Finite Group Databasebeta

← #28743166

Subgroups — #28743166

Order 15104.

738 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837840C211yesmaximal, characteristic, complemented
2#21433677C211yesmaximal, characteristic, complemented
3C2×#16837638C211yesmaximal, characteristic, complemented
4#21426209C212yesmaximal, complemented
5#21426199C212yesmaximal, complemented
6#21426192C212yesmaximal, complemented
7#21426182C212yesmaximal, complemented
8#21425884C211yesmaximal, characteristic, complemented
9#21425813C211yesmaximal, characteristic, complemented
10#21425809C211yesmaximal, characteristic, complemented
11C22×M32×C59C211yesmaximal, characteristic, complemented
12C2×#14273749C2211yescharacteristic
13C22×D472C2211yescharacteristic
14#16837913—21no—
15C2×#14273701—21no—
16#16837840—24no—
17C2×#14273672C2212yescomplemented
18C2×D944C2212yescomplemented
19#16837643C2211yescharacteristic, complemented
20#16837643C2211yescharacteristic, complemented
21#16837638—21no—
22#16837638—21no—
23#16837638C2212yes—
24#16837638C2212yes—
25#16837638C2211yescharacteristic, complemented
26#16837638C2211yescharacteristic, complemented
27#16837631—21no—
28#16837631—21no—
29#16837440C2211yescharacteristic
30C2×#14273661C2211yescharacteristic
31#16837399—21no—
32C2×#14273649—21no—
33#16837230C2212yescomplemented
34#16837228C2212yescomplemented
35#16837207C2212yescomplemented
36#16837204C2212yescomplemented
37#16837204C2212yescomplemented
38#16837204C2212yescomplemented
39C23×C8×C59C2211yescharacteristic
40C2×M32×C59—24no—
41C2×M32×C59C2211yescharacteristic
42C2×M32×C59C2211yescharacteristic
43C2×M32×C59C2212yescomplemented
44C22×C16×C59C2212yescomplemented
45C2×#11745685—21no—
46C22×D236—21no—
47C22×C22⋊D59—41no—
48#14273749—22no—
49#14273749—22no—
50C2×Dic236C2311yescharacteristic
51C2×#11745660—21no—
52C2×D472—22no—
53C2×D472—22no—
54C2×D472—21no—
55C2×D472C2312yes—
56C2×D472C2311yescharacteristic
57#14273710—41no—
58#14273706—21no—
59#14273706—21no—
60#14273701—22no—
61#14273701—22no—
62#14273701—21no—
63#14273701—21no—
64#14273701—41no—
65#14273701—41no—
66#14273696—41no—
67C4×C22⋊D59—41no—
68C4×C22⋊D59—41no—
69#14273686—21no—
70C2×C4⋊Dic59—21no—
71C2×(C2×D59)⋊C4—41no—
72#14273672—24no—
73D944—24no—
74#14273661—21no—
75#14273661C2312yes—
76#14273661C2311yescharacteristic
77#14273661C2311yescharacteristic
78#14273659—21no—
79#14273652—21no—
80#14273652—21no—
81#14273649—22no—
82#14273649—22no—
83#14273649—21no—
84#14273649—21no—
85C23×C4×C59D411yescharacteristic
86C22×C8×C59—22no—
87C22×C8×C59—22no—
88C22×C8×C59D411yescharacteristic
89C22×C8×C59D411yescharacteristic
90C22×C8×C59D411yescharacteristic
91C22×C8×C59C2311yescharacteristic
92C22×C8×C59C2311yescharacteristic
93C22×C8×C59C2311yescharacteristic
94M32×C59—22no—
95M32×C59—24no—
96M32×C59D414yescomplemented
97C2×C16×C59—24no—
98C2×C16×C59C2312yes—
99C2×C16×C59C2312yes—
100C24×C59D811yescharacteristic
101C23×D59—41no—
102C22×C4×C59—22no—
103C22×C4×C59—22no—
104C22×C4×C59D811yescharacteristic
105C22×C4×C59D811yescharacteristic
106C22×C4×C59D811yescharacteristic
107C22×C4×C59C2×D411yescharacteristic
108C22×C4×C59C2×D411yescharacteristic
109C22×C4×C59C2×D411yescharacteristic
110C2×C22⋊D59—41no—
111C2×C22⋊D59—41no—
112C2×C22⋊D59—41no—
113C2×C22⋊D59—42no—
114C2×C22⋊D59—42no—
115C2×C22⋊D59—84no—
116C22×Dic59—41no—
117#11745685—42no—
118#11745685—42no—
119C2×D236—22no—
120C2×D236—21no—
121C2×D236—42no—
122C2×D236—42no—
123C2×D236—41no—
124C2×C4×D59—41no—
125C2×Dic118—21no—
126C2×C8×C59—22no—
127C2×C8×C59—22no—
128C2×C8×C59—22no—
129C2×C8×C59—21no—
130C2×C8×C59—21no—
131C2×C8×C59—21no—
