Finite Group Databasebeta

← #14539596

Subgroups — #14539596

Order 2176.

466 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763316C211yesmaximal, characteristic, complemented
2#121×D17C211yesmaximal, characteristic, complemented
3#11763138C211yesmaximal, characteristic, complemented
4#11763134C211yesmaximal, characteristic, complemented
5#11763130C211yesmaximal, characteristic, complemented
6#11763121C211yesmaximal, characteristic, complemented
7#11763061C211yesmaximal, characteristic, complemented
8#11763059C211yesmaximal, characteristic, complemented
9#11763058C211yesmaximal, characteristic, complemented
10#11763052C211yesmaximal, characteristic, complemented
11#11762958C211yesmaximal, characteristic, complemented
12#11762954C211yesmaximal, characteristic, complemented
13#11762941C211yesmaximal, characteristic, complemented
14#11762771C211yesmaximal, characteristic, complemented
15C17×#449C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic, complemented
17C2×D4×D17—21no—
18C2×D4×D17C2211yescharacteristic
19C22×C4×D17C2211yescharacteristic
20C2×SD16×C17C2211yescharacteristic, complemented
21C2×D8×C17C2211yescharacteristic, complemented
22C17×#121C2211yescharacteristic
23#10587632C2211yescharacteristic
24#10587632C2211yescharacteristic, complemented
25C2×#63230C2211yescharacteristic, complemented
26#10587625—21no—
27#10587624C2211yescharacteristic
28#10587624C2211yescharacteristic
29#10587624C2211yescharacteristic, complemented
30#10587623—21no—
31#10587621—21no—
32C2×#63228C2211yescharacteristic, complemented
33#10587616C2211yescharacteristic
34C4×C22⋊D17—21no—
35C4×C22⋊D17—21no—
36#10587612C2211yescharacteristic, complemented
37C2×#63218C2211yescharacteristic, complemented
38C2×#63218C2211yescharacteristic, complemented
39#10587581C2211yescharacteristic
40C4⋊C4×D17C2211yescharacteristic
41#10587570—21no—
42C22⋊C4×D17—21no—
43C17×#103C2211yescharacteristic, complemented
44C17×#102C2211yescharacteristic, complemented
45C17×C22⋊C8C2211yescharacteristic, complemented
46#10587506C2211yescharacteristic, complemented
47#10587503C2211yescharacteristic, complemented
48#10587503C2211yescharacteristic, complemented
49#10587503C2211yescharacteristic, complemented
50C22⋊#3622C2211yescharacteristic, complemented
51#10587492C2211yescharacteristic, complemented
52#10587492C2211yescharacteristic, complemented
53#10587491C2211yescharacteristic
54#10587491C2211yescharacteristic
55#10587489C2211yescharacteristic, complemented
56#10587488C2211yescharacteristic, complemented
57#10587479C2211yescharacteristic, complemented
58#10587478C2211yescharacteristic, complemented
59C23×D17—21no—
60C23×D17D411yescharacteristic
61#41×C17—41no—
62#41×C17—41no—
63C2×Q8×C17C2311yescharacteristic
64C2×D4×C17—21no—
65C2×D4×C17—21no—
66C2×D4×C17C2311yescharacteristic
67C2×D4×C17C2311yescharacteristic
68C22×C4×C17—21no—
69C22×C4×C17C2311yescharacteristic
70C2×C22⋊D17—21no—
71C2×C22⋊D17—21no—
72C2×C22⋊D17—21no—
73C2×C22⋊D17—21no—
74C2×C22⋊D17—21no—
75C22×Dic17D411yescharacteristic
76D4×D17—41no—
77D4×D17—41no—
78D4×D17—44no—
79C2×D68—21no—
80C2×D68C2311yescharacteristic
81C2×C4×D17—21no—
82C2×C4×D17—41no—
83C2×C4×D17—41no—
84C2×C4×D17D411yescharacteristic
85C2×C4×D17D411yescharacteristic
86C2×C4×D17C2311yescharacteristic
87C2×C4×D17C2311yescharacteristic
88SD16×C17—41no—
89D8×C17—41no—
90C2×C8×C17C2311yescharacteristic
91C2×C8×C17C2311yescharacteristic
92C4⋊C4×C17C2311yescharacteristic
93C22⋊C4×C17—21no—
94C22⋊Dic17—21no—
95C22⋊Dic17—21no—
96C22⋊Dic17—21no—
97C22⋊Dic17—21no—
98#63230—41no—
99#63228—41no—
100#63227—21no—
101#63227—21no—
102#63227—21no—
