Finite Group Databasebeta

← #28746201

Subgroups — #28746201

Order 15104.

488 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436519C211yesmaximal, characteristic, complemented
2#21436499C211yesmaximal, characteristic, complemented
3#21426382C211yesmaximal, characteristic, complemented
4#21426288C211yesmaximal, characteristic, complemented
5#21426288C211yesmaximal, characteristic, complemented
6#21426288C211yesmaximal, characteristic, complemented
7#21426277C211yesmaximal, characteristic, complemented
8#21426254C211yesmaximal, characteristic, complemented
9#21426234C211yesmaximal, characteristic, complemented
10#21426199C211yesmaximal, characteristic, complemented
11#21426182C211yesmaximal, characteristic, complemented
12#21425973C211yesmaximal, characteristic, complemented
13#21425884C211yesmaximal, characteristic, complemented
14#21425880C211yesmaximal, characteristic, complemented
15C59×#2168C211yesmaximal, characteristic, complemented
16#16838117—21no—
17#16838098—21no—
18C2×D944C2211yescharacteristic
19#16837807C2211yescharacteristic
20#16837804C2211yescharacteristic
21#16837727—21no—
22#16837727—21no—
23#16837727—21no—
24#16837721—21no—
25#16837703—21no—
26#16837693—21no—
27#16837638C2211yescharacteristic
28#16837638C2211yescharacteristic
29#16837638C2211yescharacteristic
30#16837638C2211yescharacteristic
31C4×D472C2211yescharacteristic
32C4×D472C2211yescharacteristic
33#16837440C2211yescharacteristic
34#16837440C2211yescharacteristic
35#16837440C2211yescharacteristic
36C2×#14273661C2211yescharacteristic
37#16837230C2211yescharacteristic
38#16837230C2211yescharacteristic
39#16837230C2211yescharacteristic
40#16837230C2211yescharacteristic
41#16837228C2211yescharacteristic
42#16837228C2211yescharacteristic
43#16837228C2211yescharacteristic
44#16837228C2211yescharacteristic
45#16837228C2211yescharacteristic
46#16837204C2211yescharacteristic
47#16837204C2211yescharacteristic
48#16837204C2211yescharacteristic
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×D472C2311yescharacteristic
60C2×D472C2311yescharacteristic
61#14273727—21no—
62#14273722—41no—
63#14273719—21no—
64#14273713—41no—
65#14273713—41no—
66#14273709—41no—
67#14273701—21no—
68#14273701—21no—
69#14273701—21no—
70#14273701—21no—
71#14273699—41no—
72#14273698—41no—
73C4×D236—21no—
74C4×D236—21no—
75(C22⋊D59)⋊C4—41no—
76#14273686—21no—
77#14273686—21no—
78#14273686—21no—
79#14273685—41no—
80C2×C4⋊Dic59—21no—
81D944—22no—
82#14273661—21no—
83#14273661—21no—
84#14273661C2311yescharacteristic
85#14273661C2311yescharacteristic
86#14273661C2311yescharacteristic
87#14273661C2311yescharacteristic
88#14273661C2311yescharacteristic
89#14273661C2311yescharacteristic
90#14273659—21no—
91#14273659—21no—
92#14273659—21no—
93#14273659—21no—
94#14273649—21no—
95#14273649—21no—
96#14273649—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—
118C2×C22⋊D59—41no—
119C2×C22⋊D59—41no—
120C22×Dic59—41no—
121C2×D236—21no—
122C2×D236—21no—
123C2×C4×D59—41no—
124C2×C4×D59—41no—
125C2×Dic118—41no—
126C2×C8×C59—21no—
127C2×C8×C59—21no—
128C2×C8×C59—21no—
129C2×C8×C59—21no—
130C2×C8×C59C2×D411yescharacteristic
131C2×C8×C59C2×D411yescharacteristic
132C2×C8×C59C2×D411yescharacteristic
133C2×C8×C59C2411yescharacteristic
134C4⋊C4×C59D811yescharacteristic
135C22⋊C4×C59D811yescharacteristic
136C22⋊C4×C59D811yescharacteristic
137C42×C59C2×D411yescharacteristic
138C22⋊Dic59—41no—
139C22⋊Dic59—41no—
140C22⋊Dic59—41no—
141(C2×D59)⋊C4—41no—
142(C2×D59)⋊C4—41no—
143(C2×D59)⋊C4—41no—