132C2×C8×C59—21no—
133C2×C8×C59C2×D411yescharacteristic
134C2×C8×C59C2×D411yescharacteristic
135C2×C8×C59C2×D411yescharacteristic
136C2×C8×C59C2×D412yes—
137C2×C8×C59#4112yes—
138C2×C8×C59C2411yescharacteristic
139C22⋊Dic59—41no—
140C22⋊Dic59—41no—
141C22⋊Dic59—41no—
142(C2×D59)⋊C4—42no—
143(C2×D59)⋊C4—42no—
144(C2×D59)⋊C4—41no—
145(C2×D59)⋊C4—41no—
146C4⋊Dic59—22no—
147C4⋊Dic59—21no—
148C4⋊Dic59—21no—
149C4⋊Dic59—41no—
150Dic59⋊C4—41no—
151Dic59⋊C4—41no—
152C4×Dic59—41no—
153Dic236—21no—
154Dic236—21no—
155D472—21no—
156D472—22no—
157D472—22no—
158D472—21no—
159D472—42no—
160#11745660—42no—
161C16×C59—22no—
162C16×C59C2×D414yescomplemented
163C23×C59—22no—
164C23×C59—22no—
165C23×C59—41no—
166C23×C59C2×D811yescharacteristic
167C23×C59C2×D811yescharacteristic
168C23×C59C2×D811yescharacteristic
169C22×D59—41no—
170C22×D59—42no—
171C22×D59—82no—
172C22×D59—82no—
173C22×D59—82no—
174C2×C4×C59—22no—
175C2×C4×C59—21no—
176C2×C4×C59—21no—
177C2×C4×C59—21no—
178C2×C4×C59—21no—
179C2×C4×C59—22no—
180C2×C4×C59—22no—
181C2×C4×C59#12112yes—
182C2×C4×C59C2×D811yescharacteristic
183C2×C4×C59C2×D811yescharacteristic
184C2×C4×C59C2×D811yescharacteristic
185C2×C4×C59#13512yes—
186C2×C4×C59C22×D411yescharacteristic
187C22⋊D59—82no—
188C22⋊D59—84no—
189C22⋊D59—82no—
190C22⋊D59—82no—
191C22⋊D59—84no—
192C2×Dic59—41no—
193C2×Dic59—41no—
194C2×Dic59—41no—
195C2×Dic59—42no—
196C2×Dic59—82no—
197D236—41no—
198D236—42no—
199D236—42no—
200D236—41no—
201D236—82no—
202C4×D59—82no—
203Dic118—41no—
204Dic118—41no—
205C8×C59—21no—
206C8×C59—21no—
207C8×C59#12112yes—
208C8×C59C22×D411yescharacteristic
209C8×C59C2×#4111yescharacteristic
210#33605—591nomaximal, Sylow
211C22×C59—21no—
212C22×C59—21no—
213C22×C59—21no—
214C22×C59—21no—
215C22×C59—21no—
216C22×C59—21no—
217C22×C59—21no—
218C22×C59—44no—
219C22×C59#46612yes—
220C22×C59#50912yescomplemented
221C22×C59C22×D811yescharacteristic
222C2×D59—82no—
223C2×D59—81no—
224C2×D59—81no—
225C2×D59—82no—
226C2×D59—81no—
227C2×D59—82no—
228C2×D59—164no—
229C4×C59—21no—
230C4×C59—21no—
231C4×C59#46612yes—
232C4×C59C2×#12111yescharacteristic
233C4×C59C2×#13511yescharacteristic
234Dic59—82no—
235Dic59—81no—
236Dic59—81no—
237Dic59—81no—
238C2×#509—591no—
239C22×M32D5911yescharacteristic, complemented
240#3050—591no—
241C2×#466—591no—
242#2220—592no—
243#2218—592no—
244#2217—592no—
245#2215—592no—
246#2159—591no—
247#2140—591no—
248#2139—591no—
249C2×C59—22no—
250C2×C59—22no—
251C2×C59—41no—
252C2×C59C2×#46611yescharacteristic
253C2×C59C2×#50911yescharacteristic
254C2×C59#341411yescharacteristic
255D59—161no—
256D59—162no—
257C2×#135—591no—
258C22×D8—591no—
259C23×C8C2×D5911yescharacteristic
260#525—1181no—
261C2×#121—1181no—
262#509—1184no—
263C2×SD32—592no—
264C2×D16—592no—
265C2×M32—24no—
266C2×M32C2×D5911yescharacteristic
267C2×M32C2×D5911yescharacteristic
268C2×M32C2×D5912yescomplemented
269C22×C16C2×D5912yescomplemented
270#467—591no—
271#467—591no—
272#466—592no—
273#466—592no—
274#466—591no—
275#466—591no—
276#466—1181no—
277#466—1181no—
278#465—1181no—
279#465—1181no—
280#427—591no—
281C2×#107—591no—
282#416—1181no—
283C2×#102—1181no—
284#367—592no—
285#366—592no—
286#358—592no—
287#357—592no—
288#357—592no—
289#357—592no—
290C59#3360511yescharacteristic, complemented, Sylow
291C2×#41—1181no—
292C22×D4—1181no—
293C22×D4—2361no—
294C23×C4D23611yescharacteristic
295#135—1182no—
296#135—1182no—
297C2×Q16—591no—
298C2×SD16—1181no—
299C2×D8—592no—
300C2×D8—591no—
301C2×D8—1181no—
302C2×D8—1182no—
303C2×D8—1182no—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8D23611yescharacteristic
307C22×C8D23611yescharacteristic
