Finite Group Databasebeta

← #28484262

Subgroups — #28484262

Order 15104.

952 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837366C211yesmaximal, characteristic
2#21432383C211yesmaximal, characteristic
3C2×#16837138C212yesmaximal, complemented
4#21421975C212yesmaximal, complemented
5C59×C22⋊#95C211yesmaximal, characteristic
6C22×#11745664—21no—
7C2×#14273706C411yescharacteristic, complemented
8C2×#14273706C411yescharacteristic, complemented
9C2×#14273706C2211yescharacteristic
10#16837366—24no—
11C2×C22⋊(C59⋊C8)—21no—
12C2×C22⋊(C59⋊C8)C2212yes—
13#16837275—21no—
14#16837275—21no—
15#16837275—21no—
16#16837262C411yescharacteristic, complemented
17#16837262C411yescharacteristic, complemented
18#16837262C2211yescharacteristic
19#16837258—21no—
20#16837138—24no—
21C24×C4×C59C2211yescharacteristic
22C22×C22⋊C4×C59—21no—
23C2×#95×C59—21no—
24C2×#95×C59C2212yes—
25C22×Dic118C2×C411yescharacteristic
26C2×#11745664—24no—
27C2×#11745664—42no—
28C2×#11745664—42no—
29C22×C59⋊C8—22no—
30#14273706—22no—
31#14273706—22no—
32#14273706—42no—
33#14273706—42no—
34C2×C4⋊Dic59C2×C411yescharacteristic
35C2×C4⋊Dic59C2×C411yescharacteristic
36C2×C4⋊Dic59C2×C411yescharacteristic
37C2×Dic59⋊C4—21no—
38C2×Dic59⋊C4—21no—
39C2×Dic59⋊C4—21no—
40C2×C22⋊Dic59—21no—
41C2×C22⋊Dic59—21no—
42C2×C22⋊Dic59—21no—
43C2×C22⋊Dic59—21no—
44C2×C22⋊Dic59—21no—
45C2×C22⋊Dic59—21no—
46C22⋊(C59⋊C8)—24no—
47C22⋊(C59⋊C8)—22no—
48C22⋊(C59⋊C8)D414yes—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C59⋊#95—21no—
52C25×C59D411yescharacteristic
53C23×C4×C59—22no—
54C23×C4×C59—22no—
55C23×C4×C59—22no—
56C23×C4×C59—42no—
57C23×C4×C59—42no—
58C23×C4×C59C2×C411yescharacteristic
59C23×C4×C59C2×C411yescharacteristic
60C23×C4×C59D411yescharacteristic
61C23×C4×C59D411yescharacteristic
62C23×C4×C59D411yescharacteristic
63C23×C4×C59C2311yescharacteristic
64C2×C22⋊C4×C59—24no—
65C2×C22⋊C4×C59—42no—
66C2×C22⋊C4×C59—42no—
67#95×C59—22no—
68#95×C59—22no—
69#95×C59—42no—
70C24×C59—22no—
71C24×C59—22no—
72C24×C59—21no—
73C24×C59—21no—
74C24×C59—22no—
75C24×C59—22no—
76C24×C59—41no—
77C24×C59—41no—
78C24×C59C22⋊C411yescharacteristic
79C24×C59C22⋊C411yescharacteristic
80C24×C59C2×D411yescharacteristic
81C22×C4×C59—22no—
82C22×C4×C59—22no—
83C22×C4×C59—21no—
84C22×C4×C59—21no—
85C22×C4×C59—22no—
86C22×C4×C59—22no—
87C22×C4×C59—22no—
88C22×C4×C59—22no—
89C22×C4×C59—22no—
90C22×C4×C59—22no—
91C22×C4×C59—21no—
92C22×C4×C59—21no—
93C22×C4×C59—24no—
94C22×C4×C59—41no—
95C22×C4×C59—41no—
96C22×C4×C59—41no—
97C22×C4×C59—41no—
98C22×C4×C59—41no—
99C22×C4×C59—41no—
100C22×C4×C59—41no—
101C22×C4×C59—48no—
102C22×C4×C59—48no—
103C22×C4×C59—42no—
104C22×C4×C59—42no—
105C22×C4×C59—42no—
