Finite Group Databasebeta

← #14539137

Subgroups — #14539137

Order 2176.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763439C211yesmaximal, characteristic, complemented
2#11763377C211yesmaximal, characteristic
3#11763175C211yesmaximal, characteristic, complemented
4#11763172C211yesmaximal, characteristic, complemented
5#11763170C211yesmaximal, characteristic, complemented
6#11763168C211yesmaximal, characteristic, complemented
7#11763165C211yesmaximal, characteristic, complemented
8#11763156C211yesmaximal, characteristic, complemented
9#11763154C211yesmaximal, characteristic, complemented
10#11762812C211yesmaximal, characteristic, complemented
11#11762809C211yesmaximal, characteristic, complemented
12#11762804C211yesmaximal, characteristic, complemented
13#102×D17C211yesmaximal, characteristic, complemented
14#11762763C211yesmaximal, characteristic, complemented
15C17×#488C211yesmaximal, characteristic, complemented
16C2×#63256C2211yescharacteristic
17C2×D4×D17C2211yescharacteristic
18C17×#125C2211yescharacteristic, complemented
19C17×#124C2211yescharacteristic
20#10587633C2211yescharacteristic
21#10587626C2211yescharacteristic
22C2×#63218C2211yescharacteristic
23C2×C8×D17C2211yescharacteristic
24#10587583—21no—
25#10587582—21no—
26#10587578C2211yescharacteristic
27C4⋊C4×D17C2211yescharacteristic
28#10587576C2211yescharacteristic, complemented
29#10587575—21no—
30#10587574—21no—
31#10587574C2211yescharacteristic, complemented
32#10587570—21no—
33#10587570—21no—
34#10587568—21no—
35#10587568—21no—
36#10587560C2211yescharacteristic
37#10587558C2211yescharacteristic
38#10587555C2211yescharacteristic, complemented
39C17×#103C2211yescharacteristic, complemented
40C17×#103C2211yescharacteristic, complemented
41C17×#102C2211yescharacteristic
42C17×#102C2211yescharacteristic
43C4×C8×C17C2211yescharacteristic, complemented
44#10587506C2211yescharacteristic, complemented
45#10587506C2211yescharacteristic, complemented
46#10587503C2211yescharacteristic
47#10587503C2211yescharacteristic
48#10587492C2211yescharacteristic
49#10587492C2211yescharacteristic
50#10587492C2211yescharacteristic, complemented
51#10587492C2211yescharacteristic, complemented
52#10587486C2211yescharacteristic, complemented
53C8×Dic17C2211yescharacteristic, complemented
54#10587480C2211yescharacteristic
55#10587480C2211yescharacteristic
56#10587480C2211yescharacteristic, complemented
57#10587480C2211yescharacteristic, complemented
58#10587474C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×Q8×C17C2311yescharacteristic
61C2×D4×C17C2311yescharacteristic
62C2×C22⋊D17—21no—
63C2×C22⋊D17—21no—
64C2×C22⋊D17—21no—
65#63256—22no—
66#63256—22no—
67D4×D17—22no—
68D4×D17—22no—
69C2×D68—21no—
70C2×D68—21no—
71C2×D68—21no—
72C2×D68C2311yescharacteristic
73C2×D68C2311yescharacteristic
74C2×C4×D17—21no—
75C2×C4×D17—21no—
76C2×C4×D17C2311yescharacteristic
77C2×Dic34—21no—
78C2×Dic34—21no—
79C2×C8×C17C2311yescharacteristic
80C2×C8×C17C2311yescharacteristic
81C4⋊C4×C17—21no—
82C4⋊C4×C17—21no—
83C4⋊C4×C17C2311yescharacteristic
84C4⋊C4×C17C2311yescharacteristic
85C22⋊C4×C17—21no—
86C22⋊C4×C17—21no—
87C42×C17C2311yescharacteristic
88#63227—21no—
89#63227—21no—
90#63227—21no—
91#63227D411yescharacteristic
