Finite Group Databasebeta

← #14540804

Subgroups — #14540804

Order 2176.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763497C211yesmaximal, characteristic, complemented
2#11763444C211yesmaximal, characteristic
3#11763131C211yesmaximal, characteristic, complemented
4#11763121C211yesmaximal, characteristic, complemented
5#11763106C211yesmaximal, characteristic, complemented
6#11763054C211yesmaximal, characteristic, complemented
7#11763037C211yesmaximal, characteristic, complemented
8#11763006C211yesmaximal, characteristic, complemented
9#11762961C211yesmaximal, characteristic, complemented
10#11762961C211yesmaximal, characteristic, complemented
11#11762911C211yesmaximal, characteristic, complemented
12#11762882C211yesmaximal, characteristic, complemented
13#11762812C211yesmaximal, characteristic, complemented
14#102×D17C211yesmaximal, characteristic, complemented
15C17×#490C211yesmaximal, characteristic, complemented
16C2×D4×D17—21no—
17C2×D4×D17C2211yescharacteristic
18C2×D4×D17C2211yescharacteristic
19C17×#127C2211yescharacteristic
20C17×#125C2211yescharacteristic, complemented
21#10587624—21no—
22#10587624C2211yescharacteristic
23#10587624C2211yescharacteristic
24#10587623—21no—
25#10587623—21no—
26C2×#63229C2211yescharacteristic, complemented
27C2×#63229C2211yescharacteristic, complemented
28C2×#63219C2211yescharacteristic
29C2×#63218C2211yescharacteristic, complemented
30C2×C8×D17C2211yescharacteristic, complemented
31#10587584—21no—
32#10587583C2211yescharacteristic, complemented
33#10587583C2211yescharacteristic, complemented
34#10587580—21no—
35#10587578—21no—
36#10587578C2211yescharacteristic, complemented
37C4⋊C4×D17C2211yescharacteristic, complemented
38#10587576—21no—
39C4×D68C2211yescharacteristic
40C4×D68C2211yescharacteristic
41C4×Dic34C2211yescharacteristic, complemented
42C17×#102C2211yescharacteristic, complemented
43C17×#102C2211yescharacteristic, complemented
44C17×#102C2211yescharacteristic, complemented
45C17×#102C2211yescharacteristic, complemented
46C17×#97C2211yescharacteristic, complemented
47#10587503C2211yescharacteristic, complemented
48#10587503C2211yescharacteristic, complemented
49#10587492C2211yescharacteristic
50#10587491C2211yescharacteristic, complemented
51#10587491C2211yescharacteristic, complemented
52#10587489C2211yescharacteristic, complemented
53#10587487C2211yescharacteristic, complemented
54#10587480C2211yescharacteristic
55#10587480C2211yescharacteristic
56#10587479C2211yescharacteristic, complemented
57#10587478C2211yescharacteristic, complemented
58C4⋊#3622C2211yescharacteristic, complemented
59C23×D17—21no—
60C23×D17—21no—
61C2×D4×C17—21no—
62C2×D4×C17—21no—
63C2×D4×C17C2311yescharacteristic
64C2×D4×C17C2311yescharacteristic
65C2×C22⋊D17—21no—
66C2×C22⋊D17—21no—
67C2×C22⋊D17—21no—
68C2×C22⋊D17—21no—
69D4×D17—22no—
70D4×D17—22no—
71D4×D17—41no—
72D4×D17—41no—
73D4×D17—44no—
74C2×D68D411yescharacteristic
75C2×D68C2311yescharacteristic
76C2×C4×D17—21no—
77C2×C4×D17—21no—
78C2×C4×D17C2311yescharacteristic
79C2×C4×D17C2311yescharacteristic
80C2×Dic34C2311yescharacteristic
81C2×C8×C17C2311yescharacteristic
82C2×C8×C17C2311yescharacteristic
83C4⋊C4×C17—21no—
84C4⋊C4×C17—21no—
85C4⋊C4×C17C2311yescharacteristic
86C4⋊C4×C17C2311yescharacteristic
87C42×C17C2311yescharacteristic
88C22⋊Dic17—21no—
89C22⋊Dic17—21no—
90#63229—41no—
91#63229—41no—
92#63227—21no—
93#63227—21no—
94#63227—21no—
95#63227—21no—
96#63227D411yescharacteristic
97#63227D411yescharacteristic
98C4⋊Dic17—21no—
