Finite Group Databasebeta

← #28598368

Subgroups — #28598368

Order 15104.

1222 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837954C211yesmaximal, characteristic
2C2×#16837452C211yesmaximal, characteristic
3C2×#16837271C211yesmaximal, characteristic
4#21425055C212yesmaximal, complemented
5#21424896C212yesmaximal, complemented
6#21424404C211yesmaximal, characteristic
7#21424403C211yesmaximal, characteristic
8#21424401C211yesmaximal, characteristic
9#21424100C212yesmaximal, complemented
10#21424099C212yesmaximal, complemented
11C59×#1872C211yesmaximal, characteristic
12#16837954—24no—
13#16837954C418yescomplemented
14C2×#14273726C2211yescharacteristic
15C2×#14273725C2211yescharacteristic
16C2×#14273717—21no—
17C2×#14273706C2211yescharacteristic
18C2×#14273704—21no—
19C2×Q8×Dic59C2211yescharacteristic
20C2×#14273692—21no—
21C2×#14273692C2211yescharacteristic
22C2×#14273683—21no—
23C22×C4⋊Dic59C2211yescharacteristic
24C22×Dic59⋊C4—21no—
25C22×Dic59⋊C4—21no—
26C22×C4×Dic59C2211yescharacteristic
27#16837452—24no—
28C2×Dic59⋊C8C2212yes—
29C2×#14273652C2212yes—
30C2×#14273651C2212yes—
31C2×#14273650C2212yes—
32C2×C22⋊(C59⋊C8)C2211yescharacteristic
33C2×C8×Dic59C2212yes—
34#16837271—21no—
35#16837271—21no—
36#16837271C2212yes—
37#16837271C2212yes—
38#16837165C2212yes—
39#16837164C2212yes—
40#16837164C2212yes—
41C2×C22⋊Q8×C59C2211yescharacteristic
42C22×C4⋊C4×C59C2211yescharacteristic
43C2×#103×C59C2212yes—
44C2×C22⋊C8×C59C2211yescharacteristic
45C59×#340C2212yes—
46C22×Dic118—21no—
47C22×Dic118C2311yescharacteristic
48C23×Dic59—21no—
49C23×Dic59D411yescharacteristic
50C23×Dic59D411yescharacteristic
51C22×C59⋊C8C2312yes—
52#14273726—22no—
53#14273726C2×C414yescomplemented
54#14273725—22no—
55#14273725C2×C414yescomplemented
56#14273717—28no—
57#14273706—22no—
58#14273706C2×C414yes—
59#14273704—28no—
60Q8×Dic59—22no—
61Q8×Dic59C2×C414yescomplemented
62#14273692—28no—
63#14273692—22no—
64#14273692C2×C414yescomplemented
65#14273683—28no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—42no—
69C2×C4⋊Dic59C2312yes—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×Dic59⋊C4—21no—
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—22no—
77C2×Dic59⋊C4—22no—
78C2×Dic59⋊C4—22no—
79C2×Dic59⋊C4—22no—
80C2×Dic59⋊C4—44no—
81C2×Dic59⋊C4—44no—
82C2×C22⋊Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59C2×C414yes—
86C2×C4×Dic59D411yescharacteristic
87C2×C4×Dic59D411yescharacteristic
88C2×C4×Dic59D414yes—
89C2×C4×Dic59C2311yescharacteristic
90C2×C4×Dic59C2311yescharacteristic
91Dic59⋊C8—24no—
92#14273652—42no—
93#14273651—42no—
94#14273650—42no—
95C22⋊(C59⋊C8)—22no—
96C8×Dic59—24no—
97C22×Q8×C59C2311yescharacteristic
98C23×C4×C59—21no—
99C23×C4×C59C2311yescharacteristic
100C22×C8×C59C2312yes—
101C22⋊Q8×C59—22no—
102C22⋊Q8×C59C2×C414yes—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—42no—
106C2×C4⋊C4×C59C2312yes—
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C22⋊C4×C59—21no—
109#103×C59—42no—
110C22⋊C8×C59—22no—
111C24×C59C2×D411yescharacteristic
112C2×Q8×C59—21no—
113C2×Q8×C59—22no—
114C2×Q8×C59—22no—
115C2×Q8×C59C22×C412yes—
116C22×C4×C59—22no—
117C22×C4×C59—22no—
