Finite Group Databasebeta

← C22×#15692282 #27185866

Subgroups — C22×#15692282 #27185866

Order 12928.

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×#13108075C211yesmaximal, characteristic, complemented
2C22×D4×D101C211yesmaximal, characteristic, complemented
3C2×#15692282C2124yesmaximal, direct product factor
4C22×#13108061C211yesmaximal, characteristic, complemented
5C22×D808C211yesmaximal, characteristic, complemented
6C22×#13108059C211yesmaximal, characteristic, complemented
7C22×#13108057C211yesmaximal, characteristic, complemented
8C22×SD16×C101C211yesmaximal, characteristic, complemented
9C24×D101—21no—
10C2×#13108075—26no—
11C2×#13108075C22112yescomplemented
12C22×D404—21no—
13C22×D404C2211yescharacteristic
14C22×D404C2211yescharacteristic, complemented
15C22×C22⋊D101—21no—
16C2×D4×D101—26no—
17C2×D4×D101C22112yescomplemented
18C22×C4×D101—21no—
19C22×C4×D101C2211yescharacteristic
20#15692282C22164yesdirect product factor
21C2×#13108061C22112yescomplemented
22C2×D808C22112yescomplemented
23C2×#13108059C22112yescomplemented
24C2×#13108057C22112yescomplemented
25C22×#11717579C2211yescharacteristic, complemented
26C22×Q8×C101C2211yescharacteristic, complemented
27C22×D4×C101C2211yescharacteristic
28C2×SD16×C101C22112yescomplemented
29C22×C8×C101C2211yescharacteristic, complemented
30C23×D101—21no—
31C23×D101—21no—
32C23×D101—212no—
33C23×D101—48no—
34C23×D101D411yescharacteristic
35#13108075—224no—
36#13108075C23116yescomplemented
37C2×D404—212no—
38C2×D404—23no—
39C2×D404—23no—
40C2×D404C2316yes—
41C2×D404C2316yescomplemented
42C2×C22⋊D101—212no—
43D4×D101—224no—
44D4×D101C23116yescomplemented
45C22×Dic101D411yescharacteristic
46C2×C4×D101—212no—
47C2×C4×D101D416yes—
48C2×C4×D101C2316yes—
49#13108061C23116yescomplemented
50D808C23116yescomplemented
51#13108059C23116yescomplemented
52#13108057C23116yescomplemented
53C2×#11717579C2316yescomplemented
54C24×C101—21no—
55C2×Q8×C101—23no—
56C2×Q8×C101C2316yescomplemented
57C2×D4×C101—23no—
58C2×D4×C101C2316yes—
59C22×C4×C101—21no—
60C22×C4×C101C2311yescharacteristic
61SD16×C101C23116yescomplemented
62C2×C8×C101C2316yescomplemented
63C23×C101—26no—
64C23×C101—44no—
65C23×C101C2×D411yescharacteristic
66C22×D101—24no—
67C22×D101—26no—
68C22×D101—26no—
69C22×D101—216no—
70C22×D101—448no—
71C22×D101—44no—
72C22×D101—44no—
73C22×D101C2×D416yes—
74Q8×C101—26no—
75Q8×C101C2414yescomplemented
76D4×C101—26no—
77D4×C101C2414yes—
78C2×C4×C101—26no—
79C2×C4×C101C2×D413yes—
80C2×C4×C101C2413yes—
81C22⋊D101—216no—
82C2×Dic101C2×D416yes—
83D404—216no—
84D404—26no—
85D404—26no—
86D404C2414yes—
87D404C2414yescomplemented
88C4×D101—216no—
89C4×D101C2×D4112yes—
90C4×D101C2414yes—
91C8×C101C2414yescomplemented
92#11717579C2414yescomplemented
93C22×C101—24no—
94C22×C101—412no—
95C22×C101AGL(1,ℤ/8)14yescomplemented
96C22×C101C22×D413yes—
97C2×D101—212no—
98C2×D101—24no—
99C2×D101—24no—
100C2×D101—432no—
101C2×D101—412no—
102C2×D101—412no—
103C2×D101C22×D414yes—
104C4×C101—24no—
105C4×C101C22×D413yes—
106C4×C101C2511yescharacteristic
107Dic101C22×D414yes—
108C2×C101—44no—
109C2×C101C2×AGL(1,ℤ/8)16yescomplemented
110C2×C101C23×D411yescharacteristic
111D101—24no—
112D101—44no—
113D101—44no—
114C22×AGL(1,ℤ/8)—1011nomaximal, Sylow
115C101C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
116C22×#41—1011no—
117C23×D4—1011no—
118C2×AGL(1,ℤ/8)—10124no—
119C22×SD16—1011no—
120C22×SD16D10111yescharacteristic, complemented
121C22×D8—1011no—
122C22×D8—1011no—
123C22×M16—1011no—
124C25—2021no—
125C2×#41—10112no—
126C2×#41—2026no—
127C22×Q8C2×D10111yescharacteristic, complemented
128C22×D4—10112no—
129C22×D4—1011no—
130C22×D4—1011no—
131C22×D4—2021no—
