Finite Group Databasebeta

← C22×#15692283 #27185867

Subgroups — C22×#15692283 #27185867

Order 12928.

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×#13108074C211yesmaximal, characteristic, complemented
2C23×D404C211yesmaximal, characteristic, complemented
3C2×#15692283C2124yesmaximal, direct product factor
4C22×#13108063C212yesmaximal, complemented
5C22×D808C212yesmaximal, complemented
6C22×M16×C101C211yesmaximal, characteristic, complemented
7C24×D101—21no—
8C2×#13108074—26no—
9C2×#13108074C22112yescomplemented
10C22×Dic202C2211yescharacteristic, complemented
11C22×D404—21no—
12C22×D404—26no—
13C22×D404C2212yes—
14C22×D404C22112yescomplemented
15C22×D404C2211yescharacteristic, complemented
16C22×C22⋊D101—21no—
17C22×C4×D101—21no—
18#15692283C22164yesdirect product factor
19C2×#13108063C22124yescomplemented
20C2×D808C22124yescomplemented
21C23×C4×C101C2211yescharacteristic
22C2×M16×C101C22112yescomplemented
23C22×C8×C101C2212yescomplemented
24C23×D101—21no—
25C23×D101—22no—
26C23×D101—212no—
27C23×D101—48no—
28#13108074—224no—
29#13108074C23116yescomplemented
30C2×Dic202—23no—
31C2×Dic202C2316yescomplemented
32C2×D404—26no—
33C2×D404—224no—
34C2×D404—212no—
35C2×D404—23no—
36C2×D404C23112yes—
37C2×D404C23116yescomplemented
38C2×D404C2316yescomplemented
39C2×C22⋊D101—212no—
40C22×Dic101—21no—
41C2×C4×D101—212no—
42#13108063C23132yescomplemented
43D808C23132yescomplemented
44C24×C101D411yescharacteristic
45C22×C4×C101D411yescharacteristic
46C22×C4×C101D416yes—
47C22×C4×C101C2311yescharacteristic
48C22×C4×C101C2316yes—
49M16×C101C23116yescomplemented
50C2×C8×C101C23112yescomplemented
51C23×C101—24no—
52C23×C101C2×D416yes—
53C23×C101C2×D411yescharacteristic
54C22×D101—26no—
55C22×D101—212no—
56C22×D101—216no—
57C22×D101—448no—
58C22×D101—44no—
59C22×D101—48no—
60C2×C4×C101C2×D416yes—
61C2×C4×C101C2×D413yes—
62C2×C4×C101C2×D4112yes—
63C2×C4×C101C2413yes—
64C2×C4×C101C2414yes—
65C22⋊D101—216no—
66C2×Dic101—26no—
67D404—212no—
68D404—216no—
69D404—26no—
70D404C2418yes—
71D404C2414yescomplemented
72C4×D101—216no—
73Dic202—26no—
74Dic202C2414yescomplemented
75C8×C101C2418yescomplemented
76C22×C101—212no—
77C22×C101AGL(1,ℤ/8)14yescomplemented
78C22×C101C22×D414yes—
79C22×C101C22×D413yes—
80C2×D101—24no—
81C2×D101—28no—
82C2×D101—432no—
83C2×D101—412no—
84C2×D101—424no—
85C4×C101C22×D413yes—
86C4×C101C22×D414yes—
87C4×C101C2511yescharacteristic
88Dic101—24no—
89C2×C101—24no—
90C2×C101C2×AGL(1,ℤ/8)16yescomplemented
91C2×C101C23×D411yescharacteristic
92D101—44no—
93D101—48no—
94C22×AGL(1,ℤ/8)—1011nomaximal, Sylow
95C101C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
96C22×#41—1011no—
97C23×D4—1011no—
98C2×AGL(1,ℤ/8)—10124no—
99C22×SD16—1012no—
100C22×D8—1012no—
101C22×M16D10111yescharacteristic, complemented
102C25—2021no—
103C2×#41—10112no—
104C2×#41—2026no—
105C22×Q8—1011no—
106C22×D4—1012no—
107C22×D4—1011no—
108C22×D4—10112no—
109C22×D4—2021no—
110C22×D4—2026no—
111C22×D4—2021no—
