648—41no—
92C22⋊(C59⋊C8)—41no—
93C22⋊(C59⋊C8)—41no—
94C4×C59⋊C8—41no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C22×Q8×C59—21no—
100C22×D4×C59—21no—
101C22×D4×C59C2311yescharacteristic
102C23×C4×C59C2311yescharacteristic
103C4⋊Q8×C59—22no—
104C4⋊Q8×C59—41no—
105#124×C59—44no—
106#121×C59—22no—
107#121×C59—41no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C22⋊C4×C59—21no—
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×C22⋊C4×C59D411yescharacteristic
119C2×C22⋊C4×C59D411yescharacteristic
120C2×C42×C59—21no—
121C2×C42×C59C2311yescharacteristic
122C2×C42×C59C2311yescharacteristic
123#95×C59—21no—
124#95×C59—21no—
125#95×C59D411yescharacteristic
126#95×C59D411yescharacteristic
127C24×C59—21no—
128C24×C59C2×D411yescharacteristic
129C2×Q8×C59—42no—
130C2×Q8×C59—42no—
131C2×Q8×C59—81no—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59—42no—
139C2×D4×C59—42no—
140C2×D4×C59—81no—
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×C59C2×D411yescharacteristic
155C22×C4×C59C2411yescharacteristic
156C22×Dic59—21no—
157C22×Dic59—21no—
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—42no—
161C22×Dic59—42no—
162C22×Dic59—81no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C2×Dic118#4112yes—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—42no—
175C4⋊C4×C59#4112yes—
176C22⋊C4×C59—41no—
177C22⋊C4×C59—42no—
178C22⋊C4×C59—42no—
179C22⋊C4×C59—42no—
180C22⋊C4×C59—42no—
181C22⋊C4×C59—42no—
182C22⋊C4×C59—81no—
183C22⋊C4×C59SD1614yescomplemented
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—21no—
188C42×C59—42no—
189C22⋊Dic59—24no—
190#11745670—44no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—42no—
196C4⋊Dic59#4112yes—
197C4⋊Dic59#4112yes—
198Dic59⋊C4—24no—
199Dic59⋊C4—42no—
200Dic59⋊C4—81no—
201C4×Dic59—42no—
202C4×Dic59—81no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
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—21no—
215C23×C59—21no—
216C23×C59—21no—
217C23×C59—21no—
218C23×C59—21no—
219C23×C59—21no—
220C23×C59—21no—
221C23×C59—41no—
222C23×C59—41no—
223C23×C59—41no—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59—41no—
227C23×C59—41no—
228C23×C59C2×SD1612yes—
229C23×C59C22×D411yescharacteristic
230Q8×C59—84no—
231D4×C59—41no—
232D4×C59—41no—
233D4×C59—41no—
234D4×C59—41no—
235D4×C59—84no—
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—22no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
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×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59C2×SD1612yes—
263C2×C4×C59C2×SD1612yes—
264C2×C4×C59C2×#4111yescharacteristic
265C2×C4×C59C2×#4111yescharacteristic
266C2×C4×C59C2×#4111yescharacteristic
267C2×C4×C59C2×#4111yescharacteristic
268C2×C4×C59H(5,𝔽2)11yescharacteristic
269C2×C4×C59H(5,𝔽2)11yescharacteristic
270C2×Dic59—22no—
271C2×Dic59—22no—
272C2×Dic59—22no—
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—84no—
283Dic118—22no—
284Dic118—22no—
285Dic118—41no—
286Dic118—41no—
287C59⋊C8—41no—
288C59⋊C8—41no—
289C59⋊C8—41no—
290C59⋊C8—41no—
291#26724—591nomaximal, Sylow
292C22×C59—22no—
293C22×C59—22no—
294C22×C59—22no—
295C22×C59—41no—
296C22×C59—41no—
297C22×C59—41no—
298C22×C59—41no—
299C22×C59—81no—
300C22×C59—81no—
301C22×C59—81no—
302C22×C59—81no—
303C22×C59—81no—
304C22×C59—81no—
305C22×C59—81no—
306C22×C59#52511yescharacteristic
307C22×C59#54611yescharacteristic
308C22×C59#54611yescharacteristic
309C22×C59C22×SD1611yescharacteristic
310C22×C59#57511yescharacteristic
311C22×C59#57611yescharacteristic
312C22×C59#57811yescharacteristic
