Finite Group Databasebeta

← #44565144

Subgroups — #44565144

Order 32128.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 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
1C2×#22548761C211yesmaximal, characteristic, complemented
2#30632495C218yesmaximal, complemented
3#30632490C218yesmaximal, complemented
4#30632485C216yesmaximal, complemented
5C2×#22548740C216yesmaximal, complemented
6#30632454C211yesmaximal, characteristic, complemented
7#585×C251C211yesmaximal, characteristic, complemented
8#22548761—24no—
9#22548761C2218yescomplemented
10C2×#16860811—21no—
11C2×#16860810C2213yescomplemented
12C22×D1004C2213yescomplemented
13C2×D4×D251—23no—
14#22548746C22112yescomplemented
15#22548745C2214yescomplemented
16#22548741C22112yescomplemented
17#22548740C22112yescomplemented
18#22548740C22124yescomplemented
19#22548735C22124yescomplemented
20C2×#16860797C22112yescomplemented
21C2×#16860795C22112yescomplemented
22#22548722C2218yescomplemented
23C2×#16860794C2216yescomplemented
242−1+4×C251—21no—
252−1+4×C251C2214yescomplemented
26H(5,𝔽2)×C251—23no—
27H(5,𝔽2)×C251C2214yescomplemented
28C2×#41×C251—24no—
29C2×#41×C251C2211yescharacteristic
30C2×#41×C251C2216yescomplemented
31C23×D251—23no—
32#16860811—28no—
33#16860810—26no—
34#16860810C23112yes—
35C2×Dic502C2311yescharacteristic
36C2×D1004—23no—
37C2×D1004—26no—
38C2×D1004C2313yes—
39C2×D1004C23112yes—
40C2×C22⋊D251—23no—
41D4×D251—224no—
42C2×C4×D251—23no—
43#16860800C2318yescomplemented
44#16860798C2318yescomplemented
45#16860797C23124yescomplemented
46#16860795C23124yescomplemented
47#16860794C2314yes—
48#16860794C23112yescomplemented
49C2×C251⋊C8C23112yescomplemented
50#41×C251—24no—
51#41×C251—212no—
52#41×C251—26no—
53#41×C251—26no—
54#41×C251D414yes—
55#41×C251C2314yes—
56#41×C251C2314yes—
57#41×C251C23112yescomplemented
58C2×Q8×C251—24no—
59C2×Q8×C251D411yescharacteristic
60C2×Q8×C251C2316yescomplemented
61C2×D4×C251—26no—
62C2×D4×C251—212no—
63C2×D4×C251D413yes—
64C2×D4×C251C2316yescomplemented
65C22×C4×C251—26no—
66C22×C4×C251C2313yes—
67C22×D251—23no—
68C22×D251—212no—
69C22×D251—86no—
70Dic502—21no—
71Dic502C2412yes—
72D1004—212no—
73D1004—23no—
74D1004C2416yes—
75C22⋊D251—212no—
76C2×Dic251—21no—
77C4×D251—212no—
78C251⋊C8C2418yes—
79C23×C251—26no—
80C23×C251C2×D413yes—
81Q8×C251—26no—
82Q8×C251C2×D414yes—
83Q8×C251C2414yes—
84D4×C251—24no—
85D4×C251—212no—
86D4×C251—26no—
87D4×C251C2×D4112yes—
88D4×C251C2414yes—
89C2×C4×C251—26no—
90C2×C4×C251—24no—
91C2×C4×C251—212no—
92C2×C4×C251C2×D416yes—
93C2×C4×C251C2×D413yes—
94C2×C4×C251C2411yescharacteristic
95C2×C4×C251C2416yes—
96C22×C251—24no—
97C22×C251—43no—
98C22×C251—86no—
99C22×C251C22×D416yes—
100C22×C251C22×D411yescharacteristic
101C2×D251—26no—
102C2×D251—46no—
103C2×D251—812no—
104C4×C251—24no—
105C4×C251C22×D411yescharacteristic
106C4×C251C22×D416yes—
107C4×C251C2511yescharacteristic
108Dic251—22no—
109C2×C251—21no—
110C2×C251—26no—
111C2×C251—44no—
112C2×C251C23×D411yescharacteristic
113D251—46no—
114C251#358511yescharacteristic, complemented, Sylow
115#3585—2511nomaximal, Sylow
116#585D25111yescharacteristic, complemented
117C2×H(5,𝔽2)—2511no—
118#577—2518no—
119#576—2518no—
120#575—2516no—
121C2×AGL(1,ℤ/8)—2516no—
122#568—2511no—
1232−1+4—21no—
1242−1+4C2×D25114yescomplemented
125H(5,𝔽2)—23no—
126H(5,𝔽2)—5024no—
127H(5,𝔽2)—2518no—
128H(5,𝔽2)C2×D25114yescomplemented
129C2×#41—24no—