118C22×C4×C59—21no—
119C22×C4×C59—21no—
120C22×C4×C59—22no—
121C22×C4×C59—42no—
122C22×C4×C59—42no—
123C22×C4×C59—42no—
124C22×C4×C59C22×C412yes—
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D412yes—
127C22×C4×C59C2411yescharacteristic
128C22×Dic59—22no—
129C22×Dic59—22no—
130C22×Dic59—21no—
131C22×Dic59—21no—
132C22×Dic59—22no—
133C22×Dic59—42no—
134C22×Dic59—42no—
135C22×Dic59—42no—
136C22×Dic59C22⋊C414yes—
137C22×Dic59C22⋊C414yes—
138C22×Dic59SD1614yes—
139C22×Dic59Q1614yes—
140C22×Dic59C2×D411yescharacteristic
141C22×Dic59C2×D411yescharacteristic
142C22×Dic59C2×D411yescharacteristic
143C22×Dic59C2×D411yescharacteristic
144C2×Dic118—22no—
145C2×Dic118—22no—
146C2×Dic118—21no—
147C2×Dic118—24no—
148C2×Dic118—42no—
149C2×Dic118—42no—
150C2×Dic118C22×C412yes—
151C2×C8×C59—22no—
152C2×C8×C59—22no—
153C2×C8×C59—22no—
154C4⋊C4×C59—24no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—22no—
157C4⋊C4×C59—22no—
158C4⋊C4×C59—42no—
159C4⋊C4×C59C22×C412yes—
160C22⋊C4×C59—24no—
161C22⋊Dic59—24no—
162C4⋊Dic59—21no—
163C4⋊Dic59—24no—
164C4⋊Dic59—22no—
165C4⋊Dic59—22no—
166C4⋊Dic59—42no—
167C4⋊Dic59C22×C412yes—
168Dic59⋊C4—24no—
169Dic59⋊C4—24no—
170Dic59⋊C4—24no—
171Dic59⋊C4—24no—
172Dic59⋊C4—24no—
173Dic59⋊C4—44no—
174Dic59⋊C4—44no—
175Dic59⋊C4—44no—
176Dic59⋊C4—44no—
177C4×Dic59—24no—
178C4×Dic59—24no—
179C4×Dic59C22⋊C414yes—
180C4×Dic59C22⋊C414yes—
181C4×Dic59C22×C412yes—
182C4×Dic59C22×C412yes—
183C4×Dic59C2×D412yes—
184C4×Dic59C2×D412yes—
185C2×C59⋊C8—22no—
186C2×C59⋊C8—22no—
187C2×C59⋊C8—22no—
188C23×C59—21no—
189C23×C59—21no—
190C23×C59—21no—
191C23×C59—21no—
192C23×C59C2×C22⋊C412yes—
193C23×C59C2×SD1612yes—
194C23×C59C2×Q1612yes—
195C23×C59C22×D411yescharacteristic
196Q8×C59—24no—
197Q8×C59—42no—
198C2×C4×C59—21no—
199C2×C4×C59—21no—
200C2×C4×C59—24no—
201C2×C4×C59—24no—
202C2×C4×C59—22no—
203C2×C4×C59—22no—
204C2×C4×C59—41no—
205C2×C4×C59—41no—
206C2×C4×C59—41no—
207C2×C4×C59—41no—
208C2×C4×C59—42no—
209C2×C4×C59—42no—
210C2×C4×C59—42no—
211C2×C4×C59—84no—
212C2×C4×C59C2×C22⋊C412yes—
213C2×C4×C59C23×C411yescharacteristic
214C2×C4×C59C22×D411yescharacteristic
215C2×C4×C59H(5,𝔽2)12yes—
216C2×C4×C592−1+412yes—
217C2×Dic59—21no—
218C2×Dic59—21no—
219C2×Dic59—28no—
220C2×Dic59—22no—
221C2×Dic59—22no—
222C2×Dic59—41no—
223C2×Dic59—41no—
224C2×Dic59—41no—
225C2×Dic59—41no—
226C2×Dic59—42no—
227C2×Dic59—42no—
228C2×Dic59—42no—
229C2×Dic59—84no—
230C2×Dic59#103116yescomplemented
231C2×Dic59C2×C22⋊C412yes—
232C2×Dic59C2×C22⋊C412yes—
233C2×Dic59C2×C22⋊C412yes—
234C2×Dic59C2×C22⋊C412yes—
235C2×Dic59C2×SD1612yes—
236C2×Dic59C2×SD1612yes—
237C2×Dic59C2×Q1612yes—
238C2×Dic59C2×Q1612yes—
239C2×Dic59AGL(1,ℤ/8)12yes—
240C2×Dic59#13712yes—
241Dic118—24no—
242Dic118—42no—
243Dic118—48no—
244C8×C59—42no—
245C59⋊C8—42no—
246#21860—591nomaximal, Sylow
247C22×C59—22no—
248C22×C59—22no—
249C22×C59—22no—
250C22×C59—22no—
251C22×C59—22no—
252C22×C59—22no—
253C22×C59C2×#10314yes—
254C22×C59C22×C22⋊C411yescharacteristic
255C22×C59#53511yescharacteristic
256C22×C59#53611yescharacteristic
257C22×C59C22×SD1611yescharacteristic
258C22×C59C22×Q1611yescharacteristic
