Finite Group Databasebeta

← #45662467

Subgroups — #45662467

Order 32896.

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#30656914C211yesmaximal, characteristic
2#30656477C211yesmaximal, characteristic, complemented
3#355×C257C211yesmaximal, characteristic, complemented
4#22551637C411yescharacteristic
5#22551623—21no—
6#22551520—21no—
7#22551509C411yescharacteristic
8#22551503—21no—
9#124×C257C2211yescharacteristic
10#101×C257—21no—
11#16861618—21no—
12#16861617—41no—
13C2×C22⋊D257—41no—
14D4×D257—41no—
15#16861608—41no—
16#16861605—41no—
17#16861602—21no—
18C2×#14523339—41no—
19C2×Q8×C257D411yescharacteristic
20C2×D4×C257C2×C411yescharacteristic
21M16×C257—41no—
22C22⋊C4×C257—41no—
23C42×C257D411yescharacteristic
24C22×D257—41no—
25Dic514—21no—
26D1028—21no—
27C22⋊D257—42no—
28C22⋊D257—41no—
29C2×Dic257—41no—
30C4×D257—41no—
31#14523339—41no—
32#14523339—42no—
33C23×C257—21no—
34Q8×C257—41no—
35D4×C257—41no—
36C2×C4×C257—21no—
37C2×C4×C257—21no—
38C2×C4×C257C22⋊C411yescharacteristic
39C8×C257—81no—
40C22×C257—41no—
41C22×C257—81no—
42C22×C257#9911yescharacteristic
43C2×D257—41no—
44C2×D257—161no—
45C4×C257—21no—
46C4×C257—41no—
47C4×C257—41no—
48Dic257—41no—
49C2×C257—21no—
50C2×C257—81no—
51C2×C257AΣL(1,𝔽16)11yescharacteristic
52D257—81no—
53C257#140311yescharacteristic, complemented, Sylow
54#1403—2571nomaximal, Sylow
55#455—2571no—
56#355D25711yescharacteristic, complemented
57#329—2571no—
58H(5,𝔽2)—2571no—
59#137—5141no—
60#124C2×D25711yescharacteristic
61#104—5141no—
62#101—21no—
63#100—2571no—
64#97—5141no—
65#41—5141no—
66#41—10281no—
67C2×Q8C22⋊D25711yescharacteristic
68C2×D4—10281no—
69C2×D4—10281no—
70C2×D4C4×D25711yescharacteristic
71Q16—10281no—
72SD16—10281no—
73M16—5141no—
74M16—41no—
75C2×C8—10281no—
76C22⋊C4—41no—
77C42D102811yescharacteristic
78C23—21no—
79C23—10281no—
80Q8—5141no—
81Q8—41no—
82D4—5141no—
83D4—41no—
84D4—10282no—
85D4—10281no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—10281no—
89C2×C4—10281no—
90C2×C4#1686159511yescharacteristic
91C8—10281no—
92C8—10282no—
93C8—81no—
94C22—41no—
95C22—10281no—
96C22—81no—
97C22—41121no—
98C22#2255150611yescharacteristic
99C4—21no—
100C4—41no—
101C4—41no—
102C4—10281no—
103C2—21no—
104C2—81no—
105C2—20561no—
106C2#3065652311yescharacteristic
subgroupG/H
centerC2#30656523
commutatorC2×D4×C257C2×C4
Frattini#124C2×D257
Fitting#355×C257C2
socleC2×C257AΣL(1,𝔽16)
prime radical#22551637C4
cosocle#124×C257C22
hypercenter#355D257
hypocenterC257#1403
LCSG ▷ C2×D4×C257 ▷ C2×C4×C257 ▷ C22×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #30656477C2semidirect
3. #355×C257C2semidirect
53. C257#1403semidirect
56. #355D257semidirect

Maximal subgroups

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

subgroupindex
#306569142
#306564772
#355×C2572
#1403257