Finite Group Databasebeta

← #14539884

Subgroups — #14539884

Order 2176.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763432C211yesmaximal, characteristic, complemented
2#11763373C211yesmaximal, characteristic, complemented
3#11763195C211yesmaximal, characteristic, complemented
4#11763194C211yesmaximal, characteristic, complemented
5#11763170C211yesmaximal, characteristic, complemented
6#11763154C211yesmaximal, characteristic, complemented
7#11763033C211yesmaximal, characteristic, complemented
8#11763027C211yesmaximal, characteristic, complemented
9#11763007C211yesmaximal, characteristic, complemented
10#11763003C211yesmaximal, characteristic, complemented
11#11762959C211yesmaximal, characteristic, complemented
12#11762954C211yesmaximal, characteristic, complemented
13#11762945C211yesmaximal, characteristic, complemented
14#11762867C211yesmaximal, characteristic, complemented
15C17×#452C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C22×D68C2211yescharacteristic
18C2×Q16×C17C2211yescharacteristic, complemented
19C2×SD16×C17C2211yescharacteristic, complemented
20C17×C22⋊Q8C2211yescharacteristic
21#10587633C2211yescharacteristic
22#10587633C2211yescharacteristic
23#10587633C2211yescharacteristic, complemented
24Q8×Dic17C2211yescharacteristic, complemented
25C2×#63230C2211yescharacteristic
26C2×#63230C2211yescharacteristic
27C2×#63230C2211yescharacteristic, complemented
28#10587626C2211yescharacteristic
29#10587626C2211yescharacteristic
30#10587623—21no—
31#10587623—21no—
32C2×#63228C2211yescharacteristic, complemented
33#10587616—21no—
34#10587616—21no—
35#10587616C2211yescharacteristic
36#10587612C2211yescharacteristic
37C2×D136C2211yescharacteristic
38C2×D136C2211yescharacteristic
39#10587584—21no—
40#10587583—21no—
41#10587579C2211yescharacteristic, complemented
42#10587574C2211yescharacteristic, complemented
43#10587571—21no—
44#10587569—21no—
45C17×#103C2211yescharacteristic, complemented
46C17×#102C2211yescharacteristic, complemented
47C17×C22⋊C8C2211yescharacteristic
48#10587506C2211yescharacteristic, complemented
49#10587503C2211yescharacteristic, complemented
50C22⋊#3622C2211yescharacteristic
51#10587492C2211yescharacteristic
52#10587492C2211yescharacteristic
53#10587486C2211yescharacteristic, complemented
54#10587485C2211yescharacteristic, complemented
55#10587485C2211yescharacteristic, complemented
56C8×Dic17C2211yescharacteristic, complemented
57#10587480C2211yescharacteristic
58#10587480C2211yescharacteristic
59C23×D17—21no—
60#41×C17—41no—
61#41×C17—41no—
62C2×Q8×C17C2311yescharacteristic
63C2×Q8×C17C2311yescharacteristic
64C2×D4×C17—21no—
65C2×D4×C17C2311yescharacteristic
66C22×C4×C17—21no—
67C22×C4×C17C2311yescharacteristic
68C2×C22⋊D17—21no—
69C2×C22⋊D17—21no—
70C2×C22⋊D17—21no—
71C2×C22⋊D17—21no—
72C2×C22⋊D17—21no—
73C2×D68—21no—
74C2×D68—41no—
75C2×D68—41no—
76C2×D68C2311yescharacteristic
77C2×D68C2311yescharacteristic
78C2×D68C2311yescharacteristic
79C2×C4×D17—21no—
80C2×Dic34—21no—
81Q16×C17—41no—
82SD16×C17—41no—
83C2×C8×C17C2311yescharacteristic
84C2×C8×C17C2311yescharacteristic
85C4⋊C4×C17—21no—
86C4⋊C4×C17C2311yescharacteristic
87C22⋊C4×C17—21no—
88C22⋊Dic17D411yescharacteristic
89C22⋊Dic17D411yescharacteristic
90#63230—41no—
91#63230—41no—
92#63230—41no—
93#63228—41no—
94#63227—21no—
