Finite Group Databasebeta

← #14532430

Subgroups — #14532430

Order 2176.

1088 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C22×D4×D17C211yesmaximal, characteristic
2C2×#10587571C212yesmaximal, complemented
3C2×#10587582C211yesmaximal, characteristic, complemented
4#11762727C211yesmaximal, characteristic, complemented
5#11762726C211yesmaximal, characteristic, complemented
6#11762723C211yesmaximal, characteristic, complemented
7#11762709C212yesmaximal, complemented
8#11762664C212yesmaximal, complemented
9#11762664C212yesmaximal, complemented
10#11762656C211yesmaximal, characteristic, complemented
11C17×#394C211yesmaximal, characteristic, complemented
12C24×D17C2212yes—
13C22×D4×C17C2211yescharacteristic
14C22×C22⋊D17C2212yes—
15C2×D4×D17—28no—
16C2×D4×D17—24no—
17C22×D68C2211yescharacteristic
18C22×C4×D17C2211yescharacteristic
19C22×Dic34C2211yescharacteristic, complemented
20C2×C4⋊C4×C17C2211yescharacteristic, complemented
21C2×C22⋊C4×C17C2212yescomplemented
22C2×C22⋊C4×C17C2212yescomplemented
23C2×C22⋊Dic17C2212yescomplemented
24C2×C22⋊Dic17C2212yescomplemented
25C2×#63227C2211yescharacteristic
26C2×#63227C2211yescharacteristic
27C2×#63227C2212yes—
28C2×#63227C2212yescomplemented
29C2×#63227C2212yescomplemented
30C2×#63227C2212yescomplemented
31C2×#63227C2212yescomplemented
32C2×#63225C2211yescharacteristic, complemented
33C2×#63225C2211yescharacteristic, complemented
34C2×C4×Dic17C2212yescomplemented
35#10587582—24no—
36#10587571—28no—
37C17×#95C2211yescharacteristic, complemented
38#10587502C2211yescharacteristic, complemented
39C24×C17C2312yes—
40C23×D17—24no—
41C23×D17—24no—
42C23×D17—24no—
43C23×D17—24no—
44C23×D17—24no—
45C23×D17—42no—
46C23×D17—42no—
47C23×D17C2312yes—
48C23×D17C2311yescharacteristic
49C2×D4×C17—24no—
50C2×D4×C17—22no—
51C22×C4×C17C2311yescharacteristic
52C22×C4×C17C2312yes—
53C22×C4×C17C2311yescharacteristic
54C22×C4×C17C2311yescharacteristic
55C2×C22⋊D17—24no—
56C2×C22⋊D17—24no—
57C2×C22⋊D17D418yes—
58C22×Dic17C2311yescharacteristic
59C22×Dic17C2312yes—
60C22×Dic17C2311yescharacteristic
61C22×Dic17C2311yescharacteristic
62D4×D17—232no—
63C2×D68—24no—
64C2×D68—22no—
65C2×C4×D17—24no—
66C2×C4×D17D414yes—
67C2×Dic34—24no—
68C2×Dic34—22no—
69C4⋊C4×C17—22no—
70C22⋊C4×C17—24no—
71C22⋊C4×C17—42no—
72C22⋊Dic17—24no—
73C22⋊Dic17—42no—
74#63227—24no—
75#63227—24no—
76#63227—22no—
77#63227—22no—
78#63227—24no—
79#63227—42no—
80#63227—42no—
81#63225—22no—
82#63225—22no—
83C4×Dic17—24no—
84C23×C17—22no—
85C23×C17—22no—
86C23×C17—22no—
87C23×C17—22no—
88C23×C17—22no—
89C23×C17—22no—
90C23×C17C2×D414yes—
91C23×C17C2411yescharacteristic
92C22×D17—22no—
93C22×D17—21no—
94C22×D17—21no—
95C22×D17—216no—
96C22×D17—22no—
97C22×D17—22no—
98C22×D17—22no—
99C22×D17—22no—
100C22×D17—22no—
101C22×D17—22no—
102C22×D17—48no—
103C22×D17—48no—
104C22×D17—48no—
