Finite Group Databasebeta

← #14540868

Subgroups — #14540868

Order 2176.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763514C211yesmaximal, characteristic, complemented
2#11763456C211yesmaximal, characteristic
3#11763167C211yesmaximal, characteristic, complemented
4#11763146C211yesmaximal, characteristic, complemented
5#11763088C211yesmaximal, characteristic, complemented
6#11763053C211yesmaximal, characteristic, complemented
7#11763051C211yesmaximal, characteristic, complemented
8#11763030C211yesmaximal, characteristic, complemented
9#11762973C211yesmaximal, characteristic, complemented
10#11762961C211yesmaximal, characteristic, complemented
11#11762908C211yesmaximal, characteristic, complemented
12#11762883C211yesmaximal, characteristic, complemented
13#11762808C211yesmaximal, characteristic, complemented
14#102×D17C211yesmaximal, characteristic, complemented
15C17×#480C211yesmaximal, characteristic, complemented
16C2×D4×D17C2211yescharacteristic
17C2×#63252C2211yescharacteristic
18C17×#121C2211yescharacteristic
19C2×C4⋊C4×C17C2211yescharacteristic, complemented
20#10587624—21no—
21#10587623—21no—
22#10587622C2211yescharacteristic
23C2×#63229C2211yescharacteristic
24C4×C22⋊D17—21no—
25C4×C22⋊D17C2211yescharacteristic
26#10587610—21no—
27#10587610C2211yescharacteristic, complemented
28C2×#63223C2211yescharacteristic, complemented
29C2×Dic68C2211yescharacteristic, complemented
30C2×#63219C2211yescharacteristic
31C2×C8×D17C2211yescharacteristic
32#10587583C2211yescharacteristic
33#10587583C2211yescharacteristic
34#10587582—21no—
35#10587579C2211yescharacteristic
36C4⋊C4×D17C2211yescharacteristic
37#10587575—21no—
38#10587574C2211yescharacteristic, complemented
39#10587573C2211yescharacteristic, complemented
40#10587569—21no—
41C22⋊C4×D17—21no—
42C17×#107C2211yescharacteristic, complemented
43C17×#107C2211yescharacteristic, complemented
44C17×#102C2211yescharacteristic
45C17×#102C2211yescharacteristic
46C17×C22⋊C8C2211yescharacteristic, complemented
47#10587503C2211yescharacteristic
48#10587492C2211yescharacteristic
49#10587491C2211yescharacteristic
50#10587487C2211yescharacteristic, complemented
51#10587486C2211yescharacteristic, complemented
52#10587485C2211yescharacteristic, complemented
53#10587481C2211yescharacteristic, complemented
54#10587481C2211yescharacteristic, complemented
55#10587481C2211yescharacteristic, complemented
56#10587480C2211yescharacteristic
57#10587480C2211yescharacteristic
58#10587478C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×D4×C17—21no—
61C2×D4×C17C2311yescharacteristic
62C22×C4×C17—21no—
63C22×C4×C17C2311yescharacteristic
64C2×C22⋊D17—21no—
65C2×C22⋊D17—21no—
66C2×C22⋊D17D411yescharacteristic
67D4×D17—22no—
68D4×D17—41no—
69#63252—22no—
70#63252—41no—
71C2×D68C2311yescharacteristic
72C2×C4×D17—21no—
73C2×C4×D17—21no—
74C2×C4×D17—21no—
75C2×C4×D17C2311yescharacteristic
76C2×Dic34—21no—
77C2×Dic34C2311yescharacteristic
78C2×Dic34C2311yescharacteristic
79C2×C8×C17C2311yescharacteristic
80C2×C8×C17C2311yescharacteristic
81C4⋊C4×C17—21no—
82C4⋊C4×C17C2311yescharacteristic
83C4⋊C4×C17C2311yescharacteristic
84C4⋊C4×C17C2311yescharacteristic
85C22⋊C4×C17—21no—
86C22⋊Dic17—21no—
87C22⋊Dic17—21no—
88C22⋊Dic17—21no—
89C22⋊Dic17D411yescharacteristic
90#63229—41no—
