Finite Group Databasebeta

← #14539798

Subgroups — #14539798

Order 2176.

490 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 113 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#127×D17C211yesmaximal, characteristic, complemented
2#11763338C211yesmaximal, characteristic, complemented
3#11763199C211yesmaximal, characteristic, complemented
4#11763174C211yesmaximal, characteristic, complemented
5#11763173C211yesmaximal, characteristic, complemented
6#11763007C212yesmaximal, complemented
7#11763002C212yesmaximal, complemented
8#11762958C212yesmaximal, complemented
9#11762941C212yesmaximal, complemented
10#11762864C211yesmaximal, characteristic, complemented
11C17×#496C211yesmaximal, characteristic, complemented
12C2×D4×D17—21no—
13C2×D4×D17—21no—
14C2×D4×D17C2212yes—
15C2×SD16×C17C2212yescomplemented
16C2×D8×C17C2212yescomplemented
17C17×#127C2211yescharacteristic
18C17×#124C2211yescharacteristic, complemented
19#10587632C2211yescharacteristic
20Q8×Dic17C2211yescharacteristic, complemented
21C2×#63230C2212yescomplemented
22#10587626—21no—
23#10587626—21no—
24#10587626C2212yes—
25#10587624C2211yescharacteristic
26D4×Dic17C2211yescharacteristic, complemented
27C2×#63228C2212yescomplemented
28#10587569—21no—
29#10587567—21no—
30#10587565—21no—
31#10587563—21no—
32#10587560C2211yescharacteristic
33C42×D17C2211yescharacteristic
34#10587556C2211yescharacteristic, complemented
35C17×#97C2211yescharacteristic, complemented
36#10587503C2212yescomplemented
37#10587503C2212yescomplemented
38#10587492C2212yescomplemented
39#10587492C2212yescomplemented
40#10587491C2212yes—
41#10587485C2212yescomplemented
42#10587474C2211yescharacteristic, complemented
43C23×D17—22no—
44C2×Q8×C17C2311yescharacteristic
45C2×D4×C17—21no—
46C2×D4×C17—21no—
47C2×D4×C17C2311yescharacteristic
48C2×D4×C17C2312yes—
49C2×C22⋊D17—21no—
50C2×C22⋊D17—22no—
51C2×C22⋊D17—22no—
52C2×C22⋊D17—22no—
53C22×Dic17—21no—
54D4×D17—42no—
55D4×D17—42no—
56D4×D17—44no—
57D4×D17—44no—
58C2×D68—21no—
59C2×D68—21no—
60C2×D68C2312yes—
61C2×C4×D17D411yescharacteristic
62C2×C4×D17D411yescharacteristic
63C2×C4×D17C2311yescharacteristic
64SD16×C17—42no—
65D8×C17—42no—
66C2×C8×C17C2312yes—
67C22⋊C4×C17—21no—
68C22⋊C4×C17—21no—
69C42×C17C2311yescharacteristic
70C22⋊Dic17—21no—
71C22⋊Dic17—21no—
72#63230—42no—
73#63228—42no—
74#63227—21no—
75C4⋊Dic17—21no—
76C4⋊Dic17C2311yescharacteristic
77C4⋊Dic17C2311yescharacteristic
78#63225—21no—
79#63225—21no—
80#63225—21no—
81C4×Dic17—21no—
82C4×Dic17D411yescharacteristic
83C4×Dic17D411yescharacteristic
84C4×Dic17C2311yescharacteristic
85C2×#3622C2312yes—
86C23×C17—21no—
87C23×C17—22no—
88C22×D17—22no—
89C22×D17—44no—
90C22×D17—44no—
91C22×D17—44no—
92C22×D17C2×D411yescharacteristic
93Q8×C17—22no—
94D4×C17—22no—
95D4×C17—22no—
96D4×C17—22no—
97D4×C17—42no—
98D4×C17—42no—
99C2×C4×C17—21no—
100C2×C4×C17C2×D411yescharacteristic
101C2×C4×C17C2×D411yescharacteristic
102C2×C4×C17C2411yescharacteristic
103C22⋊D17—44no—
104C22⋊D17—44no—
105C22⋊D17—42no—
106C22⋊D17—44no—
107C2×Dic17—21no—
108C2×Dic17—21no—
109C2×Dic17—42no—
110C2×Dic17C2×D411yescharacteristic
111C2×Dic17C2×D411yescharacteristic
112C2×Dic17C2×D411yescharacteristic
113D68—22no—
114D68—22no—
115D68—42no—
116D68—42no—
117C4×D17—22no—
118C4×D17—22no—
119C4×D17—22no—
120C8×C17—22no—
121#3622—22no—
122#3217—171nomaximal, Sylow
123C22×C17—42no—
124C22×C17—41no—
125C22×C17—41no—
126C22×C17—42no—
127C22×C17—42no—
128C22×C17—41no—
129C22×C17C22×D411yescharacteristic
130C2×D17—21no—
131C2×D17—21no—
132C2×D17—42no—
133C2×D17—42no—
134C2×D17—42no—
135C2×D17—88no—
136C2×D17AGL(1,ℤ/8)12yes—
137C4×C17—21no—
138C4×C17—21no—
139C4×C17—41no—
140C4×C17H(5,𝔽2)12yes—
141Dic17—21no—
142Dic17—21no—
143Dic17—41no—
144Dic17—41no—
145Dic17AGL(1,ℤ/8)12yes—
146C2×#127—171no—
147#529—171no—
148#496—171no—
149#496D1711yescharacteristic, complemented
150#490—171no—
151#490—171no—
152#460—172no—
153#459—172no—
154#450—172no—
