Finite Group Databasebeta

← #28482766

Subgroups — #28482766

Order 15104.

658 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426796C211yesmaximal, characteristic
2#21424411C211yesmaximal, characteristic
3#21424288C211yesmaximal, characteristic, complemented
4C2×#16837135C211yesmaximal, characteristic
5#21422353C211yesmaximal, characteristic, complemented
6#21421938C211yesmaximal, characteristic, complemented
7C59×#1294C211yesmaximal, characteristic, complemented
8C2×#14273701C2211yescharacteristic
9C2×(C22⋊D59)⋊C4—21no—
10C22×C4⋊Dic59C2211yescharacteristic
11C2×#14273658—21no—
12C2×#14273649—21no—
13C2×C22⋊(C59⋊C8)C2211yescharacteristic
14#16837292—21no—
15#16837285—21no—
16#16837280—21no—
17#16837279—21no—
18#16837273—21no—
19C2×C59⋊#95—21no—
20C2×C59⋊#95—21no—
21C2×C59⋊#95C2211yescharacteristic
22#16837165—21no—
23#16837164—21no—
24#16837135C414yescomplemented
25C2×C22⋊C8×C59C2211yescharacteristic
26C59×#380C2211yescharacteristic
27C2×#95×C59C2211yescharacteristic
28C22×D236D411yescharacteristic
29C22×C22⋊D59—21no—
30C23×Dic59—21no—
31C23×Dic59D411yescharacteristic
32C23×Dic59D411yescharacteristic
33C22×C4×D59—21no—
34C22×C59⋊C8—21no—
35#14273701—21no—
36#14273701—21no—
37#14273701C2×C412yes—
38#14273701Q812yes—
39(C22⋊D59)⋊C4—42no—
40(C22⋊D59)⋊C4—42no—
41C2×C4⋊Dic59—21no—
42C2×C4⋊Dic59—21no—
43C2×C4⋊Dic59—42no—
44C2×C4⋊Dic59D411yescharacteristic
45C2×Dic59⋊C4—21no—
46C2×(C2×D59)⋊C4—21no—
47C2×(C2×D59)⋊C4—21no—
48C2×(C2×D59)⋊C4—21no—
49C2×(C2×D59)⋊C4—21no—
50C2×C4×Dic59—21no—
51#14273658—42no—
52#14273649—42no—
53C22⋊(C59⋊C8)—21no—
54C22⋊(C59⋊C8)C2×C412yescomplemented
55C59⋊#95—21no—
56C59⋊#95—21no—
57C59⋊#95—21no—
58C59⋊#95—21no—
59C59⋊#95—42no—
60C59⋊#95—42no—
61C23×C4×C59—21no—
62C23×C4×C59C2311yescharacteristic
63C22×C8×C59—21no—
64C2×C4⋊C4×C59D411yescharacteristic
65C2×C4⋊C4×C59D411yescharacteristic
66C2×C22⋊C4×C59—21no—
67C22⋊C8×C59—21no—
68C22⋊C8×C59—21no—
69#95×C59—21no—
70#95×C59—21no—
71#95×C59C2×C412yescomplemented
72C24×C59C2×D411yescharacteristic
73C23×D59—21no—
74C22×C4×C59—22no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—41no—
78C22×C4×C59—41no—
79C22×C4×C59—41no—
80C22×C4×C59—41no—
81C22×C4×C59—41no—
82C22×C4×C59—41no—
83C22×C4×C59—41no—
84C22×C4×C59—41no—
85C22×C4×C59C22×C411yescharacteristic
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59C2×Q811yescharacteristic
89C22×C4×C59#4111yescharacteristic
90C22×C4×C59#4111yescharacteristic
91C2×C22⋊D59—22no—
92C2×C22⋊D59—22no—
93C2×C22⋊D59—41no—
94C2×C22⋊D59—41no—
95C2×C22⋊D59—41no—
96C2×C22⋊D59—41no—
97C22×Dic59—21no—
98C22×Dic59—21no—
99C22×Dic59—21no—
100C22×Dic59—22no—
101C22×Dic59—21no—
102C22×Dic59—21no—
103C22×Dic59—21no—
104C22×Dic59—41no—
105C22×Dic59—41no—
106C22×Dic59—41no—
107C22×Dic59—41no—
108C22×Dic59—41no—
109C22×Dic59—41no—
110C22×Dic59—41no—
111C22×Dic59—41no—
112C22×Dic59—41no—
113C22×Dic59—41no—
114C22×Dic59SD1612yes—
115C2×D236—21no—