132C22×D4—2026no—
133C22×D4—2021no—
134C22×D4C2×D10111yescharacteristic
135C23×C4—1011no—
136C23×C4—2021no—
137AGL(1,ℤ/8)—10164no—
138C2×SD16—10112no—
139C2×SD16C2×D101112yescomplemented
140C2×D8—10112no—
141C2×D8—10112no—
142C2×M16—10112no—
143C22×C8—1011no—
144C22×C8C2×D10111yescharacteristic, complemented
145C24—21no—
146C24—1011no—
147C24—4048no—
148C24—20212no—
149C24—2021no—
150C24—2021no—
151#41—10116no—
152#41—20224no—
153C2×Q8—23no—
154C2×Q8C22×D10116yescomplemented
155C2×D4—23no—
156C2×D4—10116no—
157C2×D4—1016no—
158C2×D4—1016no—
159C2×D4—20212no—
160C2×D4—20224no—
161C2×D4—20212no—
162C2×D4—2023no—
163C2×D4—2023no—
164C2×D4C22×D10116yes—
165C22×C4—21no—
166C22×C4—1011no—
167C22×C4—1016no—
168C22×C4—1016no—
169C22×C4—20212no—
170C22×C4C22×D10111yescharacteristic
171SD16—10116no—
172SD16C22×D101116yescomplemented
173D8—10116no—
174D8—10116no—
175M16—10116no—
176C2×C8—1016no—
177C2×C8C22×D10116yescomplemented
178C23—26no—
179C23—44no—
180C23—1016no—
181C23—40448no—
182C23—4044no—
183C23—4044no—
184C23—20216no—
185C23—2026no—
186C23—2026no—
187C23—2024no—
188C23D4×D10111yescharacteristic
189Q8—26no—
190Q8C23×D10114yescomplemented
191D4—26no—
192D4—1014no—
193D4—1014no—
194D4—20216no—
195D4—20216no—
196D4—2026no—
197D4—2026no—
198D4C23×D10114yes—
199C2×C4—26no—
200C2×C4—1016no—
201C2×C4—10112no—
202C2×C4—1014no—
203C2×C4—20216no—
204C2×C4D4×D10113yes—
205C2×C4C23×D10113yes—
206C8—1014no—
207C8C23×D10114yescomplemented
208C22—24no—
209C22—412no—
210C22—1014no—
211C22—40432no—
212C22—40412no—
213C22—40412no—
214C22—2024no—
215C22—2024no—
216C22—20212no—
217C22#1569228214yesdirect product factor
218C22C2×D4×D10113yes—
219C4—24no—
220C4—1014no—
221C4C2×D4×D10113yes—
222C4C24×D10111yescharacteristic
223C2—44no—
224C2—4044no—
225C2—4044no—
226C2—2024no—
227C2C2×#1569228216yesdirect product factor
228C2C22×D4×D10111yescharacteristic
subgroupG/H
centerC23D4×D101
commutator, cosocleC4×C101C25
FrattiniC4C24×D101
FittingC22×SD16×C101C2
socleC23×C101C2×D4
hypercenterC22×SD16D101
hypocenterC101C22×AGL(1,ℤ/8)
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#13108075C2semidirect
2. C22×D4×D101C2semidirect
3. C2×#15692282C2direct (N × H)
4. C22×#13108061C2semidirect
5. C22×D808C2semidirect
6. C22×#13108059C2semidirect
7. C22×#13108057C2semidirect
8. C22×SD16×C101C2semidirect
11. C2×#13108075C22semidirect
14. C22×D404C22semidirect
17. C2×D4×D101C22semidirect
20. #15692282C22direct (N × H)
21. C2×#13108061C22semidirect
22. C2×D808C22semidirect
23. C2×#13108059C22semidirect
24. C2×#13108057C22semidirect
25. C22×#11717579C22semidirect
26. C22×Q8×C101C22semidirect
28. C2×SD16×C101C22semidirect
29. C22×C8×C101C22semidirect
36. #13108075C23semidirect
41. C2×D404C23semidirect
44. D4×D101C23semidirect
49. #13108061C23semidirect
50. D808C23semidirect
51. #13108059C23semidirect
52. #13108057C23semidirect
53. C2×#11717579C23semidirect
56. C2×Q8×C101C23semidirect
61. SD16×C101C23semidirect
62. C2×C8×C101C23semidirect
75. Q8×C101C24semidirect
87. D404C24semidirect
91. C8×C101C24semidirect
92. #11717579C24semidirect
95. C22×C101AGL(1,ℤ/8)semidirect
109. C2×C101C2×AGL(1,ℤ/8)semidirect
115. C101C22×AGL(1,ℤ/8)semidirect
120. C22×SD16D101semidirect
127. C22×Q8C2×D101semidirect
139. C2×SD16C2×D101semidirect
144. C22×C8C2×D101semidirect
154. C2×Q8C22×D101semidirect
172. SD16C22×D101semidirect
177. C2×C8C22×D101semidirect
190. Q8C23×D101semidirect
207. C8C23×D101semidirect
217. C22#15692282direct (N × H)
227. C2C2×#15692282direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131080752
C22×D4×D1012
C2×#156922822
C22×#131080612
C22×D8082
C22×#131080592
C22×#131080572
C22×SD16×C1012
C22×AGL(1,ℤ/8)101