Finite Group Databasebeta

← #28743827

Subgroups — #28743827

Order 15104.

534 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434755C211yesmaximal, characteristic, complemented
2#21426298C211yesmaximal, characteristic, complemented
3#21426293C211yesmaximal, characteristic, complemented
4#21426260C211yesmaximal, characteristic, complemented
5#21426253C211yesmaximal, characteristic, complemented
6#21426146C211yesmaximal, characteristic, complemented
7#21426141C211yesmaximal, characteristic, complemented
8#21426036C211yesmaximal, characteristic
9#21426036C211yesmaximal, characteristic
10#21425965C211yesmaximal, characteristic, complemented
11#21425960C211yesmaximal, characteristic, complemented
12C4×#14273673C211yesmaximal, characteristic, complemented
13C4×#14273671C211yesmaximal, characteristic, complemented
14#21425903C211yesmaximal, characteristic, complemented
15C59×#2951C211yesmaximal, characteristic
16#16838014—21no—
17C2×#14273673C2211yescharacteristic
18C2×#14273673C2211yescharacteristic
19C2×#14273671C2211yescharacteristic
20C2×#14273671C2211yescharacteristic
21#16837804C2211yescharacteristic
22#16837782C2211yescharacteristic
23#16837771—21no—
24#16837660—21no—
25#16837643C2211yescharacteristic
26#16837643C2211yescharacteristic
27#16837510—21no—
28C4×Dic236C2211yescharacteristic
29C4×Dic236C2211yescharacteristic
30#16837229C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837227C2211yescharacteristic
33#16837227C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35C4⋊(C59⋊C16)C2211yescharacteristic
36#16837206C2211yescharacteristic
37#16837206C2211yescharacteristic
38#16837205C2211yescharacteristic
39#16837205C2211yescharacteristic
40#16837205C2211yescharacteristic
41#16837205C2211yescharacteristic
42#16837203C2211yescharacteristic
43#16837203C2211yescharacteristic
44C22⋊(C59⋊C16)C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46#16837178C2211yescharacteristic
47C4×C59⋊C16C2211yescharacteristic
48C2×#135×C59C2211yescharacteristic
49C59×#520—21no—
50C59×#440—21no—
51C59×#440—21no—
52C4×Q16×C59C2211yescharacteristic
53C4×Q16×C59C2211yescharacteristic
54C4×D8×C59C2211yescharacteristic
55C4×D8×C59C2211yescharacteristic
56C2×#107×C59C2211yescharacteristic
57C59×#417—21no—
58C59×#405C2211yescharacteristic
59C2×Dic236—21no—
60C2×Dic236C2311yescharacteristic
61C2×Dic236C2311yescharacteristic
62#14273727—21no—
63#14273720—41no—
64#14273720—41no—
65#14273719—21no—
66#14273717—41no—
67#14273717—41no—
68#14273706—21no—
69#14273706—21no—
70C4×Dic118—21no—
71C4×Dic118—21no—
72#14273673—22no—
73#14273673—22no—
74#14273671—22no—
75#14273671—22no—
76C2×C59⋊C16C2311yescharacteristic
77C2×C59⋊C16C2311yescharacteristic
78C2×C59⋊C16C2311yescharacteristic
79C2×C59⋊C16C2311yescharacteristic
80#14273661C2311yescharacteristic
81#14273661C2311yescharacteristic
82#14273659—21no—
83#14273652—21no—
84#14273652—21no—
85#14273652—21no—
86#14273652—21no—
87C2×#41×C59—21no—
