Finite Group Databasebeta

← #28634291

Subgroups — #28634291

Order 15104.

812 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×#16837743C211yesmaximal, characteristic, complemented
2C2×#16837743C211yesmaximal, characteristic, complemented
3#21429005C211yesmaximal, characteristic
4#21425532C211yesmaximal, characteristic, complemented
5#21425527C211yesmaximal, characteristic, complemented
6#21425299C211yesmaximal, characteristic, complemented
7#21425292C211yesmaximal, characteristic, complemented
8#21424895C211yesmaximal, characteristic
9#21424864C211yesmaximal, characteristic, complemented
10#21424295C211yesmaximal, characteristic, complemented
11#21424295C211yesmaximal, characteristic, complemented
12#21424262C211yesmaximal, characteristic, complemented
13#21424098C211yesmaximal, characteristic, complemented
14#21423929C211yesmaximal, characteristic, complemented
15C59×#2354C211yesmaximal, characteristic
16C2×#14273706C2211yescharacteristic
17C2×#14273706C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19#16837743—24no—
20#16837743—24no—
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273650C2211yescharacteristic
31C2×#14273650C2211yescharacteristic
32C2×C22⋊(C59⋊C8)C2211yescharacteristic
33C2×C22⋊(C59⋊C8)C2211yescharacteristic
34C2×C22⋊(C59⋊C8)C2211yescharacteristic
35C2×C22⋊(C59⋊C8)C2211yescharacteristic
36C2×#14273638C2211yescharacteristic
37#16837333—21no—
38#16837316—21no—
39#16837290—21no—
40#16837290—21no—
41#16837268C2211yescharacteristic
42#16837262C2211yescharacteristic
43#16837262C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50C2×C4×D4×C59C2211yescharacteristic
51C2×#117×C59C2211yescharacteristic
52C2×#117×C59C2211yescharacteristic
53C59×#390—21no—
54C59×#388—21no—
55C59×#388—21no—
56C59×#384C2211yescharacteristic
57C59×#380C2211yescharacteristic
58C59×#380C2211yescharacteristic
59C4×C4⋊C4×C59—21no—
60C4×C4⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66#14273706—22no—
67#14273706—22no—
68#14273706—41no—
69#14273706—41no—
70C4×Dic118—22no—
71C4×Dic118—22no—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C22⋊Dic59—21no—
82C2×C4×Dic59—21no—
83#14273660—22no—
84#14273660—22no—
85#14273660—41no—
86#14273658—22no—
87#14273658—22no—
88#14273658—41no—
89C4⋊(C59⋊C8)—22no—
90#14273650—22no—
91#14273650—22no—
92#14273650—22no—
93#14273650D414yescomplemented
94C22⋊(C59⋊C8)—22no—
95C22⋊(C59⋊C8)—22no—
96C22⋊(C59⋊C8)—22no—
97C22⋊(C59⋊C8)—22no—
98#14273638—22no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×D4×C59D411yescharacteristic
104C23×C4×C59C2311yescharacteristic
105C23×C4×C59C2311yescharacteristic
106C4×D4×C59—22no—
107C4×D4×C59—22no—
108#117×C59—22no—
109#117×C59—22no—
110#117×C59—41no—
111#117×C59—41no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59D411yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59—21no—
121C2×C22⋊C4×C59—21no—
122C2×C22⋊C4×C59—21no—
123C2×C22⋊C4×C59D411yescharacteristic
124C2×C22⋊C4×C59D411yescharacteristic
125C2×C42×C59—21no—
126C2×C42×C59—21no—
127C2×C42×C59—21no—
128C2×C42×C59C2311yescharacteristic
129#95×C59—21no—
130#95×C59—21no—
131#95×C59—21no—
