Finite Group Databasebeta

← #28745744

Subgroups — #28745744

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
1D4×Dic236C211yesmaximal, characteristic
2#21426343C211yesmaximal, characteristic, complemented
3#21426304C211yesmaximal, characteristic, complemented
4#21426293C211yesmaximal, characteristic, complemented
5#21426257C211yesmaximal, characteristic, complemented
6#21426202C211yesmaximal, characteristic, complemented
7#21426185C211yesmaximal, characteristic, complemented
8#21426146C211yesmaximal, characteristic, complemented
9#21426029C211yesmaximal, characteristic, complemented
10#21426029C211yesmaximal, characteristic, complemented
11#21425965C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic
13#21425815C211yesmaximal, characteristic, complemented
14C2×#16837205C211yesmaximal, characteristic, complemented
15C59×#3336C211yesmaximal, characteristic
16C22×Dic236C2211yescharacteristic
17C22×Dic236C2211yescharacteristic
18D4×Dic118—21no—
19C2×#14273673C2211yescharacteristic
20C2×#14273673C2211yescharacteristic
21C2×#14273673C2211yescharacteristic
22C2×#14273667C2211yescharacteristic
23C22×C59⋊C16C2211yescharacteristic
24#16837782C2211yescharacteristic
25#16837643C2211yescharacteristic
26#16837643C2211yescharacteristic
27#16837613—21no—
28#16837560—21no—
29#16837560—21no—
30C4×Dic236C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837229C2211yescharacteristic
33#16837227C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35#16837206C2211yescharacteristic
36#16837205C2211yescharacteristic
37#16837205C2211yescharacteristic
38#16837205C2211yescharacteristic
39#16837205C2211yescharacteristic
40#16837205C2211yescharacteristic
41#16837205C2211yescharacteristic
42#16837205C2211yescharacteristic
43#16837205C2211yescharacteristic
44#16837205C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic
47#16837178C2211yescharacteristic
48C59×#552—21no—
49C59×#499C2211yescharacteristic
50C59×#485—21no—
51C59×#483—21no—
52C59×#479—21no—
53C59×#467C2211yescharacteristic
54C59×#467C2211yescharacteristic
55C4×Q16×C59C2211yescharacteristic
56C8×D4×C59C2211yescharacteristic
57C59×#427C2211yescharacteristic
58C2×#107×C59C2211yescharacteristic
59C22×Dic118—21no—
60C22×Dic118—21no—
61C2×Dic236—22no—
62C2×Dic236—22no—
63C2×Dic236—21no—
64C2×Dic236—21no—
65C2×Dic236—21no—
66C2×Dic236—41no—
67C2×Dic236—41no—
68C2×Dic236C2311yescharacteristic
69C2×Dic236C2311yescharacteristic
70C2×Dic236C2311yescharacteristic
71#14273727—21no—
72#14273726—41no—
73#14273706—21no—
74#14273706—21no—
75#14273704—41no—
76#14273704—41no—
77C4×Dic118—21no—
78D4×Dic59—41no—
79#14273673—22no—
80#14273673—41no—
81#14273673—41no—
82#14273667—41no—
83C2×C59⋊C16—41no—
84C2×C59⋊C16C2311yescharacteristic
85C2×C59⋊C16C2311yescharacteristic
86C2×C59⋊C16C2311yescharacteristic
87C2×C59⋊C16C2311yescharacteristic
88#14273661C2311yescharacteristic
89#14273652—21no—
90#14273652—21no—
91#14273652—21no—
92C2×Q16×C59C2311yescharacteristic
93C22×C8×C59C2311yescharacteristic
94C22×C8×C59C2311yescharacteristic
95#126×C59—41no—
96#125×C59—21no—
97#125×C59—41no—
98#123×C59—41no—
99C22⋊Q8×C59—21no—
100C22⋊Q8×C59—21no—
101C4×Q8×C59—21no—
102C4×D4×C59D411yescharacteristic
103#117×C59—21no—
104#117×C59—41no—
105C2×C4⋊C4×C59—21no—
106#107×C59—21no—
