Finite Group Databasebeta

← #36374405

Subgroups — #36374405

Order 22336.

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#26013669C211yesmaximal, characteristic, complemented
2C2×#19150221C211yesmaximal, characteristic, complemented
3#26013646C211yesmaximal, characteristic, complemented
4#26013574C211yesmaximal, characteristic, complemented
5#26013557C211yesmaximal, characteristic, complemented
6#26013556C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C349C211yesmaximal, characteristic
8C2×Dic698C2211yescharacteristic
9#19150223—21no—
10#19150221—21no—
11#19150221—21no—
12#19150221—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
23Dic698—21no—
24Dic698—21no—
25C2×Dic349—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
43C4×C349—21no—
44C4×C349C2×D411yescharacteristic
45C4×C349C2×D411yescharacteristic
46Dic349—41no—
47C2×C349—21no—
48C2×C349—41no—
49C2×C349—81no—
50C2×C349#12111yescharacteristic
51C349#47111yescharacteristic, complemented, Sylow
52#471—3491nomaximal, Sylow
53#137—3491no—
54AGL(1,ℤ/8)D34911yescharacteristic
55C2×SD16—3491no—
56#131—3491no—
57#108—3491no—
58#101—3491no—
59#100—3491no—
60#41C2×D34911yescharacteristic
61C2×Q8—3491no—
62C2×D4C2×D34911yescharacteristic
63Q16—6981no—
64SD16—21no—
65SD16—6981no—
66SD16—6981no—
67SD16—6981no—
68D8—21no—
69M16—3491no—
70M16—3491no—
71M16—6981no—
72M16C2×D34911yescharacteristic
73C2×C8—3491no—
74C2×C8—6981no—
75C23—21no—
76Q8—6981no—
77Q8—6981no—
78Q8C22⋊D34911yescharacteristic
79D4—21no—
80D4—21no—
81D4—21no—
82D4C22⋊D34911yescharacteristic
83C2×C4—21no—
84C2×C4—6981no—
85C2×C4C22×D34911yescharacteristic
86C8—21no—
87C8—3491no—
88C8—3491no—
89C8—6981no—
90C8—6981no—
91C22—21no—
92C22—41no—
93C22—81no—
94C22#1915023211yescharacteristic
95C4—21no—
96C4—13961no—
97C4D4×D34911yescharacteristic
98C4C2×C22⋊D34911yescharacteristic
99C2—21no—
100C2—41no—
101C2—81no—
102C2#2601361111yescharacteristic
subgroupG/H
centerC2#26013611
commutator, cosocleC2×C4×C349C23
FrattiniC2×C4C22×D349
Fitting, prime radicalAGL(1,ℤ/8)×C349C2
socleC2×C349#121
hypercenterAGL(1,ℤ/8)D349
hypocenterC349#471
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. #26013669C2semidirect
2. C2×#19150221C2semidirect
3. #26013646C2semidirect
4. #26013574C2semidirect
5. #26013557C2semidirect
6. #26013556C2semidirect
51. C349#471semidirect

Maximal subgroups

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

subgroupindex
#260136692
C2×#191502212
#260136462
#260135742
#260135572
#260135562
AGL(1,ℤ/8)×C3492
#471349