Finite Group Databasebeta

← #14540745

Subgroups — #14540745

Order 2176.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763463C211yesmaximal, characteristic, complemented
2#11763168C211yesmaximal, characteristic, complemented
3#11763137C211yesmaximal, characteristic, complemented
4#11763125C211yesmaximal, characteristic, complemented
5#11763099C211yesmaximal, characteristic, complemented
6#11763060C211yesmaximal, characteristic, complemented
7#11763040C211yesmaximal, characteristic, complemented
8#11763021C211yesmaximal, characteristic, complemented
9#11762954C211yesmaximal, characteristic, complemented
10#11762943C211yesmaximal, characteristic, complemented
11#11762919C211yesmaximal, characteristic, complemented
12#11762881C211yesmaximal, characteristic, complemented
13#11762810C211yesmaximal, characteristic, complemented
14C2×#10587503C211yesmaximal, characteristic, complemented
15C17×#546C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C22×D4×C17C2211yescharacteristic
18C2×#63254—21no—
19C17×#124C2211yescharacteristic
20C17×#123—21no—
21C17×C22⋊Q8C2211yescharacteristic
22C17×#121—21no—
23C17×#121C2211yescharacteristic
24C17×#120—21no—
25C4×D4×C17C2211yescharacteristic
26C4×D4×C17C2211yescharacteristic
27C2×C22⋊C4×C17—21no—
28C2×#63230C2211yescharacteristic, complemented
29C2×#63228C2211yescharacteristic
30C2×#63228C2211yescharacteristic
31#10587616C2211yescharacteristic
32#10587616C2211yescharacteristic, complemented
33C2×C4⋊Dic17C2211yescharacteristic
34C2×C4⋊Dic17C2211yescharacteristic, complemented
35C2×#63223C2211yescharacteristic, complemented
36C22×#3622C2211yescharacteristic
37#10587583—21no—
38#10587572—21no—
39#10587572—21no—
40C4×D68C2211yescharacteristic, complemented
41#10587555C2211yescharacteristic, complemented
42#10587506C2211yescharacteristic, complemented
43#10587506C2211yescharacteristic, complemented
44#10587506C2211yescharacteristic, complemented
45#10587503C2211yescharacteristic
46#10587503C2211yescharacteristic
47#10587503C2211yescharacteristic
48#10587503C2211yescharacteristic
49#10587503C2211yescharacteristic
50#10587503C2211yescharacteristic
51C22⋊#3622C2211yescharacteristic
52C22⋊#3622C2211yescharacteristic, complemented
53#10587480C2211yescharacteristic, complemented
54#10587479C2211yescharacteristic, complemented
55#10587479C2211yescharacteristic, complemented
56#10587479C2211yescharacteristic, complemented
57C4⋊#3622C2211yescharacteristic, complemented
58#10587474C2211yescharacteristic, complemented
59C24×C17—21no—
60#41×C17—22no—
61#41×C17—41no—
62C2×Q8×C17C2311yescharacteristic
63C2×D4×C17—22no—
64C2×D4×C17—21no—
65C2×D4×C17—21no—
66C2×D4×C17—21no—
67C2×D4×C17—41no—
68C2×D4×C17D411yescharacteristic
69C2×D4×C17C2311yescharacteristic
70C2×D4×C17C2311yescharacteristic
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17C2311yescharacteristic
74C22×C4×C17C2311yescharacteristic
75C2×C22⋊D17—21no—
76C2×C22⋊D17—21no—
77C22×Dic17—21no—
78C22×Dic17—21no—
79#63254—44no—
80C2×D68C2311yescharacteristic
81C2×C4×D17—21no—
82C2×Dic34—21no—
83C4⋊C4×C17—21no—
84C4⋊C4×C17D411yescharacteristic
85C4⋊C4×C17C2311yescharacteristic
86C22⋊C4×C17—21no—
87C22⋊C4×C17—21no—
88C22⋊C4×C17—21no—
89C22⋊C4×C17—21no—
90C22⋊C4×C17—21no—