99C4⋊Dic17—21no—
100C4⋊Dic17D411yescharacteristic
101C4⋊Dic17C2311yescharacteristic
102C4⋊Dic17C2311yescharacteristic
103#63225—21no—
104#63225—21no—
105C4×Dic17—21no—
106C4×Dic17—21no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109#63219—22no—
110#63218—41no—
111C8×D17—41no—
112C23×C17—21no—
113C23×C17—21no—
114C22×D17—21no—
115C22×D17—42no—
116C22×D17—42no—
117C22×D17—42no—
118C22×D17—42no—
119C22×D17—81no—
120C22×D17—81no—
121C22×D17C2×D411yescharacteristic
122C22×D17C2×D411yescharacteristic
123D4×C17—21no—
124D4×C17—21no—
125D4×C17—21no—
126D4×C17—21no—
127D4×C17—42no—
128D4×C17—81no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17C2×D411yescharacteristic
132C2×C4×C17C2×D411yescharacteristic
133C2×C4×C17C2411yescharacteristic
134C22⋊D17—42no—
135C22⋊D17—42no—
136C22⋊D17—42no—
137C22⋊D17—42no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17C2×D411yescharacteristic
142C2×Dic17C2×D411yescharacteristic
143D68—22no—
144D68—21no—
145D68#4112yes—
146C4×D17—21no—
147C4×D17—21no—
148C4×D17—21no—
149C4×D17—21no—
150C4×D17—42no—
151C4×D17—42no—
152Dic34—21no—
153Dic34—21no—
154C8×C17—21no—
155C8×C17—21no—
156#3622—21no—
157#3622—21no—
158#3310—171nomaximal, Sylow
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17C22×D411yescharacteristic
166C2×D17—21no—
167C2×D17—21no—
168C2×D17—21no—
169C2×D17—22no—
170C2×D17—21no—
171C2×D17—21no—
172C2×D17—41no—
173C2×D17—41no—
174C2×D17—84no—
175C2×D17—84no—
176C2×D17AGL(1,ℤ/8)12yes—
177C4×C17—21no—
178C4×C17—21no—
179C4×C17—41no—
180C4×C17—41no—
181C4×C17C2×#4111yescharacteristic
182C4×C17H(5,𝔽2)11yescharacteristic
183Dic17—21no—
184Dic17—21no—
185Dic17—41no—
186Dic17—41no—
187Dic17—41no—
188#552—171no—
189D42—171no—
190#490D1711yescharacteristic, complemented
191#482—171no—
192#480—171no—
193#478—171no—
194#468—171no—
195#465—171no—
196#460—171no—
197#450—171no—
198#450—171no—
199#440—171no—
200#435—171no—
201#418—171no—
202C2×#102—171no—
203C2×C17—81no—
204C2×C17—81no—
205C2×C17#54611yescharacteristic
206C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
207C2×C17#57711yescharacteristic
208D17—22no—
209D17—41no—
210D17—42no—
211C22×D4—171no—
212C22×D4—171no—
213C22×D4—341no—
214C2×SD16—171no—
215C2×SD16—171no—
216C2×SD16—171no—
217C2×M16—171no—
218C22×C8—171no—
219#127C2×D1711yescharacteristic
220#126—341no—
221#125—341no—
222#125C2×D1711yescharacteristic, complemented
223#123—341no—
224C22⋊Q8—171no—
225C22⋊Q8—171no—
226#121—171no—
227#121—171no—
228#121—341no—
229#120—341no—
230#120—341no—
231C4×Q8—171no—
232C4×D4—171no—
233C4×D4—171no—
234#117—171no—
235#117—341no—
236C2×C4⋊C4—171no—
237#107—171no—
238#107—171no—
239#106—171no—
240C4⋊C8—171no—
241#103—171no—
242#102—171no—
243#102—171no—
244#102—171no—
245#102—171no—
246#102—171no—
247#102C2×D1711yescharacteristic, complemented
248#102C2×D1711yescharacteristic, complemented
249#102C2×D1711yescharacteristic, complemented
250#102C2×D1711yescharacteristic, complemented
251C22⋊C8—171no—
252C22⋊C8—171no—
253#97C2×D1711yescharacteristic, complemented
254C17#331011yescharacteristic, complemented, Sylow
255C24—341no—
256C24—341no—
257C2×Q8—171no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—171no—
261C2×D4—171no—
262C2×D4—341no—
263C2×D4—341no—
264C2×D4—342no—
265C2×D4—342no—
266C2×D4—341no—
267C2×D4—341no—
268C2×D4—681no—
269C2×D4—681no—
270C2×D4—684no—
271C2×D4C22×D1711yescharacteristic
272C2×D4C22×D1711yescharacteristic
273C22×C4—171no—
274C22×C4—171no—
275C22×C4—341no—
276C22×C4—341no—
277SD16—342no—
