Finite Group Databasebeta

← C22×#22548742 #44565099

Subgroups — C22×#22548742 #44565099

Order 32128.

1720 conjugacy classes of subgroups; only the 525 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 525 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
1C22×#16860811C211yesmaximal, characteristic, complemented
2C22×D4×D251C211yesmaximal, characteristic, complemented
3C2×#22548742C2124yesmaximal, direct product factor
4C22×#16860797C211yesmaximal, characteristic, complemented
5C22×D2008C211yesmaximal, characteristic, complemented
6C22×#16860795C211yesmaximal, characteristic, complemented
7C22×#16860793C211yesmaximal, characteristic, complemented
8C22×SD16×C251C211yesmaximal, characteristic, complemented
9C24×D251—21no—
10C2×#16860811—26no—
11C2×#16860811C22112yescomplemented
12C22×D1004—21no—
13C22×D1004C2211yescharacteristic
14C22×D1004C2211yescharacteristic, complemented
15C22×C22⋊D251—21no—
16C2×D4×D251—26no—
17C2×D4×D251C22112yescomplemented
18C22×C4×D251—21no—
19C22×C4×D251C2211yescharacteristic
20#22548742C22164yesdirect product factor
21C2×#16860797C22112yescomplemented
22C2×D2008C22112yescomplemented
23C2×#16860795C22112yescomplemented
24C2×#16860793C22112yescomplemented
25C22×C251⋊C8C2211yescharacteristic, complemented
26C22×Q8×C251C2211yescharacteristic, complemented
27C22×D4×C251C2211yescharacteristic
28C2×SD16×C251C22112yescomplemented
29C22×C8×C251C2211yescharacteristic, complemented
30C23×D251—212no—
31C23×D251—21no—
32C23×D251—21no—
33C23×D251—48no—
34C23×D251D411yescharacteristic
35#16860811—224no—
36#16860811C23116yescomplemented
37C2×D1004—212no—
38C2×D1004—23no—
39C2×D1004—23no—
40C2×D1004C2316yes—
41C2×D1004C2316yescomplemented
42C2×C22⋊D251—212no—
43D4×D251—224no—
44D4×D251C23116yescomplemented
45C22×Dic251D411yescharacteristic
46C2×C4×D251—212no—
47C2×C4×D251D416yes—
48C2×C4×D251C2316yes—
49#16860797C23116yescomplemented
50D2008C23116yescomplemented
51#16860795C23116yescomplemented
52#16860793C23116yescomplemented
53C2×C251⋊C8C2316yescomplemented
54C24×C251—21no—
55C2×Q8×C251—23no—
56C2×Q8×C251C2316yescomplemented
57C2×D4×C251—23no—
58C2×D4×C251C2316yes—
59C22×C4×C251—21no—
60C22×C4×C251C2311yescharacteristic
61SD16×C251C23116yescomplemented
62C2×C8×C251C2316yescomplemented
63C22×D251—24no—
64C22×D251—26no—
65C22×D251—26no—
66C22×D251—216no—
67C22×D251—448no—
68C22×D251—44no—
69C22×D251—44no—
70C22×D251C2×D416yes—
71D1004—216no—
72D1004—26no—
73D1004—26no—
74D1004C2414yes—
75D1004C2414yescomplemented
76C22⋊D251—216no—
77C2×Dic251C2×D416yes—
78C4×D251—216no—
79C4×D251C2×D4112yes—
80C4×D251C2414yes—
81C251⋊C8C2414yescomplemented
82C23×C251—26no—
83C23×C251—44no—
84C23×C251C2×D411yescharacteristic
85Q8×C251—26no—
86Q8×C251C2414yescomplemented
87D4×C251—26no—
88D4×C251C2414yes—
89C2×C4×C251—26no—
90C2×C4×C251C2×D413yes—
91C2×C4×C251C2413yes—
92C8×C251C2414yescomplemented
93C22×C251—24no—
94C22×C251—412no—
95C22×C251AGL(1,ℤ/8)14yescomplemented
96C22×C251C22×D413yes—
97C2×D251—212no—
98C2×D251—24no—
99C2×D251—24no—
100C2×D251—432no—
101C2×D251—412no—
102C2×D251—412no—
103C2×D251C22×D414yes—
104C4×C251—24no—
105C4×C251C22×D413yes—
106C4×C251C2511yescharacteristic
107Dic251C22×D414yes—
108C2×C251—44no—
109C2×C251C2×AGL(1,ℤ/8)16yescomplemented
110C2×C251C23×D411yescharacteristic
111D251—24no—
112D251—44no—
113D251—44no—
114C251C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
115C22×AGL(1,ℤ/8)—2511nomaximal, Sylow
116C22×#41—2511no—
117C23×D4—2511no—
118C2×AGL(1,ℤ/8)—25124no—
119C22×SD16—2511no—
120C22×SD16D25111yescharacteristic, complemented
121C22×D8—2511no—
122C22×D8—2511no—
123C22×M16—2511no—
124C25—5021no—
125C2×#41—5026no—
126C2×#41—25112no—
127C22×Q8C2×D25111yescharacteristic, complemented
128C22×D4—5021no—
