Finite Group Databasebeta

← #14540799

Subgroups — #14540799

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
1D4×D68C211yesmaximal, characteristic
2#11763175C211yesmaximal, characteristic, complemented
3#11763133C211yesmaximal, characteristic, complemented
4#11763122C211yesmaximal, characteristic, complemented
5#11763106C211yesmaximal, characteristic, complemented
6#11763057C211yesmaximal, characteristic, complemented
7#11763040C211yesmaximal, characteristic, complemented
8#11763009C211yesmaximal, characteristic, complemented
9#11762959C211yesmaximal, characteristic, complemented
10#11762959C211yesmaximal, characteristic, complemented
11#11762907C211yesmaximal, characteristic, complemented
12#11762881C211yesmaximal, characteristic, complemented
13#11762813C211yesmaximal, characteristic, complemented
14C2×#10587480C211yesmaximal, characteristic, complemented
15C17×#552C211yesmaximal, characteristic, complemented
16C2×D4×D17—21no—
17C22×D68C2211yescharacteristic
18C22×D68C2211yescharacteristic
19C17×#126—21no—
20C17×#125—21no—
21C17×#125C2211yescharacteristic, complemented
22C17×#123—21no—
23C17×C22⋊Q8C2211yescharacteristic, complemented
24C17×C22⋊Q8C2211yescharacteristic, complemented
25C4×Q8×C17C2211yescharacteristic, complemented
26C4×D4×C17C2211yescharacteristic
27C17×#117—21no—
28C17×#117C2211yescharacteristic, complemented
29C2×C4⋊C4×C17C2211yescharacteristic, complemented
30C2×#63230C2211yescharacteristic
31C2×#63230C2211yescharacteristic, complemented
32C2×#63230C2211yescharacteristic, complemented
33#10587616C2211yescharacteristic
34#10587616C2211yescharacteristic
35C2×#63223C2211yescharacteristic, complemented
36C22×#3622C2211yescharacteristic, complemented
37#10587581—21no—
38#10587568—21no—
39#10587568—21no—
40#10587560C2211yescharacteristic
41C4×D68C2211yescharacteristic
42#10587506C2211yescharacteristic, complemented
43C22⋊#3622C2211yescharacteristic, complemented
44C22⋊#3622C2211yescharacteristic, complemented
45#10587480C2211yescharacteristic
46#10587480C2211yescharacteristic
47#10587480C2211yescharacteristic
48#10587480C2211yescharacteristic, complemented
49#10587480C2211yescharacteristic, complemented
50#10587480C2211yescharacteristic, complemented
51#10587480C2211yescharacteristic, complemented
52#10587480C2211yescharacteristic, complemented
53#10587480C2211yescharacteristic, complemented
54#10587479C2211yescharacteristic, complemented
55#10587478C2211yescharacteristic, complemented
56#10587478C2211yescharacteristic, complemented
57C4⋊#3622C2211yescharacteristic, complemented
58#10587474C2211yescharacteristic, complemented
59C23×D17—21no—
60C23×D17—21no—
61C2×Q8×C17C2311yescharacteristic
62C2×D4×C17D411yescharacteristic
63C22×C4×C17—21no—
64C22×C4×C17C2311yescharacteristic
65C22×C4×C17C2311yescharacteristic
66C2×C22⋊D17—21no—
67C2×C22⋊D17—21no—
68D4×D17—44no—
69C2×D68—22no—
70C2×D68—22no—
71C2×D68—21no—
72C2×D68—21no—
73C2×D68—21no—
74C2×D68—21no—
75C2×D68—41no—
76C2×D68—41no—
77C2×D68C2311yescharacteristic
78C2×D68C2311yescharacteristic
79C2×D68C2311yescharacteristic
80C2×C4×D17—21no—
81C4⋊C4×C17—21no—
82C4⋊C4×C17—21no—
83C4⋊C4×C17—21no—
84C4⋊C4×C17—21no—
85C4⋊C4×C17—21no—
86C4⋊C4×C17—21no—
87C4⋊C4×C17D411yescharacteristic
88C4⋊C4×C17C2311yescharacteristic
89C4⋊C4×C17C2311yescharacteristic
