Finite Group Databasebeta

← #28746446

Subgroups — #28746446

Order 15104.

494 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#21436964C211yesmaximal, characteristic, complemented
2#21426337C211yesmaximal, characteristic, complemented
3#21426332C211yesmaximal, characteristic, complemented
4#21426327C211yesmaximal, characteristic, complemented
5#21426260C211yesmaximal, characteristic, complemented
6#21426253C211yesmaximal, characteristic, complemented
7#21426233C211yesmaximal, characteristic, complemented
8#21426148C211yesmaximal, characteristic, complemented
9#21426148C211yesmaximal, characteristic, complemented
10#21426128C211yesmaximal, characteristic
11#21426005C211yesmaximal, characteristic, complemented
12#21425970C211yesmaximal, characteristic, complemented
13#21425958C211yesmaximal, characteristic, complemented
14C4×#14273668C211yesmaximal, characteristic, complemented
15C59×#3396C211yesmaximal, characteristic, complemented
16#16838112—21no—
17C2×#14273670C2211yescharacteristic
18C2×#14273668C2211yescharacteristic
19C2×#14273668C2211yescharacteristic
20#16837807C2211yescharacteristic
21#16837804C2211yescharacteristic
22#16837804C2211yescharacteristic
23#16837703—21no—
24#16837703—21no—
25#16837693—21no—
26C4×D472C2211yescharacteristic
27C4×D472C2211yescharacteristic
28C4×D472C2211yescharacteristic
29#16837229C2211yescharacteristic
30#16837227C2211yescharacteristic
31#16837227C2211yescharacteristic
32C4⋊(C59⋊C16)C2211yescharacteristic
33C4⋊(C59⋊C16)C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35#16837206C2211yescharacteristic
36#16837206C2211yescharacteristic
37#16837206C2211yescharacteristic
38#16837203C2211yescharacteristic
39#16837203C2211yescharacteristic
40#16837203C2211yescharacteristic
41#16837203C2211yescharacteristic
42#16837203C2211yescharacteristic
43#16837203C2211yescharacteristic
44#16837202C2211yescharacteristic
45#16837178C2211yescharacteristic
46#16837178C2211yescharacteristic
47C4×C59⋊C16C2211yescharacteristic
48C59×#559—21no—
49C59×#495C2211yescharacteristic
50C59×#495C2211yescharacteristic
51C59×#493C2211yescharacteristic
52C59×#463—21no—
53C59×#463—21no—
54C59×#460—21no—
55C8×Q8×C59C2211yescharacteristic
56C4×Q16×C59C2211yescharacteristic
57C4×D8×C59C2211yescharacteristic
58C4×D8×C59C2211yescharacteristic
59C2×D472C2311yescharacteristic
60#14273727—21no—
61#14273719—21no—
62#14273719—21no—
63#14273705—41no—
64#14273705—41no—
65#14273703—41no—
66C4×D236—21no—
67C4×D236—21no—
68C4×D236—21no—
69C4⋊C4×D59—41no—
70#14273670—22no—
71#14273668—22no—
72#14273668—22no—
73C2×C59⋊C16C2311yescharacteristic
74C2×C59⋊C16C2311yescharacteristic
75C2×C59⋊C16C2311yescharacteristic
76C2×C59⋊C16C2311yescharacteristic
77#14273661—21no—
78#14273661C2311yescharacteristic
79#14273661C2311yescharacteristic
80#14273661C2311yescharacteristic
81#14273659—21no—
82#14273659—21no—
83#14273649—21no—
84#14273649—21no—
85#14273649—21no—
