Finite Group Databasebeta

← #28746186

Subgroups — #28746186

Order 15104.

484 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436499C211yesmaximal, characteristic
2#21436489C211yesmaximal, characteristic, complemented
3#21426382C211yesmaximal, characteristic, complemented
4#21426305C211yesmaximal, characteristic, complemented
5#21426299C211yesmaximal, characteristic, complemented
6#21426282C211yesmaximal, characteristic, complemented
7#21426277C211yesmaximal, characteristic, complemented
8#21426261C211yesmaximal, characteristic, complemented
9#21426234C211yesmaximal, characteristic, complemented
10#21426209C211yesmaximal, characteristic, complemented
11#21426192C211yesmaximal, characteristic, complemented
12#21425962C211yesmaximal, characteristic, complemented
13#21425884C211yesmaximal, characteristic, complemented
14C2×#16837228C211yesmaximal, characteristic, complemented
15C59×#2168C211yesmaximal, characteristic, complemented
16#16838109—21no—
17#16838098—21no—
18C2×#14273672C2211yescharacteristic
19#16837807C2211yescharacteristic
20#16837804C2211yescharacteristic
21#16837738—21no—
22#16837738—21no—
23#16837727—21no—
24#16837721—21no—
25#16837703—21no—
26#16837698—21no—
27#16837643C2211yescharacteristic
28#16837643C2211yescharacteristic
29#16837638C2211yescharacteristic
30#16837638C2211yescharacteristic
31C4×Dic236C2211yescharacteristic
32C4×D472C2211yescharacteristic
33#16837440C2211yescharacteristic
34C2×#14273661C2211yescharacteristic
35C2×#14273661C2211yescharacteristic
36C2×#14273661C2211yescharacteristic
37#16837230C2211yescharacteristic
38#16837230C2211yescharacteristic
39#16837230C2211yescharacteristic
40#16837228C2211yescharacteristic
41#16837228C2211yescharacteristic
42#16837228C2211yescharacteristic
43#16837228C2211yescharacteristic
44#16837228C2211yescharacteristic
45#16837228C2211yescharacteristic
46#16837207C2211yescharacteristic
47#16837207C2211yescharacteristic
48#16837207C2211yescharacteristic
49#16837204C2211yescharacteristic
50#16837204C2211yescharacteristic
51#16837204C2211yescharacteristic
52C2×M32×C59C2211yescharacteristic
53C22×C16×C59C2211yescharacteristic
54C8×D4×C59C2211yescharacteristic
55C59×C4⋊C16C2211yescharacteristic
56C59×C22⋊C16C2211yescharacteristic
57C59×C22⋊C16C2211yescharacteristic
58C59×#346C2211yescharacteristic
59C2×Dic236C2311yescharacteristic
60C2×D472C2311yescharacteristic
61#14273727—21no—
62#14273722—41no—
63#14273719—21no—
64#14273718—41no—
65#14273709—41no—
66#14273706—21no—
67#14273706—21no—
68#14273704—41no—
69#14273704—41no—
70#14273701—21no—
71#14273701—21no—
72#14273699—41no—
73C4×Dic118—21no—
74C4×D236—21no—
75D4×Dic59—41no—
76(C22⋊D59)⋊C4—41no—
77#14273686—21no—
78C2×C4⋊Dic59—21no—
79C2×C4⋊Dic59—21no—
80C2×C4⋊Dic59—21no—
81#14273672—22no—
82#14273661—21no—
83#14273661—21no—
84#14273661—21no—
85#14273661—21no—
86#14273661C2311yescharacteristic
87#14273661C2311yescharacteristic
88#14273661C2311yescharacteristic
