Finite Group Databasebeta

← #30674280

Subgroups — #30674280

Order 16768.

582 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C22×C4×Dic131C211yesmaximal, characteristic, complemented
2#22553200C214yesmaximal, complemented
3#22553193C214yesmaximal, complemented
4#22553192C212yesmaximal
5#22553184C212yesmaximal, complemented
6#22553175C211yesmaximal, characteristic, complemented
7C2×#123×C131C211yesmaximal, characteristic
8C23×Dic131C2212yes—
9C2×Dic131⋊C4C2214yes—
10C2×C22⋊Dic131C2212yes—
11C2×C22⋊Dic131C2214yes—
12C2×C4×Dic131—24no—
13C2×C4×Dic131C2214yes—
14C131⋊#95C2214yes—
15C131⋊#95C2214yes—
16C131⋊#95C2214yes—
17C22×D4×C131C2211yescharacteristic
18C23×C4×C131C2211yescharacteristic
19#123×C131—24no—
20C2×C4⋊C4×C131C2212yes—
21C2×C22⋊C4×C131C2211yescharacteristic
22C2×C22⋊C4×C131C2212yes—
23C22×Dic131—28no—
24C22×Dic131—24no—
25C22×Dic131C2314yes—
26C22×Dic131C2314yes—
27Dic131⋊C4—28no—
28C22⋊Dic131—28no—
29C22⋊Dic131—24no—
30C4×Dic131—28no—
31C24×C131C2311yescharacteristic
32C24×C131C2311yescharacteristic
33C2×D4×C131—22no—
34C2×D4×C131—24no—
35C22×C4×C131—24no—
36C22×C4×C131D414yes—
37C22×C4×C131C2312yes—
38C22×C4×C131C2311yescharacteristic
39C22×C4×C131C2312yes—
40C4⋊C4×C131—24no—
41C22⋊C4×C131—22no—
42C22⋊C4×C131—24no—
43C2×Dic131—216no—
44C2×Dic131—24no—
45C2×Dic131—24no—
46C2×Dic131—24no—
47C2×Dic131—24no—
48C2×Dic131—24no—
49C2×Dic131#4118yes—
50C23×C131—24no—
51C23×C131—21no—
52C23×C131—22no—
53C23×C131—24no—
54C23×C131C2×D412yes—
55C23×C131#4114yes—
56C23×C131C2411yescharacteristic
57D4×C131—44no—
58C2×C4×C131—28no—
59C2×C4×C131—24no—
60C2×C4×C131—22no—
61C2×C4×C131—21no—
62C2×C4×C131—22no—
63C2×C4×C131—24no—
64C2×C4×C131C2×D414yes—
65C22×C131—24no—
66C22×C131—28no—
67C22×C131—41no—
68C22×C131—42no—
69C22×C131—44no—
70C22×C131#12314yes—
71C22×C131C22×D411yescharacteristic
72C22×C131C2×#4114yes—
73C22×C131C2×#4112yes—
74C4×C131—24no—
75C4×C131—41no—
76C4×C131—42no—
77Dic131—28no—
78Dic131—44no—
79C2×C131—24no—
80C2×C131—81no—
81C2×C131C2×#12311yescharacteristic
82C2×C131#52512yes—
83C2×C131#52914yes—
84C131#285311yescharacteristic, complemented, Sylow
85#2853—1311nomaximal, Sylow
86C2×#123D13111yescharacteristic
87C22×C42—1311no—
88#388—1314no—
89#387—1314no—
90#386—1312no—
91#385—1312no—
92#381—1311no—
93C22×D4C2×D13111yescharacteristic
94C23×C4—1312no—
95C23×C4C2×D13111yescharacteristic
96#123—24no—
97C2×C4⋊C4—1314no—
98C2×C4⋊C4C2×D13112yes—
99C2×C22⋊C4—1312no—
100C2×C22⋊C4—1314no—
101C2×C22⋊C4C2×D13111yescharacteristic
102C2×C22⋊C4C2×D13112yes—
103C2×C42—2624no—
104C2×C42—1314no—
105#95—1314no—
106#95—1314no—
107#95—1314no—
108C24C22×D13111yescharacteristic
109C24C22×D13111yescharacteristic
110C2×D4—22no—
111C2×D4—24no—
112C22×C4—24no—
113C22×C4—2624no—
114C22×C4—2628no—
115C22×C4—1314no—
116C22×C4—1314no—
117C22×C4C22⋊D13114yes—
118C22×C4C22×D13111yescharacteristic
119C22×C4C22×D13112yes—
120C22×C4C22×D13112yes—
121C4⋊C4—24no—
122C4⋊C4—2628no—
123C22⋊C4—22no—
124C22⋊C4—24no—
125C22⋊C4—2624no—
126C22⋊C4—2628no—
127C42—2628no—
128C23—24no—
129C23—21no—
130C23—24no—
131C23—22no—
132C23C2×C22⋊D13112yes—
133C23#1452351214yes—
134C23C23×D13111yescharacteristic
135D4—44no—
136C2×C4—28no—
137C2×C4—24no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—24no—
141C2×C4—22no—
142C2×C4—26216no—
143C2×C4—2624no—
144C2×C4—2624no—
145C2×C4—2624no—
146C2×C4—2624no—
147C2×C4—2624no—
148C2×C4—1318no—
149C2×C4C2×C22⋊D13114yes—
150C22—24no—
151C22—28no—
152C22—41no—
153C22—42no—
154C22—44no—
155C22#1686222714yes—
156C22C22×C22⋊D13111yescharacteristic
157C22#41×D13114yes—
158C22C2×#1452351212yes—
159C4—24no—
160C4—41no—
161C4—42no—
162C4—2628no—
163C4—5244no—
164C2—24no—
165C2—81no—
166C2C2×#1686222711yescharacteristic
167C2#2255381912yes—
168C2#2255383914yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC2×#123×C131C2
prime radicalC22×D4×C131C22
hypercenterC2×#123D131
hypocenterC131#2853
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic131C2semidirect
2. #22553200C2semidirect
3. #22553193C2semidirect
5. #22553184C2semidirect
6. #22553175C2semidirect
84. C131#2853semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1312
#225532002
#225531932
#225531922
#225531842
#225531752
C2×#123×C1312
#2853131