116C2×D236—41no—
117C2×D236—41no—
118C2×D236C4⋊C412yes—
119C2×C4×D59—42no—
120C2×C4×D59—81no—
121C2×C4×D59—81no—
122C2×C8×C59—41no—
123C2×C8×C59—41no—
124C2×C8×C59—41no—
125C4⋊C4×C59—41no—
126C4⋊C4×C59—41no—
127C22⋊C4×C59—41no—
128C22⋊C4×C59—41no—
129(C2×D59)⋊C4—22no—
130(C2×D59)⋊C4—22no—
131(C2×D59)⋊C4—42no—
132(C2×D59)⋊C4—81no—
133(C2×D59)⋊C4—81no—
134C4⋊Dic59—21no—
135C4⋊Dic59—22no—
136C4⋊Dic59—22no—
137C4⋊Dic59—41no—
138C4⋊Dic59—41no—
139C4⋊Dic59C4⋊C412yes—
140Dic59⋊C4—42no—
141C4×Dic59—42no—
142C2×C59⋊C8—22no—
143C2×C59⋊C8—41no—
144C2×C59⋊C8—41no—
145C23×C59—21no—
146C23×C59—21no—
147C23×C59—21no—
148C23×C59—21no—
149C23×C59C4×D411yescharacteristic
150C23×C59#12011yescharacteristic
151C23×C59C22⋊Q811yescharacteristic
152C23×C59C2×SD1611yescharacteristic
153C23×C59#13511yescharacteristic
154C23×C59AGL(1,ℤ/8)11yescharacteristic
155C23×C59AGL(1,ℤ/8)11yescharacteristic
156C22×D59—22no—
157C22×D59—41no—
158C22×D59—41no—
159C22×D59—41no—
160C22×D59—41no—
161C22×D59—41no—
162C22×D59—41no—
163C2×C4×C59—21no—
164C2×C4×C59—21no—
165C2×C4×C59—21no—
166C2×C4×C59—21no—
167C2×C4×C59—41no—
168C2×C4×C59—41no—
169C2×C4×C59—41no—
170C2×C4×C59—41no—
171C2×C4×C59—42no—
172C2×C4×C59—42no—
173C2×C4×C59—42no—
174C2×C4×C59—41no—
175C2×C4×C59—41no—
176C2×C4×C59—41no—
177C2×C4×C59—41no—
178C2×C4×C59—41no—
179C2×C4×C59—41no—
180C2×C4×C59—81no—
181C2×C4×C59—81no—
182C2×C4×C59—81no—
183C2×C4×C59—81no—
184C2×C4×C59C2×C4⋊C411yescharacteristic
185C2×C4×C59C4×D411yescharacteristic
186C2×C4×C59C22⋊Q811yescharacteristic
187C2×C4×C59#12311yescharacteristic
188C22⋊D59—42no—
189C22⋊D59—42no—
190C22⋊D59—42no—
191C22⋊D59—42no—
192C2×Dic59—22no—
193C2×Dic59—22no—
194C2×Dic59—41no—
195C2×Dic59—41no—
196C2×Dic59—41no—
197C2×Dic59—41no—
198C2×Dic59—42no—
199C2×Dic59—42no—
200C2×Dic59—42no—
201C2×Dic59—42no—
202C2×Dic59—41no—
203C2×Dic59—41no—
204C2×Dic59—42no—
205C2×Dic59—41no—
206C2×Dic59—81no—
207C2×Dic59—81no—
208C2×Dic59—81no—
209C2×Dic59—81no—
210C2×Dic59—81no—
211C2×Dic59—81no—
212D236—42no—
213D236—81no—
214C4×D59—84no—
215C8×C59—81no—
216C59⋊C8—42no—
217#8870—591nomaximal, Sylow
218C22×C59—21no—
219C22×C59—21no—
220C22×C59—21no—
221C22×C59—21no—
222C22×C59—21no—
223C22×C59—21no—
224C22×C59—21no—
225C22×C59—21no—
226C22×C59—21no—
227C22×C59—21no—
228C22×C59—21no—
229C22×C59—21no—
230C22×C59#38511yescharacteristic
231C22×C59C4×SD1611yescharacteristic
232C22×C59#44211yescharacteristic
233C22×C59#44911yescharacteristic
234C22×C59#45011yescharacteristic
235C22×C59#45411yescharacteristic
236C22×C59#45511yescharacteristic
237C22×C59B(4,𝔽2)11yescharacteristic
238C22×C59#45811yescharacteristic
239C22×C59#47611yescharacteristic
240C22×C59#47811yescharacteristic
241C2×D59—42no—
242C2×D59—42no—
243C2×D59—42no—
244C2×D59—81no—
245C2×D59—81no—
246C2×D59—81no—
247C2×D59—81no—
248C4×C59—21no—
249C4×C59—21no—
250C4×C59—21no—
251C4×C59—21no—
252C4×C59—82no—
253C4×C59—81no—