259C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C59C2×#13711yescharacteristic
261C4×C59—22no—
262C4×C59—42no—
263C4×C59—42no—
264C4×C59—82no—
265C4×C59#52012yes—
266C4×C59#53712yes—
267Dic59—22no—
268Dic59—42no—
269Dic59—42no—
270Dic59—82no—
271Dic59C2×#10314yes—
272Dic59C2×#10314yes—
273Dic59#41714yes—
274C2×#533—591no—
275C2×#431—591no—
276C2×#384—591no—
277#1990—592no—
278#1956—592no—
279#1872—591no—
280#1872—591no—
281#1872—591no—
282#1872D5911yescharacteristic
283#1803—592no—
284#1803—592no—
285C2×C59—24no—
286C2×C59#231111yescharacteristic
287C2×C59C22×#10311yescharacteristic
288C2×C59C2×#41711yescharacteristic
289C2×C59#320811yescharacteristic
290C2×C59#320911yescharacteristic
291C2×C59#322911yescharacteristic
292C2×C59#323011yescharacteristic
293#533—598no—
294#533—1184no—
295C2×C4⋊Q8—591no—
296C2×C4⋊Q8—591no—
297C2×#125—1181no—
298C2×C22⋊Q8—591no—
299C2×C22⋊Q8—1181no—
300C2×C22⋊Q8C2×D5911yescharacteristic
301C2×C4×Q8—591no—
302C2×C4×Q8—591no—
303C2×C4×Q8—1181no—
304C2×#117—1181no—
305C22×C4⋊C4—591no—
306C22×C4⋊C4—1181no—
307C22×C4⋊C4—1181no—
308C22×C4⋊C4C2×D5911yescharacteristic
309C22×C42—591no—
310#431—1184no—
311C2×C4⋊C8—592no—
312C2×#103—592no—
313C2×#103—592no—
314C2×#103—592no—
315C2×#103C2×D5912yes—
316C2×C22⋊C8—591no—
317C2×C22⋊C8C2×D5911yescharacteristic
318C2×C4×C8—592no—
319#384—592no—
320#384—592no—
321#384—1181no—
322#384—1181no—
323#340—592no—
324#340—592no—
325#340—592no—
326#340C2×D5912yes—
327C59#2186011yescharacteristic, complemented, Sylow
328C22×Q8—591no—
329C22×Q8—1181no—
330C22×Q8C22×D5911yescharacteristic
331C23×C4—21no—
332C23×C4—591no—
333C23×C4—591no—
334C23×C4—1181no—
335C23×C4C22×D5911yescharacteristic
336C22×C8—592no—
337C22×C8C22×D5912yes—
338C4⋊Q8—594no—
339C4⋊Q8—594no—
340C4⋊Q8—1182no—
341C4⋊Q8—1182no—
342#125—1188no—
343C22⋊Q8—22no—
344C22⋊Q8—594no—
345C22⋊Q8—1188no—
346C22⋊Q8—1182no—
347C22⋊Q8C4×D5914yes—
348C4×Q8—594no—
349C4×Q8—594no—
350C4×Q8—1182no—
351C4×Q8—1188no—
352C4×Q8—1182no—
353#117—1188no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—42no—
357C2×C4⋊C4—591no—
358C2×C4⋊C4—592no—
359C2×C4⋊C4—1181no—
360C2×C4⋊C4—1181no—
361C2×C4⋊C4—1181no—
362C2×C4⋊C4—1181no—
363C2×C4⋊C4—1181no—
364C2×C4⋊C4—1181no—
365C2×C4⋊C4—1182no—
366C2×C4⋊C4—1182no—
367C2×C4⋊C4—1182no—
368C2×C4⋊C4—1182no—
369C2×C4⋊C4—1181no—
370C2×C4⋊C4—2362no—
371C2×C4⋊C4—2364no—
372C2×C4⋊C4—2364no—
373C2×C4⋊C4C22×D5912yes—
374C2×C4⋊C4C22×D5911yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—1181no—
377C2×C42—594no—
378C2×C42—591no—
379C2×C42—591no—
380C2×C42—594no—
381C2×C42—591no—
382C2×C42—591no—
383C2×C42—1181no—
384C2×C42—1181no—
385C4⋊C8—1184no—
386#103—42no—
387#103—2362no—
388#103—2362no—
389#103—2362no—
390C22⋊C8—22no—
391C22⋊C8—1182no—
392C4×C8—1184no—
393C24D4×D5911yescharacteristic
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—592no—
398C2×Q8—1184no—
399C2×Q8—1181no—
400C2×Q8—1182no—
401C2×Q8—1182no—
402C2×Q8—2362no—
403C2×Q8—2362no—
404C2×Q8C2×C4×D5912yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—594no—
414C22×C4—594no—
415C22×C4—594no—
416C22×C4—591no—
417C22×C4—591no—
418C22×C4—591no—
419C22×C4—594no—
420C22×C4—591no—
421C22×C4—1182no—
