Finite Group Databasebeta

← #28743492

Subgroups — #28743492

Order 15104.

590 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433678C211yesmaximal, characteristic
2#427×D59C211yesmaximal, characteristic, complemented
3#21426307C211yesmaximal, characteristic, complemented
4#21426306C211yesmaximal, characteristic, complemented
5#21426300C211yesmaximal, characteristic, complemented
6#21426295C211yesmaximal, characteristic, complemented
7#21426290C211yesmaximal, characteristic, complemented
8#21426289C211yesmaximal, characteristic, complemented
9#21426283C211yesmaximal, characteristic, complemented
10#21426278C211yesmaximal, characteristic, complemented
11#21425878C211yesmaximal, characteristic, complemented
12#21425804C211yesmaximal, characteristic, complemented
13#21425801C211yesmaximal, characteristic, complemented
14C2×(C2×D59)⋊C16C211yesmaximal, characteristic, complemented
15C59×#2158C211yesmaximal, characteristic, complemented
16C2×#14273749C2211yescharacteristic
17C22×C8×D59C2211yescharacteristic
18#16837914—21no—
19#117×D59—21no—
20C22×C59⋊C16C2211yescharacteristic
21#16837644C2211yescharacteristic
22#16837644C2211yescharacteristic
23#16837639C2211yescharacteristic
24#16837639C2211yescharacteristic
25#16837632—21no—
26#16837632—21no—
27#16837440C2211yescharacteristic
28#16837439C2211yescharacteristic
29#16837439C2211yescharacteristic
30#16837438—21no—
31#16837437—21no—
32#107×D59C2211yescharacteristic
33#107×D59C2211yescharacteristic
34#106×D59—21no—
35#16837396—21no—
36#16837229C2211yescharacteristic, complemented
37#16837229C2211yescharacteristic, complemented
38#16837227C2211yescharacteristic, complemented
39#16837227C2211yescharacteristic, complemented
40#16837207C2211yescharacteristic, complemented
41#16837207C2211yescharacteristic, complemented
42#16837205C2211yescharacteristic, complemented
43#16837205C2211yescharacteristic, complemented
44#16837204C2211yescharacteristic, complemented
45#16837204C2211yescharacteristic, complemented
46#16837202C2211yescharacteristic, complemented
47#16837202C2211yescharacteristic, complemented
48(C2×D59)⋊C16C2211yescharacteristic
49(C2×D59)⋊C16C2211yescharacteristic
50(C2×D59)⋊C16C2211yescharacteristic
51(C2×D59)⋊C16C2211yescharacteristic
52C22×C16×C59C2211yescharacteristic
53C59×#427C2211yescharacteristic
54C59×#427C2211yescharacteristic
55C59×#367C2211yescharacteristic, complemented
56C59×#367C2211yescharacteristic, complemented
57C59×#366C2211yescharacteristic, complemented
58C59×#366C2211yescharacteristic, complemented
59C2×#11745685—21no—
60C22×C4×D59D411yescharacteristic
61#14273749—22no—
62#14273749—22no—
63C2×Dic236C2311yescharacteristic
64C2×#11745660—21no—
65C2×D472C2311yescharacteristic
66C22×C59⋊C8D411yescharacteristic
67C2×C8×D59—22no—
68C2×C8×D59—22no—
69C2×C8×D59D411yescharacteristic
70C2×C8×D59D411yescharacteristic
71C2×C8×D59C2311yescharacteristic
72C2×C8×D59C2311yescharacteristic
73#14273711—41no—
74#14273707—21no—
75#14273707—21no—
76#14273702—21no—
77#14273702—21no—
78#14273697—41no—
79C4×D236—41no—
80C4×D236—41no—
81#14273686—21no—
82#14273685—21no—
83#14273685—21no—
84#14273684—41no—
85#14273683—41no—
86C4⋊C4×D59—21no—
87C4⋊C4×D59—21no—
88C22⋊C4×D59—41no—
89C42×D59—41no—
90C2×C59⋊C16—22no—
91C2×C59⋊C16C2311yescharacteristic
92C2×C59⋊C16C2311yescharacteristic
93#14273661C2311yescharacteristic
94#14273661C2311yescharacteristic
95#14273660—21no—
96#14273660—21no—
97#14273659—21no—
98#14273658—21no—
99#14273658—21no—
100#14273650—21no—
101#14273650—21no—
102#14273647—21no—
103#14273647—21no—
104C22×C8×C59C2311yescharacteristic
105#117×C59—21no—
106#117×C59—21no—
107C2×C16×C59—22no—
108C2×C16×C59C2311yescharacteristic
109C2×C16×C59C2311yescharacteristic
110#107×C59C2311yescharacteristic
111#107×C59C2311yescharacteristic
112#107×C59C2311yescharacteristic
113#107×C59C2311yescharacteristic
114#106×C59—21no—
115#106×C59—21no—
116C23×D59D811yescharacteristic
117C22×C4×C59C2×D411yescharacteristic
