Finite Group Databasebeta

← #14532135

Subgroups — #14532135

Order 2176.

1232 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 171 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, complemented
2C2×#10587570C212yesmaximal, complemented
3C2×#10587568C212yesmaximal, complemented
4C2×#10587579C211yesmaximal, characteristic, complemented
5#11762726C211yesmaximal, characteristic, complemented
6#11762719C211yesmaximal, characteristic, complemented
7#11762717C211yesmaximal, characteristic, complemented
8#11762709C212yesmaximal, complemented
9#11762668C211yesmaximal, characteristic, complemented
10#11762664C212yesmaximal, complemented
11C17×#394C211yesmaximal, characteristic, complemented
12C24×D17C2212yes—
13C22×D4×C17C2211yescharacteristic
14C22×C22⋊D17C2212yes—
15C22×C22⋊D17C2212yescomplemented
16C2×D4×D17—28no—
17C2×D4×D17—24no—
18C22×D68C2211yescharacteristic
19C22×D68C2211yescharacteristic
20C22×D68C2212yescomplemented
21C22×C4×D17C2211yescharacteristic
22C22×C4×D17C2211yescharacteristic, complemented
23C2×C4⋊C4×C17C2211yescharacteristic, complemented
24C2×C22⋊C4×C17C2212yescomplemented
25C2×C22⋊C4×C17C2212yescomplemented
26C2×C22⋊Dic17C2212yescomplemented
27C2×#63227C2211yescharacteristic
28C2×#63227C2212yes—
29C2×#63227C2212yescomplemented
30C2×#63227C2211yescharacteristic, complemented
31C2×#63227C2212yescomplemented
32C2×#63227C2212yescomplemented
33C2×#63225C2212yescomplemented
34C2×C4×Dic17C2211yescharacteristic, complemented
35#10587579—24no—
36#10587570—28no—
37#10587568—28no—
38C17×#95C2211yescharacteristic, complemented
39#10587502C2211yescharacteristic, complemented
40C24×C17C2312yes—
41C23×D17—24no—
42C23×D17—24no—
43C23×D17—24no—
44C23×D17—24no—
45C23×D17—24no—
46C23×D17—42no—
47C23×D17—42no—
48C23×D17C2312yes—
49C23×D17C2311yescharacteristic
50C23×D17C2311yescharacteristic
51C2×D4×C17—24no—
52C2×D4×C17—22no—
53C22×C4×C17C2311yescharacteristic
54C22×C4×C17C2312yes—
55C22×C4×C17C2311yescharacteristic
56C22×C4×C17C2311yescharacteristic
57C2×C22⋊D17—24no—
58C2×C22⋊D17—24no—
59C2×C22⋊D17—24no—
60C2×C22⋊D17—24no—
61C2×C22⋊D17—42no—
62C2×C22⋊D17D418yescomplemented
63C22×Dic17C2312yes—
64C22×Dic17C2311yescharacteristic
65C22×Dic17C2311yescharacteristic
66D4×D17—232no—
67C2×D68—24no—
68C2×D68—22no—
69C2×D68—22no—
70C2×D68—22no—
71C2×D68—24no—
72C2×D68—24no—
73C2×D68—42no—
74C2×D68D414yescomplemented
75C2×C4×D17—24no—
76C2×C4×D17—24no—
77C2×C4×D17—22no—
78C2×C4×D17D414yescomplemented
79C4⋊C4×C17—22no—
80C22⋊C4×C17—24no—
81C22⋊C4×C17—24no—
82C22⋊Dic17—42no—
83#63227—24no—
84#63227—24no—
85#63227—22no—
86#63227—22no—
87#63227—24no—
88#63227—42no—
89#63225—24no—
90C4×Dic17—22no—
91C23×C17—22no—
92C23×C17—22no—
93C23×C17—22no—
94C23×C17—22no—
95C23×C17—22no—
96C23×C17—22no—
97C23×C17C2×D414yes—
98C23×C17C2411yescharacteristic
99C22×D17—22no—
100C22×D17—22no—
101C22×D17—216no—
102C22×D17—22no—
103C22×D17—21no—
104C22×D17—22no—
105C22×D17—22no—
106C22×D17—22no—
107C22×D17—21no—
108C22×D17—22no—
109C22×D17—22no—
110C22×D17—22no—
111C22×D17—22no—
112C22×D17—48no—
113C22×D17—48no—
114C22×D17—48no—
115C22×D17—48no—
116C22×D17—48no—
