Finite Group Databasebeta

← #14540639

Subgroups — #14540639

Order 2176.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763509C211yesmaximal, characteristic, complemented
2#11763454C211yesmaximal, characteristic, complemented
3#11763164C211yesmaximal, characteristic, complemented
4#11763146C211yesmaximal, characteristic, complemented
5#11763136C211yesmaximal, characteristic, complemented
6#11763109C211yesmaximal, characteristic, complemented
7#11763044C211yesmaximal, characteristic, complemented
8#11763037C211yesmaximal, characteristic, complemented
9#11763024C211yesmaximal, characteristic, complemented
10#11762951C211yesmaximal, characteristic, complemented
11#11762941C211yesmaximal, characteristic, complemented
12C8×D68C211yesmaximal, characteristic, complemented
13#11762802C211yesmaximal, characteristic, complemented
14#102×D17C211yesmaximal, characteristic, complemented
15C4×D8×C17C211yesmaximal, characteristic, complemented
16C2×#63254C2211yescharacteristic
17C2×D4×D17C2211yescharacteristic
18C2×D8×C17C2211yescharacteristic
19C4×D4×C17C2211yescharacteristic
20C4×D4×C17C2211yescharacteristic, complemented
21#10587624C2211yescharacteristic
22#10587624C2211yescharacteristic, complemented
23C2×#63229C2211yescharacteristic, complemented
24C2×#63228C2211yescharacteristic
25#10587616—21no—
26#10587615—21no—
27C4×C22⋊D17—21no—
28C2×#63227—21no—
29C2×C8×D17C2211yescharacteristic
30C2×C8×D17C2211yescharacteristic, complemented
31#10587583C2211yescharacteristic
32#10587581C2211yescharacteristic
33#10587578C2211yescharacteristic, complemented
34C4⋊C4×D17C2211yescharacteristic
35#10587572—21no—
36#10587570—21no—
37#10587568—21no—
38#10587565—21no—
39#10587561C2211yescharacteristic
40C4×D68C2211yescharacteristic
41#10587556C2211yescharacteristic, complemented
42C17×#107C2211yescharacteristic, complemented
43C17×#102C2211yescharacteristic
44C17×#102C2211yescharacteristic, complemented
45C4×C8×C17C2211yescharacteristic, complemented
46#10587503C2211yescharacteristic
47#10587503C2211yescharacteristic
48#10587492C2211yescharacteristic
49#10587492C2211yescharacteristic
50#10587491C2211yescharacteristic
51#10587491C2211yescharacteristic, complemented
52#10587487C2211yescharacteristic, complemented
53#10587487C2211yescharacteristic, complemented
54#10587481C2211yescharacteristic, complemented
55#10587480C2211yescharacteristic
56#10587479C2211yescharacteristic, complemented
57#10587478C2211yescharacteristic, complemented
58C4⋊#3622C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×D4×C17C2311yescharacteristic
61C2×D4×C17C2311yescharacteristic
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C2×C22⋊D17—21no—
65C2×C22⋊D17—21no—
66C2×C22⋊D17—21no—
67C2×C22⋊D17—21no—
68C22×Dic17—21no—
69#63254—22no—
70#63254—41no—
71D4×D17—22no—
72D4×D17—41no—
73C2×D68—21no—
74C2×D68D411yescharacteristic
75C2×D68C2311yescharacteristic
76C2×C4×D17—21no—
77C2×C4×D17C2311yescharacteristic
78C2×C4×D17C2311yescharacteristic
79C2×Dic34C2311yescharacteristic
80D8×C17—22no—
81C2×C8×C17C2311yescharacteristic
82C2×C8×C17C2311yescharacteristic
83C4⋊C4×C17C2311yescharacteristic
84C4⋊C4×C17C2311yescharacteristic
85C22⋊C4×C17—21no—
86C22⋊C4×C17—21no—
87C42×C17C2311yescharacteristic
88C22⋊Dic17—21no—
89C22⋊Dic17—21no—
90#63229—41no—
91#63228—41no—
92#63227—21no—
93#63227—21no—