112C23×C4—2021no—
113C23×C4C2×D10111yescharacteristic
114AGL(1,ℤ/8)—10164no—
115C2×SD16—10124no—
116C2×D8—10124no—
117C2×M16C2×D101112yescomplemented
118C22×C8C2×D10112yescomplemented
119C24—4048no—
120C24—20212no—
121C24—2021no—
122C24—2022no—
123C24D40411yescharacteristic
124#41—10116no—
125#41—20224no—
126C2×Q8—1016no—
127C2×Q8—2023no—
128C2×D4—10112no—
129C2×D4—1016no—
130C2×D4—10116no—
131C2×D4—20212no—
132C2×D4—20212no—
133C2×D4—2026no—
134C2×D4—20224no—
135C2×D4—2023no—
136C22×C4—2021no—
137C22×C4—20212no—
138C22×C4D40416yes—
139C22×C4D40411yescharacteristic
140C22×C4C22×D10116yes—
141C22×C4C22×D10111yescharacteristic
142SD16—10132no—
143D8—10132no—
144M16C22×D101116yescomplemented
145C2×C8C22×D101112yescomplemented
146C23—24no—
147C23—40448no—
148C23—4048no—
149C23—4044no—
150C23—2026no—
151C23—20216no—
152C23—20212no—
153C23C2×D40416yes—
154C23C2×D40411yescharacteristic
155Q8—1014no—
156Q8—2026no—
157D4—1014no—
158D4—1018no—
159D4—20216no—
160D4—20216no—
161D4—20212no—
162D4—2026no—
163C2×C4—2026no—
164C2×C4—20216no—
165C2×C4C2×D40416yes—
166C2×C4C2×D40413yes—
167C2×C4C2×D404112yes—
168C2×C4C23×D10113yes—
169C2×C4C23×D10114yes—
170C8C23×D10118yescomplemented
171C22—212no—
172C22—40432no—
173C22—40424no—
174C22—40412no—
175C22—2024no—
176C22—2028no—
177C22#1569228314yesdirect product factor
178C22C22×D40414yes—
179C22C22×D40413yes—
180C4—2024no—
181C4C22×D40414yes—
182C4C22×D40413yes—
183C4C24×D10111yescharacteristic
184C2—24no—
185C2—4048no—
186C2—4044no—
187C2C2×#1569228316yesdirect product factor
188C2C23×D40411yescharacteristic
subgroupG/H
centerC23C2×D404
commutator, cosocleC4×C101C25
FrattiniC4C24×D101
FittingC22×M16×C101C2
socleC23×C101C2×D4
hypercenterC22×M16D101
hypocenterC101C22×AGL(1,ℤ/8)
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C22×#13108074C2semidirect
2. C23×D404C2semidirect
3. C2×#15692283C2direct (N × H)
4. C22×#13108063C2semidirect
5. C22×D808C2semidirect
6. C22×M16×C101C2semidirect
9. C2×#13108074C22semidirect
10. C22×Dic202C22semidirect
14. C22×D404C22semidirect
15. C22×D404C22semidirect
18. #15692283C22direct (N × H)
19. C2×#13108063C22semidirect
20. C2×D808C22semidirect
22. C2×M16×C101C22semidirect
23. C22×C8×C101C22semidirect
29. #13108074C23semidirect
31. C2×Dic202C23semidirect
37. C2×D404C23semidirect
38. C2×D404C23semidirect
42. #13108063C23semidirect
43. D808C23semidirect
49. M16×C101C23semidirect
50. C2×C8×C101C23semidirect
71. D404C24semidirect
74. Dic202C24semidirect
75. C8×C101C24semidirect
77. C22×C101AGL(1,ℤ/8)semidirect
90. C2×C101C2×AGL(1,ℤ/8)semidirect
95. C101C22×AGL(1,ℤ/8)semidirect
101. C22×M16D101semidirect
117. C2×M16C2×D101semidirect
118. C22×C8C2×D101semidirect
144. M16C22×D101semidirect
145. C2×C8C22×D101semidirect
170. C8C23×D101semidirect
177. C22#15692283direct (N × H)
187. C2C2×#15692283direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131080742
C23×D4042
C2×#156922832
C22×#131080632
C22×D8082
C22×M16×C1012
C22×AGL(1,ℤ/8)101