308C22×C8D23611yescharacteristic
309C22×C8C22×D5911yescharacteristic
310C22×C8C22×D5911yescharacteristic
311C22×C8C22×D5911yescharacteristic
312#123—2361no—
313C22⋊Q8—1181no—
314C22⋊Q8—1181no—
315#121—1182no—
316#121—1182no—
317#121—1181no—
318#121—1181no—
319#121—2361no—
320#121—2361no—
321#120—2361no—
322C4×D4—2361no—
323C4×D4—2361no—
324#117—1181no—
325C2×C4⋊C4—1181no—
326C2×C22⋊C4—2361no—
327SD32—1184no—
328D16—1184no—
329M32—22no—
330M32—24no—
331M32C22⋊D5914yescomplemented
332C2×C16—24no—
333C2×C16C22×D5912yes—
334C2×C16C22×D5912yes—
335#107—591no—
336#107—591no—
337#107—592no—
338#107—1181no—
339#106—1181no—
340#103—1181no—
341#103—1181no—
342#102—1182no—
343#102—1182no—
344#102—1181no—
345#102—1181no—
346C24—2361no—
347C24D47211yescharacteristic
348#41—2362no—
349#41—2362no—
350C2×Q8—1181no—
351C2×D4—1182no—
352C2×D4—1181no—
353C2×D4—4724no—
354C2×D4—2361no—
355C2×D4—2361no—
356C2×D4—2361no—
357C2×D4—2362no—
358C2×D4—2362no—
359C2×D4—2362no—
360C2×D4—2361no—
361C2×D4—2362no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—2361no—
365C22×C4—2361no—
366C22×C4D47211yescharacteristic
367C22×C4D47211yescharacteristic
368C22×C4D47211yescharacteristic
369C22×C4C2×D23611yescharacteristic
370C22×C4C2×D23611yescharacteristic
371C22×C4C2×D23611yescharacteristic
372Q16—1181no—
373Q16—1181no—
374SD16—2362no—
375D8—1181no—
376D8—1182no—
377D8—1182no—
378D8—1181no—
379D8—2362no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8C2×D23611yescharacteristic
388C2×C8C2×D23611yescharacteristic
389C2×C8C2×D23611yescharacteristic
390C2×C8#1174568512yes—
391C2×C8C2×C22⋊D5912yes—
392C2×C8C23×D5911yescharacteristic
393C4⋊C4—1182no—
394C4⋊C4—1181no—
395C4⋊C4—1181no—
396C4⋊C4—2361no—
397C4⋊C4—2361no—
398C4⋊C4—2361no—
399C22⋊C4—2361no—
400C22⋊C4—2361no—
401C22⋊C4—2361no—
402C22⋊C4—2362no—
403C22⋊C4—2362no—
404C22⋊C4—2361no—
405C22⋊C4—2361no—
406C42—2361no—
407C16—22no—
408C16C2×C22⋊D5914yescomplemented
409C23—22no—
410C23—22no—
411C23—41no—
412C23—4722no—
413C23—4722no—
414C23—4722no—
415C23—2361no—
416C23—2362no—
417C23C2×D47211yescharacteristic
418C23C2×D47211yescharacteristic
419C23C2×D47211yescharacteristic
420Q8—2361no—
421Q8—2361no—
422D4—4724no—
423D4—4722no—
424D4—4722no—
425D4—4724no—
426D4—4722no—
427D4—4722no—
428D4—2362no—
429D4—2361no—
430D4—2362no—
431D4—2361no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—4722no—
440C2×C4—4722no—
441C2×C4—2361no—
442C2×C4—2361no—
443C2×C4—2362no—
444C2×C4—2361no—
445C2×C4#1427370112yes—
446C2×C4C2×D47211yescharacteristic
447C2×C4C2×D47211yescharacteristic
448C2×C4C2×D47211yescharacteristic
449C2×C4#1427374912yes—
450C2×C4C22×D23611yescharacteristic
451C8—21no—
452C8—21no—
453C8#1427370112yes—
454C8C22×C22⋊D5911yescharacteristic
455C8C2×#1174568511yescharacteristic
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—44no—
464C22—9444no—
465C22—4722no—
466C22—4721no—
467C22—4721no—
468C22—4722no—
469C22—4721no—
470C22—4722no—
471C22#1683763812yes—
472C22#1683784012yescomplemented
473C22C22×D47211yescharacteristic
474C4—21no—
475C4—21no—
476C4—4722no—
477C4—4721no—
478C4—4721no—
479C4—4721no—
480C4#1683763812yes—
481C4C2×#1427370111yescharacteristic
482C4C2×#1427374911yescharacteristic
483C2—22no—
484C2—22no—
485C2—41no—
486C2—9442no—
487C2—9441no—
488C2C2×#1683763811yescharacteristic
489C2C2×#1683784011yescharacteristic
490C2#2143723211yescharacteristic
subgroupG/H
centerC22C22×D472
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC22×M32×C59C2
socleC22×C59C22×D8
hypercenterC22×M32D59
hypocenterC59#33605
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C22×M32

