Finite Group Databasebeta

← #14539539

Subgroups — #14539539

Order 2176.

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

normal subgroups only

This group has 468 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
1C2×#10587632C211yesmaximal, characteristic, complemented
2#121×D17C211yesmaximal, characteristic
3#11763138C212yesmaximal, complemented
4#11763128C211yesmaximal, characteristic, complemented
5#11763125C211yesmaximal, characteristic, complemented
6#11763041C212yesmaximal, complemented
7#11763035C212yesmaximal, complemented
8#11762958C212yesmaximal, complemented
9#11762948C211yesmaximal, characteristic, complemented
10C22⋊C8×D17C211yesmaximal, characteristic, complemented
11C17×#448C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic, complemented
13C2×D4×D17—21no—
14C2×D4×D17C2211yescharacteristic
15C22×C4×D17C2211yescharacteristic
16C2×SD16×C17C2212yescomplemented
17C17×#121C2211yescharacteristic
18#10587632—21no—
19#10587632—21no—
20#10587632C2212yes—
21#10587632C2212yescomplemented
22C2×#63230C2212yescomplemented
23#10587625—21no—
24#10587624C2211yescharacteristic
25#10587616C2211yescharacteristic
26C2×#63227—21no—
27C2×C4⋊Dic17C2211yescharacteristic, complemented
28C2×#63225—21no—
29C2×C8×D17C2212yescomplemented
30#10587581C2211yescharacteristic
31C4⋊C4×D17C2211yescharacteristic
32#10587570—21no—
33C22⋊C4×D17—21no—
34C17×#103C2212yescomplemented
35C17×C22⋊C8C2211yescharacteristic, complemented
36#10587506C2212yescomplemented
37#10587503C2212yescomplemented
38C22⋊#3622C2211yescharacteristic, complemented
39#10587492C2212yescomplemented
40#10587491C2212yes—
41#10587488C2212yescomplemented
42#10587479C2212yescomplemented
43C23×D17—21no—
44C23×D17D411yescharacteristic
45C2×Q8×C17—21no—
46C2×Q8×C17—42no—
47C2×Q8×C17C2312yes—
48C2×D4×C17—21no—
49C2×D4×C17C2311yescharacteristic
50C22×C4×C17—21no—
51C22×C4×C17C2311yescharacteristic
52C2×C22⋊D17—21no—
53C2×C22⋊D17—21no—
54C2×C22⋊D17—21no—
55C22×Dic17—21no—
56C22×Dic17D411yescharacteristic
57D4×D17—42no—
58D4×D17—44no—
59C2×D68—21no—
60C2×D68C2311yescharacteristic
61C2×C4×D17—21no—
62C2×C4×D17—42no—
63C2×C4×D17D411yescharacteristic
64C2×C4×D17D411yescharacteristic
65C2×C4×D17C2311yescharacteristic
66C2×C4×D17C2311yescharacteristic
67SD16×C17—42no—
68C2×C8×C17C2312yes—
69C4⋊C4×C17C2311yescharacteristic
70C22⋊C4×C17—21no—
71C22⋊Dic17—21no—
72#63230—42no—
73#63227—22no—
74#63227—22no—
75#63227—21no—
76#63227—21no—
77C4⋊Dic17—21no—
78C4⋊Dic17C2312yes—
79C4⋊Dic17C2311yescharacteristic
80#63225—21no—
81#63225—22no—
82#63225—22no—
83C2×#3622C2312yes—
84C8×D17—42no—
85C23×C17—21no—
86C23×C17C2×D411yescharacteristic
87C22×D17—21no—
88C22×D17—41no—
89C22×D17—41no—
90C22×D17—42no—
91C22×D17—42no—
92C22×D17—42no—
93C22×D17SD1614yescomplemented
94C22×D17C2×D411yescharacteristic
95C22×D17C2×D411yescharacteristic
96Q8×C17—22no—
97Q8×C17—22no—
98Q8×C17—42no—
99D4×C17—22no—
100D4×C17—42no—
101C2×C4×C17—22no—
102C2×C4×C17—21no—
103C2×C4×C17—22no—
104C2×C4×C17—42no—
105C2×C4×C17C2×D411yescharacteristic
106C2×C4×C17C2411yescharacteristic
107C22⋊D17—42no—
108C22⋊D17—42no—
109C22⋊D17—42no—
110C2×Dic17—21no—
111C2×Dic17—22no—
112C2×Dic17—41no—
113C2×Dic17—42no—
114C2×Dic17C2×D411yescharacteristic
115C2×Dic17C2×D411yescharacteristic
116D68—22no—
117D68—42no—
118C4×D17—22no—
119C4×D17—22no—
120C4×D17—41no—
121C4×D17—41no—
122C4×D17—42no—
123C8×C17—22no—
124#3622—22no—
125#3187—171nomaximal, Sylow
126C22×C17—21no—
127C22×C17—21no—
128C22×C17—41no—
129C22×C17—41no—
130C22×C17—41no—
131C22×C17C2×SD1612yes—
132C22×C17C22×D411yescharacteristic
133C2×D17—21no—
134C2×D17—21no—
135C2×D17—21no—
136C2×D17—21no—
137C2×D17—44no—
138C2×D17—41no—
139C2×D17—41no—
140C2×D17—41no—
141C2×D17—84no—
142C2×D17C2×SD1612yes—
143C2×D17C2×SD1612yes—
144C4×C17—21no—
145C4×C17—42no—
146C4×C17—41no—
147C4×C17H(5,𝔽2)12yes—
148Dic17—21no—
149Dic17—21no—
150Dic17—41no—
151Dic17—42no—
152C2×C22⋊Q8—171no—
153C2×#121—171no—
154#483—172no—
155#481—171no—
156#481—171no—
157#465—172no—
158#465—172no—
159#450—172no—
160#448—171no—
