Finite Group Databasebeta

← #14539769

Subgroups — #14539769

Order 2176.

444 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763356C211yesmaximal, characteristic, complemented
2#124×D17C211yesmaximal, characteristic, complemented
3#11763197C211yesmaximal, characteristic, complemented
4#11763195C211yesmaximal, characteristic, complemented
5#11763170C211yesmaximal, characteristic, complemented
6#11763167C211yesmaximal, characteristic, complemented
7#11763025C211yesmaximal, characteristic, complemented
8#11763024C211yesmaximal, characteristic, complemented
9#11763021C211yesmaximal, characteristic, complemented
10#11762949C211yesmaximal, characteristic, complemented
11#11762946C211yesmaximal, characteristic, complemented
12#11762944C211yesmaximal, characteristic, complemented
13#11762941C211yesmaximal, characteristic, complemented
14#11762865C211yesmaximal, characteristic, complemented
15C17×#463C211yesmaximal, characteristic, complemented
16C2×Q8×D17C2211yescharacteristic
17C2×D4×D17C2211yescharacteristic
18C2×#63252—21no—
19C2×SD16×C17C2211yescharacteristic, complemented
20C2×D8×C17C2211yescharacteristic
21C17×#124C2211yescharacteristic
22C4×D4×C17C2211yescharacteristic, complemented
23#10587633C2211yescharacteristic
24C2×#63230C2211yescharacteristic, complemented
25#10587626C2211yescharacteristic, complemented
26#10587624C2211yescharacteristic
27#10587622C2211yescharacteristic
28C2×#63229C2211yescharacteristic, complemented
29C2×#63228C2211yescharacteristic
30C2×#63228C2211yescharacteristic
31C4×C22⋊D17—21no—
32C2×D136C2211yescharacteristic
33C2×#63219C2211yescharacteristic, complemented
34#10587581C2211yescharacteristic
35#10587574C2211yescharacteristic, complemented
36#10587571—21no—
37#10587571—21no—
38#10587571—21no—
39#10587570—21no—
40C22⋊C4×D17—21no—
41C22⋊C4×D17—21no—
42#10587561C2211yescharacteristic
43C4×D68C2211yescharacteristic, complemented
44C42×D17C2211yescharacteristic
45C17×C4⋊C8C2211yescharacteristic, complemented
46C17×#103C2211yescharacteristic, complemented
47C17×#102C2211yescharacteristic
48#10587506C2211yescharacteristic, complemented
49#10587503C2211yescharacteristic
50#10587492C2211yescharacteristic
51#10587491C2211yescharacteristic
52#10587491C2211yescharacteristic
53#10587487C2211yescharacteristic, complemented
54#10587486C2211yescharacteristic, complemented
55#10587486C2211yescharacteristic, complemented
56#10587481C2211yescharacteristic, complemented
57#10587480C2211yescharacteristic
58C4⋊#3622C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×Q8×C17C2311yescharacteristic
61C2×D4×C17C2311yescharacteristic
62C2×D4×C17C2311yescharacteristic
63C22×C4×C17—21no—
64C2×C22⋊D17—21no—
65C2×C22⋊D17—21no—
66C2×C22⋊D17—21no—
67C2×C22⋊D17—21no—
68Q8×D17—41no—
69Q8×D17—41no—
70D4×D17—41no—
71D4×D17—41no—
72#63252—44no—
73C2×D68—21no—
74C2×D68C2311yescharacteristic
75C2×D68C2311yescharacteristic
76C2×C4×D17—21no—
77C2×C4×D17—21no—
78C2×C4×D17D411yescharacteristic
79C2×C4×D17D411yescharacteristic
80C2×C4×D17C2311yescharacteristic
81C2×Dic34—21no—
82C2×Dic34—21no—
83C2×Dic34C2311yescharacteristic
84SD16×C17—41no—
85D8×C17—41no—
86C2×C8×C17C2311yescharacteristic
87C2×C8×C17C2311yescharacteristic
88C4⋊C4×C17C2311yescharacteristic
89C22⋊C4×C17—21no—
90C22⋊C4×C17—21no—
91C22⋊C4×C17—21no—
92C42×C17C2311yescharacteristic
93C22⋊Dic17—21no—
94C22⋊Dic17—21no—
95C22⋊Dic17—21no—
96#63230—41no—
