Finite Group Databasebeta

← #14540922

Subgroups — #14540922

Order 2176.

462 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763512C211yesmaximal, characteristic, complemented
2D4×D68C211yesmaximal, characteristic, complemented
3#11763175C211yesmaximal, characteristic, complemented
4#11763136C211yesmaximal, characteristic, complemented
5#11763129C211yesmaximal, characteristic, complemented
6#11763063C211yesmaximal, characteristic, complemented
7#11763046C211yesmaximal, characteristic, complemented
8#11763005C211yesmaximal, characteristic, complemented
9#11762944C211yesmaximal, characteristic, complemented
10#11762944C211yesmaximal, characteristic, complemented
11#11762919C211yesmaximal, characteristic, complemented
12#11762879C211yesmaximal, characteristic, complemented
13#11762812C211yesmaximal, characteristic, complemented
14#102×D17C211yesmaximal, characteristic, complemented
15C17×#478C211yesmaximal, characteristic, complemented
16C2×D4×D17C2211yescharacteristic
17C2×D4×D17C2211yescharacteristic
18C22×D68—21no—
19C17×#125C2211yescharacteristic, complemented
20C4×D4×C17C2211yescharacteristic
21C2×#63228C2211yescharacteristic
22#10587616—21no—
23C2×D136C2211yescharacteristic, complemented
24C2×D136C2211yescharacteristic, complemented
25C2×#63218C2211yescharacteristic, complemented
26C2×C8×D17C2211yescharacteristic, complemented
27#10587584—21no—
28#10587581—21no—
29#10587581C2211yescharacteristic
30#10587581C2211yescharacteristic
31#10587581C2211yescharacteristic, complemented
32#10587581C2211yescharacteristic, complemented
33#10587580—21no—
34#10587579—21no—
35#10587578C2211yescharacteristic, complemented
36C4⋊C4×D17C2211yescharacteristic, complemented
37#10587568—21no—
38#10587568—21no—
39#10587560C2211yescharacteristic
40C4×D68C2211yescharacteristic
41C4×D68C2211yescharacteristic, complemented
42C17×#107C2211yescharacteristic, complemented
43C17×#106C2211yescharacteristic, complemented
44C17×C4⋊C8C2211yescharacteristic, complemented
45C17×#102C2211yescharacteristic
46C17×#102C2211yescharacteristic
47#10587503C2211yescharacteristic
48#10587492C2211yescharacteristic, complemented
49#10587492C2211yescharacteristic, complemented
50#10587491C2211yescharacteristic, complemented
51#10587491C2211yescharacteristic, complemented
52#10587480C2211yescharacteristic, complemented
53#10587480C2211yescharacteristic, complemented
54#10587480C2211yescharacteristic, complemented
55#10587480C2211yescharacteristic, complemented
56#10587480C2211yescharacteristic, complemented
57#10587479C2211yescharacteristic, complemented
58#10587474C2211yescharacteristic, complemented
59C23×D17—21no—
60C23×D17—21no—
61C2×D4×C17C2311yescharacteristic
62C22×C4×C17—21no—
63C2×C22⋊D17—21no—
64C2×C22⋊D17—21no—
65D4×D17—22no—
66D4×D17—22no—
67D4×D17—41no—
68D4×D17—41no—
69C2×D68—21no—
70C2×D68—21no—
71C2×D68—21no—
72C2×D68—21no—
73C2×D68—21no—
74C2×D68—21no—
75C2×D68—44no—
76C2×D68D411yescharacteristic
77C2×D68C2311yescharacteristic
78C2×D68C2311yescharacteristic
79C2×D68C2311yescharacteristic
80C2×C4×D17—21no—
81C2×C4×D17—21no—
82C2×C4×D17C2311yescharacteristic
83C2×C4×D17C2311yescharacteristic
84C2×C8×C17C2311yescharacteristic
85C2×C8×C17C2311yescharacteristic
86C4⋊C4×C17—21no—
87C4⋊C4×C17—21no—
88C4⋊C4×C17C2311yescharacteristic
89C4⋊C4×C17C2311yescharacteristic
90C4⋊C4×C17C2311yescharacteristic
91C22⋊C4×C17—21no—
92C42×C17C2311yescharacteristic
93#63228—22no—
94#63227—21no—
95#63227—21no—
96#63227—21no—
97#63227—21no—
98#63227—21no—
99#63227—21no—