278SD16—681no—
279SD16—681no—
280M16—681no—
281C2×C8—171no—
282C2×C8—171no—
283C2×C8—681no—
284C2×C8C22×D1711yescharacteristic
285C2×C8C22×D1711yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—171no—
289C4⋊C4—171no—
290C4⋊C4—171no—
291C4⋊C4—341no—
292C4⋊C4—341no—
293C4⋊C4—341no—
294C4⋊C4—341no—
295C4⋊C4C22×D1711yescharacteristic
296C4⋊C4C22×D1711yescharacteristic
297C22⋊C4—171no—
298C22⋊C4—171no—
299C22⋊C4—341no—
300C22⋊C4—341no—
301C22⋊C4—341no—
302C22⋊C4—341no—
303C22⋊C4—341no—
304C22⋊C4—341no—
305C42—341no—
306C42—341no—
307C42C22×D1711yescharacteristic
308C23—21no—
309C23—21no—
310C23—171no—
311C23—171no—
312C23—341no—
313C23—682no—
314C23—682no—
315C23—682no—
316C23—682no—
317C23—1361no—
318C23—1361no—
319Q8—341no—
320Q8—341no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—42no—
326D4—81no—
327D4—172no—
328D4—342no—
329D4—341no—
330D4—682no—
331D4—682no—
332D4—682no—
333D4—682no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—171no—
337C2×C4—171no—
338C2×C4—341no—
339C2×C4—341no—
340C2×C4—341no—
341C2×C4—341no—
342C2×C4—341no—
343C2×C4—341no—
344C2×C4—341no—
345C2×C4—682no—
346C2×C4—682no—
347C2×C4D4×D1711yescharacteristic
348C2×C4D4×D1711yescharacteristic
349C2×C4C23×D1711yescharacteristic
350C8—21no—
351C8—21no—
352C8—341no—
353C8—341no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—172no—
361C22—341no—
362C22—341no—
363C22—341no—
364C22—341no—
365C22—341no—
366C22—342no—
367C22—681no—
368C22—681no—
369C22—1364no—
370C22—1364no—
371C22C2×D4×D1711yescharacteristic
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—341no—
377C4—341no—
378C4—681no—
379C4—681no—
380C4—681no—
381C4#41×D1711yescharacteristic
382C4#1058769611yescharacteristic
383C2—81no—
384C2—81no—
385C2—342no—
386C2—681no—
387C2—682no—
388C2#1176345211yescharacteristic
389C2AGL(1,ℤ/8)×D1711yescharacteristic
390C2#1176361911yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#490C2
socleC22×C17C22×D4
prime radical#11763444C2
hypercenter#490D17
hypocenterC17#3310
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #11763497C2semidirect
3. #11763131C2semidirect
4. #11763121C2semidirect
5. #11763106C2semidirect
6. #11763054C2semidirect
7. #11763037C2semidirect
8. #11763006C2semidirect
9. #11762961C2semidirect
10. #11762961C2semidirect
11. #11762911C2semidirect
12. #11762882C2semidirect
13. #11762812C2semidirect
14. #102×D17C2semidirect
15. C17×#490C2semidirect
20. C17×#125C22semidirect
26. C2×#63229C22semidirect
27. C2×#63229C22semidirect
29. C2×#63218C22semidirect
30. C2×C8×D17C22semidirect
32. #10587583C22semidirect
33. #10587583C22semidirect
36. #10587578C22semidirect
37. C4⋊C4×D17C22semidirect
41. C4×Dic34C22semidirect
42. C17×#102C22semidirect
43. C17×#102C22semidirect
44. C17×#102C22semidirect
45. C17×#102C22semidirect
46. C17×#97C22semidirect
47. #10587503C22semidirect
48. #10587503C22semidirect
50. #10587491C22semidirect
51. #10587491C22semidirect
52. #10587489C22semidirect
53. #10587487C22semidirect
56. #10587479C22semidirect
57. #10587478C22semidirect
58. C4⋊#3622C22semidirect
190. #490D17semidirect
222. #125C2×D17semidirect
247. #102C2×D17semidirect
248. #102C2×D17semidirect
249. #102C2×D17semidirect
250. #102C2×D17semidirect
253. #97C2×D17semidirect
254. C17#3310semidirect

Maximal subgroups

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

subgroupindex
#117634972
#117634442
#117631312
#117631212
#117631062
#117630542
#117630372
#117630062
#117629612
#117629112
#117628822
#117628122
#102×D172
C17×#4902
#331017