Finite Group Databasebeta

← #28656911

Subgroups — #28656911

Order 15104.

804 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
1#21427121C211yesmaximal, characteristic, complemented
2#21425413C211yesmaximal, characteristic, complemented
3#21425299C211yesmaximal, characteristic, complemented
4#21425234C211yesmaximal, characteristic, complemented
5#21425099C211yesmaximal, characteristic
6#21425045C211yesmaximal, characteristic, complemented
7#21424782C211yesmaximal, characteristic, complemented
8#21424713C211yesmaximal, characteristic, complemented
9#21424704C211yesmaximal, characteristic, complemented
10#21424667C211yesmaximal, characteristic, complemented
11#21424454C211yesmaximal, characteristic, complemented
12#21424419C211yesmaximal, characteristic
13#21424188C211yesmaximal, characteristic, complemented
14C4×#14273650C211yesmaximal, characteristic, complemented
15C59×#2491C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C2×#14273706C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
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#16837296—21no—
34#16837290—21no—
35#16837284—21no—
36C4×C4⋊Dic59C2211yescharacteristic
37#16837253—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
60C2×#11745670—22no—
61C2×#11745670—41no—
62C2×#11745670—41no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273706—22no—
68#14273706—22no—
69C4×Dic118—22no—
70C4×Dic118—41no—
71#14273686—22no—
72#14273686—41no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84#14273660—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—
160C2×Dic118—21no—
161C2×Dic118—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118#4112yes—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—42no—
172C4⋊C4×C59#4112yes—
173C22⋊C4×C59—22no—
174C22⋊C4×C59—22no—
175C22⋊C4×C59—42no—
176C22⋊C4×C59—42no—
177C22⋊C4×C59—42no—
178C22⋊C4×C59—42no—
179C22⋊C4×C59—42no—
180C22⋊C4×C59—81no—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—42no—
186C22⋊Dic59—24no—
187C22⋊Dic59—42no—
188#11745670—44no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59#4112yes—
194C4⋊Dic59#4112yes—
195Dic59⋊C4—24no—
196Dic59⋊C4—42no—
197Dic59⋊C4—42no—
198C4×Dic59—42no—
199C4×Dic59—42no—
200C4×Dic59—42no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
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—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—21no—
216C23×C59—21no—
217C23×C59—21no—
218C23×C59—21no—
219C23×C59—21no—
220C23×C59—41no—
221C23×C59—41no—
222C23×C59—41no—
223C23×C59—41no—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59—41no—
227C23×C59C22×D411yescharacteristic
228Q8×C59—84no—
229D4×C59—41no—
230D4×C59—41no—
231D4×C59—41no—
232D4×C59—41no—
233D4×C59—84no—
234C2×C4×C59—21no—
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—22no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
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#13512yes—
261C2×C4×C59AGL(1,ℤ/8)12yes—
262C2×C4×C59C2×#4111yescharacteristic
263C2×C4×C59C2×#4111yescharacteristic
264C2×C4×C59C2×#4111yescharacteristic
265C2×C4×C59C2×#4111yescharacteristic
266C2×C4×C59H(5,𝔽2)11yescharacteristic
267C2×C4×C59H(5,𝔽2)11yescharacteristic
268C2×Dic59—22no—
269C2×Dic59—22no—
270C2×Dic59—22no—
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—
280Dic118—22no—
281Dic118—22no—
282Dic118—41no—
283Dic118—41no—
284C59⋊C8—41no—
285C59⋊C8—41no—
286C59⋊C8—41no—
287C59⋊C8—41no—
288#26733—591nomaximal, Sylow
289C22×C59—41no—
290C22×C59—41no—
291C22×C59—41no—
292C22×C59—41no—
293C22×C59—41no—
294C22×C59—41no—
295C22×C59—41no—
