Finite Group Databasebeta

← C22×#16813650 #28405309

Subgroups — C22×#16813650 #28405309

Order 14464.

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×#14268647C211yesmaximal, characteristic, complemented
2C23×D452C211yesmaximal, characteristic, complemented
3C2×#16813650C2124yesmaximal, direct product factor
4C22×#14268636C212yesmaximal, complemented
5C22×D904C212yesmaximal, complemented
6C22×M16×C113C211yesmaximal, characteristic, complemented
7C24×D113—21no—
8C2×#14268647—26no—
9C2×#14268647C22112yescomplemented
10C22×Dic226C2211yescharacteristic, complemented
11C22×D452—21no—
12C22×D452—26no—
13C22×D452C2212yes—
14C22×D452C22112yescomplemented
15C22×D452C2211yescharacteristic, complemented
16C22×C22⋊D113—21no—
17C22×C4×D113—21no—
18#16813650C22164yesdirect product factor
19C2×#14268636C22124yescomplemented
20C2×D904C22124yescomplemented
21C23×C4×C113C2211yescharacteristic
22C2×M16×C113C22112yescomplemented
23C22×C8×C113C2212yescomplemented
24C23×D113—21no—
25C23×D113—22no—
26C23×D113—212no—
27C23×D113—48no—
28#14268647—224no—
29#14268647C23116yescomplemented
30C2×Dic226—23no—
31C2×Dic226C2316yescomplemented
32C2×D452—26no—
33C2×D452—224no—
34C2×D452—212no—
35C2×D452—23no—
36C2×D452C23112yes—
37C2×D452C23116yescomplemented
38C2×D452C2316yescomplemented
39C2×C22⋊D113—212no—
40C22×Dic113—21no—
41C2×C4×D113—212no—
42#14268636C23132yescomplemented
43D904C23132yescomplemented
44C24×C113D411yescharacteristic
45C22×C4×C113D411yescharacteristic
46C22×C4×C113D416yes—
47C22×C4×C113C2311yescharacteristic
48C22×C4×C113C2316yes—
49M16×C113C23116yescomplemented
50C2×C8×C113C23112yescomplemented
51C23×C113—24no—
52C23×C113C2×D416yes—
53C23×C113C2×D411yescharacteristic
54C22×D113—26no—
55C22×D113—212no—
56C22×D113—216no—
57C22×D113—448no—
58C22×D113—44no—
59C22×D113—48no—
60C2×C4×C113C2×D416yes—
61C2×C4×C113C2×D413yes—
62C2×C4×C113C2×D4112yes—
63C2×C4×C113C2413yes—
64C2×C4×C113C2414yes—
65C22⋊D113—216no—
66C2×Dic113—26no—
67D452—212no—
68D452—216no—
69D452—26no—
70D452C2418yes—
71D452C2414yescomplemented
72C4×D113—216no—
73Dic226—26no—
74Dic226C2414yescomplemented
75C8×C113C2418yescomplemented
76C22×C113—212no—
77C22×C113AGL(1,ℤ/8)14yescomplemented
78C22×C113C22×D414yes—
79C22×C113C22×D413yes—
80C2×D113—24no—
81C2×D113—28no—
82C2×D113—432no—
83C2×D113—412no—
84C2×D113—424no—
85C4×C113C22×D413yes—
86C4×C113C22×D414yes—
87C4×C113C2511yescharacteristic
88Dic113—24no—
89C2×C113—24no—
90C2×C113C2×AGL(1,ℤ/8)16yescomplemented
91C2×C113C23×D411yescharacteristic
92D113—44no—
93D113—48no—
94C22×AGL(1,ℤ/8)—1131nomaximal, Sylow
95C113C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
96C22×#41—1131no—
97C23×D4—1131no—
98C2×AGL(1,ℤ/8)—11324no—
99C22×SD16—1132no—
100C22×D8—1132no—
101C22×M16D11311yescharacteristic, complemented
102C25—2261no—
103C2×#41—11312no—
104C2×#41—2266no—
105C22×Q8—1131no—
106C22×D4—1131no—
107C22×D4—1132no—
108C22×D4—11312no—
109C22×D4—2261no—
110C22×D4—2266no—
111C22×D4—2261no—