103#63227—21no—
104C4⋊Dic17C2311yescharacteristic
105C4⋊Dic17C2311yescharacteristic
106C4⋊Dic17C2311yescharacteristic
107#63225—21no—
108#63225—21no—
109#63225—21no—
110C4×Dic17—21no—
111C2×#3622C2311yescharacteristic
112C2×#3622C2311yescharacteristic
113#63218—41no—
114#63218—41no—
115C23×C17—21no—
116C23×C17—21no—
117C23×C17C2×D411yescharacteristic
118C22×D17—22no—
119C22×D17—21no—
120C22×D17—41no—
121C22×D17—41no—
122C22×D17—42no—
123C22×D17—42no—
124C22×D17—42no—
125C22×D17C2×D411yescharacteristic
126C22×D17C2×D411yescharacteristic
127Q8×C17—21no—
128Q8×C17—21no—
129D4×C17—21no—
130D4×C17—21no—
131D4×C17—21no—
132D4×C17—21no—
133D4×C17—42no—
134D4×C17—42no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—42no—
140C2×C4×C17C2×D411yescharacteristic
141C2×C4×C17C2411yescharacteristic
142C22⋊D17—42no—
143C22⋊D17—42no—
144C22⋊D17—42no—
145C22⋊D17—42no—
146C22⋊D17—42no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—41no—
151C2×Dic17C2×D411yescharacteristic
152C2×Dic17C2×D411yescharacteristic
153D68—21no—
154D68—21no—
155D68—42no—
156C4×D17—21no—
157C4×D17—21no—
158C4×D17—21no—
159C4×D17—21no—
160C4×D17—41no—
161C4×D17—41no—
162C4×D17—42no—
163C8×C17—21no—
164C8×C17—21no—
165#3622—21no—
166#3622—21no—
167#3194—171nomaximal, Sylow
168C22×C17—21no—
169C22×C17—21no—
170C22×C17—21no—
171C22×C17—41no—
172C22×C17—41no—
173C22×C17—41no—
174C22×C17—41no—
175C22×C17—41no—
176C22×C17—41no—
177C22×C17C22×D411yescharacteristic
178C2×D17—21no—
179C2×D17—21no—
180C2×D17—21no—
181C2×D17—21no—
182C2×D17—21no—
183C2×D17—44no—
184C2×D17—41no—
185C2×D17—41no—
186C2×D17—41no—
187C2×D17—84no—
188C2×D17AGL(1,ℤ/8)12yes—
189C4×C17—21no—
190C4×C17—41no—
191C4×C17—41no—
192C4×C17H(5,𝔽2)11yescharacteristic
193C4×C17H(5,𝔽2)11yescharacteristic
194Dic17—21no—
195Dic17—21no—
196Dic17—41no—
197Dic17—41no—
198Dic17—41no—
199#525—171no—
200C2×#121—171no—
201#483—171no—
202#482—171no—
203#482—171no—
204#480—171no—
205#469—171no—
206#469—171no—
207#468—171no—
208#468—171no—
209#450—171no—
210#449—171no—
211#449D1711yescharacteristic, complemented
212#447—171no—
213#407—171no—
214C2×C17—41no—
215C2×C17—81no—
216C2×C17—81no—
217C2×C17#53411yescharacteristic
218C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C17#57511yescharacteristic
220D17—22no—
221D17—41no—
222D17—81no—
223C2×#41C2×D1711yescharacteristic, complemented
224C22×D4—171no—
225C22×D4—341no—
226C23×C4—171no—
227C2×SD16—171no—
228C2×SD16C2×D1711yescharacteristic, complemented
229C2×D8—171no—
230C2×D8C2×D1711yescharacteristic, complemented
231C2×M16—171no—
232C2×M16—171no—
233#123—341no—
234C22⋊Q8—171no—
235C22⋊Q8—171no—
236#121—171no—
237#121—171no—
238#121—171no—
239#121—171no—
240#121—171no—
241#121—341no—
242#121—341no—
243#121C2×D1711yescharacteristic
244#120—341no—
245C4×D4—341no—
246C4×D4—341no—
247#117—171no—
248C2×C4⋊C4—171no—
249C2×C22⋊C4—341no—
250#107—171no—
251#107—171no—
252#106—171no—
253#106—171no—
254#103—171no—
255#103C2×D1711yescharacteristic, complemented
256#102—171no—
257#102—171no—
258#102—171no—
259#102—171no—
260#102—171no—
261#102C2×D1711yescharacteristic, complemented
262C22⋊C8—171no—
263C22⋊C8—171no—
264C22⋊C8—171no—
265C22⋊C8C2×D1711yescharacteristic, complemented
266C17#319411yescharacteristic, complemented, Sylow
267C24—171no—
268C24—341no—
269#41—41no—
270#41—41no—
271C2×Q8C22×D1711yescharacteristic
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—171no—
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×D4—684no—
284C2×D4C22×D1711yescharacteristic
285C2×D4C22×D1711yescharacteristic
286C22×C4—21no—
287C22×C4—171no—
288C22×C4—171no—
289C22×C4—171no—
290C22×C4—171no—
