Finite Group Databasebeta

← #28745750

Subgroups — #28745750

Order 15104.

536 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436465C211yesmaximal, characteristic
2#21426341C211yesmaximal, characteristic, complemented
3#21426301C211yesmaximal, characteristic, complemented
4#21426291C211yesmaximal, characteristic, complemented
5#21426260C211yesmaximal, characteristic, complemented
6#21426202C211yesmaximal, characteristic, complemented
7#21426185C211yesmaximal, characteristic, complemented
8#21426143C211yesmaximal, characteristic, complemented
9#21426030C211yesmaximal, characteristic, complemented
10#21426030C211yesmaximal, characteristic, complemented
11#21425965C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic
13#21425816C211yesmaximal, characteristic, complemented
14C2×#16837206C211yesmaximal, characteristic, complemented
15D4×Q16×C59C211yesmaximal, characteristic
16#16838092—21no—
17C2×#14273673C2211yescharacteristic
18C2×#14273673C2211yescharacteristic
19C2×#14273673C2211yescharacteristic
20C2×#14273667C2211yescharacteristic
21C22×C59⋊C16C2211yescharacteristic
22#16837804C2211yescharacteristic
23#16837744—21no—
24#16837733—21no—
25#16837710—21no—
26#16837643C2211yescharacteristic
27#16837643C2211yescharacteristic
28C4×Dic236C2211yescharacteristic
29#16837440C2211yescharacteristic
30C2×#14273661C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837229C2211yescharacteristic
33#16837227C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35#16837206C2211yescharacteristic
36#16837206C2211yescharacteristic
37#16837206C2211yescharacteristic
38#16837206C2211yescharacteristic
39#16837206C2211yescharacteristic
40#16837206C2211yescharacteristic
41#16837206C2211yescharacteristic
42#16837206C2211yescharacteristic
43#16837206C2211yescharacteristic
44#16837205C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic
47#16837178C2211yescharacteristic
48C22×Q16×C59C2211yescharacteristic
49C22×Q16×C59C2211yescharacteristic
50D4×Q8×C59—21no—
51C59×#494C2211yescharacteristic
52C59×#467C2211yescharacteristic
53C59×#467C2211yescharacteristic
54C59×#462—21no—
55C59×#451—21no—
56C59×#451—21no—
57C4×Q16×C59C2211yescharacteristic
58C8×D4×C59C2211yescharacteristic
59C2×Dic236C2311yescharacteristic
60#14273720—41no—
61#14273719—21no—
62#14273717—41no—
63#14273708—41no—
64#14273706—21no—
65#14273706—21no—
66C4×Dic118—21no—
67#14273686—21no—
68#14273683—41no—
69C2×C4⋊Dic59—21no—
70#14273673—22no—
71#14273673—41no—
72#14273673—41no—
73#14273667—41no—
74C2×C59⋊C16—41no—
75C2×C59⋊C16C2311yescharacteristic
76C2×C59⋊C16C2311yescharacteristic
77C2×C59⋊C16C2311yescharacteristic
78C2×C59⋊C16C2311yescharacteristic
79#14273661—21no—
80#14273661C2311yescharacteristic
81#14273661C2311yescharacteristic
82#14273661C2311yescharacteristic
83#14273659—21no—
84#14273659—21no—
85#14273652—21no—
86#14273652—21no—
87#14273652—21no—
88C22×Q8×C59—21no—