254C4×C59—81no—
255Dic59—42no—
256Dic59—42no—
257Dic59—82no—
258Dic59—81no—
259#2325—591no—
260#1874—591no—
261#1853—591no—
262C2×#327—591no—
263#1442—591no—
264#1294D5911yescharacteristic, complemented
265#1280—591no—
266C2×C59—21no—
267C2×C59—21no—
268C2×C59—21no—
269C2×C59—21no—
270C2×C59#159711yescharacteristic
271C2×C59#160111yescharacteristic
272C2×C59#161211yescharacteristic
273C2×C59#161611yescharacteristic
274C2×C59#186011yescharacteristic
275C2×C59#186411yescharacteristic
276C2×C59#186711yescharacteristic
277D59—82no—
278C2×#121—591no—
279C2×C4×D4—1181no—
280C22×C4⋊C4—591no—
281C2×#106—1181no—
282C2×#102—1181no—
283C2×C22⋊C8—591no—
284C2×C22⋊C8C2×D5911yescharacteristic
285#388—1181no—
286#387—1181no—
287#386—1181no—
288#386—1181no—
289#385—1181no—
290#380C2×D5911yescharacteristic
291C2×#95—591no—
292C2×#95—1181no—
293C2×#95—1181no—
294C2×#95C2×D5911yescharacteristic
295#340—1181no—
296#340—1181no—
297#327—594no—
298C59#887011yescharacteristic, complemented, Sylow
299C22×D4—591no—
300C22×D4—1181no—
301C23×C4—21no—
302C23×C4—591no—
303C23×C4—591no—
304C23×C4—1181no—
305C23×C4—1181no—
306C23×C4C22×D5911yescharacteristic
307C22×C8—21no—
308C22×C8—1181no—
309#121—592no—
310#121—592no—
311#121—1181no—
312#121—1181no—
313C4×D4—2362no—
314C4×D4—2362no—
315C2×C4⋊C4—591no—
316C2×C4⋊C4—1181no—
317C2×C4⋊C4—1181no—
318C2×C4⋊C4—1181no—
319C2×C4⋊C4—2362no—
320C2×C4⋊C4D23611yescharacteristic
321C2×C4⋊C4D23611yescharacteristic
322C2×C22⋊C4—21no—
323C2×C22⋊C4—1181no—
324C2×C22⋊C4—1181no—
325C2×C22⋊C4—1181no—
326C2×C22⋊C4—1181no—
327C2×C42—1181no—
328#106—2362no—
329#102—2362no—
330C22⋊C8—21no—
331C22⋊C8—21no—
332C22⋊C8—592no—
333C22⋊C8—1181no—
334#95—21no—
335#95—21no—
336#95—1181no—
337#95—1181no—
338#95—1181no—
339#95—1181no—
340#95—2362no—
341#95—2362no—
342#95C4×D5912yescomplemented
343C24—1181no—
344C24D4×D5911yescharacteristic
345C2×D4—592no—
346C2×D4—1182no—
347C2×D4—1182no—
348C2×D4—1181no—
349C2×D4—2361no—
350C2×D4—2361no—
351C2×D4—2361no—
352C2×D4—2361no—
353C2×D4—2361no—
354C2×D4—2361no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—592no—
367C22×C4—1181no—
368C22×C4—1181no—
369C22×C4—1181no—
370C22×C4—1181no—
371C22×C4—1182no—
372C22×C4—1181no—
373C22×C4—1181no—
374C22×C4—4721no—
375C22×C4—4721no—
376C22×C4—2362no—
377C22×C4—2361no—
378C22×C4—2361no—
379C22×C4—2361no—
380C22×C4—2361no—
381C22×C4—2361no—
382C22×C4—2361no—
383C22×C4—2361no—
384C22×C4—2361no—
385C22×C4—2361no—
386C22×C4—2361no—
387C22×C4C2×C4×D5911yescharacteristic
388C22×C4C2×D23611yescharacteristic
389C22×C4#1174568511yescharacteristic
390C22×C4D4×D5911yescharacteristic
391C22×C4#1174568711yescharacteristic
392C22×C4Q8×D5911yescharacteristic
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—1182no—
397C2×C8—2361no—
398C2×C8—2361no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—592no—
402C4⋊C4—1182no—
403C4⋊C4—1182no—
404C4⋊C4—1181no—
405C4⋊C4—2361no—
