Finite Group Databasebeta

← #27169281

Subgroups — #27169281

Order 12928.

112 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 23 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#20265354C211yesmaximal, characteristic
2#20264974C211yesmaximal, characteristic, complemented
3#329×C101C211yesmaximal, characteristic, complemented
4#15692300C411yescharacteristic
5#15692286—21no—
6#15692183—21no—
7#15692174—21no—
8#15692172C411yescharacteristic
9#124×C101C2211yescharacteristic
10#97×C101—21no—
11#13108074—21no—
12#13108073—41no—
13C2×C22⋊D101—41no—
14D4×D101—41no—
15#13108064—41no—
16#13108061—41no—
17#13108058—21no—
18#13108058—41no—
19C2×Q8×C101D411yescharacteristic
20C2×D4×C101C2×C411yescharacteristic
21C2×C8×C101—41no—
22C22⋊C4×C101—41no—
23C42×C101D411yescharacteristic
24C23×C101—21no—
25C22×D101—41no—
26Q8×C101—41no—
27D4×C101—41no—
28C2×C4×C101—21no—
29C2×C4×C101—21no—
30C2×C4×C101C22⋊C411yescharacteristic
31C22⋊D101—42no—
32C22⋊D101—41no—
33C2×Dic101—41no—
34D404—21no—
35C4×D101—41no—
36Dic202—21no—
37C8×C101—42no—
38#11717579—41no—
39#11717579—81no—
40C22×C101—41no—
41C22×C101—81no—
42C22×C101#9911yescharacteristic
43C2×D101—41no—
44C2×D101—161no—
45C4×C101—21no—
46C4×C101—41no—
47C4×C101—41no—
48Dic101—41no—
49C2×C101—21no—
50C2×C101—81no—
51C2×C101AΣL(1,𝔽16)11yescharacteristic
52D101—81no—
53#1403—1011nomaximal, Sylow
54C101#140311yescharacteristic, complemented, Sylow
55#455—1011no—
56#355—1011no—
57#329D10111yescharacteristic, complemented
58H(5,𝔽2)—1011no—
59#137—2021no—
60#124C2×D10111yescharacteristic
61#104—2021no—
62#101—2021no—
63#100—1011no—
64#97—21no—
65#41—4041no—
66#41—2021no—
67C2×Q8D40411yescharacteristic
68C2×D4—4041no—
69C2×D4—4041no—
70C2×D4C4×D10111yescharacteristic
71Q16—4041no—
72SD16—4041no—
73M16—4041no—
74M16—2021no—
75C2×C8—41no—
76C22⋊C4—41no—
77C42C22⋊D10111yescharacteristic
78C23—21no—
79C23—4041no—
80Q8—41no—
81Q8—2021no—
82D4—41no—
83D4—4042no—
84D4—4041no—
85D4—2021no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—4041no—
89C2×C4—4041no—
90C2×C4#1310805111yescharacteristic
91C8—42no—
92C8—8081no—
93C8—4041no—
94C22—41no—
95C22—81no—
96C22—16161no—
97C22—4041no—
98C22#1569216911yescharacteristic
99C4—21no—
100C4—41no—
101C4—41no—
102C4—4041no—
103C2—21no—
104C2—81no—
105C2—8081no—
106C2#2026496511yescharacteristic
subgroupG/H
centerC2#20264965
commutatorC2×D4×C101C2×C4
Frattini#124C2×D101
Fitting#329×C101C2
socleC2×C101AΣL(1,𝔽16)
prime radical#15692300C4
cosocle#124×C101C22
hypercenter#329D101
hypocenterC101#1403
LCSG ▷ C2×D4×C101 ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #20264974C2semidirect
3. #329×C101C2semidirect
54. C101#1403semidirect
57. #329D101semidirect

Maximal subgroups

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

subgroupindex
#202653542
#202649742
#329×C1012
#1403101