89C22×Q8×C59—21no—
90C2×Q16×C59—22no—
91C2×Q16×C59—22no—
92C2×Q16×C59—21no—
93C2×Q16×C59—21no—
94C2×Q16×C59—21no—
95C2×Q16×C59—41no—
96C2×Q16×C59—41no—
97C2×Q16×C59C2311yescharacteristic
98C2×Q16×C59C2311yescharacteristic
99C2×Q16×C59C2311yescharacteristic
100C22×C8×C59C2311yescharacteristic
101C22×C8×C59C2311yescharacteristic
102C4⋊Q8×C59—21no—
103C4⋊Q8×C59—41no—
104C22⋊Q8×C59—21no—
105C22⋊Q8×C59—21no—
106C22⋊Q8×C59—41no—
107C22⋊Q8×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#103×C59—21no—
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119C2×Q8×C59—21no—
120C2×Q8×C59—21no—
121C2×Q8×C59—21no—
122C2×Q8×C59—42no—
123C2×Q8×C59—42no—
124C2×Q8×C59—41no—
125C2×Q8×C59—41no—
126C2×Q8×C59—81no—
127C2×Q8×C59—81no—
128C2×D4×C59D811yescharacteristic
129C22×C4×C59—41no—
130C22×C4×C59—41no—
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×Dic59—41no—
134C2×Dic118—21no—
135Q16×C59—21no—
136Q16×C59—21no—
137Q16×C59—21no—
138Q16×C59—21no—
139Q16×C59—21no—
140Q16×C59—42no—
141Q16×C59—42no—
142Q16×C59—81no—
143Q16×C59#4112yes—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59C2×D411yescharacteristic
149C2×C8×C59C2×D411yescharacteristic
150C2×C8×C59C2×D411yescharacteristic
151C2×C8×C59C2411yescharacteristic
152C4⋊C4×C59—21no—
153C4⋊C4×C59—41no—
154C4⋊C4×C59—41no—
155C4⋊C4×C59—41no—
156C4⋊C4×C59—41no—
157C4⋊C4×C59—41no—
158C4⋊C4×C59D811yescharacteristic
159C22⋊C4×C59—41no—
160C22⋊C4×C59—41no—
161C22⋊C4×C59D811yescharacteristic
162C22⋊C4×C59D811yescharacteristic
163C42×C59—41no—
164C42×C59C2×D411yescharacteristic
165C22⋊Dic59—41no—
166C22⋊Dic59—41no—
167C22⋊Dic59—41no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—41no—
172Dic59⋊C4—41no—
173Dic59⋊C4—41no—
174Dic59⋊C4—41no—
175Dic59⋊C4—41no—
176Dic59⋊C4—41no—
177C4×Dic59—41no—
178C4×Dic59—41no—
179Dic236—21no—
180Dic236—21no—
181C59⋊C16—21no—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C23×C59C2×D811yescharacteristic
186C23×C59C2×D811yescharacteristic
187Q8×C59—22no—
188Q8×C59—41no—
189Q8×C59—41no—
190Q8×C59—41no—
191Q8×C59—41no—
192Q8×C59—41no—
193Q8×C59—82no—
194Q8×C59—82no—
195D4×C59—22no—
196C2×C4×C59—21no—
197C2×C4×C59—21no—
198C2×C4×C59—21no—
199C2×C4×C59—21no—
200C2×C4×C59—41no—
201C2×C4×C59—41no—
202C2×C4×C59—41no—
203C2×C4×C59—41no—
204C2×C4×C59—82no—
205C2×C4×C59—82no—
206C2×C4×C59C2×D811yescharacteristic
207C2×C4×C59C2×D811yescharacteristic
208C2×C4×C59C2×D811yescharacteristic
209C2×C4×C59C2×D811yescharacteristic
210C2×C4×C59C22×D411yescharacteristic
211C2×Dic59—41no—
212C2×Dic59—41no—
213C2×Dic59—41no—
214C2×Dic59—41no—
215C2×Dic59—82no—
216Dic118—41no—
217Dic118—41no—
218C8×C59—21no—
219C8×C59—21no—
220C8×C59—21no—
221C8×C59C2×#4111yescharacteristic
222C8×C59H(5,𝔽2)11yescharacteristic
223#33879—591nomaximal, Sylow
224C22×C59—21no—
225C22×C59—21no—
226C22×C59—21no—
227C22×C59—21no—
228C22×C59—21no—
229C22×C59#51012yes—
230C22×C59C22×D811yescharacteristic
