Finite Group Databasebeta

← #30675882

Subgroups — #30675882

Order 16768.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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#22553819C211yesmaximal, characteristic
2C2×C4⋊C4×D131C211yesmaximal, characteristic, complemented
3#22553646C212yesmaximal, complemented
4#22553641C212yesmaximal, complemented
5#22553629C212yesmaximal, complemented
6#22553624C212yesmaximal, complemented
7#22553354C211yesmaximal, characteristic, complemented
8#22553352C211yesmaximal, characteristic, complemented
9#22553313C211yesmaximal, characteristic, complemented
10#22553275C211yesmaximal, characteristic, complemented
11#417×C131C211yesmaximal, characteristic, complemented
12C22×C4×D131—21no—
13C22×C4×D131C2211yescharacteristic
14C2×#14523497C2211yescharacteristic
15#16862227—21no—
16#16862222C2211yescharacteristic
17#16862222C2211yescharacteristic
18#16862217C2211yescharacteristic
19#16862217C2211yescharacteristic
20#16862214—21no—
21C4×C22⋊D131—21no—
22C4×C22⋊D131—21no—
23#16862205C2211yescharacteristic
24C2×C4⋊Dic131C2211yescharacteristic
25C2×Dic131⋊C4—21no—
26C4⋊C4×D131—21no—
27C4⋊C4×D131—21no—
28C4⋊C4×D131C2212yes—
29C4⋊C4×D131C2212yes—
30#16862180C2212yescomplemented
31#16862179C2212yescomplemented
32#16862178C2212yescomplemented
33#16862177C2212yescomplemented
34#16862170C2212yescomplemented
35#16862167C2212yescomplemented
36(C2×D131)⋊C8C2212yes—
37(C2×D131)⋊C8C2212yes—
38C2×#41×C131C2211yescharacteristic
39C2×M16×C131C2211yescharacteristic
40C2×C4⋊C4×C131C2211yescharacteristic
41#103×C131C2212yescomplemented
42#102×C131C2212yescomplemented
43C23×D131D411yescharacteristic
44C2×C22⋊D131—21no—
45C2×C22⋊D131—21no—
46C22×Dic131—21no—
47C22×Dic131D411yescharacteristic
48C2×C4×D131—22no—
49C2×C4×D131—22no—
50C2×C4×D131—22no—
51C2×C4×D131—22no—
52C2×C4×D131—44no—
53C2×C4×D131D411yescharacteristic
54C2×C4×D131D411yescharacteristic
55C2×C4×D131C2311yescharacteristic
56C2×C4×D131C2311yescharacteristic
57#14523497—22no—
58C2×C131⋊C8C2312yes—
59C4⋊Dic131—21no—
60C4⋊Dic131C2311yescharacteristic
61C4⋊Dic131C2311yescharacteristic
62C4⋊Dic131C2312yes—
63Dic131⋊C4—21no—
64Dic131⋊C4—21no—
65Dic131⋊C4—22no—
66Dic131⋊C4—22no—
67C22⋊Dic131—21no—
68C22⋊Dic131—21no—
69C22⋊Dic131—21no—
70(C2×D131)⋊C4—21no—
71(C2×D131)⋊C4—21no—
72C4×Dic131—21no—
73#41×C131—22no—
74#41×C131—22no—
75C2×Q8×C131C2311yescharacteristic
76C2×D4×C131—21no—
77C2×D4×C131C2311yescharacteristic
78C22×C4×C131—21no—
79C22×C4×C131—21no—
80C22×C4×C131C2311yescharacteristic
81M16×C131—22no—
82C2×C8×C131C2312yes—
83C4⋊C4×C131—21no—
84C4⋊C4×C131C2312yes—
85C22×D131—22no—
86C22×D131—22no—
87C22×D131—22no—
88C22×D131C2×D411yescharacteristic
89C22×D131C2×D411yescharacteristic
90C22⋊D131—42no—
91C22⋊D131—42no—
92C2×Dic131—22no—
93C2×Dic131—21no—
94C2×Dic131—21no—
95C2×Dic131—22no—
96C2×Dic131—42no—
97C2×Dic131C2×D411yescharacteristic
98C2×Dic131C2×D411yescharacteristic
99C4×D131—22no—
100C4×D131—22no—
101C4×D131—21no—
102C4×D131—21no—
103C4×D131—21no—
104C4×D131—21no—
105C4×D131—44no—
106C4×D131—44no—
107C131⋊C8—22no—
108C23×C131—21no—
109C23×C131C2×D411yescharacteristic
110Q8×C131—21no—
111Q8×C131—21no—
112D4×C131—21no—
113D4×C131—21no—
114D4×C131—42no—
115C2×C4×C131—22no—
116C2×C4×C131—21no—
117C2×C4×C131—42no—
118C2×C4×C131—42no—
119C2×C4×C131C2×D411yescharacteristic
120C2×C4×C131#4112yes—
121C2×C4×C131#4112yes—
122C2×C4×C131C2411yescharacteristic
123C8×C131—22no—
124C22×C131—21no—
125C22×C131—21no—
126C22×C131—41no—
127C22×C131—41no—
128C22×C131—41no—
129C22×C131#12312yes—
130C22×C131C22×D411yescharacteristic
131C2×D131—28no—
132C2×D131—21no—
133C2×D131—21no—
134C2×D131—21no—
135C2×D131—21no—
136C2×D131AGL(1,ℤ/8)12yes—
137C2×D131#13712yes—
138C4×C131—41no—
139C4×C131—42no—
140C4×C131#12312yes—
141C4×C131C2×#4111yescharacteristic
142C4×C131C2×#4111yescharacteristic
143Dic131—21no—
144Dic131—21no—
145Dic131—42no—
146Dic131—41no—
147Dic131—41no—
148C2×C131—22no—
149C2×C131—81no—
150C2×C131C2×#12311yescharacteristic
