Finite Group Databasebeta

← #14533938

Subgroups — #14533938

Order 2176.

672 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 139 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
2C2×C22⋊C4×D17C211yesmaximal, characteristic
3#11762736C211yesmaximal, characteristic, complemented
4#11762734C211yesmaximal, characteristic, complemented
5#11762720C211yesmaximal, characteristic, complemented
6#11762698C212yesmaximal, complemented
7#11762697C212yesmaximal, complemented
8#11762671C212yesmaximal, complemented
9#11762664C212yesmaximal
10#11762656C211yesmaximal, characteristic, complemented
11C17×#393C211yesmaximal, characteristic, complemented
12C24×D17C2211yescharacteristic
13C22×Q8×C17C2211yescharacteristic, complemented
14C22×C4×D17C2211yescharacteristic
15C22×C4×D17C2211yescharacteristic
16C2×C22⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2212yes—
18C2×C22⋊Dic17C2211yescharacteristic
19C2×C22⋊Dic17C2212yes—
20#10587632—24no—
21C2×#63227C2211yescharacteristic
22C2×#63227C2212yes—
23C2×#63227C2211yescharacteristic
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×#63227C2212yes—
27C2×C4⋊Dic17C2211yescharacteristic, complemented
28C2×#63225C2212yescomplemented
29C2×C4×Dic17C2212yescomplemented
30C22⋊C4×D17—24no—
31C17×#95C2212yescomplemented
32C17×#95C2211yescharacteristic, complemented
33#10587502C2212yescomplemented
34#10587502C2212yescomplemented
35#10587502C2212yescomplemented
36#10587502C2211yescharacteristic, complemented
37C24×C17C2311yescharacteristic
38C23×D17—22no—
39C23×D17—22no—
40C23×D17—22no—
41C23×D17—24no—
42C23×D17—42no—
43C23×D17C2311yescharacteristic
44C23×D17C2311yescharacteristic
45C2×Q8×C17—22no—
46C2×Q8×C17—42no—
47C22×C4×C17C2311yescharacteristic
48C22×C4×C17C2312yes—
49C22×C4×C17C2312yes—
50C22×C4×C17C2311yescharacteristic
51C22×Dic17C2311yescharacteristic
52C22×Dic17C2312yes—
53C22×Dic17C2312yes—
54C22×Dic17C2311yescharacteristic
55C2×C4×D17—22no—
56C2×C4×D17—24no—
57C2×C4×D17—42no—
58C2×C4×D17D414yes—
59C22⋊C4×C17—22no—
60C22⋊C4×C17—42no—
61C22⋊Dic17—24no—
62C22⋊Dic17—22no—
63#63227—24no—
64#63227—24no—
65#63227—24no—
66#63227—22no—
67#63227—22no—
68#63227—42no—
69C4⋊Dic17—22no—
70#63225—24no—
71C4×Dic17—42no—
72C23×C17—21no—
73C23×C17—21no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17#4112yes—
77C23×C17C2411yescharacteristic
78C22×D17—21no—
79C22×D17—21no—
80C22×D17—22no—
81C22×D17—28no—
82C22×D17—21no—
83C22×D17—21no—
84C22×D17—22no—
85C22×D17—22no—
86C22×D17—48no—
87C22×D17—48no—
88C22×D17—44no—
89C22×D17—44no—
90C22×D17C2×D412yes—
91C22×D17#4112yes—
92C22×D17#4114yes—
93Q8×C17—44no—
94C2×C4×C17—22no—
95C2×C4×C17—21no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—22no—
103C2×C4×C17C2×D412yes—
104C2×Dic17—22no—
105C2×Dic17—21no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17C2×D412yes—
114C4×D17—44no—
115C4×D17—44no—
116#2561—171nomaximal, Sylow
117C22×C17—22no—
118C22×C17—24no—
119C22×C17—41no—
120C22×C17—41no—
121C22×C17—42no—
122C22×C17C22×D411yescharacteristic
123C22×C17C2×#4111yescharacteristic
124C22×C17C2×#4111yescharacteristic
125C22×C17C2×#4112yes—
126C22×C17H(5,𝔽2)12yes—
127C2×D17—22no—
128C2×D17—22no—
129C2×D17—24no—
130C2×D17—24no—
131C2×D17—22no—
132C2×D17—24no—
133C2×D17—416no—
134C2×D17—41no—
135C2×D17—41no—
136C2×D17—42no—
137C2×D17#12414yes—
138C4×C17—22no—
139C4×C17—41no—
140C4×C17—42no—
141C4×C17—42no—
142Dic17—22no—
143Dic17—42no—
144Dic17—41no—
145Dic17—42no—
146C2×C22⋊Q8—171no—
147C22×C22⋊C4—171no—
148#397—171no—
149#397—171no—
150#393—171no—
151#393D1711yescharacteristic, complemented
152#388—172no—
153#388—172no—
154#381—172no—
155#379—172no—
156C4×C22⋊C4—171no—
157C2×C17—42no—
