Finite Group Databasebeta

← C22×SD16×D251 #44565082

Subgroups — C22×SD16×D251 #44565082

Order 32128.

1720 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 565 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×Q8×D251C211yesmaximal, characteristic, complemented
2C22×D4×D251C211yesmaximal, characteristic
3C22×#16860799C211yesmaximal, characteristic, complemented
4C22×#16860798C211yesmaximal, characteristic, complemented
5C22×#16860797C211yesmaximal, characteristic, complemented
6C2×SD16×D251C2124yesmaximal, direct product factor
7C22×C8×D251C211yesmaximal, characteristic, complemented
8C22×SD16×C251C211yesmaximal, characteristic, complemented
9C24×D251—21no—
10C22×Dic502—21no—
11C22×Dic502C2211yescharacteristic, complemented
12C2×Q8×D251—26no—
13C2×Q8×D251C22112yescomplemented
14C22×D1004C2211yescharacteristic
15C22×C22⋊D251—21no—
16C2×D4×D251—26no—
17C2×D4×D251C22112yes—
18C22×C4×D251—21no—
19C22×C4×D251C2211yescharacteristic
20C2×#16860799C22112yescomplemented
21C2×#16860798C22112yescomplemented
22C2×#16860797C22112yescomplemented
23SD16×D251C22164yesdirect product factor
24C22×C251⋊C8C2211yescharacteristic, complemented
25C2×C8×D251C22112yescomplemented
26C22×Q8×C251C2211yescharacteristic, complemented
27C22×D4×C251C2211yescharacteristic
28C2×SD16×C251C22112yescomplemented
29C22×C8×C251C2211yescharacteristic, complemented
30C23×D251—212no—
31C23×D251—21no—
32C23×D251—48no—
33C23×D251D411yescharacteristic
34C2×Dic502—212no—
35C2×Dic502—23no—
36C2×Dic502C2316yescomplemented
37Q8×D251—224no—
38Q8×D251C23116yescomplemented
39C2×D1004—23no—
40C2×D1004C2316yes—
41C2×C22⋊D251—212no—
42D4×D251—224no—
43D4×D251C23116yes—
44C22×Dic251—21no—
45C22×Dic251D411yescharacteristic
46C2×C4×D251—212no—
47C2×C4×D251D416yes—
48C2×C4×D251C2316yes—
49#16860799C23116yescomplemented
50#16860798C23116yescomplemented
51#16860797C23116yescomplemented
52C2×C251⋊C8C2316yescomplemented
53C8×D251C23116yescomplemented
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—26no—
64C22×D251—216no—
65C22×D251—44no—
66C22×D251—448no—
67C22×D251SD1618yesdirect product factor
68C22×D251C2×D416yes—
69Dic502—26no—
70Dic502—216no—
71Dic502C2414yescomplemented
72D1004—26no—
73D1004C2414yes—
74C22⋊D251—216no—
75C2×Dic251—26no—
76C2×Dic251C2×D416yes—
77C4×D251—216no—
78C4×D251C2×D4112yes—
79C4×D251C2414yes—
80C251⋊C8C2414yescomplemented
81C23×C251—26no—
82C23×C251—44no—
83C23×C251C2×D411yescharacteristic
84Q8×C251—26no—
85Q8×C251C2414yescomplemented
86D4×C251—26no—
87D4×C251C2414yes—
88C2×C4×C251—26no—
89C2×C4×C251C2×D413yes—
90C2×C4×C251C2413yes—
91C8×C251C2414yescomplemented
92C22×C251—24no—
93C22×C251—412no—
94C22×C251C2×SD1614yescomplemented
95C22×C251C22×D413yes—
96C2×D251—24no—
97C2×D251—412no—
98C2×D251—432no—
99C2×D251C2×SD16124yesdirect product factor
100C2×D251C22×D414yes—
101C4×C251—24no—
102C4×C251C22×D413yes—
103C4×C251C2511yescharacteristic
104Dic251—24no—
105Dic251C22×D414yes—
106C2×C251—44no—
107C2×C251C22×SD1616yescomplemented
108C2×C251C23×D411yescharacteristic
109D251—44no—
110D251C22×SD1618yesdirect product factor
111C251C23×SD1611yescharacteristic, complemented, Sylow
112C23×SD16—2511nomaximal, Sylow
113C23×Q8—2511no—
114C23×D4—2511no—
115C22×SD16—2511no—
116C22×SD16—2511no—
117C22×SD16—25124no—
118C22×SD16—2511no—
119C22×SD16D25111yescharacteristic, direct product factor
120C23×C8—2511no—
121C25—5021no—
122C22×Q8—5021no—
123C22×Q8—5026no—
124C22×Q8—25112no—
125C22×Q8—2511no—
126C22×Q8C2×D25111yescharacteristic, complemented
127C22×D4—5026no—
128C22×D4—5021no—
129C22×D4—25112no—
130C22×D4—2511no—
131C22×D4C2×D25111yescharacteristic
