Finite Group Databasebeta

← #27139900

Subgroups — #27139900

Order 12416.

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

normal subgroups only

This group has 110 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#20261107C211yesmaximal, characteristic
2#20260528C211yesmaximal, characteristic
3#353×C97C211yesmaximal, characteristic
4C2×#13095499—21no—
5C2×#13095499—21no—
6#15690810C411yescharacteristic, complemented
7#15690725—21no—
8C4×#11715895C411yescharacteristic, complemented
9#127×C97C2211yescharacteristic
10#99×C97—21no—
11C2×Dic194—21no—
12#13095499—42no—
13#13095499—42no—
14C2×#11715895—21no—
15C4⋊Dic97—41no—
16C22⋊Dic97—41no—
17C2×D4×C97—41no—
18C2×D4×C97D411yescharacteristic
19C2×D4×C97D411yescharacteristic
20C22⋊C4×C97—41no—
21C42×C97C2×C411yescharacteristic
22C23×C97—21no—
23C23×C97—21no—
24D4×C97—41no—
25D4×C97—41no—
26D4×C97—82no—
27C2×C4×C97—21no—
28C2×C4×C97—81no—
29C2×C4×C97C22⋊C411yescharacteristic
30C2×Dic97—41no—
31C2×Dic97—81no—
32Dic194—41no—
33Dic194—41no—
34#11715895—22no—
35C22×C97—41no—
36C22×C97—41no—
37C22×C97—81no—
38C22×C97—81no—
39C22×C97#9911yescharacteristic
40C4×C97—22no—
41C4×C97—21no—
42C4×C97—161no—
43Dic97—81no—
44Dic97—161no—
45C2×C97—21no—
46C2×C97—81no—
47C2×C97—81no—
48C2×C97#35311yescharacteristic
49#1409—971nomaximal, Sylow
50C97#140911yescharacteristic, complemented, Sylow
51#492—971no—
52#353—971no—
53#353D9711yescharacteristic
54C2×SD16—1941no—
55C2×SD16—1941no—
56C4⋊Q8—971no—
57#127C2×D9711yescharacteristic
58#99—21no—
59#99—1941no—
60C4×C8—971no—
61C2×Q8—1941no—
62C2×D4—41no—
63C2×D4D38811yescharacteristic
64C2×D4C22⋊D9711yescharacteristic
65SD16—3882no—
66SD16—3882no—
67C2×C8—1941no—
68C4⋊C4—3881no—
69C22⋊C4—41no—
70C22⋊C4—3881no—
71C42C4×D9711yescharacteristic
72C23—21no—
73C23—21no—
74Q8—3881no—
75Q8—3881no—
76D4—41no—
77D4—41no—
78D4—82no—
79C2×C4—21no—
80C2×C4—81no—
81C2×C4—7761no—
82C2×C4—3881no—
83C2×C4#1309548811yescharacteristic
84C8—1942no—
85C22—41no—
86C22—41no—
87C22—81no—
88C22—81no—
89C22#1569072411yescharacteristic
90C4—22no—
91C4—21no—
92C4—7761no—
93C4—161no—
94C4—15521no—
95C2—21no—
96C2—81no—
97C2—81no—
98C2#2026052711yescharacteristic
subgroupG/H
centerC2#20260527
commutatorC42×C97C2×C4
Frattini#127C2×D97
Fitting#353×C97C2
socleC2×C97#353
prime radical, cosocle#127×C97C22
hypercenter#353D97
hypocenterC97#1409
LCSG ▷ C42×C97 ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #353

Complements

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

NH ≅ G/Ntype
6. #15690810C4semidirect
8. C4×#11715895C4semidirect
50. C97#1409semidirect

Maximal subgroups

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

subgroupindex
#202611072
#202605282
#353×C972
#140997