Finite Group Databasebeta

← #14533055

Subgroups — #14533055

Order 2176.

672 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10587563C211yesmaximal, characteristic, complemented
2C22×C22⋊Dic17C211yesmaximal, characteristic
3#11762745C212yesmaximal, complemented
4#11762737C212yesmaximal, complemented
5#11762703C212yesmaximal, complemented
6#11762702C212yesmaximal, complemented
7#11762684C211yesmaximal, characteristic, complemented
8#11762681C211yesmaximal, characteristic, complemented
9#11762670C211yesmaximal, characteristic, complemented
10#11762665C211yesmaximal, characteristic
11C17×#379C211yesmaximal, characteristic
12C25×C17C2211yescharacteristic
13C23×Dic17C2211yescharacteristic
14C23×Dic17C2211yescharacteristic
15C2×C22⋊C4×C17C2212yes—
16C2×C22⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊C4×C17C2212yes—
19C2×C22⋊Dic17—24no—
20C2×C22⋊Dic17C2212yes—
21C2×C22⋊Dic17C2212yes—
22C2×C22⋊Dic17C2212yes—
23C2×C22⋊Dic17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2212yes—
26C2×C4⋊Dic17C2212yescomplemented
27C2×#63225C2211yescharacteristic, complemented
28C2×#63225C2211yescharacteristic, complemented
29C2×#63225C2212yescomplemented
30C2×C4×Dic17C2211yescharacteristic, complemented
31C2×C4×Dic17C2211yescharacteristic, complemented
32#10587563—24no—
33#10587502C2212yescomplemented
34#10587502C2212yescomplemented
35#10587502C2212yescomplemented
36#10587502C2212yescomplemented
37C24×C17—22no—
38C24×C17—24no—
39C24×C17—22no—
40C24×C17—22no—
41C24×C17—42no—
42C24×C17C2312yes—
43C24×C17C2311yescharacteristic
44C22×C4×C17C2312yes—
45C22×C4×C17C2311yescharacteristic
46C22×C4×C17C2311yescharacteristic
47C22×Dic17—22no—
48C22×Dic17—22no—
49C22×Dic17—22no—
50C22×Dic17—42no—
51C22×Dic17D414yes—
52C22×Dic17C2312yes—
53C22×Dic17C2312yes—
54C22×Dic17C2311yescharacteristic
55C22×Dic17C2311yescharacteristic
56C22×Dic17C2311yescharacteristic
57C22×Dic17C2311yescharacteristic
58C22⋊C4×C17—22no—
59C22⋊C4×C17—24no—
60C22⋊C4×C17—22no—
61C22⋊C4×C17—42no—
62C22⋊Dic17—22no—
63C22⋊Dic17—22no—
64C22⋊Dic17—24no—
65C22⋊Dic17—24no—
66C22⋊Dic17—24no—
67C22⋊Dic17—42no—
68C4⋊Dic17—42no—
69#63225—22no—
70#63225—22no—
71#63225—42no—
72C4×Dic17—22no—
73C4×Dic17—22no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—28no—
77C23×C17—22no—
78C23×C17—22no—
79C23×C17—22no—
80C23×C17—22no—
81C23×C17—22no—
82C23×C17—22no—
83C23×C17—48no—
84C23×C17—48no—
85C23×C17—44no—
86C23×C17—44no—
87C23×C17C2×D412yes—
88C23×C17#4112yes—
89C23×C17#4114yes—
90C23×C17#4112yes—
91C23×C17C2411yescharacteristic
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×Dic17—28no—
100C2×Dic17—21no—
101C2×Dic17—21no—
102C2×Dic17—21no—
103C2×Dic17—21no—
104C2×Dic17—22no—
105C2×Dic17—21no—
106C2×Dic17—21no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—44no—
115C2×Dic17C2×D412yes—
116C2×Dic17C2×D412yes—
117#2482—171nomaximal, Sylow
118C22×C17—22no—
119C22×C17—22no—
120C22×C17—24no—
121C22×C17—24no—
122C22×C17—24no—
123C22×C17—24no—
124C22×C17—24no—
125C22×C17—416no—
126C22×C17—42no—
127C22×C17—42no—
128C22×C17—42no—
129C22×C17—42no—
130C22×C17#12314yes—
131C22×C17C22×D411yescharacteristic
132C22×C17C2×#4111yescharacteristic
133C22×C17C2×#4111yescharacteristic
134C22×C17C2×#4112yes—
135C22×C17H(5,𝔽2)12yes—
136C4×C17—41no—
137C4×C17—41no—
138C4×C17—42no—
139Dic17—22no—
140Dic17—22no—
141Dic17—41no—
142Dic17—41no—
143Dic17—42no—
144Dic17—42no—
145C2×#117—171no—
146C22×C22⋊C4—171no—
147#399—172no—
148#397—172no—
149#388—172no—
150#388—172no—
151#385—171no—
152#385—171no—
153#381—171no—
154#379—171no—
