Finite Group Databasebeta

← #14540570

Subgroups — #14540570

Order 2176.

454 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 125 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#11763617C212yesmaximal, complemented
2#11763611C211yesmaximal, characteristic, complemented
3#11763194C211yesmaximal, characteristic, complemented
4#11763081C211yesmaximal, characteristic, complemented
5#11763079C211yesmaximal, characteristic, complemented
6#11763074C212yesmaximal, complemented
7#11762985C212yesmaximal, complemented
8#11762982C212yesmaximal, complemented
9#11762926C211yesmaximal, characteristic, complemented
10#11762924C211yesmaximal, characteristic, complemented
11C17×#578C211yesmaximal, characteristic, complemented
12C17×2−1+4C2212yescomplemented
13#10587696C2212yes—
14C17×#137—22no—
15C17×#137C2212yescomplemented
16C17×#135—21no—
17C17×#135C2211yescharacteristic
18C2×Q16×C17—21no—
19C2×Q16×C17C2211yescharacteristic, complemented
20C17×#131C2211yescharacteristic, complemented
21#10587636—22no—
22#10587636C2211yescharacteristic
23#10587635—22no—
24#10587635C2212yes—
25#10587635C2212yescomplemented
26#10587634C2212yescomplemented
27#10587633C2212yescomplemented
28#10587628—22no—
29#10587628C2212yes—
30C2×#63230—22no—
31C2×#63230C2212yescomplemented
32#10587605—21no—
33#10587600—22no—
34D8×D17—21no—
35C2×D136C2211yescharacteristic
36#10587587C2211yescharacteristic
37#10587508C2211yescharacteristic, complemented
38#10587508C2212yescomplemented
39#10587507C2212yescomplemented
40#10587496C2211yescharacteristic
41#10587494C2212yescomplemented
42#10587493C2211yescharacteristic, complemented
43#10587490C2211yescharacteristic, complemented
44C8×Dic17C2211yescharacteristic, complemented
45#41×C17—22no—
46#41×C17—22no—
47#41×C17—42no—
48#41×C17C2311yescharacteristic
49#41×C17C2312yes—
50C2×Q8×C17—22no—
51C2×Q8×C17—22no—
52C2×Q8×C17C2312yes—
53#63256—22no—
54D4×D17—22no—
55D4×D17—42no—
56#63252—22no—
57#63252C2311yescharacteristic
58C2×D68—22no—
59C2×D68C2312yes—
60Q16×C17—22no—
61Q16×C17—21no—
62Q16×C17—21no—
63Q16×C17D411yescharacteristic
64SD16×C17—22no—
65SD16×C17D412yes—
66D8×C17D411yescharacteristic
67M16×C17—21no—
68M16×C17C2311yescharacteristic
69C2×C8×C17—21no—
70C2×C8×C17C2311yescharacteristic
71#63231—22no—
72#63231D411yescharacteristic
73#63230—22no—
74#63230—22no—
75#63230—22no—
76#63230—22no—
77#63230—22no—
78#63230D411yescharacteristic
79#63229D411yescharacteristic
80#63228—22no—
81#63228—22no—
82#63228D411yescharacteristic
83#63227—42no—
84C4×Dic17C2311yescharacteristic
85#63223—22no—
86#63223C2312yes—
87#63223C2311yescharacteristic
88C2×#3622—22no—
89C2×#3622C2311yescharacteristic
90D136—21no—
91D136—21no—
92#63218—21no—
93C8×D17—21no—
94C22×D17—22no—
95C22×D17—42no—
96Q8×C17—22no—
97Q8×C17—22no—
98Q8×C17—42no—
99Q8×C17C2×D411yescharacteristic
100Q8×C17C2×D412yes—
101D4×C17—22no—
102D4×C17—21no—
103D4×C17—42no—
104D4×C17C2×D411yescharacteristic
105D4×C17C2×D412yes—
106C2×C4×C17—21no—
107C2×C4×C17—22no—
108C2×C4×C17—22no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17C2411yescharacteristic
112C22⋊D17—21no—
113C22⋊D17—42no—
114C2×Dic17—21no—
115D68—22no—
116D68—22no—
117D68—42no—
118D68C2×D411yescharacteristic
119C4×D17—21no—
120C4×D17—42no—
121Dic34C2×D411yescharacteristic
122C8×C17—21no—
123C8×C17C2×D411yescharacteristic
124C8×C17C2×D411yescharacteristic
125#3622—21no—
126#3622—22no—
127#3622C2×D411yescharacteristic
128#3622C2×D411yescharacteristic
129#3290—171nomaximal, Sylow
130C22×C17—22no—
131C22×C17—21no—
132C22×C17H(5,𝔽2)11yescharacteristic
133C2×D17—21no—
134C2×D17—42no—
135C2×D17—82no—
136C2×D17—162no—
137C4×C17—22no—
138C4×C17—21no—
139C4×C17—42no—
140C4×C17C22×D411yescharacteristic
141C4×C17C22×D411yescharacteristic
142Dic17—21no—
143Dic17—41no—
144#578D1711yescharacteristic, complemented
145#577—172no—
146#576—171no—
147#495—171no—
