Finite Group Databasebeta

← #27173030

Subgroups — #27173030

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#20265612C211yesmaximal, characteristic
2#20265290C211yesmaximal, characteristic, complemented
3#20265001C211yesmaximal, characteristic, complemented
4#20264989C211yesmaximal, characteristic, complemented
5#20264987C211yesmaximal, characteristic, complemented
6#20264955C211yesmaximal, characteristic, complemented
7#444×C101C211yesmaximal, characteristic, complemented
8C2×D808C2211yescharacteristic
9C4×D404—21no—
10#15692198C412yescomplemented
11#15692196C412yescomplemented
12#15692195C2211yescharacteristic
13#15692195C2211yescharacteristic
14#15692187C2211yescharacteristic
15#15692177—21no—
16#135×C101C2211yescharacteristic
17#131×C101—21no—
18#108×C101C2211yescharacteristic
19#104×C101—21no—
20#97×C101C2211yescharacteristic
21C2×D404—21no—
22C2×C4×D101—41no—
23D808—21no—
24D808C2×C412yes—
25C4⋊Dic101—21no—
26#13108051—41no—
27#13108049—21no—
28#13108049C2×C412yescomplemented
29#41×C101—21no—
30Q16×C101C2×C411yescharacteristic, complemented
31SD16×C101—21no—
32D8×C101C2×C411yescharacteristic, complemented
33M16×C101—21no—
34M16×C101—41no—
35C2×C8×C101—41no—
36C2×C8×C101D411yescharacteristic
37C2×C8×C101C2311yescharacteristic
38C42×C101D411yescharacteristic
39C22×D101—41no—
40Q8×C101—21no—
41D4×C101—21no—
42D4×C101—41no—
43C2×C4×C101—21no—
44C2×C4×C101—41no—
45C2×C4×C101C2×D411yescharacteristic
46C2×Dic101—41no—
47D404—22no—
48D404—41no—
49C4×D101—82no—
50C8×C101—21no—
51C8×C101—41no—
52C8×C101C22×C411yescharacteristic
53C8×C101#4111yescharacteristic
54C22×C101—41no—
55C22×C101AGL(1,ℤ/8)11yescharacteristic
56C2×D101—42no—
57C2×D101—81no—
58C2×D101—81no—
59C4×C101—41no—
60C4×C101—41no—
61C4×C101C4×D411yescharacteristic
62C4×C101AGL(1,ℤ/8)11yescharacteristic
63Dic101—81no—
64C2×C101—21no—
65C2×C101—81no—
66C2×C101#44211yescharacteristic
67D101—82no—
68AGL(1,ℤ/16)—1011nomaximal, Sylow
69C101AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
70#509—1011no—
71#444D10111yescharacteristic, complemented
72#442—1011no—
73#365—1011no—
74#361—1011no—
75#360—1011no—
76#347—1011no—
77#135C2×D10111yescharacteristic
78C2×D8—1011no—
79#131—21no—
80C4×D4—2021no—
81SD32—1012no—
82D16—1012no—
83M32—1011no—
84M32—1011no—
85#108C2×D10111yescharacteristic
86#106—1011no—
87#104—21no—
88#102—2021no—
89#97C2×D10111yescharacteristic
90#41—21no—
91C2×D4—2021no—
92C22×C4—4041no—
93Q16C4×D10111yescharacteristic, complemented
94SD16—21no—
95D8—1012no—
96D8—2021no—
97D8C4×D10111yescharacteristic, complemented
98M16—21no—
99M16—41no—
100C2×C8—41no—
101C2×C8C22⋊D10111yescharacteristic
102C2×C8C22×D10111yescharacteristic
103C4⋊C4—2021no—
104C22⋊C4—4041no—
105C42C22⋊D10111yescharacteristic
106C16—1012no—
107C16—2021no—
108C23—4041no—
109Q8—21no—
110D4—21no—
111D4—41no—
112D4—4041no—
113D4—2022no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—8082no—
117C2×C4—4041no—
118C2×C4C2×C22⋊D10111yescharacteristic
119C8—21no—
120C8—41no—
121C8C2×C4×D10111yescharacteristic
122C8#1310807411yescharacteristic
123C22—41no—
124C22—8081no—
125C22—8081no—
126C22—4042no—
127C22#1569228011yescharacteristic
128C4—41no—
129C4—41no—
130C4—8081no—
131C4C4×C22⋊D10111yescharacteristic
132C4#1569228111yescharacteristic
133C2—21no—
134C2—81no—
135C2—8082no—
136C2#2026528711yescharacteristic
subgroupG/H
centerC2#20265287
commutatorC8×C101C22×C4
FrattiniC2×C8C22×D101
Fitting#444×C101C2
socleC2×C101#442
prime radical#20265612C2
cosocleC2×C8×C101C23
hypercenter#444D101
hypocenterC101AGL(1,ℤ/16)
LCSG ▷ C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #444

Complements

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

NH ≅ G/Ntype
2. #20265290C2semidirect
3. #20265001C2semidirect
4. #20264989C2semidirect
5. #20264987C2semidirect
6. #20264955C2semidirect
7. #444×C101C2semidirect
10. #15692198C4semidirect
11. #15692196C4semidirect
28. #13108049C2×C4semidirect
30. Q16×C101C2×C4semidirect
32. D8×C101C2×C4semidirect
69. C101AGL(1,ℤ/16)semidirect
71. #444D101semidirect
93. Q16C4×D101semidirect
97. D8C4×D101semidirect

Maximal subgroups

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

subgroupindex
#202656122
#202652902
#202650012
#202649892
#202649872
#202649552
#444×C1012
AGL(1,ℤ/16)101