Finite Group Databasebeta

← #28745781

Subgroups — #28745781

Order 15104.

570 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436486C211yesmaximal, characteristic, complemented
2#21435775C211yesmaximal, characteristic
3#21426334C211yesmaximal, characteristic, complemented
4#21426297C211yesmaximal, characteristic, complemented
5#21426276C211yesmaximal, characteristic, complemented
6#21426231C211yesmaximal, characteristic, complemented
7#21426209C211yesmaximal, characteristic, complemented
8#21426186C211yesmaximal, characteristic, complemented
9#21426161C211yesmaximal, characteristic, complemented
10#21426036C211yesmaximal, characteristic, complemented
11#21426026C211yesmaximal, characteristic, complemented
12#21425957C211yesmaximal, characteristic, complemented
13#21425919C211yesmaximal, characteristic, complemented
14#357×D59C211yesmaximal, characteristic, complemented
15C59×#2139C211yesmaximal, characteristic, complemented
16C2×D8×D59C2211yescharacteristic
17#16838107—21no—
18#16838040—21no—
19C2×#14273673C2211yescharacteristic, complemented
20C2×#14273672C2211yescharacteristic
21C2×#14273671C2211yescharacteristic
22C2×C16×D59C2211yescharacteristic
23#16837806C2211yescharacteristic, complemented
24#16837787C2211yescharacteristic
25#16837739—21no—
26#16837737—21no—
27#16837696—21no—
28#16837644C2211yescharacteristic
29#16837643C2211yescharacteristic, complemented
30#16837640C2211yescharacteristic
31#16837638C2211yescharacteristic
32#16837619—21no—
33#16837551—21no—
34#16837547—21no—
35#16837496C2211yescharacteristic, complemented
36D8×Dic59C2211yescharacteristic
37C8×C22⋊D59C2211yescharacteristic
38#107×D59C2211yescharacteristic
39#16837228C2211yescharacteristic, complemented
40#16837227C2211yescharacteristic, complemented
41#16837227C2211yescharacteristic, complemented
42Dic59⋊C16C2211yescharacteristic, complemented
43#16837207C2211yescharacteristic, complemented
44#16837206C2211yescharacteristic, complemented
45#16837204C2211yescharacteristic
46#16837203C2211yescharacteristic
47#16837202C2211yescharacteristic
48#16837202C2211yescharacteristic
49C22⋊(C59⋊C16)C2211yescharacteristic, complemented
50(C2×D59)⋊C16C2211yescharacteristic
51#16837177C2211yescharacteristic, complemented
52C2×#135×C59C2211yescharacteristic
53C2×M32×C59C2211yescharacteristic, complemented
54C2×#107×C59C2211yescharacteristic, complemented
55C59×#358C2211yescharacteristic, complemented
56C59×#358C2211yescharacteristic, complemented
57C59×#357C2211yescharacteristic
58C59×#357C2211yescharacteristic
59C2×D4×D59—21no—
60C2×Dic236C2311yescharacteristic
61C2×#11745671—21no—
62C2×D472C2311yescharacteristic
63C2×#11745669—21no—
64D8×D59—22no—
65D8×D59—41no—
66C2×C8×D59C2311yescharacteristic
67#14273726—21no—
68#14273717—41no—
69#14273716—21no—
70#14273710—41no—
71#14273707—21no—
72#14273706—21no—
73#14273706—41no—
74#14273705—41no—
75#14273703—21no—
76#14273701—21no—
77#14273699—41no—
78#14273695—41no—
79#14273692—21no—
80D4×Dic59—21no—
81C4×C22⋊D59—41no—
82C4×C22⋊D59D411yescharacteristic
83C4⋊C4×D59—21no—
84C2×(C2×D59)⋊C4—41no—
85#14273673—41no—
86#14273672—41no—
87#14273671—22no—
88C2×C59⋊C16C2311yescharacteristic
89C2×C59⋊C16C2311yescharacteristic
90C16×D59—41no—
91#14273661C2311yescharacteristic
92#14273661C2311yescharacteristic
93#14273660—21no—
94#14273660—21no—
95Dic59⋊C8D411yescharacteristic
96#14273652—21no—
97#14273651—21no—
98#14273650—21no—
99#14273650—21no—
100#14273649—21no—
101#14273648—21no—
102C22⋊(C59⋊C8)D411yescharacteristic
103(C2×D59)⋊C8D411yescharacteristic
104C8×Dic59C2311yescharacteristic
105C2×#41×C59—21no—
106#135×C59—22no—
107#135×C59—41no—
108C2×Q16×C59C2311yescharacteristic
109C2×SD16×C59—21no—
110C2×D8×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112C2×C4⋊C4×C59—21no—
113M32×C59—41no—
114C2×C16×C59C2311yescharacteristic
115C2×C16×C59C2311yescharacteristic
116#107×C59—21no—
117#107×C59C2311yescharacteristic
118#107×C59C2311yescharacteristic
119C23×D59—41no—
120#41×C59—42no—
121#41×C59—81no—