148#473—171no—
149#472—171no—
150#471—172no—
151#455—172no—
152#454—172no—
153#443—171no—
154#443—171no—
155C2×C17—21no—
156C2×C17—42no—
157C2×C17—41no—
158C2×C17D4211yescharacteristic
159D17—41no—
160D17—82no—
1612−1+4C2×D1712yescomplemented
162H(5,𝔽2)—172no—
163#137—22no—
164#137—172no—
165#137—342no—
166#137C2×D1712yescomplemented
167AGL(1,ℤ/8)—172no—
168AGL(1,ℤ/8)—172no—
169AGL(1,ℤ/8)—342no—
170AGL(1,ℤ/8)—342no—
171#135—21no—
172#135—171no—
173#135—342no—
174#135—341no—
175#135C2×D1711yescharacteristic
176C2×Q16—21no—
177C2×Q16C2×D1711yescharacteristic, complemented
178C2×SD16—172no—
179C2×SD16—342no—
180C2×D8—171no—
181C2×D8—341no—
182#131—172no—
183#131—171no—
184#131C2×D1711yescharacteristic, complemented
185#124—172no—
186#108—171no—
187#108—171no—
188#104—171no—
189#104—172no—
190#104—171no—
191#101—172no—
192#100—172no—
193C4×C8—171no—
194C17#329011yescharacteristic, complemented, Sylow
195#41—22no—
196#41—22no—
197#41—42no—
198#41—171no—
199#41—342no—
200#41—342no—
201#41C22×D1711yescharacteristic
202#41C22×D1712yes—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8C22×D1712yes—
206C2×D4—172no—
207C2×D4—342no—
208C2×D4—342no—
209C2×D4—682no—
210Q16—22no—
211Q16—21no—
212Q16—21no—
213Q16—171no—
214Q16—342no—
215Q16C22⋊D1711yescharacteristic
216SD16—22no—
217SD16—171no—
218SD16—171no—
219SD16—342no—
220SD16—342no—
221SD16—342no—
222SD16—342no—
223SD16—342no—
224SD16C22⋊D1712yes—
225D8—171no—
226D8—342no—
227D8—342no—
228D8—341no—
229D8—341no—
230D8C22⋊D1711yescharacteristic
231M16—21no—
232M16—172no—
233M16—171no—
234M16—342no—
235M16—341no—
236M16C22×D1711yescharacteristic
237C2×C8—21no—
238C2×C8—171no—
239C2×C8—342no—
240C2×C8—341no—
241C2×C8C22×D1711yescharacteristic
242C22⋊C4—682no—
243C42—171no—
244C23—342no—
245C23—682no—
246Q8—22no—
247Q8—22no—
248Q8—42no—
249Q8—171no—
250Q8D4×D1711yescharacteristic
251Q8C2×C22⋊D1712yes—
252D4—22no—
253D4—21no—
254D4—42no—
255D4—171no—
256D4—341no—
257D4—342no—
258D4—342no—
259D4—682no—
260D4—682no—
261D4D4×D1711yescharacteristic
262D4C2×C22⋊D1712yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—341no—
269C2×C4—341no—
270C2×C4—682no—
271C2×C4C23×D1711yescharacteristic
272C8—21no—
273C8—171no—
274C8—171no—
275C8—342no—
276C8—341no—
277C8C2×C22⋊D1711yescharacteristic
278C8C2×C22⋊D1711yescharacteristic
279C22—21no—
280C22—22no—
281C22—2722no—
282C22—341no—
283C22—682no—
284C22—1362no—
285C22#1058769111yescharacteristic
286C4—21no—
287C4—22no—
288C4—42no—
289C4—341no—
290C4—681no—
291C4C2×D4×D1711yescharacteristic
292C4C22×C22⋊D1711yescharacteristic
293C2—21no—
294C2—42no—
295C2—41no—
296C2—681no—
297C2—1362no—
298C2D4×C22⋊D1711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#578C2
socleC2×C17D42
hypercenter#578D17
hypocenterC17#3290
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #11763617C2semidirect
2. #11763611C2semidirect
3. #11763194C2semidirect
4. #11763081C2semidirect
5. #11763079C2semidirect
6. #11763074C2semidirect
7. #11762985C2semidirect
8. #11762982C2semidirect
9. #11762926C2semidirect
10. #11762924C2semidirect
11. C17×#578C2semidirect
12. C17×2−1+4C22semidirect
15. C17×#137C22semidirect
19. C2×Q16×C17C22semidirect
20. C17×#131C22semidirect
25. #10587635C22semidirect
26. #10587634C22semidirect
27. #10587633C22semidirect
31. C2×#63230C22semidirect
37. #10587508C22semidirect
38. #10587508C22semidirect
39. #10587507C22semidirect
41. #10587494C22semidirect
42. #10587493C22semidirect
43. #10587490C22semidirect
44. C8×Dic17C22semidirect
144. #578D17semidirect
161. 2−1+4C2×D17semidirect
166. #137C2×D17semidirect
177. C2×Q16C2×D17semidirect
184. #131C2×D17semidirect
194. C17#3290semidirect

Maximal subgroups

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

subgroupindex
#117636172
#117636112
#117631942
#117630812
#117630792
#117630742
#117629852
#117629822
#117629262
#117629242
C17×#5782
#329017