105C22×D17—48no—
106C22×D17—48no—
107C22×D17—48no—
108C22×D17C2×D414yes—
109C22×D17C2×D412yes—
110C22×D17#4114yes—
111D4×C17—28no—
112C2×C4×C17—22no—
113C2×C4×C17—22no—
114C2×C4×C17—22no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—22no—
118C2×C4×C17—21no—
119C2×C4×C17—22no—
120C2×C4×C17—21no—
121C2×C4×C17C2×D412yes—
122C22⋊D17—216no—
123C2×Dic17—22no—
124C2×Dic17—22no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—22no—
128C2×Dic17—21no—
129C2×Dic17—22no—
130C2×Dic17—21no—
131C2×Dic17C2×D414yes—
132C2×Dic17C2×D412yes—
133D68—28no—
134C4×D17—28no—
135Dic34—44no—
136#2435—171nomaximal, Sylow
137C22×C17—24no—
138C22×C17—24no—
139C22×C17—24no—
140C22×C17—42no—
141C22×C17—42no—
142C22×C17—42no—
143C22×C17—42no—
144C22×C17C22×D411yescharacteristic
145C22×C17C22×D412yes—
146C22×C17C2×#4112yes—
147C22×C17H(5,𝔽2)11yescharacteristic
148C22×C17H(5,𝔽2)11yescharacteristic
149C2×D17—22no—
150C2×D17—22no—
151C2×D17—24no—
152C2×D17—24no—
153C2×D17—24no—
154C2×D17—24no—
155C2×D17—24no—
156C2×D17—432no—
157C2×D17—42no—
158C2×D17—42no—
159C2×D17—42no—
160C2×D17—42no—
161C2×D17#12414yescomplemented
162C4×C17—22no—
163C4×C17—41no—
164C4×C17—41no—
165C4×C17—42no—
166Dic17—41no—
167Dic17—41no—
168Dic17—42no—
169Dic17#12014yes—
170C23×D4—171no—
171C2×#124—172no—
172C2×C22⋊Q8—171no—
173#394—171no—
174#394—171no—
175#394—171no—
176#394D1711yescharacteristic, complemented
177#390—172no—
178#379—172no—
179#379—172no—
180C4×C22⋊C4—171no—
181C2×C17—44no—
182C2×C17C2×#12011yescharacteristic
183C2×C17C2×#12411yescharacteristic
184C2×C17#53511yescharacteristic
185C2×C17#54612yes—
186C2×C17#54612yes—
187D17—24no—
188D17—44no—
189C25—172no—
190C22×Q8—171no—
191C22×D4—172no—
192C22×D4—171no—
193C22×D4—344no—
194C22×D4—348no—
195C22×D4C2×D1711yescharacteristic
196C23×C4—171no—
197#124—348no—
198C22⋊Q8—344no—
199C2×C4⋊C4—171no—
200C2×C4⋊C4—171no—
201C2×C4⋊C4C2×D1711yescharacteristic, complemented
202C2×C22⋊C4—172no—
203C2×C22⋊C4—171no—
204C2×C22⋊C4—171no—
205C2×C22⋊C4—172no—
206C2×C22⋊C4—172no—
207C2×C22⋊C4—172no—
208C2×C22⋊C4—172no—
209C2×C22⋊C4—172no—
210C2×C22⋊C4—172no—
211C2×C22⋊C4C2×D1712yescomplemented
212C2×C22⋊C4C2×D1712yescomplemented
213C2×C42—172no—
214#95—171no—
215#95C2×D1711yescharacteristic, complemented
216C17#243511yescharacteristic, complemented, Sylow
217C24—171no—
218C24—172no—
219C24—344no—
220C24—344no—
221C24—344no—
222C24—344no—
223C24—344no—
224C24—682no—
225C24—682no—
226C24C22×D1712yes—
227C2×Q8—342no—
228C2×Q8—344no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—178no—
232C2×D4—344no—
233C2×D4—344no—
234C2×D4—344no—
235C2×D4—3432no—
236C2×D4—342no—
237C22×C4—171no—
238C22×C4—174no—
239C22×C4—172no—
240C22×C4—171no—
241C22×C4—171no—
242C22×C4—344no—
243C22×C4C22×D1712yes—
244C22×C4C22×D1711yescharacteristic
245C22×C4C22×D1711yescharacteristic
