Finite Group Databasebeta

← #14539754

Subgroups — #14539754

Order 2176.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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#11763345C211yesmaximal, characteristic, complemented
2C2×#10587633C211yesmaximal, characteristic, complemented
3#11763194C212yesmaximal, complemented
4#11763165C212yesmaximal, complemented
5#11763033C212yesmaximal, complemented
6#11763027C212yesmaximal, complemented
7#11762959C211yesmaximal, characteristic, complemented
8#11762948C211yesmaximal, characteristic, complemented
9#11762945C211yesmaximal, characteristic
10#11762859C211yesmaximal, characteristic, complemented
11C17×#451C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C22×D68C2211yescharacteristic
14C2×Q16×C17C2212yescomplemented
15C17×C22⋊Q8C2211yescharacteristic
16#10587633—21no—
17#10587633—21no—
18#10587633C2212yes—
19#10587633C2212yes—
20#10587633C2211yescharacteristic, complemented
21Q8×Dic17C2211yescharacteristic, complemented
22C2×#63230C2212yes—
23C2×#63230C2212yescomplemented
24#10587616C2211yescharacteristic
25C2×#63227—21no—
26C2×#63227—21no—
27C2×C4×Dic17C2211yescharacteristic
28C2×D136C2212yes—
29#10587584—21no—
30#10587579C2211yescharacteristic, complemented
31#10587578—21no—
32#10587576C2211yescharacteristic, complemented
33#10587572—21no—
34#10587567—21no—
35C17×#103C2212yescomplemented
36C17×C22⋊C8C2211yescharacteristic
37#10587506C2212yescomplemented
38C22⋊#3622C2211yescharacteristic
39#10587492C2212yes—
40#10587485C2212yescomplemented
41C8×Dic17C2212yescomplemented
42#10587480C2212yes—
43C23×D17—21no—
44C2×Q8×C17—21no—
45C2×Q8×C17—42no—
46C2×Q8×C17C2311yescharacteristic
47C2×Q8×C17C2312yes—
48C22×C4×C17—21no—
49C22×C4×C17C2311yescharacteristic
50C2×C22⋊D17—21no—
51C22×Dic17D411yescharacteristic
52C22×Dic17D411yescharacteristic
53C2×D68—21no—
54C2×D68—42no—
55C2×D68C2312yes—
56C2×D68C2311yescharacteristic
57C2×C4×D17—21no—
58Q16×C17—42no—
59C2×C8×C17C2312yes—
60C4⋊C4×C17—21no—
61C4⋊C4×C17C2311yescharacteristic
62C22⋊C4×C17—21no—
63#63230—42no—
64#63230—42no—
65#63227—21no—
66#63227—21no—
67#63227—21no—
68#63227—21no—
69#63227—22no—
70#63227—22no—
71#63227—22no—
72#63227—22no—
73C4⋊Dic17—21no—
74C4⋊Dic17—21no—
75C4⋊Dic17C2311yescharacteristic
76#63225—21no—
77C4×Dic17—21no—
78C4×Dic17D411yescharacteristic
79C4×Dic17D411yescharacteristic
80C4×Dic17C2311yescharacteristic
81C4×Dic17C2311yescharacteristic
82C2×#3622C2312yes—
83D136—42no—
84C23×C17C2×D411yescharacteristic
85C22×D17—21no—
86C22×D17—22no—
87C22×D17—42no—
88C22×D17—42no—
89C22×D17—42no—
90Q8×C17—22no—
91Q8×C17—22no—
92Q8×C17—22no—
93Q8×C17—42no—
94C2×C4×C17—22no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—22no—
98C2×C4×C17—42no—
99C2×C4×C17C2×D411yescharacteristic
100C2×C4×C17C2411yescharacteristic
101C22⋊D17—42no—
102C2×Dic17—22no—
103C2×Dic17—21no—
104C2×Dic17—41no—
105C2×Dic17C2×D411yescharacteristic
106C2×Dic17C2×D411yescharacteristic
107C2×Dic17C2×D411yescharacteristic
108C2×Dic17C2×D411yescharacteristic
109D68—22no—
110D68—22no—
111D68—22no—
112D68—42no—
113C4×D17—42no—
114C8×C17—22no—
115#3622—22no—
116#3211—171nomaximal, Sylow
117C22×C17—21no—
118C22×C17—21no—
119C22×C17#13512yes—
120C22×C17C22×D411yescharacteristic
121C2×D17—42no—
122C2×D17—41no—
123C2×D17—41no—
124C2×D17—42no—
125C2×D17—41no—
126C2×D17—42no—
127C2×D17—84no—
128C4×C17—21no—
129C4×C17—42no—
130C4×C17—41no—
131C4×C17—41no—
132C4×C17H(5,𝔽2)12yes—
133Dic17—21no—
134Dic17—21no—
135Dic17—21no—
136Dic17—21no—
137Dic17—41no—
138#529—171no—
139C2×#124—171no—
140#495—172no—
141#488—172no—
142#464—172no—
