Finite Group Databasebeta

← C22×#16813647 #28405306

Subgroups — C22×#16813647 #28405306

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
1C23×D452C211yesmaximal, characteristic, complemented
2C2×#16813647C2124yesmaximal, direct product factor
3C22×#14268634C212yesmaximal, complemented
4C22×#14268632C212yesmaximal, complemented
5C22×#14268631C211yesmaximal, characteristic, complemented
6C22×#41×C113C211yesmaximal, characteristic, complemented
7C24×D113—21no—
8C22×D452—21no—
9C22×D452—26no—
10C22×D452C2212yes—
11C22×D452C22112yescomplemented
12#16813647C22164yesdirect product factor
13C2×#14268634C22124yescomplemented
14C2×#14268632C22124yescomplemented
15C2×#14268631C22112yescomplemented
16C22×#11744751C2212yescomplemented
17C2×#41×C113—26no—
18C2×#41×C113C22112yescomplemented
19C22×Q8×C113C2211yescharacteristic, complemented
20C22×D4×C113—21no—
21C22×D4×C113C2211yescharacteristic, complemented
22C23×C4×C113—21no—
23C23×C4×C113C2211yescharacteristic
24C23×D113—22no—
25C23×D113—212no—
26C23×D113—48no—
27C2×D452—26no—
28C2×D452—224no—
29C2×D452—212no—
30C2×D452C23112yes—
31C2×D452C23116yescomplemented
32#14268634C23132yescomplemented
33#14268632C23132yescomplemented
34#14268631C23116yescomplemented
35C2×#11744751C23112yescomplemented
36C24×C113—21no—
37C24×C113D411yescharacteristic
38#41×C113—224no—
39#41×C113C23116yescomplemented
40C2×Q8×C113—23no—
41C2×Q8×C113C2316yescomplemented
42C2×D4×C113—212no—
43C2×D4×C113—23no—
44C2×D4×C113C2316yescomplemented
45C22×C4×C113—21no—
46C22×C4×C113—212no—
47C22×C4×C113D411yescharacteristic
48C22×C4×C113D416yes—
49C22×C4×C113C2311yescharacteristic
50C22×C4×C113C2316yes—
51C23×C113—24no—
52C23×C113—26no—
53C23×C113—44no—
54C23×C113C2×D416yes—
55C23×C113C2×D411yescharacteristic
56C22×D113—212no—
57C22×D113—216no—
58C22×D113—448no—
59C22×D113—48no—
60Q8×C113—26no—
61Q8×C113C2414yescomplemented
62D4×C113—216no—
63D4×C113—26no—
64D4×C113C2414yescomplemented
65C2×C4×C113—26no—
66C2×C4×C113—216no—
67C2×C4×C113C2×D416yes—
68C2×C4×C113C2×D413yes—
69C2×C4×C113C2×D4112yes—
70C2×C4×C113C2413yes—
71C2×C4×C113C2414yes—
72D452—212no—
73D452—216no—
74D452C2418yes—
75#11744751C2418yescomplemented
76C22×C113—212no—
77C22×C113—24no—
78C22×C113—412no—
79C22×C113AGL(1,ℤ/8)14yescomplemented
80C22×C113C22×D414yes—
81C22×C113C22×D413yes—
82C2×D113—28no—
83C2×D113—432no—
84C2×D113—424no—
85C4×C113—24no—
86C4×C113C22×D414yes—
87C4×C113C22×D413yes—
88C4×C113C2511yescharacteristic
89C2×C113—24no—
90C2×C113—44no—
91C2×C113C2×AGL(1,ℤ/8)16yescomplemented
92C2×C113C23×D411yescharacteristic
93D113—48no—
94C22×AGL(1,ℤ/8)—1131nomaximal, Sylow
95C113C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
96C22×#41D11311yescharacteristic, complemented
97C23×D4—1131no—
98C2×AGL(1,ℤ/8)—11324no—
99C22×SD16—1132no—
100C22×D8—1132no—
101C22×M16—1131no—
102C25—2261no—
103C2×#41—26no—
104C2×#41C2×D113112yescomplemented
105C22×Q8C2×D11311yescharacteristic, complemented
106C22×D4—21no—
107C22×D4—1132no—
108C22×D4—11312no—
109C22×D4—2266no—
110C22×D4—2261no—
111C22×D4C2×D11311yescharacteristic, complemented
112C23×C4—21no—
