Finite Group Databasebeta

← #14539587

Subgroups — #14539587

Order 2176.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C22⋊Q8×D17C211yesmaximal, characteristic, complemented
2C2×#10587624C211yesmaximal, characteristic
3#11763145C211yesmaximal, characteristic, complemented
4#11763142C211yesmaximal, characteristic, complemented
5#11763121C212yesmaximal, complemented
6#11763054C212yesmaximal, complemented
7#11763052C212yesmaximal, complemented
8#11762960C211yesmaximal, characteristic, complemented
9#11762946C212yesmaximal, complemented
10#11762771C211yesmaximal, characteristic, complemented
11C17×#450C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C22×C22⋊D17—21no—
14C2×Q8×D17C2211yescharacteristic, complemented
15C22×C4×D17C2211yescharacteristic
16C2×SD16×C17C2212yescomplemented
17C17×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Dic17—21no—
19#10587632C2211yescharacteristic, complemented
20#10587624—21no—
21#10587624—21no—
22#10587624C2212yes—
23#10587624C2212yes—
24C2×#63229C2212yescomplemented
25C2×C4⋊Dic17C2211yescharacteristic
26#10587610C2211yescharacteristic
27C2×#63218C2212yescomplemented
28#10587583C2211yescharacteristic, complemented
29#10587582—21no—
30C4⋊C4×D17—21no—
31C4⋊C4×D17C2211yescharacteristic, complemented
32C22⋊C4×D17—21no—
33#10587564—21no—
34C17×#102C2212yescomplemented
35C17×C22⋊C8C2211yescharacteristic
36#10587506C2212yescomplemented
37#10587503C2212yescomplemented
38C22⋊#3622C2211yescharacteristic
39#10587491C2212yescomplemented
40#10587489C2212yescomplemented
41#10587487C2212yescomplemented
42#10587478C2212yescomplemented
43C24×C17—21no—
44C23×D17D411yescharacteristic
45C2×Q8×C17C2311yescharacteristic
46C2×D4×C17—21no—
47C2×D4×C17—42no—
48C2×D4×C17C2312yes—
49C22×C4×C17C2311yescharacteristic
50C2×C22⋊D17—22no—
51C2×C22⋊D17—22no—
52C2×C22⋊D17—44no—
53C22×Dic17—21no—
54C22×Dic17D411yescharacteristic
55Q8×D17—42no—
56C2×C4×D17—21no—
57C2×C4×D17—21no—
58C2×C4×D17—42no—
59C2×C4×D17D411yescharacteristic
60C2×C4×D17D411yescharacteristic
61C2×C4×D17C2311yescharacteristic
62C2×C4×D17C2311yescharacteristic
63C2×Dic34—21no—
64C2×Dic34C2311yescharacteristic
65SD16×C17—42no—
66C2×C8×C17C2312yes—
67C4⋊C4×C17—21no—
68C4⋊C4×C17C2311yescharacteristic
69C22⋊C4×C17—21no—
70C22⋊Dic17—21no—
71C22⋊Dic17—22no—
72C22⋊Dic17—22no—
73#63229—42no—
74#63227—21no—
75#63227—21no—
76C4⋊Dic17—21no—
77C4⋊Dic17—21no—
78C4⋊Dic17C2312yes—
79C4⋊Dic17C2311yescharacteristic
80#63225—21no—
81#63225—21no—
82#63225—21no—
83C2×#3622C2312yes—
84#63218—42no—
85C23×C17—22no—
86C23×C17—42no—
87C23×C17—42no—
88C23×C17—42no—
89C23×C17C2×D411yescharacteristic
90C22×D17—22no—
91C22×D17—22no—
92C22×D17—41no—
93C22×D17C2×D411yescharacteristic
94C22×D17C2×D411yescharacteristic
95Q8×C17—22no—
96D4×C17—22no—
97D4×C17—22no—
98D4×C17—42no—
99C2×C4×C17—22no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17C2×D411yescharacteristic
103C2×C4×C17C2411yescharacteristic
104C22⋊D17—44no—
105C22⋊D17—44no—
106C2×Dic17—22no—
107C2×Dic17—21no—
108C2×Dic17—21no—
109C2×Dic17—41no—
110C2×Dic17—42no—
111C2×Dic17C2×D411yescharacteristic
112C2×Dic17C2×D411yescharacteristic
113C4×D17—22no—
114C4×D17—22no—
115C4×D17—41no—
116C4×D17—41no—
117C4×D17—42no—
118C4×D17—42no—
119Dic34—22no—
120Dic34—42no—
121C8×C17—22no—
122#3622—22no—
123#3193—171nomaximal, Sylow
124C22×C17—21no—
125C22×C17—21no—
126C22×C17—42no—
127C22×C17—42no—
128C22×C17—42no—
129C22×C17—84no—
130C22×C17#13712yes—
131C22×C17C22×D411yescharacteristic
132C2×D17—21no—
133C2×D17—21no—
134C2×D17—21no—
135C2×D17—21no—
136C2×D17—21no—
137C2×D17—44no—
138C2×D17AGL(1,ℤ/8)12yes—
139C4×C17—21no—
140C4×C17—41no—
141C4×C17—41no—
142C4×C17H(5,𝔽2)12yes—
143Dic17—21no—
144Dic17—21no—
145Dic17—41no—
146Dic17—41no—
147Dic17—42no—
148C2×C22⋊Q8—171no—
149C2×#121—171no—
150#484—171no—
151#484—171no—
152#480—172no—
153#468—172no—
154#468—172no—
155#450—171no—
156#450D1711yescharacteristic, complemented