86C2×Q16×C59C2311yescharacteristic
87C2×SD16×C59—21no—
88C2×SD16×C59—21no—
89C2×D8×C59—21no—
90C2×D8×C59C2311yescharacteristic
91C2×D8×C59C2311yescharacteristic
92#127×C59—21no—
93#124×C59—21no—
94#124×C59—21no—
95#124×C59—41no—
96#123×C59—41no—
97#123×C59—41no—
98C4×Q8×C59—21no—
99C4×Q8×C59D411yescharacteristic
100C4×D4×C59—21no—
101C4×D4×C59—21no—
102C4×D4×C59—41no—
103#107×C59C2311yescharacteristic
104C4⋊C8×C59D411yescharacteristic
105C4⋊C8×C59D411yescharacteristic
106C4⋊C8×C59D411yescharacteristic
107#103×C59—21no—
108#102×C59—21no—
109#102×C59—21no—
110C4×C8×C59C2311yescharacteristic
111C4×C8×C59C2311yescharacteristic
112C4×C8×C59C2311yescharacteristic
113C2×Q8×C59—21no—
114C2×Q8×C59D811yescharacteristic
115C2×D4×C59—21no—
116C2×D4×C59—21no—
117C2×D4×C59—41no—
118C2×D4×C59—41no—
119C22×C4×C59—41no—
120C22×C4×C59—41no—
121C2×D236—21no—
122C2×C4×D59—41no—
123C2×C4×D59—41no—
124C2×C4×D59—41no—
125C2×Dic118—41no—
126Q16×C59—21no—
127Q16×C59—21no—
128SD16×C59—81no—
129SD16×C59—81no—
130D8×C59—21no—
131D8×C59—21no—
132D8×C59—21no—
133D8×C59—21no—
134D8×C59—81no—
135C2×C8×C59C2×D411yescharacteristic
136C2×C8×C59C2×D411yescharacteristic
137C2×C8×C59C2×D411yescharacteristic
138C2×C8×C59C2411yescharacteristic
139C4⋊C4×C59—21no—
140C4⋊C4×C59—41no—
141C4⋊C4×C59D811yescharacteristic
142C4⋊C4×C59D811yescharacteristic
143C4⋊C4×C59D811yescharacteristic
144C22⋊C4×C59—41no—
145C22⋊C4×C59—41no—
146C22⋊C4×C59—41no—
147C22⋊C4×C59—41no—
148C22⋊C4×C59—41no—
149C22⋊C4×C59—41no—
150C42×C59—41no—
151C42×C59C2×D411yescharacteristic
152C42×C59C2×D411yescharacteristic
153C42×C59C2×D411yescharacteristic
154(C2×D59)⋊C4—41no—
155(C2×D59)⋊C4—41no—
156(C2×D59)⋊C4—41no—
157C4⋊Dic59—21no—
158C4⋊Dic59—21no—
159C4⋊Dic59—21no—
160C4⋊Dic59—41no—
161Dic59⋊C4—41no—
162Dic59⋊C4—41no—
163Dic59⋊C4—41no—
164Dic59⋊C4—41no—
165D472—21no—
166D472#4112yes—
167C59⋊C16—21no—
168C59⋊C16—21no—
169C59⋊C16—21no—
170C59⋊C16—21no—
171C23×C59—41no—
172C23×C59—41no—
173C22×D59—41no—
174Q8×C59—41no—
175Q8×C59—41no—
176Q8×C59—41no—
177D4×C59—41no—
178D4×C59—41no—
179D4×C59—41no—
180D4×C59—41no—
181D4×C59—82no—
182D4×C59—161no—
183C2×C4×C59—41no—
184C2×C4×C59—41no—
185C2×C4×C59—161no—
186C2×C4×C59—161no—
187C2×C4×C59C2×D811yescharacteristic
188C2×C4×C59C2×D811yescharacteristic
189C2×C4×C59C2×D811yescharacteristic
190C2×C4×C59C2×D811yescharacteristic
191C2×C4×C59C2×D811yescharacteristic
192C2×C4×C59C2×D811yescharacteristic
193C2×C4×C59C22×D411yescharacteristic
194C2×Dic59—41no—
195C2×Dic59—41no—
196C2×Dic59—41no—
197D236—22no—
198D236—41no—
199C4×D59—82no—
200C4×D59—161no—
201C4×D59—161no—
202Dic118—161no—
203C8×C59—21no—
204C8×C59—21no—
205C8×C59—21no—
206C8×C59C2×#4111yescharacteristic
207C8×C592−1+411yescharacteristic
208#33937—591nomaximal, Sylow
209C22×C59—81no—
210C22×C59—81no—
211C22×C59—81no—
212C22×C59—81no—
213C22×C59—81no—
214C22×C59—81no—
215C22×C59C22×D811yescharacteristic
216C2×D59—42no—
