Finite Group Databasebeta

← #14538698

Subgroups — #14538698

Order 2176.

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

normal subgroups only

This group has 420 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
1#11763322C211yesmaximal, characteristic
2C2×#10587578C211yesmaximal, characteristic, complemented
3#11763148C212yesmaximal, complemented
4#11763142C211yesmaximal, characteristic, complemented
5#11763135C212yesmaximal, complemented
6#11763128C211yesmaximal, characteristic, complemented
7#11763122C211yesmaximal, characteristic, complemented
8#11762844C212yesmaximal, complemented
9#11762804C212yesmaximal, complemented
10C22⋊C8×D17C211yesmaximal, characteristic, complemented
11C17×#483C211yesmaximal, characteristic, complemented
12C2×#63256C2211yescharacteristic
13C22×C4×D17C2211yescharacteristic
14C17×C22⋊Q8C2211yescharacteristic
15C2×C4⋊C4×C17C2211yescharacteristic
16#10587632C2211yescharacteristic
17#10587616C2211yescharacteristic
18C2×#63227—21no—
19C2×C4⋊Dic17C2211yescharacteristic
20C2×C4×Dic17—21no—
21C2×C8×D17C2212yes—
22#10587581C2211yescharacteristic
23#10587580—21no—
24#10587579—21no—
25#10587578—21no—
26#10587578—21no—
27#10587578C2212yes—
28#10587578C2212yes—
29#10587578C2211yescharacteristic
30#10587568—21no—
31#10587567—21no—
32C17×#106C2212yescomplemented
33C17×#103C2212yescomplemented
34C17×C22⋊C8C2211yescharacteristic
35#10587506C2212yescomplemented
36C22⋊#3622C2211yescharacteristic
37#10587492C2212yescomplemented
38#10587491C2212yes—
39#10587488C2212yescomplemented
40#10587480C2212yescomplemented
41#10587478C2212yescomplemented
42#10587478C2212yescomplemented
43C23×D17D411yescharacteristic
44C2×Q8×C17C2311yescharacteristic
45C22×C4×C17—21no—
46C22×C4×C17C2311yescharacteristic
47C2×C22⋊D17—21no—
48C22×Dic17—21no—
49C22×Dic17D411yescharacteristic
50#63256—24no—
51C2×D68—21no—
52C2×D68—21no—
53C2×D68C2311yescharacteristic
54C2×C4×D17—24no—
55C2×C4×D17—21no—
56C2×C4×D17D411yescharacteristic
57C2×C4×D17D411yescharacteristic
58C2×C4×D17C2311yescharacteristic
59C2×C4×D17C2311yescharacteristic
60C2×C8×C17C2312yes—
61C4⋊C4×C17—21no—
62C4⋊C4×C17—21no—
63C4⋊C4×C17C2311yescharacteristic
64C4⋊C4×C17C2312yes—
65C22⋊C4×C17—21no—
66#63227—21no—
67#63227—21no—
68#63227—22no—
69#63227—22no—
70#63227—21no—
71#63227—21no—
72C4⋊Dic17—21no—
73C4⋊Dic17C2311yescharacteristic
74C4⋊Dic17C2312yes—
75#63225—21no—
76C4×Dic17—21no—
77C4×Dic17—22no—
78C4×Dic17—22no—
79C2×#3622C2312yes—
80C8×D17—24no—
81C23×C17C2×D411yescharacteristic
82C22×D17—22no—
83C22×D17—22no—
84C22×D17—21no—
85C22×D17—41no—
86C22×D17C2×D411yescharacteristic
87C22×D17C2×D411yescharacteristic
88Q8×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—21no—
91C2×C4×C17—21no—
92C2×C4×C17—22no—
93C2×C4×C17—42no—
94C2×C4×C17C2×D411yescharacteristic
95C2×C4×C17C2411yescharacteristic
96C22⋊D17—42no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—21no—
100C2×Dic17—42no—
101C2×Dic17C2×D411yescharacteristic
102C2×Dic17C2×D411yescharacteristic
103D68—22no—
104D68—42no—
105D68—42no—
106C4×D17—22no—
107C4×D17—22no—
108C4×D17—22no—
109C4×D17—42no—
110C4×D17#4114yes—
111C8×C17—22no—
112#3622—22no—
113#3091—171nomaximal, Sylow
114C22×C17—21no—
115C22×C17—21no—
116C22×C17#13512yes—
117C22×C17C22×D411yescharacteristic
118C2×D17—21no—
119C2×D17—21no—
120C2×D17—21no—
121C2×D17—21no—
122C2×D17—21no—
123C2×D17—44no—
124C2×D17—41no—
125C2×D17—41no—
126C2×D17—41no—
127C2×D17#12312yes—
128C4×C17—21no—
129C4×C17—41no—
130C4×C17—42no—
131C4×C17—41no—
132C4×C17C2×#4112yes—
133Dic17—21no—
134Dic17—41no—
135Dic17—42no—
136Dic17#12312yes—
137#525—171no—
138C2×#117—171no—
139#485—172no—