100#63227D411yescharacteristic
101#63227D411yescharacteristic
102C4⋊Dic17D411yescharacteristic
103C4⋊Dic17C2311yescharacteristic
104#63225—21no—
105C4×Dic17—21no—
106C2×#3622C2311yescharacteristic
107C2×#3622C2311yescharacteristic
108D136—41no—
109D136—41no—
110#63218—41no—
111C8×D17—41no—
112C23×C17—21no—
113C22×D17—21no—
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
123C22×D17C2×D411yescharacteristic
124D4×C17—21no—
125D4×C17#4112yes—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—42no—
130C2×C4×C17C2×D411yescharacteristic
131C2×C4×C17C2×D411yescharacteristic
132C2×C4×C17C2411yescharacteristic
133C22⋊D17—42no—
134C22⋊D17—42no—
135C2×Dic17—21no—
136C2×Dic17C2×D411yescharacteristic
137C2×Dic17C2×D411yescharacteristic
138D68—22no—
139D68—21no—
140D68—21no—
141D68—21no—
142D68—21no—
143D68—21no—
144D68—21no—
145D68—42no—
146D68—42no—
147D68—42no—
148D68—42no—
149D68—81no—
150D68—81no—
151C4×D17—21no—
152C4×D17—21no—
153C4×D17—21no—
154C4×D17—21no—
155C4×D17—81no—
156C4×D17—81no—
157C8×C17—21no—
158C8×C17—21no—
159#3622—21no—
160#3622—21no—
161#3318—171nomaximal, Sylow
162C22×C17—22no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17C22×D411yescharacteristic
166C2×D17—21no—
167C2×D17—21no—
168C2×D17—21no—
169C2×D17—21no—
170C2×D17—21no—
171C2×D17—41no—
172C2×D17—41no—
173C2×D17—41no—
174C2×D17—41no—
175C2×D17—41no—
176C2×D17—41no—
177C2×D17—41no—
178C2×D17—41no—
179C2×D17—41no—
180C2×D17—84no—
181C2×D17—84no—
182C2×D17AGL(1,ℤ/8)12yes—
183C4×C17—21no—
184C4×C17—21no—
185C4×C17—41no—
186C4×C17—41no—
187C4×C17—41no—
188C4×C17C2×#4111yescharacteristic
189C4×C17H(5,𝔽2)11yescharacteristic
190Dic17—21no—
191Dic17—21no—
192Dic17—41no—
193#553—171no—
194D42—171no—
195#490—171no—
196#482—171no—
197#481—171no—
198#478D1711yescharacteristic, complemented
199#469—171no—
200#466—171no—
201#459—171no—
202#447—171no—
203#447—171no—
204#442—171no—
205#435—171no—
206#418—171no—
207C2×#102—171no—
208C2×C17—42no—
209C2×C17#54611yescharacteristic
210C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
211C2×C17#57611yescharacteristic
212D17—22no—
213D17—41no—
214D17—81no—
215D17—81no—
216D17—81no—
217C22×D4—171no—
218C22×D4—171no—
219C22×D4—341no—
220C2×D8—171no—
221C2×D8—171no—
222C2×D8—171no—
223C2×M16—171no—
224C22×C8—171no—
225#127—171no—
226#126—341no—
227#125C2×D1711yescharacteristic, complemented
228#123—341no—
229#121—171no—
230#121—171no—
231#121—171no—
232#121—171no—
233#121—341no—
234#121—341no—
235#120—341no—
236#120—341no—
237C4×D4—171no—
238C4×D4—171no—
239C4×D4—341no—
240C4×D4C2×D1711yescharacteristic
241#117—171no—
242C2×C4⋊C4—171no—
243#107C2×D1711yescharacteristic, complemented
244#106—171no—
245#106C2×D1711yescharacteristic, complemented
246C4⋊C8C2×D1711yescharacteristic, complemented
247#102—171no—
248#102—171no—
249#102—171no—
250#102—171no—
251#102—171no—
252#102—171no—
253#102—171no—
254#102—171no—
255#102C2×D1711yescharacteristic
256#102C2×D1711yescharacteristic
257C22⋊C8—171no—
258C22⋊C8—171no—
259#97—171no—
260C17#331811yescharacteristic, complemented, Sylow
261C24—341no—
262C24—341no—
263C2×D4—171no—
264C2×D4—171no—
265C2×D4—171no—
266C2×D4—171no—
267C2×D4—341no—
268C2×D4—341no—
269C2×D4—341no—
270C2×D4—341no—
271C2×D4—341no—
272C2×D4—341no—
273C2×D4—342no—
274C2×D4—342no—
275C2×D4—341no—
276C2×D4—341no—
277C2×D4—684no—
278C2×D4—681no—
279C2×D4—681no—
280C2×D4C22×D1711yescharacteristic
281C22×C4—21no—
282C22×C4—171no—
283C22×C4—171no—
284C22×C4—341no—
285C22×C4—341no—
286D8—342no—
287D8—681no—
288D8—681no—
289M16—681no—
