Finite Group Databasebeta

← #28606440

Subgroups — #28606440

Order 15104.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837444C211yesmaximal, characteristic
2#21426991C211yesmaximal, characteristic
3#21426967C211yesmaximal, characteristic
4#21424832C211yesmaximal, characteristic, complemented
5#21424827C211yesmaximal, characteristic, complemented
6#21424687C211yesmaximal, characteristic, complemented
7#21424680C211yesmaximal, characteristic, complemented
8#21424673C211yesmaximal, characteristic, complemented
9#21424666C211yesmaximal, characteristic, complemented
10#21424188C211yesmaximal, characteristic
11#21424184C211yesmaximal, characteristic
12#21423995C211yesmaximal, characteristic, complemented
13#106×Dic59C211yesmaximal, characteristic, complemented
14#21423877C211yesmaximal, characteristic
15C59×#1809C211yesmaximal, characteristic
16C2×#14273686C2211yescharacteristic
17C2×#14273683—21no—
18C22×C4⋊Dic59C2211yescharacteristic
19#16837444C418yescomplemented
20C2×#14273661C2211yescharacteristic
21C2×#14273661C2211yescharacteristic
22C2×#14273659C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×C22⋊(C59⋊C8)C2211yescharacteristic
25C2×#14273637C2211yescharacteristic
26C2×C8×Dic59C2211yescharacteristic
27#16837298C2211yescharacteristic
28#16837298C2211yescharacteristic
29#16837298C2211yescharacteristic
30#16837298C2211yescharacteristic
31#16837290—21no—
32#16837290—21no—
33#16837285—21no—
34#16837285—21no—
35#16837269—21no—
36#16837263—21no—
37C4⋊C4×Dic59C2211yescharacteristic
38C4⋊C4×Dic59C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C4⋊C4×Dic59C2211yescharacteristic
41#16837253—21no—
42C4×C22⋊Dic59C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C22×M16×C59C2211yescharacteristic
54C2×#107×C59C2211yescharacteristic
55C2×#106×C59C2211yescharacteristic
56C59×#380C2211yescharacteristic
57C59×#380C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C23×Dic59—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63#14273686—22no—
64#14273686C2×C414yes—
65#14273683—44no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59—22no—
68C2×C4⋊Dic59C2×C414yes—
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59D411yescharacteristic
79C2×C22⋊Dic59D411yescharacteristic
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59C2311yescharacteristic
86C2×C4×Dic59C2311yescharacteristic
87#14273661C2×C414yes—
88#14273661C2×C414yes—
89#14273659C2×C414yes—
90#14273659C2×C414yes—
91C22⋊(C59⋊C8)—22no—
92#14273637—22no—
93C8×Dic59—22no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95D411yescharacteristic
103C59⋊#95D411yescharacteristic
104C23×C4×C59C2311yescharacteristic
105C2×M16×C59—22no—
106C2×M16×C59—22no—
107C2×M16×C59C2×C414yes—
108C22×C8×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116#107×C59—22no—
117#106×C59—22no—
118#95×C59—21no—
119#95×C59—21no—
120C24×C59C2×D411yescharacteristic
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59C22×C412yes—
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59#4112yes—
132C22×C4×C59C2411yescharacteristic
133C22×Dic59—24no—
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—42no—
139C22×Dic59—42no—
140C22×Dic59C2×D411yescharacteristic
141C22×Dic59C2×D411yescharacteristic
142C22×Dic59C2×D411yescharacteristic
143C22×Dic59C2×D411yescharacteristic
144M16×C59—28no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59C22×C412yes—
148C2×C8×C59C22×C412yes—
149C2×C8×C59#4112yes—
150C2×C8×C59#4112yes—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C22⋊C4×C59—42no—
160C22⋊C4×C59—81no—
161C22⋊Dic59—24no—
162C22⋊Dic59—22no—
163C22⋊Dic59—41no—
164C4⋊Dic59—24no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59C22×C412yes—
170C4⋊Dic59C22×C412yes—
171C4⋊Dic59C22×C412yes—
172C4⋊Dic59C22×C412yes—
173Dic59⋊C4—42no—
174Dic59⋊C4—42no—
175Dic59⋊C4—42no—
176Dic59⋊C4—42no—
177C4×Dic59—24no—
178C4×Dic59—21no—
179C4×Dic59—21no—
180C4×Dic59—21no—
181C4×Dic59—21no—
182C4×Dic59—42no—