95#63227—21no—
96#63227—21no—
97#63227—21no—
98#63227—21no—
99#63227—21no—
100#63227—21no—
101#63227—21no—
102C4⋊Dic17—21no—
103C4⋊Dic17D411yescharacteristic
104C4⋊Dic17D411yescharacteristic
105C4⋊Dic17C2311yescharacteristic
106#63225—21no—
107C4×Dic17—21no—
108C4×Dic17C2311yescharacteristic
109C4×Dic17C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112D136—41no—
113D136—41no—
114C23×C17—21no—
115C23×C17C2×D411yescharacteristic
116C22×D17—21no—
117C22×D17—21no—
118C22×D17—21no—
119C22×D17—42no—
120C22×D17—81no—
121C22×D17—81no—
122Q8×C17—21no—
123Q8×C17—21no—
124Q8×C17—21no—
125Q8×C17—21no—
126D4×C17—21no—
127D4×C17—21no—
128D4×C17—42no—
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×C17C2411yescharacteristic
137C22⋊D17—81no—
138C22⋊D17—81no—
139C22⋊D17—81no—
140C22⋊D17—81no—
141C22⋊D17—81no—
142C2×Dic17—21no—
143C2×Dic17C2×D411yescharacteristic
144C2×Dic17C2×D411yescharacteristic
145C2×Dic17C2×D411yescharacteristic
146C2×Dic17C2×D411yescharacteristic
147D68—21no—
148D68—21no—
149D68—21no—
150D68—21no—
151D68—21no—
152D68—21no—
153D68—42no—
154C4×D17—81no—
155Dic34—81no—
156C8×C17—21no—
157C8×C17—21no—
158#3622—21no—
159#3622—21no—
160#3224—171nomaximal, Sylow
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17C22×D411yescharacteristic
168C2×D17—41no—
169C2×D17—41no—
170C2×D17—41no—
171C2×D17—41no—
172C2×D17—41no—
173C2×D17—41no—
174C2×D17—41no—
175C2×D17—41no—
176C2×D17—41no—
177C2×D17—84no—
178C4×C17—21no—
179C4×C17—41no—
180C4×C17—41no—
181C4×C17—41no—
182C4×C17H(5,𝔽2)11yescharacteristic
183C4×C17H(5,𝔽2)11yescharacteristic
184Dic17—21no—
185Dic17—21no—
186Dic17—21no—
187Dic17—21no—
188Dic17—41no—
189#542—171no—
190#535—171no—
191#496—171no—
192#495—171no—
193#489—171no—
194#486—171no—
195#464—171no—
196#463—171no—
197#460—171no—
198#459—171no—
199#452D1711yescharacteristic, complemented
200#450—171no—
201#449—171no—
202#447—171no—
203#433—171no—
204C2×C17—41no—
205C2×C17—81no—
206C2×C17#53511yescharacteristic
207C2×C17#57611yescharacteristic
208C2×C17#57711yescharacteristic
209D17—81no—
210D17—81no—
211D17—81no—
212C2×#41C2×D1711yescharacteristic
213C22×D4—171no—
214C2×Q16C2×D1711yescharacteristic, complemented
215C2×SD16—171no—
216C2×SD16—171no—
217C2×SD16—171no—
218C2×SD16C2×D1711yescharacteristic, complemented
219C2×D8—171no—
220C2×D8—171no—
221C2×D8—171no—
222C4⋊Q8—171no—
223#127—171no—
224#127—171no—
225#126—341no—
226#124—171no—
227#124—171no—
228#124—171no—
229#124—341no—
230#123—341no—
231C22⋊Q8—341no—
232C22⋊Q8C2×D1711yescharacteristic
233#121—171no—
234#121—341no—
235#121—341no—
236#120—341no—
237#120—341no—
238C4×Q8—171no—
239C4×D4—171no—
240#117—171no—
241C4⋊C8—171no—
242C4⋊C8—171no—
243#103—171no—
244#103C2×D1711yescharacteristic, complemented
245#102—171no—
246#102—171no—
247#102—171no—
248#102—171no—
249#102—171no—
250#102C2×D1711yescharacteristic, complemented
251C22⋊C8—171no—
252C22⋊C8C2×D1711yescharacteristic
253#97—171no—
254C4×C8—171no—
255C17#322411yescharacteristic, complemented, Sylow
256C24—341no—
257#41—41no—
258#41—41no—
259C2×Q8—341no—
260C2×Q8C22×D1711yescharacteristic
261C2×Q8C22×D1711yescharacteristic
262C2×D4—21no—
263C2×D4—171no—
264C2×D4—171no—
265C2×D4—171no—
266C2×D4—341no—
267C2×D4—341no—
268C2×D4—341no—
269C2×D4—341no—
270C2×D4—341no—
271C2×D4—341no—
272C2×D4—681no—
