Finite Group Databasebeta

← #27139885

Subgroups — #27139885

Order 12416.

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#20260912C211yesmaximal, characteristic
2#20260532C211yesmaximal, characteristic, complemented
3#329×C97C211yesmaximal, characteristic, complemented
4#15690855C411yescharacteristic
5#15690841—21no—
6#15690738—21no—
7#15690729—21no—
8#15690727C411yescharacteristic
9#124×C97C2211yescharacteristic
10#97×C97—21no—
11#13095511—21no—
12#13095510—41no—
13C2×C22⋊D97—41no—
14D4×D97—41no—
15#13095501—41no—
16#13095498—41no—
17#13095495—21no—
18#13095495—41no—
19C2×Q8×C97D411yescharacteristic
20C2×D4×C97C2×C411yescharacteristic
21C2×C8×C97—41no—
22C22⋊C4×C97—41no—
23C42×C97D411yescharacteristic
24C23×C97—21no—
25C22×D97—41no—
26Q8×C97—41no—
27D4×C97—41no—
28C2×C4×C97—21no—
29C2×C4×C97—21no—
30C2×C4×C97C22⋊C411yescharacteristic
31C22⋊D97—42no—
32C22⋊D97—41no—
33C2×Dic97—41no—
34D388—21no—
35C4×D97—41no—
36Dic194—21no—
37C8×C97—42no—
38#11715895—41no—
39#11715895—81no—
40C22×C97—41no—
41C22×C97—81no—
42C22×C97#9911yescharacteristic
43C2×D97—41no—
44C2×D97—161no—
45C4×C97—21no—
46C4×C97—41no—
47C4×C97—41no—
48Dic97—41no—
49C2×C97—21no—
50C2×C97—81no—
51C2×C97AΣL(1,𝔽16)11yescharacteristic
52D97—81no—
53#1403—971nomaximal, Sylow
54C97#140311yescharacteristic, complemented, Sylow
55#455—971no—
56#355—971no—
57#329D9711yescharacteristic, complemented
58H(5,𝔽2)—971no—
59#137—1941no—
60#124C2×D9711yescharacteristic
61#104—1941no—
62#101—1941no—
63#100—971no—
64#97—21no—
65#41—3881no—
66#41—1941no—
67C2×Q8D38811yescharacteristic
68C2×D4—3881no—
69C2×D4—3881no—
70C2×D4C4×D9711yescharacteristic
71Q16—3881no—
72SD16—3881no—
73M16—3881no—
74M16—1941no—
75C2×C8—41no—
76C22⋊C4—41no—
77C42C22⋊D9711yescharacteristic
78C23—21no—
79C23—3881no—
80Q8—41no—
81Q8—1941no—
82D4—41no—
83D4—3882no—
84D4—3881no—
85D4—1941no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—3881no—
89C2×C4—3881no—
90C2×C4#1309548811yescharacteristic
91C8—42no—
92C8—7761no—
93C8—3881no—
94C22—41no—
95C22—81no—
96C22—15521no—
97C22—3881no—
98C22#1569072411yescharacteristic
99C4—21no—
100C4—41no—
101C4—41no—
102C4—3881no—
103C2—21no—
104C2—81no—
105C2—7761no—
106C2#2026052311yescharacteristic
subgroupG/H
centerC2#20260523
commutatorC2×D4×C97C2×C4
Frattini#124C2×D97
Fitting#329×C97C2
socleC2×C97AΣL(1,𝔽16)
prime radical#15690855C4
cosocle#124×C97C22
hypercenter#329D97
hypocenterC97#1403
LCSG ▷ C2×D4×C97 ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #20260532C2semidirect
3. #329×C97C2semidirect
54. C97#1403semidirect
57. #329D97semidirect

Maximal subgroups

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

subgroupindex
#202609122
#202605322
#329×C972
#140397