Finite Group Databasebeta

← #14539638

Subgroups — #14539638

Order 2176.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763387C211yesmaximal, characteristic, complemented
2#11763373C211yesmaximal, characteristic, complemented
3#11763151C211yesmaximal, characteristic, complemented
4#11763134C211yesmaximal, characteristic, complemented
5#11763128C211yesmaximal, characteristic, complemented
6#11763123C211yesmaximal, characteristic, complemented
7#11763062C211yesmaximal, characteristic, complemented
8#11763055C211yesmaximal, characteristic, complemented
9#11763045C211yesmaximal, characteristic, complemented
10#11763038C211yesmaximal, characteristic, complemented
11#11762962C211yesmaximal, characteristic, complemented
12#11762956C211yesmaximal, characteristic, complemented
13#11762956C211yesmaximal, characteristic, complemented
14#11762945C211yesmaximal, characteristic, complemented
15C17×#408C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C2×#63252C2211yescharacteristic, complemented
18C22×D68C2211yescharacteristic
19C2×M16×C17C2211yescharacteristic, complemented
20C22×C8×C17C2211yescharacteristic, complemented
21#10587633—21no—
22#10587632—21no—
23#10587626—21no—
24#10587625—21no—
25#10587625—21no—
26#10587624—21no—
27#10587623—21no—
28#10587623—21no—
29#10587616C2211yescharacteristic
30#10587616C2211yescharacteristic
31#10587616C2211yescharacteristic
32#10587616C2211yescharacteristic
33#10587616C2211yescharacteristic
34#10587616C2211yescharacteristic, complemented
35#10587612C2211yescharacteristic
36C2×C4⋊Dic17C2211yescharacteristic, complemented
37#10587610C2211yescharacteristic
38#10587610C2211yescharacteristic, complemented
39C2×D136C2211yescharacteristic
40C2×D136C2211yescharacteristic
41C2×#63219C2211yescharacteristic
42C2×#63219C2211yescharacteristic
43C17×C22⋊C8C2211yescharacteristic
44C17×C22⋊C8C2211yescharacteristic
45C17×C22⋊C8C2211yescharacteristic, complemented
46C17×C22⋊C8C2211yescharacteristic, complemented
47#10587492C2211yescharacteristic
48#10587492C2211yescharacteristic
49#10587492C2211yescharacteristic
50#10587492C2211yescharacteristic
51#10587492C2211yescharacteristic, complemented
52#10587492C2211yescharacteristic, complemented
53#10587489C2211yescharacteristic, complemented
54#10587489C2211yescharacteristic, complemented
55#10587488C2211yescharacteristic, complemented
56#10587488C2211yescharacteristic, complemented
57#10587487C2211yescharacteristic, complemented
58#10587487C2211yescharacteristic, complemented
59C23×D17—21no—
60#41×C17—41no—
61#41×C17—41no—
62C2×Q8×C17D411yescharacteristic
63C2×D4×C17D411yescharacteristic
64C2×D4×C17D411yescharacteristic
65C2×D4×C17D411yescharacteristic
66C22×C4×C17C2311yescharacteristic
67C22×C4×C17C2311yescharacteristic
68C22×C4×C17C2311yescharacteristic
69C2×C22⋊D17—21no—
70C2×C22⋊D17—21no—
71C2×C22⋊D17—21no—
72C2×C22⋊D17—21no—
73C2×C22⋊D17—21no—
74C2×C22⋊D17—21no—
75C2×C22⋊D17—21no—
76C22×Dic17—21no—
77#63252—41no—
78#63252—41no—
79C2×D68—21no—
80C2×D68—41no—
81C2×D68—41no—
82C2×D68C2311yescharacteristic
83C2×D68C2311yescharacteristic
84C2×D68C2311yescharacteristic
85C2×C4×D17—21no—
86C2×Dic34C2311yescharacteristic
87M16×C17—41no—
88C2×C8×C17—41no—
89C2×C8×C17C2311yescharacteristic
90C2×C8×C17C2311yescharacteristic
91C2×C8×C17C2311yescharacteristic
92C2×C8×C17C2311yescharacteristic
93C22⋊Dic17—21no—
94C22⋊Dic17—21no—
95C22⋊Dic17—21no—
96#63227—21no—
97#63227—21no—
98#63227—21no—
99#63227—21no—
100#63227—21no—