88#135×C59—22no—
89#135×C59—22no—
90C2×Q16×C59C2311yescharacteristic
91C2×SD16×C59—21no—
92C2×D8×C59C2311yescharacteristic
93C2×M16×C59D411yescharacteristic
94C22×C8×C59C2311yescharacteristic
95C4×Q8×C59—21no—
96C4×Q8×C59—21no—
97C4×D4×C59—21no—
98C4×D4×C59—21no—
99C4×D4×C59—41no—
100C4×D4×C59—41no—
101#117×C59—41no—
102#117×C59D411yescharacteristic
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—41no—
105#107×C59—21no—
106#107×C59C2311yescharacteristic
107#107×C59C2311yescharacteristic
108#103×C59—21no—
109#103×C59—21no—
110#102×C59—21no—
111#102×C59—21no—
112#97×C59D411yescharacteristic
113#97×C59D411yescharacteristic
114C4×C8×C59C2311yescharacteristic
115C4×C8×C59C2311yescharacteristic
116#41×C59—24no—
117#41×C59—42no—
118C2×Q8×C59—21no—
119C2×D4×C59—21no—
120C2×D4×C59—41no—
121C22×C4×C59—41no—
122C22×C4×C59—41no—
123C22×C4×C59—41no—
124C22×C4×C59—41no—
125C22×C4×C59C2×D411yescharacteristic
126C2×Dic118—21no—
127C2×Dic118—21no—
128Q16×C59—21no—
129Q16×C59#4112yes—
130SD16×C59—24no—
131D8×C59—21no—
132D8×C59#4112yes—
133M16×C59—41no—
134C2×C8×C59—21no—
135C2×C8×C59—21no—
136C2×C8×C59C2×D411yescharacteristic
137C2×C8×C59C2×D411yescharacteristic
138C2×C8×C59C2×D411yescharacteristic
139C2×C8×C59C2411yescharacteristic
140C4⋊C4×C59—21no—
141C4⋊C4×C59—21no—
142C4⋊C4×C59—41no—
143C4⋊C4×C59—41no—
144C4⋊C4×C59—41no—
145C4⋊C4×C59D811yescharacteristic
146C4⋊C4×C59D811yescharacteristic
147C22⋊C4×C59—41no—
148C22⋊C4×C59—41no—
149C22⋊C4×C59D811yescharacteristic
150C22⋊C4×C59D811yescharacteristic
151C42×C59—41no—
152C42×C59—41no—
153C42×C59C2×D411yescharacteristic
154C42×C59C2×D411yescharacteristic
155C22⋊Dic59—41no—
156C22⋊Dic59—41no—
157C4⋊Dic59—21no—
158C4⋊Dic59—21no—
159C4⋊Dic59—41no—
160C4⋊Dic59—41no—
161Dic59⋊C4—41no—
162Dic59⋊C4—41no—
163Dic59⋊C4—41no—
164Dic59⋊C4—41no—
165Dic59⋊C4—41no—
166Dic59⋊C4—41no—
167C4×Dic59—41no—
168C4×Dic59—41no—
169Dic236—21no—
170Dic236—21no—
171Dic236—21no—
172Dic236—21no—
173Dic236—42no—
174C59⋊C16—21no—
175C59⋊C16—21no—
176C59⋊C16—21no—
177C59⋊C16—21no—
178C23×C59—41no—
179C23×C59C2×D811yescharacteristic
180Q8×C59—22no—
181Q8×C59—41no—
182D4×C59—22no—
183D4×C59—44no—
184D4×C59—41no—
185C2×C4×C59—21no—
186C2×C4×C59—21no—
187C2×C4×C59—44no—
188C2×C4×C59—41no—
189C2×C4×C59—41no—
190C2×C4×C59—41no—
191C2×C4×C59—161no—
192C2×C4×C59—161no—
193C2×C4×C59—161no—
194C2×C4×C59C2×D811yescharacteristic
195C2×C4×C59C2×D811yescharacteristic
196C2×C4×C59C2×D811yescharacteristic
197C2×C4×C59C2×D811yescharacteristic
198C2×C4×C59C2×D811yescharacteristic
199C2×C4×C59C22×D411yescharacteristic
200C2×Dic59—41no—
201C2×Dic59—41no—
202C2×Dic59—41no—
203C2×Dic59—41no—
204Dic118—41no—
205Dic118—41no—
206Dic118—41no—
207Dic118—41no—
208C8×C59—21no—
209C8×C59—21no—
210C8×C59—21no—
211C8×C59C2×#4111yescharacteristic
212C8×C59C2×#4111yescharacteristic
213#33680—591nomaximal, Sylow
214C22×C59—21no—
215C22×C59—21no—
216C22×C59—21no—
217C22×C59—42no—
