Finite Group Databasebeta

← #28743371

Subgroups — #28743371

Order 15104.

556 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#21434629C211yesmaximal, characteristic, complemented
2#21432943C211yesmaximal, characteristic
3#21426256C211yesmaximal, characteristic, complemented
4#21426254C211yesmaximal, characteristic, complemented
5#21426253C211yesmaximal, characteristic, complemented
6#21426251C211yesmaximal, characteristic, complemented
7#21426241C211yesmaximal, characteristic, complemented
8#21426240C211yesmaximal, characteristic, complemented
9#21426232C211yesmaximal, characteristic, complemented
10#21425881C211yesmaximal, characteristic, complemented
11#21425877C211yesmaximal, characteristic, complemented
12#21425853C211yesmaximal, characteristic, complemented
13#21425810C211yesmaximal, characteristic
14#357×D59C211yesmaximal, characteristic
15C59×#2224C211yesmaximal, characteristic, complemented
16C2×D8×D59C2211yescharacteristic
17#16838029—21no—
18#16837874—21no—
19C2×#14273666C2211yescharacteristic
20C2×C16×D59C2211yescharacteristic
21#16837810—21no—
22#16837808—21no—
23#16837806C2211yescharacteristic, complemented
24#16837804C2211yescharacteristic, complemented
25#16837803C2211yescharacteristic, complemented
26#16837512—21no—
27#16837511—21no—
28C4×D472C2211yescharacteristic
29D8×Dic59C2211yescharacteristic
30#16837484C2211yescharacteristic
31#16837443—21no—
32#16837439C2211yescharacteristic
33#16837439C2211yescharacteristic
34#107×D59C2211yescharacteristic
35#16837405—21no—
36#16837356C2211yescharacteristic
37#16837230C2211yescharacteristic, complemented
38#16837229C2211yescharacteristic, complemented
39#16837229C2211yescharacteristic, complemented
40#16837229C2211yescharacteristic, complemented
41#16837228C2211yescharacteristic, complemented
42#16837227C2211yescharacteristic, complemented
43C4⋊(C59⋊C16)C2211yescharacteristic, complemented
44Dic59⋊C16C2211yescharacteristic, complemented
45Dic59⋊C16C2211yescharacteristic, complemented
46#16837204C2211yescharacteristic
47#16837204C2211yescharacteristic
48#16837203C2211yescharacteristic
49#16837203C2211yescharacteristic
50#16837202C2211yescharacteristic
51#16837202C2211yescharacteristic
52C59×#500C2211yescharacteristic, complemented
53C4×D8×C59C2211yescharacteristic
54C59×#366C2211yescharacteristic, complemented
55C59×#366C2211yescharacteristic, complemented
56C59×C4⋊C16C2211yescharacteristic, complemented
57C59×#357C2211yescharacteristic
58C59×#357C2211yescharacteristic
59C2×D4×D59—21no—
60C2×D472C2311yescharacteristic
61C2×#11745669—21no—
62D8×D59—22no—
63D8×D59Q814yescomplemented
64C2×#11745659D411yescharacteristic
65C2×C8×D59C2311yescharacteristic
66#14273726—21no—
67#14273725—41no—
68#14273719—21no—
69#14273718—21no—
70#14273717—41no—
71#14273705—41no—
72#14273703—41no—
73C4×D236—21no—
74D4×Dic59—21no—
75#14273689—21no—
76C4×C22⋊D59—41no—
77(C22⋊D59)⋊C4—41no—
78#14273685—21no—
79#14273685—21no—
80#14273684D411yescharacteristic
81C4⋊C4×D59—21no—
82C2×(C2×D59)⋊C4—41no—
83C22⋊C4×D59—41no—
84#14273666—22no—
85C2×C59⋊C16C2311yescharacteristic
86C2×C59⋊C16C2311yescharacteristic
87C16×D59—22no—
88#14273661C2311yescharacteristic
89#14273661C2311yescharacteristic
90#14273661C2311yescharacteristic
91#14273660—21no—
92#14273660—21no—
93#14273659—21no—
94#14273658—21no—
95#14273658—21no—
96#14273658—21no—
97#14273649—21no—
98#14273648—21no—
99#14273647—21no—
100#14273638D411yescharacteristic
101#14273637D411yescharacteristic
102C8×Dic59C2311yescharacteristic
103C2×D8×C59C2311yescharacteristic
104C4⋊Q8×C59—21no—
105C4×D4×C59—21no—
106C2×C16×C59C2311yescharacteristic
107C2×C16×C59C2311yescharacteristic
108#107×C59—21no—
109#107×C59C2311yescharacteristic
