Finite Group Databasebeta

← #14540803

Subgroups — #14540803

Order 2176.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763494C211yesmaximal, characteristic, complemented
2#11763443C211yesmaximal, characteristic
3#11763172C211yesmaximal, characteristic, complemented
4#11763136C211yesmaximal, characteristic, complemented
5#11763112C211yesmaximal, characteristic, complemented
6#11763056C211yesmaximal, characteristic, complemented
7#11763036C211yesmaximal, characteristic, complemented
8#11763008C211yesmaximal, characteristic, complemented
9#11762962C211yesmaximal, characteristic, complemented
10#11762961C211yesmaximal, characteristic, complemented
11#11762908C211yesmaximal, characteristic, complemented
12#11762877C211yesmaximal, characteristic, complemented
13#11762812C211yesmaximal, characteristic, complemented
14C2×#10587480C211yesmaximal, characteristic, complemented
15C17×#480C211yesmaximal, characteristic, complemented
16C2×D4×D17—21no—
17C2×D4×D17C2211yescharacteristic
18C22×D68C2211yescharacteristic
19C17×#121C2211yescharacteristic
20C2×C4⋊C4×C17C2211yescharacteristic, complemented
21#10587626C2211yescharacteristic
22#10587623—21no—
23C2×#63229C2211yescharacteristic, complemented
24#10587615—21no—
25C4×C22⋊D17C2211yescharacteristic
26#10587612—21no—
27#10587610C2211yescharacteristic, complemented
28C22×#3622C2211yescharacteristic, complemented
29C2×#63219C2211yescharacteristic
30C2×#63219C2211yescharacteristic, complemented
31C2×#63218C2211yescharacteristic, complemented
32#10587584—21no—
33#10587583C2211yescharacteristic, complemented
34#10587581C2211yescharacteristic
35#10587579C2211yescharacteristic
36#10587578C2211yescharacteristic, complemented
37#10587576C2211yescharacteristic, complemented
38#10587575—21no—
39#10587573C2211yescharacteristic, complemented
40#10587570—21no—
41#10587568—21no—
42C17×#107C2211yescharacteristic, complemented
43C17×#107C2211yescharacteristic, complemented
44C17×#102C2211yescharacteristic, complemented
45C17×#102C2211yescharacteristic, complemented
46C17×C22⋊C8C2211yescharacteristic, complemented
47#10587503C2211yescharacteristic, complemented
48#10587492C2211yescharacteristic
49#10587492C2211yescharacteristic, complemented
50#10587491C2211yescharacteristic, complemented
51#10587486C2211yescharacteristic, complemented
52#10587485C2211yescharacteristic, complemented
53#10587481C2211yescharacteristic, complemented
54#10587480C2211yescharacteristic
55#10587480C2211yescharacteristic
56#10587480C2211yescharacteristic, complemented
57#10587480C2211yescharacteristic, complemented
58#10587479C2211yescharacteristic, complemented
59C23×D17—21no—
60C23×D17—21no—
61C2×D4×C17—21no—
62C2×D4×C17C2311yescharacteristic
63C22×C4×C17—21no—
64C22×C4×C17C2311yescharacteristic
65C2×C22⋊D17—21no—
66C2×C22⋊D17—21no—
67C2×C22⋊D17—21no—
68C2×C22⋊D17D411yescharacteristic
69D4×D17—22no—
70D4×D17—41no—
71D4×D17—44no—
72C2×D68—22no—
73C2×D68—21no—
74C2×D68—21no—
75C2×D68—41no—
76C2×D68C2311yescharacteristic
77C2×D68C2311yescharacteristic
78C2×C4×D17—21no—
79C2×C4×D17C2311yescharacteristic
80C2×Dic34C2311yescharacteristic
81C2×C8×C17C2311yescharacteristic
82C2×C8×C17C2311yescharacteristic
83C4⋊C4×C17—21no—
84C4⋊C4×C17C2311yescharacteristic
85C4⋊C4×C17C2311yescharacteristic
86C4⋊C4×C17C2311yescharacteristic
87C22⋊C4×C17—21no—