Complements

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

NH ≅ G/Ntype
1. C2×#16837840C2semidirect
2. #21433677C2semidirect
3. C2×#16837638C2semidirect
4. #21426209C2semidirect
5. #21426199C2semidirect
6. #21426192C2semidirect
7. #21426182C2semidirect
8. #21425884C2semidirect
9. #21425813C2semidirect
10. #21425809C2semidirect
11. C22×M32×C59C2semidirect
17. C2×#14273672C22semidirect
18. C2×D944C22semidirect
19. #16837643C22semidirect
20. #16837643C22semidirect
25. #16837638C22semidirect
26. #16837638C22semidirect
33. #16837230C22semidirect
34. #16837228C22semidirect
35. #16837207C22semidirect
36. #16837204C22semidirect
37. #16837204C22semidirect
38. #16837204C22semidirect
43. C2×M32×C59C22semidirect
44. C22×C16×C59C22semidirect
96. M32×C59D4semidirect
162. C16×C59C2×D4semidirect
220. C22×C59#509semidirect
239. C22×M32D59semidirect
268. C2×M32C2×D59semidirect
269. C22×C16C2×D59semidirect
290. C59#33605semidirect
331. M32C22⋊D59semidirect
408. C16C2×C22⋊D59semidirect
472. C22#16837840semidirect

Maximal subgroups

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

subgroupindex
C2×#168378402
#214336772
C2×#168376382
#214262092
#214261992
#214261922
#214261822
#214258842
#214258132
#214258092
C22×M32×C592
#3360559