Finite Group Databasebeta

← #27173027

Subgroups — #27173027

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#20265612C211yesmaximal, characteristic
2#20265299C211yesmaximal, characteristic, complemented
3#20265288C211yesmaximal, characteristic, complemented
4#20264999C211yesmaximal, characteristic, complemented
5#20264991C211yesmaximal, characteristic, complemented
6#20264954C211yesmaximal, characteristic, complemented
7#360×C101C211yesmaximal, characteristic, complemented
8C2×D808C2211yescharacteristic
9#15692262—21no—
10#15692217—21no—
11#15692198C412yescomplemented
12#15692196C412yescomplemented
13#15692195C2211yescharacteristic
14#15692191C2211yescharacteristic
15#15692182—21no—
16#15692175—21no—
17#15692165C2211yescharacteristic
18#135×C101C2211yescharacteristic
19M32×C101C2211yescharacteristic
20#106×C101C2211yescharacteristic
21C2×D404—21no—
22C2×C4×D101—41no—
23D808—21no—
24D808C2×C412yes—
25#13108058—21no—
26#13108058—41no—
27C2×#11717579—41no—
28C2×#11717579D411yescharacteristic
29#13108051—41no—
30C4×Dic101D411yescharacteristic
31#13108049C2×C412yescomplemented
32#41×C101—21no—
33Q16×C101C2×C411yescharacteristic, complemented
34SD16×C101—21no—
35D8×C101C2×C411yescharacteristic, complemented
36C2×C8×C101C2311yescharacteristic
37C4⋊C4×C101—21no—
38C16×C101—21no—
39C22×D101—41no—
40Q8×C101—21no—
41D4×C101—21no—
42D4×C101—41no—
43C2×C4×C101—41no—
44C2×C4×C101—41no—
45C2×C4×C101C2×D411yescharacteristic
46C2×Dic101—21no—
47D404—22no—
48D404—41no—
49C4×D101—82no—
50C8×C101C22×C411yescharacteristic
51C8×C101#4111yescharacteristic
52#11717579—21no—
53#11717579—41no—
54C22×C101—41no—
55C22×C101AGL(1,ℤ/8)11yescharacteristic
56C2×D101—42no—
57C2×D101—81no—
58C2×D101—81no—
59C4×C101—41no—
60C4×C101—81no—
61C4×C101C4×D411yescharacteristic
62C4×C101AGL(1,ℤ/8)11yescharacteristic
63Dic101—41no—
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#444—1011no—
72#442—1011no—
73#365—1011no—
74#361—1011no—
75#360D10111yescharacteristic, complemented
76#347—1011no—
77#135C2×D10111yescharacteristic
78C2×D8—1011no—
79#131—2021no—
80C4×D4—2021no—
81SD32—1012no—
82D16—1012no—
83M32—1011no—
84M32C2×D10111yescharacteristic
85#108—1011no—
86#106C2×D10111yescharacteristic
87#104—2021no—
88#102—2021no—
89#97—1011no—
90#41—21no—
91C2×D4—2021no—
92C22×C4—4041no—
93Q16C4×D10111yescharacteristic, complemented
94SD16—21no—
95D8—1012no—
96D8—2021no—
97D8C4×D10111yescharacteristic, complemented
98M16—4041no—
99M16—2021no—
100C2×C8—1011no—
101C2×C8—4041no—
102C2×C8C22×D10111yescharacteristic
103C4⋊C4—21no—
104C22⋊C4—4041no—
105C42—1011no—
106C16—21no—
107C16—1012no—
108C23—4041no—
109Q8—21no—
110D4—21no—
111D4—41no—
112D4—4041no—
113D4—2022no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—8082no—
117C2×C4—2021no—
118C2×C4D4×D10111yescharacteristic
119C8—4041no—
120C8—2021no—
121C8C2×C4×D10111yescharacteristic
122C8#1310807311yescharacteristic
123C22—41no—
124C22—8081no—
125C22—8081no—
126C22—4042no—
127C22#1569227911yescharacteristic
128C4—41no—
129C4—81no—
130C4—4041no—
131C4#1569221511yescharacteristic
132C4#1569228211yescharacteristic
133C2—21no—
134C2—81no—
135C2—8082no—
136C2#2026528611yescharacteristic
subgroupG/H
centerC2#20265286
commutatorC8×C101C22×C4
FrattiniC2×C8C22×D101
Fitting#360×C101C2
socleC2×C101#442
prime radical#20265612C2
cosocleC2×C8×C101C23
hypercenter#360D101
hypocenterC101AGL(1,ℤ/16)
LCSG ▷ C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
2. #20265299C2semidirect
3. #20265288C2semidirect
4. #20264999C2semidirect
5. #20264991C2semidirect
6. #20264954C2semidirect
7. #360×C101C2semidirect
11. #15692198C4semidirect
12. #15692196C4semidirect
31. #13108049C2×C4semidirect
33. Q16×C101C2×C4semidirect
35. D8×C101C2×C4semidirect
69. C101AGL(1,ℤ/16)semidirect
75. #360D101semidirect
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
#202652992
#202652882
#202649992
#202649912
#202649542
#360×C1012
AGL(1,ℤ/16)101