91#63227—21no—
92#63227—21no—
93#63227—21no—
94#63227D411yescharacteristic
95C4⋊Dic17—21no—
96C4⋊Dic17—21no—
97C4⋊Dic17C2311yescharacteristic
98#63225—21no—
99#63225—21no—
100#63225—21no—
101#63225—21no—
102#63225D411yescharacteristic
103C4×Dic17—21no—
104C4×Dic17C2311yescharacteristic
105#63223—41no—
106C2×#3622C2311yescharacteristic
107C2×#3622C2311yescharacteristic
108Dic68—41no—
109#63219—22no—
110C8×D17—41no—
111C23×C17—21no—
112C23×C17C2×D411yescharacteristic
113C22×D17—21no—
114C22×D17—42no—
115C22×D17—42no—
116C22×D17—81no—
117C22×D17C2×D411yescharacteristic
118D4×C17—21no—
119D4×C17—21no—
120D4×C17—81no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—81no—
127C2×C4×C17C2×D411yescharacteristic
128C2×C4×C17C2411yescharacteristic
129C22⋊D17—22no—
130C22⋊D17—42no—
131C22⋊D17—42no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17C2×D411yescharacteristic
136C2×Dic17C2×D411yescharacteristic
137C2×Dic17C2×D411yescharacteristic
138D68—21no—
139D68#4112yes—
140C4×D17—21no—
141C4×D17—21no—
142C4×D17—42no—
143C4×D17—81no—
144C4×D17—81no—
145Dic34—21no—
146Dic34—21no—
147Dic34—21no—
148Dic34—21no—
149Dic34—81no—
150C8×C17—21no—
151C8×C17—21no—
152#3622—21no—
153#3622—21no—
154#3314—171nomaximal, Sylow
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17C22×D411yescharacteristic
162C2×D17—21no—
163C2×D17—21no—
164C2×D17—22no—
165C2×D17—41no—
166C2×D17—41no—
167C2×D17—84no—
168C2×D17#13712yes—
169C4×C17—21no—
170C4×C17—41no—
171C4×C17—41no—
172C4×C17—41no—
173C4×C17C2×#4111yescharacteristic
174C4×C17H(5,𝔽2)11yescharacteristic
175Dic17—21no—
176Dic17—21no—
177Dic17—21no—
178Dic17—41no—
179Dic17—41no—
180Dic17—41no—
181#554—171no—
182#546—171no—
183#489—171no—
184#484—171no—
185#480D1711yescharacteristic, complemented
186#474—171no—
187#468—171no—
188#467—171no—
189#464—171no—
190#452—171no—
191#450—171no—
192#440—171no—
193#435—171no—
194#417—171no—
195C2×#102—171no—
196C2×C17—41no—
197C2×C17—81no—
198C2×C17#54611yescharacteristic
199C2×C17C2×#13711yescharacteristic
200C2×C17#57611yescharacteristic
201D17—22no—
202D17—42no—
203C2×#41—171no—
204C22×D4—171no—
205C2×Q16—171no—
206C2×SD16—171no—
207C2×SD16—171no—
208C2×M16—171no—
209C22×C8—171no—
210C4⋊Q8—171no—
211#125—341no—
212#124—171no—
213#123—341no—
214C22⋊Q8—171no—
215C22⋊Q8—171no—
216C22⋊Q8—171no—
217C22⋊Q8—341no—
218C22⋊Q8—341no—
219#121—341no—
220#121C2×D1711yescharacteristic
221#120—341no—
222C4×Q8—171no—
223C4×D4—171no—
224C4×D4—171no—
225C4×D4—341no—
226C2×C4⋊C4—171no—
227C2×C4⋊C4C2×D1711yescharacteristic, complemented
228C2×C22⋊C4—341no—
229#107—171no—
230#107C2×D1711yescharacteristic, complemented
231#107C2×D1711yescharacteristic, complemented
232C4⋊C8—171no—
233#103—171no—
234#103—171no—
235#103—171no—
236#103—171no—
237#102—171no—
238#102—171no—
239#102—171no—
240#102—171no—
241#102C2×D1711yescharacteristic
242#102C2×D1711yescharacteristic
243C22⋊C8—171no—
244C22⋊C8C2×D1711yescharacteristic, complemented
245#97—171no—
246C17#331411yescharacteristic, complemented, Sylow
247C24—341no—
248#41—342no—
249#41—681no—
250C2×Q8—171no—
251C2×Q8—171no—
252C2×Q8—341no—
253C2×D4—21no—
254C2×D4—171no—
255C2×D4—171no—
256C2×D4—341no—
257C2×D4—342no—
258C2×D4—341no—
259C2×D4—681no—
260C2×D4C22×D1711yescharacteristic
261C22×C4—21no—