110#107×C59C2311yescharacteristic
111#107×C59C2311yescharacteristic
112#102×C59—21no—
113C4×C8×C59C2311yescharacteristic
114C23×D59—41no—
115C2×Q8×C59—41no—
116C2×D4×C59—21no—
117C22×C4×C59—41no—
118C2×C22⋊D59—41no—
119C22×Dic59—41no—
120D4×D59—24no—
121D4×D59—42no—
122C2×D236—21no—
123C2×C4×D59—41no—
124C2×C4×D59—41no—
125C2×C4×D59—41no—
126C2×C4×D59C2×D411yescharacteristic
127C2×Dic118—41no—
128D8×C59—21no—
129D8×C59C2×Q812yes—
130C2×C8×C59C2×D411yescharacteristic
131C2×C8×C59C2411yescharacteristic
132C4⋊C4×C59—21no—
133C4⋊C4×C59—21no—
134C4⋊C4×C59—21no—
135C4⋊C4×C59—41no—
136C22⋊C4×C59—41no—
137C42×C59C2×D411yescharacteristic
138C22⋊Dic59—41no—
139#11745669—24no—
140(C2×D59)⋊C4—41no—
141(C2×D59)⋊C4—41no—
142(C2×D59)⋊C4—41no—
143(C2×D59)⋊C4—41no—
144(C2×D59)⋊C4D811yescharacteristic
145(C2×D59)⋊C4D811yescharacteristic
146C4⋊Dic59—21no—
147C4⋊Dic59—21no—
148C4⋊Dic59—21no—
149C4⋊Dic59—41no—
150Dic59⋊C4—41no—
151Dic59⋊C4—41no—
152Dic59⋊C4—41no—
153Dic59⋊C4—41no—
154Dic59⋊C4—41no—
155Dic59⋊C4D811yescharacteristic
156Dic59⋊C4D811yescharacteristic
157C4×Dic59—41no—
158C4×Dic59—41no—
159C4×Dic59C2×D411yescharacteristic
160C2×C59⋊C8C2×D411yescharacteristic
161C2×C59⋊C8C2×D411yescharacteristic
162D472—21no—
163D472C2×Q812yes—
164#11745659—22no—
165C8×D59C2×Q812yes—
166C8×D59#4112yes—
167C16×C59—21no—
168C16×C59—21no—
169C59⋊C16—21no—
170C59⋊C16—21no—
171C23×C59—41no—
172C22×D59—41no—
173C22×D59—44no—
174C22×D59—161no—
175C22×D59—161no—
176C22×D59C2×D811yescharacteristic
177Q8×C59—161no—
178D4×C59—22no—
179D4×C59—41no—
180C2×C4×C59—41no—
181C2×C4×C59—41no—
182C2×C4×C59—41no—
183C2×C4×C59—161no—
184C2×C4×C59C2×D811yescharacteristic
185C2×C4×C59C2×D811yescharacteristic
186C2×C4×C59C22×D411yescharacteristic
187C22⋊D59—44no—
188C2×Dic59—41no—
189C2×Dic59—41no—
190C2×Dic59—41no—
191C2×Dic59—161no—
192C2×Dic59C2×D811yescharacteristic
193C2×Dic59C2×D811yescharacteristic
194C2×Dic59C2×D811yescharacteristic
195D236—22no—
196D236—41no—
197C4×D59—161no—
198C4×D59—161no—
199C4×D59—161no—
200C4×D59C22⋊Q812yes—
201C4×D59#13712yes—
202Dic118—161no—
203C8×C59—21no—
204C8×C59C22×Q811yescharacteristic
205C8×C59C2×#4111yescharacteristic
206C59⋊C8—21no—
207C59⋊C8C22⋊Q812yes—
208#33631—591nomaximal, Sylow
209C22×C59—42no—
210C22×C59—81no—
211C22×C59—81no—
212C22×C59C22×D811yescharacteristic
213C2×D59—21no—
214C2×D59—21no—
215C2×D59—42no—
216C2×D59—81no—
217C2×D59—81no—
218C2×D59—164no—
219C2×D59#47412yes—
220C4×C59—21no—
221C4×C59—21no—
222C4×C59—81no—
223C4×C59—81no—
224C4×C59—81no—
225C4×C59C2×C22⋊Q811yescharacteristic
226C4×C59C2×#13711yescharacteristic
227Dic59—21no—
228Dic59—21no—
229Dic59—81no—
230Dic59—81no—
231Dic59—81no—
232Dic59#47412yes—
233#3164—591no—
234#2948—591no—
235#2228—591no—
236#2228—591no—
237#2228—591no—
238#2228—591no—
239#2226—591no—
240#2226—591no—
241#2224—591no—
242#2224D5911yescharacteristic, complemented
243#2159—591no—
244#2158—591no—
245#2151—591no—
246#2140—591no—
247C2×#357—591no—
248C2×C59—82no—
249C2×C59C2×#47411yescharacteristic
250C2×C59#341511yescharacteristic
251C2×C59#341611yescharacteristic
252D59—22no—
253D59—82no—
254C22×D8—591no—
255#544—1181no—
256#518—1181no—
257C2×M32—591no—
258C22×C16—591no—
259#501—1181no—
260#501—1181no—
261#500—591no—
262#500C2×D5911yescharacteristic, complemented
263#499—591no—
264#499—591no—
265#442—1181no—
266#442—1181no—
267C4×D8—591no—
268C4×D8—591no—
269C4×D8—591no—
270C4×D8C2×D5911yescharacteristic
271#428—1181no—
272#427—591no—
273#427—591no—
274C2×#107—591no—
275#418—1181no—
276#405—591no—
277#367—591no—
278#367—591no—
279#367—591no—
280#367—591no—
281#366—591no—
282#366—591no—
283#366C2×D5911yescharacteristic, complemented
284#366C2×D5911yescharacteristic, complemented
