Finite Group Databasebeta

← #30676005

Subgroups — #30676005

Order 16768.

454 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 119 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#22553843C211yesmaximal, characteristic, complemented
2C2×#16862208C211yesmaximal, characteristic, complemented
3#22553647C211yesmaximal, characteristic, complemented
4#22553611C212yesmaximal, complemented
5#22553526C212yesmaximal, complemented
6#22553465C211yesmaximal, characteristic, complemented
7#22553445C211yesmaximal, characteristic
8#22553399C212yesmaximal, complemented
9#22553392C212yesmaximal, complemented
10#22553362C211yesmaximal, characteristic, complemented
11#481×C131C211yesmaximal, characteristic, complemented
12C22×D524C2211yescharacteristic
13C22×C4×D131—21no—
14C2×#14523502C2212yes—
15C2×#14523498C2212yes—
16#16862239—21no—
17#16862237C2211yescharacteristic, complemented
18#16862233C2211yescharacteristic, complemented
19#16862227—21no—
20#16862225C2211yescharacteristic
21#16862211C2211yescharacteristic, complemented
22D4×Dic131—21no—
23D4×Dic131C2211yescharacteristic, complemented
24#16862208—21no—
25#16862208—21no—
26#16862208C2212yes—
27#16862208C2212yes—
28#16862202—21no—
29C2×(C2×D131)⋊C4—21no—
30C2×C4×Dic131C2211yescharacteristic
31#16862179C2212yescomplemented
32Dic131⋊C8C2212yescomplemented
33#16862169C2212yescomplemented
34#16862168C2212yes—
35#16862166C2212yes—
36C22⋊(C131⋊C8)C2211yescharacteristic
37C8×Dic131C2212yescomplemented
38#121×C131C2211yescharacteristic
39C2×C4⋊C4×C131C2211yescharacteristic
40#106×C131C2212yescomplemented
41#102×C131C2212yescomplemented
42C22⋊C8×C131C2211yescharacteristic
43C23×D131—21no—
44C2×Dic262C2311yescharacteristic
45C2×D524—24no—
46C2×D524—21no—
47C2×D524C2312yes—
48C22×Dic131—21no—
49C22×Dic131D411yescharacteristic
50C22×Dic131D411yescharacteristic
51C2×C4×D131—22no—
52C2×C4×D131—22no—
53C2×C4×D131—44no—
54#14523502—24no—
55#14523498—24no—
56C2×C131⋊C8C2312yes—
57C4⋊Dic131—21no—
58C4⋊Dic131C2311yescharacteristic
59Dic131⋊C4—21no—
60Dic131⋊C4—21no—
61Dic131⋊C4—21no—
62C22⋊Dic131—21no—
63C22⋊Dic131—21no—
64C22⋊Dic131—21no—
65C22⋊Dic131—21no—
66(C2×D131)⋊C4—22no—
67(C2×D131)⋊C4—22no—
68C4×Dic131—21no—
69C4×Dic131D411yescharacteristic
70C4×Dic131D411yescharacteristic
71C4×Dic131C2311yescharacteristic
72C4×Dic131C2311yescharacteristic
73C2×D4×C131—21no—
74C2×D4×C131C2311yescharacteristic
75C22×C4×C131—21no—
76C22×C4×C131C2311yescharacteristic
77C2×C8×C131C2312yes—
78C4⋊C4×C131—21no—
79C4⋊C4×C131C2312yes—
80C4⋊C4×C131C2311yescharacteristic
81C22⋊C4×C131—21no—
82C22×D131—24no—
83C22×D131—22no—
84C22×D131—42no—
85C22×D131—42no—
86Dic262—22no—
87D524—24no—
88D524—22no—
89D524#4114yes—
90C2×Dic131—22no—
91C2×Dic131—22no—
92C2×Dic131—21no—
93C2×Dic131—21no—
94C2×Dic131—42no—
95C2×Dic131C2×D411yescharacteristic
96C2×Dic131C2×D411yescharacteristic
97C2×Dic131C2×D411yescharacteristic
98C2×Dic131C2×D411yescharacteristic
99C4×D131—44no—
100C4×D131—44no—
101C131⋊C8—22no—
102C23×C131—21no—
103C23×C131C2×D411yescharacteristic
104D4×C131—22no—
105D4×C131—42no—
106C2×C4×C131—22no—
107C2×C4×C131—21no—
108C2×C4×C131—22no—
109C2×C4×C131—42no—
110C2×C4×C131C2×D411yescharacteristic
111C2×C4×C131C2411yescharacteristic
112C8×C131—22no—
113C22×C131—21no—
114C22×C131—21no—
115C22×C131—41no—
116C22×C131—41no—
117C22×C131—41no—
118C22×C131#13512yes—
119C22×C131C22×D411yescharacteristic
120C2×D131—24no—
121C2×D131—48no—
122C2×D131—42no—
123C2×D131—42no—
124C4×C131—21no—
125C4×C131—41no—
126C4×C131—42no—
127C4×C131C2×#4112yes—
128Dic131—21no—
129Dic131—21no—
130Dic131—21no—
131Dic131—41no—
132Dic131—41no—
133Dic131AGL(1,ℤ/8)12yes—
134C2×C131—22no—
135C2×C131—81no—
136C2×C131#52511yescharacteristic
