Finite Group Databasebeta

← #14540811

Subgroups — #14540811

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#11763501C211yesmaximal, characteristic, complemented
2#11763443C211yesmaximal, characteristic
3#11763172C211yesmaximal, characteristic, complemented
4#11763135C211yesmaximal, characteristic, complemented
5#11763120C211yesmaximal, characteristic, complemented
6#11763108C211yesmaximal, characteristic, complemented
7#11763058C211yesmaximal, characteristic, complemented
8#11763007C211yesmaximal, characteristic, complemented
9#11762959C211yesmaximal, characteristic, complemented
10#11762958C211yesmaximal, characteristic, complemented
11#11762912C211yesmaximal, characteristic, complemented
12#11762880C211yesmaximal, characteristic, complemented
13#11762812C211yesmaximal, characteristic, complemented
14C2×#10587492C211yesmaximal, characteristic, complemented
15C17×#465C211yesmaximal, characteristic, complemented
16C2×D4×D17—21no—
17C2×D4×D17C2211yescharacteristic
18C22×D68C2211yescharacteristic
19C2×SD16×C17C2211yescharacteristic, complemented
20C22×C8×C17C2211yescharacteristic, complemented
21C17×C22⋊Q8C2211yescharacteristic, complemented
22C17×#121C2211yescharacteristic
23#10587632C2211yescharacteristic, complemented
24Q8×Dic17C2211yescharacteristic, complemented
25C2×#63230C2211yescharacteristic
26C2×#63230C2211yescharacteristic, complemented
27#10587626C2211yescharacteristic
28#10587623—21no—
29C4×C22⋊D17C2211yescharacteristic
30C2×C4⋊Dic17C2211yescharacteristic, complemented
31C2×#63218C2211yescharacteristic, complemented
32#10587584—21no—
33#10587581C2211yescharacteristic
34#10587579C2211yescharacteristic
35#10587578C2211yescharacteristic, complemented
36#10587576—21no—
37#10587576C2211yescharacteristic, complemented
38#10587572—21no—
39#10587570—21no—
40#10587568—21no—
41#10587563—21no—
42C17×#106C2211yescharacteristic, complemented
43C17×#103C2211yescharacteristic, complemented
44C17×#102C2211yescharacteristic, complemented
45#10587503C2211yescharacteristic, complemented
46#10587503C2211yescharacteristic, complemented
47C22⋊#3622C2211yescharacteristic, complemented
48#10587492C2211yescharacteristic
49#10587492C2211yescharacteristic
50#10587492C2211yescharacteristic, complemented
51#10587492C2211yescharacteristic, complemented
52#10587491C2211yescharacteristic, complemented
53#10587486C2211yescharacteristic, complemented
54#10587485C2211yescharacteristic, complemented
55#10587480C2211yescharacteristic
56#10587480C2211yescharacteristic, complemented
57#10587478C2211yescharacteristic, complemented
58#10587478C2211yescharacteristic, complemented
59C23×D17—21no—
60C23×D17—21no—
61C2×Q8×C17C2311yescharacteristic
62C2×D4×C17—21no—
63C2×D4×C17C2311yescharacteristic
64C22×C4×C17C2311yescharacteristic
65C2×C22⋊D17—21no—
66C2×C22⋊D17—21no—
67C2×C22⋊D17—21no—
68C2×C22⋊D17D411yescharacteristic
69C22×Dic17—21no—
70D4×D17—22no—
71D4×D17—41no—
72D4×D17—44no—
73C2×D68—22no—
74C2×D68—21no—
75C2×D68—21no—
76C2×D68—41no—
77C2×D68C2311yescharacteristic
78C2×D68C2311yescharacteristic
79C2×C4×D17—21no—
80C2×C4×D17C2311yescharacteristic
81SD16×C17—41no—
82C2×C8×C17—41no—
83C2×C8×C17C2311yescharacteristic
84C2×C8×C17C2311yescharacteristic
85C4⋊C4×C17—21no—
86C4⋊C4×C17C2311yescharacteristic
87C4⋊C4×C17C2311yescharacteristic
88C22⋊C4×C17—21no—
89C22⋊C4×C17—21no—
90C22⋊Dic17D411yescharacteristic
91#63230—22no—
92#63230—41no—
93#63227—21no—
94#63227—21no—
95#63227—21no—
96#63227—21no—