262C22×C4—171no—
263C22×C4—341no—
264C22×C4—341no—
265C22×C4—341no—
266C22×C4C22×D1711yescharacteristic
267Q16—681no—
268SD16—342no—
269SD16—681no—
270M16—681no—
271C2×C8—171no—
272C2×C8—171no—
273C2×C8—681no—
274C2×C8C22×D1711yescharacteristic
275C2×C8C22×D1711yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—171no—
278C4⋊C4—171no—
279C4⋊C4—341no—
280C4⋊C4—341no—
281C4⋊C4—341no—
282C4⋊C4—341no—
283C4⋊C4—341no—
284C4⋊C4—341no—
285C4⋊C4C22×D1711yescharacteristic
286C4⋊C4C22×D1711yescharacteristic
287C4⋊C4C22×D1711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—171no—
290C22⋊C4—171no—
291C22⋊C4—341no—
292C22⋊C4—341no—
293C22⋊C4—341no—
294C22⋊C4—341no—
295C22⋊C4—341no—
296C22⋊C4—341no—
297C42—171no—
298C42—341no—
299C23—21no—
300C23—171no—
301C23—341no—
302C23—682no—
303C23—682no—
304C23—1361no—
305C23D4×D1711yescharacteristic
306Q8—341no—
307Q8—341no—
308Q8—341no—
309Q8—341no—
310Q8—1361no—
311D4—21no—
312D4—21no—
313D4—81no—
314D4—172no—
315D4—342no—
316D4—341no—
317D4—682no—
318D4—682no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—81no—
325C2×C4—171no—
326C2×C4—171no—
327C2×C4—171no—
328C2×C4—341no—
329C2×C4—341no—
330C2×C4—341no—
331C2×C4—341no—
332C2×C4—341no—
333C2×C4—682no—
334C2×C4—1361no—
335C2×C4—1361no—
336C2×C4D4×D1711yescharacteristic
337C2×C4C23×D1711yescharacteristic
338C8—21no—
339C8—21no—
340C8—341no—
341C8—341no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—172no—
349C22—341no—
350C22—341no—
351C22—342no—
352C22—681no—
353C22—681no—
354C22—1364no—
355C22C2×D4×D1711yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—341no—
361C4—341no—
362C4—341no—
363C4—681no—
364C4—681no—
365C4—681no—
366C4#41×D1711yescharacteristic
367C4#1058769611yescharacteristic
368C2—41no—
369C2—81no—
370C2—342no—
371C2—682no—
372C2#1176345111yescharacteristic
373C2#137×D1711yescharacteristic
374C2#1176361011yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#480C2
socleC22×C17C22×D4
prime radical#11763456C2
hypercenter#480D17
hypocenterC17#3314
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #11763514C2semidirect
3. #11763167C2semidirect
4. #11763146C2semidirect
5. #11763088C2semidirect
6. #11763053C2semidirect
7. #11763051C2semidirect
8. #11763030C2semidirect
9. #11762973C2semidirect
10. #11762961C2semidirect
11. #11762908C2semidirect
12. #11762883C2semidirect
13. #11762808C2semidirect
14. #102×D17C2semidirect
15. C17×#480C2semidirect
19. C2×C4⋊C4×C17C22semidirect
27. #10587610C22semidirect
28. C2×#63223C22semidirect
29. C2×Dic68C22semidirect
38. #10587574C22semidirect
39. #10587573C22semidirect
42. C17×#107C22semidirect
43. C17×#107C22semidirect
46. C17×C22⋊C8C22semidirect
50. #10587487C22semidirect
51. #10587486C22semidirect
52. #10587485C22semidirect
53. #10587481C22semidirect
54. #10587481C22semidirect
55. #10587481C22semidirect
58. #10587478C22semidirect
185. #480D17semidirect
227. C2×C4⋊C4C2×D17semidirect
230. #107C2×D17semidirect
231. #107C2×D17semidirect
244. C22⋊C8C2×D17semidirect
246. C17#3314semidirect

Maximal subgroups

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

subgroupindex
#117635142
#117634562
#117631672
#117631462
#117630882
#117630532
#117630512
#117630302
#117629732
#117629612
#117629082
#117628832
#117628082
#102×D172
C17×#4802
#331417