129C22×D4—5026no—
130C22×D4—5021no—
131C22×D4—25112no—
132C22×D4—2511no—
133C22×D4—2511no—
134C22×D4C2×D25111yescharacteristic
135C23×C4—5021no—
136C23×C4—2511no—
137AGL(1,ℤ/8)—25164no—
138C2×SD16—25112no—
139C2×SD16C2×D251112yescomplemented
140C2×D8—25112no—
141C2×D8—25112no—
142C2×M16—25112no—
143C22×C8—2511no—
144C22×C8C2×D25111yescharacteristic, complemented
145C24—21no—
146C24—10048no—
147C24—50212no—
148C24—5021no—
149C24—5021no—
150C24—2511no—
151#41—50224no—
152#41—25116no—
153C2×Q8—23no—
154C2×Q8C22×D25116yescomplemented
155C2×D4—23no—
156C2×D4—50212no—
157C2×D4—50224no—
158C2×D4—50212no—
159C2×D4—5023no—
160C2×D4—5023no—
161C2×D4—25116no—
162C2×D4—2516no—
163C2×D4—2516no—
164C2×D4C22×D25116yes—
165C22×C4—21no—
166C22×C4—50212no—
167C22×C4—2511no—
168C22×C4—2516no—
169C22×C4—2516no—
170C22×C4C22×D25111yescharacteristic
171SD16—25116no—
172SD16C22×D251116yescomplemented
173D8—25116no—
174D8—25116no—
175M16—25116no—
176C2×C8—2516no—
177C2×C8C22×D25116yescomplemented
178C23—26no—
179C23—44no—
180C23—100448no—
181C23—10044no—
182C23—10044no—
183C23—50216no—
184C23—5026no—
185C23—5026no—
186C23—5024no—
187C23—2516no—
188C23D4×D25111yescharacteristic
189Q8—26no—
190Q8C23×D25114yescomplemented
191D4—26no—
192D4—50216no—
193D4—50216no—
194D4—5026no—
195D4—5026no—
196D4—2514no—
197D4—2514no—
198D4C23×D25114yes—
199C2×C4—26no—
200C2×C4—50216no—
201C2×C4—2516no—
202C2×C4—25112no—
203C2×C4—2514no—
204C2×C4D4×D25113yes—
205C2×C4C23×D25113yes—
206C8—2514no—
207C8C23×D25114yescomplemented
208C22—24no—
209C22—412no—
210C22—100432no—
211C22—100412no—
212C22—100412no—
213C22—5024no—
214C22—5024no—
215C22—50212no—
216C22—2514no—
217C22#2254874214yesdirect product factor
218C22C2×D4×D25113yes—
219C4—24no—
220C4—2514no—
221C4C2×D4×D25113yes—
222C4C24×D25111yescharacteristic
223C2—44no—
224C2—10044no—
225C2—10044no—
226C2—5024no—
227C2C2×#2254874216yesdirect product factor
228C2C22×D4×D25111yescharacteristic
subgroupG/H
centerC23D4×D251
commutator, cosocleC4×C251C25
FrattiniC4C24×D251
FittingC22×SD16×C251C2
socleC23×C251C2×D4
hypercenterC22×SD16D251
hypocenterC251C22×AGL(1,ℤ/8)
LCSG ▷ C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#16860811C2semidirect
2. C22×D4×D251C2semidirect
3. C2×#22548742C2direct (N × H)
4. C22×#16860797C2semidirect
5. C22×D2008C2semidirect
6. C22×#16860795C2semidirect
7. C22×#16860793C2semidirect
8. C22×SD16×C251C2semidirect
11. C2×#16860811C22semidirect
14. C22×D1004C22semidirect
17. C2×D4×D251C22semidirect
20. #22548742C22direct (N × H)
21. C2×#16860797C22semidirect
22. C2×D2008C22semidirect
23. C2×#16860795C22semidirect
24. C2×#16860793C22semidirect
25. C22×C251⋊C8C22semidirect
26. C22×Q8×C251C22semidirect
28. C2×SD16×C251C22semidirect
29. C22×C8×C251C22semidirect
36. #16860811C23semidirect
41. C2×D1004C23semidirect
44. D4×D251C23semidirect
49. #16860797C23semidirect
50. D2008C23semidirect
51. #16860795C23semidirect
52. #16860793C23semidirect
53. C2×C251⋊C8C23semidirect
56. C2×Q8×C251C23semidirect
61. SD16×C251C23semidirect
62. C2×C8×C251C23semidirect
75. D1004C24semidirect
81. C251⋊C8C24semidirect
86. Q8×C251C24semidirect
92. C8×C251C24semidirect
95. C22×C251AGL(1,ℤ/8)semidirect
109. C2×C251C2×AGL(1,ℤ/8)semidirect
114. C251C22×AGL(1,ℤ/8)semidirect
120. C22×SD16D251semidirect
127. C22×Q8C2×D251semidirect
139. C2×SD16C2×D251semidirect
144. C22×C8C2×D251semidirect
154. C2×Q8C22×D251semidirect
172. SD16C22×D251semidirect
177. C2×C8C22×D251semidirect
190. Q8C23×D251semidirect
207. C8C23×D251semidirect
217. C22#22548742direct (N × H)
227. C2C2×#22548742direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#168608112
C22×D4×D2512
C2×#225487422
C22×#168607972
C22×D20082
C22×#168607952
C22×#168607932
C22×SD16×C2512
C22×AGL(1,ℤ/8)251