158C2×C17#52511yescharacteristic
159C2×C17C2×#12411yescharacteristic
160C2×C17#53411yescharacteristic
161C2×C17#55112yes—
162C2×C17#55112yes—
163D17—24no—
164D17—42no—
165C25—171no—
166C22×Q8C2×D1711yescharacteristic, complemented
167C23×C4—171no—
168C23×C4—171no—
169C22⋊Q8—344no—
170C2×C4⋊C4—172no—
171C2×C4⋊C4—171no—
172C2×C22⋊C4—171no—
173C2×C22⋊C4—172no—
174C2×C22⋊C4—171no—
175C2×C22⋊C4—171no—
176C2×C22⋊C4—172no—
177C2×C22⋊C4—172no—
178C2×C22⋊C4—172no—
179C2×C22⋊C4—172no—
180C2×C22⋊C4—344no—
181C2×C22⋊C4C2×D1711yescharacteristic
182C2×C22⋊C4C2×D1712yes—
183C2×C42—172no—
184#95—171no—
185#95—172no—
186#95—172no—
187#95—172no—
188#95C2×D1711yescharacteristic, complemented
189#95C2×D1712yescomplemented
190C17#256111yescharacteristic, complemented, Sylow
191C24—171no—
192C24—171no—
193C24—342no—
194C24—342no—
195C24—344no—
196C24—342no—
197C24—682no—
198C24C22×D1711yescharacteristic
199C2×Q8—22no—
200C2×Q8—42no—
201C22×C4—171no—
202C22×C4—174no—
203C22×C4—171no—
204C22×C4—172no—
205C22×C4—172no—
206C22×C4—342no—
207C22×C4—344no—
208C22×C4—682no—
209C22×C4C22×D1711yescharacteristic
210C22×C4C22×D1712yes—
211C22×C4C22×D1712yes—
212C22×C4C22×D1711yescharacteristic
213C4⋊C4—344no—
214C4⋊C4—342no—
215C22⋊C4—22no—
216C22⋊C4—42no—
217C22⋊C4—342no—
218C22⋊C4—344no—
219C22⋊C4—344no—
220C22⋊C4—344no—
221C22⋊C4—342no—
222C22⋊C4—344no—
223C22⋊C4—342no—
224C22⋊C4—682no—
225C42—682no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—21no—
230C23—172no—
231C23—174no—
232C23—172no—
233C23—348no—
234C23—342no—
235C23—341no—
236C23—341no—
237C23—341no—
238C23—342no—
239C23—341no—
240C23—342no—
241C23—688no—
242C23—684no—
243C23—688no—
244C23—684no—
245C23#6325612yes—
246C23C23×D1711yescharacteristic
247Q8—44no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—172no—
258C2×C4—342no—
259C2×C4—341no—
260C2×C4—342no—
261C2×C4—342no—
262C2×C4—342no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—342no—
266C2×C4—342no—
267C2×C4—684no—
268C2×C4—684no—
269C2×C4D4×D1712yes—
270C22—24no—
271C22—22no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—174no—
276C22—344no—
277C22—344no—
278C22—342no—
279C22—342no—
280C22—344no—
281C22—342no—
282C22—6816no—
283C22—682no—
284C22—681no—
285C22—681no—
286C22C2×D4×D1711yescharacteristic
287C22#1058769112yes—
288C22C2×#6325611yescharacteristic
289C22#41×D1712yes—
290C22#41×D1711yescharacteristic
291C4—22no—
292C4—42no—
293C4—41no—
294C4—42no—
295C4—342no—
296C4—682no—
297C4—682no—
298C4—681no—
299C2—42no—
300C2—344no—
301C2—682no—
302C2#1176332211yescharacteristic
303C2#124×D1711yescharacteristic
304C2#1176336911yescharacteristic
305C2#1176348612yes—
306C2#1176348912yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#393C2
prime radicalC24×D17C22
hypercenter#393D17
hypocenterC17#2561
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#10587632C2semidirect
3. #11762736C2semidirect
4. #11762734C2semidirect
5. #11762720C2semidirect
6. #11762698C2semidirect
7. #11762697C2semidirect
8. #11762671C2semidirect
10. #11762656C2semidirect
11. C17×#393C2semidirect
13. C22×Q8×C17C22semidirect
27. C2×C4⋊Dic17C22semidirect
28. C2×#63225C22semidirect
29. C2×C4×Dic17C22semidirect
31. C17×#95C22semidirect
32. C17×#95C22semidirect
33. #10587502C22semidirect
34. #10587502C22semidirect
35. #10587502C22semidirect
36. #10587502C22semidirect
151. #393D17semidirect
166. C22×Q8C2×D17semidirect
188. #95C2×D17semidirect
189. #95C2×D17semidirect
190. C17#2561semidirect

Maximal subgroups

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

subgroupindex
C2×#105876322
C2×C22⋊C4×D172
#117627362
#117627342
#117627202
#117626982
#117626972
#117626712
#117626642
#117626562
C17×#3932
#256117