Finite Group Databasebeta

← #28624770

Subgroups — #28624770

Order 15104.

1150 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837959C211yesmaximal, characteristic, complemented
2C2×#16837692C211yesmaximal, characteristic, complemented
3C2×#16837682C211yesmaximal, characteristic, complemented
4#21425434C212yesmaximal, complemented
5#21425242C212yesmaximal, complemented
6#21425048C211yesmaximal, characteristic, complemented
7#21425044C211yesmaximal, characteristic, complemented
8#21424419C212yesmaximal
9#21424293C212yesmaximal, complemented
10#21424076C211yesmaximal, characteristic, complemented
11C4×#121×C59C211yesmaximal, characteristic
12#16837959—24no—
13#16837959—24no—
14C2×#14273727C2211yescharacteristic
15C2×#14273727C2211yescharacteristic
16C2×#14273719C2212yes—
17C2×#14273706—21no—
18C2×#14273706—21no—
19C2×C4×Dic118—21no—
20C2×C4×Dic118—21no—
21C2×#14273686C2212yes—
22#16837692—24no—
23#16837682—24no—
24C2×#14273660C2212yes—
25C2×#14273658C2212yes—
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273648C2212yes—
29C2×#14273648C2212yes—
30C2×C22⋊(C59⋊C8)C2212yes—
31#16837164C2212yes—
32#16837164C2212yes—
33#16837164C2212yes—
34#16837164C2212yes—
35#16837157C2212yes—
36C2×#121×C59C2211yescharacteristic
37C2×C4×D4×C59—21no—
38C2×C4×D4×C59C2211yescharacteristic
39C22×C42×C59C2211yescharacteristic
40C59×#390C2211yescharacteristic
41C59×#389C2211yescharacteristic
42C59×#388—21no—
43C59×#386—21no—
44C59×#380C2211yescharacteristic
45C4×C4⋊C4×C59C2211yescharacteristic
46C4×C22⋊C4×C59—21no—
47C22×Dic118—21no—
48C22×Dic118—21no—
49C22×C59⋊C8C2312yes—
50C22×C59⋊C8C2312yes—
51#14273727—22no—
52#14273727—22no—
53#14273727D414yescomplemented
54#14273727D414yescomplemented
55#14273719—24no—
56#14273719—24no—
57#14273706—28no—
58#14273706—28no—
59C4×Dic118—28no—
60C4×Dic118—28no—
61#14273686—24no—
62#14273686—24no—
63C2×C4⋊Dic59—21no—
64C2×C4⋊Dic59—21no—
65C2×C4⋊Dic59C2312yes—
66C2×C4⋊Dic59C2312yes—
67C2×Dic59⋊C4—22no—
68C2×Dic59⋊C4—22no—
69C2×C22⋊Dic59—22no—
70C2×C4×Dic59—22no—
71#14273660—24no—
72#14273658—24no—
73C4⋊(C59⋊C8)—22no—
74C4⋊(C59⋊C8)—22no—
75#14273648—24no—
76#14273648—24no—
77C22⋊(C59⋊C8)—24no—
78C22×D4×C59—21no—
79C22×D4×C59C2311yescharacteristic
80C23×C4×C59—21no—
81C23×C4×C59D411yescharacteristic
82C23×C4×C59D411yescharacteristic
83C23×C4×C59C2311yescharacteristic
84#121×C59—22no—
85#121×C59Q814yescomplemented
86C4×D4×C59—22no—
87C4×D4×C59—22no—
88C4×D4×C59—44no—
89C2×C4⋊C4×C59—21no—
90C2×C4⋊C4×C59C2311yescharacteristic
91C2×C4⋊C4×C59C2311yescharacteristic
92C2×C4⋊C4×C59C2311yescharacteristic
93C2×C22⋊C4×C59—21no—
94C2×C22⋊C4×C59—21no—
95C2×C22⋊C4×C59—21no—
96C2×C22⋊C4×C59—21no—
97C2×C22⋊C4×C59—21no—
98C2×C42×C59—22no—
99C2×C42×C59—22no—
100C2×C42×C59—21no—
101C2×C42×C59D411yescharacteristic
102C2×C42×C59D411yescharacteristic
103C2×C42×C59C2311yescharacteristic
104C2×C42×C59C2311yescharacteristic
105#95×C59—21no—
106#95×C59—21no—
107#95×C59—21no—
108C24×C59—21no—
109C24×C59C2×D411yescharacteristic
110C2×D4×C59—24no—
111C2×D4×C59—21no—
112C2×D4×C59—21no—
113C2×D4×C59—21no—
