Finite Group Databasebeta

← #30665270

Subgroups — #30665270

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#22554062C211yesmaximal, characteristic
2#22553718C212yesmaximal, complemented
3#22553053C212yesmaximal, complemented
4#22553052C211yesmaximal, characteristic
5#497×C131C211yesmaximal, characteristic
6#16862244—21no—
7#16862239—22no—
8#16862238C2212yes—
9#16862227—21no—
10#16862223—22no—
11#16862179—22no—
12#16862177—22no—
13#16862157C2212yes—
14C2×Q16×C131—21no—
15C2×SD16×C131—21no—
16C4⋊Q8×C131C2211yescharacteristic
17#124×C131C2211yescharacteristic
18#97×C131C2211yescharacteristic
19C2×Dic262—21no—
20C22×Dic131—21no—
21C2×C131⋊C8—22no—
22C4⋊Dic131D412yes—
23C4⋊Dic131D412yes—
24Dic131⋊C4—22no—
25Dic131⋊C4—22no—
26Dic131⋊C4—22no—
27C22⋊Dic131—22no—
28C22⋊Dic131—21no—
29C4×Dic131—21no—
30C2×Q8×C131—21no—
31C2×Q8×C131D411yescharacteristic
32C2×D4×C131D411yescharacteristic
33Q16×C131—42no—
34SD16×C131—42no—
35C2×C8×C131—21no—
36C4⋊C4×C131—22no—
37C22⋊C4×C131—22no—
38C42×C131C2311yescharacteristic
39Dic262—81no—
40C2×Dic131—22no—
41C2×Dic131—22no—
42C2×Dic131—81no—
43C131⋊C8—24no—
44C23×C131—21no—
45Q8×C131—41no—
46Q8×C131—41no—
47Q8×C131—41no—
48D4×C131—41no—
49C2×C4×C131—21no—
50C2×C4×C131—41no—
51C2×C4×C131C2×D412yes—
52C2×C4×C131C2×D411yescharacteristic
53C8×C131—22no—
54C22×C131—41no—
55C22×C131—42no—
56C22×C131#12011yescharacteristic
57C4×C131—21no—
58C4×C131—22no—
59C4×C131—41no—
60C4×C131—81no—
61Dic131—42no—
62Dic131—42no—
63C2×C131—81no—
64C2×C131#45311yescharacteristic
65C2×C131#45612yes—
66C131#166011yescharacteristic, complemented, Sylow
67#1660—1311nomaximal, Sylow
68#563—1311no—
69#501—1312no—
70#497D13111yescharacteristic
71#330—1312no—
72#329—1311no—
73C2×Q16—21no—
74C2×SD16—21no—
75C4⋊Q8—2621no—
76C4⋊Q8C2×D13111yescharacteristic
77#126—2622no—
78#125—1312no—
79#124C2×D13111yescharacteristic
80#123—2621no—
81C22⋊Q8—2622no—
82#107—2622no—
83#106—2622no—
84#97—1312no—
85#97C2×D13111yescharacteristic
86C2×Q8—21no—
87C2×Q8—2621no—
88C2×Q8C22⋊D13111yescharacteristic
89C2×D4C22⋊D13111yescharacteristic
90C22×C4—2621no—
91Q16—42no—
92SD16—42no—
93C2×C8—21no—
94C2×C8—2622no—
95C4⋊C4—22no—
96C4⋊C4—2622no—
97C4⋊C4—2622no—
98C4⋊C4—2622no—
99C4⋊C4—1312no—
100C4⋊C4—1312no—
101C22⋊C4—22no—
102C22⋊C4—2622no—
103C22⋊C4—2621no—
104C42—2621no—
105C42C22×D13111yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—41no—
109Q8—41no—
110Q8—10481no—
111D4—41no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—2622no—
115C2×C4—2622no—
116C2×C4—10481no—
117C2×C4D4×D13112yes—
118C2×C4C2×C22⋊D13111yescharacteristic
119C8—22no—
120C8—2624no—
121C22—41no—
122C22—42no—
123C22#1686221411yescharacteristic
124C4—21no—
125C4—22no—
126C4—41no—
127C4—81no—
128C4—5242no—
129C4—5242no—
130C2—81no—
131C2#2255347911yescharacteristic
132C2#2255349312yes—
subgroupG/H
centerC22#16862214
commutator, cosocleC42×C131C23
FrattiniC42C22×D131
Fitting#497×C131C2
socleC22×C131#120
prime radicalC2×D4×C131D4
hypercenter#497D131
hypocenterC131#1660
LCSG ▷ C42×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #22553718C2semidirect
3. #22553053C2semidirect
66. C131#1660semidirect

Maximal subgroups

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

subgroupindex
#225540622
#225537182
#225530532
#225530522
#497×C1312
#1660131