217C2×D59—81no—
218C2×D59—81no—
219C4×C59—21no—
220C4×C59—21no—
221C4×C59—21no—
222C4×C59—21no—
223C4×C59—21no—
224C4×C59—21no—
225C4×C59—81no—
226C4×C59—81no—
227C4×C59#54811yescharacteristic
228C4×C59#57711yescharacteristic
229Dic59—81no—
230Dic59—81no—
231Dic59—81no—
232#3396D5911yescharacteristic, complemented
233#3380—591no—
234#2244—591no—
235#2243—591no—
236#2242—591no—
237#2229—591no—
238#2228—591no—
239#2224—591no—
240#2210—591no—
241#2210—591no—
242#2206—591no—
243#2183—591no—
244#2177—591no—
245#2175—591no—
246C4×D16—591no—
247C2×C59—161no—
248C2×C59—161no—
249C2×C59#335811yescharacteristic
250C2×C59#341411yescharacteristic
251C2×C59#341511yescharacteristic
252D59—82no—
253#559—21no—
254#554—1181no—
255C2×SD32—591no—
256C2×D16—591no—
257C2×D16—591no—
258#500—591no—
259#499—591no—
260#499—591no—
261#495C2×D5911yescharacteristic
262#495C2×D5911yescharacteristic
263#493C2×D5911yescharacteristic
264#478—1181no—
265#478—1181no—
266#476—1181no—
267#463—21no—
268#463—21no—
269#460—21no—
270C8×Q8C2×D5911yescharacteristic
271C4×Q16C2×D5911yescharacteristic
272C4×D8—591no—
273C4×D8—591no—
274C4×D8—591no—
275C4×D8C2×D5911yescharacteristic
276C4×D8C2×D5911yescharacteristic
277#367—591no—
278#366—591no—
279#366—591no—
280C4⋊C16—591no—
281C4⋊C16—591no—
282C4⋊C16—591no—
283#358—591no—
284#358—591no—
285#358—591no—
286#357—591no—
287#357—591no—
288#357—591no—
289#357—591no—
290#357—591no—
291#357—591no—
292#357—591no—
293#346—591no—
294#346—591no—
295C4×C16—591no—
296C59#3393711yescharacteristic, complemented, Sylow
297C2×Q16C22×D5911yescharacteristic
298C2×SD16—21no—
299C2×SD16—21no—
300C2×D8—21no—
301C2×D8—591no—
302C2×D8C22×D5911yescharacteristic
303C2×D8C22×D5911yescharacteristic
304C4⋊Q8—1181no—
305#127—21no—
306#125—1181no—
307#125—1181no—
308#124—21no—
309#124—21no—
310#124—41no—
311#123—41no—
312#123—41no—
313C22⋊Q8—2361no—
314C22⋊Q8—2361no—
315C22⋊Q8—2361no—
316C4×Q8—21no—
317C4×Q8C22⋊D5911yescharacteristic
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—41no—
321C4×D4—1181no—
322C4×D4—1181no—
323C4×D4—1181no—
324C2×C4⋊C4—2361no—
325SD32—1182no—
326D16—1182no—
327D16—1182no—
328C2×C16—591no—
329C2×C16—591no—
330C2×C16—591no—
331C2×C16—591no—
332#107—591no—
333#107—591no—
334#107—591no—
335#107—1181no—
336#107C22×D5911yescharacteristic
337#106—1181no—
338#106—1181no—
339C4⋊C8C22⋊D5911yescharacteristic
340C4⋊C8C22⋊D5911yescharacteristic
341C4⋊C8C22⋊D5911yescharacteristic
342#103—21no—
343#102—21no—
344#102—21no—
345#102—1181no—
346#102—1181no—
347#102—1181no—
348C4×C8C22×D5911yescharacteristic
349C4×C8C22×D5911yescharacteristic
350C4×C8C22×D5911yescharacteristic
351C2×Q8—21no—
352C2×Q8—2361no—
353C2×Q8#1174566911yescharacteristic
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—1181no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—2361no—
362C22×C4—2361no—
363C22×C4—2361no—
364Q16—21no—
365Q16—21no—
366SD16—81no—
367SD16—81no—
368D8—21no—