106C22×C4×C59—42no—
107C22×C4×C59—42no—
108C22×C4×C59—42no—
109C22×C4×C59C22⋊C411yescharacteristic
110C22×C4×C59C22⋊C411yescharacteristic
111C22×C4×C59C22×C411yescharacteristic
112C22×C4×C59C2×D411yescharacteristic
113C22×C4×C59C2×D412yes—
114C22×C4×C59#4112yes—
115C22×Dic59—21no—
116C22×Dic59—21no—
117C22×Dic59—21no—
118C22×Dic59—21no—
119C2×Dic118—22no—
120C2×Dic118—41no—
121C2×Dic118—41no—
122C22⋊C4×C59—48no—
123C22⋊Dic59—42no—
124C22⋊Dic59—42no—
125C22⋊Dic59—42no—
126C22⋊Dic59—42no—
127C22⋊Dic59—42no—
128C22⋊Dic59—41no—
129C22⋊Dic59—41no—
130C4⋊Dic59—22no—
131C4⋊Dic59—41no—
132C4⋊Dic59—41no—
133Dic59⋊C4—42no—
134Dic59⋊C4—42no—
135Dic59⋊C4—41no—
136Dic59⋊C4—41no—
137#11745664—48no—
138C2×C59⋊C8—24no—
139C2×C59⋊C8—42no—
140C2×C59⋊C8—42no—
141C23×C59—21no—
142C23×C59—21no—
143C23×C59—21no—
144C23×C59—21no—
145C23×C59—24no—
146C23×C59—24no—
147C23×C59—21no—
148C23×C59—21no—
149C23×C59—21no—
150C23×C59—21no—
151C23×C59—21no—
152C23×C59—21no—
153C23×C59—21no—
154C23×C59—21no—
155C23×C59—21no—
156C23×C59—21no—
157C23×C59—21no—
158C23×C59—21no—
159C23×C59—44no—
160C23×C59—44no—
161C23×C59—44no—
162C23×C59—44no—
163C23×C59—42no—
164C23×C59—42no—
165C23×C59—42no—
166C23×C59—42no—
167C23×C59#9912yes—
168C23×C59#10412yes—
169C23×C59C2×C22⋊C411yescharacteristic
170C23×C59#12112yes—
171C23×C59AGL(1,ℤ/8)11yescharacteristic
172C23×C59#13712yes—
173C23×C59#13711yescharacteristic
174C2×C4×C59—24no—
175C2×C4×C59—24no—
176C2×C4×C59—21no—
177C2×C4×C59—21no—
178C2×C4×C59—21no—
179C2×C4×C59—21no—
180C2×C4×C59—21no—
181C2×C4×C59—21no—
182C2×C4×C59—21no—
183C2×C4×C59—21no—
184C2×C4×C59—21no—
185C2×C4×C59—42no—
186C2×C4×C59—42no—
187C2×C4×C59—44no—
188C2×C4×C59—44no—
189C2×C4×C59—44no—
190C2×C4×C59—42no—
191C2×C4×C59—42no—
192C2×C4×C59—44no—
193C2×C4×C59—44no—
194C2×C4×C59—44no—
195C2×C4×C59—82no—
196C2×C4×C59—82no—
197C2×C4×C59—82no—
198C2×C4×C59—82no—
199C2×C4×C59#9912yes—
200C2×C4×C59C4×D412yes—
201C2×C4×C59#12011yescharacteristic
202C2×C4×C59#12311yescharacteristic
203C2×Dic59—41no—
204C2×Dic59—41no—
205C2×Dic59—41no—
206C2×Dic59—41no—
207C2×Dic59—41no—
208C2×Dic59—41no—
209C2×Dic59—41no—
210C2×Dic59—41no—
211C2×Dic59—41no—
212C2×Dic59—41no—
213C2×Dic59—41no—
214C2×Dic59—41no—
215Dic118—82no—
216C59⋊C8—44no—
217#9115—591nomaximal, Sylow
218C22×C59—22no—
219C22×C59—22no—
220C22×C59—21no—
221C22×C59—21no—
222C22×C59—22no—
223C22×C59—22no—
224C22×C59—21no—
225C22×C59—21no—
226C22×C59—22no—
227C22×C59—22no—
228C22×C59—22no—
229C22×C59—22no—
230C22×C59—22no—
231C22×C59—22no—
232C22×C59—48no—
233C22×C59—48no—
234C22×C59—41no—
235C22×C59—41no—
236C22×C59—41no—
237C22×C59—41no—
238C22×C59—41no—
239C22×C59—41no—
240C22×C59—41no—
