Finite Group Databasebeta

← #28629133

Subgroups — #28629133

Order 15104.

842 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837621C211yesmaximal, characteristic
2C2×#16837507C211yesmaximal, characteristic, complemented
3#21427051C211yesmaximal, characteristic, complemented
4#21425418C211yesmaximal, characteristic, complemented
5#21425413C211yesmaximal, characteristic
6#21425196C211yesmaximal, characteristic, complemented
7#21425190C211yesmaximal, characteristic
8#21425029C211yesmaximal, characteristic
9#21425020C211yesmaximal, characteristic, complemented
10#21424913C211yesmaximal, characteristic
11#21424820C211yesmaximal, characteristic, complemented
12#21424727C211yesmaximal, characteristic, complemented
13#21424720C211yesmaximal, characteristic
14#21424101C211yesmaximal, characteristic, complemented
15C59×#2545C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C22×#11745672C2211yescharacteristic
18C22×#11745670C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20#16837621—24no—
21#16837507—24no—
22C2×#14273658C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×#14273651C2211yescharacteristic
30C2×#14273651C2211yescharacteristic
31C2×#14273651C2211yescharacteristic
32C2×#14273650C2211yescharacteristic
33C2×#14273648C2211yescharacteristic
34C2×#14273648C2211yescharacteristic
35C2×#14273648C2211yescharacteristic
36C2×#14273638C2211yescharacteristic
37#16837308C2211yescharacteristic
38#16837308C2211yescharacteristic
39#16837289—21no—
40#16837289—21no—
41#16837272C2211yescharacteristic
42#16837267—21no—
43C4⋊C4×Dic59—21no—
44C4×C4⋊Dic59C2211yescharacteristic
45C4×C4⋊Dic59C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C4⋊Q8×C59C2211yescharacteristic
51C2×#124×C59C2211yescharacteristic
52C2×#124×C59C2211yescharacteristic
53C2×C4×Q8×C59C2211yescharacteristic
54C59×#396—21no—
55C59×#396—21no—
56C59×#391C2211yescharacteristic
57C59×#390C2211yescharacteristic
58C59×#388—21no—
59C59×#388—21no—
60C4×C4⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C2×#11745672—22no—
63C2×#11745672—22no—
64C2×#11745672—22no—
65C2×#11745672—22no—
66C2×#11745672—41no—
67C2×#11745672—41no—
68C2×#11745670—22no—
69C2×#11745670—41no—
70C2×#11745670—41no—
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73C22×C59⋊C8C2311yescharacteristic
74C22×C59⋊C8C2311yescharacteristic
75C4×Dic118—22no—
76C4×Dic118D414yes—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59—21no—
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×Dic59⋊C4—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86#14273658—22no—
87C4⋊(C59⋊C8)—22no—
88C4⋊(C59⋊C8)—41no—
89C4⋊(C59⋊C8)—41no—
90#14273651—22no—
91#14273651—22no—
92#14273651—22no—
93#14273651—41no—
94#14273651—41no—
95#14273651—41no—
96#14273650—22no—
97#14273648—22no—
98#14273648—22no—
99#14273648—41no—
100#14273638—22no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C22×Q8×C59C2311yescharacteristic
108C22×Q8×C59C2311yescharacteristic
109C22×D4×C59C2311yescharacteristic
110C4⋊Q8×C59—22no—
111C4⋊Q8×C59—41no—
112#124×C59—22no—
113#124×C59—22no—
114#124×C59—22no—
115#124×C59—41no—
116C4×Q8×C59—22no—
117C4×Q8×C59—22no—
118C2×C4⋊C4×C59—21no—
119C2×C4⋊C4×C59—21no—
120C2×C4⋊C4×C59—21no—
121C2×C4⋊C4×C59D411yescharacteristic
122C2×C4⋊C4×C59D411yescharacteristic
123C2×C4⋊C4×C59C2311yescharacteristic
124C2×C22⋊C4×C59—21no—
125C2×C22⋊C4×C59—21no—
126C2×C22⋊C4×C59—21no—
127C2×C22⋊C4×C59—21no—
128C2×C22⋊C4×C59—21no—
129C2×C22⋊C4×C59—21no—
130C2×C42×C59—21no—
