Finite Group Databasebeta

← #14538303

Subgroups — #14538303

Order 2176.

490 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×#10587604C211yesmaximal, characteristic, complemented
2C2×SD16×D17C211yesmaximal, characteristic, complemented
3#11763377C211yesmaximal, characteristic
4#11763285C211yesmaximal, characteristic, complemented
5#11763034C211yesmaximal, characteristic, complemented
6#11763033C211yesmaximal, characteristic, complemented
7#11763028C211yesmaximal, characteristic, complemented
8#11763010C211yesmaximal, characteristic, complemented
9#11763009C211yesmaximal, characteristic, complemented
10#11763008C211yesmaximal, characteristic, complemented
11#11763007C211yesmaximal, characteristic, complemented
12#11762834C211yesmaximal, characteristic, complemented
13#11762812C211yesmaximal, characteristic, complemented
14#11762804C211yesmaximal, characteristic, complemented
15C17×#464C211yesmaximal, characteristic, complemented
16C2×#63256C2211yescharacteristic
17C2×Q8×D17C2211yescharacteristic
18C2×D4×D17C2211yescharacteristic
19C2×Q16×C17C2211yescharacteristic, complemented
20C2×SD16×C17C2211yescharacteristic
21C17×#124C2211yescharacteristic
22C4×Q8×C17C2211yescharacteristic, complemented
23#10587633C2211yescharacteristic
24Q8×Dic17C2211yescharacteristic, complemented
25C2×#63231C2211yescharacteristic, complemented
26C2×#63230C2211yescharacteristic
27C2×#63230C2211yescharacteristic, complemented
28#10587626C2211yescharacteristic
29C2×#63229C2211yescharacteristic
30#10587604—24no—
31SD16×D17—24no—
32C2×#63219C2211yescharacteristic
33C2×#63219C2211yescharacteristic, complemented
34C2×#63218C2211yescharacteristic
35C2×C8×D17C2211yescharacteristic
36#10587578—21no—
37#10587578C2211yescharacteristic
38#10587573—21no—
39#10587573C2211yescharacteristic, complemented
40#10587570—21no—
41#10587570—21no—
42#10587568—21no—
43#10587568—21no—
44#10587560C2211yescharacteristic
45#10587558—21no—
46#10587558C2211yescharacteristic
47C4×Dic34—21no—
48C4×Dic34C2211yescharacteristic, complemented
49C17×C4⋊C8C2211yescharacteristic, complemented
50C17×#103C2211yescharacteristic, complemented
51C17×#102C2211yescharacteristic
52#10587506C2211yescharacteristic, complemented
53#10587503C2211yescharacteristic
54#10587492C2211yescharacteristic
55#10587492C2211yescharacteristic, complemented
56#10587485C2211yescharacteristic, complemented
57#10587485C2211yescharacteristic, complemented
58#10587480C2211yescharacteristic
59#10587480C2211yescharacteristic, complemented
60C4⋊#3622C2211yescharacteristic, complemented
61C23×D17—21no—
62C2×Q8×C17C2311yescharacteristic
63C2×Q8×C17C2311yescharacteristic
64C2×D4×C17C2311yescharacteristic
65C2×C22⋊D17—21no—
66C2×C22⋊D17—21no—
67C2×C22⋊D17—21no—
68#63256—22no—
69#63256—22no—
70Q8×D17—22no—
71Q8×D17D414yescomplemented
72D4×D17—22no—
73D4×D17—22no—
74C2×D68—21no—
75C2×D68—21no—
76C2×D68—21no—
77C2×D68C2311yescharacteristic
78C2×D68C2311yescharacteristic
79C2×C4×D17—21no—
80C2×C4×D17—21no—
81C2×C4×D17C2311yescharacteristic
82C2×Dic34—21no—
83C2×Dic34C2311yescharacteristic
84Q16×C17—22no—
85SD16×C17—22no—
86C2×C8×C17C2311yescharacteristic
87C2×C8×C17C2311yescharacteristic
88C4⋊C4×C17—21no—
89C4⋊C4×C17C2311yescharacteristic
90C22⋊C4×C17—21no—
91C22⋊C4×C17—21no—
92C42×C17—21no—
93C42×C17C2311yescharacteristic
94#63231—22no—
95#63230—22no—
96#63230—22no—
97#63229—22no—
98#63227—21no—
99#63227—21no—
100#63227—21no—
101#63227D411yescharacteristic