422C22×C4—1182no—
423C22×C4—1181no—
424C22×C4—1181no—
425C22×C4—1182no—
426C22×C4—2362no—
427C22×C4—2362no—
428C22×C4—2362no—
429C22×C4C2×C4×D5912yes—
430C22×C4D4×D5911yescharacteristic
431C22×C4D4×D5912yes—
432C22×C4C23×D5911yescharacteristic
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—1182no—
437C2×C8—1182no—
438C2×C8—1182no—
439C4⋊C4—24no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—592no—
445C4⋊C4—1184no—
446C4⋊C4—1184no—
447C4⋊C4—1184no—
448C4⋊C4—1184no—
449C4⋊C4—1184no—
450C4⋊C4—1184no—
451C4⋊C4—1181no—
452C4⋊C4—1182no—
453C4⋊C4—1182no—
454C4⋊C4—2362no—
455C4⋊C4—2364no—
456C4⋊C4—2364no—
457C4⋊C4—2364no—
458C4⋊C4—2364no—
459C4⋊C4C2×C4×D5912yes—
460C22⋊C4—24no—
461C22⋊C4—1184no—
462C42—594no—
463C42—594no—
464C42—592no—
465C42—592no—
466C42—592no—
467C42—592no—
468C42—1184no—
469C42—1184no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23C22⋊C4×D5912yes—
475C23SD16×D5912yes—
476C23Q16×D5912yes—
477C23C2×D4×D5911yescharacteristic
478Q8—24no—
479Q8—42no—
480Q8—1184no—
481Q8—2368no—
482Q8—2362no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—84no—
497C2×C4—5916no—
498C2×C4—592no—
499C2×C4—592no—
500C2×C4—592no—
501C2×C4—592no—
502C2×C4—592no—
503C2×C4—592no—
504C2×C4—592no—
505C2×C4—592no—
506C2×C4—592no—
507C2×C4—592no—
508C2×C4—1188no—
509C2×C4—1181no—
510C2×C4—1181no—
511C2×C4—1182no—
512C2×C4—1182no—
513C2×C4—4724no—
514C2×C4—2362no—
515C2×C4—2362no—
516C2×C4—2362no—
517C2×C4—2361no—
518C2×C4—2361no—
519C2×C4—2361no—
520C2×C4—2361no—
521C2×C4C22⋊C4×D5912yes—
522C2×C4C22×C4×D5911yescharacteristic
523C2×C4C2×D4×D5911yescharacteristic
524C2×C4#1427377212yes—
525C2×C4#1427377412yes—
526C8—42no—
527C8—2362no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22#103×D5914yes—
535C22C2×C22⋊C4×D5911yescharacteristic
536C22#1683796711yescharacteristic
537C22#1683797411yescharacteristic
538C22C2×SD16×D5911yescharacteristic
539C22C2×Q16×D5911yescharacteristic
540C22AGL(1,ℤ/8)×D5911yescharacteristic
541C22#137×D5911yescharacteristic
542C4—22no—
543C4—42no—
544C4—42no—
545C4—82no—
546C4—594no—
547C4—594no—
548C4—594no—
549C4—1182no—
550C4—4722no—
551C4—2362no—
552C4—2362no—
553C4#1683788112yes—
554C4#1683797912yes—
555C2—24no—
556C2#2142668311yescharacteristic
557C2C2×#103×D5911yescharacteristic
558C2#417×D5911yescharacteristic
559C2#2143503511yescharacteristic
560C2#2143504211yescharacteristic
561C2#2143523811yescharacteristic
562C2#2143524511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1872C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#1872D59
hypocenterC59#21860
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
4. #21425055C2semidirect
5. #21424896C2semidirect
9. #21424100C2semidirect
10. #21424099C2semidirect
13. #16837954C4semidirect
53. #14273726C2×C4semidirect
55. #14273725C2×C4semidirect
61. Q8×Dic59C2×C4semidirect
64. #14273692C2×C4semidirect
230. C2×Dic59#103semidirect
327. C59#21860semidirect

Maximal subgroups

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

subgroupindex
C2×#168379542
C2×#168374522
C2×#168372712
#214250552
#214248962
#214244042
#214244032
#214244012
#214241002
#214240992
C59×#18722
#2186059