296C22×C59—81no—
297C22×C59—81no—
298C22×C59—81no—
299C22×C59—81no—
300C22×C59—81no—
301C22×C59—81no—
302C22×C59—81no—
303C22×C59#52511yescharacteristic
304C22×C59#54611yescharacteristic
305C22×C59#54611yescharacteristic
306C22×C59C2×#13511yescharacteristic
307C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C59#57711yescharacteristic
309C22×C59#57811yescharacteristic
310C4×C59—22no—
311C4×C59—22no—
312C4×C59—41no—
313C4×C59—41no—
314C4×C59—41no—
315C4×C59—42no—
316C4×C59—81no—
317C4×C59—81no—
318C4×C59#52911yescharacteristic
319C4×C59#53811yescharacteristic
320C4×C59#55511yescharacteristic
321C4×C59#55911yescharacteristic
322Dic59—42no—
323Dic59—42no—
324Dic59—42no—
325Dic59—81no—
326#2491D5911yescharacteristic
327#2362—591no—
328#2055—591no—
329#2036—591no—
330#2024—591no—
331#2000—591no—
332#1987—591no—
333#1934—591no—
334#1925—591no—
335#1923—591no—
336#1918—591no—
337#1880—591no—
338#1875—591no—
339#1825—591no—
340C4×#103—591no—
341C2×C59—81no—
342C2×C59—161no—
343C2×C59#245911yescharacteristic
344C2×C59#310611yescharacteristic
345C2×C59#312111yescharacteristic
346C2×C59#330011yescharacteristic
347C2×C59#330311yescharacteristic
348C2×C59#330811yescharacteristic
349C2×C59#332411yescharacteristic
350C22×SD16—591no—
351C2×C4⋊Q8C2×D5911yescharacteristic
352C2×#124—21no—
353C2×C22⋊Q8—591no—
354C2×#121C2×D5911yescharacteristic
355C2×C4×Q8—591no—
356C2×#117—591no—
357C2×#107—591no—
358C2×C4⋊C8—591no—
359C2×#103—591no—
360C2×#103—591no—
361C2×#103—591no—
362C2×#102—591no—
363C2×#102—591no—
364C2×#102—591no—
365C2×C22⋊C8—591no—
366C2×C22⋊C8—591no—
367C2×C4×C8—591no—
368#394—21no—
369#393—21no—
370#391—591no—
371#390C2×D5911yescharacteristic
372#390C2×D5911yescharacteristic
373#389—591no—
374#388—21no—
375#388—1181no—
376#388—1181no—
377#387—1181no—
378#380C2×D5911yescharacteristic
379C4×C4⋊C4—591no—
380C4×C4⋊C4—1181no—
381C4×C4⋊C4C2×D5911yescharacteristic
382C4×C22⋊C4C2×D5911yescharacteristic
383#340—591no—
384#340—591no—
385#340—591no—
386#340—591no—
387#340—591no—
388#340—591no—
389#340—591no—
390#340—591no—
391#336—591no—
392#336—591no—
393C59#2673311yescharacteristic, complemented, Sylow
394C22×Q8—21no—
395C22×Q8—591no—
396C22×D4—21no—
397C22×D4C22×D5911yescharacteristic
398C23×C4C22×D5911yescharacteristic
399C2×SD16—1182no—
400C2×SD16—2361no—
401C2×SD16—2361no—
402C22×C8—591no—
403C22×C8—591no—
404C22×C8—591no—
405C22×C8—591no—
406C4⋊Q8—22no—
407C4⋊Q8—41no—
408#124—44no—
409C22⋊Q8—1182no—
410C22⋊Q8—1182no—
411#121—22no—
412#121—41no—
413C4×Q8—1182no—
414C4×Q8—2361no—
415#117—1182no—
416#117—2361no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—591no—
419C2×C4⋊C4—591no—
420C2×C4⋊C4—591no—
421C2×C4⋊C4—1181no—
422C2×C4⋊C4—1181no—
423C2×C4⋊C4—1181no—
424C2×C4⋊C4C22×D5911yescharacteristic
425C2×C4⋊C4C22×D5911yescharacteristic
426C2×C4⋊C4C22×D5911yescharacteristic
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—1181no—
434C2×C22⋊C4—1181no—
435C2×C22⋊C4C22⋊D5911yescharacteristic
436C2×C22⋊C4C22⋊D5911yescharacteristic
437C2×C42—21no—
438C2×C42—1181no—
439C2×C42—1181no—
440C2×C42—1181no—
441C2×C42C22×D5911yescharacteristic
442C2×C42C22×D5911yescharacteristic
443#107—1182no—
444C4⋊C8—2361no—
445#103—1182no—
446#103—1182no—
447#103—2361no—
448#102—1182no—
449#102—1182no—
450#102—2361no—
451C22⋊C8—2361no—
452C22⋊C8—2361no—
453C4×C8—2361no—
454#95—21no—
455#95—21no—
456#95—1181no—
457#95—1181no—
458#95—1181no—
459#95—1181no—
460#95C22⋊D5911yescharacteristic
461#95C22⋊D5911yescharacteristic
462C24—21no—
463C24C2×C22⋊D5911yescharacteristic
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—81no—
