Finite Group Databasebeta

← #24909346

Subgroups — #24909346

Order 10624.

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

normal subgroups only

This group has 144 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#19128131C211yesmaximal, characteristic
2#19127818C211yesmaximal, characteristic, complemented
3#19127817C211yesmaximal, characteristic, complemented
4#19127530C211yesmaximal, characteristic, complemented
5#19127509C211yesmaximal, characteristic, complemented
6#19127476C211yesmaximal, characteristic, complemented
7#359×C83C211yesmaximal, characteristic, complemented
8#14581814C2211yescharacteristic
9#14581800—21no—
10#14581799—21no—
11#14581738C411yescharacteristic, complemented
12#14581736C411yescharacteristic, complemented
13#14581735C411yescharacteristic, complemented
14#14581733C411yescharacteristic, complemented
15C2×C83⋊C16C2211yescharacteristic
16#14581728C2211yescharacteristic
17#14581719—21no—
18#14581718—21no—
19C8×Dic83C2211yescharacteristic
20#135×C83C2211yescharacteristic
21C2×C16×C83C2211yescharacteristic
22#108×C83C2211yescharacteristic
23#13050590—21no—
24Dic332C2×C411yescharacteristic
25#13050579—21no—
26D664C2×C411yescharacteristic
27#13050574—21no—
28#13050574—41no—
29#13050573—41no—
30C2×C83⋊C8—41no—
31C2×C83⋊C8D411yescharacteristic
32C8×D83—41no—
33C4×Dic83D411yescharacteristic
34C83⋊C16C2×C411yescharacteristic
35C83⋊C16C2×C411yescharacteristic
36#41×C83—21no—
37Q16×C83C2×C411yescharacteristic
38SD16×C83—21no—
39D8×C83C2×C411yescharacteristic
40M16×C83—21no—
41C2×C8×C83C2311yescharacteristic
42C16×C83—21no—
43Q8×C83—21no—
44D4×C83—21no—
45D4×C83—41no—
46C2×C4×C83—41no—
47C2×C4×C83C2×D411yescharacteristic
48C22⋊D83—41no—
49C2×Dic83—21no—
50D332—21no—
51C4×D83—41no—
52Dic166—21no—
53C8×C83—41no—
54C8×C83C22×C411yescharacteristic
55C8×C83#4111yescharacteristic
56C83⋊C8—41no—
57C83⋊C8D811yescharacteristic
58C83⋊C8D811yescharacteristic
59C22×C83—41no—
60C22×C83#13511yescharacteristic
61C2×D83—41no—
62C4×C83—41no—
63C4×C83C4×D411yescharacteristic
64C4×C83C2×D811yescharacteristic
65Dic83—41no—
66Dic83—41no—
67C2×C83—21no—
68C2×C83—81no—
69C2×C83C4×D811yescharacteristic
70D83—81no—
71#2179—831nomaximal, Sylow
72C83#217911yescharacteristic, complemented, Sylow
73#508—831no—
74#443—831no—
75#443—831no—
76#368—831no—
77#359—831no—
78#359D8311yescharacteristic, complemented
79#346—831no—
80#135—831no—
81#135C2×D8311yescharacteristic
82#131—1661no—
83#131—1661no—
84Q32—831no—
85SD32—831no—
86SD32—831no—
87D16—831no—
88C2×C16—831no—
89C2×C16C2×D8311yescharacteristic
90#108—831no—
91#108C2×D8311yescharacteristic
92#104—1661no—
93#104—1661no—
94C4×C8—831no—
95#41—21no—
96#41—1661no—
97Q16—831no—
98Q16C4×D8311yescharacteristic
99SD16—21no—
100SD16—1661no—
101D8—831no—
102D8C4×D8311yescharacteristic
103M16—21no—
104M16—3321no—
105M16—3321no—
106M16—1661no—
107C2×C8—3321no—
108C2×C8—3321no—
109C2×C8—831no—
110C2×C8C22×D8311yescharacteristic
111C42—831no—
112C16—21no—
113C16—831no—
114C16—831no—
115Q8—21no—
116Q8—1661no—
117D4—21no—
118D4—41no—
119D4—3321no—
120D4—1661no—
121C2×C4—41no—
122C2×C4—3321no—
123C2×C4—1661no—
124C2×C4D4×D8311yescharacteristic
125C8—41no—
126C8—3321no—
127C8—831no—
128C8—831no—
129C8C2×C4×D8311yescharacteristic
130C8#1305058911yescharacteristic
131C22—41no—
132C22—3321no—
133C22#1458181311yescharacteristic
134C4—41no—
135C4—3321no—
136C4—3321no—
137C4(C22⋊D83)⋊C411yescharacteristic
138C4D8×D8311yescharacteristic
139C2—21no—
140C2—81no—
141C2—6641no—
142C2#1912778111yescharacteristic
subgroupG/H
centerC4D8×D83
commutatorC8×C83C22×C4
FrattiniC2×C8C22×D83
Fitting#359×C83C2
socleC2×C83C4×D8
prime radical#19128131C2
cosocleC2×C8×C83C23
hypercenter#359D83
hypocenterC83#2179
LCSG ▷ C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #359

Complements

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

NH ≅ G/Ntype
2. #19127818C2semidirect
3. #19127817C2semidirect
4. #19127530C2semidirect
5. #19127509C2semidirect
6. #19127476C2semidirect
7. #359×C83C2semidirect
11. #14581738C4semidirect
12. #14581736C4semidirect
13. #14581735C4semidirect
14. #14581733C4semidirect
72. C83#2179semidirect
78. #359D83semidirect

Maximal subgroups

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

subgroupindex
#191281312
#191278182
#191278172
#191275302
#191275092
#191274762
#359×C832
#217983