132C24×C59C2×D411yescharacteristic
133C24×C59C2×D411yescharacteristic
134C2×D4×C59—22no—
135C2×D4×C59—41no—
136C2×D4×C59—41no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
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×C59—21no—
151C22×C4×C59—21no—
152C22×C4×C59—21no—
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2411yescharacteristic
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118C2×D412yes—
168C4⋊C4×C59—22no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—42no—
177C4⋊C4×C59—81no—
178C4⋊C4×C59C2×D412yes—
179C22⋊C4×C59—22no—
180C22⋊C4×C59—22no—
181C22⋊C4×C59—42no—
182C22⋊C4×C59—42no—
183C22⋊C4×C59—81no—
184C22⋊C4×C59—81no—
185C42×C59—42no—
186C42×C59—42no—
187C42×C59—42no—
188C42×C59#4112yes—
189C42×C59#4112yes—
190C22⋊Dic59—42no—
191C22⋊Dic59—42no—
192C22⋊Dic59—42no—
193C22⋊Dic59—42no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200Dic59⋊C4—42no—
201Dic59⋊C4—42no—
202Dic59⋊C4—42no—
203C4×Dic59—42no—
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⋊C8C2×D412yes—
215C2×C59⋊C8#4112yes—
216C23×C59—21no—
217C23×C59—21no—
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×C59C22×D411yescharacteristic
231D4×C59—44no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
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—22no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
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—41no—
261C2×C4×C59—41no—
262C2×C4×C59#12312yes—
263C2×C4×C59#12312yes—
264C2×C4×C59C22×D411yescharacteristic
265C2×C4×C59C2×#4111yescharacteristic
266C2×C4×C59C2×#4111yescharacteristic
267C2×C4×C59C2×#4111yescharacteristic
268C2×C4×C59H(5,𝔽2)11yescharacteristic
269C2×C4×C592−1+411yescharacteristic
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—
280C2×Dic59—41no—
281C2×Dic59—41no—
282Dic118—22no—
283Dic118—22no—
284Dic118—41no—
285Dic118—41no—
286C59⋊C8—22no—
287C59⋊C8—22no—
288C59⋊C8—41no—
289C59⋊C8—41no—
290#25025—591nomaximal, Sylow
291C22×C59—41no—
292C22×C59—41no—
293C22×C59—41no—
294C22×C59—41no—
295C22×C59—41no—
296C22×C59—41no—
297C22×C59—41no—
298C22×C59—41no—
299C22×C59—41no—
300C22×C59—41no—
301C22×C59—41no—
302C22×C59—41no—
303C22×C59—41no—
304C22×C59—41no—
305C22×C59C2×#12311yescharacteristic
306C22×C59D4211yescharacteristic
307C22×C59#54711yescharacteristic
308C22×C59#57511yescharacteristic
309C22×C59#57511yescharacteristic
310C22×C59#57711yescharacteristic
311C22×C59#57811yescharacteristic
312C4×C59—22no—
313C4×C59—22no—
314C4×C59—41no—
315C4×C59—41no—
316C4×C59—42no—
317C4×C59—81no—
318C4×C59—81no—
319C4×C59—81no—
320C4×C59#53211yescharacteristic
321C4×C59#54211yescharacteristic
322C4×C59#54711yescharacteristic
323C4×C59#55311yescharacteristic
324Dic59—42no—
325Dic59—81no—
326Dic59—81no—
327Dic59—81no—
328C2×#485—591no—
329C2×#485—591no—
330#2536—591no—
331#2354D5911yescharacteristic
332#2077—591no—
333#2076—591no—
334#2036—591no—
335#2035—591no—
336#1955—591no—
337#1949—591no—
338#1854—591no—
339#1854—591no—
340#1847—591no—
341#1803—591no—
342#1763—591no—
343C2×C59—81no—
344C2×C59—81no—
345C2×C59#242211yescharacteristic
346C2×C59#309311yescharacteristic