117C22×D17—48no—
118C22×D17C2×D414yes—
119C22×D17C2×D412yes—
120C22×D17C2×D412yes—
121C22×D17C2×D412yescomplemented
122C22×D17#4112yes—
123D4×C17—28no—
124C2×C4×C17—21no—
125C2×C4×C17—22no—
126C2×C4×C17—21no—
127C2×C4×C17—22no—
128C2×C4×C17—22no—
129C2×C4×C17—22no—
130C2×C4×C17—21no—
131C2×C4×C17—22no—
132C2×C4×C17C2×D412yes—
133C2×C4×C17C2×D412yescomplemented
134C22⋊D17—216no—
135C22⋊D17—48no—
136C2×Dic17—22no—
137C2×Dic17—22no—
138C2×Dic17—21no—
139C2×Dic17—22no—
140C2×Dic17—22no—
141C2×Dic17C2×D414yes—
142C2×Dic17C2×D412yes—
143D68—28no—
144D68—28no—
145D68—48no—
146C4×D17—28no—
147C4×D17—44no—
148#2408—171nomaximal, Sylow
149C22×C17—24no—
150C22×C17—24no—
151C22×C17—24no—
152C22×C17—42no—
153C22×C17—42no—
154C22×C17—42no—
155C22×C17—42no—
156C22×C17C22×D411yescharacteristic
157C22×C17C22×D412yes—
158C22×C17C22×D411yescharacteristic
159C22×C17C2×#4111yescharacteristic
160C22×C17H(5,𝔽2)12yes—
161C2×D17—22no—
162C2×D17—22no—
163C2×D17—24no—
164C2×D17—24no—
165C2×D17—24no—
166C2×D17—22no—
167C2×D17—24no—
168C2×D17—24no—
169C2×D17—24no—
170C2×D17—432no—
171C2×D17—41no—
172C2×D17—42no—
173C2×D17—41no—
174C2×D17—42no—
175C2×D17—42no—
176C2×D17—42no—
177C2×D17—42no—
178C2×D17#12114yescomplemented
179C4×C17—22no—
180C4×C17—22no—
181C4×C17—41no—
182C4×C17—42no—
183Dic17—41no—
184Dic17—42no—
185Dic17#12014yescomplemented
186C23×D4—171no—
187C2×#121—172no—
188C2×#120—172no—
189C2×C4×D4—171no—
190#394—171no—
191#394D1711yescharacteristic, complemented
192#392—171no—
193#392—171no—
194#390—172no—
195#380—171no—
196#379—172no—
197C2×C17—44no—
198C2×C17C2×#12011yescharacteristic
199C2×C17C2×#12111yescharacteristic
200C2×C17#53511yescharacteristic
201C2×C17D4212yes—
202C2×C17#54612yes—
203D17—24no—
204D17—42no—
205D17—44no—
206C25—172no—
207C22×D4—172no—
208C22×D4—172no—
209C22×D4—171no—
210C22×D4—171no—
211C22×D4—172no—
212C22×D4—344no—
213C22×D4—348no—
214C22×D4C2×D1711yescharacteristic
215C23×C4—171no—
216C23×C4—171no—
217#121—348no—
218#120—348no—
219C4×D4—344no—
220C2×C4⋊C4—172no—
221C2×C4⋊C4C2×D1711yescharacteristic, complemented
222C2×C22⋊C4—171no—
223C2×C22⋊C4—171no—
224C2×C22⋊C4—172no—
225C2×C22⋊C4—172no—
226C2×C22⋊C4—172no—
227C2×C22⋊C4—172no—
228C2×C22⋊C4—172no—
229C2×C22⋊C4C2×D1712yescomplemented
230C2×C22⋊C4C2×D1712yescomplemented
231C2×C42—171no—
232#95—171no—
233#95C2×D1711yescharacteristic, complemented
234C17#240811yescharacteristic, complemented, Sylow
235C24—171no—
236C24—172no—
237C24—171no—
238C24—344no—
239C24—344no—
240C24—344no—
241C24—344no—
242C24—344no—
243C24—682no—
244C24—682no—
245C24C22×D1712yes—
246C2×D4—24no—
247C2×D4—22no—
248C2×D4—178no—
249C2×D4—174no—
250C2×D4—342no—
251C2×D4—342no—
252C2×D4—344no—
253C2×D4—344no—
254C2×D4—344no—
255C2×D4—344no—
256C2×D4—344no—
257C2×D4—3432no—
258C2×D4—344no—
259C2×D4—344no—
260C2×D4—342no—
261C2×D4—682no—
262C2×D4—682no—
263C22×C4—171no—
264C22×C4—174no—
265C22×C4—172no—
266C22×C4—171no—
267C22×C4—344no—
268C22×C4—342no—
269C22×C4—344no—
270C22×C4C22×D1712yes—
271C22×C4C22×D1711yescharacteristic
272C22×C4C22×D1711yescharacteristic
273C22×C4C22×D1711yescharacteristic
274C4⋊C4—22no—
