Finite Group Databasebeta

← #14540867

Subgroups — #14540867

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#11763516C211yesmaximal, characteristic, complemented
2#11763455C211yesmaximal, characteristic
3#11763169C211yesmaximal, characteristic, complemented
4#11763145C211yesmaximal, characteristic, complemented
5#11763123C211yesmaximal, characteristic, complemented
6#11763090C211yesmaximal, characteristic, complemented
7#11763055C211yesmaximal, characteristic, complemented
8#11763050C211yesmaximal, characteristic, complemented
9#11763032C211yesmaximal, characteristic, complemented
10#11762973C211yesmaximal, characteristic, complemented
11#11762962C211yesmaximal, characteristic, complemented
12#11762911C211yesmaximal, characteristic, complemented
13#11762811C211yesmaximal, characteristic, complemented
14C2×#10587492C211yesmaximal, characteristic, complemented
15C17×#435C211yesmaximal, characteristic, complemented
16C2×#63252C2211yescharacteristic
17C22×D68C2211yescharacteristic
18C2×M16×C17C2211yescharacteristic, complemented
19C22×C8×C17C2211yescharacteristic
20C4×D4×C17C2211yescharacteristic
21#10587623—21no—
22D4×Dic17—21no—
23#10587616C2211yescharacteristic
24C2×C4⋊Dic17C2211yescharacteristic
25C2×C4⋊Dic17C2211yescharacteristic, complemented
26#10587610C2211yescharacteristic
27#10587610C2211yescharacteristic
28#10587610C2211yescharacteristic, complemented
29C2×Dic68C2211yescharacteristic, complemented
30C2×#63219C2211yescharacteristic
31C2×#63219C2211yescharacteristic
32#10587580—21no—
33#10587576—21no—
34#10587570—21no—
35C22⋊C4×D17—21no—
36#10587564—21no—
37#10587564—21no—
38#10587561C2211yescharacteristic
39C4×D68C2211yescharacteristic
40#10587555C2211yescharacteristic, complemented
41C4×Dic34C2211yescharacteristic, complemented
42C17×C4⋊C8C2211yescharacteristic, complemented
43C17×C22⋊C8C2211yescharacteristic
44C17×C22⋊C8C2211yescharacteristic, complemented
45C17×#97C2211yescharacteristic, complemented
46#10587492C2211yescharacteristic
47#10587492C2211yescharacteristic
48#10587492C2211yescharacteristic
49#10587492C2211yescharacteristic
50#10587492C2211yescharacteristic
51#10587492C2211yescharacteristic
52#10587489C2211yescharacteristic, complemented
53#10587489C2211yescharacteristic, complemented
54#10587489C2211yescharacteristic, complemented
55#10587487C2211yescharacteristic, complemented
56#10587487C2211yescharacteristic, complemented
57#10587487C2211yescharacteristic, complemented
58#10587487C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×D4×C17D411yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C22×C4×C17C2311yescharacteristic
63C2×C22⋊D17—21no—
64C2×C22⋊D17—21no—
65C22×Dic17—21no—
66C22×Dic17—21no—
67#63252—22no—
68#63252—41no—
69C2×D68—22no—
70C2×D68—21no—
71C2×D68—41no—
72C2×D68C2311yescharacteristic
73C2×D68C2311yescharacteristic
74C2×C4×D17—21no—
75C2×C4×D17—21no—
76C2×Dic34—21no—
77C2×Dic34C2311yescharacteristic
78C2×Dic34C2311yescharacteristic
79M16×C17—41no—
80C2×C8×C17—41no—
81C2×C8×C17C2311yescharacteristic
82C2×C8×C17C2311yescharacteristic
83C2×C8×C17C2311yescharacteristic
84C2×C8×C17C2311yescharacteristic
85C4⋊C4×C17D411yescharacteristic
86C22⋊C4×C17D411yescharacteristic
87C22⋊C4×C17D411yescharacteristic
88C42×C17C2311yescharacteristic
89C22⋊Dic17—21no—