112C23×C4—2261no—
113C23×C4C2×D11311yescharacteristic
114AGL(1,ℤ/8)—11364no—
115C2×SD16—11324no—
116C2×D8—11324no—
117C2×M16C2×D113112yescomplemented
118C22×C8C2×D11312yescomplemented
119C24—4528no—
120C24—22612no—
121C24—2261no—
122C24—2262no—
123C24D45211yescharacteristic
124#41—11316no—
125#41—22624no—
126C2×Q8—1136no—
127C2×Q8—2263no—
128C2×D4—1136no—
129C2×D4—11312no—
130C2×D4—11316no—
131C2×D4—22612no—
132C2×D4—22612no—
133C2×D4—2266no—
134C2×D4—22624no—
135C2×D4—2263no—
136C22×C4—2261no—
137C22×C4—22612no—
138C22×C4D45216yes—
139C22×C4D45211yescharacteristic
140C22×C4C22×D11316yes—
141C22×C4C22×D11311yescharacteristic
142SD16—11332no—
143D8—11332no—
144M16C22×D113116yescomplemented
145C2×C8C22×D113112yescomplemented
146C23—24no—
147C23—45248no—
148C23—4528no—
149C23—4524no—
150C23—2266no—
151C23—22616no—
152C23—22612no—
153C23C2×D45216yes—
154C23C2×D45211yescharacteristic
155Q8—1134no—
156Q8—2266no—
157D4—1134no—
158D4—1138no—
159D4—22616no—
160D4—22616no—
161D4—22612no—
162D4—2266no—
163C2×C4—2266no—
164C2×C4—22616no—
165C2×C4C2×D45216yes—
166C2×C4C2×D45213yes—
167C2×C4C2×D452112yes—
168C2×C4C23×D11313yes—
169C2×C4C23×D11314yes—
170C8C23×D11318yescomplemented
171C22—212no—
172C22—45232no—
173C22—45224no—
174C22—45212no—
175C22—2264no—
176C22—2268no—
177C22#1681365014yesdirect product factor
178C22C22×D45214yes—
179C22C22×D45213yes—
180C4—2264no—
181C4C22×D45214yes—
182C4C22×D45213yes—
183C4C24×D11311yescharacteristic
184C2—24no—
185C2—4528no—
186C2—4524no—
187C2C2×#1681365016yesdirect product factor
188C2C23×D45211yescharacteristic
subgroupG/H
centerC23C2×D452
commutator, cosocleC4×C113C25
FrattiniC4C24×D113
FittingC22×M16×C113C2
socleC23×C113C2×D4
hypercenterC22×M16D113
hypocenterC113C22×AGL(1,ℤ/8)
LCSG ▷ C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C22×#14268647C2semidirect
2. C23×D452C2semidirect
3. C2×#16813650C2direct (N × H)
4. C22×#14268636C2semidirect
5. C22×D904C2semidirect
6. C22×M16×C113C2semidirect
9. C2×#14268647C22semidirect
10. C22×Dic226C22semidirect
14. C22×D452C22semidirect
15. C22×D452C22semidirect
18. #16813650C22direct (N × H)
19. C2×#14268636C22semidirect
20. C2×D904C22semidirect
22. C2×M16×C113C22semidirect
23. C22×C8×C113C22semidirect
29. #14268647C23semidirect
31. C2×Dic226C23semidirect
37. C2×D452C23semidirect
38. C2×D452C23semidirect
42. #14268636C23semidirect
43. D904C23semidirect
49. M16×C113C23semidirect
50. C2×C8×C113C23semidirect
71. D452C24semidirect
74. Dic226C24semidirect
75. C8×C113C24semidirect
77. C22×C113AGL(1,ℤ/8)semidirect
90. C2×C113C2×AGL(1,ℤ/8)semidirect
95. C113C22×AGL(1,ℤ/8)semidirect
101. C22×M16D113semidirect
117. C2×M16C2×D113semidirect
118. C22×C8C2×D113semidirect
144. M16C22×D113semidirect
145. C2×C8C22×D113semidirect
170. C8C23×D113semidirect
177. C22#16813650direct (N × H)
187. C2C2×#16813650direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#142686472
C23×D4522
C2×#168136502
C22×#142686362
C22×D9042
C22×M16×C1132
C22×AGL(1,ℤ/8)113