Finite Group Databasebeta

← #42122079

Subgroups — #42122079

Order 27712.

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#28357245C211yesmaximal, characteristic, complemented
2C2×#20294082C211yesmaximal, characteristic, complemented
3#28357227C211yesmaximal, characteristic, complemented
4#28357155C211yesmaximal, characteristic, complemented
5#28357137C211yesmaximal, characteristic, complemented
6#28357136C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C433C211yesmaximal, characteristic, complemented
8C2×D1732C2211yescharacteristic
9#20294084—21no—
10#20294082—21no—
11#20294082—21no—
12#20294082—21no—
13#20294081—21no—
14#20294081C2211yescharacteristic
15#20294081C2211yescharacteristic
16C2×#16812358—21no—
17C2×#16812358C2211yescharacteristic
18#41×C433C2211yescharacteristic
19C2×D4×C433C2211yescharacteristic
20SD16×C433—21no—
21D8×C433—21no—
22M16×C433C2211yescharacteristic
23C22×D433—21no—
24D1732—21no—
25D1732—21no—
26#16812358—21no—
27#16812358—21no—
28#16812358D411yescharacteristic
29#16812358D411yescharacteristic
30C23×C433—21no—
31Q8×C433D411yescharacteristic
32D4×C433—21no—
33D4×C433—21no—
34D4×C433—21no—
35D4×C433D411yescharacteristic
36C2×C4×C433—21no—
37C2×C4×C433C2311yescharacteristic
38C8×C433—21no—
39C22×C433—21no—
40C22×C433—41no—
41C22×C433—81no—
42C22×C433#4111yescharacteristic
43C2×D433—41no—
44C2×D433—81no—
45C4×C433—21no—
46C4×C433C2×D411yescharacteristic
47C4×C433C2×D411yescharacteristic
48C2×C433—21no—
49C2×C433—41no—
50C2×C433—81no—
51C2×C433#12111yescharacteristic
52D433—81no—
53C433#47211yescharacteristic, complemented, Sylow
54#472—4331nomaximal, Sylow
55AGL(1,ℤ/8)—4331no—
56AGL(1,ℤ/8)D43311yescharacteristic, complemented
57C2×D8—4331no—
58#131—4331no—
59#108—4331no—
60#100—4331no—
61#100—4331no—
62#41C2×D43311yescharacteristic
63C2×D4—4331no—
64C2×D4C2×D43311yescharacteristic
65SD16—21no—
66SD16—8661no—
67D8—21no—
68D8—8661no—
69D8—8661no—
70D8—8661no—
71M16—8661no—
72M16—4331no—
73M16—4331no—
74M16C2×D43311yescharacteristic
75C2×C8—8661no—
76C2×C8—4331no—
77C23—21no—
78C23—8661no—
79Q8C22⋊D43311yescharacteristic
80D4—21no—
81D4—21no—
82D4—21no—
83D4—8661no—
84D4—8661no—
85D4C22⋊D43311yescharacteristic
86C2×C4—21no—
87C2×C4C22×D43311yescharacteristic
88C8—21no—
89C8—8661no—
90C8—8661no—
91C8—4331no—
92C8—4331no—
93C22—21no—
94C22—41no—
95C22—81no—
96C22—34641no—
97C22—17321no—
98C22#2029409611yescharacteristic
99C4—21no—
100C4D4×D43311yescharacteristic
101C4C2×C22⋊D43311yescharacteristic
102C2—21no—
103C2—41no—
104C2—81no—
105C2—34641no—
106C2#2835719211yescharacteristic
subgroupG/H
centerC2#28357192
commutator, cosocleC2×C4×C433C23
FrattiniC2×C4C22×D433
FittingAGL(1,ℤ/8)×C433C2
socleC2×C433#121
hypercenterAGL(1,ℤ/8)D433
hypocenterC433#472
LCSG ▷ C2×C4×C433 ▷ C2×C433 ▷ C433
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #28357245C2semidirect
2. C2×#20294082C2semidirect
3. #28357227C2semidirect
4. #28357155C2semidirect
5. #28357137C2semidirect
6. #28357136C2semidirect
7. AGL(1,ℤ/8)×C433C2semidirect
53. C433#472semidirect
56. AGL(1,ℤ/8)D433semidirect

Maximal subgroups

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

subgroupindex
#283572452
C2×#202940822
#283572272
#283571552
#283571372
#283571362
AGL(1,ℤ/8)×C4332
#472433