Finite Group Databasebeta

← C22×#16813646 #28405305

Subgroups — C22×#16813646 #28405305

Order 14464.

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×#14268646C211yesmaximal, characteristic, complemented
2C22×D4×D113C211yesmaximal, characteristic, complemented
3C2×#16813646C2124yesmaximal, direct product factor
4C22×#14268636C211yesmaximal, characteristic, complemented
5C22×#14268635C211yesmaximal, characteristic, complemented
6C22×#14268632C211yesmaximal, characteristic, complemented
7C22×#14268630C211yesmaximal, characteristic, complemented
8C22×D8×C113C211yesmaximal, characteristic, complemented
9C24×D113—21no—
10C2×#14268646—26no—
11C2×#14268646C22112yescomplemented
12C22×Dic226C2211yescharacteristic, complemented
13C22×D452C2211yescharacteristic
14C22×C22⋊D113—21no—
15C22×C22⋊D113—21no—
16C2×D4×D113—26no—
17C2×D4×D113C22112yescomplemented
18C23×Dic113—21no—
19C22×C4×D113C2211yescharacteristic
20#16813646C22164yesdirect product factor
21C2×#14268636C22112yescomplemented
22C2×#14268635C22112yescomplemented
23C2×#14268632C22112yescomplemented
24C2×#14268630C22112yescomplemented
25C22×#11744751C2211yescharacteristic, complemented
26C22×D4×C113C2211yescharacteristic
27C22×D4×C113C2211yescharacteristic, complemented
28C2×D8×C113C22112yescomplemented
29C22×C8×C113C2211yescharacteristic, complemented
30C23×D113—21no—
31C23×D113—212no—
32C23×D113—48no—
33C23×D113D411yescharacteristic
34#14268646—224no—
35#14268646C23116yescomplemented
36C2×Dic226—23no—
37C2×Dic226C2316yescomplemented
38C2×D452—23no—
39C2×D452C2316yes—
40C2×C22⋊D113—212no—
41C2×C22⋊D113—212no—
42D4×D113—224no—
43D4×D113C23116yescomplemented
44C22×Dic113—21no—
45C22×Dic113—212no—
46C22×Dic113D411yescharacteristic
47C2×C4×D113D416yes—
48C2×C4×D113C2316yes—
49#14268636C23116yescomplemented
50#14268635C23116yescomplemented
51#14268632C23116yescomplemented
52#14268630C23116yescomplemented
53C2×#11744751C2316yescomplemented
54C24×C113—21no—
55C24×C113—21no—
56C2×D4×C113—23no—
57C2×D4×C113—23no—
58C2×D4×C113C2316yes—
59C2×D4×C113C2316yescomplemented
60C22×C4×C113C2311yescharacteristic
61D8×C113C23116yescomplemented
62C2×C8×C113C2316yescomplemented
63C23×C113—26no—
64C23×C113—26no—
65C23×C113—44no—
66C23×C113—44no—
67C23×C113C2×D411yescharacteristic
68C22×D113—24no—
69C22×D113—26no—
70C22×D113—216no—
71C22×D113—448no—
72C22×D113—44no—
73C22×D113C2×D416yes—
74D4×C113—26no—
75D4×C113—26no—
76D4×C113C2414yes—
77D4×C113C2414yescomplemented
78C2×C4×C113C2×D413yes—
79C2×C4×C113C2413yes—
80C22⋊D113—216no—
81C22⋊D113—216no—
82C2×Dic113—26no—
83C2×Dic113—216no—
84C2×Dic113C2×D416yes—
85D452—26no—
86D452C2414yes—
87C4×D113C2×D4112yes—
88C4×D113C2414yes—
89Dic226—26no—
90Dic226C2414yescomplemented
91C8×C113C2414yescomplemented
92#11744751C2414yescomplemented
93C22×C113—24no—
94C22×C113—24no—
95C22×C113—412no—
96C22×C113—412no—
97C22×C113AGL(1,ℤ/8)14yescomplemented
98C22×C113C22×D413yes—
99C2×D113—212no—
100C2×D113—24no—
101C2×D113—432no—
102C2×D113—412no—
103C2×D113C22×D414yes—
104C4×C113C22×D413yes—
105C4×C113C2511yescharacteristic
106Dic113—24no—
107Dic113C22×D414yes—
108C2×C113—44no—
109C2×C113—44no—
110C2×C113C2×AGL(1,ℤ/8)16yescomplemented
111C2×C113C23×D411yescharacteristic
112D113—24no—
113D113—44no—
114C22×AGL(1,ℤ/8)—1131nomaximal, Sylow
115C113C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
116C22×#41—1131no—
117C23×D4—1131no—
118C2×AGL(1,ℤ/8)—11324no—
119C22×SD16—1131no—
120C22×SD16—1131no—
121C22×D8—1131no—
122C22×D8D11311yescharacteristic, complemented
123C22×M16—1131no—
124C25—2261no—
125C2×#41—11312no—
126C2×#41—2266no—
127C22×Q8—1131no—
128C22×D4—11312no—
129C22×D4—1131no—
130C22×D4—2266no—
131C22×D4—2261no—
132C22×D4—2261no—
