Finite Group Databasebeta

← #28743342

Subgroups — #28743342

Order 15104.

598 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434609C211yesmaximal, characteristic, complemented
2C4×D8×D59C211yesmaximal, characteristic
3#21426254C211yesmaximal, characteristic, complemented
4#21426253C211yesmaximal, characteristic, complemented
5#21426250C211yesmaximal, characteristic, complemented
6#21426242C211yesmaximal, characteristic, complemented
7#21426241C211yesmaximal, characteristic, complemented
8#21426232C211yesmaximal, characteristic, complemented
9#21426231C211yesmaximal, characteristic, complemented
10#21425883C211yesmaximal, characteristic, complemented
11#21425881C211yesmaximal, characteristic, complemented
12#21425850C211yesmaximal, characteristic, complemented
13#21425810C211yesmaximal, characteristic
14#21425810C211yesmaximal, characteristic
15C59×#2228C211yesmaximal, characteristic, complemented
16C2×D8×D59C2211yescharacteristic
17#16837958—21no—
18C4×D4×D59—21no—
19C2×#14273666C2211yescharacteristic
20C2×#14273666C2211yescharacteristic
21#16837806C2211yescharacteristic, complemented
22#16837806C2211yescharacteristic, complemented
23#16837804C2211yescharacteristic, complemented
24#16837801—21no—
25#16837799—21no—
26C4×D472C2211yescharacteristic
27D8×Dic59C2211yescharacteristic
28#16837484C2211yescharacteristic
29C4×#11745669—21no—
30#16837482—21no—
31#16837439C2211yescharacteristic
32#16837439C2211yescharacteristic
33#16837437—21no—
34#107×D59C2211yescharacteristic
35#102×D59—21no—
36C4×C8×D59C2211yescharacteristic
37#16837230C2211yescharacteristic, complemented
38#16837229C2211yescharacteristic, complemented
39#16837229C2211yescharacteristic, complemented
40#16837228C2211yescharacteristic, complemented
41#16837227C2211yescharacteristic, 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×#499C2211yescharacteristic, complemented
53C4×D8×C59C2211yescharacteristic
54C59×#367C2211yescharacteristic, complemented
55C59×#366C2211yescharacteristic, complemented
56C59×C4⋊C16C2211yescharacteristic, complemented
57C59×#357C2211yescharacteristic
58C59×#357C2211yescharacteristic
59C2×D4×D59—21no—
60C22×C4×D59—41no—
61C2×D472C2311yescharacteristic
62C2×#11745669—21no—
63D8×D59—22no—
64D8×D59Q814yescomplemented
65C2×C8×D59D411yescharacteristic
66C2×C8×D59C2311yescharacteristic
67#14273726—21no—
68#14273726—21no—
69#14273719—21no—
70#14273705—41no—
71#14273705—41no—
72#14273692—41no—
73#14273692—41no—
74C4×D236—21no—
75D4×Dic59—21no—
76#14273689—21no—
77C4×C22⋊D59—41no—
78(C22⋊D59)⋊C4—41no—
79#14273685—21no—
80#14273685—21no—
81C4⋊C4×D59—21no—
82C22⋊C4×D59—41no—
83C42×D59D411yescharacteristic
84#14273666—22no—
85#14273666—22no—
86C2×C59⋊C16C2311yescharacteristic
87C2×C59⋊C16C2311yescharacteristic
88#14273661C2311yescharacteristic
89#14273661C2311yescharacteristic
90#14273661C2311yescharacteristic
91#14273660—21no—
92#14273660—21no—
93#14273660—21no—
94#14273659—21no—
95#14273658—21no—
96#14273658—21no—
97#14273649—21no—
98#14273648—21no—
99#14273647—21no—
100C4×C59⋊C8D411yescharacteristic
101C8×Dic59D411yescharacteristic
102C8×Dic59C2311yescharacteristic
103C2×D8×C59C2311yescharacteristic
104#125×C59—21no—
105C4×D4×C59—21no—
106C2×C16×C59C2311yescharacteristic
107C2×C16×C59C2311yescharacteristic
108#107×C59C2311yescharacteristic
109#107×C59C2311yescharacteristic
110#107×C59C2311yescharacteristic
111#106×C59—21no—
112#102×C59—21no—
113C4×C8×C59C2311yescharacteristic
114C23×D59—41no—
115C2×D4×C59—21no—
116C22×C4×C59—41no—
117C2×C22⋊D59—41no—
118C22×Dic59—41no—
119D4×D59—24no—