89#14273661C2311yescharacteristic
90#14273661C2311yescharacteristic
91#14273661C2311yescharacteristic
92#14273659—21no—
93#14273659—21no—
94#14273652—21no—
95#14273652—21no—
96#14273652—21no—
97#14273649—21no—
98#14273649—21no—
99#14273649—21no—
100C22×C8×C59C2311yescharacteristic
101C22×C8×C59C2311yescharacteristic
102C4×D4×C59D411yescharacteristic
103M32×C59—22no—
104C2×C16×C59—22no—
105C2×C16×C59C2311yescharacteristic
106C2×C16×C59C2311yescharacteristic
107C2×C16×C59C2311yescharacteristic
108C2×C16×C59C2311yescharacteristic
109C4⋊C8×C59D411yescharacteristic
110C22⋊C8×C59D411yescharacteristic
111C22⋊C8×C59D411yescharacteristic
112C4×C8×C59C2311yescharacteristic
113C2×D4×C59D811yescharacteristic
114C22×C4×C59C2×D411yescharacteristic
115C22×C4×C59C2×D411yescharacteristic
116C2×C22⋊D59—41no—
117C2×C22⋊D59—41no—
118C22×Dic59—41no—
119C22×Dic59—41no—
120C22×Dic59—41no—
121C2×D236—21no—
122C2×C4×D59—41no—
123C2×Dic118—21no—
124C2×Dic118—41no—
125C2×C8×C59—21no—
126C2×C8×C59—21no—
127C2×C8×C59—21no—
128C2×C8×C59—21no—
129C2×C8×C59C2×D411yescharacteristic
130C2×C8×C59C2×D411yescharacteristic
131C2×C8×C59C2×D411yescharacteristic
132C2×C8×C59C2411yescharacteristic
133C4⋊C4×C59D811yescharacteristic
134C22⋊C4×C59D811yescharacteristic
135C22⋊C4×C59D811yescharacteristic
136C42×C59C2×D411yescharacteristic
137C22⋊Dic59—41no—
138C22⋊Dic59—41no—
139C22⋊Dic59—41no—
140(C2×D59)⋊C4—41no—
141(C2×D59)⋊C4—41no—
142(C2×D59)⋊C4—41no—
143C4⋊Dic59—21no—
144C4⋊Dic59—21no—
145C4⋊Dic59—21no—
146C4⋊Dic59—21no—
147C4⋊Dic59—21no—
148C4⋊Dic59—21no—
149C4⋊Dic59—41no—
150C4⋊Dic59—41no—
151C4⋊Dic59—41no—
152C4⋊Dic59—41no—
153Dic59⋊C4—41no—
154Dic59⋊C4—41no—
155Dic59⋊C4—41no—
156Dic59⋊C4—41no—
157Dic59⋊C4—41no—
158C4×Dic59—41no—
159C4×Dic59—41no—
160Dic236—21no—
161Dic236—21no—
162D472—21no—
163D472—21no—
164C16×C59—21no—
165C16×C59—21no—
166C16×C59—21no—
167C16×C59#4112yes—
168C23×C59C2×D811yescharacteristic
169C23×C59C2×D811yescharacteristic
170C22×D59—41no—
171D4×C59—22no—
172C2×C4×C59—21no—
173C2×C4×C59—21no—
174C2×C4×C59—21no—
175C2×C4×C59—21no—
176C2×C4×C59C2×D811yescharacteristic
177C2×C4×C59C2×D811yescharacteristic
178C2×C4×C59C2×D811yescharacteristic
179C2×C4×C59C2×D811yescharacteristic
180C2×C4×C59C22×D411yescharacteristic
181C22⋊D59—82no—
182C22⋊D59—161no—
183C2×Dic59—41no—
184C2×Dic59—41no—
185C2×Dic59—41no—
186C2×Dic59—41no—
187C2×Dic59—41no—
188C2×Dic59—41no—
189C2×Dic59—41no—
190C2×Dic59—161no—
191C2×Dic59—161no—
192C2×Dic59—161no—
193D236—41no—
194D236—41no—
195C4×D59—161no—
196Dic118—41no—
197Dic118—41no—
198Dic118—161no—
199C8×C59—21no—
200C8×C59—21no—
201C8×C59—21no—
202C8×C59C2×#4111yescharacteristic
203C8×C592−1+411yescharacteristic
204#33913—591nomaximal, Sylow
205C22×C59—21no—
206C22×C59—21no—
207C22×C59—21no—
208C22×C59—21no—
209C22×C59—21no—
210C22×C59#51012yes—
211C22×C59C22×D811yescharacteristic
212C2×D59—81no—
213C2×D59—81no—
214C2×D59—81no—
215C4×C59—21no—
216C4×C59—21no—