90C22⋊Dic17—21no—
91C22⋊Dic17—21no—
92#63227—21no—
93#63227—21no—
94#63227—21no—
95#63227—21no—
96C4⋊Dic17—21no—
97C4⋊Dic17—21no—
98C4⋊Dic17—21no—
99C4⋊Dic17C2311yescharacteristic
100C4⋊Dic17C2311yescharacteristic
101C4⋊Dic17C2311yescharacteristic
102C4⋊Dic17C2311yescharacteristic
103#63225—21no—
104#63225—21no—
105#63225—21no—
106#63225—21no—
107C4×Dic17—21no—
108Dic68—41no—
109#63219—22no—
110#63219—41no—
111C23×C17C2×D411yescharacteristic
112C23×C17C2×D411yescharacteristic
113C22×D17—21no—
114C22×D17—21no—
115C22×D17—42no—
116C22×D17—42no—
117C22×D17—81no—
118D4×C17—22no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17C2×D411yescharacteristic
124C2×C4×C17C2×D411yescharacteristic
125C2×C4×C17C2×D411yescharacteristic
126C2×C4×C17C2×D411yescharacteristic
127C2×C4×C17C2411yescharacteristic
128C22⋊D17—42no—
129C22⋊D17—81no—
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—81no—
137C2×Dic17—81no—
138D68—21no—
139D68—21no—
140D68—21no—
141D68—42no—
142D68#4112yes—
143C4×D17—42no—
144C4×D17—81no—
145Dic34—21no—
146Dic34—21no—
147Dic34—21no—
148Dic34—21no—
149Dic34—81no—
150C8×C17—21no—
151C8×C17—21no—
152C8×C17—21no—
153C8×C17—21no—
154#3314—171nomaximal, Sylow
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—21no—
160C22×C17#13712yes—
161C22×C17C22×D411yescharacteristic
162C2×D17—22no—
163C2×D17—41no—
164C2×D17—41no—
165C2×D17—41no—
166C2×D17—41no—
167C2×D17—41no—
168C2×D17—84no—
169C4×C17—21no—
170C4×C17—21no—
171C4×C17—21no—
172C4×C17—21no—
173C4×C17C2×#4111yescharacteristic
174C4×C17H(5,𝔽2)11yescharacteristic
175Dic17—41no—
176Dic17—41no—
177Dic17—41no—
178Dic17—41no—
179Dic17—41no—
180Dic17—41no—
181#554—171no—
182#546—171no—
183#489—171no—
184#484—171no—
185#480—171no—
186#474—171no—
187#468—171no—
188#467—171no—
189#464—171no—
190#452—171no—
191#450—171no—
192#440—171no—
193#435D1711yescharacteristic, complemented
194#417—171no—
195C2×#102—171no—
196C2×C17—22no—
197C2×C17—41no—
198C2×C17#54611yescharacteristic
199C2×C17C2×#13711yescharacteristic
200C2×C17#57611yescharacteristic
201D17—42no—
202D17—81no—
203C2×#41—171no—
204C22×D4—171no—
205C2×Q16—171no—
206C2×SD16—171no—
207C2×SD16—171no—
208C2×M16C2×D1711yescharacteristic, complemented
209C22×C8C2×D1711yescharacteristic
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—171no—
220#121—341no—
221#120—341no—
222C4×Q8—171no—
223C4×D4—171no—
224C4×D4—341no—
225C4×D4C2×D1711yescharacteristic
226C2×C4⋊C4—171no—
227C2×C4⋊C4—171no—
228C2×C22⋊C4—341no—
229#107—171no—
230#107—171no—
231#107—171no—
232C4⋊C8C2×D1711yescharacteristic, complemented
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#102—171no—
242#102—171no—
243C22⋊C8C2×D1711yescharacteristic
244C22⋊C8C2×D1711yescharacteristic, complemented
245#97C2×D1711yescharacteristic, complemented
246C17#331411yescharacteristic, complemented, Sylow
247C24—341no—
248#41—342no—
249#41—681no—
250C2×Q8—171no—
251C2×Q8—171no—
252C2×Q8—341no—
253C2×D4—171no—
254C2×D4—171no—
255C2×D4—342no—
256C2×D4—341no—
257C2×D4—341no—
258C2×D4—341no—
259C2×D4—681no—
260C2×D4D6811yescharacteristic
261C22×C4—341no—
262C22×C4—341no—
263C22×C4—341no—
264C22×C4—341no—
265C22×C4C22×D1711yescharacteristic
266C22×C4C22×D1711yescharacteristic
267Q16—681no—
268SD16—342no—
269SD16—681no—
270M16—41no—
271C2×C8—41no—