157#448—172no—
158#407—171no—
159C2×C17—22no—
160C2×C17—82no—
161C2×C17#53411yescharacteristic
162C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
163C2×C17C2×#13711yescharacteristic
164D17—22no—
165D17—41no—
166C22×Q8—171no—
167C22×D4—341no—
168C22×D4C2×D1711yescharacteristic
169C23×C4—171no—
170C2×SD16—172no—
171C2×SD16C2×D1712yescomplemented
172C2×M16—172no—
173C22⋊Q8—171no—
174C22⋊Q8—171no—
175C22⋊Q8—171no—
176C22⋊Q8—341no—
177C22⋊Q8—341no—
178C22⋊Q8C2×D1711yescharacteristic
179#121—172no—
180#121—172no—
181#121—341no—
182#121—341no—
183C2×C4⋊C4—171no—
184C2×C4⋊C4—171no—
185C2×C4⋊C4—341no—
186C2×C22⋊C4—341no—
187C2×C22⋊C4—341no—
188#107—172no—
189#107—172no—
190#103—172no—
191#103—172no—
192#102—172no—
193#102C2×D1712yescomplemented
194C22⋊C8—171no—
195C22⋊C8—172no—
196C22⋊C8C2×D1711yescharacteristic
197C17#319311yescharacteristic, complemented, Sylow
198C24—21no—
199C24—171no—
200C2×Q8—171no—
201C2×Q8—341no—
202C2×Q8—682no—
203C2×Q8C22×D1711yescharacteristic
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—342no—
207C2×D4—342no—
208C2×D4—684no—
209C2×D4C22×D1712yes—
210C22×C4—171no—
211C22×C4—171no—
212C22×C4—171no—
213C22×C4—171no—
214C22×C4—171no—
215C22×C4—341no—
216C22×C4—341no—
217C22×C4—341no—
218C22×C4—682no—
219C22×C4C22×D1711yescharacteristic
220SD16—42no—
221SD16—682no—
222M16—682no—
223C2×C8—172no—
224C2×C8C22×D1712yes—
225C4⋊C4—21no—
226C4⋊C4—171no—
227C4⋊C4—172no—
228C4⋊C4—341no—
229C4⋊C4—341no—
230C4⋊C4—341no—
231C4⋊C4—341no—
232C4⋊C4—341no—
233C4⋊C4C22×D1711yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—341no—
236C22⋊C4—341no—
237C22⋊C4—342no—
238C22⋊C4—342no—
239C22⋊C4—341no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—171no—
245C23—171no—
246C23—342no—
247C23—342no—
248C23—681no—
249C23D4×D1711yescharacteristic
250Q8—22no—
251Q8—342no—
252Q8—682no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—684no—
257D4—684no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—171no—
262C2×C4—171no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—341no—
266C2×C4—342no—
267C2×C4—341no—
268C2×C4—681no—
269C2×C4—681no—
270C2×C4—682no—
271C2×C4—682no—
272C2×C4—681no—
273C2×C4—682no—
274C2×C4D4×D1711yescharacteristic
275C2×C4C23×D1711yescharacteristic
276C8—22no—
277C8—342no—
278C22—21no—
279C22—21no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—84no—
284C22—172no—
285C22—341no—
286C22—341no—
287C22—341no—
288C22—341no—
289C22—341no—
290C22—684no—
291C22#1058760112yes—
292C22C2×D4×D1711yescharacteristic
293C4—21no—
294C4—41no—
295C4—41no—
296C4—341no—
297C4—341no—
298C4—682no—
299C4—681no—
300C4—681no—
301C4#1058769112yes—
302C2—22no—
303C2—82no—
304C2—342no—
305C2—681no—
306C2#1176336911yescharacteristic
307C2AGL(1,ℤ/8)×D1711yescharacteristic
308C2C2×#1058760111yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#450C2
socleC22×C17C22×D4
prime radicalC2×#10587624C2
hypercenter#450D17
hypocenterC17#3193
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D17C2semidirect
3. #11763145C2semidirect
4. #11763142C2semidirect
5. #11763121C2semidirect
6. #11763054C2semidirect
7. #11763052C2semidirect
8. #11762960C2semidirect
9. #11762946C2semidirect
10. #11762771C2semidirect
11. C17×#450C2semidirect
14. C2×Q8×D17C22semidirect
16. C2×SD16×C17C22semidirect
19. #10587632C22semidirect
24. C2×#63229C22semidirect
27. C2×#63218C22semidirect
28. #10587583C22semidirect
31. C4⋊C4×D17C22semidirect
34. C17×#102C22semidirect
36. #10587506C22semidirect
37. #10587503C22semidirect
39. #10587491C22semidirect
40. #10587489C22semidirect
41. #10587487C22semidirect
42. #10587478C22semidirect
156. #450D17semidirect
171. C2×SD16C2×D17semidirect
193. #102C2×D17semidirect
197. C17#3193semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D172
C2×#105876242
#117631452
#117631422
#117631212
#117630542
#117630522
#117629602
#117629462
#117627712
C17×#4502
#319317