Finite Group Databasebeta

← #30675907

Subgroups — #30675907

Order 16768.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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×#16862222C211yesmaximal, characteristic
2#117×D131C211yesmaximal, characteristic
3#22553652C212yesmaximal, complemented
4#22553651C212yesmaximal, complemented
5#22553646C212yesmaximal, complemented
6#22553641C212yesmaximal, complemented
7#22553355C211yesmaximal, characteristic, complemented
8#22553354C211yesmaximal, characteristic, complemented
9#22553321C211yesmaximal, characteristic, complemented
10#22553275C211yesmaximal, characteristic
11#419×C131C211yesmaximal, characteristic, complemented
12C22×C4×D131C2211yescharacteristic
13C2×#14523497C2211yescharacteristic
14#16862222—21no—
15#16862222—21no—
16#16862222C2212yes—
17#16862222C2212yes—
18#16862205C2211yescharacteristic
19#16862204C2211yescharacteristic
20#16862204C2211yescharacteristic
21#16862203—21no—
22#16862202—21no—
23C2×C4⋊Dic131C2211yescharacteristic
24C2×Dic131⋊C4—21no—
25C4⋊C4×D131C2211yescharacteristic
26C4⋊C4×D131C2211yescharacteristic
27C2×(C2×D131)⋊C4—21no—
28C22⋊C4×D131—21no—
29C42×D131—21no—
30#16862180C2212yescomplemented
31#16862179C2212yescomplemented
32#16862178C2212yescomplemented
33#16862177C2212yescomplemented
34#16862170C2212yescomplemented
35#16862170C2212yescomplemented
36(C2×D131)⋊C8C2212yes—
37(C2×D131)⋊C8C2212yes—
38C22×Q8×C131C2211yescharacteristic
39C2×M16×C131C2211yescharacteristic
40#117×C131C2211yescharacteristic
41#103×C131C2212yescomplemented
42#103×C131C2212yescomplemented
43C23×D131D411yescharacteristic
44C22×Dic131—21no—
45C22×Dic131D411yescharacteristic
46C2×C4×D131—21no—
47C2×C4×D131—21no—
48C2×C4×D131—22no—
49C2×C4×D131—22no—
50C2×C4×D131D411yescharacteristic
51C2×C4×D131D411yescharacteristic
52C2×C4×D131C2311yescharacteristic
53C2×C4×D131C2311yescharacteristic
54#14523497—22no—
55C2×C131⋊C8C2312yes—
56C4⋊Dic131—21no—
57C4⋊Dic131C2312yes—
58C4⋊Dic131C2311yescharacteristic
59C4⋊Dic131C2311yescharacteristic
60Dic131⋊C4—22no—
61Dic131⋊C4—22no—
62Dic131⋊C4—21no—
63Dic131⋊C4—21no—
64C22⋊Dic131—21no—
65(C2×D131)⋊C4—22no—
66(C2×D131)⋊C4—22no—
67(C2×D131)⋊C4—21no—
68(C2×D131)⋊C4—21no—
69C4×Dic131—21no—
70C4×Dic131—21no—
71C4×Dic131—21no—
72C2×Q8×C131—21no—
73C2×Q8×C131—22no—
74C2×Q8×C131—22no—
75C2×Q8×C131C2312yes—
76C22×C4×C131—21no—
77C22×C4×C131C2311yescharacteristic
78M16×C131—22no—
79C2×C8×C131C2312yes—
80C4⋊C4×C131C2311yescharacteristic
81C4⋊C4×C131C2311yescharacteristic
82C22⋊C4×C131—21no—
83C42×C131—21no—
84C22×D131—22no—
85C22×D131—22no—
86C22×D131—41no—
87C22×D131C2×D411yescharacteristic
88C22×D131C2×D411yescharacteristic
89C2×Dic131—22no—
90C2×Dic131—21no—
91C2×Dic131—21no—
92C2×Dic131—22no—
93C2×Dic131—42no—
94C2×Dic131C2×D411yescharacteristic
95C2×Dic131C2×D411yescharacteristic
96C4×D131—22no—
97C4×D131—22no—
98C4×D131—21no—
99C4×D131—21no—
100C4×D131—21no—
101C4×D131—21no—
102C4×D131—42no—
103C4×D131—42no—
104C131⋊C8—22no—
105C23×C131C2×D411yescharacteristic
106Q8×C131—22no—
107Q8×C131—22no—
108Q8×C131—42no—
109C2×C4×C131—21no—
110C2×C4×C131—21no—
111C2×C4×C131—22no—
112C2×C4×C131—42no—
113C2×C4×C131C2×D411yescharacteristic
114C2×C4×C131#4112yes—
115C2×C4×C131#4112yes—
116C2×C4×C131C2411yescharacteristic
117C8×C131—22no—
118C22×C131—21no—
119C22×C131—21no—
120C22×C131#12312yes—
121C22×C131C22×D411yescharacteristic
122C2×D131—21no—
123C2×D131—21no—
124C2×D131—21no—
125C2×D131—21no—
126C2×D131—21no—
127C2×D131—44no—
128C2×D131#13712yes—
129C4×C131—42no—
130C4×C131—41no—
131C4×C131—41no—
132C4×C131#12312yes—
133C4×C131C2×#4111yescharacteristic
134C4×C131C2×#4111yescharacteristic
135Dic131—21no—
136Dic131—21no—
137Dic131—41no—
138Dic131—41no—
139Dic131—42no—
140C2×C131—22no—
141C2×C131C2×#12311yescharacteristic
