Finite Group Databasebeta

← #28571267

Subgroups — #28571267

Order 15104.

684 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434100C211yesmaximal, characteristic, complemented
2#21434092C211yesmaximal, characteristic
3C2×#16837481C211yesmaximal, characteristic, complemented
4#21423668C212yesmaximal, complemented
5#21423654C212yesmaximal, complemented
6#21423640C212yesmaximal, complemented
7#21423626C212yesmaximal, complemented
8#21422844C211yesmaximal, characteristic, complemented
9#21422753C211yesmaximal, characteristic, complemented
10#21422745C211yesmaximal, characteristic, complemented
11C59×#3131C211yesmaximal, characteristic, complemented
12C22×#11745659—21no—
13C2×C4×D236C2211yescharacteristic
14#16837730—21no—
15#16837713—21no—
16#16837689C2212yes—
17#16837679C2212yes—
18#16837604C2212yes—
19#16837594C2212yes—
20#16837552—21no—
21#16837535—21no—
22C4×#11745671—22no—
23C4×#11745669—22no—
24#16837481—21no—
25#16837481—21no—
26#16837481C2212yes—
27#16837481C2212yes—
28#16837455C2211yescharacteristic
29#16837455C2211yescharacteristic
30C2×C4⋊(C59⋊C8)C2211yescharacteristic
31C2×(C2×D59)⋊C8—21no—
32#16837155C2212yescomplemented
33#16837153C2212yescomplemented
34#16837151C2212yescomplemented
35#16837148C2212yescomplemented
36#16837148C2212yescomplemented
37#16837146C2212yescomplemented
38C59×#533C2211yescharacteristic
39C59×#532C2211yescharacteristic
40C59×#488—21no—
41C59×#488—21no—
42C59×#487C2211yescharacteristic, complemented
43C59×#487C2211yescharacteristic, complemented
44C59×#486C2211yescharacteristic, complemented
45C59×#486C2211yescharacteristic, complemented
46C59×#416—21no—
47C59×#416—21no—
48C2×C4×C8×C59C2211yescharacteristic
49C22×D236D411yescharacteristic
50C22×C4×D59—21no—
51C2×#11745671—22no—
52C2×#11745669—22no—
53C2×#11745659—22no—
54C2×#11745659—22no—
55C2×#11745659—44no—
56C22×C59⋊C8—21no—
57C4×D236—21no—
58C4×D236—21no—
59C4×D236D412yes—
60C4×D236C2312yes—
61C2×C4⋊Dic59D411yescharacteristic
62C2×(C2×D59)⋊C4—21no—
63#14273660—22no—
64#14273658—22no—
65C4⋊(C59⋊C8)—21no—
66C4⋊(C59⋊C8)C2312yes—
67C4⋊(C59⋊C8)C2312yes—
68C4⋊(C59⋊C8)C2312yes—
69#14273651—22no—
70#14273648—22no—
71#14273647—22no—
72#14273647—22no—
73C22⋊(C59⋊C8)—21no—
74C22⋊(C59⋊C8)—21no—
75(C2×D59)⋊C8—22no—
76(C2×D59)⋊C8—22no—
77C4×C59⋊C8—21no—
78C4×C59⋊C8—21no—
79C2×#41×C59—21no—
80C22×C8×C59D411yescharacteristic
81C22×C8×C59D411yescharacteristic
82C4⋊Q8×C59C2311yescharacteristic
83C4⋊Q8×C59C2311yescharacteristic
84C4⋊Q8×C59C2311yescharacteristic
85#127×C59C2311yescharacteristic
86#125×C59—21no—
87#124×C59—21no—
88C22⋊Q8×C59—21no—
89C22⋊Q8×C59—21no—
90#121×C59—21no—
91#121×C59—21no—
92C4×Q8×C59—21no—
93C4×Q8×C59—21no—
94C4×D4×C59—21no—
95C4×D4×C59—21no—
96#117×C59—21no—
97C2×C42×C59C2311yescharacteristic
98#103×C59—21no—
99#103×C59—21no—
100#103×C59—21no—
101#103×C59—21no—
102#102×C59—21no—
103#102×C59—21no—
104#102×C59—21no—
