Finite Group Databasebeta

← #34094318

Subgroups — #34094318

Order 20032.

168 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 39 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
1#23756543C211yesmaximal, characteristic, complemented
2#23756523C211yesmaximal, characteristic, complemented
3#23756493C211yesmaximal, characteristic, complemented
4#23756424C211yesmaximal, characteristic, complemented
5#23756423C211yesmaximal, characteristic, complemented
6#23756413C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C313C211yesmaximal, characteristic, complemented
8#19097562—21no—
9#19097561—21no—
10#19097561C2211yescharacteristic
11C2×D1252—21no—
12C2×D1252C2211yescharacteristic
13C2×C22⋊D313—22no—
14D4×D313—22no—
15D4×D313—21no—
16#19097549—21no—
17#19097546—21no—
18#19097545C2211yescharacteristic, complemented
19C4×Dic313C2211yescharacteristic, complemented
20#41×C313C2211yescharacteristic
21C2×D4×C313C2211yescharacteristic, complemented
22SD16×C313—21no—
23D8×C313—21no—
24M16×C313C2211yescharacteristic, complemented
25C22×D313—22no—
26C22×D313D412yes—
27Dic626D411yescharacteristic, complemented
28D1252—22no—
29D1252—21no—
30D1252—21no—
31D1252D411yescharacteristic, complemented
32C22⋊D313—22no—
33C22⋊D313—21no—
34C22⋊D313—44no—
35C2×Dic313—21no—
36C4×D313—22no—
37C4×D313—21no—
38#14578809—21no—
39C23×C313—21no—
40Q8×C313D411yescharacteristic, complemented
41D4×C313—21no—
42D4×C313—21no—
43D4×C313—21no—
44D4×C313D411yescharacteristic, complemented
45C2×C4×C313—21no—
46C2×C4×C313C2311yescharacteristic
47C8×C313—21no—
48C22×C313—21no—
49C22×C313—42no—
50C22×C313—41no—
51C22×C313C2×D411yescharacteristic
52C2×D313—22no—
53C2×D313—21no—
54C2×D313—42no—
55C2×D313—82no—
56C4×C313—21no—
57C4×C313C2×D411yescharacteristic
58C4×C313C2×D411yescharacteristic
59Dic313—22no—
60Dic313—21no—
61C2×C313—21no—
62C2×C313—41no—
63C2×C313—81no—
64C2×C313#12011yescharacteristic
65D313—42no—
66D313—41no—
67C313#45311yescharacteristic, complemented, Sylow
68#453—3131nomaximal, Sylow
69H(5,𝔽2)—3131no—
70AGL(1,ℤ/8)—3131no—
71AGL(1,ℤ/8)D31311yescharacteristic, complemented
72#127—3131no—
73#104—3131no—
74#104—3131no—
75#100—3131no—
76#41—3131no—
77#41—6261no—
78#41—6261no—
79#41C2×D31311yescharacteristic
80C2×D4—3131no—
81C2×D4—6261no—
82C2×D4—6262no—
83C2×D4—6261no—
84C2×D4—6262no—
85C2×D4C2×D31311yescharacteristic, complemented
86SD16—21no—
87SD16—6261no—
88D8—21no—
89D8—6261no—
90M16—3131no—
91M16C2×D31311yescharacteristic, complemented
92C42—3131no—
93C23—21no—
94C23—3132no—
95C23—6262no—
96Q8—3131no—
97Q8C22⋊D31311yescharacteristic, complemented
98D4—21no—
99D4—21no—
100D4—21no—
101D4—3131no—
102D4—6262no—
103D4—6262no—
104D4—6261no—
105D4—6261no—
106D4—6261no—
107D4—12524no—
108D4C22⋊D31311yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—6262no—
111C2×C4—6261no—
112C2×C4—6261no—
113C2×C4C22×D31311yescharacteristic
114C8—21no—
115C8—6261no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—6261no—
120C22—6262no—
121C22—25042no—
122C22—12522no—
123C22D4×D31311yescharacteristic
124C4—21no—
125C4—6262no—
126C4—6261no—
127C4D4×D31311yescharacteristic
128C4C2×C22⋊D31311yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—12521no—
133C2—12522no—
134C2#2375646511yescharacteristic
subgroupG/H
centerC2#23756465
commutator, cosocleC2×C4×C313C23
FrattiniC2×C4C22×D313
FittingAGL(1,ℤ/8)×C313C2
socleC2×C313#120
hypercenterAGL(1,ℤ/8)D313
hypocenterC313#453
LCSG ▷ C2×C4×C313 ▷ C2×C313 ▷ C313
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #23756543C2semidirect
2. #23756523C2semidirect
3. #23756493C2semidirect
4. #23756424C2semidirect
5. #23756423C2semidirect
6. #23756413C2semidirect
7. AGL(1,ℤ/8)×C313C2semidirect
18. #19097545C22semidirect
19. C4×Dic313C22semidirect
21. C2×D4×C313C22semidirect
24. M16×C313C22semidirect
27. Dic626D4semidirect
31. D1252D4semidirect
40. Q8×C313D4semidirect
44. D4×C313D4semidirect
67. C313#453semidirect
71. AGL(1,ℤ/8)D313semidirect
85. C2×D4C2×D313semidirect
91. M16C2×D313semidirect
97. Q8C22⋊D313semidirect
108. D4C22⋊D313semidirect

Maximal subgroups

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

subgroupindex
#237565432
#237565232
#237564932
#237564242
#237564232
#237564132
AGL(1,ℤ/8)×C3132
#453313