Finite Group Databasebeta

← #14540866

Subgroups — #14540866

Order 2176.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763514C211yesmaximal, characteristic, complemented
2#11763447C211yesmaximal, characteristic
3#11763170C211yesmaximal, characteristic, complemented
4#11763143C211yesmaximal, characteristic, complemented
5#11763122C211yesmaximal, characteristic, complemented
6#11763087C211yesmaximal, characteristic, complemented
7#11763055C211yesmaximal, characteristic, complemented
8#11763049C211yesmaximal, characteristic, complemented
9#11763034C211yesmaximal, characteristic, complemented
10#11762971C211yesmaximal, characteristic, complemented
11#11762961C211yesmaximal, characteristic, complemented
12#11762906C211yesmaximal, characteristic, complemented
13#11762878C211yesmaximal, characteristic, complemented
14#102×D17C211yesmaximal, characteristic, complemented
15C17×#417C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C2×D4×D17C2211yescharacteristic
18C2×M16×C17C2211yescharacteristic, complemented
19C2×C4⋊C4×C17C2211yescharacteristic, complemented
20#10587633C2211yescharacteristic
21#10587632C2211yescharacteristic
22C2×#63231C2211yescharacteristic, complemented
23#10587625—21no—
24#10587623—21no—
25D4×Dic17C2211yescharacteristic
26C2×#63229C2211yescharacteristic
27#10587616C2211yescharacteristic
28#10587615—21no—
29C4×C22⋊D17—21no—
30C4×C22⋊D17C2211yescharacteristic
31C2×#63227—21no—
32#10587610—21no—
33#10587610C2211yescharacteristic, complemented
34C2×#63219C2211yescharacteristic
35C2×C8×D17C2211yescharacteristic
36#10587583C2211yescharacteristic
37#10587582—21no—
38C4⋊C4×D17C2211yescharacteristic
39#10587575—21no—
40#10587574C2211yescharacteristic, complemented
41#10587573C2211yescharacteristic, complemented
42C17×#103C2211yescharacteristic, complemented
43C17×#103C2211yescharacteristic, complemented
44C17×#102C2211yescharacteristic
45C17×#102C2211yescharacteristic
46#10587506C2211yescharacteristic, complemented
47#10587503C2211yescharacteristic
48C22⋊#3622C2211yescharacteristic, complemented
49#10587492C2211yescharacteristic
50#10587491C2211yescharacteristic
51#10587489C2211yescharacteristic, complemented
52#10587487C2211yescharacteristic, complemented
53#10587486C2211yescharacteristic, complemented
54#10587485C2211yescharacteristic, complemented
55#10587480C2211yescharacteristic
56#10587480C2211yescharacteristic
57#10587478C2211yescharacteristic, complemented
58#10587478C2211yescharacteristic, complemented
59C23×D17—21no—
60#41×C17—22no—
61#41×C17—41no—
62C2×Q8×C17C2311yescharacteristic
63C2×D4×C17—21no—
64C2×D4×C17C2311yescharacteristic
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17C2311yescharacteristic
68C2×C22⋊D17—21no—
69C2×C22⋊D17—21no—
70C2×C22⋊D17D411yescharacteristic
71C22×Dic17—21no—
72D4×D17—22no—
73D4×D17—41no—
74C2×D68C2311yescharacteristic
75C2×C4×D17—21no—
76C2×C4×D17C2311yescharacteristic
77C2×Dic34—21no—
78C2×Dic34C2311yescharacteristic
79M16×C17—41no—
80C2×C8×C17C2311yescharacteristic
81C2×C8×C17C2311yescharacteristic
82C4⋊C4×C17—21no—
83C4⋊C4×C17C2311yescharacteristic
84C4⋊C4×C17C2311yescharacteristic
85C22⋊Dic17—21no—
86C22⋊Dic17—21no—
87C22⋊Dic17D411yescharacteristic
88#63231—41no—
89#63229—22no—
90#63227—21no—
91#63227—21no—
92#63227—21no—
