Finite Group Databasebeta

← #14539607

Subgroups — #14539607

Order 2176.

410 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 115 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×#10587583C211yesmaximal, characteristic
2C22⋊Q8×D17C211yesmaximal, characteristic
3#11763141C212yesmaximal, complemented
4#11763139C211yesmaximal, characteristic, complemented
5#11763068C212yesmaximal, complemented
6#11763066C212yesmaximal, complemented
7#11762967C211yesmaximal, characteristic, complemented
8#11762966C212yesmaximal, complemented
9#11762964C211yesmaximal, characteristic, complemented
10#11762771C211yesmaximal, characteristic
11C17×#483C211yesmaximal, characteristic, complemented
12C2×Q8×D17C2211yescharacteristic
13C22×C4×D17C2211yescharacteristic
14C22×Dic34C2211yescharacteristic
15C17×C22⋊Q8C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17#10587632C2211yescharacteristic
18C2×#63231C2212yescomplemented
19C2×#63227—21no—
20C2×C4⋊Dic17—21no—
21#10587610C2211yescharacteristic
22C2×Dic68C2212yescomplemented
23C2×#63218C2212yes—
24#10587583—21no—
25#10587583—21no—
26#10587583C2212yes—
27#10587583C2211yescharacteristic
28#10587583C2212yes—
29#10587582—21no—
30C4⋊C4×D17—21no—
31C4⋊C4×D17C2211yescharacteristic
32C22⋊C4×D17—21no—
33#10587564—21no—
34C17×#106C2212yescomplemented
35C17×#103C2212yescomplemented
36C17×C22⋊C8C2211yescharacteristic
37C22⋊#3622C2211yescharacteristic
38#10587491C2212yes—
39#10587487C2212yescomplemented
40#10587481C2212yescomplemented
41#10587481C2212yescomplemented
42#10587479C2212yescomplemented
43C23×D17D411yescharacteristic
44C2×Q8×C17C2311yescharacteristic
45C22×C4×C17—21no—
46C22×C4×C17C2311yescharacteristic
47C22×Dic17—21no—
48C22×Dic17D411yescharacteristic
49Q8×D17—42no—
50C2×C4×D17—21no—
51C2×C4×D17—21no—
52C2×C4×D17—42no—
53C2×C4×D17D411yescharacteristic
54C2×C4×D17D411yescharacteristic
55C2×C4×D17C2311yescharacteristic
56C2×C4×D17C2311yescharacteristic
57C2×Dic34—21no—
58C2×Dic34—21no—
59C2×Dic34—42no—
60C2×Dic34C2312yes—
61C2×Dic34C2311yescharacteristic
62C2×C8×C17C2312yes—
63C4⋊C4×C17—21no—
64C4⋊C4×C17—21no—
65C4⋊C4×C17C2311yescharacteristic
66C4⋊C4×C17C2312yes—
67C22⋊C4×C17—21no—
68C22⋊Dic17—21no—
69#63231—42no—
70#63227—21no—
71#63227—21no—
72#63227—22no—
73#63227—22no—
74C4⋊Dic17—21no—
75C4⋊Dic17—22no—
76C4⋊Dic17—22no—
77C4⋊Dic17C2311yescharacteristic
78#63225—21no—
79#63225—21no—
80#63225—21no—
81C2×#3622C2312yes—
82Dic68—42no—
83#63218—42no—
84C23×C17C2×D411yescharacteristic
85C22×D17—22no—
86C22×D17—22no—
87C22×D17—41no—
88C22×D17C2×D411yescharacteristic
89C22×D17C2×D411yescharacteristic
90Q8×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—22no—
95C2×C4×C17—42no—
96C2×C4×C17C2×D411yescharacteristic
97C2×C4×C17C2411yescharacteristic
98C2×Dic17—21no—
99C2×Dic17—21no—
100C2×Dic17—22no—
101C2×Dic17—41no—
102C2×Dic17—42no—
103C2×Dic17C2×D411yescharacteristic
104C2×Dic17C2×D411yescharacteristic
105C4×D17—22no—
106C4×D17—22no—
107C4×D17—41no—
108C4×D17—41no—
109C4×D17—42no—
110C4×D17—42no—
111Dic34—22no—
112Dic34—22no—
113Dic34—22no—
114Dic34—42no—
115Dic34—42no—
116C8×C17—22no—
117#3622—22no—
118#3195—171nomaximal, Sylow
119C22×C17—21no—
120C22×C17—21no—
121C22×C17#13712yes—
122C22×C17C22×D411yescharacteristic
123C2×D17—21no—
124C2×D17—21no—
125C2×D17—21no—
126C2×D17—21no—
127C2×D17—21no—
128C2×D17—44no—
129C2×D17#13712yes—
130C4×C17—21no—
131C4×C17—41no—
132C4×C17—41no—
133C4×C17—42no—
134C4×C17H(5,𝔽2)12yes—
135Dic17—21no—
136Dic17—21no—
137Dic17—41no—
138Dic17—42no—
139Dic17—41no—
140C2×C22⋊Q8—171no—
141C2×C22⋊Q8—171no—
142#483—171no—
143#483—172no—