140#484—171no—
141#483D1711yescharacteristic, complemented
142#482—172no—
143#481—171no—
144#480—171no—
145#427—172no—
146#416—172no—
147C2×C22⋊C8—171no—
148C2×C17—22no—
149C2×C17C2×#12311yescharacteristic
150C2×C17C2×#13511yescharacteristic
151C2×C17#57511yescharacteristic
152D17—22no—
153D17—41no—
154D17—81no—
155C2×#41—171no—
156C23×C4—171no—
157C22×C8—172no—
158#123—341no—
159C22⋊Q8—171no—
160C22⋊Q8C2×D1711yescharacteristic
161#121—171no—
162#121—171no—
163#120—341no—
164C4×D4—341no—
165C4×D4—341no—
166#117—172no—
167#117—172no—
168#117—171no—
169#117—341no—
170#117—341no—
171C2×C4⋊C4—171no—
172C2×C4⋊C4C2×D1711yescharacteristic
173C2×C22⋊C4—341no—
174C2×C42—341no—
175#107—172no—
176#107—172no—
177#106—172no—
178#106C2×D1712yescomplemented
179#103—172no—
180#103C2×D1712yescomplemented
181#102—172no—
182#102—172no—
183C22⋊C8—171no—
184C22⋊C8—172no—
185C22⋊C8C2×D1711yescharacteristic
186C17#309111yescharacteristic, complemented, Sylow
187C24—171no—
188#41—344no—
189C2×Q8C22×D1711yescharacteristic
190C2×D4—171no—
191C2×D4—341no—
192C2×D4—341no—
193C2×D4—341no—
194C22×C4—21no—
195C22×C4—171no—
196C22×C4—171no—
197C22×C4—171no—
198C22×C4—171no—
199C22×C4—171no—
200C22×C4—344no—
201C22×C4—341no—
202C22×C4—341no—
203C22×C4C22×D1711yescharacteristic
204C2×C8—172no—
205C2×C8—344no—
206C2×C8C22×D1712yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—172no—
210C4⋊C4—171no—
211C4⋊C4—341no—
212C4⋊C4—341no—
213C4⋊C4C22×D1712yes—
214C4⋊C4C22×D1711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—341no—
217C22⋊C4—342no—
218C22⋊C4—342no—
219C22⋊C4—341no—
220C22⋊C4—341no—
221C22⋊C4—341no—
222C42—342no—
223C42—342no—
224C42—341no—
225C23—171no—
226C23—171no—
227C23—342no—
228C23—342no—
229C23—341no—
230C23—681no—
231C23D4×D1711yescharacteristic
232Q8—22no—
233D4—342no—
234D4—682no—
235D4—682no—
236D4—682no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—171no—
243C2×C4—174no—
244C2×C4—171no—
245C2×C4—342no—
246C2×C4—342no—
247C2×C4—342no—
248C2×C4—342no—
249C2×C4—342no—
250C2×C4—341no—
251C2×C4—682no—
252C2×C4—682no—
253C2×C4D4×D1711yescharacteristic
254C2×C4C23×D1711yescharacteristic
255C8—22no—
256C8—342no—
257C22—21no—
258C22—21no—
259C22—172no—
260C22—341no—
261C22—341no—
262C22—341no—
263C22—341no—
264C22—341no—
265C22—684no—
266C22—681no—
267C22—681no—
268C22—681no—
269C22#1058760212yes—
270C22C2×D4×D1711yescharacteristic
271C4—21no—
272C4—41no—
273C4—42no—
274C4—41no—
275C4—172no—
276C4—341no—
277C4—681no—
278C4—682no—
279C4#41×D1712yes—
280C2—22no—
281C2—342no—
282C2—681no—
283C2—1361no—
284C2#123×D1711yescharacteristic
285C2C2×#1058760211yescharacteristic
286C2#1176360811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#483C2
socleC22×C17C22×D4
prime radical#11763322C2
hypercenter#483D17
hypocenterC17#3091
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
2. C2×#10587578C2semidirect
3. #11763148C2semidirect
4. #11763142C2semidirect
5. #11763135C2semidirect
6. #11763128C2semidirect
7. #11763122C2semidirect
8. #11762844C2semidirect
9. #11762804C2semidirect
10. C22⋊C8×D17C2semidirect
11. C17×#483C2semidirect
32. C17×#106C22semidirect
33. C17×#103C22semidirect
35. #10587506C22semidirect
37. #10587492C22semidirect
39. #10587488C22semidirect
40. #10587480C22semidirect
41. #10587478C22semidirect
42. #10587478C22semidirect
141. #483D17semidirect
178. #106C2×D17semidirect
180. #103C2×D17semidirect
186. C17#3091semidirect

Maximal subgroups

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

subgroupindex
#117633222
C2×#105875782
#117631482
#117631422
#117631352
#117631282
#117631222
#117628442
#117628042
C22⋊C8×D172
C17×#4832
#309117