Finite Group Databasebeta

← #14539593

Subgroups — #14539593

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#11763319C211yesmaximal, characteristic, complemented
2#121×D17C211yesmaximal, characteristic, complemented
3#11763141C211yesmaximal, characteristic, complemented
4#11763135C211yesmaximal, characteristic, complemented
5#11763131C211yesmaximal, characteristic, complemented
6#11763121C211yesmaximal, characteristic, complemented
7#11763060C211yesmaximal, characteristic, complemented
8#11763055C211yesmaximal, characteristic, complemented
9#11763053C211yesmaximal, characteristic, complemented
10#11762961C211yesmaximal, characteristic, complemented
11#11762955C211yesmaximal, characteristic, complemented
12#11762951C211yesmaximal, characteristic, complemented
13#11762941C211yesmaximal, characteristic, complemented
14#11762772C211yesmaximal, characteristic, complemented
15C17×#469C211yesmaximal, characteristic, complemented
16C2×#63254C2211yescharacteristic, complemented
17C2×D4×D17—21no—
18C2×D4×D17C2211yescharacteristic
19C22×C4×D17C2211yescharacteristic
20C2×D8×C17C2211yescharacteristic, complemented
21C2×M16×C17C2211yescharacteristic, complemented
22C17×#121C2211yescharacteristic
23C17×#121C2211yescharacteristic, complemented
24#10587625—21no—
25#10587624C2211yescharacteristic
26#10587624C2211yescharacteristic
27#10587623—21no—
28D4×Dic17—21no—
29C2×#63229C2211yescharacteristic, complemented
30C2×#63228C2211yescharacteristic, complemented
31#10587616C2211yescharacteristic
32#10587610C2211yescharacteristic, complemented
33C2×#63223C2211yescharacteristic, complemented
34C2×#63219C2211yescharacteristic, complemented
35#10587583C2211yescharacteristic
36#10587581C2211yescharacteristic
37#10587578C2211yescharacteristic, complemented
38C4⋊C4×D17C2211yescharacteristic
39#10587570—21no—
40#10587569—21no—
41#10587567—21no—
42C22⋊C4×D17—21no—
43C17×#106C2211yescharacteristic, complemented
44C17×#102C2211yescharacteristic, complemented
45C17×#102C2211yescharacteristic, complemented
46#10587503C2211yescharacteristic, complemented
47#10587503C2211yescharacteristic, complemented
48#10587492C2211yescharacteristic, complemented
49#10587491C2211yescharacteristic
50#10587491C2211yescharacteristic
51#10587491C2211yescharacteristic, complemented
52#10587491C2211yescharacteristic, complemented
53#10587489C2211yescharacteristic, complemented
54#10587487C2211yescharacteristic, complemented
55#10587481C2211yescharacteristic, complemented
56#10587480C2211yescharacteristic, complemented
57#10587479C2211yescharacteristic, complemented
58#10587478C2211yescharacteristic, complemented
59C23×D17—21no—
60C23×D17D411yescharacteristic
61C2×D4×C17—21no—
62C2×D4×C17—21no—
63C2×D4×C17C2311yescharacteristic
64C2×D4×C17C2311yescharacteristic
65C22×C4×C17C2311yescharacteristic
66C2×C22⋊D17—21no—
67C2×C22⋊D17—21no—
68C2×C22⋊D17—21no—
69C2×C22⋊D17—21no—
70C2×C22⋊D17—21no—
71C22×Dic17—21no—
72C22×Dic17D411yescharacteristic
73#63254—41no—
74#63254—41no—
75D4×D17—41no—
76D4×D17—41no—
77D4×D17—44no—
78C2×D68—21no—
79C2×D68C2311yescharacteristic
80C2×C4×D17—21no—
81C2×C4×D17—41no—
82C2×C4×D17—41no—
83C2×C4×D17D411yescharacteristic
84C2×C4×D17D411yescharacteristic
85C2×C4×D17C2311yescharacteristic
86C2×C4×D17C2311yescharacteristic
87C2×Dic34C2311yescharacteristic
88D8×C17—41no—
89M16×C17—41no—
90C2×C8×C17C2311yescharacteristic
91C2×C8×C17C2311yescharacteristic
92C4⋊C4×C17C2311yescharacteristic
93C4⋊C4×C17C2311yescharacteristic
94C22⋊C4×C17—21no—
95C22⋊C4×C17—21no—
96C22⋊Dic17—21no—
97C22⋊Dic17—21no—
98C22⋊Dic17—21no—
99#63229—41no—
100#63228—41no—
101#63227—21no—
102#63227—21no—
103#63227—21no—
104#63227—21no—
105C4⋊Dic17—21no—
106C4⋊Dic17C2311yescharacteristic
107C4⋊Dic17C2311yescharacteristic
108#63225—21no—
109#63225—21no—
110C4×Dic17—21no—
111#63223—41no—
112C2×#3622C2311yescharacteristic
113C2×#3622C2311yescharacteristic
114#63219—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
127D4×C17—21no—
128D4×C17—21no—
129D4×C17—21no—
130D4×C17—21no—
131D4×C17—42no—
132D4×C17—42no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17C2×D411yescharacteristic
138C2×C4×C17C2411yescharacteristic
139C22⋊D17—42no—
140C22⋊D17—42no—
141C22⋊D17—42no—
142C22⋊D17—42no—
143C22⋊D17—42no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—41no—
