Finite Group Databasebeta

← #28334613

Subgroups — #28334613

Order 13696.

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#20292799C211yesmaximal, characteristic
2#20292422C211yesmaximal, characteristic, complemented
3#329×C107C211yesmaximal, characteristic, complemented
4#16812157C411yescharacteristic, complemented
5#16812135—21no—
6#16812037—21no—
7#16812028C411yescharacteristic, complemented
8#16812024—21no—
9#124×C107C2211yescharacteristic
10#97×C107—21no—
11#13133819—41no—
12#13133818—21no—
13#13133818—41no—
14Q8×D107—41no—
15#13133806—41no—
16#13133803—41no—
17#13133802—21no—
18C22⋊Dic107—41no—
19C2×Q8×C107C2×C411yescharacteristic
20C2×D4×C107D411yescharacteristic
21C2×C8×C107—41no—
22C22⋊C4×C107—41no—
23C42×C107D411yescharacteristic
24C23×C107—21no—
25Q8×C107—41no—
26D4×C107—41no—
27C2×C4×C107—21no—
28C2×C4×C107—21no—
29C2×C4×C107C22⋊C411yescharacteristic
30C22⋊D107—41no—
31C2×Dic107—81no—
32D428—21no—
33D428—41no—
34C4×D107—41no—
35C4×D107—42no—
36Dic214—21no—
37Dic214—41no—
38C8×C107—42no—
39C107⋊C8—41no—
40C22×C107—41no—
41C22×C107—81no—
42C22×C107#9911yescharacteristic
43C2×D107—41no—
44C4×C107—21no—
45C4×C107—41no—
46C4×C107—41no—
47Dic107—41no—
48Dic107—161no—
49C2×C107—21no—
50C2×C107—81no—
51C2×C107AΣL(1,𝔽16)11yescharacteristic
52D107—81no—
53#1404—1071nomaximal, Sylow
54C107#140411yescharacteristic, complemented, Sylow
55#454—1071no—
56#354—1071no—
57#329D10711yescharacteristic, complemented
582−1+4—1071no—
59AGL(1,ℤ/8)—2141no—
60#124C2×D10711yescharacteristic
61#104—2141no—
62#101—1071no—
63#99—2141no—
64#97—21no—
65#41—4281no—
66#41—4281no—
67#41—2141no—
68C2×Q8—4281no—
69C2×Q8C4×D10711yescharacteristic
70C2×D4D42811yescharacteristic
71SD16—4281no—
72D8—4281no—
73M16—2141no—
74C2×C8—41no—
75C22⋊C4—41no—
76C22⋊C4—4281no—
77C42C22⋊D10711yescharacteristic
78C23—21no—
79Q8—41no—
80Q8—4281no—
81Q8—2141no—
82D4—41no—
83D4—4281no—
84D4—4281no—
85D4—2141no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—8561no—
89C2×C4—4281no—
90C2×C4—4282no—
91C2×C4(C2×D107)⋊C411yescharacteristic
92C8—42no—
93C8—4281no—
94C22—41no—
95C22—81no—
96C22—4281no—
97C22#1681202311yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—4281no—
102C4—17121no—
103C2—21no—
104C2—81no—
105C2—8561no—
106C2#2029241511yescharacteristic
subgroupG/H
centerC2#20292415
commutatorC2×Q8×C107C2×C4
Frattini#124C2×D107
Fitting#329×C107C2
socleC2×C107AΣL(1,𝔽16)
prime radical#20292799C2
cosocle#124×C107C22
hypercenter#329D107
hypocenterC107#1404
LCSG ▷ C2×Q8×C107 ▷ C2×C4×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #20292422C2semidirect
3. #329×C107C2semidirect
4. #16812157C4semidirect
7. #16812028C4semidirect
54. C107#1404semidirect
57. #329D107semidirect

Maximal subgroups

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

subgroupindex
#202927992
#202924222
#329×C1072
#1404107