291C22×C4—171no—
292C22×C4—341no—
293C22×C4—681no—
294C22×C4—681no—
295C22×C4C22×D1711yescharacteristic
296SD16—41no—
297SD16—681no—
298D8—41no—
299D8—681no—
300M16—681no—
301M16—681no—
302C2×C8—171no—
303C2×C8—171no—
304C2×C8C22×D1711yescharacteristic
305C2×C8C22×D1711yescharacteristic
306C4⋊C4—171no—
307C4⋊C4—171no—
308C4⋊C4—171no—
309C4⋊C4—341no—
310C4⋊C4—341no—
311C4⋊C4—341no—
312C4⋊C4C22×D1711yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—341no—
315C22⋊C4—341no—
316C22⋊C4—341no—
317C22⋊C4—341no—
318C22⋊C4—341no—
319C22⋊C4—341no—
320C22⋊C4—341no—
321C22⋊C4—341no—
322C42—341no—
323C23—21no—
324C23—21no—
325C23—171no—
326C23—171no—
327C23—342no—
328C23—341no—
329C23—681no—
330C23—681no—
331C23—682no—
332C23—682no—
333C23—682no—
334C23D4×D1711yescharacteristic
335Q8—21no—
336Q8—21no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—42no—
342D4—42no—
343D4—341no—
344D4—341no—
345D4—682no—
346D4—682no—
347D4—682no—
348D4—682no—
349D4—682no—
350D4—682no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—42no—
356C2×C4—171no—
357C2×C4—171no—
358C2×C4—341no—
359C2×C4—341no—
360C2×C4—341no—
361C2×C4—341no—
362C2×C4—341no—
363C2×C4—341no—
364C2×C4—341no—
365C2×C4—681no—
366C2×C4—681no—
367C2×C4—682no—
368C2×C4—681no—
369C2×C4D4×D1711yescharacteristic
370C2×C4C23×D1711yescharacteristic
371C8—21no—
372C8—21no—
373C8—341no—
374C8—341no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—172no—
385C22—341no—
386C22—341no—
387C22—341no—
388C22—341no—
389C22—341no—
390C22—684no—
391C22—681no—
392C22—681no—
393C22—681no—
394C22—1364no—
395C22C2×D4×D1711yescharacteristic
396C4—21no—
397C4—41no—
398C4—41no—
399C4—341no—
400C4—341no—
401C4—681no—
402C4—681no—
403C4—681no—
404C4#1058769111yescharacteristic
405C4#1058769111yescharacteristic
406C2—41no—
407C2—81no—
408C2—81no—
409C2—342no—
410C2—681no—
411C2—1361no—
412C2#1176336911yescharacteristic
413C2AGL(1,ℤ/8)×D1711yescharacteristic
414C2#1176360611yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#449C2
socleC22×C17C22×D4
hypercenter#449D17
hypocenterC17#3194
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #11763316C2semidirect
2. #121×D17C2semidirect
3. #11763138C2semidirect
4. #11763134C2semidirect
5. #11763130C2semidirect
6. #11763121C2semidirect
7. #11763061C2semidirect
8. #11763059C2semidirect
9. #11763058C2semidirect
10. #11763052C2semidirect
11. #11762958C2semidirect
12. #11762954C2semidirect
13. #11762941C2semidirect
14. #11762771C2semidirect
15. C17×#449C2semidirect
16. C2×#41×C17C22semidirect
20. C2×SD16×C17C22semidirect
21. C2×D8×C17C22semidirect
24. #10587632C22semidirect
25. C2×#63230C22semidirect
29. #10587624C22semidirect
32. C2×#63228C22semidirect
36. #10587612C22semidirect
37. C2×#63218C22semidirect
38. C2×#63218C22semidirect
43. C17×#103C22semidirect
44. C17×#102C22semidirect
45. C17×C22⋊C8C22semidirect
46. #10587506C22semidirect
47. #10587503C22semidirect
48. #10587503C22semidirect
49. #10587503C22semidirect
50. C22⋊#3622C22semidirect
51. #10587492C22semidirect
52. #10587492C22semidirect
55. #10587489C22semidirect
56. #10587488C22semidirect
57. #10587479C22semidirect
58. #10587478C22semidirect
211. #449D17semidirect
223. C2×#41C2×D17semidirect
228. C2×SD16C2×D17semidirect
230. C2×D8C2×D17semidirect
255. #103C2×D17semidirect
261. #102C2×D17semidirect
265. C22⋊C8C2×D17semidirect
266. C17#3194semidirect

Maximal subgroups

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

subgroupindex
#117633162
#121×D172
#117631382
#117631342
#117631302
#117631212
#117630612
#117630592
#117630582
#117630522
#117629582
#117629542
#117629412
#117627712
C17×#4492
#319417