88C22⋊Dic17—21no—
89C22⋊Dic17D411yescharacteristic
90#63229—41no—
91#63227—21no—
92#63227—21no—
93#63227—21no—
94#63227—21no—
95#63227D411yescharacteristic
96C4⋊Dic17—21no—
97C4⋊Dic17—21no—
98C4⋊Dic17C2311yescharacteristic
99#63225—21no—
100#63225—21no—
101#63225—21no—
102#63225D411yescharacteristic
103C4×Dic17—21no—
104C4×Dic17—21no—
105C4×Dic17C2311yescharacteristic
106C2×#3622—41no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109#63219—22no—
110#63219—41no—
111#63218—41no—
112C23×C17—21no—
113C23×C17C2×D411yescharacteristic
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—42no—
117C22×D17—42no—
118C22×D17—42no—
119C22×D17—42no—
120C22×D17—81no—
121C22×D17—81no—
122C22×D17C2×D411yescharacteristic
123D4×C17—21no—
124D4×C17—21no—
125D4×C17—42no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—42no—
132C2×C4×C17C2×D411yescharacteristic
133C2×C4×C17C2411yescharacteristic
134C22⋊D17—22no—
135C22⋊D17—42no—
136C22⋊D17—42no—
137C22⋊D17—81no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17C2×D411yescharacteristic
141C2×Dic17C2×D411yescharacteristic
142C2×Dic17C2×D411yescharacteristic
143D68—21no—
144D68—21no—
145D68—21no—
146D68—42no—
147D68—42no—
148D68#4112yes—
149C4×D17—21no—
150C4×D17—21no—
151C4×D17—42no—
152Dic34—21no—
153Dic34—21no—
154C8×C17—21no—
155C8×C17—21no—
156#3622—21no—
157#3622—21no—
158#3310—171nomaximal, Sylow
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17AGL(1,ℤ/8)12yes—
165C22×C17C22×D411yescharacteristic
166C2×D17—21no—
167C2×D17—22no—
168C2×D17—21no—
169C2×D17—21no—
170C2×D17—41no—
171C2×D17—41no—
172C2×D17—41no—
173C2×D17—41no—
174C2×D17—41no—
175C2×D17—84no—
176C2×D17—84no—
177C4×C17—21no—
178C4×C17—41no—
179C4×C17—41no—
180C4×C17—41no—
181C4×C17C2×#4111yescharacteristic
182C4×C17H(5,𝔽2)11yescharacteristic
183Dic17—21no—
184Dic17—21no—
185Dic17—21no—
186Dic17—41no—
187Dic17—41no—
188#552—171no—
189D42—171no—
190#490—171no—
191#482—171no—
192#480D1711yescharacteristic, complemented
193#478—171no—
194#468—171no—
195#465—171no—
196#460—171no—
197#450—171no—
198#450—171no—
199#440—171no—
200#435—171no—
201#418—171no—
202C2×#102—171no—
203C2×C17—22no—
204C2×C17—81no—
205C2×C17#54611yescharacteristic
206C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
207C2×C17#57711yescharacteristic
208D17—41no—
209D17—42no—
210D17—81no—
211C22×D4—171no—
212C22×D4—171no—
213C22×D4—341no—
214C2×SD16—171no—
215C2×SD16—171no—
216C2×SD16—171no—
217C2×M16—171no—
218C22×C8—171no—
219#127—171no—
220#126—341no—
221#125—171no—
222#125—341no—
223#123—341no—
224C22⋊Q8—171no—
225C22⋊Q8—171no—
226#121—171no—
227#121—341no—
228#121C2×D1711yescharacteristic
229#120—341no—
230#120—341no—
231C4×Q8—171no—
232C4×D4—171no—
233C4×D4—171no—
234#117—171no—
235#117—341no—
236C2×C4⋊C4C2×D1711yescharacteristic, complemented
237#107C2×D1711yescharacteristic, complemented
238#107C2×D1711yescharacteristic, complemented
239#106—171no—
240C4⋊C8—171no—
241#103—171no—
242#102—171no—
243#102—171no—
244#102—171no—
245#102—171no—
246#102—171no—
247#102—171no—
248#102—171no—
249#102C2×D1711yescharacteristic, complemented
250#102C2×D1711yescharacteristic, complemented
251C22⋊C8—171no—
252C22⋊C8C2×D1711yescharacteristic, complemented
253#97—171no—
254C17#331011yescharacteristic, complemented, Sylow
255C24—341no—
256C24—341no—
