Finite Group Databasebeta

← #28647542

Subgroups — #28647542

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×#16837443C211yesmaximal, characteristic
2#21427805C211yesmaximal, characteristic
3#21427794C211yesmaximal, characteristic
4#21425575C211yesmaximal, characteristic, complemented
5#21425570C211yesmaximal, characteristic, complemented
6#21425431C211yesmaximal, characteristic, complemented
7#21425426C211yesmaximal, characteristic, complemented
8#21424834C211yesmaximal, characteristic, complemented
9#21424830C211yesmaximal, characteristic, complemented
10#21424292C211yesmaximal, characteristic
11#21424287C211yesmaximal, characteristic
12#21424251C211yesmaximal, characteristic, complemented
13#21424126C211yesmaximal, characteristic
14#21423881C211yesmaximal, characteristic
15C2×C59×#501C211yesmaximal, characteristic, complemented
16C22×#11745659C2211yescharacteristic
17C2×#14273703—21no—
18C2×#14273685C2211yescharacteristic
19C2×C4⋊C4×D59C2211yescharacteristic
20#16837443—24no—
21C2×#14273661C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×(C2×D59)⋊C8C2211yescharacteristic
28C2×C8×Dic59C2211yescharacteristic
29#16837337—21no—
30#16837335—21no—
31#16837324—21no—
32#16837321—21no—
33#16837301C2211yescharacteristic
34#16837301C2211yescharacteristic
35#16837300C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837298—21no—
38#16837283—21no—
39#16837273—21no—
40#16837271C2211yescharacteristic
41#16837271C2211yescharacteristic
42#16837259C2211yescharacteristic
43#16837259C2211yescharacteristic
44C4×(C2×D59)⋊C4C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837157C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×C4⋊Q8×C59C2211yescharacteristic
54C2×#125×C59C2211yescharacteristic
55C59×#501—24no—
56C2×#107×C59C2211yescharacteristic
57C2×#107×C59C2211yescharacteristic
58C2×#106×C59C2211yescharacteristic
59C2×#106×C59C2211yescharacteristic
60C2×#97×C59C2211yescharacteristic
61C22×C4×D59—21no—
62C22×C4×D59C2311yescharacteristic
63C2×#11745659—22no—
64C2×#11745659—22no—
65C2×#11745659Q814yescomplemented
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273703—44no—
69#14273685—22no—
70#14273685—41no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C4⋊C4×D59—22no—
85C4⋊C4×D59—41no—
86C2×(C2×D59)⋊C4—21no—
87C2×(C2×D59)⋊C4—21no—
88C2×(C2×D59)⋊C4—21no—
89C2×(C2×D59)⋊C4D411yescharacteristic
90C2×(C2×D59)⋊C4D411yescharacteristic
91C2×C4×Dic59—21no—
92C2×C4×Dic59C2311yescharacteristic
93#14273661—22no—
94#14273660—22no—
95#14273659—22no—
96#14273658—22no—
97#14273658—41no—
98#14273658—41no—
99(C2×D59)⋊C8Q814yescomplemented
100C8×Dic59—22no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95D411yescharacteristic
108C59⋊#95D411yescharacteristic
109C22×Q8×C59—21no—
110C22×C8×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112C4⋊Q8×C59—22no—
113C4⋊Q8×C59—41no—
114#125×C59—22no—
115#125×C59—41no—
116C2×C4⋊C4×C59—21no—
117C2×C4⋊C4×C59—21no—
118C2×C4⋊C4×C59—21no—
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C4⋊C4×C59C2311yescharacteristic
121C2×C4⋊C4×C59C2311yescharacteristic
122C2×C4⋊C4×C59C2311yescharacteristic
123C2×C42×C59C2311yescharacteristic
124#107×C59—22no—
125#107×C59D414yes—
126#106×C59—22no—
127#106×C59—22no—
128#97×C59—22no—
129C23×D59C2×D411yescharacteristic
130C2×Q8×C59—42no—
131C2×Q8×C59—42no—
132C2×Q8×C59—81no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2411yescharacteristic
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59C2×D411yescharacteristic
145C22×Dic59C2×D411yescharacteristic
146C22×Dic59C2×D411yescharacteristic
147C2×C4×D59—21no—
148C2×C4×D59—21no—
149C2×C4×D59—21no—
150C2×C4×D59—21no—
151C2×C4×D59—42no—
152C2×C4×D59—42no—
153C2×C4×D59—81no—
154C2×C4×D59C2×Q812yes—
155C2×C4×D59C2×Q812yes—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59C2×D412yes—
159C2×C8×C59C2×Q812yes—