130C2×#41—5021no—
131C2×#41—2513no—
132C2×#41C2×D25111yescharacteristic
133C2×#41C2×D25116yescomplemented
134C22×D4—5023no—
135C22×D4—2513no—
136#137—2514no—
137#137—25112no—
138AGL(1,ℤ/8)—25112no—
139AGL(1,ℤ/8)—25124no—
140AGL(1,ℤ/8)—25112no—
141#135—25124no—
142C2×SD16—25112no—
143C2×D8—25112no—
144#131—2518no—
145C2×M16—2516no—
146C24—5023no—
147#41—24no—
148#41—212no—
149#41—26no—
150#41—26no—
151#41—5028no—
152#41—5026no—
153#41—25112no—
154#41C22⋊D25114yes—
155#41C22×D25114yes—
156#41C22×D25114yes—
157#41C22×D251112yescomplemented
158C2×Q8—24no—
159C2×Q8—2511no—
160C2×Q8C22⋊D25111yescharacteristic
161C2×Q8C22×D25116yescomplemented
162C2×D4—26no—
163C2×D4—212no—
164C2×D4—5023no—
165C2×D4—50224no—
166C2×D4—5023no—
167C2×D4—5026no—
168C2×D4—25112no—
169C2×D4—2513no—
170C2×D4C22⋊D25113yes—
171C2×D4C22×D25116yescomplemented
172C22×C4—26no—
173C22×C4—5023no—
174C22×C4C22×D25113yes—
175Q16—2518no—
176SD16—25124no—
177SD16—2518no—
178D8—25124no—
179M16—2514no—
180M16—25112no—
181C2×C8—25112no—
182C23—26no—
183C23—20086no—
184C23—50212no—
185C23—5023no—
186C23C2×C22⋊D25113yes—
187Q8—26no—
188Q8—5021no—
189Q8—2512no—
190Q8C2×C22⋊D25114yes—
191Q8C23×D25114yes—
192D4—24no—
193D4—212no—
194D4—26no—
195D4—50212no—
196D4—5023no—
197D4—50212no—
198D4—2516no—
199D4C2×C22⋊D251112yes—
200D4C23×D25114yes—
201C2×C4—26no—
202C2×C4—24no—
203C2×C4—212no—
204C2×C4—5021no—
205C2×C4—50212no—
206C2×C4C2×C22⋊D25116yes—
207C2×C4C2×C22⋊D25113yes—
208C2×C4C23×D25116yes—
209C2×C4C23×D25111yescharacteristic
210C8—2518no—
211C22—24no—
212C22—43no—
213C22—86no—
214C22—200812no—
215C22—10046no—
216C22—5026no—
217C22C22×C22⋊D25116yes—
218C22C22×C22⋊D25111yescharacteristic
219C4—24no—
220C4—5022no—
221C4C22×C22⋊D25111yescharacteristic
222C4C22×C22⋊D25116yes—
223C4C24×D25111yescharacteristic
224C2—21no—
225C2—26no—
226C2—44no—
227C2—10046no—
228C2C23×C22⋊D25111yescharacteristic
subgroupG/H
centerC2C23×C22⋊D251
commutator, cosocleC4×C251C25
FrattiniC4C24×D251
Fitting#585×C251C2
socleC2×C251C23×D4
hypercenter#585D251
hypocenterC251#3585
LCSG ▷ C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×#41 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#22548761C2semidirect
2. #30632495C2semidirect
3. #30632490C2semidirect
4. #30632485C2semidirect
5. C2×#22548740C2semidirect
6. #30632454C2semidirect
7. #585×C251C2semidirect
9. #22548761C22semidirect
11. C2×#16860810C22semidirect
12. C22×D1004C22semidirect
14. #22548746C22semidirect
15. #22548745C22semidirect
16. #22548741C22semidirect
17. #22548740C22semidirect
18. #22548740C22semidirect
19. #22548735C22semidirect
20. C2×#16860797C22semidirect
21. C2×#16860795C22semidirect
22. #22548722C22semidirect
23. C2×#16860794C22semidirect
25. 2−1+4×C251C22semidirect
27. H(5,𝔽2)×C251C22semidirect
30. C2×#41×C251C22semidirect
43. #16860800C23semidirect
44. #16860798C23semidirect
45. #16860797C23semidirect
46. #16860795C23semidirect
48. #16860794C23semidirect
49. C2×C251⋊C8C23semidirect
57. #41×C251C23semidirect
60. C2×Q8×C251C23semidirect
64. C2×D4×C251C23semidirect
114. C251#3585semidirect
116. #585D251semidirect
124. 2−1+4C2×D251semidirect
128. H(5,𝔽2)C2×D251semidirect
133. C2×#41C2×D251semidirect
157. #41C22×D251semidirect
161. C2×Q8C22×D251semidirect
171. C2×D4C22×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487612
#306324952
#306324902
#306324852
C2×#225487402
#306324542
#585×C2512
#3585251