Finite Group Databasebeta

← #24909358

Subgroups — #24909358

Order 10624.

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

normal subgroups only

This group has 154 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#19128136C211yesmaximal, characteristic
2#19127823C211yesmaximal, characteristic, complemented
3#19127812C211yesmaximal, characteristic, complemented
4#19127523C211yesmaximal, characteristic, complemented
5#19127515C211yesmaximal, characteristic, complemented
6#19127478C211yesmaximal, characteristic, complemented
7#360×C83C211yesmaximal, characteristic, complemented
8C2×D664C2211yescharacteristic
9#14581799—21no—
10#14581754—21no—
11#14581735C412yescomplemented
12#14581733C412yescomplemented
13#14581732C2211yescharacteristic
14#14581728C2211yescharacteristic
15#14581719—21no—
16#14581712—21no—
17#14581702C2211yescharacteristic
18#135×C83C2211yescharacteristic
19M32×C83C2211yescharacteristic
20#106×C83C2211yescharacteristic
21C2×D332—21no—
22C2×C4×D83—41no—
23D664—21no—
24D664C2×C412yes—
25#13050574—21no—
26#13050574—41no—
27C2×C83⋊C8—41no—
28C2×C83⋊C8D411yescharacteristic
29(C2×D83)⋊C4—41no—
30C4×Dic83D411yescharacteristic
31C83⋊C16C2×C412yescomplemented
32#41×C83—21no—
33Q16×C83C2×C411yescharacteristic, complemented
34SD16×C83—21no—
35D8×C83C2×C411yescharacteristic, complemented
36C2×C8×C83C2311yescharacteristic
37C4⋊C4×C83—21no—
38C16×C83—21no—
39C22×D83—41no—
40Q8×C83—21no—
41D4×C83—21no—
42D4×C83—41no—
43C2×C4×C83—41no—
44C2×C4×C83—41no—
45C2×C4×C83C2×D411yescharacteristic
46C2×Dic83—21no—
47D332—22no—
48D332—41no—
49C4×D83—82no—
50C8×C83C22×C411yescharacteristic
51C8×C83#4111yescharacteristic
52C83⋊C8—21no—
53C83⋊C8—41no—
54C22×C83—41no—
55C22×C83AGL(1,ℤ/8)11yescharacteristic
56C2×D83—42no—
57C2×D83—81no—
58C2×D83—81no—
59C4×C83—41no—
60C4×C83—81no—
61C4×C83C4×D411yescharacteristic
62C4×C83AGL(1,ℤ/8)11yescharacteristic
63Dic83—41no—
64C2×C83—21no—
65C2×C83—81no—
66C2×C83#44211yescharacteristic
67D83—82no—
68AGL(1,ℤ/16)—831nomaximal, Sylow
69C83AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
70#509—831no—
71#444—831no—
72#442—831no—
73#365—831no—
74#361—831no—
75#360D8311yescharacteristic, complemented
76#347—831no—
77#135C2×D8311yescharacteristic
78C2×D8—831no—
79#131—1661no—
80C4×D4—1661no—
81SD32—832no—
82D16—832no—
83M32—831no—
84M32C2×D8311yescharacteristic
85#108—831no—
86#106C2×D8311yescharacteristic
87#104—1661no—
88#102—1661no—
89#97—831no—
90#41—21no—
91C2×D4—1661no—
92C22×C4—3321no—
93Q16C4×D8311yescharacteristic, complemented
94SD16—21no—
95D8—832no—
96D8—1661no—
97D8C4×D8311yescharacteristic, complemented
98M16—3321no—
99M16—1661no—
100C2×C8—3321no—
101C2×C8—831no—
102C2×C8C22×D8311yescharacteristic
103C4⋊C4—21no—
104C22⋊C4—3321no—
105C42—831no—
106C16—21no—
107C16—832no—
108C23—3321no—
109Q8—21no—
110D4—21no—
111D4—41no—
112D4—3321no—
113D4—1662no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—6642no—
117C2×C4—1661no—
118C2×C4D4×D8311yescharacteristic
119C8—3321no—
120C8—1661no—
121C8C2×C4×D8311yescharacteristic
122C8#1305058911yescharacteristic
123C22—41no—
124C22—3322no—
125C22—6641no—
126C22—6641no—
127C22#1458181611yescharacteristic
128C4—41no—
129C4—81no—
130C4—3321no—
131C4(C22⋊D83)⋊C411yescharacteristic
132C4#1458181911yescharacteristic
133C2—21no—
134C2—81no—
135C2—6642no—
136C2#1912781011yescharacteristic
subgroupG/H
centerC2#19127810
commutatorC8×C83C22×C4
FrattiniC2×C8C22×D83
Fitting#360×C83C2
socleC2×C83#442
prime radical#19128136C2
cosocleC2×C8×C83C23
hypercenter#360D83
hypocenterC83AGL(1,ℤ/16)
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
2. #19127823C2semidirect
3. #19127812C2semidirect
4. #19127523C2semidirect
5. #19127515C2semidirect
6. #19127478C2semidirect
7. #360×C83C2semidirect
11. #14581735C4semidirect
12. #14581733C4semidirect
31. C83⋊C16C2×C4semidirect
33. Q16×C83C2×C4semidirect
35. D8×C83C2×C4semidirect
69. C83AGL(1,ℤ/16)semidirect
75. #360D83semidirect
93. Q16C4×D83semidirect
97. D8C4×D83semidirect

Maximal subgroups

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

subgroupindex
#191281362
#191278232
#191278122
#191275232
#191275152
#191274782
#360×C832
AGL(1,ℤ/16)83