Finite Group Databasebeta

← #28334615

Subgroups — #28334615

Order 13696.

106 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 106 conjugacy classes of subgroups — too many to draw. Only its 25 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#20292426C211yesmaximal, characteristic, complemented
2#20292376C211yesmaximal, characteristic, complemented
3#456×C107C211yesmaximal, characteristic
4C4⋊(C107⋊C8)—21no—
5#16812028—21no—
62−1+4×C107C411yescharacteristic
7#137×C107—21no—
8C4⋊Q8×C107C2211yescharacteristic
9#104×C107—21no—
10#101×C107C411yescharacteristic
11#13133802—41no—
12C2×C107⋊C8—41no—
13#41×C107—21no—
14#41×C107—41no—
15#41×C107—41no—
16C2×Q8×C107—41no—
17C2×Q8×C107C2×C411yescharacteristic
18C2×Q8×C107D411yescharacteristic
19Q16×C107—41no—
20SD16×C107—41no—
21M16×C107—21no—
22C4⋊C4×C107—41no—
23C42×C107D411yescharacteristic
24Q8×C107—21no—
25Q8×C107—41no—
26Q8×C107—41no—
27Q8×C107—41no—
28D4×C107—21no—
29D4×C107—41no—
30D4×C107—41no—
31C2×C4×C107—21no—
32C2×C4×C107—21no—
33C2×C4×C107—21no—
34C2×C4×C107—41no—
35C2×C4×C107—42no—
36C2×C4×C107C22⋊C411yescharacteristic
37C8×C107—41no—
38C107⋊C8—81no—
39C107⋊C8—81no—
40C22×C107—41no—
41C22×C107#9911yescharacteristic
42C4×C107—21no—
43C4×C107—21no—
44C4×C107—21no—
45C4×C107—41no—
46C4×C107—41no—
47C4×C107—41no—
48C2×C107—21no—
49C2×C107—81no—
50C2×C107AΣL(1,𝔽16)11yescharacteristic
51#1405—1071nomaximal, Sylow
52C107#140511yescharacteristic, complemented, Sylow
53#456D10711yescharacteristic
54#356—1071no—
55#333—1071no—
562−1+4Dic10711yescharacteristic
57#137—21no—
58C4⋊Q8C2×D10711yescharacteristic
59C4⋊C8—2141no—
60#104—21no—
61#101—2141no—
62#101Dic10711yescharacteristic
63#41—21no—
64#41—41no—
65#41—41no—
66C2×Q8—41no—
67C2×Q8C2×Dic10711yescharacteristic
68C2×Q8C22⋊D10711yescharacteristic
69Q16—41no—
70SD16—41no—
71M16—21no—
72M16—4281no—
73C2×C8—4281no—
74C4⋊C4—41no—
75C42C22⋊D10711yescharacteristic
76Q8—21no—
77Q8—41no—
78Q8—41no—
79Q8—41no—
80D4—21no—
81D4—41no—
82D4—41no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—41no—
87C2×C4—42no—
88C2×C4C22⋊Dic10711yescharacteristic
89C8—41no—
90C8—8561no—
91C8—8561no—
92C22—41no—
93C22#1681202411yescharacteristic
94C4—21no—
95C4—21no—
96C4—21no—
97C4—41no—
98C4—41no—
99C4—41no—
100C2—21no—
101C2—81no—
102C2#2029241411yescharacteristic
subgroupG/H
centerC2#20292414
commutatorC2×Q8×C107C2×C4
FrattiniC4⋊Q8C2×D107
Fitting#456×C107C2
socleC2×C107AΣL(1,𝔽16)
prime radical2−1+4×C107C4
cosocleC4⋊Q8×C107C22
hypercenter#456D107
hypocenterC107#1405
LCSG ▷ C2×Q8×C107 ▷ C2×C4×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #20292426C2semidirect
2. #20292376C2semidirect
52. C107#1405semidirect

Maximal subgroups

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

subgroupindex
#202924262
#202923762
#456×C1072
#1405107