132C23×C4—5021no—
133C23×C4—2511no—
134C2×SD16—25112no—
135C2×SD16—25112no—
136C2×SD16—25164no—
137C2×SD16—25112no—
138C2×SD16C2×D251112yesdirect product factor
139C22×C8—2511no—
140C22×C8—25112no—
141C22×C8C2×D25111yescharacteristic, complemented
142C24—21no—
143C24—10048no—
144C24—50212no—
145C24—5021no—
146C24—2511no—
147C2×Q8—23no—
148C2×Q8—50212no—
149C2×Q8—50224no—
150C2×Q8—5023no—
151C2×Q8—25116no—
152C2×Q8—2516no—
153C2×Q8C22×D25116yescomplemented
154C2×D4—23no—
155C2×D4—50224no—
156C2×D4—50212no—
157C2×D4—5023no—
158C2×D4—25116no—
159C2×D4—2516no—
160C2×D4C22×D25116yes—
161C22×C4—21no—
162C22×C4—50212no—
163C22×C4—5021no—
164C22×C4—2511no—
165C22×C4—2516no—
166C22×C4—2516no—
167C22×C4C22×D25111yescharacteristic
168SD16—25116no—
169SD16—25116no—
170SD16—25116no—
171SD16C22×D251116yesdirect product factor
172C2×C8—2516no—
173C2×C8—25116no—
174C2×C8C22×D25116yescomplemented
175C23—26no—
176C23—44no—
177C23—100448no—
178C23—10044no—
179C23—50216no—
180C23—5026no—
181C23—2516no—
182C23—2518no—
183C23D4×D25111yescharacteristic
184Q8—26no—
185Q8—50216no—
186Q8—5026no—
187Q8—2514no—
188Q8C23×D25114yescomplemented
189D4—26no—
190D4—50216no—
191D4—5026no—
192D4—2514no—
193D4C23×D25114yes—
194C2×C4—26no—
195C2×C4—50216no—
196C2×C4—5026no—
197C2×C4—2516no—
198C2×C4—25112no—
199C2×C4—2514no—
200C2×C4D4×D25113yes—
201C2×C4C23×D25113yes—
202C8—2514no—
203C8C23×D25114yescomplemented
204C22—24no—
205C22—412no—
206C22—100432no—
207C22—100412no—
208C22—5024no—
209C22—25124no—
210C22—2514no—
211C22SD16×D25114yesdirect product factor
212C22C2×D4×D25113yes—
213C4—24no—
214C4—5024no—
215C4—2514no—
216C4C2×D4×D25113yes—
217C4C24×D25111yescharacteristic
218C2—44no—
219C2—10044no—
220C2—2518no—
221C2C2×SD16×D25116yesdirect product factor
222C2C22×D4×D25111yescharacteristic
subgroupG/H
centerC23D4×D251
commutator, cosocleC4×C251C25
FrattiniC4C24×D251
FittingC22×SD16×C251C2
socleC23×C251C2×D4
prime radicalC22×D4×D251C2
hypercenterC22×SD16D251
hypocenterC251C23×SD16
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×Q8×D251C2semidirect
3. C22×#16860799C2semidirect
4. C22×#16860798C2semidirect
5. C22×#16860797C2semidirect
6. C2×SD16×D251C2direct (N × H)
7. C22×C8×D251C2semidirect
8. C22×SD16×C251C2semidirect
11. C22×Dic502C22semidirect
13. C2×Q8×D251C22semidirect
20. C2×#16860799C22semidirect
21. C2×#16860798C22semidirect
22. C2×#16860797C22semidirect
23. SD16×D251C22direct (N × H)
24. C22×C251⋊C8C22semidirect
25. C2×C8×D251C22semidirect
26. C22×Q8×C251C22semidirect
28. C2×SD16×C251C22semidirect
29. C22×C8×C251C22semidirect
36. C2×Dic502C23semidirect
38. Q8×D251C23semidirect
49. #16860799C23semidirect
50. #16860798C23semidirect
51. #16860797C23semidirect
52. C2×C251⋊C8C23semidirect
53. C8×D251C23semidirect
56. C2×Q8×C251C23semidirect
61. SD16×C251C23semidirect
62. C2×C8×C251C23semidirect
67. C22×D251SD16direct (N × H)
71. Dic502C24semidirect
80. C251⋊C8C24semidirect
85. Q8×C251C24semidirect
91. C8×C251C24semidirect
94. C22×C251C2×SD16semidirect
99. C2×D251C2×SD16direct (N × H)
107. C2×C251C22×SD16semidirect
110. D251C22×SD16direct (N × H)
111. C251C23×SD16semidirect
119. C22×SD16D251direct (N × H)
126. C22×Q8C2×D251semidirect
138. C2×SD16C2×D251direct (N × H)
141. C22×C8C2×D251semidirect
153. C2×Q8C22×D251semidirect
171. SD16C22×D251direct (N × H)
174. C2×C8C22×D251semidirect
188. Q8C23×D251semidirect
203. C8C23×D251semidirect
211. C22SD16×D251direct (N × H)
221. C2C2×SD16×D251direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×D2512
C22×D4×D2512
C22×#168607992
C22×#168607982
C22×#168607972
C2×SD16×D2512
C22×C8×D2512
C22×SD16×C2512
C23×SD16251