Finite Group Databasebeta

← #14538397

Subgroups — #14538397

Order 2176.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×M16×D17C211yesmaximal, characteristic, complemented
2#11763316C211yesmaximal, characteristic, complemented
3C2×#10587624C211yesmaximal, characteristic, complemented
4#11763059C212yesmaximal, complemented
5#11763052C212yesmaximal, complemented
6#11763042C212yesmaximal, complemented
7#11763035C212yesmaximal, complemented
8#11762851C211yesmaximal, characteristic, complemented
9#11762814C211yesmaximal, characteristic, complemented
10#11762810C211yesmaximal, characteristic, complemented
11C2×C17×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
12C2×#41×C17C2211yescharacteristic
13C22×D4×C17C2211yescharacteristic
14C22×C22⋊D17—21no—
15C22×C4×D17C2211yescharacteristic
16C17×AGL(1,ℤ/8)—24no—
17C2×SD16×C17C2212yescomplemented
18C2×D8×C17C2212yescomplemented
19C2×M16×C17C2211yescharacteristic
20C2×C22⋊Dic17—21no—
21#10587632C2211yescharacteristic, complemented
22#10587632C2211yescharacteristic, complemented
23#10587624—21no—
24#10587624—21no—
25#10587624C2212yes—
26#10587624C2212yes—
27#10587624C2211yescharacteristic, complemented
28#10587624C2211yescharacteristic, complemented
29#10587623—21no—
30#10587621—21no—
31C4×C22⋊D17—21no—
32C4×C22⋊D17—21no—
33#10587612C2211yescharacteristic
34C2×C4⋊Dic17C2211yescharacteristic
35C2×#63223C2211yescharacteristic
36M16×D17—24no—
37C2×#63218C2212yescomplemented
38C2×C8×D17C2212yescomplemented
39#10587506C2212yescomplemented
40#10587503C2212yescomplemented
41#10587503C2212yescomplemented
42#10587503C2212yescomplemented
43#10587489C2212yescomplemented
44#10587488C2212yescomplemented
45C24×C17—21no—
46C23×D17D411yescharacteristic
47#41×C17—22no—
48#41×C17—22no—
49C2×Q8×C17C2311yescharacteristic
50C2×D4×C17—21no—
51C2×D4×C17—22no—
52C2×D4×C17—21no—
53C2×D4×C17—22no—
54C2×D4×C17C2311yescharacteristic
55C2×D4×C17C2312yes—
56C22×C4×C17—21no—
57C22×C4×C17C2311yescharacteristic
58C2×C22⋊D17—22no—
59C2×C22⋊D17—22no—
60C2×C22⋊D17—21no—
61C2×C22⋊D17—21no—
62C2×C22⋊D17—44no—
63C22×Dic17—21no—
64C22×Dic17D411yescharacteristic
65C2×C4×D17—22no—
66C2×C4×D17—22no—
67C2×C4×D17D411yescharacteristic
68C2×C4×D17D411yescharacteristic
69C2×C4×D17C2311yescharacteristic
70C2×C4×D17C2311yescharacteristic
71SD16×C17—24no—
72D8×C17—24no—
73M16×C17D414yescomplemented
74C2×C8×C17C2312yes—
75C22⋊Dic17—21no—
76C22⋊Dic17—21no—
77C22⋊Dic17—21no—
78C22⋊Dic17—22no—
79C22⋊Dic17—22no—
80#63227—21no—
81#63227—21no—
82C4⋊Dic17—21no—
83C4⋊Dic17C2311yescharacteristic
84C4⋊Dic17C2312yes—
85C4⋊Dic17C2311yescharacteristic
86#63225—21no—
87#63225—21no—
88C4×Dic17—21no—
89#63223—22no—
90C2×#3622C2312yes—
91#63218—24no—
92C8×D17—24no—
93C23×C17—21no—
94C23×C17—22no—
95C23×C17—42no—
96C23×C17—42no—
97C23×C17—42no—
98C23×C17C2×D411yescharacteristic
99C22×D17—22no—
100C22×D17—22no—
101C22×D17—41no—
102C22×D17C2×D411yescharacteristic