105#102×C59—21no—
106C4×C8×C59D411yescharacteristic
107C4×C8×C59D411yescharacteristic
108C4×C8×C59C2311yescharacteristic
109C4×C8×C59C2311yescharacteristic
110C23×D59—21no—
111#41×C59—42no—
112#41×C59—42no—
113C2×Q8×C59—21no—
114C2×Q8×C59—21no—
115C2×Q8×C59—21no—
116C2×D4×C59—21no—
117C2×D4×C59—21no—
118C2×D4×C59—21no—
119C2×D4×C59—41no—
120C22×C4×C59—41no—
121C22×C4×C59C2×D411yescharacteristic
122C22×C4×C59C2×D411yescharacteristic
123C22×C4×C59#4111yescharacteristic
124C22×Dic59—21no—
125C2×D236—21no—
126C2×D236—42no—
127C2×D236C2×D412yes—
128C2×C4×D59—22no—
129C2×C4×D59—22no—
130C2×C4×D59—42no—
131C2×C4×D59—42no—
132C2×C8×C59—22no—
133C2×C8×C59—22no—
134C2×C8×C59C2×D411yescharacteristic
135C2×C8×C59C2×D411yescharacteristic
136C2×C8×C59C2×D411yescharacteristic
137C2×C8×C59C2×D411yescharacteristic
138C4⋊C4×C59—21no—
139C4⋊C4×C59—21no—
140C4⋊C4×C59—21no—
141C4⋊C4×C59—21no—
142C4⋊C4×C59—21no—
143C4⋊C4×C59—21no—
144C4⋊C4×C59—41no—
145C4⋊C4×C59—41no—
146C22⋊C4×C59—41no—
147C22⋊C4×C59—41no—
148C22⋊C4×C59—41no—
149C42×C59—41no—
150C42×C59C2×D411yescharacteristic
151C42×C59C2×D411yescharacteristic
152C42×C59#4111yescharacteristic
153C42×C59C2411yescharacteristic
154#11745671—44no—
155#11745669—44no—
156(C2×D59)⋊C4—22no—
157(C2×D59)⋊C4—22no—
158C4⋊Dic59—21no—
159C4⋊Dic59C2×D412yes—
160C2×C59⋊C8—22no—
161C2×C59⋊C8—22no—
162C2×C59⋊C8—22no—
163C2×C59⋊C8—42no—
164#11745659—44no—
165#11745659—44no—
166C23×C59—41no—
167C23×C59#12111yescharacteristic
168C22×D59—22no—
169C22×D59—42no—
170C22×D59—42no—
171C22×D59—42no—
172Q8×C59—41no—
173Q8×C59—41no—
174Q8×C59—42no—
175Q8×C59—81no—
176D4×C59—41no—
177D4×C59—41no—
178D4×C59—42no—
179D4×C59—81no—
180D4×C59—82no—
181C2×C4×C59—22no—
182C2×C4×C59—41no—
183C2×C4×C59—41no—
184C2×C4×C59—41no—
185C2×C4×C59—41no—
186C2×C4×C59—82no—
187C2×C4×C59#12111yescharacteristic
188C2×C4×C59#12111yescharacteristic
189C2×C4×C59#12111yescharacteristic
190C2×C4×C59#13512yes—
191C2×C4×C59AGL(1,ℤ/8)12yes—
192C2×C4×C59C22×D411yescharacteristic
193C2×C4×C59C22×D411yescharacteristic
194C2×C4×C59C2×#4111yescharacteristic
195C2×Dic59—22no—
196C2×Dic59—42no—
197D236—22no—
198D236—22no—
199D236—42no—
200C4×D59—42no—
201C4×D59—42no—
202C4×D59—44no—
203C8×C59—21no—
204C8×C59—21no—
205C8×C59—21no—
206C8×C59—21no—
207C59⋊C8—42no—
208C59⋊C8—42no—
209C59⋊C8—42no—
210#19142—591nomaximal, Sylow
211C22×C59—21no—
212C22×C59—21no—
213C22×C59—81no—
214C22×C59—81no—
215C22×C59—81no—
216C22×C59#46312yes—
217C22×C59C2×#12111yescharacteristic
218C2×D59—42no—
219C2×D59—42no—
220C2×D59—42no—
221C2×D59—84no—
222C4×C59—21no—
223C4×C59—21no—
224C4×C59—21no—
225C4×C59—81no—
226C4×C59—81no—
227C4×C59—81no—
228C4×C59#46312yes—
229C4×C59C2×#13511yescharacteristic
230C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
231C4×C59#57512yes—
232Dic59—42no—
233#3131D5911yescharacteristic, complemented
234#3101—591no—
235#3100—591no—
