Finite Group Databasebeta

← #14539724

Subgroups — #14539724

Order 2176.

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

normal subgroups only

This group has 488 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
1#11763363C211yesmaximal, characteristic, complemented
2#127×D17C211yesmaximal, characteristic
3#11763180C212yesmaximal, complemented
4#11763154C212yesmaximal, complemented
5#11763010C211yesmaximal, characteristic, complemented
6#11763009C211yesmaximal, characteristic, complemented
7#11763006C211yesmaximal, characteristic, complemented
8#11762961C212yesmaximal, complemented
9#11762958C212yesmaximal, complemented
10#11762861C211yesmaximal, characteristic, complemented
11C17×#460C211yesmaximal, characteristic, complemented
12C2×D4×D17—21no—
13C2×D4×D17—21no—
14C2×D4×D17C2212yes—
15C2×SD16×C17C2212yescomplemented
16C17×#127C2211yescharacteristic
17C4×Q8×C17C2211yescharacteristic, complemented
18#10587632C2211yescharacteristic
19#10587630C2211yescharacteristic, complemented
20C2×#63230C2212yescomplemented
21#10587626—21no—
22#10587626—21no—
23#10587626C2212yes—
24C2×#63229C2212yescomplemented
25C2×#63219C2212yescomplemented
26#10587583C2211yescharacteristic
27#10587582—21no—
28#10587575—21no—
29#10587574C2211yescharacteristic, complemented
30#10587560C2211yescharacteristic
31#10587558—21no—
32C42×D17C2211yescharacteristic
33C4×Dic34—21no—
34C4×Dic34C2211yescharacteristic, complemented
35C17×C4⋊C8C2211yescharacteristic, complemented
36C17×#102C2212yescomplemented
37#10587503C2212yescomplemented
38#10587492C2212yescomplemented
39#10587491C2212yes—
40C8×Dic17C2212yescomplemented
41#10587480C2212yescomplemented
42C4⋊#3622C2211yescharacteristic, complemented
43C23×D17—22no—
44C2×Q8×C17C2311yescharacteristic
45C2×D4×C17—21no—
46C2×D4×C17—21no—
47C2×D4×C17C2312yes—
48C2×C22⋊D17—22no—
49C2×C22⋊D17—22no—
50C2×C22⋊D17—22no—
51D4×D17—42no—
52D4×D17—42no—
53D4×D17—44no—
54D4×D17—44no—
55C2×D68—21no—
56C2×D68—21no—
57C2×D68C2312yes—
58C2×C4×D17D411yescharacteristic
59C2×C4×D17D411yescharacteristic
60C2×C4×D17C2311yescharacteristic
61C2×Dic34—21no—
62C2×Dic34C2311yescharacteristic
63SD16×C17—42no—
64C2×C8×C17C2312yes—
65C4⋊C4×C17—21no—
66C4⋊C4×C17C2311yescharacteristic
67C42×C17—21no—
68C42×C17C2311yescharacteristic
69#63230—42no—
70#63229—42no—
71#63227—21no—
72#63227—21no—
73C4⋊Dic17—21no—
74C4⋊Dic17C2311yescharacteristic
75#63225—21no—
76#63225—21no—
77#63225—21no—
78#63225—21no—
79#63225—21no—
80C4×Dic17—21no—
81C4×Dic17D411yescharacteristic
82C4×Dic17D411yescharacteristic
83C4×Dic17C2311yescharacteristic
84C2×#3622C2312yes—
85#63219—42no—
86C23×C17—22no—
87C22×D17—22no—
88C22×D17—44no—
89C22×D17—44no—
90C22×D17—44no—
91C22×D17C2×D411yescharacteristic
92Q8×C17—21no—
93Q8×C17—21no—
94D4×C17—22no—
95D4×C17—22no—
96D4×C17—42no—
97D4×C17—42no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17C2×D411yescharacteristic
101C2×C4×C17C2×D411yescharacteristic
102C2×C4×C17C2411yescharacteristic
103C22⋊D17—44no—
104C22⋊D17—44no—
105C22⋊D17—44no—
106C2×Dic17—21no—
107C2×Dic17—21no—
108C2×Dic17C2×D411yescharacteristic
109C2×Dic17C2×D411yescharacteristic
110C2×Dic17C2×D411yescharacteristic
111D68—22no—
112D68—22no—
113D68—42no—
114D68—42no—
115C4×D17—21no—
116C4×D17—21no—
117C4×D17—22no—
118C4×D17SD1614yescomplemented
119Dic34—21no—
120Dic34—21no—
121Dic34—42no—
122C8×C17—22no—
123#3622—22no—
124#3208—171nomaximal, Sylow
125C22×C17—42no—
126C22×C17—42no—
127C22×C17—42no—
128C22×C17C22×D411yescharacteristic
129C2×D17—21no—
130C2×D17—21no—
131C2×D17—42no—
132C2×D17—42no—
133C2×D17—42no—
134C2×D17—88no—
135C2×D17C2×SD1612yes—
136C4×C17—21no—
137C4×C17—41no—
138C4×C17—41no—
139C4×C17C2×SD1612yes—
140C4×C17H(5,𝔽2)11yescharacteristic
141C4×C17H(5,𝔽2)11yescharacteristic
142Dic17—21no—
143Dic17—41no—
144Dic17—41no—
145Dic17C2×SD1612yes—
146Dic17AGL(1,ℤ/8)12yes—
147#533—171no—
148C2×#127—171no—
149#492—172no—
150#486—172no—
151#460—171no—
152#460—171no—
153#460—171no—
154#460D1711yescharacteristic, complemented
