Finite Group Databasebeta

← #27139887

Subgroups — #27139887

Order 12416.

112 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 25 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#20260530C211yesmaximal, characteristic, complemented
2#20260475C211yesmaximal, characteristic, complemented
3#454×C97C211yesmaximal, characteristic
4#15690725—21no—
5#15690721—21no—
62−1+4×C97C411yescharacteristic, complemented
7AGL(1,ℤ/8)×C97—21no—
8#124×C97C2211yescharacteristic
9#104×C97—21no—
10#101×C97C411yescharacteristic, complemented
11C2×#11715895—41no—
12C22⋊Dic97—41no—
13#41×C97—21no—
14#41×C97—41no—
15#41×C97—41no—
16C2×Q8×C97—41no—
17C2×Q8×C97C2×C411yescharacteristic
18C2×D4×C97D411yescharacteristic
19SD16×C97—41no—
20D8×C97—41no—
21M16×C97—21no—
22C22⋊C4×C97—41no—
23C42×C97D411yescharacteristic
24C23×C97—21no—
25Q8×C97—21no—
26Q8×C97—41no—
27Q8×C97—41no—
28D4×C97—21no—
29D4×C97—41no—
30D4×C97—41no—
31D4×C97—41no—
32C2×C4×C97—21no—
33C2×C4×C97—21no—
34C2×C4×C97—41no—
35C2×C4×C97—42no—
36C2×C4×C97C22⋊C411yescharacteristic
37C2×Dic97—81no—
38C8×C97—41no—
39#11715895—42no—
40C22×C97—41no—
41C22×C97—41no—
42C22×C97—81no—
43C22×C97#9911yescharacteristic
44C4×C97—21no—
45C4×C97—41no—
46C4×C97—41no—
47C4×C97—41no—
48Dic97—161no—
49C2×C97—21no—
50C2×C97—81no—
51C2×C97—81no—
52C2×C97AΣL(1,𝔽16)11yescharacteristic
53#1404—971nomaximal, Sylow
54C97#140411yescharacteristic, complemented, Sylow
55#454D9711yescharacteristic
56#354—971no—
57#329—971no—
582−1+4Dic9711yescharacteristic, complemented
59AGL(1,ℤ/8)—21no—
60#124C2×D9711yescharacteristic
61#104—21no—
62#101Dic9711yescharacteristic, complemented
63#99—1941no—
64#97—1941no—
65#41—21no—
66#41—41no—
67#41—41no—
68C2×Q8—41no—
69C2×Q8C2×Dic9711yescharacteristic
70C2×D4C22⋊D9711yescharacteristic
71SD16—41no—
72D8—41no—
73M16—21no—
74C2×C8—3881no—
75C22⋊C4—41no—
76C22⋊C4—3881no—
77C42C22⋊D9711yescharacteristic
78C23—21no—
79Q8—21no—
80Q8—41no—
81Q8—41no—
82D4—21no—
83D4—41no—
84D4—41no—
85D4—41no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—41no—
89C2×C4—42no—
90C2×C4—7761no—
91C2×C4C22⋊Dic9711yescharacteristic
92C8—41no—
93C8—3882no—
94C22—41no—
95C22—41no—
96C22—81no—
97C22#1569072511yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—41no—
102C4—15521no—
103C2—21no—
104C2—81no—
105C2—81no—
106C2#2026052211yescharacteristic
subgroupG/H
centerC2#20260522
commutatorC2×Q8×C97C2×C4
Frattini#124C2×D97
Fitting, prime radical#454×C97C2
socleC2×C97AΣL(1,𝔽16)
cosocle#124×C97C22
hypercenter#454D97
hypocenterC97#1404
LCSG ▷ C2×Q8×C97 ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #20260530C2semidirect
2. #20260475C2semidirect
6. 2−1+4×C97C4semidirect
10. #101×C97C4semidirect
54. C97#1404semidirect
58. 2−1+4Dic97semidirect
62. #101Dic97semidirect

Maximal subgroups

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

subgroupindex
#202605302
#202604752
#454×C972
#140497