Finite Group Databasebeta

← #28745810

Subgroups — #28745810

Order 15104.

548 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436489C211yesmaximal, characteristic, complemented
2#21426332C211yesmaximal, characteristic, complemented
3#21426291C211yesmaximal, characteristic, complemented
4#21426279C211yesmaximal, characteristic, complemented
5#21426238C211yesmaximal, characteristic, complemented
6#21426202C211yesmaximal, characteristic, complemented
7#21426190C211yesmaximal, characteristic, complemented
8#21426155C211yesmaximal, characteristic, complemented
9#21426036C211yesmaximal, characteristic, complemented
10#21426030C211yesmaximal, characteristic, complemented
11#21425965C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic, complemented
13#21425808C211yesmaximal, characteristic, complemented
14C2×#16837206C211yesmaximal, characteristic, complemented
15C59×#3285C211yesmaximal, characteristic
16#16838109—21no—
17C2×#14273673C2211yescharacteristic
18C2×#14273673C2211yescharacteristic
19C2×#14273671C2211yescharacteristic
20C2×#14273667C2211yescharacteristic
21C22×C59⋊C16C2211yescharacteristic
22#16837807C2211yescharacteristic
23#16837738—21no—
24#16837738—21no—
25#16837698—21no—
26#16837643C2211yescharacteristic
27#16837643C2211yescharacteristic
28C4×Dic236C2211yescharacteristic
29C2×#14273661C2211yescharacteristic
30C2×#14273661C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837229C2211yescharacteristic
33#16837229C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35#16837206C2211yescharacteristic
36#16837206C2211yescharacteristic
37#16837206C2211yescharacteristic
38#16837206C2211yescharacteristic
39#16837206C2211yescharacteristic
40#16837206C2211yescharacteristic
41#16837205C2211yescharacteristic
42#16837203C2211yescharacteristic
43#16837203C2211yescharacteristic
44#16837203C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic
47#16837178C2211yescharacteristic
48C2×#135×C59C2211yescharacteristic
49C22×Q16×C59C2211yescharacteristic
50C59×#548—21no—
51C59×#495C2211yescharacteristic
52C59×#467C2211yescharacteristic
53C59×#466C2211yescharacteristic
54C59×#464—21no—
55C59×#452—21no—
56C59×#448—21no—
57C4×Q16×C59C2211yescharacteristic
58C8×D4×C59C2211yescharacteristic
59C2×Dic236C2311yescharacteristic
60#14273727—21no—
61#14273718—41no—
62#14273706—21no—
63#14273706—21no—
64#14273704—41no—
65#14273704—41no—
66C4×Dic118—21no—
67D4×Dic59—41no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70#14273673—22no—
71#14273673—41no—
72#14273671—41no—
73#14273667—41no—
74C2×C59⋊C16—41no—
75C2×C59⋊C16C2311yescharacteristic
76C2×C59⋊C16C2311yescharacteristic
77C2×C59⋊C16C2311yescharacteristic
78C2×C59⋊C16C2311yescharacteristic
79#14273661—21no—
80#14273661—21no—
81#14273661—21no—
82#14273661C2311yescharacteristic
83#14273661C2311yescharacteristic
84#14273661C2311yescharacteristic
85#14273652—21no—
86#14273652—21no—
87#14273652—21no—
88C2×#41×C59—21no—
89C22×Q8×C59—21no—
90#135×C59—22no—
91#135×C59—41no—
92C2×Q16×C59—22no—
93C2×Q16×C59—21no—
