Finite Group Databasebeta

← #27173028

Subgroups — #27173028

Order 12928.

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#20265613C211yesmaximal, characteristic
2#20265300C211yesmaximal, characteristic, complemented
3#20264999C211yesmaximal, characteristic, complemented
4#20264990C211yesmaximal, characteristic, complemented
5#20264986C211yesmaximal, characteristic, complemented
6#20264955C211yesmaximal, characteristic, complemented
7#442×C101C211yesmaximal, characteristic, complemented
8#15692277C2211yescharacteristic
9#15692261—21no—
10#15692199C412yescomplemented
11#15692196C412yescomplemented
12#15692195C2211yescharacteristic
13#15692195C2211yescharacteristic
14#15692191C2211yescharacteristic
15#15692183—21no—
16C2×D8×C101C2211yescharacteristic
17C4×D4×C101—21no—
18#106×C101C2211yescharacteristic
19#102×C101—21no—
20#97×C101C2211yescharacteristic
21#13108074—21no—
22Dic404C2×C411yescharacteristic, complemented
23#13108063—21no—
24D808C2×C411yescharacteristic, complemented
25#13108058—21no—
26#13108057—41no—
27C8×D101—41no—
28#13108049—21no—
29#13108049C2×C412yescomplemented
30C2×D4×C101—21no—
31C22×C4×C101—41no—
32D8×C101—21no—
33D8×C101C2×C412yes—
34C2×C8×C101D411yescharacteristic
35C2×C8×C101C2311yescharacteristic
36C4⋊C4×C101—21no—
37C22⋊C4×C101—41no—
38C42×C101D411yescharacteristic
39C23×C101—41no—
40D4×C101—22no—
41D4×C101—41no—
42C2×C4×C101—21no—
43C2×C4×C101—41no—
44C2×C4×C101—82no—
45C2×C4×C101C2×D411yescharacteristic
46C22⋊D101—41no—
47D404—21no—
48C4×D101—41no—
49Dic202—21no—
50C8×C101—21no—
51C8×C101C22×C411yescharacteristic
52C8×C101#4111yescharacteristic
53#11717579—41no—
54C22×C101—42no—
55C22×C101—81no—
56C22×C101—81no—
57C22×C101AGL(1,ℤ/8)11yescharacteristic
58C2×D101—41no—
59C4×C101—41no—
60C4×C101—81no—
61C4×C101C4×D411yescharacteristic
62C4×C101AGL(1,ℤ/8)11yescharacteristic
63Dic101—41no—
64C2×C101—21no—
65C2×C101—82no—
66C2×C101#44211yescharacteristic
67D101—81no—
68AGL(1,ℤ/16)—1011nomaximal, Sylow
69C101AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
70#509—1011no—
71#444—1011no—
72#442D10111yescharacteristic, complemented
73#365—1011no—
74#361—1011no—
75#360—1011no—
76#347—1011no—
77#135—1011no—
78C2×D8C2×D10111yescharacteristic
79#131—2021no—
80C4×D4—21no—
81SD32—1012no—
82D16—1012no—
83M32—1011no—
84M32—1011no—
85#108—1011no—
86#106C2×D10111yescharacteristic
87#104—2021no—
88#102—21no—
89#97C2×D10111yescharacteristic
90#41—2021no—
91C2×D4—21no—
92C22×C4—41no—
93Q16—1011no—
94SD16—2021no—
95D8—21no—
96D8—1011no—
97D8C4×D10112yes—
98M16—4041no—
99M16—2021no—
100C2×C8—4041no—
101C2×C8C22⋊D10111yescharacteristic
102C2×C8C22×D10111yescharacteristic
103C4⋊C4—21no—
104C22⋊C4—41no—
105C42C22⋊D10111yescharacteristic
106C16—1012no—
107C16—2021no—
108C23—41no—
109Q8—2021no—
110D4—22no—
111D4—41no—
112D4—4041no—
113D4—2021no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—82no—
117C2×C4—4041no—
118C2×C4C2×C22⋊D10111yescharacteristic
119C8—21no—
120C8—4041no—
121C8C2×C4×D10111yescharacteristic
122C8#1310807411yescharacteristic
123C22—42no—
124C22—81no—
125C22—81no—
126C22—4041no—
127C22#1569228111yescharacteristic
128C4—41no—
129C4—81no—
130C4—4041no—
131C4C4×C22⋊D10111yescharacteristic
132C4#1569228011yescharacteristic
133C2—21no—
134C2—82no—
135C2—8081no—
136C2#2026528711yescharacteristic
subgroupG/H
centerC2#20265287
commutatorC8×C101C22×C4
FrattiniC2×C8C22×D101
Fitting#442×C101C2
socleC2×C101#442
prime radical#20265613C2
cosocleC2×C8×C101C23
hypercenter#442D101
hypocenterC101AGL(1,ℤ/16)
LCSG ▷ C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #20265300C2semidirect
3. #20264999C2semidirect
4. #20264990C2semidirect
5. #20264986C2semidirect
6. #20264955C2semidirect
7. #442×C101C2semidirect
10. #15692199C4semidirect
11. #15692196C4semidirect
22. Dic404C2×C4semidirect
24. D808C2×C4semidirect
29. #13108049C2×C4semidirect
69. C101AGL(1,ℤ/16)semidirect
72. #442D101semidirect

Maximal subgroups

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

subgroupindex
#202656132
#202653002
#202649992
#202649902
#202649862
#202649552
#442×C1012
AGL(1,ℤ/16)101