114C2×D4×C59—21no—
115C2×D4×C59—21no—
116C2×D4×C59—42no—
117C2×D4×C59—42no—
118C2×D4×C59C2×Q812yes—
119C22×C4×C59—21no—
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—22no—
127C22×C4×C59—22no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—41no—
131C22×C4×C59—41no—
132C22×C4×C59—42no—
133C22×C4×C59—42no—
134C22×C4×C59—42no—
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×Q812yes—
141C22×C4×C59#4112yes—
142C22×C4×C59C2411yescharacteristic
143C22×Dic59—22no—
144C22×Dic59—22no—
145C2×Dic118—24no—
146C2×Dic118—24no—
147C2×Dic118—42no—
148C2×Dic118—42no—
149C2×Dic118—42no—
150C2×Dic118—42no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—42no—
157C4⋊C4×C59C2×Q812yes—
158C22⋊C4×C59—24no—
159C22⋊C4×C59—42no—
160C22⋊C4×C59—42no—
161C22⋊C4×C59—42no—
162C22⋊C4×C59—42no—
163C42×C59—22no—
164C42×C59—22no—
165C42×C59—21no—
166C42×C59—21no—
167C42×C59—42no—
168C42×C59C2×D412yes—
169C42×C59C2×D412yes—
170C22⋊Dic59—28no—
171C4⋊Dic59—24no—
172C4⋊Dic59—24no—
173C4⋊Dic59—22no—
174C4⋊Dic59—22no—
175C4⋊Dic59C2×D414yes—
176C4⋊Dic59C2×D414yes—
177Dic59⋊C4—28no—
178Dic59⋊C4—28no—
179C4×Dic59—28no—
180C2×C59⋊C8—22no—
181C2×C59⋊C8—22no—
182C2×C59⋊C8—22no—
183C2×C59⋊C8—22no—
184C2×C59⋊C8—22no—
185C2×C59⋊C8—22no—
186C23×C59—21no—
187C23×C59—21no—
188C23×C59—21no—
189C23×C59—21no—
190C23×C59—21no—
191C23×C59—21no—
192C23×C59—22no—
193C23×C59—41no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59—41no—
199C23×C59C22⋊Q812yes—
200C23×C59C22×D411yescharacteristic
201D4×C59—22no—
202D4×C59—22no—
203D4×C59—48no—
204D4×C59—41no—
205D4×C59—41no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—22no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—22no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—44no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—44no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—84no—
232C2×C4×C59#12012yes—
233C2×C4×C59#12012yes—
234C2×C4×C59C22⋊Q812yes—
235C2×C4×C59#13512yes—
236C2×C4×C59AGL(1,ℤ/8)12yes—
237C2×C4×C59C22×D411yescharacteristic
238C2×C4×C59C22×D411yescharacteristic
239C2×C4×C59C22×Q811yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C592−1+411yescharacteristic
242C2×C4×C592−1+411yescharacteristic
243C2×Dic59—24no—
244C2×Dic59—24no—
245C2×Dic59—42no—
246C2×Dic59—42no—
247C2×Dic59—42no—
248C2×Dic59—42no—
249Dic118—48no—
250Dic118—48no—
251C59⋊C8—42no—
252C59⋊C8—42no—
253#24300—591nomaximal, Sylow
254C22×C59—22no—
255C22×C59—22no—
256C22×C59—22no—
257C22×C59—41no—
258C22×C59—41no—
259C22×C59—41no—
260C22×C59—41no—
261C22×C59—42no—
262C22×C59—42no—
263C22×C59—42no—
264C22×C59—81no—
265C22×C59—81no—
266C22×C59—81no—
267C22×C59—81no—
268C22×C59C2×#12011yescharacteristic
269C22×C59C2×C22⋊Q811yescharacteristic
270C22×C59#53611yescharacteristic
271C22×C59C2×#13511yescharacteristic
272C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C59#57511yescharacteristic
274C22×C59#57511yescharacteristic
275C4×C59—22no—
276C4×C59—22no—
277C4×C59—42no—
278C4×C59—41no—
279C4×C59—41no—
280C4×C59—81no—
281C4×C59—81no—
282C4×C59#44914yes—
283C4×C59#54811yescharacteristic