97#63229—41no—
98#63228—41no—
99#63228—41no—
100#63227—21no—
101#63227—21no—
102#63227—21no—
103#63227—21no—
104#63227—21no—
105#63227—21no—
106C4⋊Dic17C2311yescharacteristic
107#63225—21no—
108C4×Dic17D411yescharacteristic
109C4×Dic17D411yescharacteristic
110C4×Dic17C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112C2×#3622C2311yescharacteristic
113D136—41no—
114#63219—41no—
115C23×C17—21no—
116C23×C17—21no—
117C22×D17—21no—
118C22×D17—21no—
119C22×D17—42no—
120C22×D17—42no—
121C22×D17—42no—
122C22×D17C2×D411yescharacteristic
123Q8×C17—21no—
124Q8×C17—21no—
125D4×C17—21no—
126D4×C17—21no—
127D4×C17—21no—
128D4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—42no—
132C2×C4×C17C2×D411yescharacteristic
133C2×C4×C17C2×D411yescharacteristic
134C2×C4×C17C2411yescharacteristic
135C22⋊D17—42no—
136C22⋊D17—42no—
137C22⋊D17—42no—
138C22⋊D17—42no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17C2×D411yescharacteristic
142C2×Dic17C2×D411yescharacteristic
143C2×Dic17C2×D411yescharacteristic
144D68—21no—
145D68—21no—
146D68—21no—
147D68—21no—
148D68—42no—
149C4×D17—22no—
150C4×D17—21no—
151C4×D17—21no—
152C4×D17—41no—
153C4×D17—42no—
154C4×D17—42no—
155Dic34—21no—
156Dic34—21no—
157Dic34—42no—
158Dic34—42no—
159C8×C17—21no—
160C8×C17—21no—
161#3622—21no—
162#3622—21no—
163#3213—171nomaximal, Sylow
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17C22×D411yescharacteristic
171C2×D17—21no—
172C2×D17—21no—
173C2×D17—41no—
174C2×D17—41no—
175C2×D17—41no—
176C2×D17—41no—
177C2×D17—41no—
178C2×D17—41no—
179C2×D17—84no—
180C2×D17AGL(1,ℤ/8)12yes—
181C4×C17—21no—
182C4×C17—21no—
183C4×C17—41no—
184C4×C17—41no—
185C4×C17H(5,𝔽2)11yescharacteristic
186C4×C17H(5,𝔽2)11yescharacteristic
187Dic17—21no—
188Dic17—21no—
189Dic17—21no—
190Dic17—41no—
191Dic17—41no—
192#532—171no—
193C2×#124—171no—
194#496—171no—
195#496—171no—
196#489—171no—
197#489—171no—
198#463—171no—
199#463—171no—
200#463—171no—
201#463D1711yescharacteristic, complemented
202#448—171no—
203#448—171no—
204#447—171no—
205#447—171no—
206#432—171no—
207C2×C17—81no—
208C2×C17—81no—
209C2×C17#53511yescharacteristic
210C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
211C2×C17#57511yescharacteristic
212D17—22no—
213D17—81no—
214D17—81no—
215C2×#41—341no—
216C22×Q8—171no—
217C22×D4—171no—
218C2×SD16—171no—
219C2×SD16—171no—
220C2×SD16—171no—
221C2×SD16C2×D1711yescharacteristic, complemented
222C2×D8—171no—
223C2×D8—171no—
224C2×D8—171no—
225C2×D8C2×D1711yescharacteristic
226C4⋊Q8—171no—
227#127—171no—
228#124—171no—
229#124—171no—
230#124—171no—
231#124—341no—
232#124—341no—
233#124—341no—
234#124C2×D1711yescharacteristic
235#121—171no—
236#121—171no—
237#121—341no—
238C4×D4—171no—
239C4×D4—341no—
240C4×D4C2×D1711yescharacteristic, complemented
241C2×C22⋊C4—341no—
242C2×C22⋊C4—341no—
243C2×C42—171no—
244C4⋊C8—171no—
245C4⋊C8C2×D1711yescharacteristic, complemented
246#103—171no—
247#103—171no—
248#103—171no—
249#103C2×D1711yescharacteristic, complemented
250#102—171no—
251#102—171no—
252#102—171no—
253#102C2×D1711yescharacteristic
254C22⋊C8—171no—
255C22⋊C8—171no—
256#97—171no—
257#97—171no—
258C17#321311yescharacteristic, complemented, Sylow
259C24—341no—
260#41—684no—
261C2×Q8—171no—
262C2×Q8—341no—
263C2×Q8—341no—
264C2×Q8—681no—
265C2×Q8—681no—
266C2×Q8C22×D1711yescharacteristic
267C2×D4—171no—
268C2×D4—171no—
269C2×D4—341no—
270C2×D4—341no—
