Finite Group Databasebeta

← #30664477

Subgroups — #30664477

Order 16768.

346 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 85 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#22553276C211yesmaximal, characteristic, complemented
2C4×C22⋊Dic131C211yesmaximal, characteristic, complemented
3#22553046C214yesmaximal, complemented
4C2×#121×C131C211yesmaximal, characteristic
5C2×#14523497—21no—
6C2×C22⋊Dic131—21no—
7C2×C4×Dic131—21no—
8C22⋊(C131⋊C8)—21no—
9C22⋊(C131⋊C8)C2212yescomplemented
10C131⋊#95C2212yescomplemented
11C22×D4×C131—21no—
12C22×D4×C131C411yescharacteristic, complemented
13C23×C4×C131C2211yescharacteristic
14#121×C131—21no—
15#121×C131—21no—
16#121×C131C412yescomplemented
17#121×C131C2212yes—
18C2×C4⋊C4×C131C411yescharacteristic, complemented
19C2×C22⋊C4×C131—21no—
20C22×Dic131—22no—
21#14523497—42no—
22C2×C131⋊C8—22no—
23C22⋊Dic131—24no—
24C4×Dic131—42no—
25C24×C131—21no—
26C24×C131D411yescharacteristic
27C2×D4×C131—22no—
28C2×D4×C131—22no—
29C2×D4×C131—21no—
30C2×D4×C131—42no—
31C2×D4×C131—44no—
32C2×D4×C131C2×C412yes—
33C22×C4×C131—21no—
34C22×C4×C131—42no—
35C22×C4×C131C2×C411yescharacteristic
36C22×C4×C131C2×C411yescharacteristic
37C22×C4×C131D411yescharacteristic
38C22×C4×C131D411yescharacteristic
39C22×C4×C131D411yescharacteristic
40C22×C4×C131C2311yescharacteristic
41C4⋊C4×C131—21no—
42C4⋊C4×C131C2×C412yes—
43C22⋊C4×C131—22no—
44C22⋊C4×C131—22no—
45C2×Dic131—24no—
46C2×Dic131—42no—
47C2×Dic131—42no—
48C131⋊C8—42no—
49C23×C131—22no—
50C23×C131—22no—
51C23×C131—22no—
52C23×C131—41no—
53C23×C131—42no—
54C23×C131—42no—
55C23×C131—42no—
56C23×C131C22⋊C411yescharacteristic
57C23×C131C22⋊C411yescharacteristic
58C23×C131C2×D411yescharacteristic
59D4×C131—42no—
60D4×C131—44no—
61D4×C131—44no—
62D4×C131—42no—
63C2×C4×C131—22no—
64C2×C4×C131—22no—
65C2×C4×C131—22no—
66C2×C4×C131—22no—
67C2×C4×C131—41no—
68C2×C4×C131—41no—
69C2×C4×C131—41no—
70C2×C4×C131—42no—
71C2×C4×C131C22⋊C411yescharacteristic
72C2×C4×C131C22⋊C411yescharacteristic
73C2×C4×C131C22×C411yescharacteristic
74C2×C4×C131C2×D411yescharacteristic
75C22×C131—21no—
76C22×C131—21no—
77C22×C131—21no—
78C22×C131—21no—
79C22×C131—21no—
80C22×C131—21no—
81C22×C131—21no—
82C22×C131—44no—
83C22×C131—42no—
84C22×C131—42no—
85C22×C131—42no—
86C22×C131—84no—
87C22×C131#10412yes—
88C22×C131C2×C22⋊C411yescharacteristic
89C22×C131AGL(1,ℤ/8)12yes—
90C4×C131—21no—
91C4×C131—21no—
92C4×C131—21no—
93C4×C131—42no—
94C4×C131#9912yes—
95Dic131—44no—
96C2×C131—22no—
97C2×C131—22no—
98C2×C131—41no—
99C2×C131—82no—
100C2×C131C2×#9911yescharacteristic
101C2×C131#41811yescharacteristic
102C2×C131C2×#10411yescharacteristic
103C131#150711yescharacteristic, complemented, Sylow
104#1507—1311nomaximal, Sylow