347C2×C59#309711yescharacteristic
348C2×C59#326911yescharacteristic
349C2×C59#327111yescharacteristic
350C2×C59#334811yescharacteristic
351C2×C59#334911yescharacteristic
352C2×C22⋊Q8—591no—
353C2×C22⋊Q8—591no—
354C2×C4×Q8—591no—
355C2×C4×D4C2×D5911yescharacteristic
356C2×#117C2×D5911yescharacteristic
357C2×#117C2×D5911yescharacteristic
358#485—1184no—
359#485—1184no—
360C2×#107—591no—
361C2×#107—591no—
362C2×#107—591no—
363C2×#106—591no—
364C2×#106—591no—
365C2×#106—591no—
366C2×C4⋊C8—591no—
367C2×#103—591no—
368C2×#103—591no—
369C2×#103—591no—
370C2×#103—591no—
371C2×C22⋊C8—591no—
372C2×C22⋊C8—591no—
373C2×C22⋊C8—591no—
374C2×C22⋊C8—591no—
375C2×#97—591no—
376#398—1181no—
377#394—1181no—
378#390—21no—
379#388—21no—
380#388—21no—
381#388—1181no—
382#388—1181no—
383#384C2×D5911yescharacteristic
384#382—591no—
385#380—591no—
386#380—591no—
387#380C2×D5911yescharacteristic
388#380C2×D5911yescharacteristic
389C4×C4⋊C4—21no—
390C4×C4⋊C4C2×D5911yescharacteristic
391#340—591no—
392#340—591no—
393#340—591no—
394#340—591no—
395#340—591no—
396#340—591no—
397C59#2502511yescharacteristic, complemented, Sylow
398C22×Q8—591no—
399C22×D4C22⋊D5911yescharacteristic
400C23×C4C22×D5911yescharacteristic
401C23×C4C22×D5911yescharacteristic
402C22×C8—591no—
403C22×C8—591no—
404C22×C8—591no—
405C22×C8—591no—
406C22⋊Q8—1182no—
407C22⋊Q8—1182no—
408C22⋊Q8—2361no—
409C22⋊Q8—2361no—
410C4×Q8—1182no—
411C4×Q8—1182no—
412C4×D4—22no—
413C4×D4—22no—
414#117—22no—
415#117—22no—
416#117—41no—
417#117—41no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—591no—
421C2×C4⋊C4—591no—
422C2×C4⋊C4—591no—
423C2×C4⋊C4—1181no—
424C2×C4⋊C4—1181no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4C22⋊D5911yescharacteristic
427C2×C4⋊C4C22×D5911yescharacteristic
428C2×C4⋊C4C22×D5911yescharacteristic
429C2×C4⋊C4C22×D5911yescharacteristic
430C2×C4⋊C4C22×D5911yescharacteristic
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—1181no—
436C2×C22⋊C4—1181no—
437C2×C22⋊C4—1181no—
438C2×C22⋊C4—1181no—
439C2×C22⋊C4C22⋊D5911yescharacteristic
440C2×C22⋊C4C22⋊D5911yescharacteristic
441C2×C42—21no—
442C2×C42—21no—
443C2×C42—21no—
444C2×C42—1181no—
445C2×C42C22×D5911yescharacteristic
446#107—1182no—
447#107—1182no—
448#107—2361no—
449#106—1182no—
450#106—1182no—
451#106—2361no—
452C4⋊C8—1182no—
453#103—594no—
454#103—1182no—
455#103—1182no—
456#103—1182no—
457C22⋊C8—1182no—
458C22⋊C8—1182no—
459C22⋊C8—1182no—
460C22⋊C8—1182no—
461#97—1182no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—1181no—
466#95—1181no—
467#95—1181no—
468#95—1181no—
469C24C2×C22⋊D5911yescharacteristic
470C24C2×C22⋊D5911yescharacteristic
471C2×Q8—592no—
472C2×Q8—1181no—
473C2×Q8—1181no—
474C2×Q8—1181no—
475C2×Q8—1181no—
476C2×Q8—1181no—
477C2×D4—22no—
478C2×D4—41no—
479C2×D4—41no—
480C22×C4—21no—
481C22×C4—21no—
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—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—1181no—
497C22×C4—1181no—
498C22×C4—1181no—
499C22×C4—1181no—
500C22×C4C2×C22⋊D5911yescharacteristic
501C22×C4C2×C22⋊D5911yescharacteristic
502C22×C4C2×C22⋊D5911yescharacteristic
503C22×C4C2×C22⋊D5911yescharacteristic