144(C2×D59)⋊C4—41no—
145(C2×D59)⋊C4—41no—
146(C2×D59)⋊C4—41no—
147C4⋊Dic59—21no—
148C4⋊Dic59—21no—
149C4⋊Dic59—21no—
150C4⋊Dic59—21no—
151C4⋊Dic59—21no—
152C4⋊Dic59—21no—
153C4⋊Dic59—41no—
154C4⋊Dic59—41no—
155Dic59⋊C4—41no—
156Dic59⋊C4—41no—
157C4×Dic59—41no—
158C4×Dic59—41no—
159C4×Dic59—41no—
160D472—21no—
161D472—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—
171C22×D59—41no—
172D4×C59—41no—
173C2×C4×C59—21no—
174C2×C4×C59—21no—
175C2×C4×C59—21no—
176C2×C4×C59—21no—
177C2×C4×C59C2×D811yescharacteristic
178C2×C4×C59C2×D811yescharacteristic
179C2×C4×C59C2×D811yescharacteristic
180C2×C4×C59C2×D811yescharacteristic
181C2×C4×C59C22×D411yescharacteristic
182C22⋊D59—82no—
183C22⋊D59—161no—
184C22⋊D59—161no—
185C22⋊D59—161no—
186C2×Dic59—41no—
187C2×Dic59—41no—
188C2×Dic59—41no—
189C2×Dic59—41no—
190C2×Dic59—41no—
191C2×Dic59—41no—
192C2×Dic59—161no—
193D236—41no—
194D236—41no—
195D236—41no—
196D236—41no—
197C4×D59—82no—
198C4×D59—161no—
199Dic118—161no—
200C8×C59—21no—
201C8×C59—21no—
202C8×C59—21no—
203C8×C59C2×#4111yescharacteristic
204C8×C592−1+411yescharacteristic
205#33914—591nomaximal, Sylow
206C22×C59—21no—
207C22×C59—21no—
208C22×C59—21no—
209C22×C59—21no—
210C22×C59—21no—
211C22×C59—21no—
212C22×C59C22×D811yescharacteristic
213C2×D59—81no—
214C2×D59—81no—
215C2×D59—81no—
216C2×D59—81no—
217C2×D59—81no—
218C2×D59—81no—
219C4×C59—21no—
220C4×C59—21no—
221C4×C59—21no—
222C4×C59—21no—
223C4×C59#54711yescharacteristic
224C4×C59#57811yescharacteristic
225Dic59—81no—
226Dic59—81no—
227Dic59—81no—
228Dic59—81no—
229Dic59—81no—
230Dic59—81no—
231#3341—591no—
232#3339—591no—
233#2255—591no—
234#2234—591no—
235#2234—591no—
236#2234—591no—
237#2232—591no—
238#2228—591no—
239#2224—591no—
240#2218—591no—
241#2215—591no—
242#2177—591no—
243#2168D5911yescharacteristic, complemented
244#2159—591no—
245#2158—591no—
246C2×C59—41no—
247C2×C59—41no—
248C2×C59#333411yescharacteristic
249C2×C59#341411yescharacteristic
250C2×C59#341511yescharacteristic
251D59—161no—
252D59—161no—
253#555—1181no—
254#553—1181no—
255C2×D16—591no—
256C2×M32C2×D5911yescharacteristic
257C22×C16C2×D5911yescharacteristic
258#500—591no—
259#499—591no—
260#482—1181no—
261#482—1181no—
262#482—1181no—
263#481—1181no—
264#478—1181no—
265#476—1181no—
266#466—591no—
267#466—591no—
268#466—591no—
269#466—591no—
270C4×D8—591no—
271C4×D8—591no—
272C8×D4C2×D5911yescharacteristic
273#427—591no—
274#427—591no—
275#427—591no—
276C2×#107—591no—
277#367—591no—
278#367—591no—
279#367—591no—
280#367—591no—
281#366—591no—
282#366—591no—
283#366—591no—
284#366—591no—
285#366—591no—
286C4⋊C16C2×D5911yescharacteristic
287#357—591no—
288#357—591no—
289#357—591no—
290#357—591no—
291#357—591no—
292#357—591no—
293C22⋊C16C2×D5911yescharacteristic
294C22⋊C16C2×D5911yescharacteristic
295#346C2×D5911yescharacteristic
296C59#3391411yescharacteristic, complemented, Sylow
297C2×D8—591no—
298C2×D8—591no—
299C22×C8C22×D5911yescharacteristic
300C22×C8C22×D5911yescharacteristic
301C4⋊Q8—1181no—
302#126—2361no—
303#125—1181no—
304#124—2361no—
305#124—2361no—
306#123—2361no—
307#121—1181no—
308#121—1181no—
309#121—1181no—
310#121—1181no—
311#121—2361no—
312#121—2361no—
313C4×D4—1181no—
314C4×D4—1181no—
315C4×D4—2361no—
316C4×D4D23611yescharacteristic
317#117—1181no—
318#117—1181no—
319#117—1181no—
320#117—2361no—
321C2×C4⋊C4—1181no—
322D16—1182no—
323M32—22no—
324C2×C16—22no—
325C2×C16C22×D5911yescharacteristic
326C2×C16C22×D5911yescharacteristic
327C2×C16C22×D5911yescharacteristic