131C2×C42×C59C2311yescharacteristic
132C2×C42×C59C2311yescharacteristic
133C2×C42×C59C2311yescharacteristic
134#95×C59—21no—
135#95×C59—21no—
136#95×C59D411yescharacteristic
137#95×C59D411yescharacteristic
138C24×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59—21no—
141C2×Q8×C59—21no—
142C2×Q8×C59—21no—
143C2×Q8×C59—21no—
144C2×Q8×C59—21no—
145C2×Q8×C59—21no—
146C2×Q8×C59—21no—
147C2×Q8×C59—21no—
148C2×Q8×C59—21no—
149C2×Q8×C59—21no—
150C2×Q8×C59#4112yes—
151C2×D4×C59—21no—
152C2×D4×C59—21no—
153C2×D4×C59—21no—
154C2×D4×C59—21no—
155C2×D4×C59—21no—
156C2×D4×C59—21no—
157C22×C4×C59—21no—
158C22×C4×C59—21no—
159C22×C4×C59—21no—
160C22×C4×C59C2×D411yescharacteristic
161C22×C4×C59C2×D411yescharacteristic
162C22×C4×C59C2×D411yescharacteristic
163C22×C4×C59C2×D411yescharacteristic
164C22×C4×C59C2×D411yescharacteristic
165C22×C4×C59C2×D411yescharacteristic
166C22×C4×C59C2411yescharacteristic
167C22×Dic59—21no—
168C22×Dic59—21no—
169C22×Dic59—21no—
170C22×Dic59—21no—
171C2×Dic118—21no—
172C2×Dic118—21no—
173C2×Dic118—21no—
174C2×Dic118—21no—
175C2×Dic118—21no—
176C2×Dic118C2×D412yes—
177C4⋊C4×C59—22no—
178C4⋊C4×C59—21no—
179C4⋊C4×C59—21no—
180C4⋊C4×C59—41no—
181C4⋊C4×C59—42no—
182C4⋊C4×C59—42no—
183C4⋊C4×C59—42no—
184C22⋊C4×C59—42no—
185C22⋊C4×C59—42no—
186C22⋊C4×C59—42no—
187C22⋊C4×C59—42no—
188C22⋊C4×C59—42no—
189C22⋊C4×C59—42no—
190C42×C59—21no—
191C42×C59—21no—
192C42×C59—21no—
193C42×C59—21no—
194C42×C59—42no—
195C42×C59C2×D412yes—
196C42×C59#4112yes—
197#11745672—28no—
198#11745672—44no—
199#11745670—44no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—21no—
203C4⋊Dic59—21no—
204C4⋊Dic59—21no—
205C4⋊Dic59—42no—
206C4⋊Dic59—81no—
207C4⋊Dic59C2×D412yes—
208Dic59⋊C4—24no—
209C4×Dic59—24no—
210C4×Dic59—81no—
211C4×Dic59—81no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8—21no—
217C2×C59⋊C8—21no—
218C2×C59⋊C8—21no—
219C2×C59⋊C8—21no—
220C2×C59⋊C8—21no—
221C2×C59⋊C8—21no—
222C2×C59⋊C8—21no—
223C2×C59⋊C8#4112yes—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59—41no—
227C23×C59—41no—
228C23×C59—41no—
229C23×C59—41no—
230C23×C59—41no—
231C23×C59C22×D411yescharacteristic
232Q8×C59—22no—
233Q8×C59—22no—
234Q8×C59—41no—
235Q8×C59—41no—
236Q8×C59—41no—
237Q8×C59—41no—
238Q8×C59—41no—
239Q8×C59—41no—
240D4×C59—41no—
241D4×C59—41no—
242D4×C59—41no—
243D4×C59—41no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—21no—
251C2×C4×C59—21no—
252C2×C4×C59—21no—
253C2×C4×C59—22no—
254C2×C4×C59—21no—
255C2×C4×C59—21no—
256C2×C4×C59—21no—
257C2×C4×C59—21no—
258C2×C4×C59—21no—
259C2×C4×C59—21no—
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#12112yes—
269C2×C4×C59#12112yes—
270C2×C4×C59#13712yes—
271C2×C4×C59C22×D411yescharacteristic
272C2×C4×C59C2×#4111yescharacteristic
273C2×C4×C59C2×#4111yescharacteristic
274C2×C4×C59C2×#4111yescharacteristic
275C2×C4×C59H(5,𝔽2)11yescharacteristic
276C2×C4×C59H(5,𝔽2)11yescharacteristic
277C2×Dic59—22no—
278C2×Dic59—22no—
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—
289Dic118—22no—
290Dic118—22no—
291Dic118—41no—
292Dic118—41no—
293C59⋊C8—22no—
294C59⋊C8—41no—
295C59⋊C8—41no—
296C59⋊C8—41no—
297#24641—591nomaximal, Sylow
298C22×C59—81no—
299C22×C59—81no—
300C22×C59—81no—
301C22×C59—81no—
302C22×C59—81no—
303C22×C59—81no—
304C22×C59—81no—
305C22×C59C2×#12111yescharacteristic
306C22×C59#54611yescharacteristic
307C22×C59#55011yescharacteristic
308C22×C59C2×#13711yescharacteristic
309C22×C59#57511yescharacteristic