313C4×C59—22no—
314C4×C59—22no—
315C4×C59—41no—
316C4×C59—42no—
317C4×C59—41no—
318C4×C59—41no—
319C4×C59—81no—
320C4×C59—81no—
321C4×C59#52911yescharacteristic
322C4×C59#53811yescharacteristic
323C4×C59#55511yescharacteristic
324C4×C59#55911yescharacteristic
325Dic59—42no—
326Dic59—42no—
327Dic59—42no—
328Dic59—81no—
329#2491D5911yescharacteristic
330#2360—591no—
331#2055—591no—
332#2035—591no—
333#2024—591no—
334#2000—591no—
335#1988—591no—
336#1934—591no—
337#1926—591no—
338#1923—591no—
339#1920—591no—
340#1880—591no—
341#1874—591no—
342#1824—591no—
343C4×#103—591no—
344C2×C59—42no—
345C2×C59—161no—
346C2×C59#245911yescharacteristic
347C2×C59#310311yescharacteristic
348C2×C59#312411yescharacteristic
349C2×C59#329511yescharacteristic
350C2×C59#329811yescharacteristic
351C2×C59#331911yescharacteristic
352C2×C59#332711yescharacteristic
353C22×SD16—591no—
354C2×C4⋊Q8C2×D5911yescharacteristic
355C2×#124—21no—
356C2×C22⋊Q8—591no—
357C2×#121C2×D5911yescharacteristic
358C2×C4×Q8—591no—
359C22×C4⋊C4—591no—
360C2×#106—591no—
361C2×C4⋊C8—591no—
362C2×#103—591no—
363C2×#103—591no—
364C2×#103—591no—
365C2×#102—591no—
366C2×#102—591no—
367C2×#102—591no—
368C2×C22⋊C8—591no—
369C2×C22⋊C8—591no—
370C2×C4×C8—591no—
371#394—21no—
372#393—21no—
373#391—591no—
374#390C2×D5911yescharacteristic
375#390C2×D5911yescharacteristic
376#389—591no—
377#389—1181no—
378#388—21no—
379#387—1181no—
380#385—1181no—
381#380C2×D5911yescharacteristic
382C4×C4⋊C4—591no—
383C4×C4⋊C4C2×D5911yescharacteristic
384C4×C22⋊C4—1181no—
385C4×C22⋊C4C2×D5911yescharacteristic
386#340—591no—
387#340—591no—
388#340—591no—
389#340—591no—
390#340—591no—
391#340—591no—
392#340—591no—
393#340—591no—
394#336—591no—
395#336—591no—
396C59#2672411yescharacteristic, complemented, Sylow
397C22×Q8—21no—
398C22×Q8—591no—
399C22×D4—21no—
400C22×D4C22×D5911yescharacteristic
401C23×C4—1181no—
402C23×C4C22×D5911yescharacteristic
403C2×SD16—1182no—
404C2×SD16—2361no—
405C2×SD16—2361no—
406C22×C8—591no—
407C22×C8—591no—
408C22×C8—591no—
409C22×C8—591no—
410C4⋊Q8—22no—
411C4⋊Q8—41no—
412#124—44no—
413C22⋊Q8—1182no—
414C22⋊Q8—1182no—
415#121—22no—
416#121—41no—
417C4×Q8—1182no—
418C4×Q8—2361no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—591no—
421C2×C4⋊C4—591no—
422C2×C4⋊C4—591no—
423C2×C4⋊C4—1182no—
424C2×C4⋊C4—1181no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4—1181no—
428C2×C4⋊C4—2361no—
429C2×C4⋊C4C22×D5911yescharacteristic
430C2×C4⋊C4C22×D5911yescharacteristic
431C2×C4⋊C4C22×D5911yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—1181no—
439C2×C22⋊C4C22⋊D5911yescharacteristic
440C2×C22⋊C4C22⋊D5911yescharacteristic
441C2×C42—21no—
442C2×C42—1181no—
443C2×C42—1181no—
444C2×C42C22×D5911yescharacteristic
445C2×C42C22×D5911yescharacteristic
446#106—1182no—
447C4⋊C8—2361no—
448#103—1182no—
449#103—1182no—
450#103—2361no—
451#102—1182no—
452#102—1182no—
453#102—2361no—
454C22⋊C8—2361no—
455C22⋊C8—2361no—
456C4×C8—2361no—
457#95—21no—
458#95—21no—
459#95—1181no—
460#95—1181no—
461#95—1181no—
462#95—1181no—
463#95C22⋊D5911yescharacteristic
464#95C22⋊D5911yescharacteristic
465C24—21no—
466C24C2×C22⋊D5911yescharacteristic
467C2×Q8—42no—
468C2×Q8—42no—
469C2×Q8—81no—
470C2×Q8—592no—
471C2×Q8—1181no—
472C2×Q8—1181no—
473C2×Q8—1181no—
474C2×Q8—1181no—
475C2×Q8—1181no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—81no—
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—1181no—
495C22×C4—1181no—
496C22×C4—1181no—
497C22×C4—1181no—
498C22×C4—4721no—
499C22×C4—2362no—
500C22×C4—2362no—
501C22×C4C2×C22⋊D5911yescharacteristic
