Finite Group Databasebeta

← #14538291

Subgroups — #14538291

Order 2176.

494 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587600C211yesmaximal, characteristic
2C2×SD16×D17C211yesmaximal, characteristic, complemented
3#11763384C211yesmaximal, characteristic, complemented
4#11763281C211yesmaximal, characteristic, complemented
5#11763027C211yesmaximal, characteristic, complemented
6#11763026C211yesmaximal, characteristic, complemented
7#11763025C211yesmaximal, characteristic, complemented
8#11763024C211yesmaximal, characteristic, complemented
9#11763013C211yesmaximal, characteristic, complemented
10#11763012C211yesmaximal, characteristic, complemented
11#11763011C211yesmaximal, characteristic, complemented
12#11762834C211yesmaximal, characteristic, complemented
13#11762816C211yesmaximal, characteristic, complemented
14#11762804C211yesmaximal, characteristic, complemented
15C17×#461C211yesmaximal, characteristic, complemented
16C2×#63256C2211yescharacteristic
17C2×Q8×D17C2211yescharacteristic
18C2×D4×D17C2211yescharacteristic
19C2×SD16×C17C2211yescharacteristic
20C2×SD16×C17C2211yescharacteristic
21C17×C4⋊Q8C2211yescharacteristic, complemented
22C4×D4×C17C2211yescharacteristic
23#10587633C2211yescharacteristic, complemented
24#10587630C2211yescharacteristic, complemented
25C2×#63230C2211yescharacteristic
26C2×#63230C2211yescharacteristic
27D4×Dic17C2211yescharacteristic
28C2×#63229C2211yescharacteristic
29C2×#63228C2211yescharacteristic
30C4×C22⋊D17—21no—
31C2×#63227—21no—
32#10587600—24no—
33SD16×D17—24no—
34C2×D136C2211yescharacteristic
35C2×#63219C2211yescharacteristic
36C2×#63218C2211yescharacteristic
37C2×C8×D17C2211yescharacteristic
38#10587582—21no—
39#10587582—21no—
40#10587580—21no—
41#10587580—21no—
42#10587579C2211yescharacteristic
43#10587578C2211yescharacteristic
44#10587567—21no—
45C22⋊C4×D17—21no—
46#10587561C2211yescharacteristic, complemented
47C4×D68C2211yescharacteristic
48#10587558C2211yescharacteristic
49C17×C4⋊C8C2211yescharacteristic, complemented
50C17×#103C2211yescharacteristic, complemented
51C17×#103C2211yescharacteristic, complemented
52#10587506C2211yescharacteristic, complemented
53#10587506C2211yescharacteristic, complemented
54#10587492C2211yescharacteristic, complemented
55#10587487C2211yescharacteristic, complemented
56#10587485C2211yescharacteristic, complemented
57#10587485C2211yescharacteristic, complemented
58#10587481C2211yescharacteristic, complemented
59#10587480C2211yescharacteristic, complemented
60C4⋊#3622C2211yescharacteristic, complemented
61C23×D17—21no—
62C2×Q8×C17C2311yescharacteristic
63C2×Q8×C17C2311yescharacteristic
64C2×D4×C17C2311yescharacteristic
65C22×C4×C17—21no—
66C2×C22⋊D17—21no—
67C22×Dic17—21no—
68#63256—22no—
69#63256—22no—
70Q8×D17—22no—
71Q8×D17—22no—
72D4×D17—22no—
73D4×D17D414yes—
74C2×D68—21no—
75C2×D68C2311yescharacteristic
76C2×D68C2311yescharacteristic
77C2×C4×D17—21no—
78C2×C4×D17—21no—
79C2×C4×D17—21no—
80C2×C4×D17—21no—
81C2×C4×D17C2311yescharacteristic
82C2×Dic34—21no—
83C2×Dic34C2311yescharacteristic
84SD16×C17—22no—
85SD16×C17—22no—
86C2×C8×C17C2311yescharacteristic
87C2×C8×C17C2311yescharacteristic
88C4⋊C4×C17—21no—
89C4⋊C4×C17—21no—
90C4⋊C4×C17C2311yescharacteristic
91C22⋊C4×C17—21no—
92C42×C17C2311yescharacteristic
93C22⋊Dic17—21no—
94#63230—22no—
95#63230—22no—
96#63229—22no—
97#63228—22no—
98#63227—21no—
99#63227—21no—
100#63227—21no—
101#63227—21no—
102#63227—21no—
103#63227—21no—
