Finite Group Databasebeta

← #14539788

Subgroups — #14539788

Order 2176.

400 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 111 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#11763364C211yesmaximal, characteristic, complemented
2#124×D17C211yesmaximal, characteristic
3#11763204C211yesmaximal, characteristic, complemented
4#11763203C211yesmaximal, characteristic, complemented
5#11763171C211yesmaximal, characteristic, complemented
6#11763034C212yesmaximal, complemented
7#11763030C212yesmaximal, complemented
8#11762966C212yesmaximal, complemented
9#11762961C212yesmaximal
10#11762864C211yesmaximal, characteristic, complemented
11C17×#489C211yesmaximal, characteristic, complemented
12C2×Q8×D17C2211yescharacteristic
13C2×D4×D17C2211yescharacteristic
14C17×C4⋊Q8C2211yescharacteristic, complemented
15C17×#124C2211yescharacteristic
16#10587633C2211yescharacteristic
17#10587632—21no—
18Q8×Dic17—21no—
19C2×#63231C2212yescomplemented
20#10587622C2211yescharacteristic
21C2×#63229C2212yes—
22C2×Dic68C2212yescomplemented
23C2×#63219C2212yes—
24#10587583C2212yes—
25#10587578—21no—
26#10587576—21no—
27#10587573C2212yescomplemented
28#10587571—21no—
29#10587571—21no—
30C22⋊C4×D17—21no—
31C22⋊C4×D17—21no—
32#10587561C2211yescharacteristic
33C42×D17C2211yescharacteristic
34#10587555C2211yescharacteristic, complemented
35C17×#103C2212yescomplemented
36C17×#102C2212yes—
37C17×#97C2211yescharacteristic, complemented
38#10587491C2212yes—
39#10587485C2212yescomplemented
40#10587481C2212yescomplemented
41#10587480C2212yes—
42#10587474C2211yescharacteristic, complemented
43C23×D17—21no—
44C2×Q8×C17—21no—
45C2×Q8×C17C2311yescharacteristic
46C2×D4×C17C2311yescharacteristic
47C2×C22⋊D17—21no—
48Q8×D17—42no—
49D4×D17—42no—
50C2×D68C2311yescharacteristic
51C2×C4×D17—21no—
52C2×C4×D17D411yescharacteristic
53C2×C4×D17D411yescharacteristic
54C2×C4×D17C2311yescharacteristic
55C2×Dic34—21no—
56C2×Dic34C2311yescharacteristic
57C2×Dic34C2312yes—
58C2×C8×C17C2312yes—
59C4⋊C4×C17—21no—
60C4⋊C4×C17C2312yes—
61C22⋊C4×C17—21no—
62C22⋊C4×C17—21no—
63C42×C17C2311yescharacteristic
64C22⋊Dic17—21no—
65C22⋊Dic17—21no—
66#63231—42no—
67#63229—42no—
68#63227—22no—
69#63227—21no—
70#63227—21no—
71#63227—21no—
72#63227—21no—
73C4⋊Dic17—21no—
74C4⋊Dic17—21no—
75C4⋊Dic17—22no—
76#63225—22no—
77C4×Dic17—22no—
78C4×Dic17D411yescharacteristic
79C4×Dic17D411yescharacteristic
80C4×Dic17C2311yescharacteristic
81C2×#3622C2312yes—
82Dic68—42no—
83#63219—42no—
84C23×C17—21no—
85C22×D17—21no—
86C22×D17—42no—
87C22×D17—42no—
88C22×D17—42no—
89C22×D17C2×D411yescharacteristic
90Q8×C17—22no—
91Q8×C17—42no—
92D4×C17—22no—
93C2×C4×C17—21no—
94C2×C4×C17—22no—
95C2×C4×C17C2×D411yescharacteristic
96C2×C4×C17C2×D411yescharacteristic
97C2×C4×C17C2411yescharacteristic
98C22⋊D17—42no—
99C2×Dic17—21no—
100C2×Dic17—22no—
101C2×Dic17C2×D411yescharacteristic
102C2×Dic17C2×D411yescharacteristic
103C2×Dic17C2×D411yescharacteristic
104D68—22no—
105C4×D17—22no—
106C4×D17—22no—
107C4×D17—41no—
108C4×D17—42no—
109Dic34—22no—
110Dic34—22no—
111Dic34—22no—
112Dic34—42no—
113C8×C17—22no—
114#3622—22no—
115#3215—171nomaximal, Sylow
116C22×C17—41no—
117C22×C17—41no—
118C22×C17—41no—
119C22×C17C22×D411yescharacteristic
120C2×D17—21no—
121C2×D17—21no—
122C2×D17—41no—
123C2×D17—41no—
124C2×D17—41no—
125C2×D17—84no—
126C2×D17#13712yes—
127C4×C17—21no—
128C4×C17—21no—
129C4×C17—41no—
130C4×C17—42no—
131C4×C17H(5,𝔽2)12yes—
132Dic17—21no—
133Dic17—21no—
134Dic17—21no—
135Dic17—41no—
136Dic17—42no—
137#533—171no—
138C2×#124—171no—
139#497—171no—
140#497—171no—
141#489—171no—
142#489D1711yescharacteristic, complemented