236C2×#436—591no—
237#1703—592no—
238#1701—592no—
239#1699—592no—
240#1697—592no—
241#1564—591no—
242#1542—591no—
243#1540—591no—
244C2×C59—22no—
245C2×C59—161no—
246C2×C59C2×#46311yescharacteristic
247C2×C59#305411yescharacteristic
248C2×C59#306711yescharacteristic
249D59—82no—
250C22×M16—1181no—
251#533C2×D5911yescharacteristic
252#532C2×D5911yescharacteristic
253C2×C4×D4—591no—
254#488—21no—
255#488—21no—
256#487C2×D5911yescharacteristic, complemented
257#487C2×D5911yescharacteristic, complemented
258#486C2×D5911yescharacteristic, complemented
259#486C2×D5911yescharacteristic, complemented
260#483—1181no—
261#480—1181no—
262#476—592no—
263#474—592no—
264#461—592no—
265#459—592no—
266#450—1181no—
267#447—1181no—
268C4×SD16—1182no—
269C4×D8—1182no—
270#436—592no—
271#436—592no—
272#436—1181no—
273#436—1181no—
274#431—591no—
275#431—591no—
276C2×C4⋊C8—591no—
277#416—21no—
278#416—21no—
279C2×C22⋊C8—1181no—
280C2×C4×C8C2×D5911yescharacteristic
281#335—592no—
282#334—592no—
283#333—592no—
284#332—592no—
285#332—592no—
286#331—592no—
287C59#1914211yescharacteristic, complemented, Sylow
288C2×#41—21no—
289C22×D4—591no—
290C23×C4—1181no—
291C2×SD16—1182no—
292C2×D8—1182no—
293C2×M16—1182no—
294C2×M16—1182no—
295C2×M16—2364no—
296C22×C8—1181no—
297C22×C8C22⋊D5911yescharacteristic
298C22×C8C22⋊D5911yescharacteristic
299C4⋊Q8C22×D5911yescharacteristic
300C4⋊Q8C22×D5911yescharacteristic
301C4⋊Q8C22×D5911yescharacteristic
302#127C22×D5911yescharacteristic
303#125—21no—
304#124—21no—
305C22⋊Q8—21no—
306C22⋊Q8—21no—
307#121—21no—
308#121—21no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—592no—
314C4×D4—592no—
315C4×D4—1181no—
316C4×D4—1181no—
317#117—21no—
318C2×C4⋊C4—591no—
319C2×C22⋊C4—1181no—
320C2×C42C22×D5911yescharacteristic
321#107—1182no—
322#106—1182no—
323C4⋊C8—592no—
324C4⋊C8—592no—
325C4⋊C8—592no—
326C4⋊C8—1181no—
327#103—21no—
328#103—21no—
329#103—21no—
330#103—21no—
331#103—1182no—
332#102—21no—
333#102—21no—
334#102—21no—
335#102—21no—
336#102—1182no—
337#102—1182no—
338#102—1182no—
339C22⋊C8—1181no—
340C22⋊C8—1181no—
341C22⋊C8—1182no—
342C22⋊C8—1182no—
343C4×C8—1181no—
344C4×C8—1181no—
345C4×C8C22⋊D5911yescharacteristic
346C4×C8C22⋊D5911yescharacteristic
347C4×C8C22×D5911yescharacteristic
348C4×C8C22×D5911yescharacteristic
349C24—1181no—
350#41—42no—
351#41—42no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—41no—
359C2×D4—592no—
360C2×D4—1181no—
361C2×D4—2362no—
362C22×C4—41no—
363C22×C4—1181no—
364C22×C4—1182no—
365C22×C4—1182no—
366C22×C4—2362no—
367C22×C4—2362no—
368C22×C4D4×D5911yescharacteristic
369C22×C4#1174568711yescharacteristic
370C22×C4C2×C22⋊D5911yescharacteristic
371SD16—2364no—
372D8—2364no—
373M16—2364no—
374M16—2364no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—1182no—
378C2×C8—1182no—
379C2×C8—1182no—
380C2×C8—2362no—
381C2×C8C2×C22⋊D5911yescharacteristic