148C2×Dic17—42no—
149C2×Dic17C2×D411yescharacteristic
150C2×Dic17C2×D411yescharacteristic
151D68—21no—
152D68—21no—
153D68—42no—
154C4×D17—21no—
155C4×D17—21no—
156C4×D17—21no—
157C4×D17—21no—
158C4×D17—41no—
159C4×D17—41no—
160C4×D17—42no—
161Dic34—21no—
162Dic34—21no—
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#469D1711yescharacteristic, complemented
207#468—171no—
208#468—171no—
209#450—171no—
210#449—171no—
211#449—171no—
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×#41—171no—
224C22×D4—171no—
225C22×D4—341no—
226C23×C4—171no—
227C2×SD16—171no—
228C2×SD16—171no—
229C2×D8—171no—
230C2×D8C2×D1711yescharacteristic, complemented
231C2×M16—171no—
232C2×M16C2×D1711yescharacteristic, complemented
233#123—341no—
234C22⋊Q8—171no—
235C22⋊Q8—171no—
236#121—171no—
237#121—171no—
238#121—171no—
239#121—171no—
240#121—341no—
241#121—341no—
242#121C2×D1711yescharacteristic
243#121C2×D1711yescharacteristic, complemented
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#106C2×D1711yescharacteristic, complemented
254#103—171no—
255#103—171no—
256#102—171no—
257#102—171no—
258#102—171no—
259#102—171no—
260#102C2×D1711yescharacteristic, complemented
261#102C2×D1711yescharacteristic, complemented
262C22⋊C8—171no—
263C22⋊C8—171no—
264C22⋊C8—171no—
265C22⋊C8—171no—
266C17#319411yescharacteristic, complemented, Sylow
267C24—171no—
268C24—341no—
269#41—681no—
270#41—681no—
271C2×Q8—171no—
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—171no—
287C22×C4—171no—
288C22×C4—171no—
289C22×C4—171no—
290C22×C4—171no—
291C22×C4—341no—
292C22×C4—341no—
293C22×C4—681no—
294C22×C4—681no—
295C22×C4C22×D1711yescharacteristic
296SD16—681no—
297SD16—681no—
298D8—41no—
299D8—681no—
300M16—41no—
301M16—681no—
302C2×C8—171no—
303C2×C8—171no—
304C2×C8C22×D1711yescharacteristic
305C2×C8C22×D1711yescharacteristic
306C4⋊C4—171no—
307C4⋊C4—171no—
308C4⋊C4—341no—
309C4⋊C4—341no—
310C4⋊C4—341no—
311C4⋊C4C22×D1711yescharacteristic
312C4⋊C4C22×D1711yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
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—341no—
336Q8—341no—
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—171no—
356C2×C4—171no—
357C2×C4—341no—
358C2×C4—341no—
359C2×C4—341no—
360C2×C4—341no—
361C2×C4—341no—
362C2×C4—341no—
363C2×C4—341no—
364C2×C4—681no—
365C2×C4—681no—
366C2×C4—682no—
367C2×C4—681no—
368C2×C4—682no—
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#1058769611yescharacteristic
406C2—41no—
407C2—81no—
408C2—81no—
409C2—342no—
410C2—681no—
411C2—1361no—
412C2#1176336711yescharacteristic
413C2AGL(1,ℤ/8)×D1711yescharacteristic
414C2#1176360511yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#469C2
socleC22×C17C22×D4
hypercenter#469D17
hypocenterC17#3194
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #11763319C2semidirect
2. #121×D17C2semidirect
3. #11763141C2semidirect
4. #11763135C2semidirect
5. #11763131C2semidirect
6. #11763121C2semidirect
7. #11763060C2semidirect
8. #11763055C2semidirect
9. #11763053C2semidirect
10. #11762961C2semidirect
11. #11762955C2semidirect
12. #11762951C2semidirect
13. #11762941C2semidirect
14. #11762772C2semidirect
15. C17×#469C2semidirect
16. C2×#63254C22semidirect
20. C2×D8×C17C22semidirect
21. C2×M16×C17C22semidirect
23. C17×#121C22semidirect
29. C2×#63229C22semidirect
30. C2×#63228C22semidirect
32. #10587610C22semidirect
33. C2×#63223C22semidirect
34. C2×#63219C22semidirect
37. #10587578C22semidirect
43. C17×#106C22semidirect
44. C17×#102C22semidirect
45. C17×#102C22semidirect
46. #10587503C22semidirect
47. #10587503C22semidirect
48. #10587492C22semidirect
51. #10587491C22semidirect
52. #10587491C22semidirect
53. #10587489C22semidirect
54. #10587487C22semidirect
55. #10587481C22semidirect
56. #10587480C22semidirect
57. #10587479C22semidirect
58. #10587478C22semidirect
206. #469D17semidirect
230. C2×D8C2×D17semidirect
232. C2×M16C2×D17semidirect
243. #121C2×D17semidirect
253. #106C2×D17semidirect
260. #102C2×D17semidirect
261. #102C2×D17semidirect
266. C17#3194semidirect

Maximal subgroups

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

subgroupindex
#117633192
#121×D172
#117631412
#117631352
#117631312
#117631212
#117630602
#117630552
#117630532
#117629612
#117629552
#117629512
#117629412
#117627722
C17×#4692
#319417