Finite Group Databasebeta

← #36374412

Subgroups — #36374412

Order 22336.

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#26013664C211yesmaximal, characteristic, complemented
2C2×#19150218C211yesmaximal, characteristic, complemented
3#26013646C211yesmaximal, characteristic, complemented
4#26013574C211yesmaximal, characteristic, complemented
5#26013556C211yesmaximal, characteristic, complemented
6#26013555C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C349C211yesmaximal, characteristic, complemented
8C2×D1396C2211yescharacteristic
9#19150220—21no—
10#19150218—21no—
11#19150218—21no—
12#19150218—21no—
13#19150217—21no—
14#19150217C2211yescharacteristic
15#19150217C2211yescharacteristic
16C2×#14584095—21no—
17C2×#14584095C2211yescharacteristic
18#41×C349C2211yescharacteristic
19C2×D4×C349C2211yescharacteristic
20SD16×C349—21no—
21D8×C349—21no—
22M16×C349C2211yescharacteristic
23C22×D349—21no—
24D1396—21no—
25D1396—21no—
26#14584095—21no—
27#14584095—21no—
28#14584095D411yescharacteristic
29#14584095D411yescharacteristic
30C23×C349—21no—
31Q8×C349D411yescharacteristic
32D4×C349—21no—
33D4×C349—21no—
34D4×C349—21no—
35D4×C349D411yescharacteristic
36C2×C4×C349—21no—
37C2×C4×C349C2311yescharacteristic
38C8×C349—21no—
39C22×C349—21no—
40C22×C349—41no—
41C22×C349—81no—
42C22×C349#4111yescharacteristic
43C2×D349—41no—
44C2×D349—81no—
45C4×C349—21no—
46C4×C349C2×D411yescharacteristic
47C4×C349C2×D411yescharacteristic
48C2×C349—21no—
49C2×C349—41no—
50C2×C349—81no—
51C2×C349#12111yescharacteristic
52D349—81no—
53C349#47211yescharacteristic, complemented, Sylow
54#472—3491nomaximal, Sylow
55AGL(1,ℤ/8)—3491no—
56AGL(1,ℤ/8)D34911yescharacteristic, complemented
57C2×D8—3491no—
58#131—3491no—
59#108—3491no—
60#100—3491no—
61#100—3491no—
62#41C2×D34911yescharacteristic
63C2×D4—3491no—
64C2×D4C2×D34911yescharacteristic
65SD16—21no—
66SD16—6981no—
67D8—21no—
68D8—6981no—
69D8—6981no—
70D8—6981no—
71M16—3491no—
72M16—3491no—
73M16—6981no—
74M16C2×D34911yescharacteristic
75C2×C8—3491no—
76C2×C8—6981no—
77C23—21no—
78C23—6981no—
79Q8C22⋊D34911yescharacteristic
80D4—21no—
81D4—21no—
82D4—21no—
83D4—6981no—
84D4—6981no—
85D4C22⋊D34911yescharacteristic
86C2×C4—21no—
87C2×C4C22×D34911yescharacteristic
88C8—21no—
89C8—3491no—
90C8—3491no—
91C8—6981no—
92C8—6981no—
93C22—21no—
94C22—41no—
95C22—81no—
96C22—13961no—
97C22—27921no—
98C22#1915023211yescharacteristic
99C4—21no—
100C4D4×D34911yescharacteristic
101C4C2×C22⋊D34911yescharacteristic
102C2—21no—
103C2—41no—
104C2—81no—
105C2—27921no—
106C2#2601361111yescharacteristic
subgroupG/H
centerC2#26013611
commutator, cosocleC2×C4×C349C23
FrattiniC2×C4C22×D349
FittingAGL(1,ℤ/8)×C349C2
socleC2×C349#121
hypercenterAGL(1,ℤ/8)D349
hypocenterC349#472
LCSG ▷ C2×C4×C349 ▷ C2×C349 ▷ C349
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #26013664C2semidirect
2. C2×#19150218C2semidirect
3. #26013646C2semidirect
4. #26013574C2semidirect
5. #26013556C2semidirect
6. #26013555C2semidirect
7. AGL(1,ℤ/8)×C349C2semidirect
53. C349#472semidirect
56. AGL(1,ℤ/8)D349semidirect

Maximal subgroups

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

subgroupindex
#260136642
C2×#191502182
#260136462
#260135742
#260135562
#260135552
AGL(1,ℤ/8)×C3492
#472349