241C22×C59—41no—
242C22×C59#38611yescharacteristic
243C22×C59C2×#9911yescharacteristic
244C22×C59C2×#9911yescharacteristic
245C22×C59#41711yescharacteristic
246C22×C59#41911yescharacteristic
247C22×C59C2×#10411yescharacteristic
248C22×C59#42111yescharacteristic
249C22×C59#46812yes—
250C22×C59#47012yes—
251C4×C59—21no—
252C4×C59—22no—
253C4×C59—22no—
254C4×C59—21no—
255C4×C59—41no—
256C4×C59—84no—
257Dic59—81no—
258Dic59—81no—
259Dic59—81no—
260Dic59—81no—
261C2×#407—591no—
262#2848—591no—
263C2×#328—592no—
264C22⋊#95D5911yescharacteristic
265#1294—592no—
266C2×C59—22no—
267C2×C59—22no—
268C2×C59—42no—
269C2×C59—42no—
270C2×C59#150411yescharacteristic
271C2×C59#150811yescharacteristic
272C2×C59#151111yescharacteristic
273C2×C59#151311yescharacteristic
274C2×C59#187111yescharacteristic
275C2×C59#189411yescharacteristic
276C2×C59#189811yescharacteristic
277C24×C4C2×D5911yescharacteristic
278C22×M16—1181no—
279C2×C22⋊Q8—591no—
280C2×C22⋊Q8—591no—
281C2×C22⋊Q8—591no—
282C22×C22⋊C4—21no—
283#407—1184no—
284C2×C22⋊C8—592no—
285C2×C22⋊C8—1181no—
286#385—1181no—
287#385—1181no—
288#385—1181no—
289#380—591no—
290#380—591no—
291#380—591no—
292#379—1181no—
293C2×#95—21no—
294C2×#95C2×D5912yes—
295#328—1184no—
296C59#911511yescharacteristic, complemented, Sylow
297C25C22⋊D5911yescharacteristic
298C22×Q8—591no—
299C23×C4—22no—
300C23×C4—22no—
301C23×C4—22no—
302C23×C4—42no—
303C23×C4—42no—
304C23×C4C4×D5911yescharacteristic
305C23×C4C4×D5911yescharacteristic
306C23×C4D23611yescharacteristic
307C23×C4D23611yescharacteristic
308C23×C4C22⋊D5911yescharacteristic
309C23×C4C22×D5911yescharacteristic
310C2×M16—1184no—
311C2×M16—2362no—
312C2×M16—2362no—
313C22×C8—1182no—
314C22⋊Q8—1182no—
315C22⋊Q8—1182no—
316C22⋊Q8—2362no—
317C22⋊Q8—2362no—
318C2×C4⋊C4—591no—
319C2×C4⋊C4—591no—
320C2×C4⋊C4—591no—
321C2×C4⋊C4—1181no—
322C2×C4⋊C4—1181no—
323C2×C4⋊C4—1181no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—42no—
326C2×C22⋊C4—42no—
327C2×C22⋊C4—1181no—
328C2×C22⋊C4—1181no—
329C2×C22⋊C4—1181no—
330C2×C22⋊C4—1181no—
331C2×C22⋊C4—1181no—
332C2×C22⋊C4—1181no—
333C22⋊C8—594no—
334C22⋊C8—1184no—
335C22⋊C8—1182no—
336#95—22no—
337#95—22no—
338#95—42no—
339#95—1181no—
340#95—1181no—
341#95—1181no—
342C24—22no—
343C24—22no—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—22no—
348C24—41no—
349C24—41no—
350C24(C2×D59)⋊C411yescharacteristic
351C24(C2×D59)⋊C411yescharacteristic
352C24C2×C22⋊D5911yescharacteristic
353C2×Q8—1182no—
354C2×Q8—2361no—
355C2×Q8—2361no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—1181no—
385C22×C4—1181no—
386C22×C4—1181no—
387C22×C4—1181no—
388C22×C4(C2×D59)⋊C411yescharacteristic
389C22×C4(C2×D59)⋊C411yescharacteristic