155#379D1711yescharacteristic
156C2×C17—24no—
157C2×C17—44no—
158C2×C17C2×#12311yescharacteristic
159C2×C17#52511yescharacteristic
160C2×C17#53511yescharacteristic
161C2×C17#55112yes—
162C2×C17#55112yes—
163C25C2×D1711yescharacteristic
164C23×C4—171no—
165C23×C4—171no—
166#117—344no—
167C2×C4⋊C4—171no—
168C2×C4⋊C4—171no—
169C2×C4⋊C4—172no—
170C2×C4⋊C4—172no—
171C2×C22⋊C4—172no—
172C2×C22⋊C4—172no—
173C2×C22⋊C4—171no—
174C2×C22⋊C4—171no—
175C2×C22⋊C4—172no—
176C2×C22⋊C4—172no—
177C2×C22⋊C4—344no—
178C2×C22⋊C4C2×D1711yescharacteristic
179C2×C22⋊C4C2×D1711yescharacteristic
180C2×C22⋊C4C2×D1712yes—
181C2×C22⋊C4C2×D1712yes—
182C2×C42—171no—
183C2×C42—171no—
184#95—172no—
185#95—172no—
186#95—172no—
187#95—172no—
188C17#248211yescharacteristic, complemented, Sylow
189C24—22no—
190C24—24no—
191C24—22no—
192C24—22no—
193C24—42no—
194C24C22×D1712yes—
195C24C22×D1711yescharacteristic
196C22×C4—174no—
197C22×C4—171no—
198C22×C4—171no—
199C22×C4—172no—
200C22×C4—171no—
201C22×C4—172no—
202C22×C4—171no—
203C22×C4—342no—
204C22×C4—342no—
205C22×C4—342no—
206C22×C4—682no—
207C22×C4C22×D1712yes—
208C22×C4C22×D1711yescharacteristic
209C22×C4C22×D1711yescharacteristic
210C4⋊C4—342no—
211C4⋊C4—342no—
212C4⋊C4—682no—
213C4⋊C4—682no—
214C22⋊C4—22no—
215C22⋊C4—24no—
216C22⋊C4—22no—
217C22⋊C4—42no—
218C22⋊C4—344no—
219C22⋊C4—344no—
220C22⋊C4—342no—
221C22⋊C4—342no—
222C22⋊C4—344no—
223C22⋊C4—682no—
224C42—342no—
225C42—342no—
226C23—22no—
227C23—22no—
228C23—28no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—48no—
236C23—48no—
237C23—44no—
238C23—44no—
239C23#6325212yes—
240C23D4×D1712yes—
241C23#6325414yes—
242C23#6325412yes—
243C23C23×D1711yescharacteristic
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—172no—
252C2×C4—172no—
253C2×C4—341no—
254C2×C4—341no—
255C2×C4—341no—
256C2×C4—341no—
257C2×C4—348no—
258C2×C4—342no—
259C2×C4—341no—
260C2×C4—341no—
261C2×C4—342no—
262C2×C4—342no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—342no—
266C2×C4—342no—
267C2×C4—342no—
268C2×C4—684no—
269C22—24no—
270C22—24no—
271C22—22no—
272C22—22no—
273C22—24no—
274C22—24no—
275C22—24no—
276C22—416no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22#1058756914yes—
282C22C2×#6325211yescharacteristic
283C22C2×D4×D1711yescharacteristic
284C22C2×#6325412yes—
285C22C2×#6325411yescharacteristic
286C22#1058769112yes—
287C4—42no—
288C4—41no—
289C4—41no—
290C4—342no—
291C4—342no—
292C4—682no—
293C4—681no—
294C4—682no—
295C4—681no—
296C2—24no—
297C2—44no—
298C2C2×#1058756911yescharacteristic
299C2#1176331511yescharacteristic
300C2#1176337411yescharacteristic
301C2#1176348512yes—
302C2#1176349012yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#379C2
prime radicalC25×C17C22
hypercenter#379D17
hypocenterC17#2482
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C2×#10587563C2semidirect
3. #11762745C2semidirect
4. #11762737C2semidirect
5. #11762703C2semidirect
6. #11762702C2semidirect
7. #11762684C2semidirect
8. #11762681C2semidirect
9. #11762670C2semidirect
26. C2×C4⋊Dic17C22semidirect
27. C2×#63225C22semidirect
28. C2×#63225C22semidirect
29. C2×#63225C22semidirect
30. C2×C4×Dic17C22semidirect
31. C2×C4×Dic17C22semidirect
33. #10587502C22semidirect
34. #10587502C22semidirect
35. #10587502C22semidirect
36. #10587502C22semidirect
188. C17#2482semidirect

Maximal subgroups

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

subgroupindex
C2×#105875632
C22×C22⋊Dic172
#117627452
#117627372
#117627032
#117627022
#117626842
#117626812
#117626702
#117626652
C17×#3792
#248217