382C2×C8C2×C22⋊D5911yescharacteristic
383C2×C8C2×C22⋊D5911yescharacteristic
384C2×C8C2×C22⋊D5911yescharacteristic
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—592no—
394C4⋊C4—1181no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—1182no—
399C22⋊C4—1182no—
400C42—41no—
401C42D4×D5911yescharacteristic
402C42#1174568711yescharacteristic
403C42C2×C22⋊D5911yescharacteristic
404C42C23×D5911yescharacteristic
405C23—41no—
406C23—1182no—
407C23—2362no—
408C23—2362no—
409C23—2362no—
410C23#1427369811yescharacteristic
411Q8—41no—
412Q8—41no—
413Q8—42no—
414Q8—81no—
415D4—41no—
416D4—41no—
417D4—42no—
418D4—81no—
419D4—82no—
420D4—1182no—
421D4—1182no—
422D4—2362no—
423C2×C4—22no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—82no—
429C2×C4—1182no—
430C2×C4—2362no—
431C2×C4—2362no—
432C2×C4—2362no—
433C2×C4—2364no—
434C2×C4#1427369811yescharacteristic
435C2×C4#1427369811yescharacteristic
436C2×C4#1427369811yescharacteristic
437C2×C4#1427374712yes—
438C2×C4#1427375212yes—
439C2×C4C2×D4×D5911yescharacteristic
440C2×C4C22×C22⋊D5911yescharacteristic
441C2×C4C2×#1174568711yescharacteristic
442C8—21no—
443C8—21no—
444C8—21no—
445C8—21no—
446C8—2362no—
447C8—2362no—
448C8—2362no—
449C22—21no—
450C22—21no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—4724no—
455C22—2362no—
456C22—2362no—
457C22—2362no—
458C22#1683761412yes—
459C22C2×#1427369811yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—81no—
464C4—81no—
465C4—81no—
466C4—2362no—
467C4#1683761412yes—
468C4C2×#1427374711yescharacteristic
469C4C2×#1427375211yescharacteristic
470C4#1683819912yes—
471C2—22no—
472C2—161no—
473C2—4722no—
474C2C2×#1683761411yescharacteristic
475C2#2143370111yescharacteristic
476C2#2143381211yescharacteristic
subgroupG/H
centerC2×C4#14273698
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC59×#3131C2
socleC22×C59C2×#121
prime radical#21434092C2
hypercenter#3131D59
hypocenterC59#19142
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
1. #21434100C2semidirect
3. C2×#16837481C2semidirect
4. #21423668C2semidirect
5. #21423654C2semidirect
6. #21423640C2semidirect
7. #21423626C2semidirect
8. #21422844C2semidirect
9. #21422753C2semidirect
10. #21422745C2semidirect
11. C59×#3131C2semidirect
32. #16837155C22semidirect
33. #16837153C22semidirect
34. #16837151C22semidirect
35. #16837148C22semidirect
36. #16837148C22semidirect
37. #16837146C22semidirect
42. C59×#487C22semidirect
43. C59×#487C22semidirect
44. C59×#486C22semidirect
45. C59×#486C22semidirect
233. #3131D59semidirect
256. #487C2×D59semidirect
257. #487C2×D59semidirect
258. #486C2×D59semidirect
259. #486C2×D59semidirect
287. C59#19142semidirect

Maximal subgroups

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

subgroupindex
#214341002
#214340922
C2×#168374812
#214236682
#214236542
#214236402
#214236262
#214228442
#214227532
#214227452
C59×#31312
#1914259