94#63227—21no—
95#63227—21no—
96#63227—21no—
97#63227D411yescharacteristic
98#63227D411yescharacteristic
99C4⋊Dic17—21no—
100C4⋊Dic17D411yescharacteristic
101C4⋊Dic17C2311yescharacteristic
102C4⋊Dic17C2311yescharacteristic
103#63225—21no—
104#63225—21no—
105#63225—21no—
106C4×Dic17—21no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109C8×D17—41no—
110C8×D17—41no—
111C23×C17—21no—
112C23×C17—21no—
113C22×D17—21no—
114C22×D17—42no—
115C22×D17—42no—
116C22×D17—81no—
117C22×D17C2×D411yescharacteristic
118C22×D17C2×D411yescharacteristic
119D4×C17—21no—
120D4×C17—21no—
121D4×C17—21no—
122D4×C17#4112yes—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—81no—
126C2×C4×C17—81no—
127C2×C4×C17C2×D411yescharacteristic
128C2×C4×C17C2×D411yescharacteristic
129C2×C4×C17C2411yescharacteristic
130C22⋊D17—42no—
131C22⋊D17—42no—
132C22⋊D17—42no—
133C22⋊D17—81no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—42no—
138C2×Dic17C2×D411yescharacteristic
139C2×Dic17C2×D411yescharacteristic
140D68—22no—
141D68—21no—
142D68—21no—
143D68—81no—
144C4×D17—21no—
145C4×D17—21no—
146C4×D17—21no—
147C4×D17—21no—
148C4×D17—81no—
149Dic34—21no—
150Dic34—21no—
151C8×C17—21no—
152C8×C17—21no—
153#3622—21no—
154#3622—21no—
155#3297—171nomaximal, Sylow
156C22×C17—22no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17C22×D411yescharacteristic
163C2×D17—21no—
164C2×D17—21no—
165C2×D17—21no—
166C2×D17—21no—
167C2×D17—21no—
168C2×D17—41no—
169C2×D17—41no—
170C2×D17—41no—
171C2×D17—84no—
172C2×D17#13512yes—
173C4×C17—21no—
174C4×C17—21no—
175C4×C17—41no—
176C4×C17—41no—
177C4×C17C2×#4111yescharacteristic
178C4×C17H(5,𝔽2)11yescharacteristic
179Dic17—21no—
180Dic17—21no—
181Dic17—41no—
182Dic17—41no—
183Dic17—41no—
184#553—171no—
185#546—171no—
186#488—171no—
187#484—171no—
188#482—171no—
189#478—171no—
190#466—171no—
191#465—171no—
192#463—171no—
193#449—171no—
194#447—171no—
195C4×D8D1711yescharacteristic, complemented
196C8×D4—171no—
197#416—171no—
198C2×#102—171no—
199C2×C17—42no—
200C2×C17—81no—
201C2×C17#54611yescharacteristic
202C2×C17C2×#13511yescharacteristic
203C2×C17#57611yescharacteristic
204D17—22no—
205D17—41no—
206D17—81no—
207C2×#41—171no—
208C22×D4—171no—
209C2×SD16—171no—
210C2×D8—171no—
211C2×D8C2×D1711yescharacteristic
212C22×C8—171no—
213C22×C8—171no—
214#126—341no—
215#125—171no—
216#124—171no—
217#123—341no—
218#123—341no—
219C22⋊Q8—171no—
220#121—171no—
221#121—171no—
222#121—171no—
223#121—341no—
224#121—341no—
225#120—341no—
226C4×D4—171no—
227C4×D4—341no—
228C4×D4C2×D1711yescharacteristic
229C4×D4C2×D1711yescharacteristic, complemented
230#117—171no—
231C2×C4⋊C4—171no—
232C2×C22⋊C4—341no—
233#107—171no—
234#107C2×D1711yescharacteristic, complemented
235#106—171no—
236C4⋊C8—171no—
237#103—171no—
238#103—171no—
239#103—171no—
240#102—171no—
241#102—171no—
242#102—171no—
243#102—171no—
244#102—171no—
245#102C2×D1711yescharacteristic
246#102C2×D1711yescharacteristic, complemented
247C22⋊C8—171no—
248C22⋊C8—171no—
249C4×C8C2×D1711yescharacteristic, complemented
250C17#329711yescharacteristic, complemented, Sylow
251C24—341no—
252#41—342no—
253#41—681no—
254C2×Q8—171no—
255C2×D4—171no—
256C2×D4—171no—
257C2×D4—341no—
258C2×D4—341no—
259C2×D4—342no—
260C2×D4—341no—
261C2×D4—341no—
262C2×D4—341no—
263C2×D4—681no—
264C2×D4C22×D1711yescharacteristic