105C2×#121D13111yescharacteristic
106#407—1311no—
107C4×C22⋊C4—1311no—
108#327—1314no—
109C22×D4—21no—
110C22×D4Dic13111yescharacteristic, complemented
111C23×C4C2×D13111yescharacteristic
112C2×M16—2621no—
113#121—21no—
114#121—21no—
115#121Dic13112yescomplemented
116#121C2×D13112yes—
117C2×C4⋊C4Dic13111yescharacteristic, complemented
118C2×C22⋊C4—21no—
119C2×C22⋊C4—2621no—
120C2×C42—2621no—
121C22⋊C8—2621no—
122C22⋊C8—1312no—
123#95—1312no—
124C24—21no—
125C24C22⋊D13111yescharacteristic
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—21no—
129C2×D4—42no—
130C2×D4—44no—
131C2×D4C2×Dic13112yes—
132C22×C4—21no—
133C22×C4—42no—
134C22×C4—2622no—
135C22×C4C2×Dic13111yescharacteristic
136C22×C4C2×Dic13111yescharacteristic
137C22×C4C22⋊D13111yescharacteristic
138C22×C4C22⋊D13111yescharacteristic
139C22×C4C22⋊D13111yescharacteristic
140C22×C4C22×D13111yescharacteristic
141M16—5242no—
142C2×C8—2622no—
143C4⋊C4—21no—
144C4⋊C4C2×Dic13112yes—
145C22⋊C4—22no—
146C22⋊C4—22no—
147C22⋊C4—2624no—
148C42—5242no—
149C23—22no—
150C23—22no—
151C23—22no—
152C23—41no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23C22⋊Dic13111yescharacteristic
157C23C22⋊Dic13111yescharacteristic
158C23C2×C22⋊D13111yescharacteristic
159D4—42no—
160D4—44no—
161D4—44no—
162D4—42no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—2624no—
172C2×C4—5242no—
173C2×C4—5242no—
174C2×C4C22⋊Dic13111yescharacteristic
175C2×C4C22⋊Dic13111yescharacteristic
176C2×C4C22×Dic13111yescharacteristic
177C2×C4C2×C22⋊D13111yescharacteristic
178C8—5242no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—44no—
187C22—42no—
188C22—42no—
189C22—42no—
190C22—84no—
191C22#1686217312yes—
192C22C2×C22⋊Dic13111yescharacteristic
193C22#1686227212yes—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—42no—
198C4—5244no—
199C4#1686216112yes—
200C2—22no—
201C2—22no—
202C2—41no—
203C2—82no—
204C2C2×#1686216111yescharacteristic
205C2#2255331711yescharacteristic
206C2C2×#1686217311yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic131
commutatorC2×C4×C131C22×C4
FrattiniC22×C4C22×D131
Fitting, prime radicalC2×#121×C131C2
socleC22×C131C2×C22⋊C4
cosocleC22×C4×C131C23
hypercenterC2×#121D131
hypocenterC131#1507
LCSG ▷ C2×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #22553276C2semidirect
2. C4×C22⋊Dic131C2semidirect
3. #22553046C2semidirect
9. C22⋊(C131⋊C8)C22semidirect
10. C131⋊#95C22semidirect
12. C22×D4×C131C4semidirect
16. #121×C131C4semidirect
18. C2×C4⋊C4×C131C4semidirect
103. C131#1507semidirect
110. C22×D4Dic131semidirect
115. #121Dic131semidirect
117. C2×C4⋊C4Dic131semidirect

Maximal subgroups

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

subgroupindex
#225532762
C4×C22⋊Dic1312
#225530462
C2×#121×C1312
#1507131