217C4×C59—21no—
218C4×C59—21no—
219C4×C59#54711yescharacteristic
220C4×C59#57811yescharacteristic
221Dic59—81no—
222Dic59—81no—
223Dic59—81no—
224Dic59—81no—
225Dic59—81no—
226Dic59—81no—
227Dic59—81no—
228#3339—591no—
229#3338—591no—
230#2255—591no—
231#2237—591no—
232#2236—591no—
233#2233—591no—
234#2232—591no—
235#2229—591no—
236#2224—591no—
237#2220—591no—
238#2217—591no—
239#2175—591no—
240#2168D5911yescharacteristic, complemented
241#2159—591no—
242C2×#366—591no—
243C2×C59—22no—
244C2×C59—41no—
245C2×C59#333411yescharacteristic
246C2×C59C2×#51011yescharacteristic
247C2×C59#341511yescharacteristic
248D59—161no—
249#554—1181no—
250#553—1181no—
251C2×SD32—591no—
252C2×M32C2×D5911yescharacteristic
253C22×C16C2×D5911yescharacteristic
254#500—591no—
255#499—591no—
256#484—1181no—
257#484—1181no—
258#482—1181no—
259#481—1181no—
260#478—1181no—
261#477—1181no—
262#467—591no—
263#467—591no—
264#466—591no—
265#466—591no—
266C4×Q16—591no—
267C4×D8—591no—
268C8×D4C2×D5911yescharacteristic
269#427—591no—
270C2×#107—591no—
271C2×#107—591no—
272C2×#107—591no—
273#367—591no—
274#367—591no—
275#367—591no—
276#366—591no—
277#366—591no—
278#366—591no—
279#366—591no—
280#366—591no—
281#366—591no—
282C4⋊C16C2×D5911yescharacteristic
283#358—591no—
284#358—591no—
285#358—591no—
286#357—591no—
287#357—591no—
288#357—591no—
289C22⋊C16C2×D5911yescharacteristic
290C22⋊C16C2×D5911yescharacteristic
291#346C2×D5911yescharacteristic
292C59#3391311yescharacteristic, complemented, Sylow
293C2×Q16—591no—
294C2×D8—591no—
295C22×C8C22×D5911yescharacteristic
296C22×C8C22×D5911yescharacteristic
297C4⋊Q8—1181no—
298#126—2361no—
299#125—1181no—
300#125—2361no—
301#123—2361no—
302C22⋊Q8—1181no—
303C22⋊Q8—1181no—
304C22⋊Q8—2361no—
305C22⋊Q8—2361no—
306#121—1181no—
307#121—1181no—
308#121—2361no—
309C4×Q8—1181no—
310C4×D4—1181no—
311C4×D4—2361no—
312C4×D4—2361no—
313C4×D4D23611yescharacteristic
314#117—1181no—
315C2×C4⋊C4—1181no—
316C2×C4⋊C4—1181no—
317C2×C4⋊C4—1181no—
318SD32—1182no—
319M32—22no—
320C2×C16—22no—
321C2×C16C22×D5911yescharacteristic
322C2×C16C22×D5911yescharacteristic
323C2×C16C22×D5911yescharacteristic
324C2×C16C22×D5911yescharacteristic
325#107—591no—
326#107—591no—
327#107—591no—
328#107—591no—
329#107—591no—
330#107—591no—
331#107—1181no—
332#107—1181no—
333#107—1181no—
334#107—1181no—
335#106—1181no—
336#106—1181no—
337C4⋊C8D23611yescharacteristic
338#103—1181no—
339#103—1181no—
340#103—1181no—
341#102—1181no—
342#102—1181no—
343#102—1181no—
344C22⋊C8D23611yescharacteristic
345C22⋊C8D23611yescharacteristic
346C4×C8C22×D5911yescharacteristic
347C2×Q8—1181no—
348C2×Q8—2361no—
349C2×D4—1181no—
350C2×D4—2361no—
351C2×D4—2361no—
352C2×D4D47211yescharacteristic
353C22×C4—2361no—
354C22×C4—2361no—
355C22×C4—2361no—
356C22×C4—2361no—
357C22×C4C2×D23611yescharacteristic
358C22×C4C2×D23611yescharacteristic
359Q16—1181no—
360Q16—1181no—
361D8—1181no—
