Finite Group Databasebeta

← #44562163

Subgroups — #44562163

Order 32128.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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#30632492C211yesmaximal, characteristic, complemented
2#30632491C212yesmaximal, complemented
3#30632488C211yesmaximal, characteristic, complemented
4#30632062C211yesmaximal, characteristic, complemented
5#30631958C211yesmaximal, characteristic, complemented
6#30631956C212yesmaximal, complemented
7#30631857C212yesmaximal, complemented
8#30631853C212yesmaximal, complemented
9#30631803C211yesmaximal, characteristic, complemented
10#30631802C211yesmaximal, characteristic, complemented
11#472×C251C211yesmaximal, characteristic, complemented
12#22548761C2212yescomplemented
13#22548760C2212yescomplemented
14#22548743—22no—
15#22548743C2212yescomplemented
16#22548743C2212yescomplemented
17#22548742—22no—
18#22548741C2212yescomplemented
19#22548739—22no—
20#22548739—22no—
21#22548737—21no—
22#22548737C2211yescharacteristic
23#22548736—22no—
24#22548735C2211yescharacteristic
25#22548734—21no—
26C2×D2008—21no—
27C2×D2008C2211yescharacteristic, complemented
28C2×#16860795C2212yescomplemented
29D8×D251—21no—
30D8×D251—22no—
31#22548723C2212yescomplemented
32#22548721C2211yescharacteristic, complemented
33#22548711C2212yescomplemented
34#22548650C2211yescharacteristic, complemented
35#22548642C2211yescharacteristic, complemented
36#22548641C2211yescharacteristic, complemented
37#22548641C2212yescomplemented
38#22548631C2212yescomplemented
39C8×Dic251C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C251C2212yescomplemented
41C2×D8×C251C2211yescharacteristic, complemented
42#131×C251C2211yescharacteristic, complemented
43#108×C251C2211yescharacteristic, complemented
44#100×C251C2212yescomplemented
45#16860811—22no—
46#16860810—22no—
47#16860810C2312yes—
48#16860810C2311yescharacteristic
49#16860809—22no—
50#16860809—22no—
51C2×D1004—22no—
52C2×D1004C2312yes—
53C2×C22⋊D251—42no—
54C2×C22⋊D251—42no—
55D4×D251—22no—
56D4×D251—22no—
57D4×D251—22no—
58D4×D251—42no—
59Dic1004D411yescharacteristic, complemented
60#16860800D411yescharacteristic, complemented
61#16860799—22no—
62#16860799—22no—
63#16860799D412yescomplemented
64#16860798—22no—
65#16860798D411yescharacteristic, complemented
66#16860797D411yescharacteristic, complemented
67D2008—22no—
68D2008—22no—
69D2008—21no—
70D2008—21no—
71D2008D411yescharacteristic, complemented
72#16860795—22no—
73#16860795—22no—
74#16860795—22no—
75#16860795D411yescharacteristic, complemented
76#16860794C2311yescharacteristic
77#16860793—22no—
78#16860793—21no—
79C2×C251⋊C8C2311yescharacteristic
80C8×D251—22no—
81C8×D251—21no—
82C4×Dic251C2311yescharacteristic
83#41×C251C2311yescharacteristic
84C2×D4×C251C2312yes—
85SD16×C251—22no—
86D8×C251—22no—
87D8×C251—21no—
88D8×C251—21no—
89M16×C251—21no—
90M16×C251C2311yescharacteristic
91M16×C251C2312yes—
92C2×C8×C251—21no—
93C2×C8×C251C2311yescharacteristic
94C22×D251—22no—
95C22×D251—42no—
96C22×D251—42no—
97Dic502C2×D412yes—
98Dic502C2×D411yescharacteristic
99D1004—22no—
100D1004—22no—
101D1004—42no—
102D1004C2×D412yes—
103D1004C2×D411yescharacteristic
104C22⋊D251—22no—
105C22⋊D251—21no—
106C22⋊D251—42no—
107C22⋊D251—42no—
108C22⋊D251—42no—
109C22⋊D251—84no—
110C2×Dic251—21no—
111C2×Dic251—42no—
112C4×D251—21no—
113C4×D251—22no—
114C4×D251—42no—
115C251⋊C8—21no—
116C251⋊C8C2×D411yescharacteristic
117C251⋊C8C2×D411yescharacteristic
118C23×C251—22no—
119Q8×C251C2×D411yescharacteristic
120D4×C251—21no—
121D4×C251—22no—
122D4×C251—22no—
123D4×C251C2×D411yescharacteristic
124C2×C4×C251—21no—
125C2×C4×C251C2411yescharacteristic
126C8×C251—21no—
127C8×C251—22no—
128C8×C251C2×D411yescharacteristic
129C8×C251C2×D411yescharacteristic
130C22×C251—21no—
131C22×C251—42no—
132C22×C251—82no—
133C22×C251H(5,𝔽2)11yescharacteristic
134C2×D251—21no—
135C2×D251—22no—
136C2×D251—42no—
137C2×D251—82no—
138C2×D251—162no—
139C2×D251—162no—
140C4×C251—21no—
141C4×C251C22×D411yescharacteristic
142C4×C251C22×D411yescharacteristic
143Dic251—22no—
144Dic251—22no—
145Dic251—21no—
146C2×C251—21no—
147C2×C251—41no—
148C2×C251—82no—
149C2×C251D4211yescharacteristic
150D251—41no—
151D251—42no—
152D251—82no—
153C251#329211yescharacteristic, complemented, Sylow
154#3292—2511nomaximal, Sylow
155#576—2512no—
156#576—2511no—