231C4×C59—21no—
232C4×C59—21no—
233C4×C59—21no—
234C4×C59—21no—
235C4×C59—42no—
236C4×C59—81no—
237C4×C59—81no—
238C4×C59—81no—
239C4×C59#54611yescharacteristic
240C4×C59#57711yescharacteristic
241Dic59—81no—
242Dic59—81no—
243Dic59—81no—
244Dic59—81no—
245#3336—591no—
246D4×Q16D5911yescharacteristic
247#2245—591no—
248#2237—591no—
249#2235—591no—
250#2229—591no—
251#2219—591no—
252#2216—591no—
253#2209—591no—
254#2188—591no—
255#2188—591no—
256#2176—591no—
257#2168—591no—
258#2141—591no—
259C2×#358—591no—
260C2×C59—22no—
261C2×C59—41no—
262C2×C59#329411yescharacteristic
263C2×C59C2×#51011yescharacteristic
264C2×C59#341711yescharacteristic
265C22×Q16C2×D5911yescharacteristic
266C22×Q16C2×D5911yescharacteristic
267#552—1181no—
268D4×Q8—21no—
269C2×Q32—591no—
270C2×Q32—591no—
271C2×Q32—591no—
272C2×M32—591no—
273C22×C16—591no—
274#499—591no—
275#494C2×D5911yescharacteristic
276#485—1181no—
277#483—1181no—
278#479—1181no—
279#467—591no—
280#467—591no—
281#467C2×D5911yescharacteristic
282#467C2×D5911yescharacteristic
283#462—21no—
284#451—21no—
285#451—21no—
286C4×Q16—591no—
287C4×Q16C2×D5911yescharacteristic
288C8×D4C2×D5911yescharacteristic
289#427—591no—
290C2×#107—591no—
291#367—591no—
292#367—591no—
293#366—591no—
294C4⋊C16—591no—
295#358—591no—
296#358—591no—
297#358—591no—
298#358—591no—
299#358—591no—
300#358—591no—
301#358—591no—
302#358—591no—
303#358—591no—
304#358—591no—
305C22⋊C16—591no—
306C22⋊C16—591no—
307#346—591no—
308C59#3387911yescharacteristic, complemented, Sylow
309C22×Q8—21no—
310C22×Q8—21no—
311C2×Q16—22no—
312C2×Q16—22no—
313C2×Q16—21no—
314C2×Q16—21no—
315C2×Q16—21no—
316C2×Q16—41no—
317C2×Q16—41no—
318C2×Q16—591no—
319C2×Q16C22×D5911yescharacteristic
320C2×Q16C22×D5911yescharacteristic
321C2×Q16C22×D5911yescharacteristic
322C22×C8C22×D5911yescharacteristic
323C22×C8C22×D5911yescharacteristic
324C4⋊Q8—21no—
325C4⋊Q8—41no—
326#126—2361no—
327#125—1181no—
328#125—2361no—
329#123—2361no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—41no—
333C22⋊Q8—41no—
334C22⋊Q8—1181no—
335C22⋊Q8—1181no—
336C4×Q8—21no—
337C4×Q8—1181no—
338C4×D4—41no—
339C4×D4C22⋊D5911yescharacteristic
340#117—1181no—
341#117—2361no—
342C2×C4⋊C4—1181no—
343Q32—1182no—
344Q32—2361no—
345Q32—2361no—
346M32—2361no—
347C2×C16—591no—
348C2×C16—591no—
349C2×C16—591no—
350C2×C16—591no—
351C2×C16—2361no—
352#107—591no—
353#107—591no—
354#107—591no—
355#107—1181no—
356#107C22×D5911yescharacteristic
357#106—1181no—
358#106—1181no—
359C4⋊C8C22⋊D5911yescharacteristic
360#103—21no—
361#103—21no—
362#103—21no—
363#103—1181no—
364#103—1181no—
365#103—1181no—
366C22⋊C8C22⋊D5911yescharacteristic
367C22⋊C8C22⋊D5911yescharacteristic
368C4×C8C22×D5911yescharacteristic
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×Q8—81no—
377C2×Q8—81no—
378C2×Q8—1181no—
379C2×D4#1174566911yescharacteristic
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—2361no—
