Finite Group Databasebeta

← #28746839

Subgroups — #28746839

Order 15104.

472 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21437234C211yesmaximal, characteristic, complemented
2#21437233C211yesmaximal, characteristic, complemented
3#21426411C211yesmaximal, characteristic, complemented
4#21426410C212yesmaximal, complemented
5#21426407C212yesmaximal, complemented
6#21426404C212yesmaximal, complemented
7#21426403C211yesmaximal, characteristic, complemented
8C2×#16837237C211yesmaximal, characteristic, complemented
9D32×D59C212yesmaximal, complemented
10#21426391C211yesmaximal, characteristic, complemented
11C59×#2263C211yesmaximal, characteristic, complemented
12#16838204—21no—
13#16838203—21no—
14#16837839—22no—
15#16837838—21no—
16#16837837—21no—
17#16837836—22no—
18#16837835C2212yescomplemented
19#16837834C2211yescharacteristic, complemented
20#16837831C2212yescomplemented
21C2×D944C2211yescharacteristic, complemented
22D16×D59C2212yescomplemented
23D16×D59C2212yescomplemented
24#16837818C2211yescharacteristic
25#16837242C2211yescharacteristic, complemented
26#16837241C2212yescomplemented
27#16837240C2211yescharacteristic, complemented
28#16837239C2211yescharacteristic, complemented
29D1888C2212yescomplemented
30#16837237C2211yescharacteristic, complemented
31#16837237C2212yescomplemented
32#16837237C2212yescomplemented
33#16837235C2212yescomplemented
34C2×C59⋊C32C2211yescharacteristic, complemented
35C32×D59C2212yescomplemented
36C59×#508C2211yescharacteristic, complemented
37C2×D16×C59C2211yescharacteristic, complemented
38C59×SD64C2212yescomplemented
39C59×D32C2212yescomplemented
40C59×M64C2211yescharacteristic, complemented
41#14273773—41no—
42#14273772—41no—
43#14273754—42no—
44#14273753—41no—
45#14273752—41no—
46#14273751—42no—
47#14273750—22no—
48#14273749—21no—
49#14273746—22no—
50C2×D472—21no—
51D8×D59—22no—
52D8×D59—22no—
53#14273733D411yescharacteristic
54Dic472C2311yescharacteristic
55#14273672—21no—
56#14273671—22no—
57#14273670—22no—
58D944—21no—
59D944C2312yes—
60D944C2311yescharacteristic
61#14273668—22no—
62#14273668—22no—
63#14273667D411yescharacteristic
64#14273666D412yes—
65C16×D59C2312yes—
66C59⋊C32C2311yescharacteristic
67C59⋊C32C2311yescharacteristic
68#135×C59—21no—
69C2×D8×C59—21no—
70Q32×C59C2311yescharacteristic
71SD32×C59—21no—
72D16×C59—21no—
73D16×C59C2311yescharacteristic
74D16×C59C2312yes—
75C2×C16×C59C2311yescharacteristic
76C32×C59C2312yes—
77#41×C59—41no—
78C2×D4×C59—41no—
79C2×C22⋊D59—82no—
80#11745689—42no—
81#11745687—42no—
82#11745687—81no—
83D4×D59—42no—
84D4×D59—42no—
85D4×D59—82no—
86D4×D59—81no—
87#11745685—41no—
88#11745685—81no—
89#11745685D811yescharacteristic
90C2×D236—41no—
91C2×D236—81no—
92Q16×C59—21no—
93SD16×C59—41no—
94D8×C59—22no—
95D8×C59—21no—
96D8×C59—41no—
97C2×C8×C59C2×D411yescharacteristic
98#11745671—42no—
99#11745670—42no—
100#11745669—42no—
101#11745669—42no—
102#11745664D811yescharacteristic
103Dic236—21no—
104D472—21no—
105D472—22no—
106D472—41no—
107#11745660—41no—
108#11745659D812yes—
109C8×D59C2×D412yes—
110C16×C59C2×D411yescharacteristic
111C16×C59C2411yescharacteristic
112C59⋊C16C2×D412yes—
113C23×C59—81no—
114C22×D59—81no—
115C22×D59—82no—
116C22×D59—82no—
117Q8×C59—41no—
118D4×C59—41no—
119D4×C59—42no—
120D4×C59—81no—
121D4×C59—81no—
122C2×C4×C59—81no—
123C2×C4×C59C2×D811yescharacteristic
124C22⋊D59—81no—
125C22⋊D59—82no—
126C22⋊D59—82no—
127C22⋊D59—82no—
128C22⋊D59D1612yes—
129C2×Dic59—82no—
130D236—41no—
131D236—42no—
132D236—82no—
133D236—81no—
134D236D1611yescharacteristic
135C4×D59—81no—
136C4×D59—82no—
137C4×D59C2×D812yes—
138Dic118—41no—
139Dic118D1611yescharacteristic
140C8×C59C2×D811yescharacteristic