97#63227D411yescharacteristic
98C4⋊Dic17—21no—
99C4⋊Dic17—21no—
100C4⋊Dic17—21no—
101C4⋊Dic17C2311yescharacteristic
102C4⋊Dic17C2311yescharacteristic
103#63225—21no—
104#63225—21no—
105#63225D411yescharacteristic
106C4×Dic17—21no—
107C4×Dic17—21no—
108C4×Dic17C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111#63218—41no—
112C23×C17—21no—
113C23×C17C2×D411yescharacteristic
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—42no—
117C22×D17—42no—
118C22×D17—42no—
119C22×D17—42no—
120C22×D17—81no—
121C22×D17—81no—
122C22×D17C2×D411yescharacteristic
123Q8×C17—21no—
124Q8×C17—21no—
125D4×C17—21no—
126D4×C17—21no—
127D4×C17—42no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17C2×D411yescharacteristic
134C2×C4×C17C2411yescharacteristic
135C22⋊D17—22no—
136C22⋊D17—42no—
137C22⋊D17—42no—
138C22⋊D17—81no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—42no—
142C2×Dic17C2×D411yescharacteristic
143C2×Dic17C2×D411yescharacteristic
144C2×Dic17C2×D411yescharacteristic
145D68—21no—
146D68—21no—
147D68—21no—
148D68—42no—
149D68—42no—
150D68#4112yes—
151C4×D17—21no—
152C4×D17—21no—
153C4×D17—42no—
154C8×C17—21no—
155C8×C17—21no—
156#3622—21no—
157#3622—21no—
158#3310—171nomaximal, Sylow
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17AGL(1,ℤ/8)12yes—
165C22×C17C22×D411yescharacteristic
166C2×D17—21no—
167C2×D17—22no—
168C2×D17—21no—
169C2×D17—21no—
170C2×D17—41no—
171C2×D17—41no—
172C2×D17—41no—
173C2×D17—41no—
174C2×D17—41no—
175C2×D17—84no—
176C2×D17—84no—
177C4×C17—21no—
178C4×C17—41no—
179C4×C17—41no—
180C4×C17—41no—
181C4×C17C2×#4111yescharacteristic
182C4×C17H(5,𝔽2)11yescharacteristic
183Dic17—21no—
184Dic17—21no—
185Dic17—21no—
186Dic17—41no—
187Dic17—41no—
188#552—171no—
189D42—171no—
190#490—171no—
191#482—171no—
192#480—171no—
193#478—171no—
194#468—171no—
195#465D1711yescharacteristic, complemented
196#460—171no—
197#450—171no—
198#450—171no—
199#440—171no—
200#435—171no—
201#418—171no—
202C2×#102—171no—
203C2×C17—22no—
204C2×C17—81no—
205C2×C17#54611yescharacteristic
206C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
207C2×C17#57711yescharacteristic
208D17—41no—
209D17—42no—
210D17—81no—
211C22×D4—171no—
212C22×D4—171no—
213C22×D4—341no—
214C2×SD16—171no—
215C2×SD16—171no—
216C2×SD16C2×D1711yescharacteristic, complemented
217C2×M16—171no—
218C22×C8C2×D1711yescharacteristic, complemented
219#127—171no—
220#126—341no—
221#125—171no—
222#125—341no—
223#123—341no—
224C22⋊Q8—171no—
225C22⋊Q8C2×D1711yescharacteristic, complemented
226#121—171no—
227#121—341no—
228#121C2×D1711yescharacteristic
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#106C2×D1711yescharacteristic, complemented
240C4⋊C8—171no—
241#103C2×D1711yescharacteristic, complemented
242#102—171no—
243#102—171no—
244#102—171no—
245#102—171no—
246#102—171no—
247#102—171no—
248#102—171no—
249#102—171no—
250#102C2×D1711yescharacteristic, complemented
251C22⋊C8—171no—
252C22⋊C8—171no—
253#97—171no—
254C17#331011yescharacteristic, complemented, Sylow
255C24—341no—
256C24—341no—
257C2×Q8C22×D1711yescharacteristic
258C2×D4—21no—
259C2×D4—171no—
260C2×D4—171no—
261C2×D4—171no—
262C2×D4—342no—
263C2×D4—341no—
264C2×D4—341no—
265C2×D4—341no—
266C2×D4—341no—
267C2×D4—342no—
268C2×D4—341no—
269C2×D4—681no—
270C2×D4—681no—
271C2×D4—684no—
272C2×D4C22×D1711yescharacteristic
273C22×C4—171no—
274C22×C4—341no—
275C22×C4—341no—
276C22×C4C22×D1711yescharacteristic
277SD16—41no—
278SD16—342no—