265C2×D4C22×D1711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—171no—
269C22×C4—171no—
270C22×C4—341no—
271C22×C4—341no—
272SD16—681no—
273D8—22no—
274D8—681no—
275C2×C8—171no—
276C2×C8—171no—
277C2×C8—681no—
278C2×C8—681no—
279C2×C8C22×D1711yescharacteristic
280C2×C8C22×D1711yescharacteristic
281C4⋊C4—171no—
282C4⋊C4—171no—
283C4⋊C4—171no—
284C4⋊C4—341no—
285C4⋊C4—341no—
286C4⋊C4—341no—
287C4⋊C4—341no—
288C4⋊C4C22×D1711yescharacteristic
289C4⋊C4C22×D1711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—171no—
293C22⋊C4—171no—
294C22⋊C4—341no—
295C22⋊C4—341no—
296C22⋊C4—341no—
297C22⋊C4—341no—
298C22⋊C4—341no—
299C22⋊C4—341no—
300C22⋊C4—341no—
301C42—341no—
302C42C22×D1711yescharacteristic
303C23—21no—
304C23—21no—
305C23—171no—
306C23—171no—
307C23—341no—
308C23—682no—
309C23—682no—
310C23—1361no—
311Q8—341no—
312Q8—341no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—342no—
317D4—341no—
318D4—341no—
319D4—682no—
320D4—682no—
321D4—682no—
322D4—1361no—
323D4—1361no—
324D4#6325212yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—171no—
330C2×C4—171no—
331C2×C4—341no—
332C2×C4—341no—
333C2×C4—341no—
334C2×C4—341no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—682no—
339C2×C4—1361no—
340C2×C4D4×D1711yescharacteristic
341C2×C4D4×D1711yescharacteristic
342C2×C4C23×D1711yescharacteristic
343C8—21no—
344C8—21no—
345C8—341no—
346C8—341no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—172no—
354C22—341no—
355C22—341no—
356C22—341no—
357C22—341no—
358C22—341no—
359C22—681no—
360C22—681no—
361C22—681no—
362C22—1364no—
363C22C2×D4×D1711yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—341no—
369C4—341no—
370C4—681no—
371C4—681no—
372C4—681no—
373C4C2×#6325211yescharacteristic
374C4#1058769611yescharacteristic
375C2—42no—
376C2—81no—
377C2—342no—
378C2—681no—
379C2—1361no—
380C2#1176345511yescharacteristic
381C2#135×D1711yescharacteristic
382C2#1176361011yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC4×D8×C17C2
socleC22×C17C22×D4
hypercenterC4×D8D17
hypocenterC17#3297
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #11763509C2semidirect
2. #11763454C2semidirect
3. #11763164C2semidirect
4. #11763146C2semidirect
5. #11763136C2semidirect
6. #11763109C2semidirect
7. #11763044C2semidirect
8. #11763037C2semidirect
9. #11763024C2semidirect
10. #11762951C2semidirect
11. #11762941C2semidirect
12. C8×D68C2semidirect
13. #11762802C2semidirect
14. #102×D17C2semidirect
15. C4×D8×C17C2semidirect
20. C4×D4×C17C22semidirect
22. #10587624C22semidirect
23. C2×#63229C22semidirect
30. C2×C8×D17C22semidirect
33. #10587578C22semidirect
41. #10587556C22semidirect
42. C17×#107C22semidirect
44. C17×#102C22semidirect
45. C4×C8×C17C22semidirect
51. #10587491C22semidirect
52. #10587487C22semidirect
53. #10587487C22semidirect
54. #10587481C22semidirect
56. #10587479C22semidirect
57. #10587478C22semidirect
58. C4⋊#3622C22semidirect
195. C4×D8D17semidirect
229. C4×D4C2×D17semidirect
234. #107C2×D17semidirect
246. #102C2×D17semidirect
249. C4×C8C2×D17semidirect
250. C17#3297semidirect

Maximal subgroups

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

subgroupindex
#117635092
#117634542
#117631642
#117631462
#117631362
#117631092
#117630442
#117630372
#117630242
#117629512
#117629412
C8×D682
#117628022
#102×D172
C4×D8×C172
#329717