94C2×Q16×C59—41no—
95C2×Q16×C59C2311yescharacteristic
96C2×Q16×C59C2311yescharacteristic
97C2×SD16×C59—21no—
98C2×SD16×C59—21no—
99C2×D8×C59C2311yescharacteristic
100C22×C8×C59C2311yescharacteristic
101C22×C8×C59C2311yescharacteristic
102#124×C59—21no—
103#124×C59—41no—
104C22⋊Q8×C59—21no—
105C22⋊Q8×C59—41no—
106#121×C59—21no—
107#121×C59—41no—
108C4×Q8×C59—21no—
109C4×D4×C59—41no—
110C4×D4×C59D411yescharacteristic
111#107×C59C2311yescharacteristic
112C4⋊C8×C59D411yescharacteristic
113#103×C59—21no—
114#103×C59—21no—
115#102×C59—21no—
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119#41×C59—42no—
120#41×C59—81no—
121C2×Q8×C59—21no—
122C2×Q8×C59—21no—
123C2×Q8×C59—42no—
124C2×Q8×C59—41no—
125C2×Q8×C59—81no—
126C2×D4×C59—21no—
127C2×D4×C59—41no—
128C2×D4×C59—41no—
129C2×D4×C59D811yescharacteristic
130C22×C4×C59—41no—
131C22×C4×C59—41no—
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×Dic59—41no—
135C22×Dic59—41no—
136C2×Dic118—21no—
137C2×Dic118—41no—
138Q16×C59—21no—
139Q16×C59—21no—
140Q16×C59—21no—
141Q16×C59—42no—
142Q16×C59#4112yes—
143SD16×C59—42no—
144SD16×C59—81no—
145D8×C59—21no—
146D8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59C2×D411yescharacteristic
152C2×C8×C59C2×D411yescharacteristic
153C2×C8×C59C2×D411yescharacteristic
154C2×C8×C59C2411yescharacteristic
155C4⋊C4×C59—21no—
156C4⋊C4×C59—41no—
157C4⋊C4×C59—41no—
158C4⋊C4×C59D811yescharacteristic
159C22⋊C4×C59—41no—
160C22⋊C4×C59—41no—
161C22⋊C4×C59—41no—
162C22⋊C4×C59—41no—
163C22⋊C4×C59D811yescharacteristic
164C22⋊C4×C59D811yescharacteristic
165C42×C59—41no—
166C42×C59C2×D411yescharacteristic
167C22⋊Dic59—41no—
168C22⋊Dic59—41no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—41no—
173C4⋊Dic59—41no—
174C4⋊Dic59—41no—
175Dic59⋊C4—41no—
176Dic59⋊C4—41no—
177Dic59⋊C4—41no—
178C4×Dic59—41no—
179Dic236—21no—
180Dic236—21no—
181C59⋊C16—21no—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C23×C59—41no—
186C23×C59C2×D811yescharacteristic
187C23×C59C2×D811yescharacteristic
188Q8×C59—22no—
189Q8×C59—41no—
190Q8×C59—41no—
191Q8×C59—41no—
192Q8×C59—82no—
193D4×C59—22no—
194D4×C59—41no—
195D4×C59—41no—
196D4×C59—82no—
197D4×C59—82no—
198C2×C4×C59—21no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—41no—
203C2×C4×C59—41no—
204C2×C4×C59—41no—
205C2×C4×C59—82no—
206C2×C4×C59—82no—
207C2×C4×C59C2×D811yescharacteristic
208C2×C4×C59C2×D811yescharacteristic
209C2×C4×C59C2×D811yescharacteristic
210C2×C4×C59C2×D811yescharacteristic
211C2×C4×C59C22×D411yescharacteristic
212C2×Dic59—41no—
213C2×Dic59—41no—
214C2×Dic59—41no—
215C2×Dic59—41no—
216C2×Dic59—161no—
217C2×Dic59—161no—
218Dic118—41no—
219Dic118—41no—
220Dic118—161no—
221C8×C59—21no—
222C8×C59—21no—
223C8×C59—21no—
224C8×C59C2×#4111yescharacteristic
225C8×C59H(5,𝔽2)11yescharacteristic
226#33883—591nomaximal, Sylow
227C22×C59—21no—
228C22×C59—21no—
229C22×C59—21no—
230C22×C59—21no—
231C22×C59—21no—
232C22×C59—81no—
233C22×C59—81no—
234C22×C59—81no—
235C22×C59#51012yes—
236C22×C59C22×D811yescharacteristic
237C4×C59—21no—