290C2×C8—171no—
291C2×C8—171no—
292C2×C8—681no—
293C2×C8C22×D1711yescharacteristic
294C2×C8C22×D1711yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—171no—
298C4⋊C4—171no—
299C4⋊C4—341no—
300C4⋊C4C22×D1711yescharacteristic
301C4⋊C4C22×D1711yescharacteristic
302C4⋊C4C22×D1711yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—171no—
305C22⋊C4—171no—
306C22⋊C4—341no—
307C22⋊C4—341no—
308C22⋊C4—341no—
309C22⋊C4—341no—
310C22⋊C4—341no—
311C22⋊C4—341no—
312C42—341no—
313C42C22×D1711yescharacteristic
314C23—21no—
315C23—171no—
316C23—171no—
317C23—341no—
318C23—341no—
319C23—341no—
320C23—682no—
321C23—682no—
322C23—682no—
323C23—682no—
324C23—1361no—
325C23—1361no—
326D4—21no—
327D4—342no—
328D4—341no—
329D4—341no—
330D4—341no—
331D4—341no—
332D4—341no—
333D4—341no—
334D4—682no—
335D4—682no—
336D4—682no—
337D4—682no—
338D4—682no—
339D4—682no—
340D4—1361no—
341D4—1361no—
342D4#6325612yes—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—171no—
348C2×C4—171no—
349C2×C4—341no—
350C2×C4—341no—
351C2×C4—341no—
352C2×C4—341no—
353C2×C4—341no—
354C2×C4—1361no—
355C2×C4—1361no—
356C2×C4D4×D1711yescharacteristic
357C2×C4D4×D1711yescharacteristic
358C2×C4C23×D1711yescharacteristic
359C8—21no—
360C8—21no—
361C8—341no—
362C8—341no—
363C22—22no—
364C22—41no—
365C22—41no—
366C22—172no—
367C22—341no—
368C22—341no—
369C22—341no—
370C22—341no—
371C22—341no—
372C22—681no—
373C22—681no—
374C22—681no—
375C22—681no—
376C22—681no—
377C22—681no—
378C22—681no—
379C22—681no—
380C22—681no—
381C22—1364no—
382C22—1364no—
383C22C2×D4×D1711yescharacteristic
384C4—21no—
385C4—21no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—341no—
390C4—341no—
391C4—681no—
392C4C2×#6325611yescharacteristic
393C4#1058769611yescharacteristic
394C2—42no—
395C2—342no—
396C2—681no—
397C2—1361no—
398C2—1361no—
399C2—1361no—
400C2#1176345011yescharacteristic
401C2AGL(1,ℤ/8)×D1711yescharacteristic
402C2#1176361111yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#478C2
socleC22×C17C22×D4
hypercenter#478D17
hypocenterC17#3318
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #11763512C2semidirect
2. D4×D68C2semidirect
3. #11763175C2semidirect
4. #11763136C2semidirect
5. #11763129C2semidirect
6. #11763063C2semidirect
7. #11763046C2semidirect
8. #11763005C2semidirect
9. #11762944C2semidirect
10. #11762944C2semidirect
11. #11762919C2semidirect
12. #11762879C2semidirect
13. #11762812C2semidirect
14. #102×D17C2semidirect
15. C17×#478C2semidirect
19. C17×#125C22semidirect
23. C2×D136C22semidirect
24. C2×D136C22semidirect
25. C2×#63218C22semidirect
26. C2×C8×D17C22semidirect
31. #10587581C22semidirect
32. #10587581C22semidirect
35. #10587578C22semidirect
36. C4⋊C4×D17C22semidirect
41. C4×D68C22semidirect
42. C17×#107C22semidirect
43. C17×#106C22semidirect
44. C17×C4⋊C8C22semidirect
48. #10587492C22semidirect
49. #10587492C22semidirect
50. #10587491C22semidirect
51. #10587491C22semidirect
52. #10587480C22semidirect
53. #10587480C22semidirect
54. #10587480C22semidirect
55. #10587480C22semidirect
56. #10587480C22semidirect
57. #10587479C22semidirect
58. #10587474C22semidirect
198. #478D17semidirect
227. #125C2×D17semidirect
243. #107C2×D17semidirect
245. #106C2×D17semidirect
246. C4⋊C8C2×D17semidirect
260. C17#3318semidirect

Maximal subgroups

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

subgroupindex
#117635122
D4×D682
#117631752
#117631362
#117631292
#117630632
#117630462
#117630052
#117629442
#117629192
#117628792
#117628122
#102×D172
C17×#4782
#331817