Finite Group Databasebeta

← #17192467

Subgroups — #17192467

Order 4352.

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

normal subgroups only

This group has 466 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
1D17×#508C212yesmaximal, complemented
2#14531103C212yesmaximal, complemented
3#14531100C212yesmaximal, complemented
4#14531098C212yesmaximal, complemented
5#14531095C212yesmaximal, complemented
6#14531093C211yesmaximal, characteristic, complemented
7#14531091C212yesmaximal, complemented
8C2×#11762642C211yesmaximal, characteristic, complemented
9C17×#2262C211yesmaximal, characteristic, complemented
10#135×D17—22no—
11#11763242C2212yes—
12#11763241C2212yescomplemented
13#11763240—22no—
14#11763239—22no—
15#11763238C2212yes—
16Q32×D17C2212yescomplemented
17SD32×D17—22no—
18D16×D17C2212yescomplemented
19C2×C16×D17C2211yescharacteristic
20Dic272C2211yescharacteristic, complemented
21#11762649C2212yescomplemented
22#11762648C2212yescomplemented
23#11762647C2212yescomplemented
24#11762646C2212yescomplemented
25D544C2211yescharacteristic, complemented
26#11762644C2212yescomplemented
27#11762642C2212yescomplemented
28#11762642C2212yescomplemented
29C2×#10587465C2211yescharacteristic, complemented
30C17×#508C2212yescomplemented
31C17×Q64C2211yescharacteristic, complemented
32C17×SD64C2212yescomplemented
33C17×D32C2211yescharacteristic, complemented
34C2×C17×C32C2211yescharacteristic, complemented
35#41×D17—42no—
36C17×#135—22no—
37#10587636—42no—
38#10587605—22no—
39Q16×D17—22no—
40#10587602—42no—
41SD16×D17—42no—
42#10587598—22no—
43D8×D17—22no—
44#10587590—22no—
45C2×C8×D17D411yescharacteristic
46C17×Q32C2312yes—
47C17×SD32—22no—
48C17×D16C2312yes—
49C2×C16×C17C2311yescharacteristic
50#10587500—22no—
51#10587499—22no—
52#10587498—22no—
53#10587497—22no—
54C2×#63214D411yescharacteristic
55Dic136C2312yes—
56#10587471—22no—
57D272C2312yes—
58C16×D17D412yes—
59C16×D17C2312yes—
60C17×C32C2311yescharacteristic
61C17×C32C2311yescharacteristic
62#10587465C2312yes—
63#41×C17—42no—
64#63256—42no—
65Q8×D17—42no—
66#63254—42no—
67#63254—82no—
68D4×D17—42no—
69D4×D17—82no—
70#63252—42no—
71#63252—82no—
72C2×C4×D17—82no—
73C2×C4×D17D811yescharacteristic
74Q16×C17—22no—
75SD16×C17—42no—
76D8×C17—22no—
77C2×C8×C17C2×D411yescharacteristic
78#63231—42no—
79#63230—42no—
80#63229—42no—
81#63228—42no—
82C2×#3622D811yescharacteristic
83Dic68—22no—
84D136—22no—
85#63219—42no—
86C8×D17D812yes—
87C8×D17C2×D412yes—
88C16×C17C2×D411yescharacteristic
89C16×C17C2411yescharacteristic
90#63214C2×D412yes—
91#33981—171nomaximal, Sylow
92C22×D17—82no—
93C22×D17D1611yescharacteristic
94Q8×C17—42no—
95D4×C17—42no—
96D4×C17—82no—
97C2×C4×C17—82no—
98C2×C4×C17C2×D811yescharacteristic
99C22⋊D17—82no—
100C22⋊D17—82no—
101C22⋊D17—82no—
102C2×Dic17—82no—
103C2×Dic17D1611yescharacteristic
104D68—42no—
105D68—82no—
106C4×D17—82no—
107C4×D17—82no—
108C4×D17—82no—
109C4×D17D1612yes—
110C4×D17C2×D812yes—
111Dic34—42no—
112Dic34—82no—
113C8×C17C2×D811yescharacteristic
114C8×C17C22×D411yescharacteristic
115#3622C2×D812yes—
116C2×#508—172no—
117#2264—172no—
118#2264—172no—
119#2263—172no—
120#2263—172no—
121#2262—172no—
122#2262—171no—
123#2262D1711yescharacteristic, complemented
124C2×M64—171no—
125C22×C17—82no—
126C22×C17C2×D1611yescharacteristic
127C2×D17—41no—
128C2×D17—82no—
129C2×D17—164no—
130C2×D17C2×D1612yes—
131C4×C17—82no—
132C4×C17C2×D1611yescharacteristic
133C4×C17C22×D811yescharacteristic
134Dic17—82no—
135Dic17C2×D1612yes—
136C2×#135—342no—
137#508—172no—
138#508—172no—
139#508—172no—
140#508—342no—
141#508—342no—
142#508C2×D1712yescomplemented
143C2×Q32—172no—
144C2×SD32—342no—
145C2×D16—172no—
146C22×C16—171no—
147Q64—172no—
148Q64—171no—
149Q64C2×D1711yescharacteristic, complemented
150SD64—172no—
151SD64—172no—
152SD64—172no—
153SD64C2×D1712yescomplemented
154D32—172no—
155D32—171no—
156D32C2×D1711yescharacteristic, complemented
157M64—172no—
158M64—172no—
159C2×C32—171no—
160C2×C32C2×D1711yescharacteristic, complemented
161C2×C17—21no—
162C2×C17—162no—
163C2×C17C22×D1611yescharacteristic
164D17—22no—
165D17—162no—
166C2×#41—682no—
167#135—22no—
168#135—342no—
169#135—342no—
170#135—342no—
171#135—682no—
172#135—682no—
173C2×Q16—342no—
174C2×SD16—682no—
175C2×D8—342no—
176C22×C8—171no—
177Q32—172no—
178Q32—342no—
179Q32C22×D1712yes—
180SD32—22no—
181SD32—342no—
182SD32—342no—
183SD32—342no—
184D16—172no—
185D16—342no—
186D16C22×D1712yes—
187C2×C16—171no—
188C2×C16—172no—
189C2×C16—172no—
190C2×C16C22×D1711yescharacteristic
191C32—172no—
192C32C22×D1711yescharacteristic
193C32C22×D1711yescharacteristic
194C17#3398111yescharacteristic, complemented, Sylow
195#41—42no—
196#41—682no—
197#41—682no—
198#41—682no—
199#41—1362no—
200#41—1362no—
201C2×Q8—682no—
202C2×D4—682no—
203C2×D4—1362no—
204C22×C4—171no—
205C22×C4—1362no—
206Q16—22no—
207Q16—342no—
208Q16—682no—
209SD16—42no—
210SD16—682no—
211SD16—682no—
212SD16—682no—
213D8—22no—
214D8—342no—
215D8—682no—
216C2×C8—171no—
217C2×C8—172no—
218C2×C8—172no—
219C2×C8D4×D1711yescharacteristic
220C16—172no—
221C16D4×D1711yescharacteristic
222C16C23×D1711yescharacteristic
223C23—171no—
224C23—1362no—
225Q8—42no—
226Q8—682no—
227Q8—1362no—
228D4—42no—
229D4—82no—
230D4—682no—
231D4—1362no—
232D4—1362no—
233D4—1362no—
234D4—1362no—
235C2×C4—82no—
236C2×C4—171no—
237C2×C4—172no—
238C2×C4—172no—
239C2×C4—1362no—
240C2×C4—1362no—
241C2×C4—1362no—
242C2×C4—1362no—
243C2×C4D8×D1711yescharacteristic
244C8—172no—
245C8D8×D1711yescharacteristic
246C8C2×D4×D1711yescharacteristic
247C22—82no—
248C22—2724no—
249C22—172no—
250C22—681no—
251C22—1362no—
252C22D16×D1711yescharacteristic
253C4—82no—
254C4—172no—
255C4—1362no—
256C4D16×D1711yescharacteristic
257C4C2×D8×D1711yescharacteristic
258C2—21no—
259C2—162no—
260C2—2722no—
261C2—342no—
262C2C2×D16×D1711yescharacteristic
subgroupG/H
centerC4D16×D17
commutator, cosocleC16×C17C24
FrattiniC16C23×D17
FittingC17×#2262C2
socleC2×C17C22×D16
hypercenter#2262D17
hypocenterC17#33981
LCSG ▷ C16×C17 ▷ C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2262

