Finite Group Databasebeta

← #24909351

Subgroups — #24909351

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#19128141C211yesmaximal, characteristic
2#19127823C211yesmaximal, characteristic
3#19127807C211yesmaximal, characteristic, complemented
4#19127523C211yesmaximal, characteristic, complemented
5#19127518C211yesmaximal, characteristic, complemented
6#19127478C211yesmaximal, characteristic, complemented
7#360×C83C211yesmaximal, characteristic
8C2×Dic332C2211yescharacteristic
9#14581799—21no—
10#14581757—21no—
11#14581738C412yescomplemented
12#14581736C412yescomplemented
13#14581732C2211yescharacteristic
14#14581728C2211yescharacteristic
15#14581719—21no—
16#14581715—21no—
17#14581702C2211yescharacteristic
18#135×C83C2211yescharacteristic
19M32×C83C2211yescharacteristic
20#106×C83C2211yescharacteristic
21C2×Dic166—21no—
22Dic332—21no—
23Dic332C2×C412yes—
24#13050574—21no—
25#13050574—41no—
26C2×C83⋊C8—41no—
27C2×C83⋊C8D411yescharacteristic
28Dic83⋊C4—41no—
29C4×Dic83—41no—
30C4×Dic83D411yescharacteristic
31C83⋊C16C2×C412yes—
32#41×C83—21no—
33Q16×C83C2×C411yescharacteristic
34SD16×C83—21no—
35D8×C83C2×C411yescharacteristic
36C2×C8×C83C2311yescharacteristic
37C4⋊C4×C83—21no—
38C16×C83—21no—
39Q8×C83—21no—
40D4×C83—21no—
41D4×C83—41no—
42C2×C4×C83—41no—
43C2×C4×C83—41no—
44C2×C4×C83C2×D411yescharacteristic
45C2×Dic83—21no—
46C2×Dic83—41no—
47Dic166—22no—
48Dic166—41no—
49C8×C83C22×C411yescharacteristic
50C8×C83#4111yescharacteristic
51C83⋊C8—21no—
52C83⋊C8—41no—
53C22×C83—41no—
54C22×C83AGL(1,ℤ/8)11yescharacteristic
55C4×C83—41no—
56C4×C83—81no—
57C4×C83C4×D411yescharacteristic
58C4×C83AGL(1,ℤ/8)11yescharacteristic
59Dic83—41no—
60Dic83—42no—
61C2×C83—21no—
62C2×C83—81no—
63C2×C83#44211yescharacteristic
64#2180—831nomaximal, Sylow
65C83#218011yescharacteristic, complemented, Sylow
66#510—831no—
67#444—831no—
68#441—831no—
69#365—831no—
70#362—831no—
71#360D8311yescharacteristic
72#347—831no—
73#135C2×D8311yescharacteristic
74C2×Q16—831no—
75#131—1661no—
76C4×Q8—1661no—
77Q32—832no—
78SD32—832no—
79M32—831no—
80M32C2×D8311yescharacteristic
81#108—831no—
82#106C2×D8311yescharacteristic
83#104—1661no—
84#103—1661no—
85#97—831no—
86#41—21no—
87C2×Q8—1661no—
88Q16—832no—
89Q16—1661no—
90Q16C4×D8311yescharacteristic
91SD16—21no—
92D8C4×D8311yescharacteristic
93M16—3321no—
94M16—1661no—
95C2×C8—3321no—
96C2×C8—831no—
97C2×C8C22×D8311yescharacteristic
98C4⋊C4—21no—
99C4⋊C4—3321no—
100C42—3321no—
101C42—831no—
102C16—21no—
103C16—832no—
104Q8—21no—
105Q8—3321no—
106Q8—1662no—
107D4—21no—
108D4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—3321no—
112C2×C4—1661no—
113C2×C4D4×D8311yescharacteristic
114C8—3321no—
115C8—1661no—
116C8C2×C4×D8311yescharacteristic
117C8#1305058911yescharacteristic
118C22—41no—
119C22#1458181611yescharacteristic
120C4—41no—
121C4—81no—
122C4—3321no—
123C4—3322no—
124C4(C22⋊D83)⋊C411yescharacteristic
125C4#1458181911yescharacteristic
126C2—21no—
127C2—81no—
128C2#1912781011yescharacteristic
subgroupG/H
centerC2#19127810
commutatorC8×C83C22×C4
FrattiniC2×C8C22×D83
Fitting#360×C83C2
socleC2×C83#442
prime radical#135×C83C22
cosocleC2×C8×C83C23
hypercenter#360D83
hypocenterC83#2180
LCSG ▷ C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
3. #19127807C2semidirect
4. #19127523C2semidirect
5. #19127518C2semidirect
6. #19127478C2semidirect
11. #14581738C4semidirect
12. #14581736C4semidirect
65. C83#2180semidirect

Maximal subgroups

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

subgroupindex
#191281412
#191278232
#191278072
#191275232
#191275182
#191274782
#360×C832
#218083