104#63227—21no—
105#63227D411yescharacteristic
106#63227D411yescharacteristic
107C4⋊Dic17C2311yescharacteristic
108#63225—21no—
109#63225—21no—
110#63225D411yescharacteristic
111#63225D411yescharacteristic
112C4×Dic17—21no—
113C4×Dic17C2311yescharacteristic
114C2×#3622C2311yescharacteristic
115C2×#3622C2311yescharacteristic
116D136—22no—
117#63219—22no—
118#63218—22no—
119C8×D17—22no—
120C23×C17—21no—
121C22×D17—21no—
122C22×D17—24no—
123C22×D17—21no—
124C22×D17—81no—
125C22×D17—81no—
126C22×D17C2×D411yescharacteristic
127Q8×C17—21no—
128Q8×C17—21no—
129Q8×C17—21no—
130Q8×C17—21no—
131D4×C17—21no—
132D4×C17C2×D412yes—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—81no—
137C2×C4×C17C2×D411yescharacteristic
138C2×C4×C17C2×D411yescharacteristic
139C2×C4×C17C2411yescharacteristic
140C22⋊D17—24no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—81no—
144C2×Dic17C2×D411yescharacteristic
145C2×Dic17C2×D411yescharacteristic
146C2×Dic17C2×D411yescharacteristic
147D68—21no—
148D68—21no—
149D68—21no—
150D68—42no—
151D68C2×D412yes—
152C4×D17—42no—
153C4×D17—42no—
154C4×D17—81no—
155C4×D17—81no—
156C4×D17C2×D412yes—
157C4×D17#4112yes—
158Dic34—21no—
159Dic34—21no—
160Dic34—42no—
161C8×C17—21no—
162C8×C17—21no—
163#3622—21no—
164#3622—21no—
165#3042—171nomaximal, Sylow
166C22×C17—22no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17C22×D411yescharacteristic
170C2×D17—21no—
171C2×D17—21no—
172C2×D17—22no—
173C2×D17—41no—
174C2×D17—41no—
175C2×D17—41no—
176C2×D17—41no—
177C2×D17—41no—
178C2×D17—84no—
179C2×D17#12112yes—
180C4×C17—21no—
181C4×C17—21no—
182C4×C17—41no—
183C4×C17—41no—
184C4×C17—41no—
185C4×C17C22×D411yescharacteristic
186C4×C17C2×#4111yescharacteristic
187Dic17—21no—
188Dic17—21no—
189Dic17—41no—
190Dic17—41no—
191Dic17#12112yes—
192C2×AGL(1,ℤ/8)—171no—
193C22×SD16—171no—
194#536—171no—
195#518—171no—
196#463—171no—
197#463—171no—
198#463—171no—
199#463—171no—
200#461—171no—
201#461—171no—
202#461—171no—
203#461D1711yescharacteristic, complemented
204#424—171no—
205#419—171no—
206#416—171no—
207C2×C17—42no—
208C2×C17C2×#12111yescharacteristic
209C2×C17#57711yescharacteristic
210C2×C17#57711yescharacteristic
211D17—22no—
212D17—42no—
213D17—81no—
214C2×#41—171no—
215C22×Q8—171no—
216C22×D4—171no—
217AGL(1,ℤ/8)—344no—
218C2×SD16—171no—
219C2×SD16—171no—
220C2×SD16—171no—
221C2×SD16—171no—
222C2×SD16—344no—
223C2×SD16C2×D1711yescharacteristic
224C2×SD16C2×D1711yescharacteristic
225C2×D8—171no—
226C2×D8—171no—
227C2×M16—171no—
228C22×C8—171no—
229C4⋊Q8—171no—
230C4⋊Q8C2×D1711yescharacteristic, complemented
231#124—171no—
232#124—171no—
233#123—341no—
234#123—341no—
235C22⋊Q8—341no—
236C22⋊Q8—341no—
237C4×D4—171no—
238C4×D4—171no—
239C4×D4—171no—
240C4×D4—341no—
241C4×D4—341no—
242C4×D4C2×D1711yescharacteristic
243#117—171no—
244#117—171no—
245C2×C22⋊C4—341no—
246C2×C22⋊C4—341no—
247C4⋊C8—171no—
248C4⋊C8—171no—
249C4⋊C8—171no—
250C4⋊C8C2×D1711yescharacteristic, complemented
251#103—171no—
252#103—171no—
253#103—171no—
254#103—171no—
255#103C2×D1711yescharacteristic, complemented
256#103C2×D1711yescharacteristic, complemented
257#102—171no—
258#102—171no—
259C17#304211yescharacteristic, complemented, Sylow
260C24—341no—
261#41—342no—
262#41—342no—
263C2×Q8—171no—
264C2×Q8—341no—
265C2×Q8—342no—
266C2×Q8—342no—
267C2×Q8C22×D1711yescharacteristic
268C2×Q8C22×D1711yescharacteristic
269C2×D4—174no—
270C2×D4—171no—
271C2×D4—171no—
272C2×D4—341no—
273C2×D4—342no—
