Finite Group Databasebeta

← #30664113

Subgroups — #30664113

Order 16768.

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#22553484C211yesmaximal, characteristic
2#22553107C211yesmaximal, characteristic, complemented
3#329×C131C211yesmaximal, characteristic, complemented
4#16862294C411yescharacteristic, complemented
5#16862272—21no—
6#16862174—21no—
7#16862165C411yescharacteristic, complemented
8#16862161—21no—
9#124×C131C2211yescharacteristic
10#97×C131—21no—
11#14523514—41no—
12#14523513—21no—
13#14523513—41no—
14Q8×D131—41no—
15#14523501—41no—
16#14523498—41no—
17#14523497—21no—
18C22⋊Dic131—41no—
19C2×Q8×C131C2×C411yescharacteristic
20C2×D4×C131D411yescharacteristic
21C2×C8×C131—41no—
22C22⋊C4×C131—41no—
23C42×C131D411yescharacteristic
24Dic262—21no—
25Dic262—41no—
26D524—21no—
27D524—41no—
28C22⋊D131—41no—
29C2×Dic131—81no—
30C4×D131—41no—
31C4×D131—42no—
32C131⋊C8—41no—
33C23×C131—21no—
34Q8×C131—41no—
35D4×C131—41no—
36C2×C4×C131—21no—
37C2×C4×C131—21no—
38C2×C4×C131C22⋊C411yescharacteristic
39C8×C131—42no—
40C22×C131—41no—
41C22×C131—81no—
42C22×C131#9911yescharacteristic
43C2×D131—41no—
44C4×C131—21no—
45C4×C131—41no—
46C4×C131—41no—
47Dic131—41no—
48Dic131—161no—
49C2×C131—21no—
50C2×C131—81no—
51C2×C131AΣL(1,𝔽16)11yescharacteristic
52D131—81no—
53C131#140411yescharacteristic, complemented, Sylow
54#1404—1311nomaximal, Sylow
55#454—1311no—
56#354—1311no—
57#329D13111yescharacteristic, complemented
582−1+4—1311no—
59AGL(1,ℤ/8)—2621no—
60#124C2×D13111yescharacteristic
61#104—2621no—
62#101—1311no—
63#99—2621no—
64#97—21no—
65#41—2621no—
66#41—5241no—
67#41—5241no—
68C2×Q8—5241no—
69C2×Q8C4×D13111yescharacteristic
70C2×D4D52411yescharacteristic
71SD16—5241no—
72D8—5241no—
73M16—2621no—
74C2×C8—41no—
75C22⋊C4—41no—
76C22⋊C4—5241no—
77C42C22⋊D13111yescharacteristic
78C23—21no—
79Q8—41no—
80Q8—2621no—
81Q8—5241no—
82D4—41no—
83D4—2621no—
84D4—5241no—
85D4—5241no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—5241no—
89C2×C4—5242no—
90C2×C4—10481no—
91C2×C4(C2×D131)⋊C411yescharacteristic
92C8—42no—
93C8—5241no—
94C22—41no—
95C22—81no—
96C22—5241no—
97C22#1686216011yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—5241no—
102C4—20961no—
103C2—21no—
104C2—81no—
105C2—10481no—
106C2#2255310011yescharacteristic
subgroupG/H
centerC2#22553100
commutatorC2×Q8×C131C2×C4
Frattini#124C2×D131
Fitting#329×C131C2
socleC2×C131AΣL(1,𝔽16)
prime radical#22553484C2
cosocle#124×C131C22
hypercenter#329D131
hypocenterC131#1404
LCSG ▷ C2×Q8×C131 ▷ C2×C4×C131 ▷ C22×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #22553107C2semidirect
3. #329×C131C2semidirect
4. #16862294C4semidirect
7. #16862165C4semidirect
53. C131#1404semidirect
57. #329D131semidirect

Maximal subgroups

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

subgroupindex
#225534842
#225531072
#329×C1312
#1404131