285C4⋊C16—591no—
286C4⋊C16—591no—
287C4⋊C16—591no—
288C4⋊C16C2×D5911yescharacteristic, complemented
289#357—591no—
290#357—591no—
291#357—591no—
292#357—591no—
293#357—591no—
294#357—591no—
295#357C2×D5911yescharacteristic
296#357C2×D5911yescharacteristic
297C59#3363111yescharacteristic, complemented, Sylow
298C22×D4—1181no—
299C2×D8—594no—
300C2×D8—591no—
301C2×D8—1182no—
302C2×D8—1181no—
303C2×D8C22×D5911yescharacteristic
304C2×M16—591no—
305C22×C8—591no—
306C4⋊Q8—21no—
307C4⋊Q8—1181no—
308C4⋊Q8—2361no—
309#125—1181no—
310#125—1181no—
311#125—2361no—
312C22⋊Q8—2361no—
313C22⋊Q8—2361no—
314C4×D4—21no—
315C4×D4—1181no—
316C4×D4—1181no—
317C4×D4—1181no—
318C4×D4—2361no—
319C4×D4—2361no—
320#117—591no—
321#117—1181no—
322#117—1181no—
323C2×C4⋊C4—1181no—
324C2×C22⋊C4—2361no—
325C2×C22⋊C4—2361no—
326M32—1182no—
327C2×C16—591no—
328C2×C16—591no—
329C2×C16—1182no—
330C2×C16C22×D5911yescharacteristic
331C2×C16C22×D5911yescharacteristic
332#107—21no—
333#107—591no—
334#107—591no—
335#107—591no—
336#107—1181no—
337#107—1181no—
338#107C22×D5911yescharacteristic
339#107C22×D5911yescharacteristic
340#107C22×D5911yescharacteristic
341#106—1181no—
342#106—1181no—
343#106—1181no—
344#106—1181no—
345#102—21no—
346#102—1181no—
347#102—1181no—
348#102—1181no—
349#97—591no—
350#97—591no—
351C4×C8—591no—
352C4×C8C22×D5911yescharacteristic
353C24—2361no—
354C2×Q8—41no—
355C2×Q8—2361no—
356C2×D4—21no—
357C2×D4—1184no—
358C2×D4—1181no—
359C2×D4—2362no—
360C2×D4—2361no—
361C22×C4—41no—
362C22×C4—591no—
363C22×C4—2361no—
364C22×C4—2361no—
365C22×C4—2361no—
366C22×C4—2361no—
367D8—21no—
368D8—592no—
369D8—1184no—
370D8—1181no—
371D8Q8×D5912yes—
372M16—1182no—
373C2×C8—591no—
374C2×C8—592no—
375C2×C8—592no—
376C2×C8—591no—
377C2×C8D4×D5911yescharacteristic
378C2×C8C23×D5911yescharacteristic
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—41no—
383C4⋊C4—591no—
384C4⋊C4—591no—
385C4⋊C4—1181no—
386C4⋊C4—1181no—
387C4⋊C4—1181no—
388C4⋊C4—2361no—
389C4⋊C4—2361no—
390C4⋊C4—2361no—
391C4⋊C4—2361no—
392C4⋊C4—2361no—
393C4⋊C4—2361no—
394C22⋊C4—41no—
395C22⋊C4—591no—
396C22⋊C4—591no—
397C22⋊C4—2361no—
398C22⋊C4—2361no—
399C22⋊C4—2361no—
400C22⋊C4—2361no—
401C22⋊C4—2361no—
402C42—591no—
403C42—2361no—
404C42—2361no—
405C42D4×D5911yescharacteristic
406C16—21no—
407C16—21no—
408C16—1181no—
409C16—1181no—
410C23—41no—
411C23—591no—
412C23—9441no—
413C23—9441no—
414C23—2364no—
415C23—2361no—
416Q8—161no—
417Q8—9441no—
418D4—22no—
419D4—41no—
420D4—1182no—
421D4—2364no—
422D4—2361no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—161no—
427C2×C4—591no—
428C2×C4—592no—
429C2×C4—592no—
430C2×C4—591no—
431C2×C4—591no—
432C2×C4—9441no—
433C2×C4—9441no—
434C2×C4—9441no—
435C2×C4—9441no—
436C2×C4—2361no—
437C2×C4—2361no—
438C2×C4—2361no—
439C2×C4D8×D5911yescharacteristic
440C2×C4D8×D5911yescharacteristic
441C2×C4C2×D4×D5911yescharacteristic
442C8—21no—
443C8—592no—
444C8—1181no—
445C8C2×Q8×D5911yescharacteristic
446C8#41×D5911yescharacteristic
447C22—42no—
448C22—81no—
449C22—81no—
450C22—592no—
451C22—1181no—
452C22—1181no—
453C22—9444no—
454C22—4721no—
455C22—4721no—
456C22—2362no—
457C22C2×D8×D5911yescharacteristic
458C4—21no—
459C4—21no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—592no—
464C4—1181no—
465C4—1181no—
466C4—4721no—
467C4—4721no—
468C4—4721no—
469C4C22⋊Q8×D5911yescharacteristic
470C4#137×D5911yescharacteristic
471C2—82no—
472C2—1182no—
473C2—4722no—
474C2#474×D5911yescharacteristic
475C2#2143723311yescharacteristic
476C2#2143724111yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2224C2
socleC22×C59C22×D8
prime radicalC2×D8×D59C22
hypercenter#2224D59
hypocenterC59#33631
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2224