91C22⋊C4×C17D411yescharacteristic
92C22⋊C4×C17D411yescharacteristic
93C42×C17C2311yescharacteristic
94#63230—41no—
95#63228—22no—
96#63228—41no—
97#63227—21no—
98#63227—21no—
99#63227—21no—
100C4⋊Dic17—21no—
101C4⋊Dic17—21no—
102C4⋊Dic17—21no—
103C4⋊Dic17C2311yescharacteristic
104C4⋊Dic17C2311yescharacteristic
105C4⋊Dic17C2311yescharacteristic
106#63223—41no—
107C2×#3622—41no—
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—42no—
115C23×C17—42no—
116C23×C17—81no—
117C23×C17C2×D411yescharacteristic
118C23×C17C2×D411yescharacteristic
119C22×D17—21no—
120Q8×C17—21no—
121Q8×C17—21no—
122D4×C17—22no—
123D4×C17—21no—
124D4×C17—21no—
125D4×C17—21no—
126D4×C17—42no—
127D4×C17—42no—
128D4×C17—81no—
129D4×C17#4112yescomplemented
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—42no—
137C2×C4×C17—81no—
138C2×C4×C17C2×D411yescharacteristic
139C2×C4×C17C2×D411yescharacteristic
140C2×C4×C17C2×D411yescharacteristic
141C2×C4×C17C2×D411yescharacteristic
142C2×C4×C17C2411yescharacteristic
143C22⋊D17—42no—
144C22⋊D17—42no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—42no—
149C2×Dic17—42no—
150D68—21no—
151D68—21no—
152C4×D17—42no—
153Dic34—42no—
154#3622—21no—
155#3622—21no—
156#3622—21no—
157#3622—21no—
158#3306—171nomaximal, Sylow
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—22no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—84no—
171C22×C17AGL(1,ℤ/8)12yes—
172C22×C17C22×D411yescharacteristic
173C2×D17—41no—
174C2×D17—41no—
175C2×D17—41no—
176C4×C17—21no—
177C4×C17—21no—
178C4×C17—21no—
179C4×C17—21no—
180C4×C17—41no—
181C4×C17—41no—
182C4×C17C2×#4111yescharacteristic
183C4×C17H(5,𝔽2)11yescharacteristic
184Dic17—41no—
185Dic17—41no—
186Dic17—41no—
187#547—171no—
188#546D1711yescharacteristic, complemented
189#489—171no—
190#483—171no—
191#481—171no—
192#476—171no—
193#469—171no—
194#465—171no—
195#463—171no—
196#449—171no—
197#447—171no—
198#442—171no—
199#435—171no—
200#417—171no—
201C2×#102—171no—
202C2×C17—22no—
203C2×C17—41no—
204C2×C17—42no—
205C2×C17—81no—
206C2×C17#54611yescharacteristic
207C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
208C2×C17#57711yescharacteristic
209D17—81no—
210C2×#41—341no—
211C2×#41C2×D1711yescharacteristic
212C22×D4C2×D1711yescharacteristic
213C2×SD16—171no—
214C2×D8—171no—
215C2×D8—171no—
216C2×M16—171no—
217C22×C8—171no—
218C4⋊Q8—171no—
219#124C2×D1711yescharacteristic
220#123—21no—
221#123—341no—
222#123—341no—
223C22⋊Q8—341no—
224C22⋊Q8C2×D1711yescharacteristic
225#121—21no—
226#121—171no—
227#121—171no—
228#121C2×D1711yescharacteristic
229#120—21no—
230C4×D4—171no—
231C4×D4C2×D1711yescharacteristic
232C4×D4C2×D1711yescharacteristic
233C2×C4⋊C4—171no—
234C2×C4⋊C4—171no—
235C2×C22⋊C4—21no—
236#106—171no—
237#106—171no—
238#106—171no—
239C4⋊C8—171no—
240#103—171no—
241#103—171no—
242#103—171no—
243#102—171no—
244#102—171no—
245#102—171no—
246#102—171no—
247#102—171no—
248#102—171no—
249#102—171no—
250C22⋊C8—171no—
251C22⋊C8—171no—
252#97—171no—
253C17#330611yescharacteristic, complemented, Sylow
254C24—21no—
255#41—22no—
256#41—41no—
257#41—684no—
258C2×Q8—341no—
259C2×Q8C22×D1711yescharacteristic
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—171no—
266C2×D4—341no—
267C2×D4—341no—
268C2×D4C22⋊D1711yescharacteristic