143#464—172no—
144#464—172no—
145#451—172no—
146#450—172no—
147#432—171no—
148C2×C17—81no—
149C2×C17#53511yescharacteristic
150C2×C17C2×#13711yescharacteristic
151C2×C17#57511yescharacteristic
152D17—22no—
153D17—81no—
154C22×Q8—171no—
155C22×D4—171no—
156C2×Q16—172no—
157C2×Q16—172no—
158C2×SD16—172no—
159C2×SD16—172no—
160C4⋊Q8—171no—
161C4⋊Q8C2×D1711yescharacteristic, complemented
162#125—341no—
163#124—171no—
164#124—171no—
165#124—171no—
166#124—341no—
167#124—341no—
168#124C2×D1711yescharacteristic
169C22⋊Q8—172no—
170C22⋊Q8—341no—
171C4×Q8—172no—
172C4×Q8—341no—
173#117—341no—
174C2×C22⋊C4—341no—
175C2×C22⋊C4—341no—
176C2×C42—171no—
177C4⋊C8—172no—
178#103—172no—
179#103C2×D1712yescomplemented
180#102—172no—
181#102C2×D1712yes—
182C22⋊C8—172no—
183#97—171no—
184#97C2×D1711yescharacteristic, complemented
185C17#321511yescharacteristic, complemented, Sylow
186C24—341no—
187C2×Q8—21no—
188C2×Q8—171no—
189C2×Q8—172no—
190C2×Q8—341no—
191C2×Q8—682no—
192C2×Q8C22×D1711yescharacteristic
193C2×D4—171no—
194C2×D4—341no—
195C2×D4—682no—
196C2×D4C22×D1711yescharacteristic
197C22×C4—171no—
198C22×C4—171no—
199C22×C4—171no—
200C22×C4—341no—
201Q16—682no—
202Q16—682no—
203SD16—682no—
204SD16—682no—
205C2×C8—172no—
206C2×C8C22×D1712yes—
207C4⋊C4—21no—
208C4⋊C4—342no—
209C4⋊C4—342no—
210C4⋊C4—341no—
211C4⋊C4—341no—
212C4⋊C4C22×D1712yes—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—341no—
216C22⋊C4—341no—
217C22⋊C4—342no—
218C22⋊C4—341no—
219C22⋊C4—341no—
220C22⋊C4—341no—
221C22⋊C4—341no—
222C42—171no—
223C42—171no—
224C42—171no—
225C42—342no—
226C42C22×D1711yescharacteristic
227C23—21no—
228C23—171no—
229C23—341no—
230C23—682no—
231C23—682no—
232C23—682no—
233Q8—22no—
234Q8—42no—
235Q8—342no—
236Q8—342no—
237Q8—342no—
238Q8—682no—
239D4—22no—
240D4—342no—
241D4—682no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—171no—
245C2×C4—171no—
246C2×C4—171no—
247C2×C4—342no—
248C2×C4—342no—
249C2×C4—342no—
250C2×C4—341no—
251C2×C4—681no—
252C2×C4—682no—
253C2×C4D4×D1711yescharacteristic
254C2×C4D4×D1711yescharacteristic
255C2×C4C23×D1711yescharacteristic
256C8—22no—
257C8—342no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—172no—
262C22—341no—
263C22—341no—
264C22—681no—
265C22—681no—
266C22—681no—
267C22—1364no—
268C22C2×D4×D1711yescharacteristic
269C4—21no—
270C4—21no—
271C4—42no—
272C4—41no—
273C4—341no—
274C4—341no—
275C4—341no—
276C4—681no—
277C4—682no—
278C4#1058769612yes—
279C2—81no—
280C2—342no—
281C2—1361no—
282C2#1176337711yescharacteristic
283C2#137×D1711yescharacteristic
284C2#1176360511yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#489C2
socleC22×C17C22×D4
prime radicalC2×D4×D17C22
hypercenter#489D17
hypocenterC17#3215
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #11763364C2semidirect
3. #11763204C2semidirect
4. #11763203C2semidirect
5. #11763171C2semidirect
6. #11763034C2semidirect
7. #11763030C2semidirect
8. #11762966C2semidirect
10. #11762864C2semidirect
11. C17×#489C2semidirect
14. C17×C4⋊Q8C22semidirect
19. C2×#63231C22semidirect
22. C2×Dic68C22semidirect
27. #10587573C22semidirect
34. #10587555C22semidirect
35. C17×#103C22semidirect
37. C17×#97C22semidirect
39. #10587485C22semidirect
40. #10587481C22semidirect
42. #10587474C22semidirect
142. #489D17semidirect
161. C4⋊Q8C2×D17semidirect
179. #103C2×D17semidirect
184. #97C2×D17semidirect
185. C17#3215semidirect

Maximal subgroups

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

subgroupindex
#117633642
#124×D172
#117632042
#117632032
#117631712
#117630342
#117630302
#117629662
#117629612
#117628642
C17×#4892
#321517