93#63227—21no—
94#63227—21no—
95#63227D411yescharacteristic
96C4⋊Dic17—21no—
97C4⋊Dic17C2311yescharacteristic
98C4⋊Dic17C2311yescharacteristic
99#63225—21no—
100#63225—21no—
101#63225—21no—
102#63225—21no—
103#63225—21no—
104#63225D411yescharacteristic
105C4×Dic17—21no—
106C4×Dic17C2311yescharacteristic
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109#63219—41no—
110C8×D17—41no—
111C23×C17—21no—
112C23×C17C2×D411yescharacteristic
113C22×D17—21no—
114C22×D17—42no—
115C22×D17—42no—
116C22×D17—81no—
117C22×D17C2×D411yescharacteristic
118Q8×C17—21no—
119Q8×C17—21no—
120D4×C17—21no—
121D4×C17—42no—
122D4×C17#4112yes—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—42no—
129C2×C4×C17—81no—
130C2×C4×C17C2×D411yescharacteristic
131C2×C4×C17C2411yescharacteristic
132C22⋊D17—22no—
133C22⋊D17—42no—
134C22⋊D17—81no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—81no—
139C2×Dic17C2×D411yescharacteristic
140C2×Dic17C2×D411yescharacteristic
141C2×Dic17C2×D411yescharacteristic
142D68—21no—
143D68—21no—
144C4×D17—21no—
145C4×D17—21no—
146C4×D17—81no—
147Dic34—21no—
148Dic34—21no—
149Dic34—81no—
150C8×C17—21no—
151C8×C17—21no—
152#3622—21no—
153#3622—21no—
154#3314—171nomaximal, Sylow
155C22×C17—22no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17C22×D411yescharacteristic
162C2×D17—21no—
163C2×D17—21no—
164C2×D17—41no—
165C2×D17—41no—
166C2×D17—41no—
167C2×D17—84no—
168C2×D17#13712yes—
169C4×C17—21no—
170C4×C17—41no—
171C4×C17—41no—
172C4×C17—41no—
173C4×C17C2×#4111yescharacteristic
174C4×C17H(5,𝔽2)11yescharacteristic
175Dic17—21no—
176Dic17—21no—
177Dic17—21no—
178Dic17—41no—
179Dic17—41no—
180Dic17—41no—
181#554—171no—
182#546—171no—
183#489—171no—
184#484—171no—
185#480—171no—
186#474—171no—
187#468—171no—
188#467—171no—
189#464—171no—
190#452—171no—
191#450—171no—
192#440—171no—
193#435—171no—
194#417D1711yescharacteristic, complemented
195C2×#102—171no—
196C2×C17—42no—
197C2×C17—41no—
198C2×C17#54611yescharacteristic
199C2×C17C2×#13711yescharacteristic
200C2×C17#57611yescharacteristic
201D17—22no—
202D17—81no—
203C2×#41C2×D1711yescharacteristic
204C22×D4—171no—
205C2×Q16—171no—
206C2×SD16—171no—
207C2×SD16—171no—
208C2×M16C2×D1711yescharacteristic, complemented
209C22×C8—171no—
210C4⋊Q8—171no—
211#125—341no—
212#124—171no—
213#123—341no—
214C22⋊Q8—171no—
215C22⋊Q8—171no—
216C22⋊Q8—171no—
217C22⋊Q8—341no—
218C22⋊Q8—341no—
219#121—171no—
220#121—341no—
221#120—341no—
222C4×Q8—171no—
223C4×D4—171no—
224C4×D4—171no—
225C4×D4—341no—
226C2×C4⋊C4—171no—
227C2×C4⋊C4C2×D1711yescharacteristic, complemented
228C2×C22⋊C4—341no—
229#107—171no—
230#107—171no—
231#107—171no—
232C4⋊C8—171no—
233#103—171no—
234#103—171no—
235#103C2×D1711yescharacteristic, complemented
236#103C2×D1711yescharacteristic, complemented
237#102—171no—
238#102—171no—
239#102—171no—
240#102—171no—
241#102C2×D1711yescharacteristic
242#102C2×D1711yescharacteristic
243C22⋊C8—171no—
244C22⋊C8—171no—
245#97—171no—
246C17#331411yescharacteristic, complemented, Sylow
247C24—341no—
248#41—22no—
249#41—41no—
250C2×Q8—171no—
251C2×Q8—341no—
252C2×Q8C22×D1711yescharacteristic
253C2×D4—21no—
254C2×D4—171no—
255C2×D4—171no—
256C2×D4—341no—
257C2×D4—342no—
258C2×D4—341no—
259C2×D4—681no—
260C2×D4C22×D1711yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—171no—
