Finite Group Databasebeta

← #14538721

Subgroups — #14538721

Order 2176.

462 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 121 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#121×D17C211yesmaximal, characteristic
2C2×#10587578C211yesmaximal, characteristic, complemented
3#11763135C212yesmaximal, complemented
4#11763131C212yesmaximal, complemented
5#11763127C211yesmaximal, characteristic, complemented
6#11763123C211yesmaximal, characteristic, complemented
7#11763120C211yesmaximal, characteristic, complemented
8#11762848C212yesmaximal, complemented
9#11762812C212yesmaximal, complemented
10#11762771C211yesmaximal, characteristic, complemented
11C17×#481C211yesmaximal, characteristic, complemented
12C2×D4×D17—21no—
13C2×D4×D17C2211yescharacteristic
14C22×C4×D17C2211yescharacteristic
15C17×#121C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic, complemented
17#10587625—21no—
18#10587624C2211yescharacteristic
19#10587616C2211yescharacteristic
20C2×#63227—21no—
21C2×C4⋊Dic17C2211yescharacteristic, complemented
22C2×C4×Dic17—21no—
23C2×#63218C2212yes—
24#10587581C2211yescharacteristic
25#10587578—21no—
26#10587578—21no—
27#10587578C2212yes—
28#10587578C2212yescomplemented
29C4⋊C4×D17C2211yescharacteristic
30#10587570—21no—
31C22⋊C4×D17—21no—
32C17×#106C2212yescomplemented
33C17×#102C2212yescomplemented
34C17×C22⋊C8C2211yescharacteristic, complemented
35#10587503C2212yescomplemented
36C22⋊#3622C2211yescharacteristic, complemented
37#10587492C2212yescomplemented
38#10587491C2212yescomplemented
39#10587489C2212yescomplemented
40#10587480C2212yescomplemented
41#10587479C2212yescomplemented
42#10587478C2212yescomplemented
43C23×D17—21no—
44C23×D17D411yescharacteristic
45C2×D4×C17—21no—
46C2×D4×C17C2311yescharacteristic
47C22×C4×C17—21no—
48C22×C4×C17C2311yescharacteristic
49C2×C22⋊D17—21no—
50C2×C22⋊D17—21no—
51C2×C22⋊D17—21no—
52C22×Dic17—21no—
53C22×Dic17D411yescharacteristic
54D4×D17—24no—
55D4×D17—44no—
56C2×D68—21no—
57C2×D68C2311yescharacteristic
58C2×C4×D17—24no—
59C2×C4×D17—21no—
60C2×C4×D17D411yescharacteristic
61C2×C4×D17D411yescharacteristic
62C2×C4×D17C2311yescharacteristic
63C2×C4×D17C2311yescharacteristic
64C2×C8×C17C2312yes—
65C4⋊C4×C17—21no—
66C4⋊C4×C17C2311yescharacteristic
67C4⋊C4×C17C2312yes—
68C22⋊C4×C17—21no—
69C22⋊Dic17—21no—
70#63227—22no—
71#63227—22no—
72#63227—21no—
73#63227—21no—
74C4⋊Dic17—21no—
75C4⋊Dic17C2312yes—
76C4⋊Dic17C2311yescharacteristic
77#63225—21no—
78C4×Dic17—22no—
79C4×Dic17—22no—
80C2×#3622C2312yes—
81#63218—24no—
82C23×C17—21no—
83C23×C17C2×D411yescharacteristic
84C22×D17—22no—
85C22×D17—22no—
86C22×D17—21no—
87C22×D17—41no—
88C22×D17—42no—
89C22×D17—42no—
90C22×D17—42no—
91C22×D17C2×D411yescharacteristic
92C22×D17C2×D411yescharacteristic
93D4×C17—22no—
94D4×C17—42no—
95C2×C4×C17—22no—
96C2×C4×C17—21no—
97C2×C4×C17—22no—
98C2×C4×C17—42no—
99C2×C4×C17C2×D411yescharacteristic
100C2×C4×C17C2411yescharacteristic
101C22⋊D17—42no—
102C22⋊D17—42no—
103C22⋊D17—42no—
104C2×Dic17—22no—
105C2×Dic17—21no—
106C2×Dic17—22no—
107C2×Dic17—42no—
108C2×Dic17C2×D411yescharacteristic
109C2×Dic17C2×D411yescharacteristic
110D68—22no—
111D68—42no—
112C4×D17—22no—
113C4×D17—22no—
114C4×D17—22no—
115C4×D17—42no—
116C4×D17#4114yes—
117C8×C17—22no—
118#3622—22no—
119#3094—171nomaximal, Sylow
120C22×C17—21no—
121C22×C17—21no—
122C22×C17—41no—
123C22×C17—41no—
124C22×C17—41no—
125C22×C17AGL(1,ℤ/8)12yes—
126C22×C17C22×D411yescharacteristic
127C2×D17—21no—
128C2×D17—21no—
129C2×D17—21no—
130C2×D17—21no—
131C2×D17—21no—
132C2×D17—44no—
133C2×D17—41no—
134C2×D17—41no—
135C2×D17—41no—
136C2×D17—84no—
137C2×D17#12312yes—
138C4×C17—21no—
139C4×C17—41no—
140C4×C17—42no—
141C4×C17C2×#4112yes—
142Dic17—21no—
143Dic17—41no—
144Dic17—42no—
145Dic17#12312yes—
146C2×#121—171no—
147C2×#117—171no—
148#482—172no—
149#482—172no—
150#481—171no—
151#481D1711yescharacteristic, complemented
152#480—171no—
153#480—171no—
154#428—172no—