122C2×Q8×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59—41no—
125C22×C4×C59—41no—
126C22×C4×C59—41no—
127C22×C4×C59C2×D411yescharacteristic
128C2×C22⋊D59—41no—
129C2×C22⋊D59—41no—
130C2×C22⋊D59D811yescharacteristic
131C22×Dic59—41no—
132D4×D59—42no—
133D4×D59—81no—
134C2×D236—21no—
135C2×C4×D59—41no—
136C2×C4×D59C2×D411yescharacteristic
137C2×Dic118—21no—
138C2×Dic118—41no—
139Q16×C59—21no—
140Q16×C59—21no—
141SD16×C59—42no—
142D8×C59—21no—
143D8×C59#4112yes—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59C2×D411yescharacteristic
147C2×C8×C59C2411yescharacteristic
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—41no—
151C22⋊Dic59—41no—
152C22⋊Dic59—41no—
153C22⋊Dic59D811yescharacteristic
154#11745671—81no—
155#11745669—42no—
156(C2×D59)⋊C4—41no—
157(C2×D59)⋊C4—41no—
158(C2×D59)⋊C4—41no—
159(C2×D59)⋊C4—41no—
160(C2×D59)⋊C4—41no—
161(C2×D59)⋊C4D811yescharacteristic
162C4⋊Dic59—21no—
163C4⋊Dic59—21no—
164C4⋊Dic59—41no—
165Dic59⋊C4—41no—
166Dic59⋊C4—41no—
167Dic59⋊C4—41no—
168Dic59⋊C4—41no—
169Dic59⋊C4—41no—
170Dic59⋊C4D811yescharacteristic
171C4×Dic59—41no—
172C4×Dic59C2×D411yescharacteristic
173C2×C59⋊C8C2×D411yescharacteristic
174C2×C59⋊C8C2×D411yescharacteristic
175Dic236—21no—
176Dic236—21no—
177D472—21no—
178D472—21no—
179C8×D59—21no—
180C8×D59—21no—
181C16×C59—21no—
182C16×C59—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C23×C59—41no—
186C23×C59C2×D811yescharacteristic
187C22×D59—41no—
188C22×D59—82no—
189C22×D59—82no—
190C22×D59—161no—
191C22×D59C2×D811yescharacteristic
192Q8×C59—41no—
193Q8×C59—41no—
194D4×C59—22no—
195D4×C59—41no—
196D4×C59—82no—
197C2×C4×C59—21no—
198C2×C4×C59—21no—
199C2×C4×C59—41no—
200C2×C4×C59—41no—
201C2×C4×C59—41no—
202C2×C4×C59—82no—
203C2×C4×C59—161no—
204C2×C4×C59C2×D811yescharacteristic
205C2×C4×C59C22×D411yescharacteristic
206C22⋊D59—22no—
207C22⋊D59—82no—
208C22⋊D59—161no—
209C2×Dic59—41no—
210C2×Dic59—41no—
211C2×Dic59—41no—
212C2×Dic59—161no—
213C2×Dic59C2×D811yescharacteristic
214C2×Dic59C2×D811yescharacteristic
215C2×Dic59C2×D811yescharacteristic
216D236—41no—
217D236—41no—
218C4×D59—21no—
219C4×D59—21no—
220C4×D59—161no—
221Dic118—41no—
222Dic118—41no—
223Dic118—161no—
224C8×C59—21no—
225C8×C59C2×#4111yescharacteristic
226C8×C59H(5,𝔽2)11yescharacteristic
227C59⋊C8—21no—
228C59⋊C8—21no—
229#33881—591nomaximal, Sylow
230C22×C59—21no—
231C22×C59—21no—
232C22×C59—21no—
233C22×C59—42no—
234C22×C59—81no—
235C22×C59—81no—
236C22×C59C22×D811yescharacteristic
237C2×D59—21no—
238C2×D59—21no—
239C2×D59—81no—
240C2×D59—81no—
241C2×D59—81no—
242C2×D59—164no—
243C2×D59#51012yes—
244C4×C59—21no—
245C4×C59—81no—
246C4×C59—81no—
247C4×C59—81no—
248C4×C59#54611yescharacteristic
249C4×C59#57711yescharacteristic
250Dic59—21no—
251Dic59—21no—
252Dic59—21no—
253Dic59—81no—
254Dic59—81no—
255Dic59—81no—
256#3338—591no—
257#3280—591no—
258#2243—591no—
259#2236—591no—
260#2232—591no—
261#2224—591no—
262#2220—591no—
263#2216—591no—
264#2211—591no—
265#2189—591no—
266#2187—591no—
267#2175—591no—
268#2168—591no—
269#2139D5911yescharacteristic, complemented
270C2×#357—591no—
271C2×C59—41no—
272C2×C59—82no—
273C2×C59#329411yescharacteristic
274C2×C59C2×#51011yescharacteristic
275C2×C59#341511yescharacteristic
276D59—22no—
277D59—161no—
278C2×#135C2×D5911yescharacteristic
279C22×D8—591no—
280#554—1181no—
281#546—1181no—
282C2×Q32—591no—
283C2×SD32—591no—
284C2×SD32—591no—
285C2×M32C2×D5911yescharacteristic, complemented
286C22×C16—591no—
287#500—591no—
288#495—591no—
289#484—1181no—
290#484—1181no—
291#477—1181no—
292#467—591no—
293#467—591no—
294#467—591no—
295#466—591no—
296#463—1181no—
297#450—1181no—
298#449—1181no—
299C4×Q16—591no—
300C4×D8—591no—
301C8×D4—591no—
302C2×#107—591no—
303C2×#107C2×D5911yescharacteristic, complemented
304#366—591no—
305#366—591no—
306#366—591no—
307C4⋊C16—591no—
308#358—591no—
309#358—591no—
310#358C2×D5911yescharacteristic, complemented
311#358C2×D5911yescharacteristic, complemented
312#357—591no—
313#357—591no—