155#447—172no—
156#432—171no—
157C2×C17—82no—
158C2×C17—81no—
159C2×C17#53511yescharacteristic
160C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
161C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
162D17—22no—
163D17—82no—
164C22×D4—172no—
165C22×D4—341no—
166C22×D4—341no—
167C2×SD16—172no—
168C2×SD16C2×D1712yescomplemented
169C2×D8—172no—
170C2×D8C2×D1712yescomplemented
171#127—172no—
172#127—171no—
173#127—341no—
174#127—341no—
175#127C2×D1711yescharacteristic
176#126—341no—
177#125—171no—
178#124C2×D1711yescharacteristic, complemented
179#123—341no—
180C22⋊Q8—171no—
181#121—171no—
182C4×Q8—171no—
183C4×D4—171no—
184C4×D4—341no—
185#117—341no—
186C2×C42—171no—
187C4⋊C8—172no—
188#102—172no—
189#102—172no—
190#102—172no—
191#102—172no—
192C22⋊C8—172no—
193#97—171no—
194#97C2×D1711yescharacteristic, complemented
195C17#321711yescharacteristic, complemented, Sylow
196C24—342no—
197C2×Q8C22×D1711yescharacteristic
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—172no—
201C2×D4—342no—
202C2×D4—342no—
203C2×D4—342no—
204C2×D4—341no—
205C2×D4—341no—
206C2×D4—341no—
207C2×D4—682no—
208C2×D4—682no—
209C2×D4—684no—
210C2×D4—684no—
211C2×D4C22×D1712yes—
212C2×D4C22×D1711yescharacteristic
213C22×C4—171no—
214C22×C4—171no—
215C22×C4—171no—
216C22×C4—341no—
217SD16—42no—
218SD16—682no—
219D8—42no—
220D8—682no—
221C2×C8—172no—
222C2×C8C22×D1712yes—
223C4⋊C4—171no—
224C4⋊C4—171no—
225C4⋊C4—341no—
226C4⋊C4—341no—
227C4⋊C4—341no—
228C4⋊C4—341no—
229C22⋊C4—21no—
230C22⋊C4—21no—
231C22⋊C4—341no—
232C22⋊C4—341no—
233C22⋊C4—341no—
234C42—171no—
235C42—171no—
236C42—171no—
237C42—341no—
238C42C22×D1711yescharacteristic
239C23—21no—
240C23—22no—
241C23—171no—
242C23—342no—
243C23—684no—
244C23—684no—
245C23—684no—
246Q8—22no—
247D4—22no—
248D4—22no—
249D4—22no—
250D4—42no—
251D4—42no—
252D4—342no—
253D4—342no—
254D4—682no—
255D4—682no—
256D4—684no—
257D4—684no—
258D4—684no—
259D4—682no—
260C2×C4—21no—
261C2×C4—171no—
262C2×C4—171no—
263C2×C4—171no—
264C2×C4—342no—
265C2×C4—342no—
266C2×C4—342no—
267C2×C4—341no—
268C2×C4—341no—
269C2×C4—682no—
270C2×C4D4×D1711yescharacteristic
271C2×C4D4×D1711yescharacteristic
272C2×C4C23×D1711yescharacteristic
273C8—22no—
274C8—342no—
275C22—41no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—42no—
280C22—41no—
281C22—172no—
282C22—341no—
283C22—341no—
284C22—682no—
285C22—682no—
286C22—682no—
287C22—1368no—
288C22C2×D4×D1711yescharacteristic
289C4—21no—
290C4—21no—
291C4—41no—
292C4—172no—
293C4—341no—
294C4—341no—
295C4—681no—
296C4—681no—
297C4#1058769112yes—
298C2—82no—
299C2—81no—
300C2—342no—
301C2—1362no—
302C2#1176337211yescharacteristic
303C2AGL(1,ℤ/8)×D1711yescharacteristic
304C2AGL(1,ℤ/8)×D1711yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#496C2
socleC22×C17C22×D4
hypercenter#496D17
hypocenterC17#3217
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #127×D17C2semidirect
2. #11763338C2semidirect
3. #11763199C2semidirect
4. #11763174C2semidirect
5. #11763173C2semidirect
6. #11763007C2semidirect
7. #11763002C2semidirect
8. #11762958C2semidirect
9. #11762941C2semidirect
10. #11762864C2semidirect
11. C17×#496C2semidirect
15. C2×SD16×C17C22semidirect
16. C2×D8×C17C22semidirect
18. C17×#124C22semidirect
20. Q8×Dic17C22semidirect
21. C2×#63230C22semidirect
26. D4×Dic17C22semidirect
27. C2×#63228C22semidirect
34. #10587556C22semidirect
35. C17×#97C22semidirect
36. #10587503C22semidirect
37. #10587503C22semidirect
38. #10587492C22semidirect
39. #10587492C22semidirect
41. #10587485C22semidirect
42. #10587474C22semidirect
149. #496D17semidirect
168. C2×SD16C2×D17semidirect
170. C2×D8C2×D17semidirect
178. #124C2×D17semidirect
194. #97C2×D17semidirect
195. C17#3217semidirect

Maximal subgroups

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

subgroupindex
#127×D172
#117633382
#117631992
#117631742
#117631732
#117630072
#117630022
#117629582
#117629412
#117628642
C17×#4962
#321717