92#63227D411yescharacteristic
93C4⋊Dic17—21no—
94C4⋊Dic17—21no—
95C4⋊Dic17C2311yescharacteristic
96C4⋊Dic17C2311yescharacteristic
97#63225—21no—
98#63225—21no—
99#63225—21no—
100#63225D411yescharacteristic
101#63225D411yescharacteristic
102C4×Dic17—21no—
103C4×Dic17C2311yescharacteristic
104C2×#3622C2311yescharacteristic
105C2×#3622C2311yescharacteristic
106#63218—22no—
107C8×D17—22no—
108C23×C17—21no—
109C22×D17—21no—
110C22×D17—21no—
111C22×D17—42no—
112C22×D17—81no—
113C22×D17—81no—
114C22×D17C2×D411yescharacteristic
115Q8×C17—21no—
116Q8×C17—21no—
117D4×C17—21no—
118D4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17C2×D411yescharacteristic
123C2×C4×C17C2×D411yescharacteristic
124C2×C4×C17C2411yescharacteristic
125C22⋊D17—42no—
126C22⋊D17—81no—
127C22⋊D17—81no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17C2×D411yescharacteristic
131C2×Dic17C2×D411yescharacteristic
132C2×Dic17C2×D411yescharacteristic
133D68—21no—
134D68—21no—
135D68—21no—
136D68—21no—
137D68—42no—
138D68—81no—
139D68—81no—
140C4×D17—42no—
141C4×D17—81no—
142C4×D17#4112yes—
143C4×D17#4112yes—
144Dic34—81no—
145Dic34—81no—
146C8×C17—21no—
147C8×C17—21no—
148#3622—21no—
149#3622—21no—
150#3136—171nomaximal, Sylow
151C22×C17—41no—
152C22×C17—41no—
153C22×C17—41no—
154C22×C17C22×D411yescharacteristic
155C2×D17—21no—
156C2×D17—21no—
157C2×D17—41no—
158C2×D17—41no—
159C2×D17—41no—
160C2×D17—41no—
161C2×D17—41no—
162C2×D17—41no—
163C2×D17—84no—
164C2×D17#12412yes—
165C4×C17—21no—
166C4×C17—21no—
167C4×C17—41no—
168C4×C17—41no—
169C4×C17—41no—
170C4×C17C2×#4111yescharacteristic
171C4×C17C2×#4111yescharacteristic
172Dic17—21no—
173Dic17—21no—
174Dic17—41no—
175Dic17—41no—
176Dic17#12412yes—
177#544—171no—
178#535—171no—
179#490—171no—
180#490—171no—
181#489—171no—
182#489—171no—
183#488—171no—
184#488D1711yescharacteristic, complemented
185#486—171no—
186#486—171no—
187#418—171no—
188#417—171no—
189#416—171no—
190C2×#102—171no—
191#405—171no—
192C2×C17—81no—
193C2×C17C2×#12411yescharacteristic
194C2×C17#57611yescharacteristic
195C2×C17#57711yescharacteristic
196D17—22no—
197D17—81no—
198D17—81no—
199C2×#41—171no—
200C22×D4—171no—
201C2×M16—171no—
202C22×C8—171no—
203C4⋊Q8—171no—
204C4⋊Q8—171no—
205C4⋊Q8—341no—
206#127—171no—
207#127—171no—
208#125—171no—
209#125—341no—
210#125C2×D1711yescharacteristic, complemented
211#124—171no—
212#124C2×D1711yescharacteristic
213C22⋊Q8—341no—
214C22⋊Q8—341no—
215#121—341no—
216#121—341no—
217#120—341no—
218#120—341no—
219#117—171no—
220#117—171no—
221C2×C4⋊C4—171no—
222#103—171no—
223#103—171no—
224#103C2×D1711yescharacteristic, complemented
225#103C2×D1711yescharacteristic, complemented
226#102—171no—
227#102—171no—
228#102—171no—
229#102—171no—
230#102—171no—
231#102—171no—
232#102—171no—
233#102—171no—
234#102—171no—
235#102—171no—
236#102C2×D1711yescharacteristic
237#102C2×D1711yescharacteristic
238#97—171no—
239#97—171no—
240C4×C8—171no—
241C4×C8C2×D1711yescharacteristic, complemented
242C17#313611yescharacteristic, complemented, Sylow
243C24—341no—
244#41—342no—
245#41—342no—
246C2×Q8—341no—
247C2×Q8—341no—
248C2×Q8C22×D1711yescharacteristic
249C2×D4—171no—
250C2×D4—171no—
251C2×D4—341no—
252C2×D4—342no—
253C2×D4—342no—
254C2×D4—341no—
255C2×D4—341no—