502C22×C4C2×C22⋊D5911yescharacteristic
503C22×C4C2×C22⋊D5911yescharacteristic
504C22×C4C2×C22⋊D5911yescharacteristic
505C22×C4C2×C22⋊D5911yescharacteristic
506C22×C4C23×D5911yescharacteristic
507SD16—2364no—
508C2×C8—1181no—
509C2×C8—1181no—
510C2×C8—1181no—
511C2×C8—1181no—
512C2×C8—1181no—
513C2×C8—1181no—
514C2×C8—1181no—
515C2×C8—1181no—
516C2×C8—1181no—
517C2×C8—1181no—
518C2×C8—1181no—
519C2×C8—1181no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—42no—
526C4⋊C4—592no—
527C4⋊C4—592no—
528C4⋊C4—1184no—
529C4⋊C4—1181no—
530C4⋊C4—1181no—
531C4⋊C4—1181no—
532C4⋊C4—1181no—
533C4⋊C4—4721no—
534C4⋊C4—2362no—
535C4⋊C4—2362no—
536C4⋊C4#1174568712yes—
537C22⋊C4—41no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—81no—
544C22⋊C4—1184no—
545C22⋊C4#1174567014yescomplemented
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—42no—
551C42—4721no—
552C42—2362no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23C2×#1174567012yes—
567C23C22×C22⋊D5911yescharacteristic
568Q8—84no—
569Q8—1182no—
570Q8—1182no—
571Q8—2361no—
572Q8—2361no—
573D4—41no—
574D4—41no—
575D4—41no—
576D4—41no—
577D4—84no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
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—22no—
592C2×C4—41no—
593C2×C4—41no—
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—1182no—
605C2×C4—1182no—
606C2×C4—1182no—
607C2×C4—4724no—
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×C4C2×#1174567012yes—
618C2×C4C2×#1174567012yes—
619C2×C4#41×D5911yescharacteristic
620C2×C4#41×D5911yescharacteristic
621C2×C4#41×D5911yescharacteristic
622C2×C4C2×#1174568711yescharacteristic
623C2×C4#1427377211yescharacteristic
624C2×C4#1427377311yescharacteristic
625C8—2361no—
626C8—2361no—
627C8—2361no—
628C8—2361no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22#1683790911yescharacteristic
644C22#1683804011yescharacteristic
645C22#1683804611yescharacteristic
646C22C22×#1174567011yescharacteristic
647C22#1683820011yescharacteristic
648C22#1683820511yescharacteristic
649C22#1683821711yescharacteristic
650C4—22no—
651C4—22no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—81no—
657C4—81no—
658C4—4721no—
659C4—2362no—
660C4—2362no—
661C4—2362no—
662C4#1683793111yescharacteristic
663C4#1683798611yescharacteristic
664C4#1683811411yescharacteristic
665C4#1683813411yescharacteristic
666C2—42no—
667C2—161no—
668C2#2142802911yescharacteristic
669C2#2143412311yescharacteristic
670C2#2143435211yescharacteristic
671C2#2143592011yescharacteristic
672C2#2143595511yescharacteristic
673C2#2143624611yescharacteristic
674C2#2143636411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2491C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2491D59
hypocenterC59#26724
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2491

Complements

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

NH ≅ G/Ntype
1. #21427099C2semidirect
2. #21425413C2semidirect
3. #21425292C2semidirect
4. #21425234C2semidirect
6. #21425049C2semidirect
7. #21424782C2semidirect
8. #21424720C2semidirect
9. #21424704C2semidirect
10. #21424681C2semidirect
11. #21424454C2semidirect
13. #21424184C2semidirect
14. C4×#14273650C2semidirect
183. C22⋊C4×C59SD16semidirect
396. C59#26724semidirect
545. C22⋊C4#11745670semidirect

Maximal subgroups

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

subgroupindex
#214270992
#214254132
#214252922
#214252342
#214250992
#214250492
#214247822
#214247202
#214247042
#214246812
#214244542
#214244142
#214241842
C4×#142736502
C59×#24912
#2672459