118C2×C22⋊D59—41no—
119C22×Dic59D811yescharacteristic
120#11745685—42no—
121#11745685—42no—
122C2×D236—21no—
123C2×D236—41no—
124C2×D236—41no—
125C2×C4×D59—22no—
126C2×C4×D59—22no—
127C2×C4×D59—41no—
128C2×C4×D59—41no—
129C2×C4×D59—41no—
130C2×C4×D59D811yescharacteristic
131C2×C4×D59D811yescharacteristic
132C2×C4×D59C2×D411yescharacteristic
133C2×C4×D59C2×D411yescharacteristic
134C2×Dic118—21no—
135C2×C8×C59C2×D411yescharacteristic
136C2×C8×C59#4112yes—
137C2×C8×C59#4112yes—
138C2×C8×C59C2411yescharacteristic
139C4⋊C4×C59—21no—
140C4⋊C4×C59—21no—
141C4⋊C4×C59—21no—
142C4⋊C4×C59—21no—
143C22⋊C4×C59—41no—
144C22⋊C4×C59—41no—
145C42×C59—41no—
146C42×C59—41no—
147C22⋊Dic59—41no—
148(C2×D59)⋊C4—41no—
149(C2×D59)⋊C4—41no—
150(C2×D59)⋊C4—41no—
151(C2×D59)⋊C4—41no—
152(C2×D59)⋊C4—41no—
153(C2×D59)⋊C4—41no—
154C4⋊Dic59—21no—
155C4⋊Dic59—21no—
156C4⋊Dic59—41no—
157C4⋊Dic59—41no—
158Dic59⋊C4—41no—
159Dic59⋊C4—41no—
160C4×Dic59—41no—
161C4×Dic59—41no—
162C4×Dic59—41no—
163C2×C59⋊C8—22no—
164C2×C59⋊C8C2×D411yescharacteristic
165C2×C59⋊C8C2×D411yescharacteristic
166Dic236—21no—
167Dic236—21no—
168D472—21no—
169D472—21no—
170#11745660—42no—
171C8×D59—22no—
172C8×D59—22no—
173C8×D59—21no—
174C8×D59—21no—
175C8×D59—21no—
176C8×D59—21no—
177C16×C59—21no—
178C16×C59—21no—
179C59⋊C16—21no—
180C59⋊C16—21no—
181C23×C59C2×D811yescharacteristic
182C22×D59—22no—
183C22×D59—22no—
184C22×D59—41no—
185C22×D59—41no—
186C22×D59C2×D811yescharacteristic
187C22×D59C2×D811yescharacteristic
188C2×C4×C59—41no—
189C2×C4×C59—41no—
190C2×C4×C59—41no—
191C2×C4×C59—41no—
192C2×C4×C59#12312yes—
193C2×C4×C59C2×D811yescharacteristic
194C2×C4×C59#13712yes—
195C2×C4×C59C22×D411yescharacteristic
196C22⋊D59—82no—
197C2×Dic59—22no—
198C2×Dic59—41no—
199C2×Dic59—41no—
200C2×Dic59—41no—
201C2×Dic59C2×D811yescharacteristic
202C2×Dic59C2×D811yescharacteristic
203D236—41no—
204D236—41no—
205D236—82no—
206D236—82no—
207C4×D59—21no—
208C4×D59—21no—
209C4×D59—21no—
210C4×D59—21no—
211C4×D59—22no—
212C4×D59—22no—
213C4×D59—82no—
214C4×D59—82no—
215C4×D59—82no—
216Dic118—41no—
217Dic118—41no—
218C8×C59#12312yes—
219C8×C59C2×#4111yescharacteristic
220C8×C59C2×#4111yescharacteristic
221C59⋊C8—21no—
222C59⋊C8—21no—
223#33645—591nomaximal, Sylow
224C22×C59—21no—
225C22×C59—21no—
226C22×C59#48012yes—
227C22×C59C22×D811yescharacteristic
228C2×D59—21no—
229C2×D59—21no—
230C2×D59—21no—
231C2×D59—21no—
232C2×D59—21no—
233C2×D59—44no—
234C2×D59—81no—
235C2×D59—81no—
236C2×D59—81no—
237C2×D59#50812yes—
238C4×C59—81no—
239C4×C59—81no—
240C4×C59—81no—
241C4×C59—81no—
242C4×C59#48012yes—
243C4×C59C2×#12311yescharacteristic
244C4×C59C2×#13711yescharacteristic
245Dic59—21no—
246Dic59—21no—
247Dic59—81no—
248Dic59—81no—
249Dic59—81no—
250#3050—591no—
251C2×#427—591no—
252#2237—591no—
253#2237—591no—
254#2236—591no—
255#2235—591no—
256#2234—591no—
257#2234—591no—
258#2233—591no—
259#2232—591no—
260#2158—591no—
261#2158D5911yescharacteristic, complemented
262#2138—591no—
263#2138—591no—
264C2×C22⋊C16—591no—
265C2×C59—22no—
266C2×C59C2×#48011yescharacteristic
267C2×C59C2×#50811yescharacteristic
268C2×C59#341411yescharacteristic
269D59—22no—
270D59—41no—
271D59—161no—
272C2×#135—591no—
273C23×C8—591no—
274#525—1181no—
275C2×#117—1181no—
276C22×C16—591no—
277C22×C16C2×D5911yescharacteristic
278#467—591no—
279#467—591no—
280#466—591no—
281#466—591no—
282#465—1181no—
283#465—1181no—
284#427—591no—
285#427—591no—
286#427—591no—
287#427—1181no—
288#427—1181no—
289#427C2×D5911yescharacteristic
290#427C2×D5911yescharacteristic
291C2×#107—591no—
292C2×#107—591no—
293C2×#106—1181no—
294#416—1181no—
295#367—591no—
296#367—591no—
297#367C2×D5911yescharacteristic, complemented
298#367C2×D5911yescharacteristic, complemented
299#366—591no—
300#366—591no—
301#366C2×D5911yescharacteristic, complemented
302#366C2×D5911yescharacteristic, complemented
303#358—591no—
304#358—591no—
305#358—591no—
306#358—591no—
307#357—591no—
308#357—591no—
309#357—591no—
