Finite Group Databasebeta

← #14539628

Subgroups — #14539628

Order 2176.

418 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763322C211yesmaximal, characteristic, complemented
2#11763321C211yesmaximal, characteristic, complemented
3#11763153C211yesmaximal, characteristic, complemented
4#11763136C211yesmaximal, characteristic, complemented
5#11763133C211yesmaximal, characteristic, complemented
6#11763066C211yesmaximal, characteristic, complemented
7#11763063C211yesmaximal, characteristic, complemented
8#11763058C211yesmaximal, characteristic, complemented
9#11763056C211yesmaximal, characteristic, complemented
10#11762974C211yesmaximal, characteristic, complemented
11#11762969C211yesmaximal, characteristic, complemented
12#11762957C211yesmaximal, characteristic, complemented
13#11762956C211yesmaximal, characteristic, complemented
14#11762771C211yesmaximal, characteristic, complemented
15C17×#485C211yesmaximal, characteristic, complemented
16C2×#63256C2211yescharacteristic
17C2×#63252C2211yescharacteristic
18C22×C4×D17C2211yescharacteristic
19C17×C22⋊Q8C2211yescharacteristic
20C17×#117C2211yescharacteristic
21#10587632C2211yescharacteristic
22C2×#63231C2211yescharacteristic, complemented
23C2×#63230C2211yescharacteristic, complemented
24#10587616C2211yescharacteristic
25C2×D136C2211yescharacteristic, complemented
26C2×#63219C2211yescharacteristic, complemented
27C2×#63218C2211yescharacteristic
28C2×#63218C2211yescharacteristic
29#10587583C2211yescharacteristic
30#10587583C2211yescharacteristic
31#10587581C2211yescharacteristic
32#10587581C2211yescharacteristic
33#10587581C2211yescharacteristic
34#10587580—21no—
35#10587579—21no—
36#10587578C2211yescharacteristic
37#10587572—21no—
38#10587568—21no—
39#10587568—21no—
40#10587567—21no—
41C4×D68—21no—
42C4×D68—21no—
43C17×#107C2211yescharacteristic, complemented
44C17×#106C2211yescharacteristic, complemented
45C17×#103C2211yescharacteristic, complemented
46C17×#103C2211yescharacteristic, complemented
47C17×C22⋊C8C2211yescharacteristic
48C22⋊#3622C2211yescharacteristic
49#10587492C2211yescharacteristic, complemented
50#10587491C2211yescharacteristic
51#10587491C2211yescharacteristic
52#10587487C2211yescharacteristic, complemented
53#10587481C2211yescharacteristic, complemented
54#10587480C2211yescharacteristic, complemented
55#10587480C2211yescharacteristic, complemented
56#10587480C2211yescharacteristic, complemented
57#10587479C2211yescharacteristic, complemented
58#10587478C2211yescharacteristic, complemented
59C23×D17D411yescharacteristic
60C2×Q8×C17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C2×C22⋊D17—21no—
63C2×C22⋊D17—21no—
64C22×Dic17D411yescharacteristic
65#63256—41no—
66#63256—41no—
67#63252—41no—
68#63252—41no—
69C2×D68—21no—
70C2×D68—21no—
71C2×D68—21no—
72C2×D68—21no—
73C2×D68C2311yescharacteristic
74C2×D68C2311yescharacteristic
75C2×C4×D17—21no—
76C2×C4×D17—21no—
77C2×C4×D17—41no—
78C2×C4×D17—41no—
79C2×C4×D17D411yescharacteristic
80C2×C4×D17D411yescharacteristic
81C2×C4×D17C2311yescharacteristic
82C2×C4×D17C2311yescharacteristic
83C2×Dic34C2311yescharacteristic
84C2×C8×C17C2311yescharacteristic
85C2×C8×C17C2311yescharacteristic
86C4⋊C4×C17—21no—
87C4⋊C4×C17C2311yescharacteristic
88C4⋊C4×C17C2311yescharacteristic
89C4⋊C4×C17C2311yescharacteristic
90C22⋊C4×C17—21no—
91C22⋊C4×C17—21no—
92C42×C17—21no—
93#63231—41no—