183C4×Dic59—42no—
184C4×Dic59—81no—
185C4×Dic59—81no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C23×C59—21no—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59C4×D412yes—
199C23×C59C22×D411yescharacteristic
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—22no—
203C2×C4×C59—22no—
204C2×C4×C59—22no—
205C2×C4×C59—22no—
206C2×C4×C59—41no—
207C2×C4×C59—41no—
208C2×C4×C59—41no—
209C2×C4×C59—41no—
210C2×C4×C59—41no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
213C2×C4×C59—41no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59C4×D412yes—
219C2×C4×C59C23×C411yescharacteristic
220C2×C4×C59C2×#4111yescharacteristic
221C2×C4×C59C2×#4111yescharacteristic
222C2×C4×C59C2×#4111yescharacteristic
223C2×C4×C592−1+411yescharacteristic
224C2×C4×C592−1+411yescharacteristic
225C2×Dic59—21no—
226C2×Dic59—21no—
227C2×Dic59—22no—
228C2×Dic59—22no—
229C2×Dic59—22no—
230C2×Dic59—22no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—48no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59AGL(1,ℤ/8)12yes—
250C8×C59—22no—
251C8×C59#11714yescomplemented
252C59⋊C8—41no—
253C59⋊C8—41no—
254#22622—591nomaximal, Sylow
255C22×C59—22no—
256C22×C59—22no—
257C22×C59—22no—
258C22×C59—41no—
259C22×C59—41no—
260C22×C59—41no—
261C22×C59—41no—
262C22×C59C2×C4×D411yescharacteristic
263C22×C59#54711yescharacteristic
264C22×C59#54711yescharacteristic
265C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C59#57511yescharacteristic
267C22×C59#57711yescharacteristic
268C22×C59#57811yescharacteristic
269C4×C59—22no—
270C4×C59—81no—
271C4×C59—81no—
272C4×C59—81no—
273C4×C59—81no—
274C4×C59C2×#11711yescharacteristic
275C4×C59#51911yescharacteristic
276C4×C59#55211yescharacteristic
277C4×C59#55211yescharacteristic
278Dic59—22no—
279Dic59—41no—
280Dic59—41no—
281Dic59—41no—
282Dic59—42no—
283Dic59—42no—
284Dic59—42no—
285Dic59—42no—
286C2×#428—591no—
287#2347—591no—
288#2344—591no—
289#1943—591no—
290#1942—591no—
291#1921—591no—
292#1920—591no—
293#1919—591no—
294#1918—591no—
295#1825—591no—
296#1824—591no—
297#1809D5911yescharacteristic
298#1776—591no—
299C4×#106—591no—
300#1753—591no—
301C2×C59—42no—
302C2×C59#234411yescharacteristic
303C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
304C2×C59#294711yescharacteristic
305C2×C59#334511yescharacteristic
306C2×C59#334911yescharacteristic
307C2×C59#335111yescharacteristic
308C2×C59#335311yescharacteristic
309C22×M16C2×D5911yescharacteristic
310C2×#117—591no—
311C2×#117—1181no—
312C22×C4⋊C4—591no—
313#428—598no—
314C2×#107—591no—
315C2×#107—591no—
316C2×#107C2×D5911yescharacteristic
317C2×#106—591no—
318C2×#106—591no—
319C2×#106C2×D5911yescharacteristic
320C2×C22⋊C8—591no—
321C2×#97—591no—
322C2×C4×C8—591no—
323#389—591no—
324#389—591no—
325#389—591no—
326#389—591no—
327#388—1181no—
328#388—1181no—
329#387—1181no—
330#387—1181no—
331#383—1181no—
332#381—1181no—
333#380C2×D5911yescharacteristic
334#380C2×D5911yescharacteristic
335C4×C4⋊C4—591no—
336C4×C4⋊C4—591no—
337C4×C4⋊C4—591no—
338C4×C4⋊C4—591no—
339C4×C4⋊C4—1181no—
340C4×C22⋊C4—591no—
341#340—591no—
342#340—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340C2×D5911yescharacteristic
350#340C2×D5911yescharacteristic
351#336—591no—
352#336—591no—
353C59#2262211yescharacteristic, complemented, Sylow
354C23×C4—1181no—
355C23×C4C22×D5911yescharacteristic
356C2×M16—22no—
357C2×M16—22no—
358C2×M16C4×D5914yes—
359C22×C8—591no—
360C22×C8—591no—
361C22×C8C22×D5911yescharacteristic
362C22×C8C22×D5911yescharacteristic
363#117—594no—
364#117—1182no—
365#117—2364no—
366C2×C4⋊C4—594no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4—1182no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4C22×D5911yescharacteristic
378C2×C4⋊C4C22×D5911yescharacteristic
379C2×C4⋊C4C22×D5911yescharacteristic
380C2×C4⋊C4C22×D5911yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—591no—
384C2×C22⋊C4—591no—
385C2×C22⋊C4—1181no—
386C2×C42—591no—
387C2×C42—591no—