102#63227D411yescharacteristic
103C4⋊Dic17—21no—
104C4⋊Dic17C2311yescharacteristic
105#63225—21no—
106#63225—21no—
107#63225D411yescharacteristic
108#63225D411yescharacteristic
109C4×Dic17—21no—
110C4×Dic17—21no—
111C4×Dic17—21no—
112C4×Dic17—21no—
113C4×Dic17C2311yescharacteristic
114C2×#3622C2311yescharacteristic
115C2×#3622C2311yescharacteristic
116#63219—22no—
117#63219—22no—
118#63218—22no—
119C8×D17—22no—
120C23×C17—21no—
121C22×D17—21no—
122C22×D17—21no—
123C22×D17—42no—
124C22×D17—81no—
125C22×D17—81no—
126C22×D17C2×D411yescharacteristic
127Q8×C17—21no—
128Q8×C17—21no—
129Q8×C17—21no—
130Q8×C17C2×D412yes—
131D4×C17—21no—
132D4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17C2×D411yescharacteristic
137C2×C4×C17C2×D411yescharacteristic
138C2×C4×C17C2411yescharacteristic
139C22⋊D17—42no—
140C22⋊D17—81no—
141C22⋊D17—81no—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17C2×D411yescharacteristic
145C2×Dic17C2×D411yescharacteristic
146C2×Dic17C2×D411yescharacteristic
147D68—21no—
148D68—21no—
149D68—21no—
150D68—21no—
151D68—42no—
152D68—81no—
153D68—81no—
154C4×D17—24no—
155C4×D17—42no—
156C4×D17C2×D412yes—
157C4×D17#4112yes—
158Dic34—21no—
159Dic34—24no—
160Dic34C2×D412yes—
161C8×C17—21no—
162C8×C17—21no—
163#3622—21no—
164#3622—21no—
165#3043—171nomaximal, Sylow
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17C22×D411yescharacteristic
170C2×D17—21no—
171C2×D17—21no—
172C2×D17—41no—
173C2×D17—41no—
174C2×D17—41no—
175C2×D17—41no—
176C2×D17—41no—
177C2×D17—41no—
178C2×D17—84no—
179C2×D17#12112yes—
180C4×C17—22no—
181C4×C17—21no—
182C4×C17—21no—
183C4×C17—41no—
184C4×C17—41no—
185C4×C17C22×D411yescharacteristic
186C4×C17C2×#4111yescharacteristic
187Dic17—22no—
188Dic17—21no—
189Dic17—21no—
190Dic17—41no—
191Dic17#12112yes—
192C2×#137—171no—
193C22×SD16—171no—
194#535—171no—
195#519—171no—
196#464—171no—
197#464—171no—
198#464—171no—
199#464D1711yescharacteristic, complemented
200#460—171no—
201#460—171no—
202#460—171no—
203#460—171no—
204#424—171no—
205#418—171no—
206#416—171no—
207C2×C17—81no—
208C2×C17C2×#12111yescharacteristic
209C2×C17#57711yescharacteristic
210C2×C17#57711yescharacteristic
211D17—22no—
212D17—81no—
213D17—81no—
214C2×#41—171no—
215C22×Q8—171no—
216C22×D4—171no—
217#137—344no—
218C2×Q16—171no—
219C2×Q16C2×D1711yescharacteristic, complemented
220C2×SD16—171no—
221C2×SD16—171no—
222C2×SD16—171no—
223C2×SD16—171no—
224C2×SD16—171no—
225C2×SD16—344no—
226C2×SD16C2×D1711yescharacteristic
227C2×M16—171no—
228C22×C8—171no—
229#127—171no—
230#127—171no—
231#124—171no—
232#124C2×D1711yescharacteristic
233#121—341no—
234#121—341no—
235#120—341no—
236#120—341no—
237C4×Q8—171no—
238C4×Q8—171no—
239C4×Q8—171no—
240C4×Q8—341no—
241C4×Q8—341no—
242C4×Q8C2×D1711yescharacteristic, complemented
243#117—171no—
244#117—171no—
245#117—341no—
246#117—341no—
247C4⋊C8—171no—
248C4⋊C8—171no—
249C4⋊C8—171no—
250C4⋊C8C2×D1711yescharacteristic, complemented
251#103—171no—
252#103C2×D1711yescharacteristic, complemented
253#102—171no—
254#102—171no—
255#102—171no—
256#102—171no—
257#102—171no—
258#102C2×D1711yescharacteristic
259C17#304311yescharacteristic, complemented, Sylow
260C24—341no—
261#41—342no—
262#41—342no—
263C2×Q8—174no—
264C2×Q8—171no—
265C2×Q8—341no—
266C2×Q8—342no—
267C2×Q8C22×D1711yescharacteristic
268C2×Q8C22×D1711yescharacteristic
269C2×D4—171no—
270C2×D4—171no—
271C2×D4—341no—
272C2×D4—342no—
273C2×D4—342no—
274C2×D4—341no—
275C2×D4—341no—