273C2×D4—681no—
274C2×D4C22×D1711yescharacteristic
275C22×C4—21no—
276C22×C4—341no—
277C22×C4C22×D1711yescharacteristic
278Q16—41no—
279SD16—41no—
280SD16—681no—
281SD16—681no—
282SD16—681no—
283D8—681no—
284D8—681no—
285D8—681no—
286C2×C8—171no—
287C2×C8—171no—
288C2×C8C22×D1711yescharacteristic
289C2×C8C22×D1711yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—171no—
292C4⋊C4—171no—
293C4⋊C4—171no—
294C4⋊C4—341no—
295C4⋊C4—341no—
296C4⋊C4C22×D1711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—171no—
299C22⋊C4—171no—
300C22⋊C4—341no—
301C22⋊C4—341no—
302C22⋊C4—341no—
303C22⋊C4—341no—
304C22⋊C4—341no—
305C22⋊C4—341no—
306C22⋊C4—341no—
307C22⋊C4—341no—
308C42—171no—
309C42—171no—
310C42—341no—
311C23—21no—
312C23—341no—
313C23—341no—
314C23—341no—
315C23—682no—
316C23—1361no—
317C23—1361no—
318C23D4×D1711yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—1361no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—341no—
328D4—341no—
329D4—341no—
330D4—341no—
331D4—341no—
332D4—341no—
333D4—682no—
334D4—1361no—
335D4—1361no—
336D4—1361no—
337D4—1361no—
338D4—1361no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—171no—
346C2×C4—171no—
347C2×C4—171no—
348C2×C4—171no—
349C2×C4—341no—
350C2×C4—1361no—
351C2×C4D4×D1711yescharacteristic
352C2×C4C23×D1711yescharacteristic
353C8—21no—
354C8—21no—
355C8—341no—
356C8—341no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—681no—
364C22—681no—
365C22—681no—
366C22—681no—
367C22—681no—
368C22—681no—
369C22—681no—
370C22—681no—
371C22—681no—
372C22—1364no—
373C22C2×D4×D1711yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—341no—
379C4—341no—
380C4—341no—
381C4—341no—
382C4—681no—
383C4#1058769111yescharacteristic
384C4#1058769111yescharacteristic
385C2—41no—
386C2—81no—
387C2—1361no—
388C2—1361no—
389C2—1361no—
390C2#1176337411yescharacteristic
391C2#1176361111yescharacteristic
392C2#1176361811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#452C2
socleC22×C17C22×D4
hypercenter#452D17
hypocenterC17#3224
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #11763432C2semidirect
2. #11763373C2semidirect
3. #11763195C2semidirect
4. #11763194C2semidirect
5. #11763170C2semidirect
6. #11763154C2semidirect
7. #11763033C2semidirect
8. #11763027C2semidirect
9. #11763007C2semidirect
10. #11763003C2semidirect
11. #11762959C2semidirect
12. #11762954C2semidirect
13. #11762945C2semidirect
14. #11762867C2semidirect
15. C17×#452C2semidirect
18. C2×Q16×C17C22semidirect
19. C2×SD16×C17C22semidirect
23. #10587633C22semidirect
24. Q8×Dic17C22semidirect
27. C2×#63230C22semidirect
32. C2×#63228C22semidirect
41. #10587579C22semidirect
42. #10587574C22semidirect
45. C17×#103C22semidirect
46. C17×#102C22semidirect
48. #10587506C22semidirect
49. #10587503C22semidirect
53. #10587486C22semidirect
54. #10587485C22semidirect
55. #10587485C22semidirect
56. C8×Dic17C22semidirect
199. #452D17semidirect
214. C2×Q16C2×D17semidirect
218. C2×SD16C2×D17semidirect
244. #103C2×D17semidirect
250. #102C2×D17semidirect
255. C17#3224semidirect

Maximal subgroups

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

subgroupindex
#117634322
#117633732
#117631952
#117631942
#117631702
#117631542
#117630332
#117630272
#117630072
#117630032
#117629592
#117629542
#117629452
#117628672
C17×#4522
#322417