151C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C131C2×#13711yescharacteristic
153D131—22no—
154D131—22no—
155C131#309211yescharacteristic, complemented, Sylow
156#3092—1311nomaximal, Sylow
157#525—1311no—
158C22×C4⋊C4—1311no—
159#484—1312no—
160#483—1312no—
161#481—1312no—
162#480—1312no—
163#428—1311no—
164#428—1311no—
165#417—1311no—
166#417D13111yescharacteristic, complemented
167#407—1311no—
168C2×#41C2×D13111yescharacteristic
169C23×C4—2621no—
170C23×C4—1311no—
171C2×M16—1311no—
172C2×M16C2×D13111yescharacteristic
173#123—2621no—
174C22⋊Q8—1311no—
175C22⋊Q8—1311no—
176#121—1311no—
177#121—1311no—
178#120—2621no—
179C4×D4—2621no—
180C4×D4—2621no—
181#117—1311no—
182C2×C4⋊C4—2621no—
183C2×C4⋊C4—2621no—
184C2×C4⋊C4—2621no—
185C2×C4⋊C4—1311no—
186C2×C4⋊C4—1312no—
187C2×C4⋊C4—1312no—
188C2×C4⋊C4C2×D13111yescharacteristic
189#107—1312no—
190#107—1312no—
191#106—1312no—
192#106—1312no—
193#103—1312no—
194#103C2×D13112yescomplemented
195#102—1312no—
196#102C2×D13112yescomplemented
197C22⋊C8—1312no—
198C22⋊C8—1312no—
199C24—1311no—
200#41—22no—
201#41—22no—
202C2×Q8C22×D13111yescharacteristic
203C2×D4—21no—
204C2×D4—2621no—
205C2×D4—2621no—
206C2×D4C22×D13111yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—2622no—
210C22×C4—2622no—
211C22×C4—2622no—
212C22×C4—2622no—
213C22×C4—2621no—
214C22×C4—5244no—
215C22×C4—1311no—
216C22×C4—1311no—
217C22×C4—1311no—
218C22×C4—1311no—
219C22×C4—1311no—
220C22×C4C22×D13111yescharacteristic
221M16—22no—
222M16—2622no—
223C2×C8—1312no—
224C2×C8C22×D13112yes—
225C4⋊C4—21no—
226C4⋊C4—2622no—
227C4⋊C4—2622no—
228C4⋊C4—2621no—
229C4⋊C4—2621no—
230C4⋊C4—2621no—
231C4⋊C4—1312no—
232C4⋊C4—1311no—
233C4⋊C4—1311no—
234C4⋊C4C22×D13112yes—
235C22⋊C4—2621no—
236C22⋊C4—2621no—
237C22⋊C4—2621no—
238C22⋊C4—2621no—
239C22⋊C4—2621no—
240C42—2621no—
241C23—21no—
242C23—2622no—
243C23—2622no—
244C23—2622no—
245C23—1311no—
246C23—1311no—
247C23D4×D13111yescharacteristic
248Q8—21no—
249Q8—21no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—5242no—
254D4—5242no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—2622no—
260C2×C4—2622no—
261C2×C4—2622no—
262C2×C4—2621no—
263C2×C4—2621no—
264C2×C4—2621no—
265C2×C4—2621no—
266C2×C4—2621no—
267C2×C4—2621no—
268C2×C4—2622no—
269C2×C4—5244no—
270C2×C4—5244no—
271C2×C4—5242no—
272C2×C4—1311no—
273C2×C4—1311no—
274C2×C4D4×D13111yescharacteristic
275C2×C4#1452351212yes—
276C2×C4#1452351312yes—
277C2×C4C23×D13111yescharacteristic
278C8—22no—
279C8—2622no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—2628no—
286C22—2621no—
287C22—2621no—
288C22—2621no—
289C22—2621no—
290C22—1312no—
291C22—1312no—
292C22#1686222812yes—
293C22C2×D4×D13111yescharacteristic
294C4—41no—
295C4—42no—
296C4—2621no—
297C4—2621no—
298C4—5242no—
299C4—5241no—
300C4—5241no—
301C4#1686222812yes—
302C4C2×#1452351211yescharacteristic
303C4C2×#1452351311yescharacteristic
304C2—22no—
305C2—2622no—
306C2—2622no—
307C2—81no—
308C2C2×#1686222811yescharacteristic
309C2AGL(1,ℤ/8)×D13111yescharacteristic
310C2#137×D13111yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#417×C131C2
socleC22×C131C22×D4
prime radical#22553819C2
hypercenter#417D131
hypocenterC131#3092
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D131C2semidirect
3. #22553646C2semidirect
4. #22553641C2semidirect
5. #22553629C2semidirect
6. #22553624C2semidirect
7. #22553354C2semidirect
8. #22553352C2semidirect
9. #22553313C2semidirect
10. #22553275C2semidirect
11. #417×C131C2semidirect
30. #16862180C22semidirect
31. #16862179C22semidirect
32. #16862178C22semidirect
33. #16862177C22semidirect
34. #16862170C22semidirect
35. #16862167C22semidirect
41. #103×C131C22semidirect
42. #102×C131C22semidirect
155. C131#3092semidirect
166. #417D131semidirect
194. #103C2×D131semidirect
196. #102C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225538192
C2×C4⋊C4×D1312
#225536462
#225536412
#225536292
#225536242
#225533542
#225533522
#225533132
#225532752
#417×C1312
#3092131