Finite Group Databasebeta

← #43252945

Subgroups — #43252945

Order 28544.

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#28383752C211yesmaximal, characteristic
2#28383375C211yesmaximal, characteristic, complemented
3#329×C223C211yesmaximal, characteristic, complemented
4#20297352C411yescharacteristic, complemented
5#20297330—21no—
6#20297232—21no—
7#20297223C411yescharacteristic, complemented
8#20297219—21no—
9#124×C223C2211yescharacteristic
10#97×C223—21no—
11#16813171—41no—
12#16813170—21no—
13#16813170—41no—
14Q8×D223—41no—
15#16813158—41no—
16#16813155—41no—
17#16813154—21no—
18C22⋊Dic223—41no—
19C2×Q8×C223C2×C411yescharacteristic
20C2×D4×C223D411yescharacteristic
21C2×C8×C223—41no—
22C22⋊C4×C223—41no—
23C42×C223D411yescharacteristic
24Dic446—21no—
25Dic446—41no—
26D892—21no—
27D892—41no—
28C22⋊D223—41no—
29C2×Dic223—81no—
30C4×D223—41no—
31C4×D223—42no—
32C223⋊C8—41no—
33C23×C223—21no—
34Q8×C223—41no—
35D4×C223—41no—
36C2×C4×C223—21no—
37C2×C4×C223—21no—
38C2×C4×C223C22⋊C411yescharacteristic
39C8×C223—42no—
40C22×C223—41no—
41C22×C223—81no—
42C22×C223#9911yescharacteristic
43C2×D223—41no—
44C4×C223—21no—
45C4×C223—41no—
46C4×C223—41no—
47Dic223—41no—
48Dic223—161no—
49C2×C223—21no—
50C2×C223—81no—
51C2×C223AΣL(1,𝔽16)11yescharacteristic
52D223—81no—
53C223#140411yescharacteristic, complemented, Sylow
54#1404—2231nomaximal, Sylow
55#454—2231no—
56#354—2231no—
57#329D22311yescharacteristic, complemented
582−1+4—2231no—
59AGL(1,ℤ/8)—4461no—
60#124C2×D22311yescharacteristic
61#104—4461no—
62#101—2231no—
63#99—4461no—
64#97—21no—
65#41—8921no—
66#41—8921no—
67#41—4461no—
68C2×Q8—8921no—
69C2×Q8C4×D22311yescharacteristic
70C2×D4D89211yescharacteristic
71SD16—8921no—
72D8—8921no—
73M16—4461no—
74C2×C8—41no—
75C22⋊C4—41no—
76C22⋊C4—8921no—
77C42C22⋊D22311yescharacteristic
78C23—21no—
79Q8—41no—
80Q8—8921no—
81Q8—4461no—
82D4—41no—
83D4—8921no—
84D4—8921no—
85D4—4461no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—8921no—
89C2×C4—8922no—
90C2×C4—17841no—
91C2×C4(C2×D223)⋊C411yescharacteristic
92C8—42no—
93C8—8921no—
94C22—41no—
95C22—81no—
96C22—8921no—
97C22#2029721811yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—8921no—
102C4—35681no—
103C2—21no—
104C2—81no—
105C2—17841no—
106C2#2838336811yescharacteristic
subgroupG/H
centerC2#28383368
commutatorC2×Q8×C223C2×C4
Frattini#124C2×D223
Fitting#329×C223C2
socleC2×C223AΣL(1,𝔽16)
prime radical#28383752C2
cosocle#124×C223C22
hypercenter#329D223
hypocenterC223#1404
LCSG ▷ C2×Q8×C223 ▷ C2×C4×C223 ▷ C22×C223 ▷ C2×C223 ▷ C223
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #28383375C2semidirect
3. #329×C223C2semidirect
4. #20297352C4semidirect
7. #20297223C4semidirect
53. C223#1404semidirect
57. #329D223semidirect

Maximal subgroups

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

subgroupindex
#283837522
#283833752
#329×C2232
#1404223