314#357—591no—
315#357—591no—
316#357C2×D5911yescharacteristic
317#357C2×D5911yescharacteristic
318C22⋊C16—591no—
319C22⋊C16—591no—
320#346—591no—
321C59#3388111yescharacteristic, complemented, Sylow
322C2×#41—21no—
323C22×D4—1181no—
324#135—22no—
325#135—41no—
326C2×Q16—591no—
327C2×Q16C22×D5911yescharacteristic
328C2×SD16—21no—
329C2×SD16—1181no—
330C2×D8—591no—
331C2×D8—1181no—
332C2×D8—1182no—
333C2×D8—2361no—
334C2×D8C22×D5911yescharacteristic
335C22×C8—591no—
336C22×C8C22×D5911yescharacteristic
337C4⋊Q8—1181no—
338#125—2361no—
339#124—1181no—
340#123—2361no—
341C22⋊Q8—1181no—
342C22⋊Q8—1181no—
343C22⋊Q8—1181no—
344C22⋊Q8—2361no—
345C22⋊Q8—2361no—
346#121—1181no—
347#121—2361no—
348#120—2361no—
349C4×Q8—1181no—
350C4×D4—591no—
351C4×D4—1181no—
352C4×D4—2361no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—1181no—
355C2×C22⋊C4—2361no—
356Q32—2361no—
357SD32—1182no—
358SD32—2361no—
359M32—41no—
360C2×C16—591no—
361C2×C16—591no—
362C2×C16—2361no—
363C2×C16C22×D5911yescharacteristic
364C2×C16C22×D5911yescharacteristic
365#107—21no—
366#107—591no—
367#107—591no—
368#107—1181no—
369#107—1181no—
370#107C22×D5911yescharacteristic
371#107C22×D5911yescharacteristic
372C4⋊C8—591no—
373#103—1181no—
374#103—1181no—
375#103—1181no—
376#103—1181no—
377#102—1181no—
378#102—1181no—
379C22⋊C8—591no—
380C22⋊C8—591no—
381C4×C8—591no—
382C24—2361no—
383#41—42no—
384#41—81no—
385C2×Q8—21no—
386C2×Q8—1181no—
387C2×Q8—2361no—
388C2×D4—21no—
389C2×D4—41no—
390C2×D4—591no—
391C2×D4—1181no—
392C2×D4—4721no—
393C2×D4—2361no—
394C2×D4—2362no—
395C2×D4—2361no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—591no—
399C22×C4—2361no—
400C22×C4—2361no—
401C22×C4D4×D5911yescharacteristic
402Q16—21no—
403Q16—21no—
404Q16—1181no—
405Q16—1181no—
406SD16—42no—
407SD16—4721no—
408D8—21no—
409D8—1181no—
410D8—1181no—
411D8—2362no—
412D8#1174568512yes—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—591no—
416C2×C8—591no—
417C2×C8—1181no—
418C2×C8—1181no—
419C2×C8D4×D5911yescharacteristic
420C2×C8C23×D5911yescharacteristic
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—41no—
424C4⋊C4—591no—
425C4⋊C4—1181no—
426C4⋊C4—1181no—
427C4⋊C4—2361no—
428C4⋊C4—2361no—
429C4⋊C4—2361no—
430C4⋊C4—2361no—
431C4⋊C4—2361no—
432C4⋊C4—2361no—
433C22⋊C4—591no—
434C22⋊C4—591no—
435C22⋊C4—2361no—
436C22⋊C4—2361no—
437C22⋊C4—2361no—
438C22⋊C4—2361no—
439C22⋊C4—2361no—
440C22⋊C4—2361no—
441C22⋊C4—2361no—
442C42—591no—
443C42—2361no—
444C16—21no—
445C16—21no—
446C16—1181no—
447C16—1181no—
448C23—41no—
449C23—591no—
450C23—9441no—
451C23—4722no—
452C23—4722no—
453C23—2361no—
454C23D8×D5911yescharacteristic
455Q8—41no—
456Q8—41no—
457Q8—9441no—
458Q8—2361no—
459Q8—2361no—
460D4—22no—
461D4—41no—
462D4—82no—
463D4—1182no—
464D4—9441no—
465D4—4722no—
466D4—2361no—
467D4—2361no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—82no—
474C2×C4—161no—
475C2×C4—591no—
476C2×C4—591no—
477C2×C4—591no—
478C2×C4—1181no—
479C2×C4—1181no—
480C2×C4—9441no—
481C2×C4—9441no—
482C2×C4—2361no—
483C2×C4—2361no—
484C2×C4—2361no—
485C2×C4D8×D5911yescharacteristic
486C2×C4C2×D4×D5911yescharacteristic
487C8—21no—
488C8—1181no—
489C8—1181no—
490C8C2×#1174568511yescharacteristic
491C8#1427377211yescharacteristic
492C22—21no—
493C22—21no—
494C22—21no—
495C22—42no—
496C22—81no—
497C22—81no—
498C22—592no—
499C22—1181no—
500C22—1181no—
501C22—9444no—
502C22—4721no—
503C22—4721no—
504C22—4721no—
505C22C2×D8×D5911yescharacteristic
506C4—21no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—1181no—
511C4—1181no—
512C4—1181no—
513C4—4721no—
514C4—4721no—
515C4—4721no—
516C4#1683804211yescharacteristic
517C4#1683820811yescharacteristic
518C2—41no—
519C2—82no—
520C2—1182no—
521C2—9441no—
522C2#2143590511yescharacteristic
523C2#510×D5911yescharacteristic
524C2#2143723611yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2139C2
socleC22×C59C22×D8
prime radical#21435775C2
hypercenter#2139D59
hypocenterC59#33881
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2139