107#107×C59C2311yescharacteristic
108#107×C59C2311yescharacteristic
109#107×C59C2311yescharacteristic
110#106×C59—21no—
111#106×C59—21no—
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×D4×C59D811yescharacteristic
121C22×C4×C59—41no—
122C22×C4×C59C2×D411yescharacteristic
123C22×C4×C59C2×D411yescharacteristic
124C22×Dic59—41no—
125C22×Dic59—41no—
126C2×Dic118—21no—
127C2×Dic118—21no—
128C2×Dic118—21no—
129C2×Dic118—42no—
130C2×Dic118—42no—
131C2×Dic118—41no—
132C2×Dic118—41no—
133C2×Dic118—81no—
134C2×Dic118—81no—
135Q16×C59—21no—
136Q16×C59—21no—
137C2×C8×C59—21no—
138C2×C8×C59—21no—
139C2×C8×C59—21no—
140C2×C8×C59—21no—
141C2×C8×C59C2×D411yescharacteristic
142C2×C8×C59C2×D411yescharacteristic
143C2×C8×C59C2×D411yescharacteristic
144C2×C8×C59C2411yescharacteristic
145C4⋊C4×C59—21no—
146C4⋊C4×C59—21no—
147C4⋊C4×C59—21no—
148C4⋊C4×C59—41no—
149C4⋊C4×C59—41no—
150C4⋊C4×C59—41no—
151C4⋊C4×C59—41no—
152C4⋊C4×C59—41no—
153C4⋊C4×C59—41no—
154C4⋊C4×C59D811yescharacteristic
155C22⋊C4×C59—41no—
156C22⋊C4×C59—41no—
157C22⋊C4×C59—41no—
158C22⋊C4×C59D811yescharacteristic
159C22⋊C4×C59D811yescharacteristic
160C42×C59—41no—
161C42×C59—41no—
162C42×C59C2×D411yescharacteristic
163C22⋊Dic59—41no—
164C22⋊Dic59—41no—
165C4⋊Dic59—21no—
166C4⋊Dic59—41no—
167C4⋊Dic59—41no—
168Dic59⋊C4—41no—
169Dic59⋊C4—41no—
170Dic59⋊C4—41no—
171C4×Dic59—41no—
172Dic236—21no—
173Dic236—21no—
174Dic236—21no—
175Dic236—21no—
176Dic236—21no—
177Dic236—42no—
178Dic236—42no—
179Dic236—81no—
180Dic236#4112yes—
181C59⋊C16—21no—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C23×C59C2×D811yescharacteristic
186C23×C59C2×D811yescharacteristic
187Q8×C59—41no—
188Q8×C59—41no—
189D4×C59—22no—
190C2×C4×C59—21no—
191C2×C4×C59—21no—
192C2×C4×C59—21no—
193C2×C4×C59—21no—
194C2×C4×C59—41no—
195C2×C4×C59—41no—
196C2×C4×C59—41no—
197C2×C4×C59—41no—
198C2×C4×C59—82no—
199C2×C4×C59C2×D811yescharacteristic
200C2×C4×C59C2×D811yescharacteristic
201C2×C4×C59C2×D811yescharacteristic
202C2×C4×C59C2×D811yescharacteristic
203C2×C4×C59C22×D411yescharacteristic
204C2×Dic59—41no—
205C2×Dic59—41no—
206C2×Dic59—41no—
207C2×Dic59—41no—
208C2×Dic59—82no—
209C2×Dic59—82no—
210Dic118—22no—
211Dic118—41no—
212Dic118—41no—
213Dic118—41no—
214Dic118—41no—
215Dic118—41no—
216Dic118—82no—
217Dic118—82no—
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—81no—
236C4×C59—81no—
237C4×C59—81no—
238C4×C59—81no—
239C4×C59#54611yescharacteristic
240C4×C59#57711yescharacteristic
241Dic59—42no—
242Dic59—81no—
243Dic59—81no—
244Dic59—81no—
245#3336D5911yescharacteristic
246D4×Q16—591no—
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×Q16—591no—
266C22×Q16—591no—
267#552—21no—
268D4×Q8—1181no—
269C2×Q32—591no—
270C2×Q32—591no—
271C2×Q32—591no—
272C2×M32—591no—
273C22×C16—591no—
274#499C2×D5911yescharacteristic
275#494—591no—
276#485—21no—
277#483—21no—
278#479—21no—
279#467—591no—
280#467—591no—
281#467C2×D5911yescharacteristic
282#467C2×D5911yescharacteristic
283#462—1181no—
284#451—1181no—
285#451—1181no—
286C4×Q16—591no—
287C4×Q16C2×D5911yescharacteristic
288C8×D4C2×D5911yescharacteristic
289#427C2×D5911yescharacteristic
290C2×#107C2×D5911yescharacteristic
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—1181no—
310C22×Q8—1181no—
311C2×Q16—591no—
