Finite Group Databasebeta

← #30665285

Subgroups — #30665285

Order 16768.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#22554060C211yesmaximal, characteristic
2#22553682C211yesmaximal, characteristic, complemented
3#22553672C212yesmaximal, complemented
4#22553051C212yesmaximal, complemented
5#330×C131C211yesmaximal, characteristic, complemented
6#16862246C2211yescharacteristic
7#16862234C2212yes—
8#16862231—22no—
9#16862224—22no—
10#16862221—22no—
11#16862171—22no—
12#16862169—22no—
13#16862168—22no—
14#16862157C2211yescharacteristic
15#125×C131C2211yescharacteristic
16#97×C131C2212yes—
17C2×Dic262—21no—
18C2×Dic262D412yes—
19C2×D524—22no—
20C2×D524D412yes—
21C2×C4×D131—22no—
22C2×C131⋊C8—21no—
23C4⋊Dic131—22no—
24Dic131⋊C4—22no—
25(C2×D131)⋊C4—22no—
26(C2×D131)⋊C4—22no—
27(C2×D131)⋊C4—22no—
28C2×C8×C131—22no—
29C4⋊C4×C131—22no—
30C4⋊C4×C131D412yes—
31C42×C131C2311yescharacteristic
32C22×D131—22no—
33Dic262—42no—
34Dic262—42no—
35D524—42no—
36D524—82no—
37C2×Dic131—22no—
38C2×Dic131—41no—
39C4×D131—82no—
40C131⋊C8—41no—
41C2×C4×C131—22no—
42C2×C4×C131C2×D412yes—
43C2×C4×C131C2×D411yescharacteristic
44C8×C131—42no—
45C22×C131#12011yescharacteristic
46C2×D131—42no—
47C2×D131—42no—
48C2×D131—42no—
49C4×C131—21no—
50C4×C131—22no—
51C4×C131—42no—
52Dic131—42no—
53Dic131—81no—
54C2×C131#45411yescharacteristic
55C2×C131#45512yes—
56D131—82no—
57C131#166211yescharacteristic, complemented, Sylow
58#1662—1311nomaximal, Sylow
59#562—1311no—
60#491—1311no—
61#489—1312no—
62#330D13111yescharacteristic, complemented
63#329—1312no—
64C4⋊Q8—1311no—
65#125C2×D13111yescharacteristic
66#124—1312no—
67#123—2622no—
68C22⋊Q8—2622no—
69#121—2622no—
70#103—2622no—
71#103—2622no—
72#102—2622no—
73#97—1311no—
74#97C2×D13112yes—
75C2×Q8—2621no—
76C2×Q8—1312no—
77C2×D4—2622no—
78C2×D4—1312no—
79C22×C4—2622no—
80C2×C8—22no—
81C2×C8—2621no—
82C4⋊C4—22no—
83C4⋊C4—2622no—
84C4⋊C4—2622no—
85C4⋊C4D52412yes—
86C22⋊C4—2622no—
87C22⋊C4—2622no—
88C22⋊C4—2622no—
89C42C22×D13111yescharacteristic
90C23—2622no—
91Q8—5242no—
92Q8—5242no—
93D4—5242no—
94D4—10482no—
95C2×C4—22no—
96C2×C4—2622no—
97C2×C4—5241no—
98C2×C4—10482no—
99C2×C4D4×D13112yes—
100C2×C4C2×D52411yescharacteristic
101C8—42no—
102C8—5241no—
103C22—5242no—
104C22—5242no—
105C22—5242no—
106C22#1686221611yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—5242no—
111C4—10481no—
112C2—10482no—
113C2#2255348711yescharacteristic
114C2#2255349112yes—
subgroupG/H
centerC22#16862216
commutator, cosocleC42×C131C23
FrattiniC42C22×D131
Fitting#330×C131C2
socleC22×C131#120
prime radical#22554060C2
hypercenter#330D131
hypocenterC131#1662
LCSG ▷ C42×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #22553682C2semidirect
3. #22553672C2semidirect
4. #22553051C2semidirect
5. #330×C131C2semidirect
57. C131#1662semidirect
62. #330D131semidirect

Maximal subgroups

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

subgroupindex
#225540602
#225536822
#225536722
#225530512
#330×C1312
#1662131