Finite Group Databasebeta

← #27173033

Subgroups — #27173033

Order 12928.

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#20265613C211yesmaximal, characteristic
2#20265298C211yesmaximal, characteristic, complemented
3#20265289C211yesmaximal, characteristic, complemented
4#20265001C211yesmaximal, characteristic, complemented
5#20264988C211yesmaximal, characteristic, complemented
6#20264954C211yesmaximal, characteristic, complemented
7#361×C101C211yesmaximal, characteristic, complemented
8#15692277C2211yescharacteristic
9#15692263—21no—
10D4×Dic101—21no—
11#15692199C412yescomplemented
12#15692196C412yescomplemented
13#15692195C2211yescharacteristic
14#15692187C2211yescharacteristic
15#15692181—21no—
16#15692176—21no—
17#15692165C2211yescharacteristic
18C2×D8×C101C2211yescharacteristic
19M32×C101C2211yescharacteristic
20#108×C101C2211yescharacteristic
21#13108074—21no—
22C22×Dic101—41no—
23Dic404C2×C411yescharacteristic, complemented
24#13108063—21no—
25D808C2×C411yescharacteristic, complemented
26#13108057—41no—
27C2×#11717579D411yescharacteristic
28C8×D101—41no—
29C4⋊Dic101—21no—
30C22⋊Dic101—41no—
31C4×Dic101D411yescharacteristic
32#13108049C2×C412yescomplemented
33C2×D4×C101—21no—
34D8×C101—21no—
35D8×C101C2×C412yes—
36M16×C101—21no—
37C2×C8×C101C2311yescharacteristic
38C16×C101—21no—
39C23×C101—41no—
40D4×C101—22no—
41D4×C101—41no—
42C2×C4×C101C2×D411yescharacteristic
43C22⋊D101—41no—
44C2×Dic101—21no—
45C2×Dic101—41no—
46C2×Dic101—82no—
47D404—21no—
48C4×D101—41no—
49Dic202—21no—
50C8×C101—41no—
51C8×C101C22×C411yescharacteristic
52C8×C101#4111yescharacteristic
53#11717579—21no—
54C22×C101—42no—
55C22×C101—81no—
56C22×C101—81no—
57C22×C101AGL(1,ℤ/8)11yescharacteristic
58C2×D101—41no—
59C4×C101C4×D411yescharacteristic
60C4×C101AGL(1,ℤ/8)11yescharacteristic
61Dic101—41no—
62Dic101—41no—
63Dic101—81no—
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#442—1011no—
73#365—1011no—
74#361D10111yescharacteristic, complemented
75#360—1011no—
76#347—1011no—
77#135—1011no—
78C2×D8C2×D10111yescharacteristic
79#131—2021no—
80C4×D4—2021no—
81SD32—1012no—
82D16—1012no—
83M32—1011no—
84M32C2×D10111yescharacteristic
85#108C2×D10111yescharacteristic
86#106—1011no—
87#104—2021no—
88#102—2021no—
89#97—1011no—
90#41—2021no—
91C2×D4—21no—
92C22×C4—4041no—
93Q16—1011no—
94SD16—2021no—
95D8—21no—
96D8—1011no—
97D8C4×D10112yes—
98M16—21no—
99M16—4041no—
100C2×C8—1011no—
101C2×C8—4041no—
102C2×C8C22×D10111yescharacteristic
103C4⋊C4—2021no—
104C22⋊C4—4041no—
105C42—1011no—
106C16—21no—
107C16—1012no—
108C23—41no—
109Q8—2021no—
110D4—22no—
111D4—41no—
112D4—4041no—
113D4—2021no—
114C2×C4—8082no—
115C2×C4—4041no—
116C2×C4—4041no—
117C2×C4—2021no—
118C2×C4D4×D10111yescharacteristic
119C8—41no—
120C8—2021no—
121C8C2×C4×D10111yescharacteristic
122C8#1310807311yescharacteristic
123C22—42no—
124C22—81no—
125C22—81no—
126C22—4041no—
127C22#1569228211yescharacteristic
128C4—8081no—
129C4—4041no—
130C4—4041no—
131C4#1569221511yescharacteristic
132C4#1569227911yescharacteristic
133C2—21no—
134C2—82no—
135C2—8081no—
136C2#2026528611yescharacteristic
subgroupG/H
centerC2#20265286
commutatorC8×C101C22×C4
FrattiniC2×C8C22×D101
Fitting#361×C101C2
socleC2×C101#442
prime radical#20265613C2
cosocleC2×C8×C101C23
hypercenter#361D101
hypocenterC101AGL(1,ℤ/16)
LCSG ▷ C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
2. #20265298C2semidirect
3. #20265289C2semidirect
4. #20265001C2semidirect
5. #20264988C2semidirect
6. #20264954C2semidirect
7. #361×C101C2semidirect
11. #15692199C4semidirect
12. #15692196C4semidirect
23. Dic404C2×C4semidirect
25. D808C2×C4semidirect
32. #13108049C2×C4semidirect
69. C101AGL(1,ℤ/16)semidirect
74. #361D101semidirect

Maximal subgroups

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

subgroupindex
#202656132
#202652982
#202652892
#202650012
#202649882
#202649542
#361×C1012
AGL(1,ℤ/16)101