Finite Group Databasebeta

← #28334616

Subgroups — #28334616

Order 13696.

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#20292809C211yesmaximal, characteristic
2#20292426C211yesmaximal, characteristic, complemented
3#333×C107C211yesmaximal, characteristic, complemented
4#16812157C411yescharacteristic
5#16812140—21no—
6#16812037—21no—
7#16812028—21no—
8#16812028C411yescharacteristic
9C4⋊Q8×C107C2211yescharacteristic
10C4⋊C8×C107—21no—
11#13133819—41no—
12#13133818—21no—
13#13133818—41no—
14Q8×D107—41no—
15#13133808—41no—
16#13133805—41no—
17#13133802—21no—
18#13133802—41no—
19C2×Q8×C107C2×C411yescharacteristic
20C2×Q8×C107D411yescharacteristic
21C2×C8×C107—41no—
22C4⋊C4×C107—41no—
23C42×C107D411yescharacteristic
24Q8×C107—41no—
25Q8×C107—41no—
26C2×C4×C107—21no—
27C2×C4×C107—21no—
28C2×C4×C107—21no—
29C2×C4×C107C22⋊C411yescharacteristic
30C22⋊D107—41no—
31D428—21no—
32D428—41no—
33C4×D107—41no—
34C4×D107—42no—
35Dic214—21no—
36Dic214—41no—
37C8×C107—81no—
38C107⋊C8—41no—
39C107⋊C8—81no—
40C22×C107#9911yescharacteristic
41C2×D107—41no—
42C4×C107—21no—
43C4×C107—21no—
44C4×C107—21no—
45C4×C107—41no—
46C4×C107—41no—
47Dic107—41no—
48C2×C107—21no—
49C2×C107AΣL(1,𝔽16)11yescharacteristic
50D107—81no—
51#1405—1071nomaximal, Sylow
52C107#140511yescharacteristic, complemented, Sylow
53#456—1071no—
54#356—1071no—
55#333D10711yescharacteristic, complemented
562−1+4—1071no—
57#137—2141no—
58C4⋊Q8C2×D10711yescharacteristic
59C4⋊C8—21no—
60#104—2141no—
61#101—1071no—
62#101—2141no—
63#41—4281no—
64#41—4281no—
65#41—2141no—
66C2×Q8—4281no—
67C2×Q8C4×D10711yescharacteristic
68C2×Q8D42811yescharacteristic
69Q16—4281no—
70SD16—4281no—
71M16—4281no—
72M16—2141no—
73C2×C8—41no—
74C4⋊C4—41no—
75C42C22⋊D10711yescharacteristic
76Q8—41no—
77Q8—41no—
78Q8—4281no—
79Q8—2141no—
80D4—4281no—
81D4—4281no—
82D4—2141no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—4281no—
87C2×C4—4282no—
88C2×C4(C2×D107)⋊C411yescharacteristic
89C8—81no—
90C8—8561no—
91C8—4281no—
92C22—4281no—
93C22#1681202311yescharacteristic
94C4—21no—
95C4—21no—
96C4—21no—
97C4—41no—
98C4—41no—
99C4—4281no—
100C2—21no—
101C2—8561no—
102C2#2029241511yescharacteristic
subgroupG/H
centerC2#20292415
commutatorC2×Q8×C107C2×C4
FrattiniC4⋊Q8C2×D107
Fitting#333×C107C2
socleC2×C107AΣL(1,𝔽16)
prime radical#16812157C4
cosocleC4⋊Q8×C107C22
hypercenter#333D107
hypocenterC107#1405
LCSG ▷ C2×Q8×C107 ▷ C2×C4×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #20292426C2semidirect
3. #333×C107C2semidirect
52. C107#1405semidirect
55. #333D107semidirect

Maximal subgroups

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

subgroupindex
#202928092
#202924262
#333×C1072
#1405107