274C2×D4—341no—
275C2×D4C22×D1711yescharacteristic
276C22×C4—21no—
277C22×C4—171no—
278C22×C4—341no—
279C22×C4—341no—
280C22×C4—341no—
281C22×C4—341no—
282C22×C4—341no—
283SD16—22no—
284SD16—22no—
285SD16—342no—
286SD16—342no—
287SD16—342no—
288SD16—342no—
289D8—342no—
290D8—342no—
291M16—342no—
292C2×C8—171no—
293C2×C8—171no—
294C2×C8—342no—
295C2×C8C22×D1711yescharacteristic
296C2×C8C22×D1711yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—171no—
300C4⋊C4—171no—
301C4⋊C4—171no—
302C4⋊C4—341no—
303C4⋊C4—341no—
304C4⋊C4C22×D1711yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—171no—
307C22⋊C4—171no—
308C22⋊C4—341no—
309C22⋊C4—341no—
310C22⋊C4—341no—
311C22⋊C4—341no—
312C22⋊C4—341no—
313C22⋊C4—341no—
314C22⋊C4—341no—
315C22⋊C4—341no—
316C42—171no—
317C42—341no—
318C42C22×D1711yescharacteristic
319C23—21no—
320C23—171no—
321C23—344no—
322C23—341no—
323C23—341no—
324C23—1361no—
325C23—1361no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—341no—
331Q8—341no—
332Q8—682no—
333D4—21no—
334D4—172no—
335D4—344no—
336D4—341no—
337D4—341no—
338D4—341no—
339D4—682no—
340D4D4×D1712yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—81no—
345C2×C4—171no—
346C2×C4—172no—
347C2×C4—172no—
348C2×C4—171no—
349C2×C4—171no—
350C2×C4—341no—
351C2×C4—341no—
352C2×C4—682no—
353C2×C4—682no—
354C2×C4—1361no—
355C2×C4—1361no—
356C2×C4—1361no—
357C2×C4D4×D1711yescharacteristic
358C2×C4D4×D1711yescharacteristic
359C2×C4C23×D1711yescharacteristic
360C8—21no—
361C8—21no—
362C8—341no—
363C8—341no—
364C22—22no—
365C22—41no—
366C22—41no—
367C22—172no—
368C22—341no—
369C22—341no—
370C22—342no—
371C22—681no—
372C22—681no—
373C22—681no—
374C22—681no—
375C22—681no—
376C22—1364no—
377C22C2×D4×D1711yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—172no—
384C4—341no—
385C4—341no—
386C4—681no—
387C4—681no—
388C4C2×D4×D1711yescharacteristic
389C4#41×D1711yescharacteristic
390C2—42no—
391C2—342no—
392C2—682no—
393C2—1361no—
394C2#121×D1711yescharacteristic
395C2#1176361511yescharacteristic
396C2#1176361811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#461C2
socleC22×C17C22×D4
prime radicalC2×#10587600C2
hypercenter#461D17
hypocenterC17#3042
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. C2×SD16×D17C2semidirect
3. #11763384C2semidirect
4. #11763281C2semidirect
5. #11763027C2semidirect
6. #11763026C2semidirect
7. #11763025C2semidirect
8. #11763024C2semidirect
9. #11763013C2semidirect
10. #11763012C2semidirect
11. #11763011C2semidirect
12. #11762834C2semidirect
13. #11762816C2semidirect
14. #11762804C2semidirect
15. C17×#461C2semidirect
21. C17×C4⋊Q8C22semidirect
23. #10587633C22semidirect
24. #10587630C22semidirect
46. #10587561C22semidirect
49. C17×C4⋊C8C22semidirect
50. C17×#103C22semidirect
51. C17×#103C22semidirect
52. #10587506C22semidirect
53. #10587506C22semidirect
54. #10587492C22semidirect
55. #10587487C22semidirect
56. #10587485C22semidirect
57. #10587485C22semidirect
58. #10587481C22semidirect
59. #10587480C22semidirect
60. C4⋊#3622C22semidirect
203. #461D17semidirect
230. C4⋊Q8C2×D17semidirect
250. C4⋊C8C2×D17semidirect
255. #103C2×D17semidirect
256. #103C2×D17semidirect
259. C17#3042semidirect

Maximal subgroups

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

subgroupindex
C2×#105876002
C2×SD16×D172
#117633842
#117632812
#117630272
#117630262
#117630252
#117630242
#117630132
#117630122
#117630112
#117628342
#117628162
#117628042
C17×#4612
#304217