Finite Group Databasebeta

← #24909359

Subgroups — #24909359

Order 10624.

154 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 53 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#19128137C211yesmaximal, characteristic
2#19127824C211yesmaximal, characteristic, complemented
3#19127523C211yesmaximal, characteristic, complemented
4#19127514C211yesmaximal, characteristic, complemented
5#19127510C211yesmaximal, characteristic, complemented
6#19127479C211yesmaximal, characteristic, complemented
7#442×C83C211yesmaximal, characteristic, complemented
8#14581814C2211yescharacteristic
9#14581798—21no—
10#14581736C412yescomplemented
11#14581733C412yescomplemented
12#14581732C2211yescharacteristic
13#14581732C2211yescharacteristic
14#14581728C2211yescharacteristic
15#14581720—21no—
16C2×D8×C83C2211yescharacteristic
17C4×D4×C83—21no—
18#106×C83C2211yescharacteristic
19#102×C83—21no—
20#97×C83C2211yescharacteristic
21#13050590—21no—
22Dic332C2×C411yescharacteristic, complemented
23#13050579—21no—
24D664C2×C411yescharacteristic, complemented
25#13050574—21no—
26#13050573—41no—
27C8×D83—41no—
28C83⋊C16—21no—
29C83⋊C16C2×C412yescomplemented
30C2×D4×C83—21no—
31C22×C4×C83—41no—
32D8×C83—21no—
33D8×C83C2×C412yes—
34C2×C8×C83D411yescharacteristic
35C2×C8×C83C2311yescharacteristic
36C4⋊C4×C83—21no—
37C22⋊C4×C83—41no—
38C42×C83D411yescharacteristic
39C23×C83—41no—
40D4×C83—22no—
41D4×C83—41no—
42C2×C4×C83—21no—
43C2×C4×C83—41no—
44C2×C4×C83—82no—
45C2×C4×C83C2×D411yescharacteristic
46C22⋊D83—41no—
47D332—21no—
48C4×D83—41no—
49Dic166—21no—
50C8×C83—21no—
51C8×C83C22×C411yescharacteristic
52C8×C83#4111yescharacteristic
53C83⋊C8—41no—
54C22×C83—42no—
55C22×C83—81no—
56C22×C83—81no—
57C22×C83AGL(1,ℤ/8)11yescharacteristic
58C2×D83—41no—
59C4×C83—41no—
60C4×C83—81no—
61C4×C83C4×D411yescharacteristic
62C4×C83AGL(1,ℤ/8)11yescharacteristic
63Dic83—41no—
64C2×C83—21no—
65C2×C83—82no—
66C2×C83#44211yescharacteristic
67D83—81no—
68AGL(1,ℤ/16)—831nomaximal, Sylow
69C83AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
70#509—831no—
71#444—831no—
72#442D8311yescharacteristic, complemented
73#365—831no—
74#361—831no—
75#360—831no—
76#347—831no—
77#135—831no—
78C2×D8C2×D8311yescharacteristic
79#131—1661no—
80C4×D4—21no—
81SD32—832no—
82D16—832no—
83M32—831no—
84M32—831no—
85#108—831no—
86#106C2×D8311yescharacteristic
87#104—1661no—
88#102—21no—
89#97C2×D8311yescharacteristic
90#41—1661no—
91C2×D4—21no—
92C22×C4—41no—
93Q16—831no—
94SD16—1661no—
95D8—21no—
96D8—831no—
97D8C4×D8312yes—
98M16—3321no—
99M16—1661no—
100C2×C8—3321no—
101C2×C8C22⋊D8311yescharacteristic
102C2×C8C22×D8311yescharacteristic
103C4⋊C4—21no—
104C22⋊C4—41no—
105C42C22⋊D8311yescharacteristic
106C16—832no—
107C16—1661no—
108C23—41no—
109Q8—1661no—
110D4—22no—
111D4—41no—
112D4—3321no—
113D4—1661no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—82no—
117C2×C4—3321no—
118C2×C4C2×C22⋊D8311yescharacteristic
119C8—21no—
120C8—3321no—
121C8C2×C4×D8311yescharacteristic
122C8#1305059011yescharacteristic
123C22—42no—
124C22—81no—
125C22—81no—
126C22—3321no—
127C22#1458181811yescharacteristic
128C4—41no—
129C4—81no—
130C4—3321no—
131C4C4×C22⋊D8311yescharacteristic
132C4#1458181711yescharacteristic
133C2—21no—
134C2—82no—
135C2—6641no—
136C2#1912781111yescharacteristic
subgroupG/H
centerC2#19127811
commutatorC8×C83C22×C4
FrattiniC2×C8C22×D83
Fitting#442×C83C2
socleC2×C83#442
prime radical#19128137C2
cosocleC2×C8×C83C23
hypercenter#442D83
hypocenterC83AGL(1,ℤ/16)
LCSG ▷ C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #19127824C2semidirect
3. #19127523C2semidirect
4. #19127514C2semidirect
5. #19127510C2semidirect
6. #19127479C2semidirect
7. #442×C83C2semidirect
10. #14581736C4semidirect
11. #14581733C4semidirect
22. Dic332C2×C4semidirect
24. D664C2×C4semidirect
29. C83⋊C16C2×C4semidirect
69. C83AGL(1,ℤ/16)semidirect
72. #442D83semidirect

Maximal subgroups

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

subgroupindex
#191281372
#191278242
#191275232
#191275142
#191275102
#191274792
#442×C832
AGL(1,ℤ/16)83