390C22×C4C2×C4×D5911yescharacteristic
391C22×C4C2×D23611yescharacteristic
392C22×C4D4×D5912yes—
393C22×C4#1174568712yes—
394M16—2368no—
395C2×C8—1184no—
396C2×C8—2362no—
397C2×C8—2362no—
398C4⋊C4—1182no—
399C4⋊C4—2361no—
400C4⋊C4—2361no—
401C4⋊C4—2362no—
402C4⋊C4—2362no—
403C4⋊C4—2361no—
404C4⋊C4—2361no—
405C22⋊C4—48no—
406C22⋊C4—2362no—
407C22⋊C4—2362no—
408C22⋊C4—2362no—
409C22⋊C4—2362no—
410C22⋊C4—2362no—
411C22⋊C4—2361no—
412C22⋊C4—2361no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—24no—
418C23—24no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23#1427364112yes—
440C23#1427365512yes—
441C23C2×(C2×D59)⋊C411yescharacteristic
442C23#1427369912yes—
443C23#1427375311yescharacteristic
444C23#1427375712yes—
445C23#1427375811yescharacteristic
446Q8—4722no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—2361no—
473C2×C4—2361no—
474C2×C4—2361no—
475C2×C4—2361no—
476C2×C4—2361no—
477C2×C4—2361no—
478C2×C4—2361no—
479C2×C4—2361no—
480C2×C4—2361no—
481C2×C4—2361no—
482C2×C4—2361no—
483C2×C4—2361no—
484C2×C4#1427364112yes—
485C2×C4(C22⋊D59)⋊C412yes—
486C2×C4#1427369711yescharacteristic
487C2×C4#1427371111yescharacteristic
488C8—2364no—
489C22—21no—
490C22—21no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—21no—
495C22—21no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—48no—
504C22—48no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22#1683727911yescharacteristic
514C22C2×#1427364111yescharacteristic
515C22C2×#1427364111yescharacteristic
516C22#1683740111yescharacteristic
517C22#1683741011yescharacteristic
518C22C2×#1427365511yescharacteristic
519C22#1683742011yescharacteristic
520C22#1683764612yes—
521C22#1683765812yes—
522C4—21no—
523C4—22no—
524C4—21no—
525C4—22no—
526C4—41no—
527C4—84no—
528C4—4721no—
529C4—4721no—
530C4—4721no—
531C4—4721no—
532C2—22no—
533C2—22no—
534C2—42no—
535C2—42no—
536C2#2142259811yescharacteristic
537C2#2142261511yescharacteristic
538C2#2142262811yescharacteristic
539C2#2142263711yescharacteristic
540C2#2142439611yescharacteristic
541C2#2142452711yescharacteristic
542C2#2142454711yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×C22⋊#95C2
socleC23×C59C2×C22⋊C4
prime radicalC25×C59D4
cosocleC23×C4×C59C23
hypercenterC22⋊#95D59
hypocenterC59#9115
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
3. C2×#16837138C2semidirect
4. #21421975C2semidirect
7. C2×#14273706C4semidirect
8. C2×#14273706C4semidirect
16. #16837262C4semidirect
17. #16837262C4semidirect
296. C59#9115semidirect

Maximal subgroups

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

subgroupindex
C2×#168373662
#214323832
C2×#168371382
#214219752
C59×C22⋊#952
#911559