141C8×C59C22×D411yescharacteristic
142C59⋊C8C2×D812yes—
143#33982—591nomaximal, Sylow
144C22×C59—82no—
145C22×C59—81no—
146C22×C59—321no—
147C22×C59C2×D1611yescharacteristic
148C2×D59—82no—
149C2×D59—81no—
150C2×D59—322no—
151C2×D59—322no—
152C2×D59—321no—
153C2×D59C2×D1612yes—
154C4×C59—81no—
155C4×C59C2×D1611yescharacteristic
156C4×C59C22×D811yescharacteristic
157Dic59—81no—
158Dic59C2×D1612yes—
159#3415—591no—
160#3415—591no—
161#2263—592no—
162#2263—592no—
163#2263—591no—
164#2263D5911yescharacteristic, complemented
165#2262—591no—
166#2262—592no—
167C2×D32—591no—
168C2×D32—592no—
169#2258—591no—
170C2×C59—21no—
171C2×C59—161no—
172C2×C59—162no—
173C2×C59C22×D1611yescharacteristic
174D59—22no—
175D59—162no—
176D59—161no—
177#576—1181no—
178#576—1181no—
179#509—1182no—
180#509—1182no—
181#509—1181no—
182#509—1181no—
183#508—592no—
184#508—592no—
185#508—591no—
186#508C2×D5911yescharacteristic, complemented
187C2×D16—592no—
188C2×D16—592no—
189C2×D16—591no—
190C2×D16C2×D5911yescharacteristic, complemented
191#504—591no—
192Q64—591no—
193SD64—591no—
194SD64—591no—
195SD64—592no—
196SD64C2×D5912yescomplemented
197D32—592no—
198D32—591no—
199D32—592no—
200D32—592no—
201D32C2×D5912yescomplemented
202M64—592no—
203M64C2×D5911yescharacteristic, complemented
204C2×C32—591no—
205C2×C32—592no—
206C59#3398211yescharacteristic, complemented, Sylow
207H(5,𝔽2)—2361no—
208H(5,𝔽2)—2361no—
209AGL(1,ℤ/8)—2362no—
210AGL(1,ℤ/8)—2362no—
211AGL(1,ℤ/8)—2361no—
212AGL(1,ℤ/8)—2361no—
213#135—21no—
214#135—1182no—
215#135—1182no—
216#135—1181no—
217C2×D8—21no—
218C2×D8—1182no—
219C2×D8—1182no—
220C2×D8—1181no—
221#131—591no—
222Q32—591no—
223Q32C22×D5911yescharacteristic
224SD32—21no—
225SD32—1182no—
226SD32—1182no—
227SD32—1181no—
228D16—21no—
229D16—592no—
230D16—591no—
231D16—1182no—
232D16—1182no—
233D16—1181no—
234D16C22×D5912yes—
235D16C22×D5911yescharacteristic
236M32—591no—
237M32—592no—
238C2×C16—592no—
239C2×C16C22×D5911yescharacteristic
240C32—591no—
241C32—591no—
242C32C22×D5912yes—
243#41—41no—
244#41—591no—
245#41—4721no—
246#41—4721no—
247#41—2362no—
248#41—2362no—
249#41—2361no—
250C2×D4—41no—
251C2×D4—4721no—
252C2×D4—4722no—
253C2×D4—4721no—
254C2×D4—4722no—
255C2×D4—2362no—
256C2×D4—2362no—
257C2×D4—2361no—
258Q16—21no—
259Q16—1181no—
260SD16—41no—
261SD16—2362no—
262SD16—2362no—
263SD16—2361no—
264D8—22no—
265D8—21no—
266D8—41no—
267D8—1181no—
268D8—1182no—
269D8—2362no—
270D8—2362no—
271D8—2361no—
272M16—591no—
273M16—592no—
274C2×C8—592no—
275C2×C8D4×D5911yescharacteristic
276C16—592no—
277C16D4×D5911yescharacteristic
278C16C23×D5911yescharacteristic
279C23—81no—
280C23—4722no—
281C23—4722no—
282C23—4721no—
283Q8—41no—
284Q8—591no—
285Q8—2361no—
286D4—42no—
287D4—41no—
288D4—81no—
289D4—81no—
290D4—592no—
291D4—591no—
292D4—4722no—
293D4—4722no—
294D4—4722no—
295D4—4722no—
296D4—4721no—
297D4—4721no—
298D4—2361no—
299D4—2362no—
300C2×C4—81no—
301C2×C4—592no—
302C2×C4—4722no—
303C2×C4—4722no—
304C2×C4—4721no—
305C2×C4D8×D5911yescharacteristic
306C8—592no—
307C8D8×D5911yescharacteristic
308C8C2×D4×D5911yescharacteristic
309C22—81no—
310C22—82no—
311C22—321no—
312C22—592no—
313C22—18881no—
314C22—18882no—
315C22—18882no—
316C22—4722no—
317C22—4721no—
318C22D16×D5911yescharacteristic
319C4—81no—
320C4—592no—
321C4—4721no—
322C4D16×D5911yescharacteristic
323C4C2×D8×D5911yescharacteristic
324C2—21no—
325C2—162no—
326C2—161no—
327C2—1182no—
328C2—9442no—
329C2—9441no—
330C2C2×D16×D5911yescharacteristic
subgroupG/H
centerC2C2×D16×D59
commutator, cosocleC16×C59C24
FrattiniC16C23×D59
FittingC59×#2263C2
socleC2×C59C22×D16
hypercenter#2263D59
hypocenterC59#33982
LCSG ▷ C16×C59 ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2263