120D4×D59—42no—
121C2×D236—21no—
122C2×C4×D59—41no—
123C2×C4×D59—41no—
124C2×C4×D59—41no—
125C2×C4×D59—84no—
126C2×C4×D59D811yescharacteristic
127C2×C4×D59D811yescharacteristic
128C2×C4×D59C2×D411yescharacteristic
129C2×Dic118—41no—
130C2×Dic118—41no—
131D8×C59—21no—
132D8×C59C2×Q812yes—
133C2×C8×C59C2×D411yescharacteristic
134C2×C8×C59C2411yescharacteristic
135C4⋊C4×C59—21no—
136C4⋊C4×C59—21no—
137C4⋊C4×C59—21no—
138C4⋊C4×C59—41no—
139C4⋊C4×C59—41no—
140C22⋊C4×C59—41no—
141C42×C59C2×D411yescharacteristic
142C22⋊Dic59—41no—
143#11745669—24no—
144(C2×D59)⋊C4—41no—
145(C2×D59)⋊C4—41no—
146(C2×D59)⋊C4—41no—
147(C2×D59)⋊C4—41no—
148C4⋊Dic59—21no—
149C4⋊Dic59—21no—
150C4⋊Dic59—21no—
151Dic59⋊C4—41no—
152Dic59⋊C4—41no—
153Dic59⋊C4—41no—
154Dic59⋊C4—41no—
155Dic59⋊C4—41no—
156C4×Dic59—41no—
157C4×Dic59—41no—
158C4×Dic59D811yescharacteristic
159C4×Dic59D811yescharacteristic
160C4×Dic59C2×D411yescharacteristic
161C2×C59⋊C8C2×D411yescharacteristic
162C2×C59⋊C8C2×D411yescharacteristic
163D472—21no—
164D472C2×Q812yes—
165C8×D59—22no—
166C8×D59C2×Q812yes—
167C8×D59#4112yes—
168C16×C59—21no—
169C16×C59—21no—
170C59⋊C16—21no—
171C59⋊C16—21no—
172C23×C59—41no—
173C22×D59—41no—
174C22×D59—44no—
175C22×D59—82no—
176C22×D59—82no—
177C22×D59C2×D811yescharacteristic
178D4×C59—22no—
179D4×C59—41no—
180C2×C4×C59—41no—
181C2×C4×C59—41no—
182C2×C4×C59—41no—
183C2×C4×C59—82no—
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—82no—
192C2×Dic59C2×D811yescharacteristic
193C2×Dic59C2×D811yescharacteristic
194C2×Dic59C2×D811yescharacteristic
195D236—22no—
196D236—41no—
197C4×D59—22no—
198C4×D59—41no—
199C4×D59—82no—
200C4×D59—82no—
201C4×D59—82no—
202C4×D59C22⋊Q812yes—
203C4×D59#13712yes—
204Dic118—82no—
205Dic118—82no—
206C8×C59—21no—
207C8×C59C22×Q811yescharacteristic
208C8×C59C2×#4111yescharacteristic
209C59⋊C8—21no—
210C59⋊C8C22⋊Q812yes—
211#33627—591nomaximal, Sylow
212C22×C59—42no—
213C22×C59—81no—
214C22×C59—81no—
215C22×C59C22×D811yescharacteristic
216C2×D59—21no—
217C2×D59—21no—
218C2×D59—42no—
219C2×D59—88no—
220C2×D59—81no—
221C2×D59—81no—
222C2×D59#47412yes—
223C4×C59—21no—
224C4×C59—21no—
225C4×C59—81no—
226C4×C59—81no—
227C4×C59—81no—
228C4×C59C2×C22⋊Q811yescharacteristic
229C4×C59C2×#13711yescharacteristic
230Dic59—21no—
231Dic59—81no—
232Dic59—81no—
233Dic59—81no—
234Dic59#47412yes—
235Dic59#50912yes—
236#3160—591no—
237C2×C4×D8—591no—
238#2228—591no—
239#2228—591no—
240#2228—591no—
241#2228D5911yescharacteristic, complemented
242#2226—591no—
243#2226—591no—
244#2224—591no—
245#2224—591no—
246#2159—591no—
247#2159—591no—
248#2150—591no—
249#2140—591no—
250#2140—591no—
251C2×C59—82no—
252C2×C59C2×#47411yescharacteristic
253C2×C59C2×#50911yescharacteristic
254C2×C59#341411yescharacteristic
255D59—22no—
256D59—82no—
257C22×D8—591no—
258#533—1181no—
259C2×C4×D4—1181no—
260C2×M32—591no—
261C2×M32—591no—
262#500—591no—
263#500—591no—
264#499—591no—
265#499—1181no—
266#499C2×D5911yescharacteristic, complemented
267#498—1181no—
268C4×D8—591no—
269C4×D8—591no—
270C4×D8—591no—
271C4×D8—1181no—
272C4×D8—1181no—
273C4×D8C2×D5911yescharacteristic
274#427—591no—
275#427—591no—
276#427—1181no—
277C2×#107—591no—
278C2×#102—1181no—
279C2×C4×C8—591no—
280#367—591no—
281#367—591no—
282#367—591no—
283#367C2×D5911yescharacteristic, complemented
284#366—591no—
285#366—591no—
286#366—591no—
287#366C2×D5911yescharacteristic, complemented
288C4⋊C16—591no—
289C4⋊C16—591no—
290C4⋊C16—591no—
291C4⋊C16C2×D5911yescharacteristic, complemented
292#357—591no—
293#357—591no—
294#357—591no—
295#357—591no—
296#357—591no—