275C4⋊C4—344no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—342no—
279C22⋊C4—344no—
280C22⋊C4—344no—
281C22⋊C4—342no—
282C22⋊C4—344no—
283C22⋊C4—682no—
284C22⋊C4—682no—
285C42—342no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—172no—
293C23—172no—
294C23—172no—
295C23—174no—
296C23—172no—
297C23—3416no—
298C23—342no—
299C23—342no—
300C23—341no—
301C23—342no—
302C23—342no—
303C23—342no—
304C23—342no—
305C23—341no—
306C23—342no—
307C23—342no—
308C23—342no—
309C23—342no—
310C23—688no—
311C23—688no—
312C23—688no—
313C23—688no—
314C23—688no—
315C23—688no—
316C23D4×D1714yes—
317C23C23×D1711yescharacteristic
318D4—28no—
319D4—348no—
320D4—348no—
321D4—3416no—
322D4—688no—
323D4—688no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—172no—
333C2×C4—174no—
334C2×C4—348no—
335C2×C4—341no—
336C2×C4—342no—
337C2×C4—342no—
338C2×C4—342no—
339C2×C4—342no—
340C2×C4—684no—
341C2×C4D4×D1712yes—
342C2×C4D4×D1712yescomplemented
343C22—24no—
344C22—24no—
345C22—24no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—174no—
351C22—342no—
352C22—342no—
353C22—344no—
354C22—344no—
355C22—344no—
356C22—342no—
357C22—344no—
358C22—344no—
359C22—344no—
360C22—6832no—
361C22—681no—
362C22—682no—
363C22—682no—
364C22—681no—
365C22—682no—
366C22—682no—
367C22—682no—
368C22C2×D4×D1711yescharacteristic
369C22C2×D4×D1712yes—
370C22C2×D4×D1711yescharacteristic
371C22#41×D1711yescharacteristic
372C22#1058769612yes—
373C4—22no—
374C4—22no—
375C4—41no—
376C4—42no—
377C4—174no—
378C4—681no—
379C4—682no—
380C2—44no—
381C2—344no—
382C2—684no—
383C2—682no—
384C2#120×D1711yescharacteristic
385C2#121×D1711yescharacteristic
386C2#1176337711yescharacteristic
387C2#1176344312yes—
388C2#1176345112yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
hypercenter#394D17
hypocenterC17#2408
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×D4×D17C2semidirect
2. C2×#10587570C2semidirect
3. C2×#10587568C2semidirect
4. C2×#10587579C2semidirect
5. #11762726C2semidirect
6. #11762719C2semidirect
7. #11762717C2semidirect
8. #11762709C2semidirect
9. #11762668C2semidirect
10. #11762664C2semidirect
11. C17×#394C2semidirect
15. C22×C22⋊D17C22semidirect
20. C22×D68C22semidirect
22. C22×C4×D17C22semidirect
23. C2×C4⋊C4×C17C22semidirect
24. C2×C22⋊C4×C17C22semidirect
25. C2×C22⋊C4×C17C22semidirect
26. C2×C22⋊Dic17C22semidirect
29. C2×#63227C22semidirect
30. C2×#63227C22semidirect
31. C2×#63227C22semidirect
32. C2×#63227C22semidirect
33. C2×#63225C22semidirect
34. C2×C4×Dic17C22semidirect
38. C17×#95C22semidirect
39. #10587502C22semidirect
62. C2×C22⋊D17D4semidirect
74. C2×D68D4semidirect
78. C2×C4×D17D4semidirect
121. C22×D17C2×D4semidirect
133. C2×C4×C17C2×D4semidirect
178. C2×D17#121semidirect
185. Dic17#120semidirect
191. #394D17semidirect
221. C2×C4⋊C4C2×D17semidirect
229. C2×C22⋊C4C2×D17semidirect
230. C2×C22⋊C4C2×D17semidirect
233. #95C2×D17semidirect
234. C17#2408semidirect
342. C2×C4D4×D17semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D172
C2×#105875702
C2×#105875682
C2×#105875792
#117627262
#117627192
#117627172
#117627092
#117626682
#117626642
C17×#3942
#240817