Finite Group Databasebeta

← #30664971

Subgroups — #30664971

Order 16768.

210 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 53 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#22553943C211yesmaximal, characteristic
2#22553643C212yesmaximal
3#22553242C211yesmaximal, characteristic
4#22553050C212yesmaximal
5#324×C131C211yesmaximal, characteristic
6#16862245—22no—
7#16862237—22no—
8#16862225—21no—
9#16862225C2212yes—
10#16862205—21no—
11C2×C4⋊Dic131C2211yescharacteristic
12C2×Dic131⋊C4—21no—
13#16862178—22no—
14#16862171—22no—
15C131⋊#95—21no—
16C131⋊#95C2211yescharacteristic
17C2×C4⋊C4×C131C2211yescharacteristic
18C22⋊C8×C131C2212yes—
19C2×Dic262—21no—
20C2×Dic262D412yes—
21C22×Dic131—21no—
22C22×Dic131—21no—
23C4⋊Dic131—21no—
24C4⋊Dic131—21no—
25C4⋊Dic131—21no—
26C4⋊Dic131—22no—
27C4⋊Dic131D412yes—
28Dic131⋊C4—22no—
29Dic131⋊C4—22no—
30Dic131⋊C4—42no—
31C22⋊Dic131—22no—
32C4×Dic131—22no—
33C22×C4×C131D411yescharacteristic
34C22×C4×C131D411yescharacteristic
35C22×C4×C131C2311yescharacteristic
36C2×C8×C131—22no—
37C4⋊C4×C131—22no—
38Dic262—42no—
39Dic262—81no—
40C2×Dic131—22no—
41C2×Dic131—22no—
42C2×Dic131—22no—
43C2×Dic131—41no—
44C2×Dic131—41no—
45C2×Dic131—41no—
46C2×Dic131—41no—
47C2×Dic131—41no—
48C23×C131C2×D411yescharacteristic
49C2×C4×C131—21no—
50C2×C4×C131—21no—
51C2×C4×C131—21no—
52C2×C4×C131—41no—
53C2×C4×C131—41no—
54C2×C4×C131SD1612yes—
55C2×C4×C131C2×D412yes—
56C8×C131—42no—
57C22×C131—21no—
58C22×C131—21no—
59C22×C131#12011yescharacteristic
60C22×C131C2×SD1611yescharacteristic
61C22×C131#13711yescharacteristic
62C4×C131—22no—
63C4×C131—22no—
64C4×C131—41no—
65Dic131—42no—
66Dic131—42no—
67Dic131—42no—
68Dic131—81no—
69C2×C131—21no—
70C2×C131—21no—
71C2×C131#44811yescharacteristic
72C2×C131#45611yescharacteristic
73C2×C131#45811yescharacteristic
74C131#161111yescharacteristic, complemented, Sylow
75#1611—1311nomaximal, Sylow
76#544—1311no—
77#483—1312no—
78#398—1311no—
79#328—1312no—
80#324D13111yescharacteristic
81C4⋊Q8—2622no—
82#125—2622no—
83C22⋊Q8—2621no—
84C22⋊Q8—1312no—
85#117—2621no—
86C2×C4⋊C4—2621no—
87C2×C4⋊C4—1311no—
88C2×C4⋊C4C2×D13111yescharacteristic
89#106—2622no—
90#103—2622no—
91C22⋊C8C2×D13112yes—
92#95—2621no—
93#95—1311no—
94C2×Q8—2621no—
95C2×Q8—1312no—
96C22×C4—2621no—
97C22×C4—2621no—
98C22×C4D52411yescharacteristic
99C22×C4D52411yescharacteristic
100C22×C4C22×D13111yescharacteristic
101C2×C8—22no—
102C4⋊C4—22no—
103C4⋊C4—2622no—
104C4⋊C4—2621no—
105C4⋊C4—2621no—
106C4⋊C4—2621no—
107C4⋊C4—2622no—
108C4⋊C4—2622no—
109C4⋊C4—5242no—
110C4⋊C4—1312no—
111C22⋊C4—2622no—
112C42—2622no—
113C23C2×D52411yescharacteristic
114Q8—5242no—
115Q8—10481no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—2622no—
122C2×C4—2622no—
123C2×C4—2622no—
124C2×C4—5241no—
125C2×C4—5241no—
126C2×C4—5241no—
127C2×C4—5241no—
128C2×C4—5241no—
129C2×C4#1452350212yes—
130C2×C4D4×D13112yes—
131C8—42no—
132C22—21no—
133C22—21no—
134C22#1686221611yescharacteristic
135C22C2×#1452350211yescharacteristic
136C22#1686227911yescharacteristic
137C4—22no—
138C4—22no—
139C4—41no—
140C4—5242no—
141C4—5242no—
142C4—5242no—
143C4—10481no—
144C2—21no—
145C2—21no—
146C2#2255345311yescharacteristic
147C2#2255349711yescharacteristic
148C2#2255350311yescharacteristic
subgroupG/H
centerC22#16862216
commutator, cosocleC22×C4×C131C23
FrattiniC22×C4C22×D131
Fitting#324×C131C2
socleC22×C131#120
prime radicalC23×C131C2×D4
hypercenter#324D131
hypocenterC131#1611
LCSG ▷ C22×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
74. C131#1611semidirect

Maximal subgroups

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

subgroupindex
#225539432
#225536432
#225532422
#225530502
#324×C1312
#1611131