Finite Group Databasebeta

← #38766997

Subgroups — #38766997

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#27164195C211yesmaximal, characteristic
2#27163758C211yesmaximal, characteristic, complemented
3#355×C199C211yesmaximal, characteristic, complemented
4#20264000C411yescharacteristic
5#20263986—21no—
6#20263883—21no—
7#20263872C411yescharacteristic
8#20263866—21no—
9#124×C199C2211yescharacteristic
10#101×C199—21no—
11#15691886—21no—
12#15691885—41no—
13C2×C22⋊D199—41no—
14D4×D199—41no—
15#15691876—41no—
16#15691873—41no—
17#15691870—21no—
18C2×C199⋊C8—41no—
19C2×Q8×C199D411yescharacteristic
20C2×D4×C199C2×C411yescharacteristic
21M16×C199—41no—
22C22⋊C4×C199—41no—
23C42×C199D411yescharacteristic
24C22×D199—41no—
25Dic398—21no—
26D796—21no—
27C22⋊D199—42no—
28C22⋊D199—41no—
29C2×Dic199—41no—
30C4×D199—41no—
31C199⋊C8—41no—
32C199⋊C8—42no—
33C23×C199—21no—
34Q8×C199—41no—
35D4×C199—41no—
36C2×C4×C199—21no—
37C2×C4×C199—21no—
38C2×C4×C199C22⋊C411yescharacteristic
39C8×C199—81no—
40C22×C199—41no—
41C22×C199—81no—
42C22×C199#9911yescharacteristic
43C2×D199—41no—
44C2×D199—161no—
45C4×C199—21no—
46C4×C199—41no—
47C4×C199—41no—
48Dic199—41no—
49C2×C199—21no—
50C2×C199—81no—
51C2×C199AΣL(1,𝔽16)11yescharacteristic
52D199—81no—
53C199#140311yescharacteristic, complemented, Sylow
54#1403—1991nomaximal, Sylow
55#455—1991no—
56#355D19911yescharacteristic, complemented
57#329—1991no—
58H(5,𝔽2)—1991no—
59#137—3981no—
60#124C2×D19911yescharacteristic
61#104—3981no—
62#101—21no—
63#100—1991no—
64#97—3981no—
65#41—7961no—
66#41—3981no—
67C2×Q8C22⋊D19911yescharacteristic
68C2×D4—7961no—
69C2×D4—7961no—
70C2×D4C4×D19911yescharacteristic
71Q16—7961no—
72SD16—7961no—
73M16—41no—
74M16—3981no—
75C2×C8—7961no—
76C22⋊C4—41no—
77C42D79611yescharacteristic
78C23—21no—
79C23—7961no—
80Q8—41no—
81Q8—3981no—
82D4—41no—
83D4—7962no—
84D4—7961no—
85D4—3981no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—7961no—
89C2×C4—7961no—
90C2×C4(C2×D199)⋊C411yescharacteristic
91C8—81no—
92C8—7961no—
93C8—7962no—
94C22—41no—
95C22—81no—
96C22—7961no—
97C22—31841no—
98C22#2026386911yescharacteristic
99C4—21no—
100C4—41no—
101C4—41no—
102C4—7961no—
103C2—21no—
104C2—81no—
105C2—15921no—
106C2#2716380411yescharacteristic
subgroupG/H
centerC2#27163804
commutatorC2×D4×C199C2×C4
Frattini#124C2×D199
Fitting#355×C199C2
socleC2×C199AΣL(1,𝔽16)
prime radical#20264000C4
cosocle#124×C199C22
hypercenter#355D199
hypocenterC199#1403
LCSG ▷ C2×D4×C199 ▷ C2×C4×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #27163758C2semidirect
3. #355×C199C2semidirect
53. C199#1403semidirect
56. #355D199semidirect

Maximal subgroups

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

subgroupindex
#271641952
#271637582
#355×C1992
#1403199