Finite Group Databasebeta

← #14540468

Subgroups — #14540468

Order 2176.

526 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 123 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×#10587605C211yesmaximal, characteristic, complemented
2C2×D8×D17C211yesmaximal, characteristic, complemented
3#11763452C212yesmaximal, complemented
4#11763047C211yesmaximal, characteristic, complemented
5#11763042C211yesmaximal, characteristic, complemented
6#11763021C212yesmaximal, complemented
7#11762958C212yesmaximal, complemented
8#11762953C212yesmaximal, complemented
9C4×D136C211yesmaximal, characteristic, complemented
10C8×D68C211yesmaximal, characteristic, complemented
11C17×#495C211yesmaximal, characteristic, complemented
12C2×#63256C2212yescomplemented
13C2×D4×D17C2212yes—
14C2×Q16×C17C2211yescharacteristic, complemented
15C2×SD16×C17C2212yescomplemented
16C2×D8×C17C2211yescharacteristic
17C17×#124C2212yescomplemented
18#10587632C2212yes—
19C2×#63230C2212yescomplemented
20#10587624C2212yescomplemented
21C2×#63228C2212yes—
22#10587605—24no—
23D8×D17—24no—
24C2×D136C2211yescharacteristic
25C2×C8×D17C2211yescharacteristic
26C2×C8×D17C2211yescharacteristic, complemented
27#10587570—22no—
28#10587569—22no—
29#10587568—22no—
30C22⋊C4×D17—22no—
31C4×D68C2211yescharacteristic
32C4×D68C2212yescomplemented
33C4×C8×C17C2211yescharacteristic, complemented
34#10587506C2212yescomplemented
35#10587503C2212yescomplemented
36#10587492C2212yescomplemented
37#10587491C2211yescharacteristic
38#10587491C2211yescharacteristic, complemented
39#10587489C2211yescharacteristic, complemented
40C4⋊#3622C2211yescharacteristic, complemented
41C23×D17—22no—
42C2×Q8×C17C2312yes—
43C2×D4×C17C2312yes—
44C2×C22⋊D17—22no—
45C2×C22⋊D17—22no—
46#63256—24no—
47#63256—42no—
48D4×D17—24no—
49D4×D17—42no—
50C2×D68—22no—
51C2×D68D411yescharacteristic
52C2×D68C2312yes—
53C2×C4×D17—22no—
54C2×C4×D17—22no—
55C2×C4×D17C2311yescharacteristic
56C2×C4×D17C2311yescharacteristic
57Q16×C17—22no—
58SD16×C17—42no—
59D8×C17—22no—
60C2×C8×C17C2311yescharacteristic
61C2×C8×C17C2311yescharacteristic
62C22⋊C4×C17—22no—
63C22⋊C4×C17—22no—
64C42×C17C2311yescharacteristic
65C22⋊Dic17—22no—
66#63230—24no—
67#63228—24no—
68#63227—22no—
69#63227—22no—
70#63227D411yescharacteristic
71#63227D411yescharacteristic
72C4⋊Dic17D411yescharacteristic
73C4⋊Dic17C2312yes—
74#63225—22no—
75C2×#3622C2311yescharacteristic
76C2×#3622C2311yescharacteristic
77D136D414yescomplemented
78C8×D17—22no—
79C8×D17—22no—
80C23×C17—22no—
81C22×D17—22no—
82C22×D17—44no—
83C22×D17—44no—
84C22×D17—82no—
85C22×D17C2×D411yescharacteristic
86C22×D17C2×D411yescharacteristic
87Q8×C17—22no—
88Q8×C17—22no—
89D4×C17—22no—
90D4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17C2×D411yescharacteristic
93C2×C4×C17C2×D411yescharacteristic
94C2×C4×C17C2411yescharacteristic
95C22⋊D17—44no—
96C22⋊D17—82no—
97C2×Dic17—22no—
98C2×Dic17C2×D411yescharacteristic
99C2×Dic17C2×D411yescharacteristic
100D68—22no—
101D68—22no—
102D68—44no—
103D68C2×D414yes—
104C4×D17—21no—
105C4×D17—21no—
106C4×D17—21no—
107C4×D17—21no—
108C4×D17—44no—
109C4×D17—82no—
110C8×C17—21no—
111C8×C17C2×D412yes—
112#3622—21no—
113#3622—21no—
114#3280—171nomaximal, Sylow
115C22×C17—42no—
116C22×C17—42no—
117C22×C17—42no—
118C22×C17C22×D411yescharacteristic
119C2×D17—21no—
120C2×D17—21no—
121C2×D17—21no—
122C2×D17—24no—
123C2×D17—21no—
124C2×D17—21no—
125C2×D17—42no—
126C2×D17—42no—
127C2×D17—88no—
128C2×D17#13512yescomplemented
129C4×C17—21no—
130C4×C17—21no—
131C4×C17—42no—
132C4×C17C22×D411yescharacteristic
133C4×C17H(5,𝔽2)11yescharacteristic
134Dic17—21no—
135Dic17—21no—
136Dic17—42no—
137C2×#135—171no—
138C22×D8—171no—
139#546—172no—
140#495D1711yescharacteristic, complemented
141#467—171no—
142#466—171no—
143#463—172no—
144#450—172no—
145#449—172no—
146C4×D8—171no—