101#63227—21no—
102C4⋊Dic17—21no—
103C4⋊Dic17C2311yescharacteristic
104C4⋊Dic17C2311yescharacteristic
105C4⋊Dic17C2311yescharacteristic
106C4⋊Dic17C2311yescharacteristic
107#63225—21no—
108#63225—21no—
109C4×Dic17—21no—
110D136—41no—
111D136—41no—
112#63219—41no—
113#63219—41no—
114C23×C17C2×D411yescharacteristic
115C23×C17C2×D411yescharacteristic
116C23×C17C2×D411yescharacteristic
117C22×D17—21no—
118C22×D17—21no—
119C22×D17—21no—
120C22×D17—42no—
121C22×D17—81no—
122C22×D17—81no—
123Q8×C17—41no—
124D4×C17—41no—
125D4×C17—41no—
126D4×C17—41no—
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×C17C2×D411yescharacteristic
134C2×C4×C17C2×D411yescharacteristic
135C2×C4×C17C2×D411yescharacteristic
136C2×C4×C17C2411yescharacteristic
137C22⋊D17—42no—
138C22⋊D17—81no—
139C22⋊D17—81no—
140C22⋊D17—81no—
141C22⋊D17—81no—
142C22⋊D17—81no—
143C22⋊D17—81no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—81no—
150D68—21no—
151D68—21no—
152D68—21no—
153D68—21no—
154D68—21no—
155D68—21no—
156D68—42no—
157C4×D17—42no—
158Dic34—21no—
159Dic34—21no—
160C8×C17—21no—
161C8×C17—21no—
162C8×C17—21no—
163C8×C17—21no—
164#3198—171nomaximal, Sylow
165C22×C17—21no—
166C22×C17—21no—
167C22×C17—21no—
168C22×C17—21no—
169C22×C17—21no—
170C22×C17—21no—
171C22×C17—21no—
172C22×C17—21no—
173C22×C17—21no—
174C22×C17C22×D411yescharacteristic
175C2×D17—41no—
176C2×D17—41no—
177C2×D17—41no—
178C2×D17—41no—
179C2×D17—41no—
180C2×D17—41no—
181C2×D17—41no—
182C2×D17—41no—
183C2×D17—41no—
184C2×D17—84no—
185C4×C17—21no—
186C4×C17—21no—
187C4×C17—21no—
188C4×C17H(5,𝔽2)11yescharacteristic
189C4×C17H(5,𝔽2)11yescharacteristic
190Dic17—41no—
191Dic17—41no—
192Dic17—41no—
193Dic17—41no—
194Dic17—41no—
195#537—171no—
196#535—171no—
197#485—171no—
198#482—171no—
199#481—171no—
200#480—171no—
201#469—171no—
202#468—171no—
203#466—171no—
204#465—171no—
205#450—171no—
206#449—171no—
207#449—171no—
208#447—171no—
209#408D1711yescharacteristic, complemented
210C2×C17—41no—
211C2×C17—41no—
212C2×C17—41no—
213C2×C17#53411yescharacteristic
214C2×C17#57611yescharacteristic
215C2×C17#57711yescharacteristic
216D17—81no—
217D17—81no—
218D17—81no—
219C2×#41—171no—
220C2×#41C2×D1711yescharacteristic
221C22×D4—171no—
222C2×SD16—171no—
223C2×SD16—171no—
224C2×D8—171no—
225C2×D8—171no—
226C2×M16C2×D1711yescharacteristic, complemented
227C22×C8C2×D1711yescharacteristic, complemented
228#127—341no—
229#124—341no—
230C22⋊Q8—171no—
231C22⋊Q8—171no—
232C22⋊Q8—341no—
233#121—171no—
234#121—171no—
235#121—171no—
236#121—171no—
237#121—171no—
238#121—171no—
239#121—341no—
240#121—341no—
241#121—341no—
242#120—341no—
243#120—341no—
244#117—171no—
245C2×C4⋊C4—171no—
246#107—171no—
247#107—171no—
248#106—171no—
249#106—171no—
250#103—171no—
251#103—171no—
252#102—171no—
253#102—171no—
254#102—171no—
255#102—171no—
256#102—171no—
257#102—171no—
258C22⋊C8C2×D1711yescharacteristic
259C22⋊C8C2×D1711yescharacteristic
260C22⋊C8C2×D1711yescharacteristic, complemented
261C22⋊C8C2×D1711yescharacteristic, complemented
262C17#319811yescharacteristic, complemented, Sylow
263C24—341no—
264#41—41no—
265#41—41no—
266#41—681no—
267#41—681no—
268C2×Q8—171no—
269C2×Q8D6811yescharacteristic
270C2×D4—171no—
271C2×D4—171no—
272C2×D4—171no—
273C2×D4—341no—
274C2×D4—341no—
275C2×D4—341no—
276C2×D4—341no—
277C2×D4—341no—
278C2×D4—341no—
279C2×D4—341no—
280C2×D4—341no—
281C2×D4—681no—
282C2×D4—681no—
283C2×D4D6811yescharacteristic
284C2×D4D6811yescharacteristic