143#463—172no—
144#451D1711yescharacteristic, complemented
145#450—171no—
146#448—171no—
147#447—171no—
148#431—171no—
149C2×C17—22no—
150C2×C17#53511yescharacteristic
151C2×C17C2×#13511yescharacteristic
152C2×C17#57511yescharacteristic
153D17—81no—
154D17—82no—
155C22×Q8C2×D1711yescharacteristic
156C22×D4—171no—
157C2×Q16C2×D1712yescomplemented
158C2×SD16—172no—
159C2×SD16—172no—
160C2×D8—172no—
161#126—341no—
162#125—171no—
163#124—172no—
164#124—172no—
165#124—171no—
166#124—341no—
167#124—341no—
168#123—341no—
169C22⋊Q8C2×D1711yescharacteristic
170#121—171no—
171C4×Q8—171no—
172C4×D4—171no—
173C4×D4—341no—
174#117—341no—
175C2×C22⋊C4—341no—
176C2×C22⋊C4—341no—
177C2×C42—171no—
178C4⋊C8—172no—
179#103—172no—
180#103C2×D1712yescomplemented
181#102—172no—
182#102—172no—
183C22⋊C8—171no—
184C22⋊C8C2×D1711yescharacteristic
185C4×C8—172no—
186C17#321111yescharacteristic, complemented, Sylow
187C24—341no—
188C2×Q8—21no—
189C2×Q8—42no—
190C2×Q8C22×D1712yes—
191C2×Q8C22×D1711yescharacteristic
192C2×D4—172no—
193C2×D4—171no—
194C2×D4—341no—
195C2×D4—341no—
196C2×D4—682no—
197C22×C4—21no—
198C22×C4—171no—
199C22×C4—171no—
200C22×C4—341no—
201C22×C4C22×D1711yescharacteristic
202Q16—42no—
203SD16—682no—
204SD16—682no—
205D8—682no—
206C2×C8—172no—
207C2×C8C22×D1712yes—
208C4⋊C4—21no—
209C4⋊C4—171no—
210C4⋊C4—341no—
211C4⋊C4—341no—
212C4⋊C4—341no—
213C4⋊C4C22×D1711yescharacteristic
214C22⋊C4—21no—
215C22⋊C4—342no—
216C22⋊C4—342no—
217C22⋊C4—342no—
218C22⋊C4—342no—
219C22⋊C4—341no—
220C22⋊C4—341no—
221C22⋊C4—341no—
222C22⋊C4—341no—
223C42—171no—
224C42—171no—
225C42—171no—
226C42—171no—
227C42—341no—
228C23—341no—
229C23—342no—
230C23—682no—
231C23—682no—
232C23—682no—
233C23D4×D1711yescharacteristic
234Q8—22no—
235Q8—22no—
236Q8—22no—
237Q8—42no—
238D4—342no—
239D4—342no—
240D4—342no—
241D4—682no—
242D4—682no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—171no—
249C2×C4—171no—
250C2×C4—171no—
251C2×C4—171no—
252C2×C4—342no—
253C2×C4—341no—
254C2×C4—682no—
255C2×C4—681no—
256C2×C4D4×D1711yescharacteristic
257C2×C4C23×D1711yescharacteristic
258C8—22no—
259C8—342no—
260C22—21no—
261C22—21no—
262C22—681no—
263C22—682no—
264C22—681no—
265C22—682no—
266C22—682no—
267C22—681no—
268C22—1364no—
269C22#1058760512yes—
270C22C2×D4×D1711yescharacteristic
271C4—21no—
272C4—41no—
273C4—42no—
274C4—41no—
275C4—341no—
276C4—341no—
277C4—341no—
278C4—341no—
279C4—681no—
280C4#1058769112yes—
281C2—22no—
282C2—1362no—
283C2—1361no—
284C2#1176337411yescharacteristic
285C2C2×#1058760511yescharacteristic
286C2#1176360811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#451C2
socleC22×C17C22×D4
prime radical#11762945C2
hypercenter#451D17
hypocenterC17#3211
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #11763345C2semidirect
2. C2×#10587633C2semidirect
3. #11763194C2semidirect
4. #11763165C2semidirect
5. #11763033C2semidirect
6. #11763027C2semidirect
7. #11762959C2semidirect
8. #11762948C2semidirect
10. #11762859C2semidirect
11. C17×#451C2semidirect
14. C2×Q16×C17C22semidirect
20. #10587633C22semidirect
21. Q8×Dic17C22semidirect
23. C2×#63230C22semidirect
30. #10587579C22semidirect
32. #10587576C22semidirect
35. C17×#103C22semidirect
37. #10587506C22semidirect
40. #10587485C22semidirect
41. C8×Dic17C22semidirect
144. #451D17semidirect
157. C2×Q16C2×D17semidirect
180. #103C2×D17semidirect
186. C17#3211semidirect

Maximal subgroups

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

subgroupindex
#117633452
C2×#105876332
#117631942
#117631652
#117630332
#117630272
#117629592
#117629482
#117629452
#117628592
C17×#4512
#321117