Finite Group Databasebeta

← #28647553

Subgroups — #28647553

Order 15104.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837810C211yesmaximal, characteristic, complemented
2C2×#16837445C211yesmaximal, characteristic
3#21427803C211yesmaximal, characteristic
4#21427799C211yesmaximal, characteristic
5#21425576C211yesmaximal, characteristic, complemented
6#21425570C211yesmaximal, characteristic, complemented
7#21425433C211yesmaximal, characteristic, complemented
8#21425427C211yesmaximal, characteristic, complemented
9#21424833C211yesmaximal, characteristic, complemented
10#21424828C211yesmaximal, characteristic, complemented
11#21424291C211yesmaximal, characteristic
12#21424251C211yesmaximal, characteristic, complemented
13#21424127C211yesmaximal, characteristic
14#21423882C211yesmaximal, characteristic
15C59×#1854C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C2×#14273726C2211yescharacteristic
18C2×#14273718C2211yescharacteristic
19C2×#14273704—21no—
20C22×C4⋊Dic59C2211yescharacteristic
21#16837810—24no—
22#16837445—24no—
23C2×#14273660C2211yescharacteristic
24C2×#14273660C2211yescharacteristic
25C2×#14273660C2211yescharacteristic
26C2×#14273659C2211yescharacteristic
27C2×#14273659C2211yescharacteristic
28C2×#14273658C2211yescharacteristic
29C2×#14273658C2211yescharacteristic
30C2×#14273658C2211yescharacteristic
31C2×#14273637C2211yescharacteristic
32C2×C8×Dic59C2211yescharacteristic
33#16837338—21no—
34#16837336—21no—
35#16837325—21no—
36#16837323—21no—
37#16837301C2211yescharacteristic
38#16837301C2211yescharacteristic
39#16837300—21no—
40#16837298C2211yescharacteristic
41#16837298C2211yescharacteristic
42#16837286—21no—
43#16837274—21no—
44#16837271C2211yescharacteristic
45#16837271C2211yescharacteristic
46#16837262C2211yescharacteristic
47C4×C22⋊Dic59C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837156C2211yescharacteristic
53#16837156C2211yescharacteristic
54C2×#117×C59C2211yescharacteristic
55C2×#107×C59C2211yescharacteristic
56C2×#106×C59C2211yescharacteristic
57C2×C22⋊C8×C59C2211yescharacteristic
58C59×#380C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C59×#340C2211yescharacteristic
61C22×Dic118—21no—
62C23×Dic59—21no—
63C2×#11745664—22no—
64C2×#11745664—22no—
65C2×#11745664Q814yescomplemented
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273726—22no—
69#14273726—41no—
70#14273718—22no—
71#14273718—41no—
72#14273704—44no—
73C2×C4⋊Dic59—22no—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59—21no—
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59—41no—
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C2×Dic59⋊C4—21no—
86C2×Dic59⋊C4—21no—
87C2×Dic59⋊C4—21no—
88C2×Dic59⋊C4—21no—
89C2×Dic59⋊C4—21no—
90C2×Dic59⋊C4—21no—
91C2×C22⋊Dic59—21no—
92C2×C22⋊Dic59D411yescharacteristic
93C2×C22⋊Dic59D411yescharacteristic
94C2×C4×Dic59C2311yescharacteristic
95C2×C4×Dic59C2311yescharacteristic
96#14273660—22no—
97#14273660—22no—
98#14273660D414yes—
99#14273659—22no—
100#14273659—41no—
101#14273658—22no—
102#14273658—22no—
103#14273658—22no—
104#14273637—22no—
105C8×Dic59—22no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C59⋊#95D411yescharacteristic
112C59⋊#95D411yescharacteristic
113C23×C4×C59C2311yescharacteristic
114C22×C8×C59C2311yescharacteristic
115C22×C8×C59C2311yescharacteristic
116#117×C59—22no—
117#117×C59—41no—
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C4⋊C4×C59C2311yescharacteristic
121C2×C4⋊C4×C59C2311yescharacteristic
122C2×C22⋊C4×C59—21no—
123C2×C22⋊C4×C59—21no—
124C2×C42×C59—21no—
125#107×C59—22no—
126#106×C59—41no—
127C22⋊C8×C59Q814yescomplemented
128#95×C59—21no—
129C24×C59C2×D411yescharacteristic
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×Q812yes—
140C22×C4×C59C2×Q812yes—
141C22×C4×C59C2411yescharacteristic
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—42no—
147C22×Dic59—42no—
148C22×Dic59—81no—
149C22×Dic59C2×D411yescharacteristic
150C22×Dic59C2×D411yescharacteristic
151C22×Dic59C2×D411yescharacteristic
152C22×Dic59C2×D411yescharacteristic
153C2×Dic118—42no—
154C2×Dic118—42no—