272C2×C8C22×D1711yescharacteristic
273C2×C8C22×D1711yescharacteristic
274C2×C8C22×D1711yescharacteristic
275C2×C8C22×D1711yescharacteristic
276C4⋊C4—171no—
277C4⋊C4—171no—
278C4⋊C4—171no—
279C4⋊C4—171no—
280C4⋊C4—341no—
281C4⋊C4—341no—
282C4⋊C4—341no—
283C4⋊C4—341no—
284C4⋊C4—341no—
285C4⋊C4—341no—
286C4⋊C4—341no—
287C4⋊C4D6811yescharacteristic
288C22⋊C4—341no—
289C22⋊C4—341no—
290C22⋊C4—341no—
291C22⋊C4—341no—
292C22⋊C4—341no—
293C22⋊C4—341no—
294C22⋊C4—341no—
295C22⋊C4D6811yescharacteristic
296C22⋊C4D6811yescharacteristic
297C42—341no—
298C42C22×D1711yescharacteristic
299C23—341no—
300C23—341no—
301C23—682no—
302C23—682no—
303C23—1361no—
304C23C2×D6811yescharacteristic
305C23C2×D6811yescharacteristic
306Q8—341no—
307Q8—341no—
308Q8—341no—
309Q8—341no—
310Q8—1361no—
311D4—22no—
312D4—172no—
313D4—341no—
314D4—341no—
315D4—341no—
316D4—682no—
317D4—682no—
318D4—1361no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—341no—
324C2×C4—341no—
325C2×C4—341no—
326C2×C4—341no—
327C2×C4—341no—
328C2×C4—341no—
329C2×C4—682no—
330C2×C4—1361no—
331C2×C4—1361no—
332C2×C4—1361no—
333C2×C4C2×D6811yescharacteristic
334C2×C4C2×D6811yescharacteristic
335C2×C4C2×D6811yescharacteristic
336C2×C4C2×D6811yescharacteristic
337C2×C4C23×D1711yescharacteristic
338C8—21no—
339C8—21no—
340C8—21no—
341C8—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—342no—
348C22—681no—
349C22—681no—
350C22—681no—
351C22—681no—
352C22—681no—
353C22—1364no—
354C22#1058759512yes—
355C22C22×D6811yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—681no—
361C4—681no—
362C4—681no—
363C4—681no—
364C4—681no—
365C4—681no—
366C4#1058769111yescharacteristic
367C4#41×D1711yescharacteristic
368C2—22no—
369C2—41no—
370C2—682no—
371C2—1361no—
372C2#1176345411yescharacteristic
373C2C2×#1058759511yescharacteristic
374C2#1176361411yescharacteristic
subgroupG/H
centerC22C22×D68
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#435C2
socleC22×C17C22×D4
prime radical#11763455C2
hypercenter#435D17
hypocenterC17#3314
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #11763516C2semidirect
3. #11763169C2semidirect
4. #11763145C2semidirect
5. #11763123C2semidirect
6. #11763090C2semidirect
7. #11763055C2semidirect
8. #11763050C2semidirect
9. #11763032C2semidirect
10. #11762973C2semidirect
11. #11762962C2semidirect
12. #11762911C2semidirect
13. #11762811C2semidirect
14. C2×#10587492C2semidirect
15. C17×#435C2semidirect
18. C2×M16×C17C22semidirect
25. C2×C4⋊Dic17C22semidirect
28. #10587610C22semidirect
29. C2×Dic68C22semidirect
40. #10587555C22semidirect
41. C4×Dic34C22semidirect
42. C17×C4⋊C8C22semidirect
44. C17×C22⋊C8C22semidirect
45. C17×#97C22semidirect
52. #10587489C22semidirect
53. #10587489C22semidirect
54. #10587489C22semidirect
55. #10587487C22semidirect
56. #10587487C22semidirect
57. #10587487C22semidirect
58. #10587487C22semidirect
193. #435D17semidirect
208. C2×M16C2×D17semidirect
232. C4⋊C8C2×D17semidirect
244. C22⋊C8C2×D17semidirect
245. #97C2×D17semidirect
246. C17#3314semidirect

Maximal subgroups

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

subgroupindex
#117635162
#117634552
#117631692
#117631452
#117631232
#117630902
#117630552
#117630502
#117630322
#117629732
#117629622
#117629112
#117628112
C2×#105874922
C17×#4352
#331417