383C22×C4C2×C22⋊D5911yescharacteristic
384C22×C4C2×C22⋊D5911yescharacteristic
385Q16—21no—
386Q16—21no—
387Q16—21no—
388Q16—21no—
389Q16—21no—
390Q16—42no—
391Q16—42no—
392Q16—81no—
393Q16—1181no—
394Q16—1181no—
395Q16#1174568712yes—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8C2×C22⋊D5911yescharacteristic
401C2×C8C2×C22⋊D5911yescharacteristic
402C2×C8C2×C22⋊D5911yescharacteristic
403C2×C8C23×D5911yescharacteristic
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—1181no—
411C4⋊C4—1181no—
412C4⋊C4—1181no—
413C4⋊C4—2361no—
414C4⋊C4—2361no—
415C4⋊C4—2361no—
416C4⋊C4—2361no—
417C4⋊C4—2361no—
418C4⋊C4—2361no—
419C4⋊C4#1174566911yescharacteristic
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—2361no—
423C22⋊C4—2361no—
424C22⋊C4—2361no—
425C22⋊C4#1174566911yescharacteristic
426C22⋊C4#1174566911yescharacteristic
427C42—41no—
428C42—2361no—
429C42—2361no—
430C42C2×C22⋊D5911yescharacteristic
431C16—1181no—
432C16—1181no—
433C16—1181no—
434C16—1181no—
435C23C2×#1174566911yescharacteristic
436C23C2×#1174566911yescharacteristic
437Q8—22no—
438Q8—41no—
439Q8—41no—
440Q8—41no—
441Q8—41no—
442Q8—41no—
443Q8—82no—
444Q8—82no—
445Q8—2361no—
446Q8—2361no—
447D4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—4722no—
459C2×C4—2361no—
460C2×C4—2361no—
461C2×C4—2361no—
462C2×C4—2361no—
463C2×C4C2×#1174566911yescharacteristic
464C2×C4C2×#1174566911yescharacteristic
465C2×C4C2×#1174566911yescharacteristic
466C2×C4C2×#1174566911yescharacteristic
467C2×C4C22×C22⋊D5911yescharacteristic
468C8—21no—
469C8—21no—
470C8—21no—
471C8C2×#1174568711yescharacteristic
472C8#1427377211yescharacteristic
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22#1683784312yes—
479C22C22×#1174566911yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—42no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4—4721no—
489C4—4721no—
490C4—4721no—
491C4—4721no—
492C4#1683804611yescharacteristic
493C4#1683821311yescharacteristic
494C2—22no—
495C2—41no—
496C2#2143590911yescharacteristic
497C2C2×#1683784311yescharacteristic
498C2#2143724711yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingD4×Q16×C59C2
socleC22×C59C22×D8
prime radicalC2×D4×C59D8
hypercenterD4×Q16D59
hypocenterC59#33879
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
2. #21426341C2semidirect
3. #21426301C2semidirect
4. #21426291C2semidirect
5. #21426260C2semidirect
6. #21426202C2semidirect
7. #21426185C2semidirect
8. #21426143C2semidirect
9. #21426030C2semidirect
10. #21426030C2semidirect
11. #21425965C2semidirect
13. #21425816C2semidirect
14. C2×#16837206C2semidirect
308. C59#33879semidirect

Maximal subgroups

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

subgroupindex
#214364652
#214263412
#214263012
#214262912
#214262602
#214262022
#214261852
#214261432
#214260302
#214259652
#214259222
#214258162
C2×#168372062
D4×Q16×C592
#3387959