Finite Group Databasebeta

← #20298764

Subgroups — #20298764

Order 7232.

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#16813668C211yesmaximal, characteristic, complemented
2#16813648C211yesmaximal, characteristic, complemented
3#16813618C211yesmaximal, characteristic, complemented
4#16813549C211yesmaximal, characteristic, complemented
5#16813548C211yesmaximal, characteristic, complemented
6#16813538C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C113C211yesmaximal, characteristic, complemented
8#14268648—21no—
9#14268647—21no—
10#14268647C2211yescharacteristic
11C2×D452—21no—
12C2×D452C2211yescharacteristic
13C2×C22⋊D113—22no—
14D4×D113—22no—
15D4×D113—21no—
16#14268635—21no—
17#14268632—21no—
18#14268631C2211yescharacteristic, complemented
19C4×Dic113C2211yescharacteristic, complemented
20#41×C113C2211yescharacteristic
21C2×D4×C113C2211yescharacteristic, complemented
22SD16×C113—21no—
23D8×C113—21no—
24M16×C113C2211yescharacteristic, complemented
25C23×C113—21no—
26C22×D113—22no—
27C22×D113D412yes—
28Q8×C113D411yescharacteristic, complemented
29D4×C113—21no—
30D4×C113—21no—
31D4×C113—21no—
32D4×C113D411yescharacteristic, complemented
33C2×C4×C113—21no—
34C2×C4×C113C2311yescharacteristic
35C22⋊D113—22no—
36C22⋊D113—21no—
37C22⋊D113—44no—
38C2×Dic113—21no—
39D452—22no—
40D452—21no—
41D452—21no—
42D452D411yescharacteristic, complemented
43C4×D113—22no—
44C4×D113—21no—
45Dic226D411yescharacteristic, complemented
46C8×C113—21no—
47#11744751—21no—
48C22×C113—21no—
49C22×C113—42no—
50C22×C113—41no—
51C22×C113C2×D411yescharacteristic
52C2×D113—22no—
53C2×D113—21no—
54C2×D113—42no—
55C2×D113—82no—
56C4×C113—21no—
57C4×C113C2×D411yescharacteristic
58C4×C113C2×D411yescharacteristic
59Dic113—22no—
60Dic113—21no—
61C2×C113—21no—
62C2×C113—41no—
63C2×C113—81no—
64C2×C113#12011yescharacteristic
65D113—42no—
66D113—41no—
67C113#45311yescharacteristic, complemented, Sylow
68#453—1131nomaximal, Sylow
69H(5,𝔽2)—1131no—
70AGL(1,ℤ/8)—1131no—
71AGL(1,ℤ/8)D11311yescharacteristic, complemented
72#127—1131no—
73#104—1131no—
74#104—1131no—
75#100—1131no—
76#41—1131no—
77#41—2261no—
78#41—2261no—
79#41C2×D11311yescharacteristic
80C2×D4—1131no—
81C2×D4—2261no—
82C2×D4—2262no—
83C2×D4—2262no—
84C2×D4—2261no—
85C2×D4C2×D11311yescharacteristic, complemented
86SD16—21no—
87SD16—2261no—
88D8—21no—
89D8—2261no—
90M16—1131no—
91M16C2×D11311yescharacteristic, complemented
92C42—1131no—
93C23—21no—
94C23—1132no—
95C23—2262no—
96Q8—1131no—
97Q8C22⋊D11311yescharacteristic, complemented
98D4—21no—
99D4—21no—
100D4—21no—
101D4—1131no—
102D4—4524no—
103D4—2262no—
104D4—2262no—
105D4—2261no—
106D4—2261no—
107D4—2261no—
108D4C22⋊D11311yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—2262no—
111C2×C4—2261no—
112C2×C4—2261no—
113C2×C4C22×D11311yescharacteristic
114C8—21no—
115C8—2261no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—9042no—
120C22—4522no—
121C22—2261no—
122C22—2262no—
123C22D4×D11311yescharacteristic
124C4—21no—
125C4—2262no—
126C4—2261no—
127C4D4×D11311yescharacteristic
128C4C2×C22⋊D11311yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—4521no—
133C2—4522no—
134C2#1681359011yescharacteristic
subgroupG/H
centerC2#16813590
commutatorC2×C4×C113C23
FrattiniC2×C4C22×D113
FittingAGL(1,ℤ/8)×C113C2
socleC2×C113#120
hypercenterAGL(1,ℤ/8)D113
hypocenterC113#453
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #16813668C2semidirect
2. #16813648C2semidirect
3. #16813618C2semidirect
4. #16813549C2semidirect
5. #16813548C2semidirect
6. #16813538C2semidirect
7. AGL(1,ℤ/8)×C113C2semidirect
18. #14268631C22semidirect
19. C4×Dic113C22semidirect
21. C2×D4×C113C22semidirect
24. M16×C113C22semidirect
28. Q8×C113D4semidirect
32. D4×C113D4semidirect
42. D452D4semidirect
45. Dic226D4semidirect
67. C113#453semidirect
71. AGL(1,ℤ/8)D113semidirect
85. C2×D4C2×D113semidirect
91. M16C2×D113semidirect
97. Q8C22⋊D113semidirect
108. D4C22⋊D113semidirect