Complements

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

NH ≅ G/Ntype
1. #21436486C2semidirect
3. #21426334C2semidirect
4. #21426297C2semidirect
5. #21426276C2semidirect
6. #21426231C2semidirect
7. #21426209C2semidirect
8. #21426186C2semidirect
9. #21426161C2semidirect
10. #21426036C2semidirect
11. #21426026C2semidirect
12. #21425957C2semidirect
13. #21425919C2semidirect
14. #357×D59C2semidirect
15. C59×#2139C2semidirect
19. C2×#14273673C22semidirect
23. #16837806C22semidirect
29. #16837643C22semidirect
35. #16837496C22semidirect
39. #16837228C22semidirect
40. #16837227C22semidirect
41. #16837227C22semidirect
42. Dic59⋊C16C22semidirect
43. #16837207C22semidirect
44. #16837206C22semidirect
49. C22⋊(C59⋊C16)C22semidirect
51. #16837177C22semidirect
53. C2×M32×C59C22semidirect
54. C2×#107×C59C22semidirect
55. C59×#358C22semidirect
56. C59×#358C22semidirect
269. #2139D59semidirect
285. C2×M32C2×D59semidirect
303. C2×#107C2×D59semidirect
310. #358C2×D59semidirect
311. #358C2×D59semidirect
321. C59#33881semidirect

Maximal subgroups

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

subgroupindex
#214364862
#214357752
#214263342
#214262972
#214262762
#214262312
#214262092
#214261862
#214261612
#214260362
#214260262
#214259572
#214259192
#357×D592
C59×#21392
#3388159