Finite Group Databasebeta

← #38767002

Subgroups — #38767002

Order 25472.

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#27164190C211yesmaximal, characteristic
2#27163813C211yesmaximal, characteristic, complemented
3#329×C199C211yesmaximal, characteristic, complemented
4#20264003C411yescharacteristic, complemented
5#20263981—21no—
6#20263883—21no—
7#20263874C411yescharacteristic, complemented
8#20263870—21no—
9#124×C199C2211yescharacteristic
10#97×C199—21no—
11#15691887—41no—
12#15691886—21no—
13#15691886—41no—
14Q8×D199—41no—
15#15691874—41no—
16#15691871—41no—
17#15691870—21no—
18C22⋊Dic199—41no—
19C2×Q8×C199C2×C411yescharacteristic
20C2×D4×C199D411yescharacteristic
21C2×C8×C199—41no—
22C22⋊C4×C199—41no—
23C42×C199D411yescharacteristic
24Dic398—21no—
25Dic398—41no—
26D796—21no—
27D796—41no—
28C22⋊D199—41no—
29C2×Dic199—81no—
30C4×D199—41no—
31C4×D199—42no—
32C199⋊C8—41no—
33C23×C199—21no—
34Q8×C199—41no—
35D4×C199—41no—
36C2×C4×C199—21no—
37C2×C4×C199—21no—
38C2×C4×C199C22⋊C411yescharacteristic
39C8×C199—42no—
40C22×C199—41no—
41C22×C199—81no—
42C22×C199#9911yescharacteristic
43C2×D199—41no—
44C4×C199—21no—
45C4×C199—41no—
46C4×C199—41no—
47Dic199—41no—
48Dic199—161no—
49C2×C199—21no—
50C2×C199—81no—
51C2×C199AΣL(1,𝔽16)11yescharacteristic
52D199—81no—
53C199#140411yescharacteristic, complemented, Sylow
54#1404—1991nomaximal, Sylow
55#454—1991no—
56#354—1991no—
57#329D19911yescharacteristic, complemented
582−1+4—1991no—
59AGL(1,ℤ/8)—3981no—
60#124C2×D19911yescharacteristic
61#104—3981no—
62#101—1991no—
63#99—3981no—
64#97—21no—
65#41—7961no—
66#41—7961no—
67#41—3981no—
68C2×Q8—7961no—
69C2×Q8C4×D19911yescharacteristic
70C2×D4D79611yescharacteristic
71SD16—7961no—
72D8—7961no—
73M16—3981no—
74C2×C8—41no—
75C22⋊C4—41no—
76C22⋊C4—7961no—
77C42C22⋊D19911yescharacteristic
78C23—21no—
79Q8—41no—
80Q8—7961no—
81Q8—3981no—
82D4—41no—
83D4—7961no—
84D4—7961no—
85D4—3981no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—7961no—
89C2×C4—7962no—
90C2×C4—15921no—
91C2×C4(C2×D199)⋊C411yescharacteristic
92C8—42no—
93C8—7961no—
94C22—41no—
95C22—81no—
96C22—7961no—
97C22#2026386911yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—7961no—
102C4—31841no—
103C2—21no—
104C2—81no—
105C2—15921no—
106C2#2716380611yescharacteristic
subgroupG/H
centerC2#27163806
commutatorC2×Q8×C199C2×C4
Frattini#124C2×D199
Fitting#329×C199C2
socleC2×C199AΣL(1,𝔽16)
prime radical#27164190C2
cosocle#124×C199C22
hypercenter#329D199
hypocenterC199#1404
LCSG ▷ C2×Q8×C199 ▷ C2×C4×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #27163813C2semidirect
3. #329×C199C2semidirect
4. #20264003C4semidirect
7. #20263874C4semidirect
53. C199#1404semidirect
57. #329D199semidirect

Maximal subgroups

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

subgroupindex
#271641902
#271638132
#329×C1992
#1404199