90C4⋊C4×C17C2311yescharacteristic
91C22⋊C4×C17—21no—
92C22⋊C4×C17—21no—
93C22⋊C4×C17—21no—
94C22⋊C4×C17D411yescharacteristic
95C22⋊C4×C17D411yescharacteristic
96C42×C17—21no—
97C42×C17—21no—
98C42×C17C2311yescharacteristic
99#63230—22no—
100#63230—41no—
101#63230—41no—
102#63227—21no—
103#63227—21no—
104#63227—21no—
105C4⋊Dic17C2311yescharacteristic
106#63223—41no—
107C2×#3622—41no—
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112C23×C17C2×D411yescharacteristic
113C23×C17C2×D411yescharacteristic
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—21no—
117C22×D17—42no—
118C22×D17—42no—
119C22×D17—42no—
120C22×D17—42no—
121C22×D17—81no—
122C22×D17—81no—
123Q8×C17—21no—
124Q8×C17—21no—
125D4×C17—22no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—42no—
135C2×C4×C17C2×D411yescharacteristic
136C2×C4×C17C2×D411yescharacteristic
137C2×C4×C17C2×D411yescharacteristic
138C2×C4×C17C2×D411yescharacteristic
139C2×C4×C17C2411yescharacteristic
140C22⋊D17—42no—
141C22⋊D17—42no—
142C2×Dic17—21no—
143D68—21no—
144D68—21no—
145D68—21no—
146D68—21no—
147D68—21no—
148D68—42no—
149D68—42no—
150D68—42no—
151D68—81no—
152D68#4112yes—
153C4×D17—42no—
154#3622—21no—
155#3622—21no—
156#3622—21no—
157#3622—21no—
158#3310—171nomaximal, Sylow
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17AGL(1,ℤ/8)12yes—
165C22×C17C22×D411yescharacteristic
166C2×D17—22no—
167C2×D17—41no—
168C2×D17—41no—
169C2×D17—41no—
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—21no—
179C4×C17—21no—
180C4×C17—21no—
181C4×C17—41no—
182C4×C17—41no—
183C4×C17—41no—
184C4×C17—41no—
185C4×C17C2×#4111yescharacteristic
186C4×C17H(5,𝔽2)11yescharacteristic
187Dic17—41no—
188#552D1711yescharacteristic, complemented
189D42—171no—
190#490—171no—
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—22no—
204C2×C17—41no—
205C2×C17#54611yescharacteristic
206C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
207C2×C17#57711yescharacteristic
208D17—42no—
209D17—81no—
210D17—81no—
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#127—171no—
220#126—21no—
221#125—21no—
222#125C2×D1711yescharacteristic, complemented
223#123—21no—
224C22⋊Q8C2×D1711yescharacteristic, complemented
225C22⋊Q8C2×D1711yescharacteristic, complemented
226#121—171no—
227#121—171no—
228#121—341no—
229#120—341no—
230#120—341no—
231C4×Q8C2×D1711yescharacteristic, complemented
232C4×D4—171no—
233C4×D4C2×D1711yescharacteristic
234#117—21no—
235#117C2×D1711yescharacteristic, complemented
236C2×C4⋊C4C2×D1711yescharacteristic, complemented
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#102—171no—
248#102—171no—
249#102—171no—
250#102—171no—
251C22⋊C8—171no—
252C22⋊C8—171no—
253#97—171no—
254C17#331011yescharacteristic, complemented, Sylow
255C24—341no—
256C24—341no—
257C2×Q8C22×D1711yescharacteristic
258C2×D4—171no—
259C2×D4—171no—
260C2×D4—171no—
261C2×D4—342no—
262C2×D4—342no—
263C2×D4—341no—
264C2×D4—341no—
265C2×D4—341no—
266C2×D4—341no—
267C2×D4—341no—
268C2×D4—341no—
269C2×D4—681no—
270C2×D4—681no—
271C2×D4—684no—
272C2×D4C22⋊D1711yescharacteristic
273C22×C4—21no—
274C22×C4—341no—
275C22×C4C22×D1711yescharacteristic
276C22×C4C22×D1711yescharacteristic
277SD16—342no—
278SD16—681no—
279SD16—681no—
280M16—681no—
281C2×C8—171no—
282C2×C8—171no—
283C2×C8—171no—
284C2×C8—171no—
