Finite Group Databasebeta

← #24909357

Subgroups — #24909357

Order 10624.

142 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 51 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#19128142C211yesmaximal, characteristic
2#19127822C211yesmaximal, characteristic
3#19127808C211yesmaximal, characteristic, complemented
4#19127525C211yesmaximal, characteristic, complemented
5#19127512C211yesmaximal, characteristic
6#19127478C211yesmaximal, characteristic, complemented
7#362×C83C211yesmaximal, characteristic, complemented
8#14581814C2211yescharacteristic
9#14581800—21no—
10Q8×Dic83—21no—
11#14581738C412yescomplemented
12#14581735C412yescomplemented
13#14581732C2211yescharacteristic
14#14581724C2211yescharacteristic
15#14581718—21no—
16#14581716—21no—
17#14581702C2211yescharacteristic
18C2×Q16×C83C2211yescharacteristic
19M32×C83C2211yescharacteristic
20#108×C83C2211yescharacteristic
21#13050590—21no—
22Dic332C2×C411yescharacteristic
23#13050579—21no—
24D664C2×C411yescharacteristic
25#13050573—41no—
26C2×C83⋊C8D411yescharacteristic
27C8×D83—41no—
28C4⋊Dic83—21no—
29C4⋊Dic83—41no—
30C4×Dic83—41no—
31C4×Dic83D411yescharacteristic
32C83⋊C16C2×C412yes—
33C2×Q8×C83—21no—
34Q16×C83—21no—
35Q16×C83C2×C412yes—
36M16×C83—21no—
37C2×C8×C83C2311yescharacteristic
38C16×C83—21no—
39Q8×C83—22no—
40Q8×C83—41no—
41C2×C4×C83—41no—
42C2×C4×C83C2×D411yescharacteristic
43C22⋊D83—41no—
44C2×Dic83—21no—
45C2×Dic83—41no—
46D332—21no—
47C4×D83—41no—
48Dic166—21no—
49C8×C83—41no—
50C8×C83C22×C411yescharacteristic
51C8×C83#4111yescharacteristic
52C83⋊C8—21no—
53C22×C83AGL(1,ℤ/8)11yescharacteristic
54C2×D83—41no—
55C4×C83—42no—
56C4×C83C4×D411yescharacteristic
57C4×C83AGL(1,ℤ/8)11yescharacteristic
58Dic83—41no—
59Dic83—41no—
60Dic83—81no—
61C2×C83—21no—
62C2×C83#44211yescharacteristic
63D83—81no—
64#2180—831nomaximal, Sylow
65C83#218011yescharacteristic, complemented, Sylow
66#510—831no—
67#444—831no—
68#441—831no—
69#365—831no—
70#362D8311yescharacteristic, complemented
71#360—831no—
72#347—831no—
73#135—831no—
74C2×Q16C2×D8311yescharacteristic
75#131—1661no—
76C4×Q8—1661no—
77Q32—832no—
78SD32—832no—
79M32—831no—
80M32C2×D8311yescharacteristic
81#108C2×D8311yescharacteristic
82#106—831no—
83#104—1661no—
84#103—1661no—
85#97—831no—
86#41—1661no—
87C2×Q8—21no—
88Q16—21no—
89Q16—831no—
90Q16C4×D8312yes—
91SD16—1661no—
92D8—831no—
93M16—21no—
94M16—3321no—
95C2×C8—3321no—
96C2×C8—831no—
97C2×C8C22×D8311yescharacteristic
98C4⋊C4—3321no—
99C4⋊C4—1661no—
100C42—3321no—
101C42—831no—
102C16—21no—
103C16—832no—
104Q8—22no—
105Q8—41no—
106Q8—1661no—
107D4—3321no—
108D4—1661no—
109C2×C4—41no—
110C2×C4—3321no—
111C2×C4—3321no—
112C2×C4—1661no—
113C2×C4D4×D8311yescharacteristic
114C8—41no—
115C8—1661no—
116C8C2×C4×D8311yescharacteristic
117C8#1305058911yescharacteristic
118C22—3321no—
119C22#1458181911yescharacteristic
120C4—42no—
121C4—3321no—
122C4—3321no—
123C4—6641no—
124C4(C22⋊D83)⋊C411yescharacteristic
125C4#1458181611yescharacteristic
126C2—21no—
127C2—6641no—
128C2#1912781011yescharacteristic
subgroupG/H
centerC2#19127810
commutatorC8×C83C22×C4
FrattiniC2×C8C22×D83
Fitting#362×C83C2
socleC2×C83#442
prime radical#14581814C22
cosocleC2×C8×C83C23
hypercenter#362D83
hypocenterC83#2180
LCSG ▷ C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #362

Complements

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

NH ≅ G/Ntype
3. #19127808C2semidirect
4. #19127525C2semidirect
6. #19127478C2semidirect
7. #362×C83C2semidirect
11. #14581738C4semidirect
12. #14581735C4semidirect
65. C83#2180semidirect
70. #362D83semidirect

Maximal subgroups

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

subgroupindex
#191281422
#191278222
#191278082
#191275252
#191275122
#191274782
#362×C832
#218083