264C22×C4—341no—
265C22×C4—341no—
266C22×C4C22×D1711yescharacteristic
267Q16—681no—
268SD16—342no—
269SD16—681no—
270M16—41no—
271C2×C8—171no—
272C2×C8—171no—
273C2×C8—681no—
274C2×C8C22×D1711yescharacteristic
275C2×C8C22×D1711yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—171no—
278C4⋊C4—171no—
279C4⋊C4—171no—
280C4⋊C4—341no—
281C4⋊C4—341no—
282C4⋊C4—341no—
283C4⋊C4—341no—
284C4⋊C4—341no—
285C4⋊C4—341no—
286C4⋊C4C22×D1711yescharacteristic
287C4⋊C4C22×D1711yescharacteristic
288C22⋊C4—171no—
289C22⋊C4—171no—
290C22⋊C4—341no—
291C22⋊C4—341no—
292C22⋊C4—341no—
293C22⋊C4—341no—
294C22⋊C4—341no—
295C22⋊C4—341no—
296C22⋊C4—341no—
297C42—171no—
298C42—341no—
299C23—21no—
300C23—171no—
301C23—341no—
302C23—682no—
303C23—682no—
304C23—1361no—
305C23D4×D1711yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—341no—
309Q8—341no—
310Q8—1361no—
311D4—21no—
312D4—42no—
313D4—342no—
314D4—341no—
315D4—341no—
316D4—682no—
317D4—1361no—
318D4#6325212yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—81no—
326C2×C4—171no—
327C2×C4—171no—
328C2×C4—171no—
329C2×C4—341no—
330C2×C4—341no—
331C2×C4—341no—
332C2×C4—341no—
333C2×C4—341no—
334C2×C4—1361no—
335C2×C4—1361no—
336C2×C4D4×D1711yescharacteristic
337C2×C4C23×D1711yescharacteristic
338C8—21no—
339C8—21no—
340C8—341no—
341C8—341no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—41no—
347C22—41no—
348C22—172no—
349C22—341no—
350C22—341no—
351C22—681no—
352C22—681no—
353C22—681no—
354C22—1364no—
355C22C2×D4×D1711yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—341no—
361C4—341no—
362C4—341no—
363C4—681no—
364C4—681no—
365C4—681no—
366C4C2×#6325211yescharacteristic
367C4#1058769111yescharacteristic
368C2—41no—
369C2—42no—
370C2—342no—
371C2—1361no—
372C2#1176344911yescharacteristic
373C2#137×D1711yescharacteristic
374C2#1176361311yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#417C2
socleC22×C17C22×D4
prime radical#11763447C2
hypercenter#417D17
hypocenterC17#3314
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #11763514C2semidirect
3. #11763170C2semidirect
4. #11763143C2semidirect
5. #11763122C2semidirect
6. #11763087C2semidirect
7. #11763055C2semidirect
8. #11763049C2semidirect
9. #11763034C2semidirect
10. #11762971C2semidirect
11. #11762961C2semidirect
12. #11762906C2semidirect
13. #11762878C2semidirect
14. #102×D17C2semidirect
15. C17×#417C2semidirect
18. C2×M16×C17C22semidirect
19. C2×C4⋊C4×C17C22semidirect
22. C2×#63231C22semidirect
33. #10587610C22semidirect
40. #10587574C22semidirect
41. #10587573C22semidirect
42. C17×#103C22semidirect
43. C17×#103C22semidirect
46. #10587506C22semidirect
48. C22⋊#3622C22semidirect
51. #10587489C22semidirect
52. #10587487C22semidirect
53. #10587486C22semidirect
54. #10587485C22semidirect
57. #10587478C22semidirect
58. #10587478C22semidirect
194. #417D17semidirect
208. C2×M16C2×D17semidirect
227. C2×C4⋊C4C2×D17semidirect
235. #103C2×D17semidirect
236. #103C2×D17semidirect
246. C17#3314semidirect

Maximal subgroups

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

subgroupindex
#117635142
#117634472
#117631702
#117631432
#117631222
#117630872
#117630552
#117630492
#117630342
#117629712
#117629612
#117629062
#117628782
#102×D172
C17×#4172
#331417