467C2×Q8—592no—
468C2×Q8—1181no—
469C2×Q8—1181no—
470C2×Q8—1181no—
471C2×Q8—1181no—
472C2×Q8—1181no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—81no—
482C22×C4—21no—
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—1181no—
492C22×C4—1181no—
493C22×C4—1181no—
494C22×C4—1181no—
495C22×C4C2×C22⋊D5911yescharacteristic
496C22×C4C2×C22⋊D5911yescharacteristic
497C22×C4C2×C22⋊D5911yescharacteristic
498C22×C4C2×C22⋊D5911yescharacteristic
499C22×C4C2×C22⋊D5911yescharacteristic
500C22×C4C23×D5911yescharacteristic
501SD16—2364no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8—1181no—
510C2×C8—1181no—
511C2×C8—1181no—
512C2×C8—1181no—
513C2×C8—1181no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—592no—
521C4⋊C4—592no—
522C4⋊C4—1184no—
523C4⋊C4—1181no—
524C4⋊C4—1181no—
525C4⋊C4—1181no—
526C4⋊C4—1181no—
527C4⋊C4—2362no—
528C4⋊C4—2362no—
529C4⋊C4#1174568712yes—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—81no—
538C22⋊C4—1184no—
539C22⋊C4—2362no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—42no—
545C42—2362no—
546C42—2362no—
547C42—2362no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23C22×C22⋊D5911yescharacteristic
563Q8—84no—
564Q8—1182no—
565Q8—1182no—
566Q8—2361no—
567Q8—2361no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572D4—84no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
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—22no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
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—1182no—
601C2×C4—1182no—
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#1427374712yes—
612C2×C4#1427375312yes—
613C2×C4#41×D5911yescharacteristic
614C2×C4#41×D5911yescharacteristic
615C2×C4#41×D5911yescharacteristic
616C2×C4C2×#1174568711yescharacteristic
617C2×C4#1427377211yescharacteristic
618C2×C4#1427377311yescharacteristic
619C8—2361no—
620C8—2361no—
621C8—2361no—
622C8—2361no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22#1683790911yescharacteristic
638C22#1683804011yescharacteristic
639C22#1683804611yescharacteristic
640C22C2×#1427374711yescharacteristic
641C22C2×#1427375311yescharacteristic
642C22#1683821311yescharacteristic
643C22#1683821711yescharacteristic
644C4—22no—
645C4—22no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—42no—
650C4—81no—
651C4—81no—
652C4—4721no—
653C4—2362no—
654C4—2362no—
655C4—2362no—
656C4#1683793111yescharacteristic
657C4#1683798611yescharacteristic
658C4#1683811411yescharacteristic
659C4#1683813411yescharacteristic
660C2—81no—
661C2—161no—
662C2#2142802911yescharacteristic
663C2#2143415011yescharacteristic
664C2#2143431911yescharacteristic
665C2#2143598311yescharacteristic
666C2#2143601811yescharacteristic
667C2#2143608111yescharacteristic
668C2#2143632711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2491C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2491D59
hypocenterC59#26733
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. #21427121C2semidirect
2. #21425413C2semidirect
3. #21425299C2semidirect
4. #21425234C2semidirect
6. #21425045C2semidirect
7. #21424782C2semidirect
8. #21424713C2semidirect
9. #21424704C2semidirect
10. #21424667C2semidirect
11. #21424454C2semidirect
13. #21424188C2semidirect
14. C4×#14273650C2semidirect
393. C59#26733semidirect

Maximal subgroups

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

subgroupindex
#214271212
#214254132
#214252992
#214252342
#214250992
#214250452
#214247822
#214247132
#214247042
#214246672
#214244542
#214244192
#214241882
C4×#142736502
C59×#24912
#2673359