310C22×C59#57711yescharacteristic
311C22×C59#57811yescharacteristic
312C4×C59—22no—
313C4×C59—22no—
314C4×C59—22no—
315C4×C59—42no—
316C4×C59—41no—
317C4×C59—41no—
318C4×C59—41no—
319C4×C59—81no—
320C4×C59—81no—
321C4×C59#52511yescharacteristic
322C4×C59#53511yescharacteristic
323C4×C59#55311yescharacteristic
324C4×C59#55511yescharacteristic
325Dic59—42no—
326Dic59—42no—
327Dic59—81no—
328Dic59—81no—
329C2×#464—591no—
330C2×#441—591no—
331#2545D5911yescharacteristic
332#2355—591no—
333#2056—591no—
334#2055—591no—
335#2016—591no—
336#2015—591no—
337#1983—591no—
338#1981—591no—
339#1959—591no—
340#1940—591no—
341#1927—591no—
342#1926—591no—
343#1803—591no—
344C2×C59—161no—
345C2×C59#241411yescharacteristic
346C2×C59#303711yescharacteristic
347C2×C59#304611yescharacteristic
348C2×C59#331311yescharacteristic
349C2×C59#331611yescharacteristic
350C2×C59#339811yescharacteristic
351C2×C59#340111yescharacteristic
352C22×Q16—591no—
353C22×Q16—591no—
354C22×SD16—591no—
355C2×C4⋊Q8C2×D5911yescharacteristic
356C2×#124C2×D5911yescharacteristic
357C2×#124C2×D5911yescharacteristic
358C2×C4×Q8—591no—
359C2×C4×Q8C2×D5911yescharacteristic
360#464—1184no—
361#441—1184no—
362C2×#106—591no—
363C2×C4⋊C8—591no—
364C2×C4⋊C8—591no—
365C2×C4⋊C8—591no—
366C2×#103—591no—
367C2×#103—591no—
368C2×#103—591no—
369C2×#103—591no—
370C2×#103—591no—
371C2×#103—591no—
372C2×#103—591no—
373C2×#102—591no—
374C2×#102—591no—
375C2×#102—591no—
376C2×#97—591no—
377#396—21no—
378#396—21no—
379#391—591no—
380#391—591no—
381#391C2×D5911yescharacteristic
382#390C2×D5911yescharacteristic
383#388—21no—
384#388—21no—
385#387—1181no—
386#387—1181no—
387#384—591no—
388#382—1181no—
389C4×C4⋊C4—591no—
390C4×C4⋊C4—591no—
391C4×C4⋊C4—1181no—
392C4×C4⋊C4C2×D5911yescharacteristic
393#340—591no—
394#340—591no—
395#336—591no—
396#336—591no—
397C59#2464111yescharacteristic, complemented, Sylow
398C22×Q8—591no—
399C22×Q8C22×D5911yescharacteristic
400C22×Q8C22×D5911yescharacteristic
401C22×D4C22×D5911yescharacteristic
402C2×Q16—1182no—
403C2×Q16—1182no—
404C2×Q16—1182no—
405C2×Q16—1182no—
406C2×Q16—2361no—
407C2×Q16—2361no—
408C2×SD16—1182no—
409C2×SD16—2361no—
410C2×SD16—2361no—
411C22×C8—591no—
412C22×C8—591no—
413C22×C8—591no—
414C22×C8—591no—
415C4⋊Q8—22no—
416C4⋊Q8—41no—
417#124—22no—
418#124—22no—
419#124—22no—
420#124—41no—
421C4×Q8—22no—
422C4×Q8—22no—
423C4×Q8—594no—
424C4×Q8—1182no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—591no—
429C2×C4⋊C4—591no—
430C2×C4⋊C4—591no—
431C2×C4⋊C4—1181no—
432C2×C4⋊C4—1181no—
433C2×C4⋊C4—1181no—
434C2×C4⋊C4C22⋊D5911yescharacteristic
435C2×C4⋊C4C22⋊D5911yescharacteristic
436C2×C4⋊C4C22×D5911yescharacteristic
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C42—21no—
444C2×C42—1181no—
445C2×C42—1181no—
446C2×C42—1181no—
447C2×C42C22×D5911yescharacteristic
448C2×C42C22×D5911yescharacteristic
449C2×C42C22×D5911yescharacteristic
450#106—1182no—
451C4⋊C8—1182no—
452C4⋊C8—2361no—
453C4⋊C8—2361no—
454#103—1182no—
455#103—1182no—
456#103—1182no—
457#103—1182no—
458#103—2361no—
459#103—2361no—
460#103—2361no—
461#102—1182no—
462#102—1182no—
463#102—2361no—
464#97—1182no—
465#95—21no—
466#95—21no—
467#95—1181no—
468#95—1181no—
469#95—1181no—
470#95—1181no—
471#95—1181no—
472#95—1181no—
473#95C22⋊D5911yescharacteristic
474#95C22⋊D5911yescharacteristic
475C24—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—592no—
488C2×Q8—1181no—
489C2×Q8—1181no—
490C2×Q8—1181no—
491C2×Q8—1181no—
492C2×Q8—1181no—
