Finite Group Databasebeta

← #14538427

Subgroups — #14538427

Order 2176.

524 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 127 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×D8×D17C211yesmaximal, characteristic, complemented
2C2×#10587587C211yesmaximal, characteristic, complemented
3#11763375C212yesmaximal, complemented
4#11763061C212yesmaximal, complemented
5#11763060C212yesmaximal, complemented
6#11763046C211yesmaximal, characteristic, complemented
7#11763045C211yesmaximal, characteristic, complemented
8#11763042C211yesmaximal, characteristic, complemented
9#11762846C211yesmaximal, characteristic, complemented
10#11762812C212yesmaximal, complemented
11C17×#466C211yesmaximal, characteristic, complemented
12C2×D4×D17C2212yes—
13C2×#63252C2211yescharacteristic
14C2×D8×C17C2211yescharacteristic
15C22×C8×C17C2211yescharacteristic, complemented
16C17×#121C2212yescomplemented
17#10587626—22no—
18#10587624C2212yescomplemented
19#10587623—22no—
20C2×#63228C2212yes—
21#10587616C2212yescomplemented
22C2×#63223C2211yescharacteristic, complemented
23D8×D17—24no—
24C2×D136C2211yescharacteristic
25#10587587—24no—
26C2×#63218C2211yescharacteristic
27C2×#63218C2211yescharacteristic, complemented
28C2×C8×D17C2211yescharacteristic
29C2×C8×D17C2211yescharacteristic, complemented
30#10587581C2212yescomplemented
31#10587578C2212yes—
32#10587571—22no—
33#10587568—22no—
34C17×#107C2211yescharacteristic, complemented
35C17×#102C2212yescomplemented
36#10587503C2212yescomplemented
37#10587492C2212yescomplemented
38#10587489C2211yescharacteristic, complemented
39#10587480C2212yescomplemented
40#10587479C2212yescomplemented
41C23×D17—22no—
42C2×D4×C17—22no—
43C2×D4×C17C2312yes—
44C22×C4×C17C2311yescharacteristic
45C2×C22⋊D17—22no—
46C2×C22⋊D17—22no—
47C2×C22⋊D17—22no—
48C2×C22⋊D17D411yescharacteristic
49C2×C22⋊D17D411yescharacteristic
50D4×D17—24no—
51D4×D17—24no—
52#63252—22no—
53#63252—22no—
54C2×D68—22no—
55C2×D68D411yescharacteristic
56C2×D68C2312yes—
57C2×C4×D17C2311yescharacteristic
58C2×C4×D17C2311yescharacteristic
59C2×Dic34D411yescharacteristic
60D8×C17—22no—
61C2×C8×C17—22no—
62C2×C8×C17C2311yescharacteristic
63C2×C8×C17C2311yescharacteristic
64C4⋊C4×C17C2312yes—
65C22⋊C4×C17—22no—
66C22⋊Dic17—22no—
67#63228—24no—
68#63227—22no—
69#63227—22no—
70#63227—22no—
71C4⋊Dic17C2312yes—
72C4×Dic17—22no—
73#63223—22no—
74C2×#3622C2311yescharacteristic
75C2×#3622C2311yescharacteristic
76D136—22no—
77#63218—22no—
78#63218—22no—
79C8×D17—22no—
80C8×D17D414yescomplemented
81C23×C17—22no—
82C23×C17C2×D411yescharacteristic
83C22×D17—22no—
84C22×D17—44no—
85C22×D17—82no—
86C22×D17—82no—
87C22×D17C2×D411yescharacteristic
88C22×D17C2×D411yescharacteristic
89D4×C17—22no—
90D4×C17—22no—
91D4×C17—82no—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—22no—
95C2×C4×C17C2×D411yescharacteristic
96C2×C4×C17C2411yescharacteristic
97C22⋊D17—22no—
98C22⋊D17—22no—
99C22⋊D17—44no—
100C22⋊D17—82no—
101C22⋊D17—82no—
102C2×Dic17—22no—
103C2×Dic17C2×D411yescharacteristic
104C2×Dic17C2×D411yescharacteristic
105D68—22no—
106D68—22no—
107D68—22no—
108D68—82no—
109C4×D17—21no—
110C4×D17—21no—
111C4×D17C2×D412yes—
112C4×D17#4112yes—
113Dic34—22no—
114C8×C17—21no—
115C8×C17C2×D412yes—
116#3622—21no—
117#3622C2×D412yes—
118#3058—171nomaximal, Sylow
119C22×C17—21no—
120C22×C17—21no—
121C22×C17—21no—
122C22×C17—42no—
123C22×C17—42no—
124C22×C17—42no—
125C22×C17C22×D411yescharacteristic
126C2×D17—21no—
127C2×D17—21no—
128C2×D17—21no—
129C2×D17—21no—
130C2×D17—21no—
131C2×D17—42no—
132C2×D17—42no—
133C2×D17—42no—
134C2×D17—88no—
135C2×D17#12112yes—
136C4×C17—21no—
137C4×C17—42no—
138C4×C17C22×D411yescharacteristic
139C4×C17C2×#4111yescharacteristic
140Dic17—21no—
141Dic17—42no—
142Dic17#12112yes—
143C22×D8—171no—
144C2×#131—171no—
145#535—172no—
146#469—172no—
147#469—172no—
148#466—171no—
149#466—171no—