504C22×C4C23×D5911yescharacteristic
505C2×C8—592no—
506C2×C8—592no—
507C2×C8—1181no—
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—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—42no—
526C4⋊C4—81no—
527C4⋊C4—1181no—
528C4⋊C4—1181no—
529C4⋊C4—1181no—
530C4⋊C4—1181no—
531C4⋊C4—1181no—
532C4⋊C4—1181no—
533C4⋊C4—2362no—
534C4⋊C4—2362no—
535C4⋊C4—2362no—
536C4⋊C4D4×D5912yes—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—81no—
542C22⋊C4—81no—
543C22⋊C4—2362no—
544C22⋊C4—2362no—
545C22⋊C4—2362no—
546C22⋊C4—2362no—
547C42—42no—
548C42—42no—
549C42—42no—
550C42—2362no—
551C42#1174568512yes—
552C42#1174568512yes—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23C22×C22⋊D5911yescharacteristic
568Q8—1182no—
569Q8—1182no—
570Q8—2361no—
571Q8—2361no—
572D4—44no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
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—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—1182no—
604C2×C4—2361no—
605C2×C4—2361no—
606C2×C4—2361no—
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#1427371012yes—
616C2×C4#1427371012yes—
617C2×C4C2×D4×D5911yescharacteristic
618C2×C4#41×D5911yescharacteristic
619C2×C4C2×#1174568511yescharacteristic
620C2×C4C2×#1174568511yescharacteristic
621C2×C4#1427377211yescharacteristic
622C2×C4#1427377411yescharacteristic
623C8—1182no—
624C8—1182no—
625C8—2361no—
626C8—2361no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22C2×#1427371011yescharacteristic
642C22D4×C22⋊D5911yescharacteristic
643C22#1683805111yescharacteristic
644C22#1683820011yescharacteristic
645C22#1683820011yescharacteristic
646C22#1683821311yescharacteristic
647C22#1683821711yescharacteristic
648C4—22no—
649C4—22no—
650C4—41no—
651C4—41no—
652C4—42no—
653C4—81no—
654C4—81no—
655C4—81no—
656C4—4721no—
657C4—4721no—
658C4—4721no—
659C4—2362no—
660C4#1683794911yescharacteristic
661C4#1683801711yescharacteristic
662C4#1683805711yescharacteristic
663C4#1683810211yescharacteristic
664C2—81no—
665C2—81no—
666C2#2142760411yescharacteristic
667C2#2143402711yescharacteristic
668C2#2143406511yescharacteristic
669C2#2143564711yescharacteristic
670C2#2143567711yescharacteristic
671C2#2143661111yescharacteristic
672C2#2143662011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2354C2
socleC23×C59C22×D4
prime radicalC22×D4×C59D4
hypercenter#2354D59
hypocenterC59#25025
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2354

Complements

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

NH ≅ G/Ntype
1. C2×#16837743C2semidirect
2. C2×#16837743C2semidirect
4. #21425532C2semidirect
5. #21425527C2semidirect
6. #21425299C2semidirect
7. #21425292C2semidirect
9. #21424864C2semidirect
10. #21424295C2semidirect
11. #21424295C2semidirect
12. #21424262C2semidirect
13. #21424098C2semidirect
14. #21423929C2semidirect
93. #14273650D4semidirect
397. C59#25025semidirect

Maximal subgroups

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

subgroupindex
C2×#168377432
#214290052
#214255322
#214255272
#214252992
#214252922
#214248952
#214248642
#214242952
#214242622
#214240982
#214239292
C59×#23542
#2502559