218C22×C59—81no—
219C22×C59—81no—
220C22×C59C22×D811yescharacteristic
221C4×C59—21no—
222C4×C59—21no—
223C4×C59—21no—
224C4×C59—21no—
225C4×C59—21no—
226C4×C59—42no—
227C4×C59—81no—
228C4×C59—81no—
229C4×C59#52511yescharacteristic
230C4×C59#57511yescharacteristic
231Dic59—81no—
232Dic59—81no—
233Dic59—81no—
234Dic59—81no—
235#3177—591no—
236#2951D5911yescharacteristic
237#2236—591no—
238#2235—591no—
239#2229—591no—
240#2228—591no—
241#2209—591no—
242#2208—591no—
243#2189—591no—
244#2189—591no—
245#2176—591no—
246#2175—591no—
247C4×Q32—591no—
248C4×SD32—591no—
249#2164—591no—
250C2×C59—41no—
251C2×C59—82no—
252C2×C59#310011yescharacteristic
253C2×C59#341611yescharacteristic
254C2×C59#341711yescharacteristic
255C2×#135C2×D5911yescharacteristic
256#541—1181no—
257#520—21no—
258C2×Q32—591no—
259C2×Q32—591no—
260C2×SD32—591no—
261C2×SD32—591no—
262#499—591no—
263#494—591no—
264#491—1181no—
265#470—1181no—
266#467—591no—
267#467—591no—
268#441—1181no—
269#440—21no—
270#440—21no—
271C4×Q16—591no—
272C4×Q16—591no—
273C4×Q16C2×D5911yescharacteristic
274C4×Q16C2×D5911yescharacteristic
275C4×D8C2×D5911yescharacteristic
276C4×D8C2×D5911yescharacteristic
277C2×#107C2×D5911yescharacteristic
278#417—21no—
279#405C2×D5911yescharacteristic
280#367—591no—
281#367—591no—
282#366—591no—
283#366—591no—
284C4⋊C16—591no—
285C4⋊C16—591no—
286#358—591no—
287#358—591no—
288#358—591no—
289#358—591no—
290#358—591no—
291#358—591no—
292#357—591no—
293#357—591no—
294C22⋊C16—591no—
295C22⋊C16—591no—
296#346—591no—
297C4×C16—591no—
298C59#3368011yescharacteristic, complemented, Sylow
299C2×#41—21no—
300#135—22no—
301#135—22no—
302C2×Q16—591no—
303C2×Q16—591no—
304C2×Q16—1181no—
305C2×Q16C22×D5911yescharacteristic
306C2×SD16—21no—
307C2×D8C22×D5911yescharacteristic
308C2×M16C22⋊D5911yescharacteristic
309C22×C8C22×D5911yescharacteristic
310C4⋊Q8—1181no—
311#126—2361no—
312#126—2361no—
313#125—1181no—
314#125—2361no—
315#125—2361no—
316C22⋊Q8—1181no—
317C22⋊Q8—1181no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8—1181no—
321C4×Q8—1181no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—41no—
325C4×D4—41no—
326#117—41no—
327#117C22⋊D5911yescharacteristic
328C2×C4⋊C4—21no—
329C2×C4⋊C4—41no—
330Q32—1182no—
331Q32—1182no—
332SD32—1182no—
333SD32—1182no—
334C2×C16—591no—
335C2×C16—591no—
336C2×C16—591no—
337C2×C16—591no—
338#107—21no—
339#107—591no—
340#107—591no—
341#107C22×D5911yescharacteristic
342#107C22×D5911yescharacteristic
343#106—1181no—
344#103—21no—
345#103—21no—
346#103—1181no—
347#103—1181no—
348#103—1181no—
349#103—1181no—
350#102—21no—
351#102—21no—
352#97C22⋊D5911yescharacteristic
353#97C22⋊D5911yescharacteristic
354C4×C8C22×D5911yescharacteristic
355C4×C8C22×D5911yescharacteristic
356#41—24no—
357#41—42no—
358C2×Q8—21no—
359C2×Q8—1181no—
360C2×Q8—1181no—
361C2×D4—21no—
362C2×D4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4C2×C22⋊D5911yescharacteristic
368Q16—21no—
369Q16—1181no—