103C22×D17C2×D411yescharacteristic
104Q8×C17—21no—
105Q8×C17—21no—
106D4×C17—21no—
107D4×C17—22no—
108D4×C17—22no—
109D4×C17—21no—
110D4×C17—42no—
111D4×C17—42no—
112C2×C4×C17—21no—
113C2×C4×C17—42no—
114C2×C4×C17C2×D411yescharacteristic
115C2×C4×C17C2×D412yes—
116C2×C4×C17#4112yes—
117C2×C4×C17C2411yescharacteristic
118C22⋊D17—42no—
119C22⋊D17—44no—
120C22⋊D17—42no—
121C22⋊D17—44no—
122C2×Dic17—22no—
123C2×Dic17—22no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—42no—
127C2×Dic17C2×D411yescharacteristic
128C2×Dic17C2×D411yescharacteristic
129C4×D17—22no—
130C4×D17—22no—
131C4×D17—21no—
132C4×D17—21no—
133C4×D17—21no—
134C4×D17—21no—
135C8×C17C2×D414yescomplemented
136#3622—22no—
137#3055—171nomaximal, Sylow
138C22×C17—21no—
139C22×C17—21no—
140C22×C17—41no—
141C22×C17—42no—
142C22×C17—42no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—42no—
146C22×C17—84no—
147C22×C17#12112yes—
148C22×C17C22×D411yescharacteristic
149C2×D17—21no—
150C2×D17—21no—
151C2×D17—21no—
152C2×D17—21no—
153C2×D17—21no—
154C2×D17—44no—
155C2×D17AGL(1,ℤ/8)12yescomplemented
156C4×C17—41no—
157C4×C17#12112yes—
158C4×C17C22×D411yescharacteristic
159C4×C17C2×#4111yescharacteristic
160Dic17—21no—
161Dic17—21no—
162Dic17—42no—
163Dic17—41no—
164Dic17—41no—
165C2×AGL(1,ℤ/8)D1711yescharacteristic, complemented
166C22×M16—171no—
167#525—171no—
168C2×#121—171no—
169#469—172no—
170#468—172no—
171#466—172no—
172#465—172no—
173#428—171no—
174#418—171no—
175#417—171no—
176C2×C17—22no—
177C2×C17—81no—
178C2×C17—82no—
179C2×C17C2×#12111yescharacteristic
180C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C17#57511yescharacteristic
182D17—22no—
183D17—41no—
184C2×#41C2×D1711yescharacteristic
185C22×D4—341no—
186C22×D4C2×D1711yescharacteristic
187C23×C4—171no—
188AGL(1,ℤ/8)—24no—
189C2×SD16C2×D1712yescomplemented
190C2×D8C2×D1712yescomplemented
191C2×M16—171no—
192C2×M16—172no—
193C2×M16—344no—
194C2×M16C2×D1711yescharacteristic
195C22×C8—172no—
196#123—341no—
197C22⋊Q8—171no—
198C22⋊Q8—171no—
199#121—172no—
200#121—172no—
201#121—171no—
202#121—171no—
203#121—341no—
204#121—341no—
205#120—341no—
206C4×D4—341no—
207C4×D4—341no—
208#117—171no—
209C2×C4⋊C4—171no—
210C2×C22⋊C4—341no—
211#107—172no—
212#106—172no—
213#103—172no—
214#102—172no—
215#102—172no—
216#102—172no—
217C17#305511yescharacteristic, complemented, Sylow
218C24—21no—
219C24—171no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D1711yescharacteristic
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—341no—
228C2×D4—341no—
229C2×D4—342no—
230C2×D4—342no—
231C2×D4—684no—
232C2×D4C22×D1711yescharacteristic
233C2×D4C22×D1712yes—
234C22×C4—21no—
235C22×C4—171no—
236C22×C4—171no—
237C22×C4—171no—
238C22×C4—171no—
239C22×C4—171no—
240C22×C4—342no—
241C22×C4—342no—
242C22×C4—341no—
243C22×C4C22×D1711yescharacteristic
244SD16—24no—
245D8—24no—
246M16—342no—
247M16—344no—
248M16C22⋊D1714yescomplemented