285C2×D4D6811yescharacteristic
286C22×C4—341no—
287C22×C4—341no—
288C22×C4C22×D1711yescharacteristic
289C22×C4C22×D1711yescharacteristic
290C22×C4C22×D1711yescharacteristic
291SD16—681no—
292SD16—681no—
293D8—681no—
294D8—681no—
295M16—41no—
296C2×C8—41no—
297C2×C8C22×D1711yescharacteristic
298C2×C8C22×D1711yescharacteristic
299C2×C8C22×D1711yescharacteristic
300C2×C8C22×D1711yescharacteristic
301C4⋊C4—171no—
302C4⋊C4—171no—
303C4⋊C4—171no—
304C4⋊C4—171no—
305C4⋊C4—341no—
306C4⋊C4—341no—
307C4⋊C4—341no—
308C22⋊C4—341no—
309C22⋊C4—341no—
310C22⋊C4—341no—
311C22⋊C4—341no—
312C22⋊C4—341no—
313C22⋊C4—341no—
314C22⋊C4—341no—
315C22⋊C4—341no—
316C22⋊C4—341no—
317C42—341no—
318C23—341no—
319C23—341no—
320C23—341no—
321C23—682no—
322C23—1361no—
323C23—1361no—
324C23C2×D6811yescharacteristic
325C23C2×D6811yescharacteristic
326C23C2×D6811yescharacteristic
327Q8—41no—
328Q8—341no—
329Q8—341no—
330D4—41no—
331D4—41no—
332D4—41no—
333D4—341no—
334D4—341no—
335D4—341no—
336D4—341no—
337D4—341no—
338D4—341no—
339D4—682no—
340D4—682no—
341D4—1361no—
342D4—1361no—
343D4—1361no—
344D4—1361no—
345D4—1361no—
346D4—1361no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—341no—
354C2×C4—341no—
355C2×C4—341no—
356C2×C4—341no—
357C2×C4—341no—
358C2×C4—682no—
359C2×C4—1361no—
360C2×C4C2×D6811yescharacteristic
361C2×C4C2×D6811yescharacteristic
362C2×C4C2×D6811yescharacteristic
363C2×C4C23×D1711yescharacteristic
364C8—21no—
365C8—21no—
366C8—21no—
367C8—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—681no—
378C22—681no—
379C22—681no—
380C22—681no—
381C22—681no—
382C22—681no—
383C22—681no—
384C22—681no—
385C22—681no—
386C22—1364no—
387C22C22×D6811yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—681no—
392C4—681no—
393C4—681no—
394C4—681no—
395C4—681no—
396C4#1058769111yescharacteristic
397C4#1058769111yescharacteristic
398C2—41no—
399C2—41no—
400C2—41no—
401C2—1361no—
402C2—1361no—
403C2—1361no—
404C2#1176337111yescharacteristic
405C2#1176361411yescharacteristic
406C2#1176362111yescharacteristic
subgroupG/H
centerC22C22×D68
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#408C2
socleC22×C17C22×D4
hypercenter#408D17
hypocenterC17#3198
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #11763387C2semidirect
2. #11763373C2semidirect
3. #11763151C2semidirect
4. #11763134C2semidirect
5. #11763128C2semidirect
6. #11763123C2semidirect
7. #11763062C2semidirect
8. #11763055C2semidirect
9. #11763045C2semidirect
10. #11763038C2semidirect
11. #11762962C2semidirect
12. #11762956C2semidirect
13. #11762956C2semidirect
14. #11762945C2semidirect
15. C17×#408C2semidirect
17. C2×#63252C22semidirect
19. C2×M16×C17C22semidirect
20. C22×C8×C17C22semidirect
34. #10587616C22semidirect
36. C2×C4⋊Dic17C22semidirect
38. #10587610C22semidirect
45. C17×C22⋊C8C22semidirect
46. C17×C22⋊C8C22semidirect
51. #10587492C22semidirect
52. #10587492C22semidirect
53. #10587489C22semidirect
54. #10587489C22semidirect
55. #10587488C22semidirect
56. #10587488C22semidirect
57. #10587487C22semidirect
58. #10587487C22semidirect
209. #408D17semidirect
226. C2×M16C2×D17semidirect
227. C22×C8C2×D17semidirect
260. C22⋊C8C2×D17semidirect
261. C22⋊C8C2×D17semidirect
262. C17#3198semidirect

Maximal subgroups

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

subgroupindex
#117633872
#117633732
#117631512
#117631342
#117631282
#117631232
#117630622
#117630552
#117630452
#117630382
#117629622
#117629562
#117629452
C17×#4082
#319817