285C2×C8—681no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—171no—
293C4⋊C4C22⋊D1711yescharacteristic
294C4⋊C4C22×D1711yescharacteristic
295C4⋊C4C22×D1711yescharacteristic
296C4⋊C4C22×D1711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—341no—
301C22⋊C4—341no—
302C22⋊C4—341no—
303C22⋊C4C22⋊D1711yescharacteristic
304C22⋊C4C22⋊D1711yescharacteristic
305C42—21no—
306C42—21no—
307C42C22×D1711yescharacteristic
308C23—341no—
309C23—341no—
310C23—341no—
311C23—682no—
312C23—682no—
313C23—682no—
314C23—682no—
315C23—1361no—
316C23—1361no—
317C23C2×C22⋊D1711yescharacteristic
318C23C2×C22⋊D1711yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—22no—
322D4—172no—
323D4—341no—
324D4—341no—
325D4—341no—
326D4—341no—
327D4—341no—
328D4—682no—
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—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—341no—
344C2×C4—682no—
345C2×C4C2×C22⋊D1711yescharacteristic
346C2×C4C2×C22⋊D1711yescharacteristic
347C2×C4C2×C22⋊D1711yescharacteristic
348C2×C4C2×C22⋊D1711yescharacteristic
349C2×C4C23×D1711yescharacteristic
350C8—341no—
351C8—341no—
352C8—341no—
353C8—341no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—342no—
360C22—681no—
361C22—681no—
362C22—681no—
363C22—681no—
364C22—681no—
365C22—681no—
366C22—681no—
367C22—681no—
368C22—1364no—
369C22—1364no—
370C22#1058763512yes—
371C22C22×C22⋊D1711yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—681no—
381C4#41×D1711yescharacteristic
382C4#1058769611yescharacteristic
383C2—22no—
384C2—41no—
385C2—682no—
386C2—1361no—
387C2—1361no—
388C2#1176344711yescharacteristic
389C2C2×#1058763511yescharacteristic
390C2#1176361711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#552C2
socleC22×C17C22×D4
prime radicalD4×D68C2
hypercenter#552D17
hypocenterC17#3310
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #11763175C2semidirect
3. #11763133C2semidirect
4. #11763122C2semidirect
5. #11763106C2semidirect
6. #11763057C2semidirect
7. #11763040C2semidirect
8. #11763009C2semidirect
9. #11762959C2semidirect
10. #11762959C2semidirect
11. #11762907C2semidirect
12. #11762881C2semidirect
13. #11762813C2semidirect
14. C2×#10587480C2semidirect
15. C17×#552C2semidirect
21. C17×#125C22semidirect
23. C17×C22⋊Q8C22semidirect
24. C17×C22⋊Q8C22semidirect
25. C4×Q8×C17C22semidirect
28. C17×#117C22semidirect
29. C2×C4⋊C4×C17C22semidirect
31. C2×#63230C22semidirect
32. C2×#63230C22semidirect
35. C2×#63223C22semidirect
36. C22×#3622C22semidirect
42. #10587506C22semidirect
43. C22⋊#3622C22semidirect
44. C22⋊#3622C22semidirect
48. #10587480C22semidirect
49. #10587480C22semidirect
50. #10587480C22semidirect
51. #10587480C22semidirect
52. #10587480C22semidirect
53. #10587480C22semidirect
54. #10587479C22semidirect
55. #10587478C22semidirect
56. #10587478C22semidirect
57. C4⋊#3622C22semidirect
58. #10587474C22semidirect
188. #552D17semidirect
222. #125C2×D17semidirect
224. C22⋊Q8C2×D17semidirect
225. C22⋊Q8C2×D17semidirect
231. C4×Q8C2×D17semidirect
235. #117C2×D17semidirect
236. C2×C4⋊C4C2×D17semidirect
254. C17#3310semidirect

Maximal subgroups

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

subgroupindex
D4×D682
#117631752
#117631332
#117631222
#117631062
#117630572
#117630402
#117630092
#117629592
#117629072
#117628812
#117628132
C2×#105874802
C17×#5522
#331017