113C23×C4C2×D11311yescharacteristic
114AGL(1,ℤ/8)—11364no—
115C2×SD16—11324no—
116C2×D8—11324no—
117C2×M16—11312no—
118C22×C8—1132no—
119C24—21no—
120C24—4528no—
121C24—22612no—
122C24—2262no—
123C24C22⋊D11311yescharacteristic
124#41—224no—
125#41C22×D113116yescomplemented
126C2×Q8—23no—
127C2×Q8C22×D11316yescomplemented
128C2×D4—212no—
129C2×D4—23no—
130C2×D4—11312no—
131C2×D4—11316no—
132C2×D4—22612no—
133C2×D4—2266no—
134C2×D4—22624no—
135C2×D4C22×D11316yescomplemented
136C22×C4—21no—
137C22×C4—212no—
138C22×C4C22⋊D11316yes—
139C22×C4C22⋊D11311yescharacteristic
140C22×C4C22×D11316yes—
141C22×C4C22×D11311yescharacteristic
142SD16—11332no—
143D8—11332no—
144M16—11316no—
145C2×C8—11312no—
146C23—24no—
147C23—26no—
148C23—44no—
149C23—45248no—
150C23—4528no—
151C23—22616no—
152C23—22612no—
153C23C2×C22⋊D11316yes—
154C23C2×C22⋊D11311yescharacteristic
155Q8—26no—
156Q8C23×D11314yescomplemented
157D4—216no—
158D4—26no—
159D4—1138no—
160D4—22616no—
161D4—22612no—
162D4C23×D11314yescomplemented
163C2×C4—26no—
164C2×C4—216no—
165C2×C4C2×C22⋊D11316yes—
166C2×C4C2×C22⋊D11313yes—
167C2×C4C2×C22⋊D113112yes—
168C2×C4C23×D11313yes—
169C2×C4C23×D11314yes—
170C8—1138no—
171C22—212no—
172C22—24no—
173C22—412no—
174C22—45232no—
175C22—45224no—
176C22—2268no—
177C22#1681364714yesdirect product factor
178C22C22×C22⋊D11314yes—
179C22C22×C22⋊D11313yes—
180C4—24no—
181C4C22×C22⋊D11314yes—
182C4C22×C22⋊D11313yes—
183C4C24×D11311yescharacteristic
184C2—24no—
185C2—44no—
186C2—4528no—
187C2C2×#1681364716yesdirect product factor
188C2C23×C22⋊D11311yescharacteristic
subgroupG/H
centerC23C2×C22⋊D113
commutator, cosocleC4×C113C25
FrattiniC4C24×D113
FittingC22×#41×C113C2
socleC23×C113C2×D4
hypercenterC22×#41D113
hypocenterC113C22×AGL(1,ℤ/8)
LCSG ▷ C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C23×D452C2semidirect
2. C2×#16813647C2direct (N × H)
3. C22×#14268634C2semidirect
4. C22×#14268632C2semidirect
5. C22×#14268631C2semidirect
6. C22×#41×C113C2semidirect
11. C22×D452C22semidirect
12. #16813647C22direct (N × H)
13. C2×#14268634C22semidirect
14. C2×#14268632C22semidirect
15. C2×#14268631C22semidirect
16. C22×#11744751C22semidirect
18. C2×#41×C113C22semidirect
19. C22×Q8×C113C22semidirect
21. C22×D4×C113C22semidirect
31. C2×D452C23semidirect
32. #14268634C23semidirect
33. #14268632C23semidirect
34. #14268631C23semidirect
35. C2×#11744751C23semidirect
39. #41×C113C23semidirect
41. C2×Q8×C113C23semidirect
44. C2×D4×C113C23semidirect
61. Q8×C113C24semidirect
64. D4×C113C24semidirect
75. #11744751C24semidirect
79. C22×C113AGL(1,ℤ/8)semidirect
91. C2×C113C2×AGL(1,ℤ/8)semidirect
95. C113C22×AGL(1,ℤ/8)semidirect
96. C22×#41D113semidirect
104. C2×#41C2×D113semidirect
105. C22×Q8C2×D113semidirect
111. C22×D4C2×D113semidirect
125. #41C22×D113semidirect
127. C2×Q8C22×D113semidirect
135. C2×D4C22×D113semidirect
156. Q8C23×D113semidirect
162. D4C23×D113semidirect
177. C22#16813647direct (N × H)
187. C2C2×#16813647direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D4522
C2×#168136472
C22×#142686342
C22×#142686322
C22×#142686312
C22×#41×C1132
C22×AGL(1,ℤ/8)113