142C2×C131C2×#13711yescharacteristic
143C2×C131#57511yescharacteristic
144D131—22no—
145D131—41no—
146C131#309511yescharacteristic, complemented, Sylow
147#3095—1311nomaximal, Sylow
148C2×C22⋊Q8—1311no—
149C2×#117—1311no—
150#485—1312no—
151#485—1312no—
152#484—1312no—
153#483—1312no—
154#428—1311no—
155#428—1311no—
156#419—1311no—
157#419D13111yescharacteristic, complemented
158#407—1311no—
159C22×Q8C2×D13111yescharacteristic
160C23×C4—1311no—
161C2×M16—1311no—
162C2×M16C2×D13111yescharacteristic
163C22⋊Q8—2621no—
164C22⋊Q8—2621no—
165C22⋊Q8—1312no—
166C22⋊Q8—1312no—
167#117—2621no—
168#117—2621no—
169#117—1311no—
170#117—1311no—
171#117—1311no—
172#117C2×D13111yescharacteristic
173C2×C4⋊C4—2621no—
174C2×C4⋊C4—1311no—
175C2×C4⋊C4—1311no—
176C2×C4⋊C4—1311no—
177C2×C22⋊C4—2621no—
178C2×C22⋊C4—2621no—
179C2×C42—2621no—
180#107—1312no—
181#107—1312no—
182#106—1312no—
183#106—1312no—
184#103—1312no—
185#103—1312no—
186#103C2×D13112yescomplemented
187#103C2×D13112yescomplemented
188C22⋊C8—1312no—
189C22⋊C8—1312no—
190C24—1311no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8C22×D13112yes—
195C22×C4—21no—
196C22×C4—2621no—
197C22×C4—2621no—
198C22×C4—2622no—
199C22×C4—2622no—
200C22×C4—2621no—
201C22×C4—1311no—
202C22×C4—1311no—
203C22×C4—1311no—
204C22×C4—1311no—
205C22×C4—1311no—
206C22×C4C22×D13111yescharacteristic
207M16—22no—
208M16—2622no—
209C2×C8—1312no—
210C2×C8C22×D13112yes—
211C4⋊C4—2622no—
212C4⋊C4—2622no—
213C4⋊C4—2621no—
214C4⋊C4—2621no—
215C4⋊C4—2621no—
216C4⋊C4—1311no—
217C4⋊C4—1311no—
218C4⋊C4—1312no—
219C4⋊C4C22×D13111yescharacteristic
220C4⋊C4C22×D13111yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—2621no—
223C22⋊C4—2621no—
224C22⋊C4—2622no—
225C22⋊C4—2622no—
226C22⋊C4—2621no—
227C42—21no—
228C42—2621no—
229C42—2621no—
230C42—2621no—
231C23—2622no—
232C23—2622no—
233C23—5241no—
234C23—1311no—
235C23—1311no—
236C23D4×D13111yescharacteristic
237Q8—22no—
238Q8—22no—
239Q8—42no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—2622no—
245C2×C4—2622no—
246C2×C4—2622no—
247C2×C4—2621no—
248C2×C4—2621no—
249C2×C4—2621no—
250C2×C4—2621no—
251C2×C4—2621no—
252C2×C4—2621no—
253C2×C4—2622no—
254C2×C4—5242no—
255C2×C4—5242no—
256C2×C4—5242no—
257C2×C4—1311no—
258C2×C4—1311no—
259C2×C4D4×D13111yescharacteristic
260C2×C4#1452351212yes—
261C2×C4#1452351312yes—
262C2×C4C23×D13111yescharacteristic
263C8—22no—
264C8—2622no—
265C22—21no—
266C22—21no—
267C22—2621no—
268C22—2621no—
269C22—2621no—
270C22—2621no—
271C22—2621no—
272C22—5244no—
273C22—1312no—
274C22#1686222812yes—
275C22C2×D4×D13111yescharacteristic
276C4—42no—
277C4—41no—
278C4—41no—
279C4—2621no—
280C4—2621no—
281C4—5241no—
282C4—5241no—
283C4—5242no—
284C4#1686222812yes—
285C4C2×#1452351211yescharacteristic
286C4C2×#1452351311yescharacteristic
287C2—22no—
288C2—2622no—
289C2—5241no—
290C2C2×#1686222811yescharacteristic
291C2#137×D13111yescharacteristic
292C2#2255411111yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#419×C131C2
socleC22×C131C22×D4
prime radicalC23×D131D4
hypercenter#419D131
hypocenterC131#3095
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #22553652C2semidirect
4. #22553651C2semidirect
5. #22553646C2semidirect
6. #22553641C2semidirect
7. #22553355C2semidirect
8. #22553354C2semidirect
9. #22553321C2semidirect
11. #419×C131C2semidirect
30. #16862180C22semidirect
31. #16862179C22semidirect
32. #16862178C22semidirect
33. #16862177C22semidirect
34. #16862170C22semidirect
35. #16862170C22semidirect
41. #103×C131C22semidirect
42. #103×C131C22semidirect
146. C131#3095semidirect
157. #419D131semidirect
186. #103C2×D131semidirect
187. #103C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622222
#117×D1312
#225536522
#225536512
#225536462
#225536412
#225533552
#225533542
#225533212
#225532752
#419×C1312
#3095131