94#63230—41no—
95#63227—21no—
96#63227—21no—
97#63227—21no—
98#63227—21no—
99#63227—21no—
100#63227—21no—
101#63227—21no—
102#63227—21no—
103C4⋊Dic17—21no—
104C4⋊Dic17—21no—
105C4⋊Dic17C2311yescharacteristic
106#63225—21no—
107C4×Dic17—21no—
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110D136—41no—
111#63219—41no—
112#63218—41no—
113#63218—41no—
114C23×C17C2×D411yescharacteristic
115C22×D17—21no—
116C22×D17—21no—
117C22×D17—41no—
118C22×D17—41no—
119C22×D17—41no—
120C22×D17C2×D411yescharacteristic
121C22×D17C2×D411yescharacteristic
122Q8×C17—21no—
123Q8×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17C2×D411yescharacteristic
131C2×C4×C17C2411yescharacteristic
132C22⋊D17—42no—
133C22⋊D17—42no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—41no—
137C2×Dic17C2×D411yescharacteristic
138C2×Dic17C2×D411yescharacteristic
139D68—21no—
140D68—21no—
141D68—21no—
142D68—21no—
143D68—42no—
144D68—42no—
145D68—42no—
146D68—42no—
147C4×D17—21no—
148C4×D17—21no—
149C4×D17—21no—
150C4×D17—21no—
151C4×D17—41no—
152C4×D17—41no—
153C4×D17—42no—
154C4×D17—42no—
155Dic34—21no—
156Dic34—21no—
157C8×C17—21no—
158C8×C17—21no—
159#3622—21no—
160#3622—21no—
161#3197—171nomaximal, Sylow
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17C22×D411yescharacteristic
166C2×D17—21no—
167C2×D17—21no—
168C2×D17—21no—
169C2×D17—21no—
170C2×D17—21no—
171C2×D17—21no—
172C2×D17—44no—
173C2×D17—41no—
174C2×D17—41no—
175C2×D17—41no—
176C2×D17—41no—
177C2×D17—41no—
178C2×D17—41no—
179C4×C17—21no—
180C4×C17—41no—
181C4×C17—41no—
182C4×C17—41no—
183C4×C17—41no—
184C4×C17H(5,𝔽2)11yescharacteristic
185C4×C17H(5,𝔽2)11yescharacteristic
186Dic17—21no—
187Dic17—21no—
188Dic17—41no—
189Dic17—41no—
190#525—171no—
191#525—171no—
192#485—171no—
193#485D1711yescharacteristic, complemented
194#482—171no—
195#482—171no—
196#470—171no—
197#469—171no—
198#468—171no—
199#468—171no—
200#452—171no—
201#452—171no—
202#449—171no—
203#449—171no—
204#407—171no—
205C2×C17—41no—
206C2×C17#53411yescharacteristic
207C2×C17#57511yescharacteristic
208C2×C17#57511yescharacteristic
209D17—41no—
210D17—41no—
211D17—81no—
212D17—81no—
213C2×#41—171no—
214C2×#41—171no—
215C23×C4—171no—
216C2×Q16—171no—
217C2×SD16—171no—
218C2×SD16—171no—
219C2×D8—171no—
220C2×M16—171no—
221C2×M16—171no—
222#123—341no—
223#123—341no—
224C22⋊Q8—171no—
225C22⋊Q8—171no—
226C22⋊Q8—171no—
227C22⋊Q8C2×D1711yescharacteristic
228#121—171no—
229#121—171no—
230#121—171no—
231#121—171no—
232#120—341no—
233#120—341no—
234C4×D4—341no—
235C4×D4—341no—
236C4×D4—341no—
237C4×D4—341no—
238#117—171no—
239#117C2×D1711yescharacteristic
240#107—171no—
241#107C2×D1711yescharacteristic, complemented
242#106—171no—
243#106C2×D1711yescharacteristic, complemented
244#103—171no—
245#103—171no—
246#103C2×D1711yescharacteristic, complemented
247#103C2×D1711yescharacteristic, complemented
248#102—171no—
249#102—171no—
250#102—171no—
251#102—171no—
252C22⋊C8—171no—
253C22⋊C8—171no—
254C22⋊C8—171no—
255C22⋊C8C2×D1711yescharacteristic
256C17#319711yescharacteristic, complemented, Sylow
257C24—171no—
258#41—681no—
259#41—681no—
260#41—681no—
261#41—681no—
262C2×Q8—171no—
263C2×Q8C22×D1711yescharacteristic
264C2×D4—171no—
265C2×D4—171no—
266C2×D4—341no—
267C2×D4—341no—
268C2×D4—341no—
269C2×D4—341no—
270C2×D4—341no—
271C2×D4—341no—
272C22×C4—171no—
273C22×C4—171no—
274C22×C4—171no—