246C22×C4C22×D1711yescharacteristic
247C4⋊C4—22no—
248C4⋊C4—342no—
249C4⋊C4—342no—
250C22⋊C4—24no—
251C22⋊C4—42no—
252C22⋊C4—342no—
253C22⋊C4—342no—
254C22⋊C4—344no—
255C22⋊C4—344no—
256C22⋊C4—344no—
257C22⋊C4—344no—
258C22⋊C4—682no—
259C22⋊C4—682no—
260C22⋊C4—682no—
261C42—344no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—172no—
269C23—174no—
270C23—174no—
271C23—3416no—
272C23—342no—
273C23—341no—
274C23—341no—
275C23—342no—
276C23—342no—
277C23—342no—
278C23—342no—
279C23—342no—
280C23—342no—
281C23—688no—
282C23—688no—
283C23—688no—
284C23—688no—
285C23—688no—
286C23—688no—
287C23D4×D1714yes—
288C23C23×D1711yescharacteristic
289Q8—684no—
290D4—28no—
291D4—348no—
292D4—3416no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—174no—
303C2×C4—172no—
304C2×C4—348no—
305C2×C4—342no—
306C2×C4—342no—
307C2×C4—341no—
308C2×C4—341no—
309C2×C4—341no—
310C2×C4—342no—
311C2×C4—341no—
312C2×C4—342no—
313C2×C4D4×D1712yes—
314C22—24no—
315C22—24no—
316C22—24no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—174no—
322C22—342no—
323C22—342no—
324C22—344no—
325C22—344no—
326C22—344no—
327C22—344no—
328C22—344no—
329C22—6832no—
330C22—682no—
331C22—682no—
332C22—682no—
333C22—682no—
334C22C2×D4×D1711yescharacteristic
335C22C2×D4×D1712yes—
336C22#1058769111yescharacteristic
337C22#41×D1712yes—
338C22#1058769611yescharacteristic
339C4—22no—
340C4—41no—
341C4—41no—
342C4—42no—
343C4—174no—
344C4—681no—
345C4—681no—
346C4—682no—
347C2—44no—
348C2—344no—
349C2—684no—
350C2#120×D1711yescharacteristic
351C2#124×D1711yescharacteristic
352C2#1176337511yescharacteristic
353C2#1176344812yes—
354C2#1176345112yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
prime radicalC22×D4×D17C2
hypercenter#394D17
hypocenterC17#2435
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#10587571C2semidirect
3. C2×#10587582C2semidirect
4. #11762727C2semidirect
5. #11762726C2semidirect
6. #11762723C2semidirect
7. #11762709C2semidirect
8. #11762664C2semidirect
9. #11762664C2semidirect
10. #11762656C2semidirect
11. C17×#394C2semidirect
19. C22×Dic34C22semidirect
20. C2×C4⋊C4×C17C22semidirect
21. C2×C22⋊C4×C17C22semidirect
22. C2×C22⋊C4×C17C22semidirect
23. C2×C22⋊Dic17C22semidirect
24. C2×C22⋊Dic17C22semidirect
28. C2×#63227C22semidirect
29. C2×#63227C22semidirect
30. C2×#63227C22semidirect
31. C2×#63227C22semidirect
32. C2×#63225C22semidirect
33. C2×#63225C22semidirect
34. C2×C4×Dic17C22semidirect
37. C17×#95C22semidirect
38. #10587502C22semidirect
161. C2×D17#124semidirect
176. #394D17semidirect
201. C2×C4⋊C4C2×D17semidirect
211. C2×C22⋊C4C2×D17semidirect
212. C2×C22⋊C4C2×D17semidirect
215. #95C2×D17semidirect
216. C17#2435semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D172
C2×#105875712
C2×#105875822
#117627272
#117627262
#117627232
#117627092
#117626642
#117626562
C17×#3942
#243517