Complements

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

NH ≅ G/Ntype
1. #21434629C2semidirect
3. #21426256C2semidirect
4. #21426254C2semidirect
5. #21426253C2semidirect
6. #21426251C2semidirect
7. #21426241C2semidirect
8. #21426240C2semidirect
9. #21426232C2semidirect
10. #21425881C2semidirect
11. #21425877C2semidirect
12. #21425853C2semidirect
15. C59×#2224C2semidirect
23. #16837806C22semidirect
24. #16837804C22semidirect
25. #16837803C22semidirect
37. #16837230C22semidirect
38. #16837229C22semidirect
39. #16837229C22semidirect
40. #16837229C22semidirect
41. #16837228C22semidirect
42. #16837227C22semidirect
43. C4⋊(C59⋊C16)C22semidirect
44. Dic59⋊C16C22semidirect
45. Dic59⋊C16C22semidirect
52. C59×#500C22semidirect
54. C59×#366C22semidirect
55. C59×#366C22semidirect
56. C59×C4⋊C16C22semidirect
63. D8×D59Q8semidirect
242. #2224D59semidirect
262. #500C2×D59semidirect
283. #366C2×D59semidirect
284. #366C2×D59semidirect
288. C4⋊C16C2×D59semidirect
297. C59#33631semidirect

Maximal subgroups

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

subgroupindex
#214346292
#214329432
#214262562
#214262542
#214262532
#214262512
#214262412
#214262402
#214262322
#214258812
#214258772
#214258532
#214258102
#357×D592
C59×#22242
#3363159