Complements

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

NH ≅ G/Ntype
1. #21437234C2semidirect
2. #21437233C2semidirect
3. #21426411C2semidirect
4. #21426410C2semidirect
5. #21426407C2semidirect
6. #21426404C2semidirect
7. #21426403C2semidirect
8. C2×#16837237C2semidirect
9. D32×D59C2semidirect
10. #21426391C2semidirect
11. C59×#2263C2semidirect
18. #16837835C22semidirect
19. #16837834C22semidirect
20. #16837831C22semidirect
21. C2×D944C22semidirect
22. D16×D59C22semidirect
23. D16×D59C22semidirect
25. #16837242C22semidirect
26. #16837241C22semidirect
27. #16837240C22semidirect
28. #16837239C22semidirect
29. D1888C22semidirect
30. #16837237C22semidirect
31. #16837237C22semidirect
32. #16837237C22semidirect
33. #16837235C22semidirect
34. C2×C59⋊C32C22semidirect
35. C32×D59C22semidirect
36. C59×#508C22semidirect
37. C2×D16×C59C22semidirect
38. C59×SD64C22semidirect
39. C59×D32C22semidirect
40. C59×M64C22semidirect
164. #2263D59semidirect
186. #508C2×D59semidirect
190. C2×D16C2×D59semidirect
196. SD64C2×D59semidirect
201. D32C2×D59semidirect
203. M64C2×D59semidirect
206. C59#33982semidirect

Maximal subgroups

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

subgroupindex
#214372342
#214372332
#214264112
#214264102
#214264072
#214264042
#214264032
C2×#168372372
D32×D592
#214263912
C59×#22632
#3398259