310#357—591no—
311C22⋊C16—591no—
312C22⋊C16—591no—
313C22⋊C16—591no—
314C22⋊C16—591no—
315C59#3364511yescharacteristic, complemented, Sylow
316C2×#41—1181no—
317C23×C4—591no—
318#135—1182no—
319#135—1182no—
320C2×Q16—591no—
321C2×SD16—1181no—
322C2×D8—591no—
323C22×C8—591no—
324C22×C8—591no—
325C22×C8—591no—
326C22×C8—591no—
327C22×C8—591no—
328C22×C8—1182no—
329C22×C8—1182no—
330C22×C8C22×D5911yescharacteristic
331#123—2361no—
332C22⋊Q8—1181no—
333C22⋊Q8—1181no—
334#121—1181no—
335#121—1181no—
336#120—2361no—
337C4×D4—2361no—
338C4×D4—2361no—
339#117—21no—
340#117—21no—
341#117—1181no—
342#117—1181no—
343#117—1181no—
344#117—2361no—
345#117—2361no—
346C2×C4⋊C4—1181no—
347C2×C4⋊C4—1181no—
348C2×C22⋊C4—2361no—
349C2×C42—2361no—
350C2×C16—22no—
351C2×C16—591no—
352C2×C16—591no—
353C2×C16—1182no—
354C2×C16C22×D5911yescharacteristic
355C2×C16C22×D5911yescharacteristic
356#107—591no—
357#107—591no—
358#107—1181no—
359#107—1181no—
360#107C22×D5911yescharacteristic
361#107C22×D5911yescharacteristic
362#107C22×D5911yescharacteristic
363#107C22×D5911yescharacteristic
364#106—21no—
365#106—21no—
366#106—1181no—
367#106—1181no—
368#106—1181no—
369#103—1181no—
370#103—1181no—
371#102—1181no—
372#102—1181no—
373C24—591no—
374#41—2362no—
375#41—2362no—
376C2×Q8—1181no—
377C2×D4—1181no—
378C2×D4—2361no—
379C2×D4—2361no—
380C2×D4—2361no—
381C22×C4—591no—
382C22×C4—591no—
383C22×C4—591no—
384C22×C4—591no—
385C22×C4—591no—
386C22×C4—1182no—
387C22×C4—1182no—
388C22×C4—2361no—
389C22×C4—2361no—
390C22×C4—2361no—
391C22×C4D4×D5911yescharacteristic
392Q16—1181no—
393Q16—1181no—
394SD16—2362no—
395D8—1181no—
396D8—1181no—
397C2×C8—591no—
398C2×C8—591no—
399C2×C8—1182no—
400C2×C8—1182no—
401C2×C8—1182no—
402C2×C8—1181no—
403C2×C8—1181no—
404C2×C8—1181no—
405C2×C8—1181no—
406C2×C8#1174568512yes—
407C2×C8D4×D5911yescharacteristic
408C2×C8#1174568912yes—
409C2×C8C23×D5911yescharacteristic
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—1181no—
415C4⋊C4—1181no—
416C4⋊C4—2361no—
417C4⋊C4—2361no—
418C4⋊C4—2361no—
419C4⋊C4—2361no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—2361no—
423C22⋊C4—2361no—
424C22⋊C4—2361no—
425C22⋊C4—2361no—
426C22⋊C4—2361no—
427C22⋊C4—2361no—
428C22⋊C4—2361no—
429C42—41no—
430C42—41no—
431C42—2361no—
432C42—2361no—
433C42—2361no—
434C16—21no—
435C16—21no—
436C16—1181no—
437C16—1181no—
438C23—591no—
439C23—591no—
440C23—1182no—
441C23—1182no—
442C23—2361no—
443C23—2361no—
444C23D8×D5911yescharacteristic
445Q8—2361no—
446Q8—2361no—
447D4—4722no—
448D4—4722no—
449D4—4722no—
450D4—2361no—
451D4—2361no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—591no—
457C2×C4—591no—
458C2×C4—1182no—
459C2×C4—1181no—
460C2×C4—1181no—
461C2×C4—1181no—
462C2×C4—1181no—
463C2×C4—1182no—
464C2×C4—1182no—
465C2×C4—4722no—
466C2×C4—4722no—
467C2×C4—4722no—
468C2×C4—2361no—
469C2×C4—2361no—
470C2×C4—2361no—
471C2×C4#1427371212yes—
472C2×C4D8×D5911yescharacteristic
473C2×C4#1427375912yes—
474C2×C4C2×D4×D5911yescharacteristic
475C8—1181no—
476C8—1181no—
477C8#1427371212yes—
478C8C2×#1174568511yescharacteristic
479C8C2×#1174568911yescharacteristic
480C22—21no—
481C22—21no—
482C22—592no—
483C22—1181no—
484C22—1181no—
485C22—1181no—
486C22—1181no—
487C22—1181no—
488C22—4721no—
489C22—4721no—
490C22—4721no—
491C22—2364no—
492C22#1683771712yes—
493C22C2×D8×D5911yescharacteristic
494C4—81no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4—1181no—
499C4—1181no—
500C4—4721no—
501C4—4721no—
502C4—4721no—
503C4#1683771712yes—
504C4C2×#1427371211yescharacteristic
505C4C2×#1427375911yescharacteristic
506C2—22no—
507C2—1182no—
508C2—9441no—
509C2—2361no—
510C2C2×#1683771711yescharacteristic
511C2#508×D5911yescharacteristic
512C2#2143722911yescharacteristic
subgroupG/H
centerC2×C4D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2158C2
socleC22×C59C22×D8
prime radical#21433678C2
hypercenter#2158D59
hypocenterC59#33645
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2158