150#466—171no—
151#466D1711yescharacteristic, complemented
152#427—171no—
153#418—172no—
154C2×C17—41no—
155C2×C17—82no—
156C2×C17C2×#12111yescharacteristic
157C2×C17#57611yescharacteristic
158C2×C17#57611yescharacteristic
159D17—22no—
160D17—41no—
161D17—82no—
162C2×#41—171no—
163C22×D4—172no—
164C2×D8—172no—
165C2×D8—171no—
166C2×D8—344no—
167C2×D8C2×D1711yescharacteristic
168#131—344no—
169C2×M16—171no—
170C2×M16—171no—
171C2×M16—171no—
172C22×C8—171no—
173C22×C8—171no—
174C22×C8C2×D1711yescharacteristic, complemented
175#127—342no—
176#124—342no—
177#121—172no—
178#121—172no—
179#121—172no—
180#121C2×D1712yescomplemented
181#120—342no—
182#120—342no—
183#117—172no—
184#107—171no—
185#107C2×D1711yescharacteristic, complemented
186#106—172no—
187#102—172no—
188#102—172no—
189#102—172no—
190#102C2×D1712yescomplemented
191C17#305811yescharacteristic, complemented, Sylow
192C24—342no—
193#41—342no—
194#41—342no—
195C2×Q8—171no—
196C2×D4—22no—
197C2×D4—171no—
198C2×D4—171no—
199C2×D4—171no—
200C2×D4—172no—
201C2×D4—342no—
202C2×D4—344no—
203C2×D4—344no—
204C2×D4—342no—
205C2×D4—342no—
206C2×D4—342no—
207C2×D4C22×D1712yes—
208C22×C4—171no—
209C22×C4—171no—
210C22×C4C22×D1711yescharacteristic
211D8—22no—
212D8—344no—
213D8—342no—
214M16—342no—
215M16—342no—
216M16—342no—
217C2×C8—22no—
218C2×C8—171no—
219C2×C8—174no—
220C2×C8—171no—
221C2×C8—342no—
222C2×C8C22×D1711yescharacteristic
223C2×C8C22×D1711yescharacteristic
224C4⋊C4—172no—
225C4⋊C4C22×D1712yes—
226C22⋊C4—22no—
227C22⋊C4—342no—
228C22⋊C4—342no—
229C22⋊C4—342no—
230C22⋊C4—342no—
231C42—342no—
232C23—22no—
233C23—171no—
234C23—171no—
235C23—342no—
236C23—684no—
237C23—1362no—
238C23—1362no—
239C23D4×D1711yescharacteristic
240Q8—342no—
241D4—22no—
242D4—22no—
243D4—82no—
244D4—342no—
245D4—342no—
246D4—342no—
247D4—342no—
248D4—342no—
249D4—684no—
250D4—1362no—
251D4—1362no—
252D4—1362no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—171no—
257C2×C4—172no—
258C2×C4—172no—
259C2×C4—171no—
260C2×C4—341no—
261C2×C4—341no—
262C2×C4—342no—
263C2×C4D4×D1711yescharacteristic
264C2×C4C23×D1711yescharacteristic
265C8—21no—
266C8—172no—
267C8—341no—
268C8D4×D1712yes—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—172no—
276C22—341no—
277C22—341no—
278C22—341no—
279C22—341no—
280C22—341no—
281C22—682no—
282C22—682no—
283C22—682no—
284C22—1368no—
285C22C2×D4×D1711yescharacteristic
286C4—21no—
287C4—42no—
288C4—172no—
289C4—341no—
290C4—682no—
291C4C2×D4×D1711yescharacteristic
292C4#41×D1711yescharacteristic
293C2—41no—
294C2—82no—
295C2—342no—
296C2—681no—
297C2—1362no—
298C2#121×D1711yescharacteristic
299C2#1176361011yescharacteristic
300C2#1176361111yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#466C2
socleC22×C17C22×D4
hypercenter#466D17
hypocenterC17#3058
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×D8×D17C2semidirect
2. C2×#10587587C2semidirect
3. #11763375C2semidirect
4. #11763061C2semidirect
5. #11763060C2semidirect
6. #11763046C2semidirect
7. #11763045C2semidirect
8. #11763042C2semidirect
9. #11762846C2semidirect
10. #11762812C2semidirect
11. C17×#466C2semidirect
15. C22×C8×C17C22semidirect
16. C17×#121C22semidirect
18. #10587624C22semidirect
21. #10587616C22semidirect
22. C2×#63223C22semidirect
27. C2×#63218C22semidirect
29. C2×C8×D17C22semidirect
30. #10587581C22semidirect
34. C17×#107C22semidirect
35. C17×#102C22semidirect
36. #10587503C22semidirect
37. #10587492C22semidirect
38. #10587489C22semidirect
39. #10587480C22semidirect
40. #10587479C22semidirect
80. C8×D17D4semidirect
151. #466D17semidirect
174. C22×C8C2×D17semidirect
180. #121C2×D17semidirect
185. #107C2×D17semidirect
190. #102C2×D17semidirect
191. C17#3058semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D172
C2×#105875872
#117633752
#117630612
#117630602
#117630462
#117630452
#117630422
#117628462
#117628122
C17×#4662
#305817