Finite Group Databasebeta

← #39909508

Subgroups — #39909508

Order 26176.

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

normal subgroups only

This group has 104 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#27189747C211yesmaximal, characteristic, complemented
2C2×#20266976C211yesmaximal, characteristic, complemented
3#27189724C211yesmaximal, characteristic, complemented
4#27189652C211yesmaximal, characteristic, complemented
5#27189635C211yesmaximal, characteristic, complemented
6#27189634C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C409C211yesmaximal, characteristic
8C2×Dic818C2211yescharacteristic
9#20266978—21no—
10#20266976—21no—
11#20266976—21no—
12#20266976—21no—
13#20266972—21no—
14#20266972C2211yescharacteristic
15#20266972C2211yescharacteristic
16C2×#15692543—21no—
17C2×#15692543C2211yescharacteristic
18#41×C409C2211yescharacteristic
19C2×D4×C409C2211yescharacteristic
20SD16×C409—21no—
21D8×C409—21no—
22M16×C409C2211yescharacteristic
23Dic818—21no—
24Dic818—21no—
25C2×Dic409—21no—
26#15692543—21no—
27#15692543—21no—
28#15692543D411yescharacteristic
29#15692543D411yescharacteristic
30C23×C409—21no—
31Q8×C409D411yescharacteristic
32D4×C409—21no—
33D4×C409—21no—
34D4×C409—21no—
35D4×C409D411yescharacteristic
36C2×C4×C409—21no—
37C2×C4×C409C2311yescharacteristic
38C8×C409—21no—
39C22×C409—21no—
40C22×C409—41no—
41C22×C409—81no—
42C22×C409#4111yescharacteristic
43C4×C409—21no—
44C4×C409C2×D411yescharacteristic
45C4×C409C2×D411yescharacteristic
46Dic409—41no—
47C2×C409—21no—
48C2×C409—41no—
49C2×C409—81no—
50C2×C409#12111yescharacteristic
51C409#47111yescharacteristic, complemented, Sylow
52#471—4091nomaximal, Sylow
53#137—4091no—
54AGL(1,ℤ/8)D40911yescharacteristic
55C2×SD16—4091no—
56#131—4091no—
57#108—4091no—
58#101—4091no—
59#100—4091no—
60#41C2×D40911yescharacteristic
61C2×Q8—4091no—
62C2×D4C2×D40911yescharacteristic
63Q16—8181no—
64SD16—21no—
65SD16—8181no—
66SD16—8181no—
67SD16—8181no—
68D8—21no—
69M16—8181no—
70M16—4091no—
71M16—4091no—
72M16C2×D40911yescharacteristic
73C2×C8—8181no—
74C2×C8—4091no—
75C23—21no—
76Q8—8181no—
77Q8—8181no—
78Q8C22⋊D40911yescharacteristic
79D4—21no—
80D4—21no—
81D4—21no—
82D4C22⋊D40911yescharacteristic
83C2×C4—21no—
84C2×C4—8181no—
85C2×C4C22×D40911yescharacteristic
86C8—21no—
87C8—8181no—
88C8—8181no—
89C8—4091no—
90C8—4091no—
91C22—21no—
92C22—41no—
93C22—81no—
94C22#2026698711yescharacteristic
95C4—21no—
96C4—16361no—
97C4D4×D40911yescharacteristic
98C4C2×C22⋊D40911yescharacteristic
99C2—21no—
100C2—41no—
101C2—81no—
102C2#2718968911yescharacteristic
subgroupG/H
centerC2#27189689
commutator, cosocleC2×C4×C409C23
FrattiniC2×C4C22×D409
Fitting, prime radicalAGL(1,ℤ/8)×C409C2
socleC2×C409#121
hypercenterAGL(1,ℤ/8)D409
hypocenterC409#471
LCSG ▷ C2×C4×C409 ▷ C2×C409 ▷ C409
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #27189747C2semidirect
2. C2×#20266976C2semidirect
3. #27189724C2semidirect
4. #27189652C2semidirect
5. #27189635C2semidirect
6. #27189634C2semidirect
51. C409#471semidirect

Maximal subgroups

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

subgroupindex
#271897472
C2×#202669762
#271897242
#271896522
#271896352
#271896342
AGL(1,ℤ/8)×C4092
#471409