Finite Group Databasebeta

← #30675902

Subgroups — #30675902

Order 16768.

462 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#16862220C211yesmaximal, characteristic
2C2×#16862205C211yesmaximal, characteristic, complemented
3#22553637C214yesmaximal, complemented
4#22553631C212yesmaximal, complemented
5#22553626C212yesmaximal, complemented
6#22553354C212yesmaximal, complemented
7#22553318C212yesmaximal, complemented
8#407×C131C211yesmaximal, characteristic, complemented
9C22×D524C2211yescharacteristic
10C22×C22⋊D131—21no—
11#16862220—21no—
12#16862220—21no—
13#16862220C2212yes—
14#16862220C2212yes—
15#16862205—21no—
16#16862205—21no—
17#16862205C2212yes—
18#16862205C2212yescomplemented
19C2×C4⋊Dic131C2211yescharacteristic
20C2×C4⋊Dic131C2211yescharacteristic, complemented
21C2×C4⋊Dic131C2211yescharacteristic, complemented
22C2×C22⋊Dic131—21no—
23C2×(C2×D131)⋊C4—21no—
24C2×C4×Dic131—21no—
25#16862180C2214yescomplemented
26#16862178C2214yescomplemented
27#16862168C2214yescomplemented
28#16862168C2214yescomplemented
29C23×C4×C131C2211yescharacteristic
30C2×M16×C131C2212yes—
31C22⋊C8×C131C2212yescomplemented
32C22⋊C8×C131C2212yescomplemented
33C23×D131—21no—
34C2×D524—21no—
35C2×D524—24no—
36C2×D524C2312yes—
37C2×C22⋊D131—22no—
38C2×C22⋊D131—22no—
39C2×C22⋊D131—44no—
40C22×Dic131—21no—
41C22×Dic131—21no—
42C22×Dic131—21no—
43C4⋊Dic131—21no—
44C4⋊Dic131—21no—
45C4⋊Dic131—21no—
46C4⋊Dic131C2312yes—
47C4⋊Dic131C2312yes—
48C4⋊Dic131C2312yes—
49C22⋊Dic131—22no—
50C22⋊Dic131—22no—
51(C2×D131)⋊C4—22no—
52(C2×D131)⋊C4—22no—
53C4×Dic131—22no—
54C4×Dic131—22no—
55C24×C131D411yescharacteristic
56C22×C4×C131—24no—
57C22×C4×C131D411yescharacteristic
58C22×C4×C131D411yescharacteristic
59C22×C4×C131D411yescharacteristic
60C22×C4×C131C2311yescharacteristic
61C22×C4×C131C2311yescharacteristic
62C22×C4×C131C2311yescharacteristic
63M16×C131—24no—
64C2×C8×C131C2314yes—
65C22×D131—22no—
66C22×D131—42no—
67C22×D131—42no—
68C22×D131—42no—
69D524—24no—
70D524—42no—
71C22⋊D131—44no—
72C22⋊D131—44no—
73C2×Dic131—22no—
74C2×Dic131—22no—
75C2×Dic131—22no—
76C2×Dic131—42no—
77C2×Dic131—42no—
78C2×Dic131—42no—
79C23×C131—22no—
80C23×C131—22no—
81C23×C131—41no—
82C23×C131C2×D411yescharacteristic
83C23×C131C2×D411yescharacteristic
84C23×C131C2×D411yescharacteristic
85C2×C4×C131—22no—
86C2×C4×C131—22no—
87C2×C4×C131—22no—
88C2×C4×C131—22no—
89C2×C4×C131—22no—
90C2×C4×C131C2×D411yescharacteristic
91C2×C4×C131C2×D411yescharacteristic
92C2×C4×C131C2×D411yescharacteristic
93C2×C4×C131#4114yes—
94C2×C4×C131C2411yescharacteristic
95C8×C131—24no—
96C22×C131—21no—
97C22×C131—21no—
98C22×C131—21no—
99C22×C131—21no—
100C22×C131—21no—
101C22×C131—21no—
102C22×C131—21no—
103C22×C131—44no—
104C22×C131#12312yes—
105C22×C131AGL(1,ℤ/8)12yes—
106C22×C131C22×D411yescharacteristic
107C2×D131—42no—
108C2×D131—42no—
109C2×D131—42no—
110C2×D131—84no—
111C4×C131—21no—
112C4×C131—21no—
113C4×C131#12312yes—
114C4×C131C2×#4112yes—
115Dic131—42no—
116Dic131—42no—
117Dic131—42no—
118C2×C131—22no—
119C2×C131—22no—
120C2×C131—41no—
121C2×C131C2×#12311yescharacteristic
122C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
123C2×C131#57511yescharacteristic
124D131—82no—
125C131#309411yescharacteristic, complemented, Sylow
126#3094—1311nomaximal, Sylow