369D8—21no—
370D8—21no—
371D8—21no—
372D8—81no—
373D8—592no—
374D8—1181no—
375C2×C8C2×C22⋊D5911yescharacteristic
376C2×C8C2×C22⋊D5911yescharacteristic
377C2×C8C2×C22⋊D5911yescharacteristic
378C2×C8C23×D5911yescharacteristic
379C4⋊C4—21no—
380C4⋊C4—41no—
381C4⋊C4—1181no—
382C4⋊C4—1181no—
383C4⋊C4—1181no—
384C4⋊C4—2361no—
385C4⋊C4—2361no—
386C4⋊C4—2361no—
387C4⋊C4—2361no—
388C4⋊C4—2361no—
389C4⋊C4#1174566911yescharacteristic
390C4⋊C4#1174566911yescharacteristic
391C4⋊C4#1174566911yescharacteristic
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—2361no—
399C22⋊C4—2361no—
400C22⋊C4—2361no—
401C42—41no—
402C42C2×C22⋊D5911yescharacteristic
403C42C2×C22⋊D5911yescharacteristic
404C42C2×C22⋊D5911yescharacteristic
405C16—1181no—
406C16—1181no—
407C16—1181no—
408C16—1181no—
409C23—41no—
410C23—41no—
411C23—2361no—
412Q8—41no—
413Q8—41no—
414Q8—41no—
415Q8—9441no—
416D4—41no—
417D4—41no—
418D4—41no—
419D4—41no—
420D4—82no—
421D4—161no—
422D4—1182no—
423D4—2361no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—161no—
427C2×C4—161no—
428C2×C4—9441no—
429C2×C4—9441no—
430C2×C4—4722no—
431C2×C4—2361no—
432C2×C4—2361no—
433C2×C4—2361no—
434C2×C4C2×#1174566911yescharacteristic
435C2×C4C2×#1174566911yescharacteristic
436C2×C4C2×#1174566911yescharacteristic
437C2×C4C2×#1174566911yescharacteristic
438C2×C4C2×#1174566911yescharacteristic
439C2×C4C2×#1174566911yescharacteristic
440C2×C4C22×C22⋊D5911yescharacteristic
441C8—21no—
442C8—21no—
443C8—21no—
444C8#41×D5911yescharacteristic
445C8#1427377411yescharacteristic
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—4721no—
453C22—4721no—
454C22—2362no—
455C22C22×#1174566911yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—81no—
463C4—81no—
464C4—4721no—
465C4—4721no—
466C4—4721no—
467C4#1683805911yescharacteristic
468C4#1683821311yescharacteristic
469C2—161no—
470C2—161no—
471C2—4722no—
472C2#2143672711yescharacteristic
473C2#2143723011yescharacteristic
474C2#2143723511yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3396C2
socleC22×C59C22×D8
prime radical#21426128C2
hypercenter#3396D59
hypocenterC59#33937
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3396

Complements

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

NH ≅ G/Ntype
1. #21436964C2semidirect
2. #21426337C2semidirect
3. #21426332C2semidirect
4. #21426327C2semidirect
5. #21426260C2semidirect
6. #21426253C2semidirect
7. #21426233C2semidirect
8. #21426148C2semidirect
9. #21426148C2semidirect
11. #21426005C2semidirect
12. #21425970C2semidirect
13. #21425958C2semidirect
14. C4×#14273668C2semidirect
15. C59×#3396C2semidirect
232. #3396D59semidirect
296. C59#33937semidirect

Maximal subgroups

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

subgroupindex
#214369642
#214263372
#214263322
#214263272
#214262602
#214262532
#214262332
#214261482
#214261282
#214260052
#214259702
#214259582
C4×#142736682
C59×#33962
#3393759