Complements

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

NH ≅ G/Ntype
1. D17×#508C2semidirect
2. #14531103C2semidirect
3. #14531100C2semidirect
4. #14531098C2semidirect
5. #14531095C2semidirect
6. #14531093C2semidirect
7. #14531091C2semidirect
8. C2×#11762642C2semidirect
9. C17×#2262C2semidirect
12. #11763241C22semidirect
16. Q32×D17C22semidirect
18. D16×D17C22semidirect
20. Dic272C22semidirect
21. #11762649C22semidirect
22. #11762648C22semidirect
23. #11762647C22semidirect
24. #11762646C22semidirect
25. D544C22semidirect
26. #11762644C22semidirect
27. #11762642C22semidirect
28. #11762642C22semidirect
29. C2×#10587465C22semidirect
30. C17×#508C22semidirect
31. C17×Q64C22semidirect
32. C17×SD64C22semidirect
33. C17×D32C22semidirect
34. C2×C17×C32C22semidirect
123. #2262D17semidirect
142. #508C2×D17semidirect
149. Q64C2×D17semidirect
153. SD64C2×D17semidirect
156. D32C2×D17semidirect
160. C2×C32C2×D17semidirect
194. C17#33981semidirect

Maximal subgroups

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

subgroupindex
D17×#5082
#145311032
#145311002
#145310982
#145310952
#145310932
#145310912
C2×#117626422
C17×#22622
#3398117