Finite Group Databasebeta

← #37526711

Subgroups — #37526711

Order 22976.

108 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 108 conjugacy classes of subgroups — too many to draw. Only its 37 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#26040040C211yesmaximal, characteristic, complemented
2C2×#19153326C211yesmaximal, characteristic, complemented
3#26040022C211yesmaximal, characteristic, complemented
4#26039950C211yesmaximal, characteristic, complemented
5#26039932C211yesmaximal, characteristic, complemented
6#26039931C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C359C211yesmaximal, characteristic, complemented
8C2×D1436C2211yescharacteristic
9#19153328—21no—
10#19153326—21no—
11#19153326—21no—
12#19153326—21no—
13#19153325—21no—
14#19153325C2211yescharacteristic
15#19153325C2211yescharacteristic
16C2×C359⋊C8—21no—
17C2×C359⋊C8C2211yescharacteristic
18#41×C359C2211yescharacteristic
19C2×D4×C359C2211yescharacteristic
20SD16×C359—21no—
21D8×C359—21no—
22M16×C359C2211yescharacteristic
23C22×D359—21no—
24D1436—21no—
25D1436—21no—
26C359⋊C8—21no—
27C359⋊C8—21no—
28C359⋊C8D411yescharacteristic
29C359⋊C8D411yescharacteristic
30C23×C359—21no—
31Q8×C359D411yescharacteristic
32D4×C359—21no—
33D4×C359—21no—
34D4×C359—21no—
35D4×C359D411yescharacteristic
36C2×C4×C359—21no—
37C2×C4×C359C2311yescharacteristic
38C8×C359—21no—
39C22×C359—21no—
40C22×C359—41no—
41C22×C359—81no—
42C22×C359#4111yescharacteristic
43C2×D359—41no—
44C2×D359—81no—
45C4×C359—21no—
46C4×C359C2×D411yescharacteristic
47C4×C359C2×D411yescharacteristic
48C2×C359—21no—
49C2×C359—41no—
50C2×C359—81no—
51C2×C359#12111yescharacteristic
52D359—81no—
53C359#47211yescharacteristic, complemented, Sylow
54#472—3591nomaximal, Sylow
55AGL(1,ℤ/8)—3591no—
56AGL(1,ℤ/8)D35911yescharacteristic, complemented
57C2×D8—3591no—
58#131—3591no—
59#108—3591no—
60#100—3591no—
61#100—3591no—
62#41C2×D35911yescharacteristic
63C2×D4—3591no—
64C2×D4C2×D35911yescharacteristic
65SD16—21no—
66SD16—7181no—
67D8—21no—
68D8—7181no—
69D8—7181no—
70D8—7181no—
71M16—3591no—
72M16—3591no—
73M16—7181no—
74M16C2×D35911yescharacteristic
75C2×C8—3591no—
76C2×C8—7181no—
77C23—21no—
78C23—7181no—
79Q8C22⋊D35911yescharacteristic
80D4—21no—
81D4—21no—
82D4—21no—
83D4—7181no—
84D4—7181no—
85D4C22⋊D35911yescharacteristic
86C2×C4—21no—
87C2×C4C22×D35911yescharacteristic
88C8—21no—
89C8—3591no—
90C8—3591no—
91C8—7181no—
92C8—7181no—
93C22—21no—
94C22—41no—
95C22—81no—
96C22—28721no—
97C22—14361no—
98C22#1915334011yescharacteristic
99C4—21no—
100C4D4×D35911yescharacteristic
101C4C2×C22⋊D35911yescharacteristic
102C2—21no—
103C2—41no—
104C2—81no—
105C2—28721no—
106C2#2603998711yescharacteristic
subgroupG/H
centerC2#26039987
commutator, cosocleC2×C4×C359C23
FrattiniC2×C4C22×D359
FittingAGL(1,ℤ/8)×C359C2
socleC2×C359#121
hypercenterAGL(1,ℤ/8)D359
hypocenterC359#472
LCSG ▷ C2×C4×C359 ▷ C2×C359 ▷ C359
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #26040040C2semidirect
2. C2×#19153326C2semidirect
3. #26040022C2semidirect
4. #26039950C2semidirect
5. #26039932C2semidirect
6. #26039931C2semidirect
7. AGL(1,ℤ/8)×C359C2semidirect
53. C359#472semidirect
56. AGL(1,ℤ/8)D359semidirect

Maximal subgroups

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

subgroupindex
#260400402
C2×#191533262
#260400222
#260399502
#260399322
#260399312
AGL(1,ℤ/8)×C3592
#472359