133C22×D4C2×D11311yescharacteristic
134C22×D4C2×D11311yescharacteristic, complemented
135C23×C4—1131no—
136C23×C4—2261no—
137AGL(1,ℤ/8)—11364no—
138C2×SD16—11312no—
139C2×SD16—11312no—
140C2×D8—11312no—
141C2×D8C2×D113112yescomplemented
142C2×M16—11312no—
143C22×C8—1131no—
144C22×C8C2×D11311yescharacteristic, complemented
145C24—21no—
146C24—21no—
147C24—1131no—
148C24—4528no—
149C24—22612no—
150C24—2261no—
151#41—11316no—
152#41—22624no—
153C2×Q8—1136no—
154C2×Q8—2263no—
155C2×D4—23no—
156C2×D4—23no—
157C2×D4—11316no—
158C2×D4—1136no—
159C2×D4—22624no—
160C2×D4—22612no—
161C2×D4—22612no—
162C2×D4—2263no—
163C2×D4C22×D11316yes—
164C2×D4C22×D11316yescomplemented
165C22×C4—1131no—
166C22×C4—1136no—
167C22×C4—1136no—
168C22×C4—22612no—
169C22×C4—2261no—
170C22×C4C22×D11311yescharacteristic
171SD16—11316no—
172SD16—11316no—
173D8—11316no—
174D8C22×D113116yescomplemented
175M16—11316no—
176C2×C8—1136no—
177C2×C8C22×D11316yescomplemented
178C23—26no—
179C23—26no—
180C23—44no—
181C23—44no—
182C23—1136no—
183C23—45248no—
184C23—4524no—
185C23—22616no—
186C23—2266no—
187C23—2264no—
188C23D4×D11311yescharacteristic
189Q8—1134no—
190Q8—2266no—
191D4—26no—
192D4—26no—
193D4—1134no—
194D4—22616no—
195D4—22616no—
196D4—2266no—
197D4C23×D11314yes—
198D4C23×D11314yescomplemented
199C2×C4—1136no—
200C2×C4—11312no—
201C2×C4—1134no—
202C2×C4—22616no—
203C2×C4—2266no—
204C2×C4D4×D11313yes—
205C2×C4C23×D11313yes—
206C8—1134no—
207C8C23×D11314yescomplemented
208C22—24no—
209C22—24no—
210C22—412no—
211C22—412no—
212C22—1134no—
213C22—45232no—
214C22—45212no—
215C22—2264no—
216C22—22612no—
217C22#1681364614yesdirect product factor
218C22C2×D4×D11313yes—
219C4—1134no—
220C4—2264no—
221C4C2×D4×D11313yes—
222C4C24×D11311yescharacteristic
223C2—44no—
224C2—44no—
225C2—4524no—
226C2—2264no—
227C2C2×#1681364616yesdirect product factor
228C2C22×D4×D11311yescharacteristic
subgroupG/H
centerC23D4×D113
commutator, cosocleC4×C113C25
FrattiniC4C24×D113
FittingC22×D8×C113C2
socleC23×C113C2×D4
hypercenterC22×D8D113
hypocenterC113C22×AGL(1,ℤ/8)
LCSG ▷ C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#14268646C2semidirect
2. C22×D4×D113C2semidirect
3. C2×#16813646C2direct (N × H)
4. C22×#14268636C2semidirect
5. C22×#14268635C2semidirect
6. C22×#14268632C2semidirect
7. C22×#14268630C2semidirect
8. C22×D8×C113C2semidirect
11. C2×#14268646C22semidirect
12. C22×Dic226C22semidirect
17. C2×D4×D113C22semidirect
20. #16813646C22direct (N × H)
21. C2×#14268636C22semidirect
22. C2×#14268635C22semidirect
23. C2×#14268632C22semidirect
24. C2×#14268630C22semidirect
25. C22×#11744751C22semidirect
27. C22×D4×C113C22semidirect
28. C2×D8×C113C22semidirect
29. C22×C8×C113C22semidirect
35. #14268646C23semidirect
37. C2×Dic226C23semidirect
43. D4×D113C23semidirect
49. #14268636C23semidirect
50. #14268635C23semidirect
51. #14268632C23semidirect
52. #14268630C23semidirect
53. C2×#11744751C23semidirect
59. C2×D4×C113C23semidirect
61. D8×C113C23semidirect
62. C2×C8×C113C23semidirect
77. D4×C113C24semidirect
90. Dic226C24semidirect
91. C8×C113C24semidirect
92. #11744751C24semidirect
97. C22×C113AGL(1,ℤ/8)semidirect
110. C2×C113C2×AGL(1,ℤ/8)semidirect
115. C113C22×AGL(1,ℤ/8)semidirect
122. C22×D8D113semidirect
134. C22×D4C2×D113semidirect
141. C2×D8C2×D113semidirect
144. C22×C8C2×D113semidirect
164. C2×D4C22×D113semidirect
174. D8C22×D113semidirect
177. C2×C8C22×D113semidirect
198. D4C23×D113semidirect
207. C8C23×D113semidirect
217. C22#16813646direct (N × H)
227. C2C2×#16813646direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#142686462
C22×D4×D1132
C2×#168136462
C22×#142686362
C22×#142686352
C22×#142686322
C22×#142686302
C22×D8×C1132
C22×AGL(1,ℤ/8)113