155#418—172no—
156#407—171no—
157C2×C17—22no—
158C2×C17—81no—
159C2×C17C2×#12311yescharacteristic
160C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
161C2×C17#57511yescharacteristic
162D17—22no—
163D17—41no—
164D17—81no—
165C22×D4—171no—
166C22×D4—341no—
167C23×C4—171no—
168C2×M16—172no—
169#121—171no—
170#121—171no—
171#121—171no—
172#121—341no—
173#121—341no—
174#121C2×D1711yescharacteristic
175#117—172no—
176#117—172no—
177#117—341no—
178#117—341no—
179C2×C4⋊C4—171no—
180C2×C4⋊C4—171no—
181C2×C4⋊C4C2×D1711yescharacteristic, complemented
182C2×C22⋊C4—341no—
183C2×C22⋊C4—341no—
184C2×C42—341no—
185#107—172no—
186#107—172no—
187#106—172no—
188#106C2×D1712yescomplemented
189#102—172no—
190#102—172no—
191#102—172no—
192#102C2×D1712yescomplemented
193C22⋊C8—171no—
194C22⋊C8—172no—
195C22⋊C8C2×D1711yescharacteristic, complemented
196C17#309411yescharacteristic, complemented, Sylow
197C24—171no—
198C24—341no—
199C2×D4—21no—
200C2×D4—171no—
201C2×D4—341no—
202C2×D4—344no—
203C2×D4—341no—
204C2×D4—341no—
205C2×D4—341no—
206C2×D4—684no—
207C2×D4C22×D1711yescharacteristic
208C22×C4—21no—
209C22×C4—171no—
210C22×C4—171no—
211C22×C4—171no—
212C22×C4—171no—
213C22×C4—171no—
214C22×C4—344no—
215C22×C4—341no—
216C22×C4—341no—
217C22×C4C22×D1711yescharacteristic
218M16—344no—
219C2×C8—172no—
220C2×C8C22×D1712yes—
221C4⋊C4—21no—
222C4⋊C4—172no—
223C4⋊C4—171no—
224C4⋊C4—341no—
225C4⋊C4—341no—
226C4⋊C4C22×D1712yes—
227C4⋊C4C22×D1711yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—341no—
230C22⋊C4—342no—
231C22⋊C4—342no—
232C22⋊C4—341no—
233C22⋊C4—341no—
234C42—342no—
235C42—342no—
236C23—21no—
237C23—171no—
238C23—171no—
239C23—342no—
240C23—342no—
241C23—341no—
242C23—681no—
243C23—682no—
244C23—682no—
245C23—682no—
246C23D4×D1711yescharacteristic
247D4—22no—
248D4—42no—
249D4—342no—
250D4—682no—
251D4—682no—
252D4—682no—
253D4—682no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—171no—
259C2×C4—174no—
260C2×C4—171no—
261C2×C4—342no—
262C2×C4—342no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—342no—
266C2×C4—341no—
267C2×C4—682no—
268C2×C4—682no—
269C2×C4D4×D1711yescharacteristic
270C2×C4C23×D1711yescharacteristic
271C8—22no—
272C8—342no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—172no—
279C22—341no—
280C22—341no—
281C22—341no—
282C22—341no—
283C22—341no—
284C22—684no—
285C22—681no—
286C22—681no—
287C22—681no—
288C22—1364no—
289C22#1058760012yes—
290C22C2×D4×D1711yescharacteristic
291C4—21no—
292C4—41no—
293C4—42no—
294C4—172no—
295C4—341no—
296C4—681no—
297C4—682no—
298C4#41×D1712yes—
299C2—22no—
300C2—81no—
301C2—342no—
302C2—681no—
303C2—1361no—
304C2#123×D1711yescharacteristic
305C2C2×#1058760011yescharacteristic
306C2#1176360511yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#481C2
socleC22×C17C22×D4
prime radical#121×D17C2
hypercenter#481D17
hypocenterC17#3094
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×#10587578C2semidirect
3. #11763135C2semidirect
4. #11763131C2semidirect
5. #11763127C2semidirect
6. #11763123C2semidirect
7. #11763120C2semidirect
8. #11762848C2semidirect
9. #11762812C2semidirect
10. #11762771C2semidirect
11. C17×#481C2semidirect
16. C2×C4⋊C4×C17C22semidirect
21. C2×C4⋊Dic17C22semidirect
28. #10587578C22semidirect
32. C17×#106C22semidirect
33. C17×#102C22semidirect
34. C17×C22⋊C8C22semidirect
35. #10587503C22semidirect
36. C22⋊#3622C22semidirect
37. #10587492C22semidirect
38. #10587491C22semidirect
39. #10587489C22semidirect
40. #10587480C22semidirect
41. #10587479C22semidirect
42. #10587478C22semidirect
151. #481D17semidirect
181. C2×C4⋊C4C2×D17semidirect
188. #106C2×D17semidirect
192. #102C2×D17semidirect
195. C22⋊C8C2×D17semidirect
196. C17#3094semidirect

Maximal subgroups

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

subgroupindex
#121×D172
C2×#105875782
#117631352
#117631312
#117631272
#117631232
#117631202
#117628482
#117628122
#117627712
C17×#4812
#309417