Finite Group Databasebeta

← #27139888

Subgroups — #27139888

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#20260907C211yesmaximal, characteristic
2#20260530C211yesmaximal, characteristic, complemented
3#329×C97C211yesmaximal, characteristic, complemented
4#15690858C411yescharacteristic, complemented
5#15690836—21no—
6#15690738—21no—
7#15690729C411yescharacteristic, complemented
8#15690725—21no—
9#124×C97C2211yescharacteristic
10#97×C97—21no—
11#13095512—41no—
12#13095511—21no—
13#13095511—41no—
14Q8×D97—41no—
15#13095499—41no—
16#13095496—41no—
17#13095495—21no—
18C22⋊Dic97—41no—
19C2×Q8×C97C2×C411yescharacteristic
20C2×D4×C97D411yescharacteristic
21C2×C8×C97—41no—
22C22⋊C4×C97—41no—
23C42×C97D411yescharacteristic
24C23×C97—21no—
25Q8×C97—41no—
26D4×C97—41no—
27C2×C4×C97—21no—
28C2×C4×C97—21no—
29C2×C4×C97C22⋊C411yescharacteristic
30C22⋊D97—41no—
31C2×Dic97—81no—
32D388—21no—
33D388—41no—
34C4×D97—41no—
35C4×D97—42no—
36Dic194—21no—
37Dic194—41no—
38C8×C97—42no—
39#11715895—41no—
40C22×C97—41no—
41C22×C97—81no—
42C22×C97#9911yescharacteristic
43C2×D97—41no—
44C4×C97—21no—
45C4×C97—41no—
46C4×C97—41no—
47Dic97—41no—
48Dic97—161no—
49C2×C97—21no—
50C2×C97—81no—
51C2×C97AΣL(1,𝔽16)11yescharacteristic
52D97—81no—
53#1404—971nomaximal, Sylow
54C97#140411yescharacteristic, complemented, Sylow
55#454—971no—
56#354—971no—
57#329D9711yescharacteristic, complemented
582−1+4—971no—
59AGL(1,ℤ/8)—1941no—
60#124C2×D9711yescharacteristic
61#104—1941no—
62#101—971no—
63#99—1941no—
64#97—21no—
65#41—3881no—
66#41—3881no—
67#41—1941no—
68C2×Q8—3881no—
69C2×Q8C4×D9711yescharacteristic
70C2×D4D38811yescharacteristic
71SD16—3881no—
72D8—3881no—
73M16—1941no—
74C2×C8—41no—
75C22⋊C4—41no—
76C22⋊C4—3881no—
77C42C22⋊D9711yescharacteristic
78C23—21no—
79Q8—41no—
80Q8—3881no—
81Q8—1941no—
82D4—41no—
83D4—3881no—
84D4—3881no—
85D4—1941no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—7761no—
89C2×C4—3881no—
90C2×C4—3882no—
91C2×C4#1309548811yescharacteristic
92C8—42no—
93C8—3881no—
94C22—41no—
95C22—81no—
96C22—3881no—
97C22#1569072411yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—15521no—
102C4—3881no—
103C2—21no—
104C2—81no—
105C2—7761no—
106C2#2026052311yescharacteristic
subgroupG/H
centerC2#20260523
commutatorC2×Q8×C97C2×C4
Frattini#124C2×D97
Fitting#329×C97C2
socleC2×C97AΣL(1,𝔽16)
prime radical#20260907C2
cosocle#124×C97C22
hypercenter#329D97
hypocenterC97#1404
LCSG ▷ C2×Q8×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. #20260530C2semidirect
3. #329×C97C2semidirect
4. #15690858C4semidirect
7. #15690729C4semidirect
54. C97#1404semidirect
57. #329D97semidirect

Maximal subgroups

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

subgroupindex
#202609072
#202605302
#329×C972
#140497