144#483D1711yescharacteristic, complemented
145#470—172no—
146#470—172no—
147#451—171no—
148#451—172no—
149#451—171no—
150#407—171no—
151C2×C17—22no—
152C2×C17#53411yescharacteristic
153C2×C17C2×#13711yescharacteristic
154C2×C17C2×#13711yescharacteristic
155D17—22no—
156D17—41no—
157C22×Q8—171no—
158C22×Q8—171no—
159C23×C4—171no—
160C2×Q16—172no—
161C2×Q16—172no—
162C2×M16—172no—
163C22⋊Q8—171no—
164C22⋊Q8—171no—
165C22⋊Q8—172no—
166C22⋊Q8—171no—
167C22⋊Q8—172no—
168C22⋊Q8—341no—
169C22⋊Q8—341no—
170C22⋊Q8—341no—
171C22⋊Q8—341no—
172C22⋊Q8C2×D1711yescharacteristic
173C2×C4⋊C4—171no—
174C2×C4⋊C4—341no—
175C2×C4⋊C4—341no—
176C2×C4⋊C4C2×D1711yescharacteristic
177C2×C22⋊C4—341no—
178C2×C22⋊C4—341no—
179#106—172no—
180#106C2×D1712yescomplemented
181#103—172no—
182#103—172no—
183#103—172no—
184#103C2×D1712yescomplemented
185C22⋊C8—171no—
186C22⋊C8—172no—
187C22⋊C8C2×D1711yescharacteristic
188C17#319511yescharacteristic, complemented, Sylow
189C24—171no—
190C2×Q8—172no—
191C2×Q8—171no—
192C2×Q8—341no—
193C2×Q8—341no—
194C2×Q8—682no—
195C2×Q8—682no—
196C2×Q8C22×D1711yescharacteristic
197C22×C4—21no—
198C22×C4—171no—
199C22×C4—171no—
200C22×C4—171no—
201C22×C4—171no—
202C22×C4—171no—
203C22×C4—341no—
204C22×C4—341no—
205C22×C4—341no—
206C22×C4—682no—
207C22×C4C22×D1711yescharacteristic
208Q16—682no—
209Q16—682no—
210M16—682no—
211C2×C8—172no—
212C2×C8C22×D1712yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—171no—
216C4⋊C4—341no—
217C4⋊C4—341no—
218C4⋊C4—341no—
219C4⋊C4—342no—
220C4⋊C4—342no—
221C4⋊C4—341no—
222C4⋊C4C22×D1711yescharacteristic
223C4⋊C4C22×D1712yes—
224C22⋊C4—21no—
225C22⋊C4—341no—
226C22⋊C4—341no—
227C22⋊C4—342no—
228C22⋊C4—342no—
229C22⋊C4—341no—
230C23—171no—
231C23—171no—
232C23—342no—
233C23—342no—
234C23—681no—
235C23D4×D1711yescharacteristic
236Q8—22no—
237Q8—342no—
238Q8—342no—
239Q8—342no—
240Q8—682no—
241Q8—682no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—171no—
248C2×C4—171no—
249C2×C4—342no—
250C2×C4—342no—
251C2×C4—341no—
252C2×C4—342no—
253C2×C4—341no—
254C2×C4—681no—
255C2×C4—681no—
256C2×C4—682no—
257C2×C4—682no—
258C2×C4—681no—
259C2×C4—682no—
260C2×C4D4×D1711yescharacteristic
261C2×C4C23×D1711yescharacteristic
262C8—22no—
263C8—342no—
264C22—21no—
265C22—21no—
266C22—172no—
267C22—341no—
268C22—341no—
269C22—341no—
270C22—341no—
271C22—341no—
272C22—684no—
273C22#1058760112yes—
274C22C2×D4×D1711yescharacteristic
275C4—21no—
276C4—41no—
277C4—42no—
278C4—41no—
279C4—341no—
280C4—341no—
281C4—681no—
282C4—681no—
283C4—682no—
284C4#1058769612yes—
285C2—22no—
286C2—342no—
287C2—681no—
288C2#1176337011yescharacteristic
289C2#137×D1711yescharacteristic
290C2C2×#1058760111yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#483C2
socleC22×C17C22×D4
prime radicalC23×D17D4
hypercenter#483D17
hypocenterC17#3195
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
3. #11763141C2semidirect
4. #11763139C2semidirect
5. #11763068C2semidirect
6. #11763066C2semidirect
7. #11762967C2semidirect
8. #11762966C2semidirect
9. #11762964C2semidirect
11. C17×#483C2semidirect
18. C2×#63231C22semidirect
22. C2×Dic68C22semidirect
34. C17×#106C22semidirect
35. C17×#103C22semidirect
39. #10587487C22semidirect
40. #10587481C22semidirect
41. #10587481C22semidirect
42. #10587479C22semidirect
144. #483D17semidirect
180. #106C2×D17semidirect
184. #103C2×D17semidirect
188. C17#3195semidirect

Maximal subgroups

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

subgroupindex
C2×#105875832
C22⋊Q8×D172
#117631412
#117631392
#117630682
#117630662
#117629672
#117629662
#117629642
#117627712
C17×#4832
#319517