249C2×C8—172no—
250C2×C8—344no—
251C2×C8C22×D1712yes—
252C4⋊C4—171no—
253C4⋊C4—171no—
254C4⋊C4—172no—
255C4⋊C4—341no—
256C4⋊C4—341no—
257C4⋊C4—341no—
258C22⋊C4—341no—
259C22⋊C4—341no—
260C22⋊C4—342no—
261C22⋊C4—342no—
262C22⋊C4—341no—
263C22⋊C4—341no—
264C22⋊C4—341no—
265C42—341no—
266C23—21no—
267C23—22no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—171no—
272C23—171no—
273C23—342no—
274C23—342no—
275C23—681no—
276C23D4×D1711yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—22no—
280D4—21no—
281D4—21no—
282D4—22no—
283D4—42no—
284D4—42no—
285D4—684no—
286D4—682no—
287D4—684no—
288D4—682no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—171no—
292C2×C4—171no—
293C2×C4—342no—
294C2×C4—341no—
295C2×C4—341no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—342no—
299C2×C4—342no—
300C2×C4—341no—
301C2×C4—341no—
302C2×C4—342no—
303C2×C4—682no—
304C2×C4D4×D1711yescharacteristic
305C2×C4#6325412yes—
306C2×C4C2×C22⋊D1712yes—
307C2×C4C23×D1711yescharacteristic
308C8—342no—
309C8C2×C22⋊D1714yescomplemented
310C22—21no—
311C22—21no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—84no—
319C22—172no—
320C22—341no—
321C22—341no—
322C22—341no—
323C22—341no—
324C22—341no—
325C22—684no—
326C22#1058762412yes—
327C22C2×D4×D1711yescharacteristic
328C4—41no—
329C4—341no—
330C4—341no—
331C4—682no—
332C4—681no—
333C4—681no—
334C4#1058762412yes—
335C4C2×#6325411yescharacteristic
336C4C22×C22⋊D1711yescharacteristic
337C2—22no—
338C2—82no—
339C2—81no—
340C2—342no—
341C2—681no—
342C2C2×#1058762411yescharacteristic
343C2AGL(1,ℤ/8)×D1711yescharacteristic
344C2#1176360511yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC2×C17×AGL(1,ℤ/8)C2
socleC22×C17C22×D4
hypercenterC2×AGL(1,ℤ/8)D17
hypocenterC17#3055
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×M16×D17C2semidirect
2. #11763316C2semidirect
3. C2×#10587624C2semidirect
4. #11763059C2semidirect
5. #11763052C2semidirect
6. #11763042C2semidirect
7. #11763035C2semidirect
8. #11762851C2semidirect
9. #11762814C2semidirect
10. #11762810C2semidirect
11. C2×C17×AGL(1,ℤ/8)C2semidirect
17. C2×SD16×C17C22semidirect
18. C2×D8×C17C22semidirect
21. #10587632C22semidirect
22. #10587632C22semidirect
27. #10587624C22semidirect
28. #10587624C22semidirect
37. C2×#63218C22semidirect
38. C2×C8×D17C22semidirect
39. #10587506C22semidirect
40. #10587503C22semidirect
41. #10587503C22semidirect
42. #10587503C22semidirect
43. #10587489C22semidirect
44. #10587488C22semidirect
73. M16×C17D4semidirect
135. C8×C17C2×D4semidirect
155. C2×D17AGL(1,ℤ/8)semidirect
165. C2×AGL(1,ℤ/8)D17semidirect
189. C2×SD16C2×D17semidirect
190. C2×D8C2×D17semidirect
217. C17#3055semidirect
248. M16C22⋊D17semidirect
309. C8C2×C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D172
#117633162
C2×#105876242
#117630592
#117630522
#117630422
#117630352
#117628512
#117628142
#117628102
C2×C17×AGL(1,ℤ/8)2
#305517