256C2×D4—341no—
257C2×D4—341no—
258C2×D4—341no—
259C2×D4C22×D1711yescharacteristic
260C22×C4—171no—
261C22×C4—341no—
262C22×C4—341no—
263M16—342no—
264C2×C8—171no—
265C2×C8—171no—
266C2×C8—342no—
267C2×C8C22×D1711yescharacteristic
268C2×C8C22×D1711yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—171no—
272C4⋊C4—171no—
273C4⋊C4—171no—
274C4⋊C4—171no—
275C4⋊C4—341no—
276C4⋊C4—341no—
277C4⋊C4—341no—
278C4⋊C4—341no—
279C4⋊C4—341no—
280C4⋊C4C22×D1711yescharacteristic
281C4⋊C4C22×D1711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—171no—
285C22⋊C4—171no—
286C22⋊C4—341no—
287C22⋊C4—341no—
288C22⋊C4—341no—
289C42—171no—
290C42—341no—
291C42C22×D1711yescharacteristic
292C23—21no—
293C23—171no—
294C23—341no—
295C23—341no—
296C23—682no—
297C23—1361no—
298C23—1361no—
299Q8—21no—
300Q8—21no—
301Q8—1361no—
302Q8—1361no—
303D4—21no—
304D4—21no—
305D4—341no—
306D4—341no—
307D4—341no—
308D4—341no—
309D4—682no—
310D4—682no—
311D4—1361no—
312D4—1361no—
313D4—1361no—
314D4—1361no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—171no—
319C2×C4—172no—
320C2×C4—172no—
321C2×C4—171no—
322C2×C4—171no—
323C2×C4—341no—
324C2×C4—341no—
325C2×C4—682no—
326C2×C4—1361no—
327C2×C4D4×D1711yescharacteristic
328C2×C4D4×D1711yescharacteristic
329C2×C4C23×D1711yescharacteristic
330C8—21no—
331C8—21no—
332C8—341no—
333C8—341no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—172no—
338C22—341no—
339C22—341no—
340C22—681no—
341C22—681no—
342C22—681no—
343C22—681no—
344C22—681no—
345C22—681no—
346C22—1364no—
347C22C2×D4×D1711yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—172no—
354C4—341no—
355C4—341no—
356C4—681no—
357C4—681no—
358C4#41×D1711yescharacteristic
359C4#41×D1711yescharacteristic
360C2—81no—
361C2—342no—
362C2—1361no—
363C2—1361no—
364C2#124×D1711yescharacteristic
365C2#1176361111yescharacteristic
366C2#1176361611yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#488C2
socleC22×C17C22×D4
prime radical#11763377C2
hypercenter#488D17
hypocenterC17#3136
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #11763439C2semidirect
3. #11763175C2semidirect
4. #11763172C2semidirect
5. #11763170C2semidirect
6. #11763168C2semidirect
7. #11763165C2semidirect
8. #11763156C2semidirect
9. #11763154C2semidirect
10. #11762812C2semidirect
11. #11762809C2semidirect
12. #11762804C2semidirect
13. #102×D17C2semidirect
14. #11762763C2semidirect
15. C17×#488C2semidirect
18. C17×#125C22semidirect
28. #10587576C22semidirect
31. #10587574C22semidirect
38. #10587555C22semidirect
39. C17×#103C22semidirect
40. C17×#103C22semidirect
43. C4×C8×C17C22semidirect
44. #10587506C22semidirect
45. #10587506C22semidirect
50. #10587492C22semidirect
51. #10587492C22semidirect
52. #10587486C22semidirect
53. C8×Dic17C22semidirect
56. #10587480C22semidirect
57. #10587480C22semidirect
58. #10587474C22semidirect
184. #488D17semidirect
210. #125C2×D17semidirect
224. #103C2×D17semidirect
225. #103C2×D17semidirect
241. C4×C8C2×D17semidirect
242. C17#3136semidirect

Maximal subgroups

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

subgroupindex
#117634392
#117633772
#117631752
#117631722
#117631702
#117631682
#117631652
#117631562
#117631542
#117628122
#117628092
#117628042
#102×D172
#117627632
C17×#4882
#313617