271C2×D4—341no—
272C2×D4—341no—
273C2×D4—341no—
274C2×D4—681no—
275C2×D4—681no—
276C2×D4C22×D1711yescharacteristic
277C2×D4C22×D1711yescharacteristic
278C22×C4—21no—
279C22×C4—171no—
280C22×C4—171no—
281C22×C4—171no—
282C22×C4—341no—
283C22×C4—341no—
284SD16—41no—
285SD16—681no—
286SD16—681no—
287SD16—681no—
288D8—41no—
289D8—681no—
290D8—681no—
291D8—681no—
292C2×C8—171no—
293C2×C8—171no—
294C2×C8C22×D1711yescharacteristic
295C2×C8C22×D1711yescharacteristic
296C4⋊C4—171no—
297C4⋊C4—341no—
298C4⋊C4C22×D1711yescharacteristic
299C22⋊C4—21no—
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—
310C22⋊C4—341no—
311C42—171no—
312C42—171no—
313C42—171no—
314C42C22×D1711yescharacteristic
315C23—21no—
316C23—21no—
317C23—171no—
318C23—341no—
319C23—341no—
320C23—682no—
321C23—682no—
322C23—682no—
323Q8—21no—
324Q8—21no—
325Q8—341no—
326Q8—341no—
327Q8—682no—
328Q8—682no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—341no—
334D4—341no—
335D4—341no—
336D4—341no—
337D4—682no—
338D4—682no—
339D4—682no—
340D4—682no—
341D4—682no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—171no—
346C2×C4—171no—
347C2×C4—171no—
348C2×C4—342no—
349C2×C4—341no—
350C2×C4—341no—
351C2×C4—341no—
352C2×C4—341no—
353C2×C4—681no—
354C2×C4—682no—
355C2×C4—682no—
356C2×C4D4×D1711yescharacteristic
357C2×C4D4×D1711yescharacteristic
358C2×C4C23×D1711yescharacteristic
359C8—21no—
360C8—21no—
361C8—341no—
362C8—341no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—172no—
370C22—341no—
371C22—341no—
372C22—681no—
373C22—681no—
374C22—681no—
375C22—681no—
376C22—681no—
377C22—681no—
378C22—1364no—
379C22C2×D4×D1711yescharacteristic
380C4—21no—
381C4—21no—
382C4—41no—
383C4—41no—
384C4—341no—
385C4—341no—
386C4—341no—
387C4—681no—
388C4—681no—
389C4#1058769111yescharacteristic
390C4#1058769611yescharacteristic
391C2—81no—
392C2—81no—
393C2—342no—
394C2—1361no—
395C2—1361no—
396C2#1176337511yescharacteristic
397C2AGL(1,ℤ/8)×D1711yescharacteristic
398C2#1176360611yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#463C2
socleC22×C17C22×D4
hypercenter#463D17
hypocenterC17#3213
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #11763356C2semidirect
2. #124×D17C2semidirect
3. #11763197C2semidirect
4. #11763195C2semidirect
5. #11763170C2semidirect
6. #11763167C2semidirect
7. #11763025C2semidirect
8. #11763024C2semidirect
9. #11763021C2semidirect
10. #11762949C2semidirect
11. #11762946C2semidirect
12. #11762944C2semidirect
13. #11762941C2semidirect
14. #11762865C2semidirect
15. C17×#463C2semidirect
19. C2×SD16×C17C22semidirect
22. C4×D4×C17C22semidirect
24. C2×#63230C22semidirect
25. #10587626C22semidirect
28. C2×#63229C22semidirect
33. C2×#63219C22semidirect
35. #10587574C22semidirect
43. C4×D68C22semidirect
45. C17×C4⋊C8C22semidirect
46. C17×#103C22semidirect
48. #10587506C22semidirect
53. #10587487C22semidirect
54. #10587486C22semidirect
55. #10587486C22semidirect
56. #10587481C22semidirect
58. C4⋊#3622C22semidirect
201. #463D17semidirect
221. C2×SD16C2×D17semidirect
240. C4×D4C2×D17semidirect
245. C4⋊C8C2×D17semidirect
249. #103C2×D17semidirect
258. C17#3213semidirect

Maximal subgroups

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

subgroupindex
#117633562
#124×D172
#117631972
#117631952
#117631702
#117631672
#117630252
#117630242
#117630212
#117629492
#117629462
#117629442
#117629412
#117628652
C17×#4632
#321317