388C2×C42—1181no—
389C2×C42—1181no—
390C2×C42—1181no—
391C2×C42—1181no—
392C2×C42—1181no—
393#107—22no—
394#107—594no—
395#107—594no—
396#106—22no—
397#106—594no—
398#106—594no—
399C22⋊C8—1182no—
400#97—1182no—
401C4×C8—1182no—
402#95—21no—
403#95—21no—
404#95—591no—
405#95—591no—
406#95—1181no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410#95—1181no—
411#95—1181no—
412#95—1181no—
413#95—1181no—
414C24D4×D5911yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—591no—
424C22×C4—591no—
425C22×C4—591no—
426C22×C4—591no—
427C22×C4—1184no—
428C22×C4—1181no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4—1181no—
432C22×C4—2362no—
433C22×C4—2362no—
434C22×C4C2×C4×D5912yes—
435C22×C4D4×D5911yescharacteristic
436C22×C4#1174568912yes—
437C22×C4C23×D5911yescharacteristic
438M16—28no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—1181no—
442C2×C8—1181no—
443C2×C8—1181no—
444C2×C8—1181no—
445C2×C8—1181no—
446C2×C8—1181no—
447C2×C8C2×C4×D5912yes—
448C2×C8C2×C4×D5912yes—
449C2×C8#1174568712yes—
450C2×C8#1174568712yes—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—592no—
460C4⋊C4—592no—
461C4⋊C4—592no—
462C4⋊C4—592no—
463C4⋊C4—1184no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—2362no—
469C4⋊C4—2362no—
470C4⋊C4—2362no—
471C4⋊C4—2362no—
472C22⋊C4—42no—
473C22⋊C4—81no—
474C22⋊C4—1182no—
475C22⋊C4—1184no—
476C22⋊C4—2361no—
477C42—1184no—
478C42—1181no—
479C42—1181no—
480C42—1181no—
481C42—1181no—
482C42—4721no—
483C42—4721no—
484C42—2362no—
485C42—2362no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23#1427368912yes—
493C23C2×D4×D5911yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—592no—
513C2×C4—1181no—
514C2×C4—1181no—
515C2×C4—1181no—
516C2×C4—1181no—
517C2×C4—1181no—
518C2×C4—1181no—
519C2×C4—1181no—
520C2×C4—1181no—
521C2×C4—1181no—
522C2×C4—1181no—
523C2×C4—1181no—
524C2×C4—1182no—
525C2×C4—1182no—
526C2×C4—1182no—
527C2×C4—1182no—
528C2×C4—2368no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4#1427368912yes—
538C2×C4C22×C4×D5911yescharacteristic
539C2×C4C2×#1174568711yescharacteristic
540C2×C4C2×#1174568711yescharacteristic
541C2×C4C2×#1174568911yescharacteristic
542C2×C4#1427377411yescharacteristic
543C2×C4#1427377411yescharacteristic
544C8—22no—
545C8—2361no—
546C8—2361no—
547C8#1427368314yescomplemented
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×#1427368911yescharacteristic
556C22#1683805211yescharacteristic
557C22#1683805211yescharacteristic
558C22AGL(1,ℤ/8)×D5911yescharacteristic
559C22#1683820111yescharacteristic
560C22#1683821211yescharacteristic
561C22#1683821811yescharacteristic
562C4—22no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—1182no—
568C4—2361no—
569C4—2362no—
570C4—2362no—
571C4—2362no—
572C4—2362no—
573C4—2361no—
574C4—2361no—
575C4C2×#1427368311yescharacteristic
576C4#1683787911yescharacteristic
577C4#1683809411yescharacteristic
578C4#1683809411yescharacteristic
579C2—42no—
580C2#2142696711yescharacteristic
581C2#2143290311yescharacteristic
582C2#2143293711yescharacteristic
583C2#2143656111yescharacteristic
584C2#2143662111yescharacteristic
585C2#2143666211yescharacteristic
586C2#2143667511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1809C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#1809D59
hypocenterC59#22622
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
4. #21424832C2semidirect
5. #21424827C2semidirect
6. #21424687C2semidirect
7. #21424680C2semidirect
8. #21424673C2semidirect
9. #21424666C2semidirect
12. #21423995C2semidirect
13. #106×Dic59C2semidirect
19. #16837444C4semidirect
251. C8×C59#117semidirect
353. C59#22622semidirect
547. C8#14273683semidirect

Maximal subgroups

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

subgroupindex
C2×#168374442
#214269912
#214269672
#214248322
#214248272
#214246872
#214246802
#214246732
#214246662
#214241882
#214241842
#214239952
#106×Dic592
#214238772
C59×#18092
#2262259