127C2×#121—1311no—
128C2×#117—1311no—
129#482—1314no—
130#481—1312no—
131#480—1312no—
132#428—1312no—
133#418—1312no—
134#407D13111yescharacteristic, complemented
135C22×D4—2621no—
136C22×D4—1311no—
137C23×C4C2×D13111yescharacteristic
138C2×M16C2×D13112yes—
139#121—2621no—
140#121—2621no—
141#121—1312no—
142#121—1312no—
143#117—2621no—
144#117—2621no—
145#117—1312no—
146#117—1312no—
147C2×C4⋊C4—1311no—
148C2×C4⋊C4—1311no—
149C2×C4⋊C4—1311no—
150C2×C22⋊C4—2621no—
151C2×C22⋊C4—2621no—
152C2×C42—2621no—
153#107—1314no—
154#106—1314no—
155#102—1314no—
156#102—1314no—
157C22⋊C8C2×D13112yescomplemented
158C22⋊C8C2×D13112yescomplemented
159C24—2621no—
160C24D52411yescharacteristic
161C2×D4—2622no—
162C2×D4—2622no—
163C2×D4—2621no—
164C2×D4—2624no—
165C2×D4—5244no—
166C2×D4—1312no—
167C22×C4—24no—
168C22×C4—2621no—
169C22×C4—2621no—
170C22×C4—2621no—
171C22×C4D52411yescharacteristic
172C22×C4D52411yescharacteristic
173C22×C4D52411yescharacteristic
174C22×C4C22×D13111yescharacteristic
175C22×C4C22×D13111yescharacteristic
176C22×C4C22×D13111yescharacteristic
177M16—24no—
178C2×C8C22×D13114yes—
179C4⋊C4—2621no—
180C4⋊C4—2621no—
181C4⋊C4—2621no—
182C4⋊C4—1312no—
183C4⋊C4—1312no—
184C4⋊C4—1312no—
185C22⋊C4—2622no—
186C22⋊C4—2622no—
187C22⋊C4—2622no—
188C22⋊C4—2622no—
189C42—2622no—
190C42—2622no—
191C23—22no—
192C23—22no—
193C23—41no—
194C23—2622no—
195C23—5242no—
196C23—5242no—
197C23—5242no—
198C23C2×D52411yescharacteristic
199C23C2×D52411yescharacteristic
200C23C2×D52411yescharacteristic
201D4—2624no—
202D4—5244no—
203D4—5244no—
204D4—5242no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—2622no—
211C2×C4—2622no—
212C2×C4—2622no—
213C2×C4—5242no—
214C2×C4—5242no—
215C2×C4—5242no—
216C2×C4C2×D52411yescharacteristic
217C2×C4C2×D52411yescharacteristic
218C2×C4C2×D52411yescharacteristic
219C2×C4#1452351214yes—
220C2×C4C23×D13111yescharacteristic
221C8—24no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—5242no—
231C22—5242no—
232C22—5242no—
233C22—10484no—
234C22#1686223012yes—
235C22#1686227412yes—
236C22C22×D52411yescharacteristic
237C4—21no—
238C4—21no—
239C4—5242no—
240C4—5242no—
241C4—5242no—
242C4#1686223012yes—
243C4C2×#1452351212yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—10482no—
248C2C2×#1686223011yescharacteristic
249C2C2×#1686227411yescharacteristic
250C2#2255411211yescharacteristic
subgroupG/H
centerC22C22×D524
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#407×C131C2
socleC22×C131C22×D4
prime radicalC2×#16862220C2
hypercenter#407D131
hypocenterC131#3094
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#16862205C2semidirect
3. #22553637C2semidirect
4. #22553631C2semidirect
5. #22553626C2semidirect
6. #22553354C2semidirect
7. #22553318C2semidirect
8. #407×C131C2semidirect
18. #16862205C22semidirect
20. C2×C4⋊Dic131C22semidirect
21. C2×C4⋊Dic131C22semidirect
25. #16862180C22semidirect
26. #16862178C22semidirect
27. #16862168C22semidirect
28. #16862168C22semidirect
31. C22⋊C8×C131C22semidirect
32. C22⋊C8×C131C22semidirect
125. C131#3094semidirect
134. #407D131semidirect
157. C22⋊C8C2×D131semidirect
158. C22⋊C8C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622202
C2×#168622052
#225536372
#225536312
#225536262
#225533542
#225533182
#407×C1312
#3094131