279SD16—681no—
280M16—681no—
281C2×C8—41no—
282C2×C8—171no—
283C2×C8—171no—
284C2×C8C22×D1711yescharacteristic
285C2×C8C22×D1711yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—171no—
288C4⋊C4—171no—
289C4⋊C4—171no—
290C4⋊C4—341no—
291C4⋊C4—341no—
292C4⋊C4—341no—
293C4⋊C4—341no—
294C4⋊C4—341no—
295C4⋊C4C22×D1711yescharacteristic
296C4⋊C4C22×D1711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—171no—
300C22⋊C4—171no—
301C22⋊C4—341no—
302C22⋊C4—341no—
303C22⋊C4—341no—
304C22⋊C4—341no—
305C42—171no—
306C42—341no—
307C42—341no—
308C23—21no—
309C23—171no—
310C23—341no—
311C23—341no—
312C23—682no—
313C23—682no—
314C23—682no—
315C23—682no—
316C23—1361no—
317C23—1361no—
318C23D4×D1711yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—172no—
325D4—342no—
326D4—341no—
327D4—341no—
328D4—341no—
329D4—682no—
330D4—682no—
331D4—682no—
332D4—682no—
333D4—1361no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—171no—
340C2×C4—171no—
341C2×C4—171no—
342C2×C4—341no—
343C2×C4—341no—
344C2×C4—341no—
345C2×C4—341no—
346C2×C4—682no—
347C2×C4—682no—
348C2×C4D4×D1711yescharacteristic
349C2×C4C23×D1711yescharacteristic
350C8—21no—
351C8—21no—
352C8—341no—
353C8—341no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—341no—
360C22—341no—
361C22—341no—
362C22—342no—
363C22—681no—
364C22—681no—
365C22—681no—
366C22—681no—
367C22—681no—
368C22—1364no—
369C22—1364no—
370C22#1058760012yes—
371C22C2×D4×D1711yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—341no—
377C4—341no—
378C4—341no—
379C4—681no—
380C4—681no—
381C4#1058769111yescharacteristic
382C4#41×D1711yescharacteristic
383C2—22no—
384C2—81no—
385C2—681no—
386C2—682no—
387C2—1361no—
388C2#1176344811yescharacteristic
389C2C2×#1058760011yescharacteristic
390C2#1176361611yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#465C2
socleC22×C17C22×D4
prime radical#11763443C2
hypercenter#465D17
hypocenterC17#3310
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #11763501C2semidirect
3. #11763172C2semidirect
4. #11763135C2semidirect
5. #11763120C2semidirect
6. #11763108C2semidirect
7. #11763058C2semidirect
8. #11763007C2semidirect
9. #11762959C2semidirect
10. #11762958C2semidirect
11. #11762912C2semidirect
12. #11762880C2semidirect
13. #11762812C2semidirect
14. C2×#10587492C2semidirect
15. C17×#465C2semidirect
19. C2×SD16×C17C22semidirect
20. C22×C8×C17C22semidirect
21. C17×C22⋊Q8C22semidirect
23. #10587632C22semidirect
24. Q8×Dic17C22semidirect
26. C2×#63230C22semidirect
30. C2×C4⋊Dic17C22semidirect
31. C2×#63218C22semidirect
35. #10587578C22semidirect
37. #10587576C22semidirect
42. C17×#106C22semidirect
43. C17×#103C22semidirect
44. C17×#102C22semidirect
45. #10587503C22semidirect
46. #10587503C22semidirect
47. C22⋊#3622C22semidirect
50. #10587492C22semidirect
51. #10587492C22semidirect
52. #10587491C22semidirect
53. #10587486C22semidirect
54. #10587485C22semidirect
56. #10587480C22semidirect
57. #10587478C22semidirect
58. #10587478C22semidirect
195. #465D17semidirect
216. C2×SD16C2×D17semidirect
218. C22×C8C2×D17semidirect
225. C22⋊Q8C2×D17semidirect
239. #106C2×D17semidirect
241. #103C2×D17semidirect
250. #102C2×D17semidirect
254. C17#3310semidirect

Maximal subgroups

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

subgroupindex
#117635012
#117634432
#117631722
#117631352
#117631202
#117631082
#117630582
#117630072
#117629592
#117629582
#117629122
#117628802
#117628122
C2×#105874922
C17×#4652
#331017