275C22×C4—171no—
276C22×C4—171no—
277C22×C4—341no—
278C22×C4—341no—
279C22×C4—681no—
280C22×C4—681no—
281C22×C4C22×D1711yescharacteristic
282Q16—681no—
283SD16—681no—
284SD16—681no—
285D8—681no—
286M16—681no—
287M16—681no—
288C2×C8—171no—
289C2×C8—171no—
290C2×C8C22×D1711yescharacteristic
291C2×C8C22×D1711yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—171no—
294C4⋊C4—341no—
295C4⋊C4—341no—
296C4⋊C4—341no—
297C4⋊C4C22×D1711yescharacteristic
298C4⋊C4C22×D1711yescharacteristic
299C4⋊C4C22×D1711yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—341no—
303C22⋊C4—341no—
304C22⋊C4—341no—
305C22⋊C4—341no—
306C22⋊C4—341no—
307C22⋊C4—341no—
308C22⋊C4—341no—
309C22⋊C4—341no—
310C42—21no—
311C42—341no—
312C23—171no—
313C23—171no—
314C23—341no—
315C23—341no—
316C23—681no—
317C23—681no—
318C23—681no—
319C23D4×D1711yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—341no—
323Q8—341no—
324D4—341no—
325D4—341no—
326D4—341no—
327D4—341no—
328D4—682no—
329D4—682no—
330D4—682no—
331D4—682no—
332D4—682no—
333D4—682no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—171no—
341C2×C4—171no—
342C2×C4—341no—
343C2×C4—341no—
344C2×C4—341no—
345C2×C4—341no—
346C2×C4—341no—
347C2×C4—341no—
348C2×C4—681no—
349C2×C4—681no—
350C2×C4—682no—
351C2×C4—682no—
352C2×C4—681no—
353C2×C4D4×D1711yescharacteristic
354C2×C4C23×D1711yescharacteristic
355C8—21no—
356C8—21no—
357C8—341no—
358C8—341no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—341no—
363C22—341no—
364C22—341no—
365C22—341no—
366C22—341no—
367C22—341no—
368C22—684no—
369C22—681no—
370C22—681no—
371C22—681no—
372C22—681no—
373C22—681no—
374C22—681no—
375C22C2×D4×D1711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—341no—
382C4—341no—
383C4—681no—
384C4—681no—
385C4#1058769611yescharacteristic
386C4#1058769611yescharacteristic
387C2—41no—
388C2—681no—
389C2—681no—
390C2—1361no—
391C2—1361no—
392C2#1176337011yescharacteristic
393C2#1176360611yescharacteristic
394C2#1176360811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#485C2
socleC22×C17C22×D4
hypercenter#485D17
hypocenterC17#3197
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #11763322C2semidirect
2. #11763321C2semidirect
3. #11763153C2semidirect
4. #11763136C2semidirect
5. #11763133C2semidirect
6. #11763066C2semidirect
7. #11763063C2semidirect
8. #11763058C2semidirect
9. #11763056C2semidirect
10. #11762974C2semidirect
11. #11762969C2semidirect
12. #11762957C2semidirect
13. #11762956C2semidirect
14. #11762771C2semidirect
15. C17×#485C2semidirect
22. C2×#63231C22semidirect
23. C2×#63230C22semidirect
25. C2×D136C22semidirect
26. C2×#63219C22semidirect
43. C17×#107C22semidirect
44. C17×#106C22semidirect
45. C17×#103C22semidirect
46. C17×#103C22semidirect
49. #10587492C22semidirect
52. #10587487C22semidirect
53. #10587481C22semidirect
54. #10587480C22semidirect
55. #10587480C22semidirect
56. #10587480C22semidirect
57. #10587479C22semidirect
58. #10587478C22semidirect
193. #485D17semidirect
241. #107C2×D17semidirect
243. #106C2×D17semidirect
246. #103C2×D17semidirect
247. #103C2×D17semidirect
256. C17#3197semidirect

Maximal subgroups

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

subgroupindex
#117633222
#117633212
#117631532
#117631362
#117631332
#117630662
#117630632
#117630582
#117630562
#117629742
#117629692
#117629572
#117629562
#117627712
C17×#4852
#319717