328C2×C16C22×D5911yescharacteristic
329#107—591no—
330#107—591no—
331#107—591no—
332#107—591no—
333#107—591no—
334#107—591no—
335#107—1181no—
336#107—1181no—
337#106—1181no—
338#106—1181no—
339#106—1181no—
340#106—1181no—
341C4⋊C8D23611yescharacteristic
342#102—1181no—
343#102—1181no—
344#102—1181no—
345#102—1181no—
346#102—1181no—
347#102—1181no—
348C22⋊C8D23611yescharacteristic
349C22⋊C8D23611yescharacteristic
350C4×C8C22×D5911yescharacteristic
351C2×Q8—2361no—
352C2×D4—1181no—
353C2×D4—1181no—
354C2×D4—2361no—
355C2×D4—2361no—
356C2×D4—2361no—
357C2×D4—2361no—
358C2×D4D47211yescharacteristic
359C22×C4—2361no—
360C22×C4—2361no—
361C22×C4—2361no—
362C22×C4C2×D23611yescharacteristic
363C22×C4C2×D23611yescharacteristic
364D8—1181no—
365D8—1181no—
366D8—1181no—
367D8—1181no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8C2×D23611yescharacteristic
373C2×C8C2×D23611yescharacteristic
374C2×C8C2×D23611yescharacteristic
375C2×C8C23×D5911yescharacteristic
376C4⋊C4—1181no—
377C4⋊C4—1181no—
378C4⋊C4—1181no—
379C4⋊C4—1181no—
380C4⋊C4—1181no—
381C4⋊C4—1181no—
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⋊C4—2361no—
394C22⋊C4—2361no—
395C22⋊C4—2361no—
396C22⋊C4D47211yescharacteristic
397C22⋊C4D47211yescharacteristic
398C42—2361no—
399C42—2361no—
400C42—2361no—
401C42C2×D23611yescharacteristic
402C16—21no—
403C16—21no—
404C16—21no—
405C16#1174568712yes—
406C23—2361no—
407C23—2361no—
408C23C2×D47211yescharacteristic
409C23C2×D47211yescharacteristic
410Q8—9441no—
411D4—41no—
412D4—9441no—
413D4—9441no—
414D4—9441no—
415D4—4722no—
416D4—2361no—
417D4—2361no—
418D4—2361no—
419D4—2361no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—9441no—
425C2×C4—9441no—
426C2×C4—4722no—
427C2×C4—2361no—
428C2×C4—2361no—
429C2×C4—2361no—
430C2×C4—2361no—
431C2×C4—2361no—
432C2×C4—2361no—
433C2×C4C2×D47211yescharacteristic
434C2×C4C2×D47211yescharacteristic
435C2×C4C2×D47211yescharacteristic
436C2×C4C2×D47211yescharacteristic
437C2×C4C22×D23611yescharacteristic
438C8—21no—
439C8—21no—
440C8—21no—
441C8C2×#1174568711yescharacteristic
442C8#1427377411yescharacteristic
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—4721no—
450C22—4721no—
451C22—4721no—
452C22—4721no—
453C22—4721no—
454C22—4721no—
455C22C22×D47211yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—4721no—
461C4—4721no—
462C4—4721no—
463C4—4721no—
464C4—4721no—
465C4—4721no—
466C4#1683805611yescharacteristic
467C4#1683821911yescharacteristic
468C2—41no—
469C2—41no—
470C2—9441no—
471C2—9441no—
472C2#2143644411yescharacteristic
473C2#2143723211yescharacteristic
474C2#2143723711yescharacteristic
subgroupG/H
centerC22C22×D472
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2168C2
socleC22×C59C22×D8
hypercenter#2168D59
hypocenterC59#33914
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
1. #21436519C2semidirect
2. #21436499C2semidirect
3. #21426382C2semidirect
4. #21426288C2semidirect
5. #21426288C2semidirect
6. #21426288C2semidirect
7. #21426277C2semidirect
8. #21426254C2semidirect
9. #21426234C2semidirect
10. #21426199C2semidirect
11. #21426182C2semidirect
12. #21425973C2semidirect
13. #21425884C2semidirect
14. #21425880C2semidirect
15. C59×#2168C2semidirect
243. #2168D59semidirect
296. C59#33914semidirect

Maximal subgroups

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

subgroupindex
#214365192
#214364992
#214263822
#214262882
#214262772
#214262542
#214262342
#214261992
#214261822
#214259732
#214258842
#214258802
C59×#21682
#3391459