Finite Group Databasebeta

← #14538430

Subgroups — #14538430

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×#10587593C211yesmaximal, characteristic, complemented
3#11763375C212yesmaximal, complemented
4#11763063C211yesmaximal, characteristic, complemented
5#11763062C211yesmaximal, characteristic, complemented
6#11763059C211yesmaximal, characteristic, complemented
7#11763044C212yesmaximal, complemented
8#11763043C212yesmaximal, complemented
9#11762844C211yesmaximal, characteristic, complemented
10#11762812C212yesmaximal, complemented
11C17×#469C211yesmaximal, characteristic, complemented
12C2×D4×D17C2212yes—
13C2×#63252C2211yescharacteristic
14C2×D8×C17C2211yescharacteristic
15C2×M16×C17C2211yescharacteristic, complemented
16C17×#121C2212yescomplemented
17#10587626—22no—
18#10587624C2212yescomplemented
19#10587623—22no—
20C2×#63228C2212yes—
21#10587616C2212yescomplemented
22C22×#3622C2211yescharacteristic, complemented
23D8×D17—24no—
24#10587593—24no—
25C2×D136C2211yescharacteristic
26C2×#63218C2211yescharacteristic
27C2×#63218C2211yescharacteristic, complemented
28C2×C8×D17C2211yescharacteristic
29C2×C8×D17C2211yescharacteristic, complemented
30#10587581C2212yescomplemented
31#10587578C2212yes—
32#10587571—22no—
33#10587568—22no—
34C17×#106C2211yescharacteristic, complemented
35C17×#102C2212yescomplemented
36#10587503C2212yescomplemented
37#10587492C2212yescomplemented
38#10587488C2211yescharacteristic, complemented
39#10587480C2212yescomplemented
40#10587478C2212yescomplemented
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—
61M16×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—
73C2×#3622—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—22no—
121C22×C17—42no—
122C22×C17—42no—
123C22×C17—42no—
124C22×C17C22×D411yescharacteristic
125C2×D17—22no—
126C2×D17—21no—
127C2×D17—22no—
128C2×D17—42no—
129C2×D17—42no—
130C2×D17—42no—
131C2×D17—88no—
132C2×D17#12112yes—
133C4×C17—21no—
134C4×C17—42no—
135C4×C17C22×D411yescharacteristic
136C4×C17C2×#4111yescharacteristic
137Dic17—21no—
138Dic17—42no—
139Dic17#12112yes—
140C22×D8—171no—
141C2×#131—171no—
142#535—172no—
143#469—171no—
144#469—171no—
145#469—171no—
146#469D1711yescharacteristic, complemented
147#466—172no—
148#466—172no—
149#427—171no—
150#418—172no—
151C2×C17—41no—
152C2×C17—82no—
153C2×C17C2×#12111yescharacteristic
154C2×C17#57612yes—
155D17—22no—
156D17—41no—
157D17—82no—
158C2×#41—171no—
159C22×D4—172no—
160C2×D8—172no—
161C2×D8—171no—
162C2×D8—344no—
163C2×D8C2×D1711yescharacteristic
164#131—344no—
165C2×M16—171no—
166C2×M16—171no—
167C2×M16C2×D1711yescharacteristic, complemented
168C22×C8—171no—
169C22×C8—171no—
170C22×C8—171no—
171#127—342no—
172#124—342no—
173#121—172no—
174#121—172no—
175#121—172no—
176#121C2×D1712yescomplemented
177#120—342no—
178#120—342no—
179#117—172no—
180#107—172no—
181#106—171no—
182#106C2×D1711yescharacteristic, complemented
183#102—172no—
184#102—172no—
185#102—172no—
186#102C2×D1712yescomplemented
187C17#305811yescharacteristic, complemented, Sylow
188C24—342no—
189#41—342no—
190#41—342no—
191C2×Q8—171no—
192C2×D4—22no—
193C2×D4—171no—
194C2×D4—171no—
195C2×D4—171no—
196C2×D4—172no—
197C2×D4—342no—
198C2×D4—344no—
199C2×D4—344no—
200C2×D4—342no—
201C2×D4—342no—
202C2×D4—342no—
203C2×D4C22×D1712yes—
204C22×C4—171no—
205C22×C4—171no—
206C22×C4C22×D1711yescharacteristic
207D8—22no—
208D8—344no—
209D8—342no—
210M16—22no—
211M16—342no—
212M16—342no—
213C2×C8—171no—
214C2×C8—174no—
215C2×C8—171no—
216C2×C8—342no—
217C2×C8—342no—
218C2×C8C22×D1711yescharacteristic
219C2×C8C22×D1711yescharacteristic
220C4⋊C4—172no—
221C4⋊C4C22×D1712yes—
222C22⋊C4—22no—
223C22⋊C4—342no—
224C22⋊C4—342no—
225C22⋊C4—342no—
226C22⋊C4—342no—
227C42—342no—
228C23—22no—
229C23—171no—
230C23—171no—
231C23—342no—
232C23—684no—
233C23—1362no—
234C23—1362no—
235C23D4×D1711yescharacteristic
236Q8—342no—
237D4—22no—
238D4—22no—
239D4—82no—
240D4—342no—
241D4—342no—
242D4—342no—
243D4—342no—
244D4—342no—
245D4—684no—
246D4—1362no—
247D4—1362no—
248D4—1362no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—171no—
253C2×C4—172no—
254C2×C4—172no—
255C2×C4—171no—
256C2×C4—341no—
257C2×C4—341no—
258C2×C4—342no—
259C2×C4D4×D1711yescharacteristic
260C2×C4C23×D1711yescharacteristic
261C8—21no—
262C8—172no—
263C8—341no—
264C8D4×D1712yes—
265C22—21no—
266C22—22no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—172no—
271C22—342no—
272C22—341no—
273C22—342no—
274C22—682no—
275C22—682no—
276C22—682no—
277C22—1368no—
278C22C2×D4×D1711yescharacteristic
279C4—21no—
280C4—42no—
281C4—172no—
282C4—341no—
283C4—682no—
284C4C2×D4×D1711yescharacteristic
285C4#41×D1711yescharacteristic
286C2—41no—
287C2—82no—
288C2—342no—
289C2—681no—
290C2—1362no—
291C2#121×D1711yescharacteristic
292C2#1176361312yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#469C2
socleC22×C17C22×D4
hypercenter#469D17
hypocenterC17#3058
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D8×D17C2semidirect
2. C2×#10587593C2semidirect
3. #11763375C2semidirect
4. #11763063C2semidirect
5. #11763062C2semidirect
6. #11763059C2semidirect
7. #11763044C2semidirect
8. #11763043C2semidirect
9. #11762844C2semidirect
10. #11762812C2semidirect
11. C17×#469C2semidirect
15. C2×M16×C17C22semidirect
16. C17×#121C22semidirect
18. #10587624C22semidirect
21. #10587616C22semidirect
22. C22×#3622C22semidirect
27. C2×#63218C22semidirect
29. C2×C8×D17C22semidirect
30. #10587581C22semidirect
34. C17×#106C22semidirect
35. C17×#102C22semidirect
36. #10587503C22semidirect
37. #10587492C22semidirect
38. #10587488C22semidirect
39. #10587480C22semidirect
40. #10587478C22semidirect
80. C8×D17D4semidirect
146. #469D17semidirect
167. C2×M16C2×D17semidirect
176. #121C2×D17semidirect
182. #106C2×D17semidirect
186. #102C2×D17semidirect
187. 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×#105875932
#117633752
#117630632
#117630622
#117630592
#117630442
#117630432
#117628442
#117628122
C17×#4692
#305817