Finite Group Databasebeta

← #45662473

Subgroups — #45662473

Order 32896.

106 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 106 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#30656919C211yesmaximal, characteristic
2#30656486C211yesmaximal, characteristic, complemented
3#356×C257C211yesmaximal, characteristic, complemented
4#22551640C411yescharacteristic
5#22551623—21no—
6C4⋊#14523339—21no—
7#22551520—21no—
8#22551511C411yescharacteristic
9C4⋊Q8×C257C2211yescharacteristic
10#101×C257—21no—
11#16861619—41no—
12#16861618—21no—
13#16861618—41no—
14Q8×D257—41no—
15#16861608—41no—
16#16861605—41no—
17#16861602—21no—
18C2×#14523339—41no—
19C2×Q8×C257C2×C411yescharacteristic
20C2×Q8×C257D411yescharacteristic
21M16×C257—41no—
22C4⋊C4×C257—41no—
23C42×C257D411yescharacteristic
24Dic514—21no—
25Dic514—41no—
26D1028—21no—
27D1028—41no—
28C22⋊D257—41no—
29C4×D257—41no—
30C4×D257—42no—
31#14523339—41no—
32#14523339—81no—
33Q8×C257—41no—
34Q8×C257—41no—
35C2×C4×C257—21no—
36C2×C4×C257—21no—
37C2×C4×C257—21no—
38C2×C4×C257C22⋊C411yescharacteristic
39C8×C257—81no—
40C22×C257#9911yescharacteristic
41C2×D257—41no—
42C4×C257—21no—
43C4×C257—21no—
44C4×C257—21no—
45C4×C257—41no—
46C4×C257—41no—
47Dic257—41no—
48C2×C257—21no—
49C2×C257AΣL(1,𝔽16)11yescharacteristic
50D257—81no—
51C257#140511yescharacteristic, complemented, Sylow
52#1405—2571nomaximal, Sylow
53#456—2571no—
54#356D25711yescharacteristic, complemented
55#333—2571no—
562−1+4—2571no—
57#137—5141no—
58C4⋊Q8C2×D25711yescharacteristic
59C4⋊C8—5141no—
60#104—5141no—
61#101—2571no—
62#101—21no—
63#41—5141no—
64#41—10281no—
65#41—10281no—
66C2×Q8—10281no—
67C2×Q8C4×D25711yescharacteristic
68C2×Q8C22⋊D25711yescharacteristic
69Q16—10281no—
70SD16—10281no—
71M16—5141no—
72M16—41no—
73C2×C8—10281no—
74C4⋊C4—41no—
75C42D102811yescharacteristic
76Q8—5141no—
77Q8—41no—
78Q8—41no—
79Q8—10281no—
80D4—5141no—
81D4—10281no—
82D4—10281no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—10281no—
87C2×C4—10282no—
88C2×C4#1686159511yescharacteristic
89C8—10281no—
90C8—81no—
91C8—20561no—
92C22—10281no—
93C22#2255150611yescharacteristic
94C4—21no—
95C4—21no—
96C4—21no—
97C4—41no—
98C4—41no—
99C4—10281no—
100C2—21no—
101C2—20561no—
102C2#3065652311yescharacteristic
subgroupG/H
centerC2#30656523
commutatorC2×Q8×C257C2×C4
FrattiniC4⋊Q8C2×D257
Fitting#356×C257C2
socleC2×C257AΣL(1,𝔽16)
prime radical#22551640C4
cosocleC4⋊Q8×C257C22
hypercenter#356D257
hypocenterC257#1405
LCSG ▷ C2×Q8×C257 ▷ C2×C4×C257 ▷ C22×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
2. #30656486C2semidirect
3. #356×C257C2semidirect
51. C257#1405semidirect
54. #356D257semidirect

Maximal subgroups

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

subgroupindex
#306569192
#306564862
#356×C2572
#1405257