370Q16—1181no—
371Q16—1181no—
372Q16—1181no—
373Q16—2362no—
374Q16#1174568512yes—
375SD16—24no—
376D8—21no—
377D8#1174568512yes—
378M16—41no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8C2×C22⋊D5911yescharacteristic
382C2×C8C2×C22⋊D5911yescharacteristic
383C2×C8C2×C22⋊D5911yescharacteristic
384C2×C8C23×D5911yescharacteristic
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—1181no—
391C4⋊C4—1181no—
392C4⋊C4—2361no—
393C4⋊C4—2361no—
394C4⋊C4—2361no—
395C4⋊C4—2361no—
396C4⋊C4—2361no—
397C4⋊C4—2361no—
398C4⋊C4—2361no—
399C4⋊C4—2361no—
400C4⋊C4#1174566911yescharacteristic
401C4⋊C4#1174566911yescharacteristic
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—2361no—
405C22⋊C4—2361no—
406C22⋊C4#1174566911yescharacteristic
407C22⋊C4#1174566911yescharacteristic
408C42—41no—
409C42—41no—
410C42—2361no—
411C42—2361no—
412C42C2×C22⋊D5911yescharacteristic
413C42C2×C22⋊D5911yescharacteristic
414C16—1181no—
415C16—1181no—
416C16—1181no—
417C16—1181no—
418C23—41no—
419C23C2×#1174566911yescharacteristic
420Q8—22no—
421Q8—41no—
422Q8—2361no—
423Q8—2361no—
424Q8—2361no—
425Q8—2361no—
426D4—22no—
427D4—44no—
428D4—41no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—44no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—161no—
436C2×C4—161no—
437C2×C4—161no—
438C2×C4—2361no—
439C2×C4—2361no—
440C2×C4—2361no—
441C2×C4—2361no—
442C2×C4C2×#1174566911yescharacteristic
443C2×C4C2×#1174566911yescharacteristic
444C2×C4C2×#1174566911yescharacteristic
445C2×C4C2×#1174566911yescharacteristic
446C2×C4C2×#1174566911yescharacteristic
447C2×C4C22×C22⋊D5911yescharacteristic
448C8—21no—
449C8—21no—
450C8—21no—
451C8C2×#1174568511yescharacteristic
452C8C2×#1174568511yescharacteristic
453C22—21no—
454C22—21no—
455C22—21no—
456C22—42no—
457C22—81no—
458C22—81no—
459C22C22×#1174566911yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—42no—
466C4—81no—
467C4—81no—
468C4—4721no—
469C4—4721no—
470C4—4721no—
471C4—4721no—
472C4#1683791311yescharacteristic
473C4#1683820011yescharacteristic
474C2—41no—
475C2—82no—
476C2#2143409611yescharacteristic
477C2#2143724311yescharacteristic
478C2#2143724711yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2951C2
socleC22×C59C22×D8
prime radicalC2×#135×C59C22
hypercenter#2951D59
hypocenterC59#33680
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
1. #21434755C2semidirect
2. #21426298C2semidirect
3. #21426293C2semidirect
4. #21426260C2semidirect
5. #21426253C2semidirect
6. #21426146C2semidirect
7. #21426141C2semidirect
10. #21425965C2semidirect
11. #21425960C2semidirect
12. C4×#14273673C2semidirect
13. C4×#14273671C2semidirect
14. #21425903C2semidirect
298. C59#33680semidirect

Maximal subgroups

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

subgroupindex
#214347552
#214262982
#214262932
#214262602
#214262532
#214261462
#214261412
#214260362
#214259652
#214259602
C4×#142736732
C4×#142736712
#214259032
C59×#29512
#3368059