493C2×Q8#1174568712yes—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—1181no—
504C22×C4—1181no—
505C22×C4—1181no—
506C22×C4—1181no—
507C22×C4C2×C22⋊D5911yescharacteristic
508C22×C4C2×C22⋊D5911yescharacteristic
509C22×C4C2×C22⋊D5911yescharacteristic
510C22×C4C2×C22⋊D5911yescharacteristic
511C22×C4C2×C22⋊D5911yescharacteristic
512C22×C4C2×C22⋊D5911yescharacteristic
513C22×C4C23×D5911yescharacteristic
514Q16—1188no—
515Q16—2364no—
516SD16—2364no—
517C2×C8—592no—
518C2×C8—1181no—
519C2×C8—1181no—
520C2×C8—1181no—
521C2×C8—1181no—
522C2×C8—1181no—
523C2×C8—1181no—
524C2×C8—1181no—
525C2×C8—1181no—
526C2×C8—1181no—
527C2×C8—1181no—
528C2×C8—1181no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—41no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—592no—
537C4⋊C4—1184no—
538C4⋊C4—1181no—
539C4⋊C4—1181no—
540C4⋊C4—1181no—
541C4⋊C4—1181no—
542C4⋊C4—1181no—
543C4⋊C4—4721no—
544C4⋊C4—2362no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—42no—
556C42—1184no—
557C42—4721no—
558C42—4721no—
559C42D4×D5912yes—
560C42#1174568712yes—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23C22×C22⋊D5911yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—41no—
572Q8—41no—
573Q8—41no—
574Q8—41no—
575Q8—41no—
576Q8—41no—
577Q8—1182no—
578Q8—1182no—
579Q8—2361no—
580Q8—2361no—
581D4—41no—
582D4—41no—
583D4—41no—
584D4—41no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
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—1182no—
610C2×C4—1182no—
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#1427369812yes—
622C2×C4#1427369812yes—
623C2×C4#1427375712yes—
624C2×C4C2×D4×D5911yescharacteristic
625C2×C4#41×D5911yescharacteristic
626C2×C4C2×#1174568711yescharacteristic
627C2×C4C2×#1174568711yescharacteristic
628C2×C4#1427377211yescharacteristic
629C2×C4#1427377311yescharacteristic
630C8—1182no—
631C8—2361no—
632C8—2361no—
633C8—2361no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22C2×#1427369811yescharacteristic
642C22#1683804611yescharacteristic
643C22#1683807111yescharacteristic
644C22C2×#1427375711yescharacteristic
645C22#1683820011yescharacteristic
646C22#1683821011yescharacteristic
647C22#1683821711yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—42no—
655C4—81no—
656C4—81no—
657C4—4721no—
658C4—4721no—
659C4—2362no—
660C4—2362no—
661C4#1683791211yescharacteristic
662C4#1683796811yescharacteristic
663C4#1683810411yescharacteristic
664C4#1683811611yescharacteristic
665C2—161no—
666C2#2142751011yescharacteristic
667C2#2143356411yescharacteristic
668C2#2143364211yescharacteristic
669C2#2143616211yescharacteristic
670C2#2143620711yescharacteristic
671C2#2143712711yescharacteristic
672C2#2143716211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2545C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2545D59
hypocenterC59#24641
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2545

Complements

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

NH ≅ G/Ntype
2. C2×#16837507C2semidirect
3. #21427051C2semidirect
4. #21425418C2semidirect
6. #21425196C2semidirect
9. #21425020C2semidirect
11. #21424820C2semidirect
12. #21424727C2semidirect
14. #21424101C2semidirect
397. C59#24641semidirect

Maximal subgroups

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

subgroupindex
C2×#168376212
C2×#168375072
#214270512
#214254182
#214254132
#214251962
#214251902
#214250292
#214250202
#214249132
#214248202
#214247272
#214247202
#214241012
C59×#25452
#2464159