157#576—2511no—
158#493—2511no—
159#472—2511no—
160#472—2512no—
161#472D25111yescharacteristic, complemented
162#453—2512no—
163#453—2512no—
164#443—2511no—
165#443—2511no—
166H(5,𝔽2)—2512no—
167H(5,𝔽2)—2512no—
168AGL(1,ℤ/8)—5022no—
169AGL(1,ℤ/8)—5022no—
170AGL(1,ℤ/8)—5022no—
171AGL(1,ℤ/8)—5022no—
172AGL(1,ℤ/8)—2512no—
173AGL(1,ℤ/8)—2512no—
174AGL(1,ℤ/8)—2512no—
175AGL(1,ℤ/8)C2×D25112yescomplemented
176#135—5021no—
177#135—5021no—
178#135—5022no—
179#135—2511no—
180#135—2511no—
181C2×D8—5021no—
182C2×D8—5022no—
183C2×D8—5021no—
184C2×D8—2512no—
185C2×D8—2511no—
186C2×D8C2×D25111yescharacteristic, complemented
187#131—2512no—
188#131—2511no—
189#131C2×D25111yescharacteristic, complemented
190#127—2512no—
191#108—2511no—
192#108C2×D25111yescharacteristic, complemented
193#104—2511no—
194#104—2512no—
195#104—2511no—
196#100—2512no—
197#100C2×D25112yescomplemented
198C4×C8—2511no—
199#41—5022no—
200#41—5022no—
201#41—5022no—
202#41—5022no—
203#41—2512no—
204#41—2511no—
205#41C22×D25111yescharacteristic
206C2×D4—10042no—
207C2×D4—10042no—
208C2×D4—10042no—
209C2×D4—5022no—
210C2×D4—5022no—
211C2×D4—5022no—
212C2×D4—5022no—
213C2×D4—2512no—
214C2×D4C22×D25112yes—
215Q16—2511no—
216Q16—2511no—
217SD16—22no—
218SD16—5022no—
219SD16—5022no—
220SD16—5022no—
221SD16—2511no—
222SD16—2511no—
223SD16—2512no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—5022no—
228D8—5022no—
229D8—5022no—
230D8—5022no—
231D8—5022no—
232D8—5021no—
233D8—5021no—
234D8—2511no—
235D8—2511no—
236M16—21no—
237M16—5022no—
238M16—5021no—
239M16—2511no—
240M16C22×D25112yes—
241M16C22×D25111yescharacteristic
242C2×C8—21no—
243C2×C8—5022no—
244C2×C8—5021no—
245C2×C8—2511no—
246C2×C8C22×D25111yescharacteristic
247C42—2511no—
248C23—22no—
249C23—10042no—
250C23—10042no—
251C23—5022no—
252Q8—2511no—
253Q8—2512no—
254Q8D4×D25111yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—20084no—
259D4—10042no—
260D4—10042no—
261D4—10042no—
262D4—10042no—
263D4—5022no—
264D4—5021no—
265D4—5022no—
266D4—5022no—
267D4—2511no—
268D4—2512no—
269D4D4×D25111yescharacteristic
270C2×C4—21no—
271C2×C4—10042no—
272C2×C4—10042no—
273C2×C4—5021no—
274C2×C4—5022no—
275C2×C4—5021no—
276C2×C4C23×D25111yescharacteristic
277C8—21no—
278C8—22no—
279C8—5021no—
280C8—2511no—
281C8—2511no—
282C8D4×D25111yescharacteristic
283C8D4×D25111yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—40162no—
288C22—40162no—
289C22—20082no—
290C22—10042no—
291C22—5022no—
292C22—5021no—
293C22#2254876111yescharacteristic
294C4—21no—
295C4—5022no—
296C4—5022no—
297C4—5021no—
298C4C2×D4×D25111yescharacteristic
299C4C2×D4×D25111yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—20082no—
304C2—10042no—
305C2—10041no—
306C2#3063232111yescharacteristic
subgroupG/H
centerC2#30632321
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#472×C251C2
socleC2×C251D42
hypercenter#472D251
hypocenterC251#3292
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #30632492C2semidirect
2. #30632491C2semidirect
3. #30632488C2semidirect
4. #30632062C2semidirect
5. #30631958C2semidirect
6. #30631956C2semidirect
7. #30631857C2semidirect
8. #30631853C2semidirect
9. #30631803C2semidirect
10. #30631802C2semidirect
11. #472×C251C2semidirect
12. #22548761C22semidirect
13. #22548760C22semidirect
15. #22548743C22semidirect
16. #22548743C22semidirect
18. #22548741C22semidirect
27. C2×D2008C22semidirect
28. C2×#16860795C22semidirect
31. #22548723C22semidirect
32. #22548721C22semidirect
33. #22548711C22semidirect
34. #22548650C22semidirect
35. #22548642C22semidirect
36. #22548641C22semidirect
37. #22548641C22semidirect
38. #22548631C22semidirect
39. C8×Dic251C22semidirect
40. AGL(1,ℤ/8)×C251C22semidirect
41. C2×D8×C251C22semidirect
42. #131×C251C22semidirect
43. #108×C251C22semidirect
44. #100×C251C22semidirect
59. Dic1004D4semidirect
60. #16860800D4semidirect
63. #16860799D4semidirect
65. #16860798D4semidirect
66. #16860797D4semidirect
71. D2008D4semidirect
75. #16860795D4semidirect
153. C251#3292semidirect
161. #472D251semidirect
175. AGL(1,ℤ/8)C2×D251semidirect
186. C2×D8C2×D251semidirect
189. #131C2×D251semidirect
192. #108C2×D251semidirect
197. #100C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306324922
#306324912
#306324882
#306320622
#306319582
#306319562
#306318572
#306318532
#306318032
#306318022
#472×C2512
#3292251