147C8×D4—171no—
148C2×C17—82no—
149C2×C17D4211yescharacteristic
150C2×C17C2×#13511yescharacteristic
151C2×C17#57611yescharacteristic
152D17—22no—
153D17—41no—
154D17—44no—
155C2×#41—172no—
156C22×D4—172no—
157#135—344no—
158C2×Q16C2×D1711yescharacteristic, complemented
159C2×SD16—172no—
160C2×SD16C2×D1712yescomplemented
161C2×D8—172no—
162C2×D8—171no—
163C2×D8—344no—
164C2×D8C2×D1711yescharacteristic
165C22×C8—171no—
166C22×C8—171no—
167#124C2×D1712yescomplemented
168#123—342no—
169C22⋊Q8—172no—
170#121—172no—
171#121—342no—
172#120—342no—
173C4×D4—171no—
174C4×D4—172no—
175C2×C22⋊C4—342no—
176#107—171no—
177C4⋊C8—171no—
178#103—172no—
179#102—172no—
180#102—172no—
181C22⋊C8—171no—
182C22⋊C8—171no—
183C4×C8C2×D1711yescharacteristic, complemented
184C17#328011yescharacteristic, complemented, Sylow
185C24—342no—
186#41—344no—
187#41—682no—
188C2×Q8C22×D1712yes—
189C2×D4—171no—
190C2×D4—172no—
191C2×D4—342no—
192C2×D4—342no—
193C2×D4—344no—
194C2×D4—342no—
195C2×D4—682no—
196C2×D4C22×D1712yes—
197C22×C4—171no—
198C22×C4—171no—
199C22×C4—342no—
200C22×C4—342no—
201Q16—22no—
202SD16—42no—
203SD16—344no—
204D8—22no—
205D8—174no—
206D8—344no—
207C2×C8—171no—
208C2×C8—171no—
209C2×C8—342no—
210C2×C8—342no—
211C2×C8C22×D1711yescharacteristic
212C2×C8C22×D1711yescharacteristic
213C4⋊C4—171no—
214C4⋊C4—172no—
215C4⋊C4—342no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—171no—
219C22⋊C4—171no—
220C22⋊C4—342no—
221C22⋊C4—342no—
222C22⋊C4—342no—
223C42C22×D1711yescharacteristic
224C23—22no—
225C23—171no—
226C23—171no—
227C23—342no—
228C23—684no—
229C23—684no—
230C23—1362no—
231Q8—22no—
232Q8—22no—
233D4—22no—
234D4—22no—
235D4—174no—
236D4—342no—
237D4—342no—
238D4—684no—
239D4—684no—
240D4—1362no—
241C2×C4—22no—
242C2×C4—171no—
243C2×C4—171no—
244C2×C4—341no—
245C2×C4—341no—
246C2×C4—341no—
247C2×C4—341no—
248C2×C4—342no—
249C2×C4—684no—
250C2×C4—1362no—
251C2×C4D4×D1711yescharacteristic
252C2×C4D4×D1711yescharacteristic
253C2×C4C23×D1711yescharacteristic
254C8—21no—
255C8—341no—
256C8—341no—
257C8D4×D1712yes—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—172no—
262C22—341no—
263C22—341no—
264C22—341no—
265C22—341no—
266C22—341no—
267C22—344no—
268C22—682no—
269C22—682no—
270C22—1368no—
271C22C2×D4×D1711yescharacteristic
272C4—21no—
273C4—21no—
274C4—42no—
275C4—341no—
276C4—341no—
277C4—682no—
278C4C2×D4×D1711yescharacteristic
279C4#1058769111yescharacteristic
280C2—82no—
281C2—342no—
282C2—681no—
283C2—684no—
284C2#1176344411yescharacteristic
285C2#135×D1711yescharacteristic
286C2#1176361111yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#495C2
socleC22×C17C22×D4
hypercenter#495D17
hypocenterC17#3280
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#10587605C2semidirect
2. C2×D8×D17C2semidirect
3. #11763452C2semidirect
4. #11763047C2semidirect
5. #11763042C2semidirect
6. #11763021C2semidirect
7. #11762958C2semidirect
8. #11762953C2semidirect
9. C4×D136C2semidirect
10. C8×D68C2semidirect
11. C17×#495C2semidirect
12. C2×#63256C22semidirect
14. C2×Q16×C17C22semidirect
15. C2×SD16×C17C22semidirect
17. C17×#124C22semidirect
19. C2×#63230C22semidirect
20. #10587624C22semidirect
26. C2×C8×D17C22semidirect
32. C4×D68C22semidirect
33. C4×C8×C17C22semidirect
34. #10587506C22semidirect
35. #10587503C22semidirect
36. #10587492C22semidirect
38. #10587491C22semidirect
39. #10587489C22semidirect
40. C4⋊#3622C22semidirect
77. D136D4semidirect
128. C2×D17#135semidirect
140. #495D17semidirect
158. C2×Q16C2×D17semidirect
160. C2×SD16C2×D17semidirect
167. #124C2×D17semidirect
183. C4×C8C2×D17semidirect
184. C17#3280semidirect

Maximal subgroups

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

subgroupindex
C2×#105876052
C2×D8×D172
#117634522
#117630472
#117630422
#117630212
#117629582
#117629532
C4×D1362
C8×D682
C17×#4952
#328017