276C2×D4—341no—
277C2×D4—341no—
278C2×D4—341no—
279C2×D4C22×D1711yescharacteristic
280C22×C4—171no—
281C22×C4—341no—
282C22×C4—341no—
283Q16—22no—
284Q16—342no—
285SD16—22no—
286SD16—342no—
287SD16—342no—
288SD16—342no—
289SD16—342no—
290SD16—342no—
291M16—342no—
292C2×C8—171no—
293C2×C8—171no—
294C2×C8—342no—
295C2×C8C22×D1711yescharacteristic
296C2×C8C22×D1711yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—171no—
299C4⋊C4—171no—
300C4⋊C4—171no—
301C4⋊C4—341no—
302C4⋊C4—341no—
303C4⋊C4—341no—
304C4⋊C4C22×D1711yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—171no—
308C22⋊C4—171no—
309C22⋊C4—341no—
310C22⋊C4—341no—
311C22⋊C4—341no—
312C42—21no—
313C42—171no—
314C42—341no—
315C42—341no—
316C42—341no—
317C42—341no—
318C42C22×D1711yescharacteristic
319C23—21no—
320C23—171no—
321C23—341no—
322C23—341no—
323C23—682no—
324C23—1361no—
325C23—1361no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—172no—
330Q8—344no—
331Q8—341no—
332Q8D4×D1712yes—
333D4—21no—
334D4—21no—
335D4—341no—
336D4—341no—
337D4—341no—
338D4—341no—
339D4—682no—
340D4—682no—
341D4—1361no—
342D4—1361no—
343D4—1361no—
344D4—1361no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—171no—
349C2×C4—172no—
350C2×C4—172no—
351C2×C4—171no—
352C2×C4—171no—
353C2×C4—344no—
354C2×C4—341no—
355C2×C4—341no—
356C2×C4—682no—
357C2×C4D4×D1711yescharacteristic
358C2×C4D4×D1711yescharacteristic
359C2×C4C23×D1711yescharacteristic
360C8—21no—
361C8—21no—
362C8—341no—
363C8—341no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—172no—
368C22—341no—
369C22—341no—
370C22—681no—
371C22—681no—
372C22—681no—
373C22—681no—
374C22—681no—
375C22—681no—
376C22—1364no—
377C22C2×D4×D1711yescharacteristic
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—172no—
384C4—342no—
385C4—341no—
386C4—341no—
387C4—681no—
388C4C2×D4×D1711yescharacteristic
389C4#41×D1711yescharacteristic
390C2—81no—
391C2—342no—
392C2—1361no—
393C2—1361no—
394C2#121×D1711yescharacteristic
395C2#1176361611yescharacteristic
396C2#1176361811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#464C2
socleC22×C17C22×D4
prime radical#11763377C2
hypercenter#464D17
hypocenterC17#3043
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#10587604C2semidirect
2. C2×SD16×D17C2semidirect
4. #11763285C2semidirect
5. #11763034C2semidirect
6. #11763033C2semidirect
7. #11763028C2semidirect
8. #11763010C2semidirect
9. #11763009C2semidirect
10. #11763008C2semidirect
11. #11763007C2semidirect
12. #11762834C2semidirect
13. #11762812C2semidirect
14. #11762804C2semidirect
15. C17×#464C2semidirect
19. C2×Q16×C17C22semidirect
22. C4×Q8×C17C22semidirect
24. Q8×Dic17C22semidirect
25. C2×#63231C22semidirect
27. C2×#63230C22semidirect
33. C2×#63219C22semidirect
39. #10587573C22semidirect
48. C4×Dic34C22semidirect
49. C17×C4⋊C8C22semidirect
50. C17×#103C22semidirect
52. #10587506C22semidirect
55. #10587492C22semidirect
56. #10587485C22semidirect
57. #10587485C22semidirect
59. #10587480C22semidirect
60. C4⋊#3622C22semidirect
71. Q8×D17D4semidirect
199. #464D17semidirect
219. C2×Q16C2×D17semidirect
242. C4×Q8C2×D17semidirect
250. C4⋊C8C2×D17semidirect
252. #103C2×D17semidirect
259. C17#3043semidirect

Maximal subgroups

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

subgroupindex
C2×#105876042
C2×SD16×D172
#117633772
#117632852
#117630342
#117630332
#117630282
#117630102
#117630092
#117630082
#117630072
#117628342
#117628122
#117628042
C17×#4642
#304317