160C2×C8×C59C2×Q812yes—
161C2×C8×C59#4112yes—
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—42no—
169C4⋊C4×C59—42no—
170C4⋊C4×C59—42no—
171C4⋊C4×C59C2×D412yes—
172C4⋊C4×C59C2×D412yes—
173C42×C59—21no—
174C42×C59—21no—
175(C2×D59)⋊C4—22no—
176(C2×D59)⋊C4—22no—
177(C2×D59)⋊C4—42no—
178(C2×D59)⋊C4—42no—
179(C2×D59)⋊C4—81no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—42no—
189C4⋊Dic59—42no—
190Dic59⋊C4—42no—
191Dic59⋊C4—42no—
192Dic59⋊C4—42no—
193Dic59⋊C4—42no—
194Dic59⋊C4—81no—
195Dic59⋊C4—81no—
196Dic59⋊C4—81no—
197C4×Dic59—21no—
198C4×Dic59—21no—
199C4×Dic59—42no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8C2×Q812yes—
205C2×C59⋊C8C2×Q812yes—
206#11745659—28no—
207C23×C59C22×D411yescharacteristic
208C22×D59—21no—
209C22×D59—21no—
210C22×D59—21no—
211C22×D59—21no—
212C22×D59—21no—
213C22×D59—21no—
214C22×D59C4⋊Q812yes—
215Q8×C59—84no—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
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×C59—41no—
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—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59C4⋊Q812yes—
259C2×Dic59AGL(1,ℤ/8)12yes—
260C2×Dic59#13712yes—
261C4×D59—22no—
262C4×D59—22no—
263C4×D59—22no—
264C4×D59—22no—
265C4×D59—84no—
266C8×C59—22no—
267C8×C59C22⋊Q814yescomplemented
268C59⋊C8—22no—
269C59⋊C8—22no—
270#26078—591nomaximal, Sylow
271C22×C59C2×C4⋊Q811yescharacteristic
272C22×C59D4211yescharacteristic
273C22×C59#54711yescharacteristic
274C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C59C2×#13711yescharacteristic
276C22×C59#57711yescharacteristic
277C22×C59#57711yescharacteristic
278C2×D59—22no—
279C2×D59—22no—
280C2×D59—22no—
281C2×D59—41no—
282C2×D59—41no—
283C2×D59—41no—
284C2×D59—41no—
285C4×C59—42no—
286C4×C59—41no—
287C4×C59—42no—
288C4×C59—41no—
289C4×C59—81no—
290C4×C59—81no—
291C4×C59C2×C22⋊Q811yescharacteristic
292C4×C59#54411yescharacteristic
293C4×C59D4×Q811yescharacteristic
294C4×C59#55411yescharacteristic
295Dic59—22no—
296Dic59—22no—
297Dic59—22no—
298Dic59—81no—
299Dic59—81no—
300Dic59—81no—
301Dic59—81no—
302C2×#501D5911yescharacteristic, complemented
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#1854—591no—
313#1853—591no—
314#1844—591no—
315#1809—591no—
316#1753—591no—
317C2×C59#244011yescharacteristic
318C2×C59#316311yescharacteristic
319C2×C59#316511yescharacteristic
320C2×C59#327211yescharacteristic
321C2×C59#327811yescharacteristic
322C2×C59#334411yescharacteristic
323C2×C59#334511yescharacteristic
324D59—42no—
325C22×M16—591no—
326C2×C4⋊Q8C2×D5911yescharacteristic
327C2×#125C2×D5911yescharacteristic
328C2×C22⋊Q8—1181no—
329C2×#117—591no—
330C22×C4⋊C4—591no—
331#501—24no—
332#428—1184no—
333C2×#107—591no—
334C2×#107—591no—
335C2×#107C2×D5911yescharacteristic
336C2×#107C2×D5911yescharacteristic
337C2×#106—591no—
338C2×#106—591no—
339C2×#106—591no—
340C2×#106—591no—
341C2×#106C2×D5911yescharacteristic
342C2×#106C2×D5911yescharacteristic
343C2×C22⋊C8—591no—
344C2×#97C2×D5911yescharacteristic
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#380—591no—
361C4×C22⋊C4—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#336—591no—
369#336—591no—
370C59#2607811yescharacteristic, complemented, Sylow
371C22×Q8—21no—
372C23×C4—591no—
373C23×C4—1181no—
374C2×M16—594no—
375C2×M16—1182no—
376C2×M16—1182no—
377C22×C8—591no—
378C22×C8—591no—
379C22×C8C22×D5911yescharacteristic
380C22×C8C22×D5911yescharacteristic
381C4⋊Q8—22no—
382C4⋊Q8—41no—
383#125—22no—
384#125—41no—
385C22⋊Q8—2364no—
386#117—1182no—
387#117—2361no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—1182no—
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—591no—
411C2×C22⋊C4—591no—
412C2×C22⋊C4—1181no—
413C2×C22⋊C4—1181no—
414C2×C22⋊C4—1181no—
415C2×C42—591no—
416C2×C42—1181no—
417C2×C42C22×D5911yescharacteristic
418#107—22no—