297#357—591no—
298#357C2×D5911yescharacteristic
299#357C2×D5911yescharacteristic
300C59#3362711yescharacteristic, complemented, Sylow
301C22×D4—1181no—
302C23×C4—2361no—
303C2×D8—594no—
304C2×D8—591no—
305C2×D8—1182no—
306C2×D8—1181no—
307C2×D8C22×D5911yescharacteristic
308C22×C8—591no—
309C22×C8—591no—
310C4⋊Q8—1181no—
311C4⋊Q8—1181no—
312#125—21no—
313#125—1181no—
314C22⋊Q8—2361no—
315C22⋊Q8—2361no—
316C4×Q8—2361no—
317C4×Q8—2361no—
318C4×D4—21no—
319C4×D4—1181no—
320C4×D4—1181no—
321C4×D4—1181no—
322C4×D4—2361no—
323C4×D4—2361no—
324#117—1181no—
325#117—1181no—
326C2×C4⋊C4—1181no—
327C2×C22⋊C4—2361no—
328C2×C42—591no—
329M32—1182no—
330M32—1182no—
331C2×C16—591no—
332C2×C16—591no—
333C2×C16C22×D5911yescharacteristic
334C2×C16C22×D5911yescharacteristic
335#107—591no—
336#107—591no—
337#107—591no—
338#107—1181no—
339#107—1181no—
340#107—1181no—
341#107C22×D5911yescharacteristic
342#107C22×D5911yescharacteristic
343#107C22×D5911yescharacteristic
344#106—21no—
345#106—1181no—
346#106—1181no—
347#106—1181no—
348#102—21no—
349#102—1181no—
350#102—1181no—
351#102—1181no—
352C4×C8—591no—
353C4×C8—591no—
354C4×C8—591no—
355C4×C8C22×D5911yescharacteristic
356C24—2361no—
357C2×Q8—2361no—
358C2×Q8—2361no—
359C2×D4—21no—
360C2×D4—1184no—
361C2×D4—1181no—
362C2×D4—2362no—
363C2×D4—2361no—
364C22×C4—41no—
365C22×C4—591no—
366C22×C4—591no—
367C22×C4—591no—
368C22×C4—4724no—
369C22×C4—2361no—
370C22×C4—2361no—
371C22×C4—2361no—
372C22×C4—2361no—
373D8—21no—
374D8—592no—
375D8—1184no—
376D8—1181no—
377D8Q8×D5912yes—
378C2×C8—591no—
379C2×C8—592no—
380C2×C8—592no—
381C2×C8—591no—
382C2×C8—1182no—
383C2×C8D4×D5911yescharacteristic
384C2×C8C23×D5911yescharacteristic
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—1181no—
391C4⋊C4—1181no—
392C4⋊C4—1181no—
393C4⋊C4—2361no—
394C4⋊C4—2361no—
395C4⋊C4—2361no—
396C4⋊C4—2361no—
397C4⋊C4—2361no—
398C22⋊C4—41no—
399C22⋊C4—2361no—
400C22⋊C4—2361no—
401C22⋊C4—2361no—
402C22⋊C4—2361no—
403C22⋊C4—2361no—
404C42—591no—
405C42—591no—
406C42—591no—
407C42—2361no—
408C42—2361no—
409C42D4×D5911yescharacteristic
410C16—21no—
411C16—21no—
412C16—1181no—
413C16—1181no—
414C23—41no—
415C23—591no—
416C23—4722no—
417C23—4722no—
418C23—2364no—
419C23—2361no—
420Q8—4722no—
421Q8—4722no—
422D4—22no—
423D4—41no—
424D4—1182no—
425D4—2364no—
426D4—2361no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—82no—
431C2×C4—591no—
432C2×C4—591no—
433C2×C4—592no—
434C2×C4—592no—
435C2×C4—591no—
436C2×C4—1182no—
437C2×C4—4722no—
438C2×C4—4722no—
439C2×C4—4722no—
440C2×C4—4722no—
441C2×C4—2361no—
442C2×C4—2361no—
443C2×C4—2361no—
444C2×C4—2361no—
445C2×C4D8×D5911yescharacteristic
446C2×C4D8×D5911yescharacteristic
447C2×C4C2×D4×D5911yescharacteristic
448C8—21no—
449C8—592no—
450C8—1181no—
451C8C2×Q8×D5911yescharacteristic
452C8#41×D5911yescharacteristic
453C22—42no—
454C22—81no—
455C22—81no—
456C22—592no—
457C22—1181no—
458C22—1181no—
459C22—4728no—
460C22—4721no—
461C22—4721no—
462C22—2362no—
463C22C2×D8×D5911yescharacteristic
464C4—21no—
465C4—21no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—592no—
470C4—592no—
471C4—1181no—
472C4—4721no—
473C4—4721no—
474C4—4721no—
475C4C22⋊Q8×D5911yescharacteristic
476C4#137×D5911yescharacteristic
477C2—82no—
478C2—1182no—
479C2—4722no—
480C2#474×D5911yescharacteristic
481C2#509×D5911yescharacteristic
482C2#2143722911yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2228C2
socleC22×C59C22×D8
prime radicalC2×D8×D59C22
hypercenter#2228D59
hypocenterC59#33627
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2228

