Finite Group Databasebeta

← #43252946

Subgroups — #43252946

Order 28544.

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#28383762C211yesmaximal, characteristic
2#28383329C211yesmaximal, characteristic, complemented
3#356×C223C211yesmaximal, characteristic, complemented
4#20297352C411yescharacteristic
5#20297335—21no—
6C4⋊(C223⋊C8)—21no—
7#20297232—21no—
8#20297223C411yescharacteristic
9C4⋊Q8×C223C2211yescharacteristic
10#101×C223—21no—
11#16813171—41no—
12#16813170—21no—
13#16813170—41no—
14Q8×D223—41no—
15#16813160—41no—
16#16813157—41no—
17#16813154—21no—
18C2×C223⋊C8—41no—
19C2×Q8×C223C2×C411yescharacteristic
20C2×Q8×C223D411yescharacteristic
21M16×C223—41no—
22C4⋊C4×C223—41no—
23C42×C223D411yescharacteristic
24Dic446—21no—
25Dic446—41no—
26D892—21no—
27D892—41no—
28C22⋊D223—41no—
29C4×D223—41no—
30C4×D223—42no—
31C223⋊C8—41no—
32C223⋊C8—81no—
33Q8×C223—41no—
34Q8×C223—41no—
35C2×C4×C223—21no—
36C2×C4×C223—21no—
37C2×C4×C223—21no—
38C2×C4×C223C22⋊C411yescharacteristic
39C8×C223—81no—
40C22×C223#9911yescharacteristic
41C2×D223—41no—
42C4×C223—21no—
43C4×C223—21no—
44C4×C223—21no—
45C4×C223—41no—
46C4×C223—41no—
47Dic223—41no—
48C2×C223—21no—
49C2×C223AΣL(1,𝔽16)11yescharacteristic
50D223—81no—
51C223#140511yescharacteristic, complemented, Sylow
52#1405—2231nomaximal, Sylow
53#456—2231no—
54#356D22311yescharacteristic, complemented
55#333—2231no—
562−1+4—2231no—
57#137—4461no—
58C4⋊Q8C2×D22311yescharacteristic
59C4⋊C8—4461no—
60#104—4461no—
61#101—21no—
62#101—2231no—
63#41—8921no—
64#41—8921no—
65#41—4461no—
66C2×Q8—8921no—
67C2×Q8C4×D22311yescharacteristic
68C2×Q8C22⋊D22311yescharacteristic
69Q16—8921no—
70SD16—8921no—
71M16—41no—
72M16—4461no—
73C2×C8—8921no—
74C4⋊C4—41no—
75C42D89211yescharacteristic
76Q8—41no—
77Q8—41no—
78Q8—8921no—
79Q8—4461no—
80D4—8921no—
81D4—8921no—
82D4—4461no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—8921no—
87C2×C4—8922no—
88C2×C4(C2×D223)⋊C411yescharacteristic
89C8—81no—
90C8—8921no—
91C8—17841no—
92C22—8921no—
93C22#2029721811yescharacteristic
94C4—21no—
95C4—21no—
96C4—21no—
97C4—41no—
98C4—41no—
99C4—8921no—
100C2—21no—
101C2—17841no—
102C2#2838336611yescharacteristic
subgroupG/H
centerC2#28383366
commutatorC2×Q8×C223C2×C4
FrattiniC4⋊Q8C2×D223
Fitting#356×C223C2
socleC2×C223AΣL(1,𝔽16)
prime radical#20297352C4
cosocleC4⋊Q8×C223C22
hypercenter#356D223
hypocenterC223#1405
LCSG ▷ C2×Q8×C223 ▷ C2×C4×C223 ▷ C22×C223 ▷ C2×C223 ▷ C223
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
2. #28383329C2semidirect
3. #356×C223C2semidirect
51. C223#1405semidirect
54. #356D223semidirect

Maximal subgroups

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

subgroupindex
#283837622
#283833292
#356×C2232
#1405223