Complements

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

NH ≅ G/Ntype
2. #427×D59C2semidirect
3. #21426307C2semidirect
4. #21426306C2semidirect
5. #21426300C2semidirect
6. #21426295C2semidirect
7. #21426290C2semidirect
8. #21426289C2semidirect
9. #21426283C2semidirect
10. #21426278C2semidirect
11. #21425878C2semidirect
12. #21425804C2semidirect
13. #21425801C2semidirect
14. C2×(C2×D59)⋊C16C2semidirect
15. C59×#2158C2semidirect
36. #16837229C22semidirect
37. #16837229C22semidirect
38. #16837227C22semidirect
39. #16837227C22semidirect
40. #16837207C22semidirect
41. #16837207C22semidirect
42. #16837205C22semidirect
43. #16837205C22semidirect
44. #16837204C22semidirect
45. #16837204C22semidirect
46. #16837202C22semidirect
47. #16837202C22semidirect
55. C59×#367C22semidirect
56. C59×#367C22semidirect
57. C59×#366C22semidirect
58. C59×#366C22semidirect
261. #2158D59semidirect
297. #367C2×D59semidirect
298. #367C2×D59semidirect
301. #366C2×D59semidirect
302. #366C2×D59semidirect
315. C59#33645semidirect

Maximal subgroups

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

subgroupindex
#214336782
#427×D592
#214263072
#214263062
#214263002
#214262952
#214262902
#214262892
#214262832
#214262782
#214258782
#214258042
#214258012
C2×(C2×D59)⋊C162
C59×#21582
#3364559