155#450—172no—
156#450—172no—
157#431—171no—
158C2×C17—82no—
159C2×C17#53511yescharacteristic
160C2×C17C22×SD1611yescharacteristic
161C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
162D17—22no—
163D17—82no—
164C22×D4—172no—
165C22×D4—341no—
166C22×D4—341no—
167C2×SD16—172no—
168C2×SD16—172no—
169C2×SD16—172no—
170C2×SD16C2×D1712yescomplemented
171C4⋊Q8—171no—
172C4⋊Q8—171no—
173#127—172no—
174#127—171no—
175#127—341no—
176#127—341no—
177#127C2×D1711yescharacteristic
178#125—341no—
179C22⋊Q8—171no—
180C22⋊Q8—171no—
181C22⋊Q8—341no—
182C4×Q8—171no—
183C4×Q8—341no—
184C4×Q8C2×D1711yescharacteristic, complemented
185#117—341no—
186C2×C42—171no—
187C4⋊C8—171no—
188C4⋊C8C2×D1711yescharacteristic, complemented
189#102—172no—
190#102—172no—
191#102—172no—
192#102C2×D1712yescomplemented
193C22⋊C8—172no—
194C4×C8—172no—
195C17#320811yescharacteristic, complemented, Sylow
196C24—342no—
197C2×Q8—171no—
198C2×Q8—341no—
199C2×Q8C22×D1711yescharacteristic
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—172no—
203C2×D4—342no—
204C2×D4—342no—
205C2×D4—342no—
206C2×D4—341no—
207C2×D4—341no—
208C2×D4—684no—
209C2×D4—682no—
210C2×D4—682no—
211C2×D4—684no—
212C2×D4C22×D1712yes—
213C22×C4—171no—
214C22×C4—171no—
215C22×C4—171no—
216SD16—42no—
217SD16—682no—
218SD16—682no—
219SD16—682no—
220C2×C8—172no—
221C2×C8C22×D1712yes—
222C4⋊C4—21no—
223C4⋊C4—171no—
224C4⋊C4—341no—
225C4⋊C4—341no—
226C4⋊C4—341no—
227C4⋊C4—341no—
228C4⋊C4—341no—
229C4⋊C4—341no—
230C4⋊C4C22×D1711yescharacteristic
231C22⋊C4—341no—
232C22⋊C4—341no—
233C42—21no—
234C42—171no—
235C42—171no—
236C42—171no—
237C42—341no—
238C42C22×D1711yescharacteristic
239C23—22no—
240C23—171no—
241C23—342no—
242C23—684no—
243C23—684no—
244C23—684no—
245Q8—21no—
246Q8—21no—
247Q8—341no—
248Q8—341no—
249Q8—682no—
250D4—22no—
251D4—22no—
252D4—42no—
253D4—42no—
254D4—342no—
255D4—342no—
256D4—682no—
257D4—684no—
258D4—684no—
259D4—684no—
260D4—682no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—171no—
264C2×C4—171no—
265C2×C4—174no—
266C2×C4—171no—
267C2×C4—342no—
268C2×C4—341no—
269C2×C4—341no—
270C2×C4—341no—
271C2×C4—341no—
272C2×C4D4×D1711yescharacteristic
273C2×C4D4×D1711yescharacteristic
274C2×C4C23×D1711yescharacteristic
275C8—22no—
276C8—342no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—172no—
281C22—341no—
282C22—341no—
283C22—682no—
284C22—682no—
285C22—682no—
286C22—1368no—
287C22C2×D4×D1711yescharacteristic
288C4—21no—
289C4—41no—
290C4—41no—
291C4—172no—
292C4—172no—
293C4—341no—
294C4—681no—
295C4—681no—
296C4SD16×D1712yes—
297C4#1058769111yescharacteristic
298C4#1058769611yescharacteristic
299C2—82no—
300C2—342no—
301C2—1362no—
302C2#1176337511yescharacteristic
303C2C2×SD16×D1711yescharacteristic
304C2AGL(1,ℤ/8)×D1711yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#460C2
socleC22×C17C22×D4
prime radical#127×D17C2
hypercenter#460D17
hypocenterC17#3208
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #11763363C2semidirect
3. #11763180C2semidirect
4. #11763154C2semidirect
5. #11763010C2semidirect
6. #11763009C2semidirect
7. #11763006C2semidirect
8. #11762961C2semidirect
9. #11762958C2semidirect
10. #11762861C2semidirect
11. C17×#460C2semidirect
15. C2×SD16×C17C22semidirect
17. C4×Q8×C17C22semidirect
19. #10587630C22semidirect
20. C2×#63230C22semidirect
24. C2×#63229C22semidirect
25. C2×#63219C22semidirect
29. #10587574C22semidirect
34. C4×Dic34C22semidirect
35. C17×C4⋊C8C22semidirect
36. C17×#102C22semidirect
37. #10587503C22semidirect
38. #10587492C22semidirect
40. C8×Dic17C22semidirect
41. #10587480C22semidirect
42. C4⋊#3622C22semidirect
118. C4×D17SD16semidirect
154. #460D17semidirect
170. C2×SD16C2×D17semidirect
184. C4×Q8C2×D17semidirect
188. C4⋊C8C2×D17semidirect
192. #102C2×D17semidirect
195. C17#3208semidirect

Maximal subgroups

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

subgroupindex
#117633632
#127×D172
#117631802
#117631542
#117630102
#117630092
#117630062
#117629612
#117629582
#117628612
C17×#4602
#320817