Finite Group Databasebeta

← #22550144

Subgroups — #22550144

Order 8128.

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#16861234C211yesmaximal, characteristic, complemented
2C2×#14523233C211yesmaximal, characteristic, complemented
3#16861216C211yesmaximal, characteristic, complemented
4#16861144C211yesmaximal, characteristic, complemented
5#16861126C211yesmaximal, characteristic, complemented
6#16861125C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
8C2×D508C2211yescharacteristic
9#14523235—21no—
10#14523233—21no—
11#14523233—21no—
12#14523233—21no—
13#14523232—21no—
14#14523232C2211yescharacteristic
15#14523232C2211yescharacteristic
16C2×C127⋊C8—21no—
17C2×C127⋊C8C2211yescharacteristic
18#41×C127C2211yescharacteristic
19C2×D4×C127C2211yescharacteristic
20SD16×C127—21no—
21D8×C127—21no—
22M16×C127C2211yescharacteristic
23C22×D127—21no—
24D508—21no—
25D508—21no—
26C127⋊C8—21no—
27C127⋊C8—21no—
28C127⋊C8D411yescharacteristic
29C127⋊C8D411yescharacteristic
30C23×C127—21no—
31Q8×C127D411yescharacteristic
32D4×C127—21no—
33D4×C127—21no—
34D4×C127—21no—
35D4×C127D411yescharacteristic
36C2×C4×C127—21no—
37C2×C4×C127C2311yescharacteristic
38C8×C127—21no—
39C22×C127—21no—
40C22×C127—41no—
41C22×C127—81no—
42C22×C127#4111yescharacteristic
43C2×D127—41no—
44C2×D127—81no—
45C4×C127—21no—
46C4×C127C2×D411yescharacteristic
47C4×C127C2×D411yescharacteristic
48C2×C127—21no—
49C2×C127—41no—
50C2×C127—81no—
51C2×C127#12111yescharacteristic
52D127—81no—
53C127#47211yescharacteristic, complemented, Sylow
54#472—1271nomaximal, Sylow
55AGL(1,ℤ/8)—1271no—
56AGL(1,ℤ/8)D12711yescharacteristic, complemented
57C2×D8—1271no—
58#131—1271no—
59#108—1271no—
60#100—1271no—
61#100—1271no—
62#41C2×D12711yescharacteristic
63C2×D4—1271no—
64C2×D4C2×D12711yescharacteristic
65SD16—21no—
66SD16—2541no—
67D8—21no—
68D8—2541no—
69D8—2541no—
70D8—2541no—
71M16—1271no—
72M16—1271no—
73M16—2541no—
74M16C2×D12711yescharacteristic
75C2×C8—1271no—
76C2×C8—2541no—
77C23—21no—
78C23—2541no—
79Q8C22⋊D12711yescharacteristic
80D4—21no—
81D4—21no—
82D4—21no—
83D4—2541no—
84D4—2541no—
85D4C22⋊D12711yescharacteristic
86C2×C4—21no—
87C2×C4C22×D12711yescharacteristic
88C8—21no—
89C8—1271no—
90C8—1271no—
91C8—2541no—
92C8—2541no—
93C22—21no—
94C22—41no—
95C22—81no—
96C22—10161no—
97C22—5081no—
98C22#1452324711yescharacteristic
99C4—21no—
100C4D4×D12711yescharacteristic
101C4C2×C22⋊D12711yescharacteristic
102C2—21no—
103C2—41no—
104C2—81no—
105C2—10161no—
106C2#1686118111yescharacteristic
subgroupG/H
centerC2#16861181
commutatorC2×C4×C127C23
FrattiniC2×C4C22×D127
FittingAGL(1,ℤ/8)×C127C2
socleC2×C127#121
hypercenterAGL(1,ℤ/8)D127
hypocenterC127#472
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #16861234C2semidirect
2. C2×#14523233C2semidirect
3. #16861216C2semidirect
4. #16861144C2semidirect
5. #16861126C2semidirect
6. #16861125C2semidirect
7. AGL(1,ℤ/8)×C127C2semidirect
53. C127#472semidirect
56. AGL(1,ℤ/8)D127semidirect