161#448D1711yescharacteristic, complemented
162C2×C22⋊C8—171no—
163C2×C17—22no—
164C2×C17—81no—
165C2×C17#53411yescharacteristic
166C2×C17C22×SD1611yescharacteristic
167C2×C17#57511yescharacteristic
168D17—22no—
169D17—22no—
170D17—81no—
171C22×Q8C2×D1711yescharacteristic, complemented
172C22×D4—171no—
173C22×D4—341no—
174C23×C4—171no—
175C2×SD16—172no—
176C2×SD16C2×D1712yescomplemented
177C22×C8—172no—
178C22⋊Q8—172no—
179C22⋊Q8—172no—
180C22⋊Q8—341no—
181C22⋊Q8—341no—
182#121—171no—
183#121—171no—
184#121—171no—
185#121—341no—
186#121—341no—
187#121C2×D1711yescharacteristic
188C2×C4⋊C4—171no—
189C2×C4⋊C4—171no—
190C2×C4⋊C4—341no—
191C2×C22⋊C4—341no—
192C2×C22⋊C4—341no—
193#106—172no—
194#106—172no—
195#103—172no—
196#103C2×D1712yescomplemented
197#102—172no—
198#102—172no—
199C22⋊C8—171no—
200C22⋊C8—172no—
201C22⋊C8C2×D1711yescharacteristic, complemented
202C17#318711yescharacteristic, complemented, Sylow
203C24—171no—
204C24—341no—
205C2×Q8—21no—
206C2×Q8—42no—
207C2×Q8C22×D1712yes—
208C2×D4—21no—
209C2×D4—171no—
210C2×D4—341no—
211C2×D4—341no—
212C2×D4—341no—
213C2×D4—341no—
214C2×D4—682no—
215C2×D4—684no—
216C2×D4C22×D1711yescharacteristic
217C22×C4—21no—
218C22×C4—171no—
219C22×C4—171no—
220C22×C4—171no—
221C22×C4—171no—
222C22×C4—171no—
223C22×C4—341no—
224C22×C4—341no—
225C22×C4—682no—
226C22×C4C22×D1711yescharacteristic
227SD16—42no—
228SD16—682no—
229C2×C8—172no—
230C2×C8—682no—
231C2×C8C22×D1712yes—
232C4⋊C4—172no—
233C4⋊C4—171no—
234C4⋊C4—341no—
235C4⋊C4—342no—
236C4⋊C4—342no—
237C4⋊C4—341no—
238C4⋊C4C22×D1711yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—341no—
241C22⋊C4—342no—
242C22⋊C4—342no—
243C22⋊C4—341no—
244C22⋊C4—341no—
245C23—21no—
246C23—171no—
247C23—171no—
248C23—174no—
249C23—341no—
250C23—681no—
251C23—681no—
252C23—682no—
253C23—682no—
254C23—682no—
255C23D4×D1711yescharacteristic
256Q8—22no—
257Q8—22no—
258Q8—42no—
259D4—22no—
260D4—42no—
261D4—342no—
262D4—682no—
263D4—682no—
264D4—682no—
265D4—682no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—171no—
271C2×C4—171no—
272C2×C4—342no—
273C2×C4—342no—
274C2×C4—342no—
275C2×C4—341no—
276C2×C4—681no—
277C2×C4—681no—
278C2×C4—682no—
279C2×C4—681no—
280C2×C4—682no—
281C2×C4D4×D1711yescharacteristic
282C2×C4C23×D1711yescharacteristic
283C8—22no—
284C8—342no—
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—172no—
291C22—172no—
292C22—341no—
293C22—341no—
294C22—341no—
295C22—341no—
296C22—684no—
297C22—681no—
298C22—681no—
299C22—681no—
300C22—1364no—
301C22SD16×D1712yes—
302C22C2×D4×D1711yescharacteristic
303C4—21no—
304C4—41no—
305C4—42no—
306C4—341no—
307C4—341no—
308C4—681no—
309C4—682no—
310C4#1058769112yes—
311C2—22no—
312C2—81no—
313C2—342no—
314C2—342no—
315C2—1361no—
316C2#1176336911yescharacteristic
317C2C2×SD16×D1711yescharacteristic
318C2#1176360811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#448C2
socleC22×C17C22×D4
prime radical#121×D17C2
hypercenter#448D17
hypocenterC17#3187
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#10587632C2semidirect
3. #11763138C2semidirect
4. #11763128C2semidirect
5. #11763125C2semidirect
6. #11763041C2semidirect
7. #11763035C2semidirect
8. #11762958C2semidirect
9. #11762948C2semidirect
10. C22⋊C8×D17C2semidirect
11. C17×#448C2semidirect
12. C22×Q8×C17C22semidirect
16. C2×SD16×C17C22semidirect
21. #10587632C22semidirect
22. C2×#63230C22semidirect
27. C2×C4⋊Dic17C22semidirect
29. C2×C8×D17C22semidirect
34. C17×#103C22semidirect
35. C17×C22⋊C8C22semidirect
36. #10587506C22semidirect
37. #10587503C22semidirect
38. C22⋊#3622C22semidirect
39. #10587492C22semidirect
41. #10587488C22semidirect
42. #10587479C22semidirect
93. C22×D17SD16semidirect
161. #448D17semidirect
171. C22×Q8C2×D17semidirect
176. C2×SD16C2×D17semidirect
196. #103C2×D17semidirect
201. C22⋊C8C2×D17semidirect
202. C17#3187semidirect

Maximal subgroups

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

subgroupindex
C2×#105876322
#121×D172
#117631382
#117631282
#117631252
#117630412
#117630352
#117629582
#117629482
C22⋊C8×D172
C17×#4482
#318717