284C4×C59#54811yescharacteristic
285C4×C59D4×Q811yescharacteristic
286C4×C59D4×Q811yescharacteristic
287Dic59—44no—
288Dic59—44no—
289C2×#533—591no—
290C2×#476—591no—
291C2×#474—591no—
292C4×#121D5911yescharacteristic
293#2059—592no—
294#2025—592no—
295#1988—591no—
296#1987—591no—
297#1875—592no—
298#1854—592no—
299#1797—591no—
300C2×C59—42no—
301C2×C59—82no—
302C2×C59#240911yescharacteristic
303C2×C59C2×#44911yescharacteristic
304C2×C59#300511yescharacteristic
305C2×C59#307611yescharacteristic
306C2×C59#307711yescharacteristic
307C2×C59#323311yescharacteristic
308C2×C59#323511yescharacteristic
309#533—1184no—
310#533—1184no—
311C2×C4⋊Q8—591no—
312C2×C4⋊Q8—591no—
313C2×#125—592no—
314C2×C22⋊Q8—1181no—
315C2×C22⋊Q8—1181no—
316C2×#121C2×D5911yescharacteristic
317C2×C4×Q8—1181no—
318C2×C4×Q8—1181no—
319C2×C4×D4—21no—
320C2×C4×D4C2×D5911yescharacteristic
321C2×#117—592no—
322C22×C42C2×D5911yescharacteristic
323#476—1184no—
324#474—1184no—
325C2×#107—592no—
326C2×#106—592no—
327C2×C4⋊C8—591no—
328C2×C4⋊C8—591no—
329C2×#102—592no—
330C2×#102—592no—
331C2×C22⋊C8—592no—
332#390C2×D5911yescharacteristic
333#389C2×D5911yescharacteristic
334#388—21no—
335#386—21no—
336#380C2×D5911yescharacteristic
337C4×C4⋊C4C2×D5911yescharacteristic
338C4×C22⋊C4—21no—
339#340—592no—
340#340—592no—
341#340—592no—
342#340—592no—
343#336—592no—
344C59#2430011yescharacteristic, complemented, Sylow
345C22×Q8—1181no—
346C22×Q8—1181no—
347C22×D4—21no—
348C22×D4C22×D5911yescharacteristic
349C23×C4—21no—
350C23×C4C22⋊D5911yescharacteristic
351C23×C4C22⋊D5911yescharacteristic
352C23×C4C22×D5911yescharacteristic
353C22×C8—592no—
354C22×C8—592no—
355C4⋊Q8—594no—
356C4⋊Q8—594no—
357C4⋊Q8—1182no—
358C4⋊Q8—1182no—
359#125—1184no—
360#125—1184no—
361C22⋊Q8—1188no—
362C22⋊Q8—1188no—
363#121—22no—
364#121Dic11814yescomplemented
365C4×Q8—1188no—
366C4×Q8—1188no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—44no—
370#117—1184no—
371#117—1184no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—592no—
374C2×C4⋊C4—592no—
375C2×C4⋊C4—1182no—
376C2×C4⋊C4—1182no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4—1181no—
379C2×C4⋊C4C22×D5911yescharacteristic
380C2×C4⋊C4C22×D5911yescharacteristic
381C2×C4⋊C4C22×D5911yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1182no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—21no—
391C2×C42—1182no—
392C2×C42C22⋊D5911yescharacteristic
393C2×C42C22⋊D5911yescharacteristic
394C2×C42C22×D5911yescharacteristic
395C2×C42C22×D5911yescharacteristic
396#107—1184no—
397#106—1184no—
398C4⋊C8—1182no—
399C4⋊C8—1182no—
400#102—1184no—
401#102—1184no—
402C22⋊C8—1184no—
403#95—21no—
404#95—21no—
405#95—21no—
406C24—21no—
407C24C2×C22⋊D5911yescharacteristic
408C2×Q8—1184no—
409C2×Q8—1184no—
410C2×Q8—2362no—
411C2×Q8—2362no—
412C2×Q8—2362no—
413C2×Q8—2362no—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4C2×Dic11812yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—1182no—
440C22×C4—1182no—
441C22×C4C2×Dic11812yes—
442C22×C4#1174568512yes—
443C22×C4C2×C22⋊D5911yescharacteristic