Complements

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

NH ≅ G/Ntype
1. #21434609C2semidirect
3. #21426254C2semidirect
4. #21426253C2semidirect
5. #21426250C2semidirect
6. #21426242C2semidirect
7. #21426241C2semidirect
8. #21426232C2semidirect
9. #21426231C2semidirect
10. #21425883C2semidirect
11. #21425881C2semidirect
12. #21425850C2semidirect
15. C59×#2228C2semidirect
21. #16837806C22semidirect
22. #16837806C22semidirect
23. #16837804C22semidirect
37. #16837230C22semidirect
38. #16837229C22semidirect
39. #16837229C22semidirect
40. #16837228C22semidirect
41. #16837227C22semidirect
42. #16837227C22semidirect
43. C4⋊(C59⋊C16)C22semidirect
44. Dic59⋊C16C22semidirect
45. Dic59⋊C16C22semidirect
52. C59×#499C22semidirect
54. C59×#367C22semidirect
55. C59×#366C22semidirect
56. C59×C4⋊C16C22semidirect
64. D8×D59Q8semidirect
241. #2228D59semidirect
266. #499C2×D59semidirect
283. #367C2×D59semidirect
287. #366C2×D59semidirect
291. C4⋊C16C2×D59semidirect
300. C59#33627semidirect

Maximal subgroups

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

subgroupindex
#214346092
C4×D8×D592
#214262542
#214262532
#214262502
#214262422
#214262412
#214262322
#214262312
#214258832
#214258812
#214258502
#214258102
C59×#22282
#3362759