238C4×C59—21no—
239C4×C59—21no—
240C4×C59—21no—
241C4×C59—42no—
242C4×C59—81no—
243C4×C59—81no—
244C4×C59#54611yescharacteristic
245C4×C59#57711yescharacteristic
246Dic59—81no—
247Dic59—81no—
248Dic59—81no—
249Dic59—81no—
250#3338—591no—
251#3285D5911yescharacteristic
252#2243—591no—
253#2235—591no—
254#2233—591no—
255#2225—591no—
256#2219—591no—
257#2217—591no—
258#2211—591no—
259#2189—591no—
260#2188—591no—
261#2176—591no—
262#2168—591no—
263#2139—591no—
264C2×#358—591no—
265C2×C59—22no—
266C2×C59—41no—
267C2×C59—161no—
268C2×C59#329411yescharacteristic
269C2×C59C2×#51011yescharacteristic
270C2×C59#341611yescharacteristic
271C2×#135C2×D5911yescharacteristic
272C22×Q16C2×D5911yescharacteristic
273#554—1181no—
274#548—21no—
275C2×Q32—591no—
276C2×Q32—591no—
277C2×SD32—591no—
278C2×M32—591no—
279C22×C16—591no—
280#500—591no—
281#495C2×D5911yescharacteristic
282#484—1181no—
283#484—1181no—
284#477—1181no—
285#467—591no—
286#467—591no—
287#467C2×D5911yescharacteristic
288#466C2×D5911yescharacteristic
289#464—21no—
290#452—21no—
291#448—21no—
292C4×Q16—591no—
293C4×Q16C2×D5911yescharacteristic
294C8×D4C2×D5911yescharacteristic
295C2×#107—591no—
296C2×#107—591no—
297#367—591no—
298#367—591no—
299#367—591no—
300C4⋊C16—591no—
301#358—591no—
302#358—591no—
303#358—591no—
304#358—591no—
305#358—591no—
306#358—591no—
307#358—591no—
308#357—591no—
309#357—591no—
310#357—591no—
311C22⋊C16—591no—
312C22⋊C16—591no—
313#346—591no—
314C59#3388311yescharacteristic, complemented, Sylow
315C2×#41—21no—
316C22×Q8—21no—
317#135—22no—
318#135—41no—
319C2×Q16—22no—
320C2×Q16—21no—
321C2×Q16—41no—
322C2×Q16—591no—
323C2×Q16C22×D5911yescharacteristic
324C2×Q16C22×D5911yescharacteristic
325C2×SD16—21no—
326C2×SD16—21no—
327C2×D8C22×D5911yescharacteristic
328C22×C8C22×D5911yescharacteristic
329C22×C8C22×D5911yescharacteristic
330C4⋊Q8—1181no—
331#125—2361no—
332#124—21no—
333#124—41no—
334C22⋊Q8—21no—
335C22⋊Q8—41no—
336C22⋊Q8—1181no—
337C22⋊Q8—1181no—
338C22⋊Q8—2361no—
339C22⋊Q8—2361no—
340#121—21no—
341#121—41no—
342C4×Q8—21no—
343C4×Q8—1181no—
344C4×D4—41no—
345C4×D4—2361no—
346C4×D4C22⋊D5911yescharacteristic
347C2×C4⋊C4—1181no—
348C2×C4⋊C4—1181no—
349Q32—1182no—
350Q32—2361no—
351SD32—2361no—
352M32—2361no—
353C2×C16—591no—
354C2×C16—591no—
355C2×C16—591no—
356C2×C16—591no—
357C2×C16—2361no—
358#107—591no—
359#107—591no—
360#107—591no—
361#107—1181no—
362#107—1181no—
363#107—1181no—
364#107C22×D5911yescharacteristic
365C4⋊C8C22⋊D5911yescharacteristic
366#103—21no—
367#103—21no—
368#103—1181no—
369#103—1181no—
370#103—1181no—
371#102—21no—
372C22⋊C8C22⋊D5911yescharacteristic
373C22⋊C8C22⋊D5911yescharacteristic
374C4×C8C22×D5911yescharacteristic
375#41—42no—
376#41—81no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—42no—
380C2×Q8—41no—
381C2×Q8—81no—
382C2×Q8—1181no—
383C2×Q8—2361no—
384C2×D4—21no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4#1174566911yescharacteristic
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—2361no—
391C22×C4—2361no—