406C4⋊C4—2361no—
407C4⋊C4—2362no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—1182no—
411C22⋊C4—1182no—
412C22⋊C4—4721no—
413C22⋊C4—4721no—
414C22⋊C4—2362no—
415C42—2362no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—1182no—
421C23—2361no—
422C23—2361no—
423C23—2361no—
424C23—2361no—
425C23—2361no—
426C23—2361no—
427C23(C22⋊D59)⋊C411yescharacteristic
428C23#1427369711yescharacteristic
429C23#1427370511yescharacteristic
430C23SD16×D5911yescharacteristic
431C23#1427375011yescharacteristic
432C23#1427375111yescharacteristic
433C23#1427375411yescharacteristic
434D4—4721no—
435D4—2362no—
436D4—2362no—
437D4—2362no—
438D4—2362no—
439D4—2362no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—1182no—
462C2×C4—1182no—
463C2×C4—4724no—
464C2×C4—4721no—
465C2×C4—4721no—
466C2×C4—4721no—
467C2×C4—4721no—
468C2×C4—4721no—
469C2×C4—4721no—
470C2×C4—2362no—
471C2×C4—2362no—
472C2×C4—2362no—
473C2×C4—2361no—
474C2×C4—2361no—
475C2×C4—2362no—
476C2×C4—2361no—
477C2×C4—2361no—
478C2×C4—2361no—
479C2×C4—2361no—
480C2×C4—2361no—
481C2×C4—2362no—
482C2×C4C4⋊C4×D5911yescharacteristic
483C2×C4C4×D23611yescharacteristic
484C2×C4#1427370711yescharacteristic
485C2×C4#1427370911yescharacteristic
486C8—81no—
487C8—2362no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—4721no—
501C22—4721no—
502C22—4721no—
503C22—4721no—
504C22—2362no—
505C22—2362no—
506C22—2362no—
507C22#1683727611yescharacteristic
508C22#1683748711yescharacteristic
509C22#1683751111yescharacteristic
510C22#1683755011yescharacteristic
511C22#1683755411yescharacteristic
512C22#1683757611yescharacteristic
513C22#1683758111yescharacteristic
514C22#1683759011yescharacteristic
515C22#1683759311yescharacteristic
516C22#1683769111yescharacteristic
517C22#1683770511yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—82no—
523C4—81no—
524C4—81no—
525C4—4721no—
526C4—4722no—
527C4—2362no—
528C4—2362no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—21no—
533C2—4722no—
534C2#2142302811yescharacteristic
535C2#2142304911yescharacteristic
536C2#2142311311yescharacteristic
537C2#2142313411yescharacteristic
538C2#2142433011yescharacteristic
539C2#2142435811yescharacteristic
540C2#2142437811yescharacteristic
subgroupG/H
centerC23#14273697
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1294C2
socleC23×C59#120
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenter#1294D59
hypocenterC59#8870
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
3. #21424288C2semidirect
5. #21422353C2semidirect
6. #21421938C2semidirect
7. C59×#1294C2semidirect
24. #16837135C4semidirect
54. C22⋊(C59⋊C8)C2×C4semidirect
71. #95×C59C2×C4semidirect
264. #1294D59semidirect
298. C59#8870semidirect
342. #95C4×D59semidirect

Maximal subgroups

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

subgroupindex
#214267962
#214244112
#214242882
C2×#168371352
#214223532
#214219382
C59×#12942
#887059