419#107—1182no—
420#107—1182no—
421#107C22⋊D5914yes—
422#106—22no—
423#106—22no—
424#106—1182no—
425#106—1182no—
426#106—2361no—
427#106—2361no—
428C22⋊C8—594no—
429#97—22no—
430C4×C8—1182no—
431#95—591no—
432#95—591no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438#95—1181no—
439C24—591no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—81no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—591no—
448C22×C4—591no—
449C22×C4—591no—
450C22×C4—592no—
451C22×C4—592no—
452C22×C4—1181no—
453C22×C4—1181no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—4721no—
461C22×C4—2362no—
462C22×C4—2362no—
463C22×C4D4×D5911yescharacteristic
464C22×C4D4×D5911yescharacteristic
465C22×C4C23×D5911yescharacteristic
466M16—1188no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—592no—
470C2×C8—592no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8Q8×D5912yes—
476C2×C8Q8×D5912yes—
477C2×C8#1174568912yes—
478C2×C8C2×C22⋊D5912yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—4721no—
497C4⋊C4—4721no—
498C4⋊C4—4721no—
499C4⋊C4—2362no—
500C4⋊C4—2362no—
501C4⋊C4—2362no—
502C4⋊C4—2362no—
503C4⋊C4—2362no—
504C4⋊C4—2362no—
505C4⋊C4C2×C22⋊D5912yes—
506C4⋊C4C2×C22⋊D5912yes—
507C22⋊C4—1182no—
508C22⋊C4—1182no—
509C22⋊C4—4721no—
510C22⋊C4—2362no—
511C22⋊C4—2362no—
512C42—21no—
513C42—21no—
514C42—1181no—
515C42—1181no—
516C42—2362no—
517C23—592no—
518C23—1181no—
519C23—1181no—
520C23—1181no—
521C23—1181no—
522C23—1181no—
523C23—1181no—
524C23C2×D4×D5911yescharacteristic
525Q8—84no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
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—592no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1182no—
554C2×C4—1182no—
555C2×C4—1182no—
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—2361no—
570C2×C4C22×C22⋊D5911yescharacteristic
571C2×C4C2×Q8×D5911yescharacteristic
572C2×C4C2×Q8×D5911yescharacteristic
573C2×C4C2×#1174568911yescharacteristic
574C2×C4#1427377211yescharacteristic
575C2×C4#1427377511yescharacteristic
576C8—22no—
577C8—1182no—
578C8—1182no—
579C8#1427370314yescomplemented
580C22—1182no—
581C22—1182no—
582C22—1182no—
583C22—2361no—
584C22—2361no—
585C22—2361no—
586C22—2361no—
587C22C4⋊Q8×D5911yescharacteristic
588C22D4×C22⋊D5911yescharacteristic
589C22#1683805511yescharacteristic
590C22AGL(1,ℤ/8)×D5911yescharacteristic
591C22#137×D5911yescharacteristic
592C22#1683821111yescharacteristic
593C22#1683821211yescharacteristic
594C4—41no—
595C4—42no—
596C4—42no—
597C4—41no—
598C4—81no—
599C4—81no—
600C4—1182no—
601C4—1182no—
602C4—1182no—
603C4—4721no—
604C4—4721no—
605C4—4721no—
606C4—4721no—
607C4C2×#1427370311yescharacteristic
608C4#1683802911yescharacteristic
609C4Q8×C22⋊D5911yescharacteristic
610C4#1683811311yescharacteristic
611C2—2362no—
612C2#2142779411yescharacteristic
613C2#2143462311yescharacteristic
614C2#2143464111yescharacteristic
615C2#2143569011yescharacteristic
616C2#2143575611yescharacteristic
617C2#2143655011yescharacteristic
618C2#2143656511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#501C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
hypercenterC2×#501D59
hypocenterC59#26078
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
4. #21425575C2semidirect
5. #21425570C2semidirect
6. #21425431C2semidirect
7. #21425426C2semidirect
8. #21424834C2semidirect
9. #21424830C2semidirect
12. #21424251C2semidirect
15. C2×C59×#501C2semidirect
65. C2×#11745659Q8semidirect
99. (C2×D59)⋊C8Q8semidirect
267. C8×C59C22⋊Q8semidirect
302. C2×#501D59semidirect
370. C59#26078semidirect
579. C8#14273703semidirect

Maximal subgroups

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

subgroupindex
C2×#168374432
#214278052
#214277942
#214255752
#214255702
#214254312
#214254262
#214248342
#214248302
#214242922
#214242872
#214242512
#214241262
#214238812
C2×C59×#5012
#2607859