137C2×C131C2×#13511yescharacteristic
138C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
139D131—44no—
140C131#310511yescharacteristic, complemented, Sylow
141#3105—1311nomaximal, Sylow
142#529—1311no—
143C2×C4×D4—1311no—
144#484—1311no—
145#481D13111yescharacteristic, complemented
146#478—1312no—
147#463—1312no—
148#450—1311no—
149#447—1311no—
150C4×SD16—1312no—
151C4×D8—1312no—
152#431—1311no—
153C22×D4—1311no—
154C23×C4—2621no—
155C2×SD16—1312no—
156C2×D8—1312no—
157#126—2621no—
158#125—1311no—
159#124—1311no—
160#123—2621no—
161C22⋊Q8—1311no—
162#121C2×D13111yescharacteristic
163C4×Q8—1311no—
164C4×D4—2621no—
165C4×D4—2621no—
166C4×D4—2621no—
167C4×D4—1311no—
168C4×D4—1312no—
169C4×D4—1312no—
170#117—2621no—
171C2×C4⋊C4C2×D13111yescharacteristic
172C2×C22⋊C4—2621no—
173C2×C42—1311no—
174#107—1312no—
175#106C2×D13112yescomplemented
176C4⋊C8—1312no—
177#103—1312no—
178#102—1312no—
179#102—1312no—
180#102C2×D13112yescomplemented
181C22⋊C8—1311no—
182C22⋊C8C2×D13111yescharacteristic
183C4×C8—1312no—
184C24—2621no—
185C2×Q8—1311no—
186C2×D4—21no—
187C2×D4—2624no—
188C2×D4—2621no—
189C2×D4—1312no—
190C2×D4C22×D13111yescharacteristic
191C22×C4—21no—
192C22×C4—2622no—
193C22×C4—2622no—
194C22×C4—2621no—
195C22×C4—5244no—
196C22×C4—1311no—
197C22×C4—1311no—
198C22×C4C22×D13111yescharacteristic
199SD16—2624no—
200D8—2624no—
201C2×C8—1312no—
202C2×C8C22×D13112yes—
203C4⋊C4—21no—
204C4⋊C4—2621no—
205C4⋊C4—2621no—
206C4⋊C4—2621no—
207C4⋊C4—2621no—
208C4⋊C4—1311no—
209C4⋊C4C22×D13111yescharacteristic
210C4⋊C4C22×D13112yes—
211C22⋊C4—21no—
212C22⋊C4—2622no—
213C22⋊C4—2622no—
214C22⋊C4—2621no—
215C22⋊C4—2621no—
216C22⋊C4—2621no—
217C22⋊C4—2621no—
218C42—2621no—
219C42—1311no—
220C42—1311no—
221C42—1311no—
222C42—1311no—
223C23—21no—
224C23—2624no—
225C23—2622no—
226C23—5242no—
227C23—5242no—
228C23D4×D13111yescharacteristic
229Q8—2622no—
230D4—22no—
231D4—42no—
232D4—2624no—
233D4—2622no—
234D4—1314no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—2622no—
240C2×C4—2622no—
241C2×C4—2621no—
242C2×C4—2621no—
243C2×C4—5244no—
244C2×C4—5244no—
245C2×C4—5242no—
246C2×C4—1311no—
247C2×C4—1311no—
248C2×C4—1311no—
249C2×C4—1311no—
250C2×C4D4×D13111yescharacteristic
251C2×C4C23×D13111yescharacteristic
252C8—22no—
253C8—2622no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—2624no—
260C22—5248no—
261C22—5242no—
262C22—5242no—
263C22#1686226712yes—
264C22C2×D4×D13111yescharacteristic
265C4—21no—
266C4—41no—
267C4—42no—
268C4—2621no—
269C4—2621no—
270C4—2621no—
271C4—5241no—
272C4—5241no—
273C4—1312no—
274C4#41×D13112yes—
275C2—22no—
276C2—81no—
277C2—5244no—
278C2#2255382011yescharacteristic
279C2C2×#1686226711yescharacteristic
280C2AGL(1,ℤ/8)×D13111yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#481×C131C2
socleC22×C131C22×D4
prime radical#22553445C2
hypercenter#481D131
hypocenterC131#3105
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #22553843C2semidirect
2. C2×#16862208C2semidirect
3. #22553647C2semidirect
4. #22553611C2semidirect
5. #22553526C2semidirect
6. #22553465C2semidirect
8. #22553399C2semidirect
9. #22553392C2semidirect
10. #22553362C2semidirect
11. #481×C131C2semidirect
17. #16862237C22semidirect
18. #16862233C22semidirect
21. #16862211C22semidirect
23. D4×Dic131C22semidirect
31. #16862179C22semidirect
32. Dic131⋊C8C22semidirect
33. #16862169C22semidirect
37. C8×Dic131C22semidirect
40. #106×C131C22semidirect
41. #102×C131C22semidirect
140. C131#3105semidirect
145. #481D131semidirect
175. #106C2×D131semidirect
180. #102C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225538432
C2×#168622082
#225536472
#225536112
#225535262
#225534652
#225534452
#225533992
#225533922
#225533622
#481×C1312
#3105131