312C2×Q16—591no—
313C2×Q16—591no—
314C2×Q16—1182no—
315C2×Q16—1182no—
316C2×Q16—1181no—
317C2×Q16—1181no—
318C2×Q16—1181no—
319C2×Q16—2361no—
320C2×Q16—2361no—
321C2×Q16C22×D5911yescharacteristic
322C22×C8C22×D5911yescharacteristic
323C22×C8C22×D5911yescharacteristic
324C4⋊Q8—1181no—
325C4⋊Q8—2361no—
326#126—41no—
327#125—21no—
328#125—41no—
329#123—41no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—1181no—
333C22⋊Q8—1181no—
334C22⋊Q8—2361no—
335C22⋊Q8—2361no—
336C4×Q8—21no—
337C4×Q8—1181no—
338C4×D4—2361no—
339C4×D4C22⋊D5911yescharacteristic
340#117—21no—
341#117—41no—
342C2×C4⋊C4—21no—
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—21no—
353#107—591no—
354#107C22×D5911yescharacteristic
355#107C22×D5911yescharacteristic
356#107C22×D5911yescharacteristic
357#106—21no—
358#106—21no—
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—1181no—
371C2×Q8—1181no—
372C2×Q8—1181no—
373C2×Q8—4721no—
374C2×Q8—4721no—
375C2×Q8—2362no—
376C2×Q8—2362no—
377C2×Q8—2361no—
378C2×Q8—2361no—
379C2×D4#1174566911yescharacteristic
380C22×C4—41no—
381C22×C4—2361no—
382C22×C4—2361no—
383C22×C4C2×C22⋊D5911yescharacteristic
384C22×C4C2×C22⋊D5911yescharacteristic
385Q16—21no—
386Q16—21no—
387Q16—592no—
388Q16—1181no—
389Q16—1181no—
390Q16—1181no—
391Q16—1181no—
392Q16—1181no—
393Q16—4721no—
394Q16—2362no—
395Q16—2362no—
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—21no—
406C4⋊C4—21no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—1181no—
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—41no—
423C22⋊C4—2361no—
424C22⋊C4—2361no—
425C22⋊C4#1174566911yescharacteristic
426C22⋊C4#1174566911yescharacteristic
427C42—41no—
428C42—41no—
429C42—2361no—
430C42C2×C22⋊D5911yescharacteristic
431C16—1181no—
432C16—1181no—
433C16—1181no—
434C16—1181no—
435C23C2×#1174566911yescharacteristic
436C23C2×#1174566911yescharacteristic
437Q8—41no—
438Q8—41no—
439Q8—1182no—
440Q8—4722no—
441Q8—4722no—
442Q8—2361no—
443Q8—2361no—
444Q8—2361no—
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—4722no—
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—
471C8#41×D5911yescharacteristic
472C8#1427377311yescharacteristic
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22#1683784212yes—
479C22C22×#1174566911yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4—4721no—
489C4—4721no—
490C4—4721no—
491C4—2362no—
492C4#1683804011yescharacteristic
493C4#1683821011yescharacteristic
494C2—22no—
495C2—41no—
496C2#2143590311yescharacteristic
497C2C2×#1683784211yescharacteristic
498C2#2143724711yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3336C2
socleC22×C59C22×D8
prime radicalC2×D4×C59D8
hypercenter#3336D59
hypocenterC59#33879
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #21426343C2semidirect
3. #21426304C2semidirect
4. #21426293C2semidirect
5. #21426257C2semidirect
6. #21426202C2semidirect
7. #21426185C2semidirect
8. #21426146C2semidirect
9. #21426029C2semidirect
10. #21426029C2semidirect
11. #21425965C2semidirect
13. #21425815C2semidirect
14. C2×#16837205C2semidirect
308. C59#33879semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2362
#214263432
#214263042
#214262932
#214262572
#214262022
#214261852
#214261462
#214260292
#214259652
#214259222
#214258152
C2×#168372052
C59×#33362
#3387959