392C22×C4C2×C22⋊D5911yescharacteristic
393C22×C4C2×C22⋊D5911yescharacteristic
394Q16—21no—
395Q16—21no—
396Q16—21no—
397Q16—42no—
398Q16—1181no—
399Q16—1181no—
400Q16#1174568712yes—
401SD16—42no—
402SD16—81no—
403D8—21no—
404D8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8C2×C22⋊D5911yescharacteristic
410C2×C8C2×C22⋊D5911yescharacteristic
411C2×C8C2×C22⋊D5911yescharacteristic
412C2×C8C23×D5911yescharacteristic
413C4⋊C4—21no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—1181no—
417C4⋊C4—1181no—
418C4⋊C4—1181no—
419C4⋊C4—2361no—
420C4⋊C4—2361no—
421C4⋊C4—2361no—
422C4⋊C4—2361no—
423C4⋊C4—2361no—
424C4⋊C4—2361no—
425C4⋊C4#1174566911yescharacteristic
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—2361no—
431C22⋊C4—2361no—
432C22⋊C4#1174566911yescharacteristic
433C22⋊C4#1174566911yescharacteristic
434C42—41no—
435C42—2361no—
436C42C2×C22⋊D5911yescharacteristic
437C16—1181no—
438C16—1181no—
439C16—1181no—
440C16—1181no—
441C23—41no—
442C23C2×#1174566911yescharacteristic
443C23C2×#1174566911yescharacteristic
444Q8—22no—
445Q8—41no—
446Q8—41no—
447Q8—41no—
448Q8—82no—
449Q8—9441no—
450Q8—2361no—
451Q8—2361no—
452D4—22no—
453D4—41no—
454D4—41no—
455D4—82no—
456D4—82no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—9441no—
467C2×C4—9441no—
468C2×C4—2361no—
469C2×C4—2361no—
470C2×C4—2361no—
471C2×C4—2361no—
472C2×C4C2×#1174566911yescharacteristic
473C2×C4C2×#1174566911yescharacteristic
474C2×C4C2×#1174566911yescharacteristic
475C2×C4C2×#1174566911yescharacteristic
476C2×C4C22×C22⋊D5911yescharacteristic
477C8—21no—
478C8—21no—
479C8—21no—
480C8C2×#1174568711yescharacteristic
481C8#1427377211yescharacteristic
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22#1683784212yes—
491C22C22×#1174566911yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—42no—
497C4—81no—
498C4—81no—
499C4—4721no—
500C4—4721no—
501C4—4721no—
502C4—4721no—
503C4#1683804611yescharacteristic
504C4#1683821311yescharacteristic
505C2—22no—
506C2—41no—
507C2—161no—
508C2#2143590911yescharacteristic
509C2C2×#1683784211yescharacteristic
510C2#2143724011yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
Fitting, prime radicalC59×#3285C2
socleC22×C59C22×D8
hypercenter#3285D59
hypocenterC59#33883
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3285

Complements

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

NH ≅ G/Ntype
1. #21436489C2semidirect
2. #21426332C2semidirect
3. #21426291C2semidirect
4. #21426279C2semidirect
5. #21426238C2semidirect
6. #21426202C2semidirect
7. #21426190C2semidirect
8. #21426155C2semidirect
9. #21426036C2semidirect
10. #21426030C2semidirect
11. #21425965C2semidirect
12. #21425922C2semidirect
13. #21425808C2semidirect
14. C2×#16837206C2semidirect
314. C59#33883semidirect

Maximal subgroups

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

subgroupindex
#214364892
#214263322
#214262912
#214262792
#214262382
#214262022
#214261902
#214261552
#214260362
#214260302
#214259652
#214259222
#214258082
C2×#168372062
C59×#32852
#3388359