362D8—1181no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8C2×D23611yescharacteristic
368C2×C8C2×D23611yescharacteristic
369C2×C8C2×D23611yescharacteristic
370C2×C8C23×D5911yescharacteristic
371C4⋊C4—1181no—
372C4⋊C4—1181no—
373C4⋊C4—1181no—
374C4⋊C4—1181no—
375C4⋊C4—1181no—
376C4⋊C4—1181no—
377C4⋊C4—2361no—
378C4⋊C4—2361no—
379C4⋊C4—2361no—
380C4⋊C4—2361no—
381C4⋊C4—2361no—
382C4⋊C4—2361no—
383C4⋊C4—2361no—
384C4⋊C4—2361no—
385C4⋊C4—2361no—
386C4⋊C4D47211yescharacteristic
387C22⋊C4—2361no—
388C22⋊C4—2361no—
389C22⋊C4—2361no—
390C22⋊C4—2361no—
391C22⋊C4—2361no—
392C22⋊C4—2361no—
393C22⋊C4D47211yescharacteristic
394C22⋊C4D47211yescharacteristic
395C42—2361no—
396C42—2361no—
397C42C2×D23611yescharacteristic
398C16—21no—
399C16—21no—
400C16—21no—
401C16#1174568712yes—
402C23—2361no—
403C23C2×D47211yescharacteristic
404C23C2×D47211yescharacteristic
405Q8—9441no—
406Q8—2361no—
407Q8—2361no—
408D4—22no—
409D4—9441no—
410D4—4722no—
411D4—2361no—
412D4—2361no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—9441no—
418C2×C4—9441no—
419C2×C4—9441no—
420C2×C4—9441no—
421C2×C4—2361no—
422C2×C4—2361no—
423C2×C4—2361no—
424C2×C4—2361no—
425C2×C4—2361no—
426C2×C4—2361no—
427C2×C4—2361no—
428C2×C4C2×D47211yescharacteristic
429C2×C4C2×D47211yescharacteristic
430C2×C4C2×D47211yescharacteristic
431C2×C4C2×D47211yescharacteristic
432C2×C4C22×D23611yescharacteristic
433C8—21no—
434C8—21no—
435C8—21no—
436C8C2×#1174568711yescharacteristic
437C8#1427377411yescharacteristic
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—4721no—
444C22—4721no—
445C22—4721no—
446C22#1683784512yes—
447C22C22×D47211yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—4721no—
453C4—4721no—
454C4—4721no—
455C4—4721no—
456C4—4721no—
457C4—4721no—
458C4—4721no—
459C4#1683805611yescharacteristic
460C4#1683821911yescharacteristic
461C2—22no—
462C2—41no—
463C2—9441no—
464C2#2143644411yescharacteristic
465C2C2×#1683784511yescharacteristic
466C2#2143723711yescharacteristic
subgroupG/H
centerC22C22×D472
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2168C2
socleC22×C59C22×D8
prime radical#21436499C2
hypercenter#2168D59
hypocenterC59#33913
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #2168

Complements

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

NH ≅ G/Ntype
2. #21436489C2semidirect
3. #21426382C2semidirect
4. #21426305C2semidirect
5. #21426299C2semidirect
6. #21426282C2semidirect
7. #21426277C2semidirect
8. #21426261C2semidirect
9. #21426234C2semidirect
10. #21426209C2semidirect
11. #21426192C2semidirect
12. #21425962C2semidirect
13. #21425884C2semidirect
14. C2×#16837228C2semidirect
15. C59×#2168C2semidirect
240. #2168D59semidirect
292. C59#33913semidirect

Maximal subgroups

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

subgroupindex
#214364992
#214364892
#214263822
#214263052
#214262992
#214262822
#214262772
#214262612
#214262342
#214262092
#214261922
#214259622
#214258842
C2×#168372282
C59×#21682
#3391359