155C2×Dic118—81no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59C2×Q812yes—
161C2×C8×C59C2×Q812yes—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59C2×D412yes—
170C22⋊C4×C59—42no—
171C22⋊C4×C59—42no—
172C42×C59—42no—
173C22⋊Dic59—22no—
174C22⋊Dic59—22no—
175C22⋊Dic59—81no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—42no—
184C4⋊Dic59—42no—
185C4⋊Dic59—42no—
186C4⋊Dic59—42no—
187C4⋊Dic59C2×D412yes—
188Dic59⋊C4—42no—
189Dic59⋊C4—42no—
190Dic59⋊C4—42no—
191Dic59⋊C4—42no—
192Dic59⋊C4—42no—
193Dic59⋊C4—81no—
194Dic59⋊C4—81no—
195Dic59⋊C4—81no—
196C4×Dic59—21no—
197C4×Dic59—21no—
198C4×Dic59—21no—
199C4×Dic59—21no—
200#11745664—28no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8C2×D412yes—
204C2×C59⋊C8C2×Q812yes—
205C2×C59⋊C8C2×Q812yes—
206C2×C59⋊C8#4112yes—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59C4⋊Q812yes—
214C23×C59C22×D411yescharacteristic
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
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—41no—
232C2×C4×C59—41no—
233C2×C4×C59C4⋊Q812yes—
234C2×C4×C59C22×D411yescharacteristic
235C2×C4×C59C22×Q811yescharacteristic
236C2×C4×C59C22×Q811yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59H(5,𝔽2)11yescharacteristic
239C2×C4×C592−1+411yescharacteristic
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—22no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—84no—
263C2×Dic59AGL(1,ℤ/8)12yes—
264C2×Dic59#13712yes—
265Dic118—84no—
266C8×C59—22no—
267C8×C59—22no—
268C59⋊C8—22no—
269C59⋊C8C22⋊Q814yescomplemented
270#26078—591nomaximal, Sylow
271C22×C59—22no—
272C22×C59—22no—
273C22×C59—22no—
274C22×C59—41no—
275C22×C59—41no—
276C22×C59—41no—
277C22×C59—41no—
278C22×C59C2×C4⋊Q811yescharacteristic
279C22×C59D4211yescharacteristic
280C22×C59#54711yescharacteristic
281C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C59C2×#13711yescharacteristic
283C22×C59#57711yescharacteristic
284C22×C59#57711yescharacteristic
285C4×C59—22no—
286C4×C59—42no—
287C4×C59—81no—
288C4×C59—81no—
289C4×C59—81no—
290C4×C59C2×C22⋊Q811yescharacteristic
291C4×C59#54411yescharacteristic
292C4×C59D4×Q811yescharacteristic
293C4×C59#55411yescharacteristic
294Dic59—22no—
295Dic59—22no—
296Dic59—41no—
297Dic59—42no—
298Dic59—41no—
299Dic59—81no—
300Dic59—81no—
301Dic59—81no—
302C2×#501—591no—
303C2×#428—591no—
304#2441—591no—
305#2440—591no—
306#2084—591no—
307#2083—591no—
308#2059—591no—
309#2058—591no—
310#1943—591no—
311#1942—591no—
312#1854D5911yescharacteristic
313#1853—591no—
314#1844—591no—
315#1809—591no—
316#1753—591no—
317C2×C59—42no—
318C2×C59#244011yescharacteristic
319C2×C59#316311yescharacteristic
320C2×C59#316511yescharacteristic
321C2×C59#327211yescharacteristic
322C2×C59#327811yescharacteristic
323C2×C59#334411yescharacteristic
324C2×C59#334511yescharacteristic
325C22×M16—591no—
326C2×C4⋊Q8—591no—
327C2×#125—591no—
328C2×C22⋊Q8—1181no—
329C2×#117C2×D5911yescharacteristic
330C22×C4⋊C4—591no—
331#501—1184no—
332#428—1184no—
333C2×#107—591no—
334C2×#107—591no—
335C2×#107—591no—
336C2×#107C2×D5911yescharacteristic
337C2×#106—591no—
338C2×#106—591no—
339C2×#106—591no—
340C2×#106—591no—
341C2×#106—591no—
342C2×#106C2×D5911yescharacteristic
343C2×C22⋊C8C2×D5911yescharacteristic
344C2×#97—591no—
345C2×C4×C8—591no—
346#399—1181no—
347#398—1181no—
348#396—1181no—
349#395—1181no—
350#389—591no—
351#389—591no—
352#389—591no—
353#389—591no—
354#389—1181no—
355#387—1181no—
356#385—1181no—
357#384—591no—
358#384—591no—
359#380—591no—
360#380C2×D5911yescharacteristic
361C4×C22⋊C4—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340C2×D5911yescharacteristic
367#340C2×D5911yescharacteristic
368#336—591no—
369#336—591no—
370C59#2607811yescharacteristic, complemented, Sylow
371C22×Q8—1181no—
372C23×C4—1181no—
373C23×C4C22×D5911yescharacteristic
374C2×M16—594no—
375C2×M16—1182no—
376C2×M16—1182no—
377C22×C8—591no—
378C22×C8—591no—
379C22×C8C22×D5911yescharacteristic
380C22×C8C22×D5911yescharacteristic
381C4⋊Q8—1182no—
382C4⋊Q8—2361no—
383#125—1182no—
384#125—2361no—
385C22⋊Q8—2364no—
386#117—22no—
387#117—41no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—1182no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4—1181no—