444C22×C4C2×C22⋊D5911yescharacteristic
445C22×C4C2×C22⋊D5911yescharacteristic
446C22×C4C2×C22⋊D5911yescharacteristic
447C22×C4C2×C22⋊D5911yescharacteristic
448C22×C4C23×D5911yescharacteristic
449C2×C8—1182no—
450C2×C8—1182no—
451C2×C8—1182no—
452C2×C8—1182no—
453C2×C8—1182no—
454C2×C8—1182no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—594no—
462C4⋊C4—594no—
463C4⋊C4—1188no—
464C4⋊C4—1188no—
465C4⋊C4—1184no—
466C4⋊C4—1184no—
467C4⋊C4—1182no—
468C4⋊C4—1182no—
469C4⋊C4C2×Dic11812yes—
470C22⋊C4—24no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—1188no—
476C42—22no—
477C42—22no—
478C42—21no—
479C42—21no—
480C42—42no—
481C42—1188no—
482C42D4×D5912yes—
483C42D4×D5912yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23#1427370612yes—
498C23C22×C22⋊D5911yescharacteristic
499Q8—2368no—
500Q8—2368no—
501D4—22no—
502D4—22no—
503D4—48no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—44no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—84no—
532C2×C4—1184no—
533C2×C4—1184no—
534C2×C4—2362no—
535C2×C4—2362no—
536C2×C4—2362no—
537C2×C4—2362no—
538C2×C4#1427369512yes—
539C2×C4#1427369512yes—
540C2×C4#1427370612yes—
541C2×C4#1427374712yes—
542C2×C4#1427375312yes—
543C2×C4C2×D4×D5911yescharacteristic
544C2×C4C2×D4×D5911yescharacteristic
545C2×C4C22×Dic11811yescharacteristic
546C2×C4C2×#1174568511yescharacteristic
547C2×C4#1427377411yescharacteristic
548C2×C4#1427377411yescharacteristic
549C8—2362no—
550C8—2362no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22C2×#1427369511yescharacteristic
566C22C2×#1427370611yescharacteristic
567C22#1683797311yescharacteristic
568C22C2×#1427374711yescharacteristic
569C22C2×#1427375311yescharacteristic
570C22#1683820011yescharacteristic
571C22#1683820011yescharacteristic
572C4—22no—
573C4—22no—
574C4—42no—
575C4—41no—
576C4—41no—
577C4—81no—
578C4—81no—
579C4—2364no—
580C4—2364no—
581C4#1683754514yes—
582C4#1683806111yescharacteristic
583C4#1683806111yescharacteristic
584C4D4×Dic11811yescharacteristic
585C4D4×Dic11811yescharacteristic
586C2—42no—
587C2—82no—
588C2#2142746111yescharacteristic
589C2C2×#1683754511yescharacteristic
590C2#2143333611yescharacteristic
591C2#2143388411yescharacteristic
592C2#2143389311yescharacteristic
593C2#2143527111yescharacteristic
594C2#2143529011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC4×#121×C59C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenterC4×#121D59
hypocenterC59#24300
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16837959C2semidirect
2. C2×#16837692C2semidirect
3. C2×#16837682C2semidirect
4. #21425434C2semidirect
5. #21425242C2semidirect
6. #21425048C2semidirect
7. #21425044C2semidirect
9. #21424293C2semidirect
10. #21424076C2semidirect
53. #14273727D4semidirect
54. #14273727D4semidirect
85. #121×C59Q8semidirect
344. C59#24300semidirect
364. #121Dic118semidirect

Maximal subgroups

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

subgroupindex
C2×#168379592
C2×#168376922
C2×#168376822
#214254342
#214252422
#214250482
#214250442
#214244192
#214242932
#214240762
C4×#121×C592
#2430059