269C2×D4C22×D1711yescharacteristic
270C2×D4C22×D1711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—341no—
274C22×C4—341no—
275C22×C4—341no—
276C22×C4C22×D1711yescharacteristic
277C22×C4C22×D1711yescharacteristic
278SD16—681no—
279D8—342no—
280D8—681no—
281M16—681no—
282C2×C8—171no—
283C2×C8—171no—
284C2×C8—171no—
285C2×C8—171no—
286C2×C8—681no—
287C4⋊C4—21no—
288C4⋊C4—171no—
289C4⋊C4—171no—
290C4⋊C4—171no—
291C4⋊C4—341no—
292C4⋊C4—341no—
293C4⋊C4—341no—
294C4⋊C4C22⋊D1711yescharacteristic
295C4⋊C4C22×D1711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—341no—
302C22⋊C4—341no—
303C22⋊C4—341no—
304C22⋊C4C22⋊D1711yescharacteristic
305C22⋊C4C22⋊D1711yescharacteristic
306C42C22×D1711yescharacteristic
307C23—21no—
308C23—21no—
309C23—42no—
310C23—42no—
311C23—81no—
312C23—341no—
313C23C2×C22⋊D1711yescharacteristic
314C23C2×C22⋊D1711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—682no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—42no—
324D4—81no—
325D4—341no—
326D4—341no—
327D4—682no—
328D4—682no—
329D4#6325412yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4—341no—
339C2×C4—341no—
340C2×C4—341no—
341C2×C4—682no—
342C2×C4—682no—
343C2×C4—682no—
344C2×C4C2×C22⋊D1711yescharacteristic
345C2×C4C2×C22⋊D1711yescharacteristic
346C2×C4C2×C22⋊D1711yescharacteristic
347C2×C4C2×C22⋊D1711yescharacteristic
348C2×C4C23×D1711yescharacteristic
349C8—341no—
350C8—341no—
351C8—341no—
352C8—341no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—681no—
366C22—681no—
367C22—681no—
368C22#1058763512yes—
369C22C22×C22⋊D1711yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—681no—
377C4—681no—
378C4—681no—
379C4C2×#6325411yescharacteristic
380C4#1058769111yescharacteristic
381C2—22no—
382C2—41no—
383C2—42no—
384C2—81no—
385C2—1361no—
386C2#1176345311yescharacteristic
387C2C2×#1058763511yescharacteristic
388C2#1176362011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#546C2
socleC22×C17C22×D4
hypercenter#546D17
hypocenterC17#3306
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #11763463C2semidirect
2. #11763168C2semidirect
3. #11763137C2semidirect
4. #11763125C2semidirect
5. #11763099C2semidirect
6. #11763060C2semidirect
7. #11763040C2semidirect
8. #11763021C2semidirect
9. #11762954C2semidirect
10. #11762943C2semidirect
11. #11762919C2semidirect
12. #11762881C2semidirect
13. #11762810C2semidirect
14. C2×#10587503C2semidirect
15. C17×#546C2semidirect
28. C2×#63230C22semidirect
32. #10587616C22semidirect
34. C2×C4⋊Dic17C22semidirect
35. C2×#63223C22semidirect
40. C4×D68C22semidirect
41. #10587555C22semidirect
42. #10587506C22semidirect
43. #10587506C22semidirect
44. #10587506C22semidirect
52. C22⋊#3622C22semidirect
53. #10587480C22semidirect
54. #10587479C22semidirect
55. #10587479C22semidirect
56. #10587479C22semidirect
57. C4⋊#3622C22semidirect
58. #10587474C22semidirect
129. D4×C17#41semidirect
188. #546D17semidirect
253. C17#3306semidirect
329. D4#63254semidirect

Maximal subgroups

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

subgroupindex
#117634632
#117631682
#117631372
#117631252
#117630992
#117630602
#117630402
#117630212
#117629542
#117629432
#117629192
#117628812
#117628102
C2×#105875032
C17×#5462
#330617