405C2×C4⋊C4—2361no—
406C2×C4⋊C4C22×D5911yescharacteristic
407C2×C4⋊C4C22×D5911yescharacteristic
408C2×C4⋊C4C22×D5911yescharacteristic
409C2×C4⋊C4C22×D5911yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—591no—
413C2×C22⋊C4—591no—
414C2×C22⋊C4—1181no—
415C2×C42—21no—
416C2×C42—591no—
417C2×C42—591no—
418#107—22no—
419#107—594no—
420#107—1182no—
421#107—1182no—
422#106—41no—
423#106—1182no—
424#106—1182no—
425#106—1182no—
426#106—1182no—
427#106—2361no—
428C22⋊C8Dic11814yescomplemented
429#97—1182no—
430C4×C8—1182no—
431#95—21no—
432#95—591no—
433#95—591no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438#95—1181no—
439C24D4×D5911yescharacteristic
440C2×Q8—4721no—
441C2×Q8—2362no—
442C2×Q8—2362no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—591no—
452C22×C4—591no—
453C22×C4—591no—
454C22×C4—591no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—4721no—
460C22×C4—2362no—
461C22×C4—2362no—
462C22×C4C2×Dic11812yes—
463C22×C4D4×D5911yescharacteristic
464C22×C4Q8×D5912yes—
465C22×C4C23×D5911yescharacteristic
466M16—1188no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—592no—
472C2×C8—592no—
473C2×C8—592no—
474C2×C8—592no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8C2×Dic11812yes—
478C2×C8C2×Dic11812yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—592no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—4721no—
495C4⋊C4—4721no—
496C4⋊C4—4721no—
497C4⋊C4—2362no—
498C4⋊C4—2362no—
499C4⋊C4—2362no—
500C4⋊C4—2362no—
501C4⋊C4—2362no—
502C4⋊C4—2362no—
503C4⋊C4—2362no—
504C4⋊C4—2362no—
505C4⋊C4—2362no—
506C4⋊C4D4×D5912yes—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—1182no—
510C22⋊C4—1182no—
511C22⋊C4—4721no—
512C42—42no—
513C42—1181no—
514C42—1181no—
515C42—1181no—
516C42—1181no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23#1427372612yes—
524C23C2×D4×D5911yescharacteristic
525Q8—4724no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—592no—
545C2×C4—592no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1182no—
557C2×C4—4724no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4#1427372612yes—
570C2×C4C2×D4×D5911yescharacteristic
571C2×C4C2×Q8×D5911yescharacteristic
572C2×C4C22×Dic11811yescharacteristic
573C2×C4#41×D5911yescharacteristic
574C2×C4#1427377211yescharacteristic
575C2×C4#1427377411yescharacteristic
576C8—22no—
577C8—22no—
578C8—594no—
579C8—1182no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#1427372611yescharacteristic
588C22#1683803411yescharacteristic
589C22#1683805011yescharacteristic
590C22AGL(1,ℤ/8)×D5911yescharacteristic
591C22#137×D5911yescharacteristic
592C22#1683820911yescharacteristic
593C22#1683820911yescharacteristic
594C4—22no—
595C4—42no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—1182no—
600C4—1182no—
601C4—4721no—
602C4—4721no—
603C4—4721no—
604C4—2361no—
605C4—2362no—
606C4—2361no—
607C4C22⋊Q8×D5911yescharacteristic
608C4#1683803111yescharacteristic
609C4D4×Dic11811yescharacteristic
610C4#1683811111yescharacteristic
611C2—42no—
612C2#2142779711yescharacteristic
613C2#2143462811yescharacteristic
614C2#2143464011yescharacteristic
615C2#2143568911yescharacteristic
616C2#2143575511yescharacteristic
617C2#2143654411yescharacteristic
618C2#2143656911yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1854C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#1854D59
hypocenterC59#26078
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
1. C2×#16837810C2semidirect
5. #21425576C2semidirect
6. #21425570C2semidirect
7. #21425433C2semidirect
8. #21425427C2semidirect
9. #21424833C2semidirect
10. #21424828C2semidirect
12. #21424251C2semidirect
65. C2×#11745664Q8semidirect
127. C22⋊C8×C59Q8semidirect
269. C59⋊C8C22⋊Q8semidirect
370. C59#26078semidirect
428. C22⋊C8Dic118semidirect

Maximal subgroups

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

subgroupindex
C2×#168378102
C2×#168374452
#214278032
#214277992
#214255762
#214255702
#214254332
#214254272
#214248332
#214248282
#214242912
#214242512
#214241272
#214238822
C59×#18542
#2607859