257C2×Q8—171no—
258C2×D4—21no—
259C2×D4—171no—
260C2×D4—171no—
261C2×D4—171no—
262C2×D4—342no—
263C2×D4—341no—
264C2×D4—341no—
265C2×D4—341no—
266C2×D4—341no—
267C2×D4—342no—
268C2×D4—341no—
269C2×D4—681no—
270C2×D4—681no—
271C2×D4—684no—
272C2×D4C22×D1711yescharacteristic
273C22×C4—21no—
274C22×C4—171no—
275C22×C4—341no—
276C22×C4C22×D1711yescharacteristic
277SD16—342no—
278SD16—681no—
279SD16—681no—
280M16—681no—
281C2×C8—171no—
282C2×C8—171no—
283C2×C8—681no—
284C2×C8C22×D1711yescharacteristic
285C2×C8C22×D1711yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—171no—
288C4⋊C4—171no—
289C4⋊C4—341no—
290C4⋊C4—341no—
291C4⋊C4—341no—
292C4⋊C4—341no—
293C4⋊C4—341no—
294C4⋊C4C22×D1711yescharacteristic
295C4⋊C4C22×D1711yescharacteristic
296C4⋊C4C22×D1711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—171no—
299C22⋊C4—171no—
300C22⋊C4—341no—
301C22⋊C4—341no—
302C22⋊C4—341no—
303C22⋊C4—341no—
304C22⋊C4—341no—
305C42—171no—
306C42—341no—
307C42—341no—
308C23—21no—
309C23—171no—
310C23—341no—
311C23—341no—
312C23—682no—
313C23—682no—
314C23—682no—
315C23—682no—
316C23—1361no—
317C23—1361no—
318C23D4×D1711yescharacteristic
319Q8—341no—
320Q8—341no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—172no—
325D4—342no—
326D4—341no—
327D4—341no—
328D4—341no—
329D4—682no—
330D4—682no—
331D4—682no—
332D4—682no—
333D4—1361no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—171no—
341C2×C4—171no—
342C2×C4—171no—
343C2×C4—341no—
344C2×C4—341no—
345C2×C4—341no—
346C2×C4—341no—
347C2×C4—682no—
348C2×C4D4×D1711yescharacteristic
349C2×C4C23×D1711yescharacteristic
350C8—21no—
351C8—21no—
352C8—341no—
353C8—341no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—341no—
360C22—341no—
361C22—341no—
362C22—342no—
363C22—681no—
364C22—681no—
365C22—681no—
366C22—681no—
367C22—681no—
368C22—1364no—
369C22—1364no—
370C22#1058759712yes—
371C22C2×D4×D1711yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—341no—
377C4—341no—
378C4—341no—
379C4—681no—
380C4—681no—
381C4#41×D1711yescharacteristic
382C4#1058769611yescharacteristic
383C2—22no—
384C2—81no—
385C2—681no—
386C2—682no—
387C2—1361no—
388C2#1176345111yescharacteristic
389C2C2×#1058759711yescharacteristic
390C2#1176361811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#480C2
socleC22×C17C22×D4
prime radical#11763443C2
hypercenter#480D17
hypocenterC17#3310
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #11763494C2semidirect
3. #11763172C2semidirect
4. #11763136C2semidirect
5. #11763112C2semidirect
6. #11763056C2semidirect
7. #11763036C2semidirect
8. #11763008C2semidirect
9. #11762962C2semidirect
10. #11762961C2semidirect
11. #11762908C2semidirect
12. #11762877C2semidirect
13. #11762812C2semidirect
14. C2×#10587480C2semidirect
15. C17×#480C2semidirect
20. C2×C4⋊C4×C17C22semidirect
23. C2×#63229C22semidirect
27. #10587610C22semidirect
28. C22×#3622C22semidirect
30. C2×#63219C22semidirect
31. C2×#63218C22semidirect
33. #10587583C22semidirect
36. #10587578C22semidirect
37. #10587576C22semidirect
39. #10587573C22semidirect
42. C17×#107C22semidirect
43. C17×#107C22semidirect
44. C17×#102C22semidirect
45. C17×#102C22semidirect
46. C17×C22⋊C8C22semidirect
47. #10587503C22semidirect
49. #10587492C22semidirect
50. #10587491C22semidirect
51. #10587486C22semidirect
52. #10587485C22semidirect
53. #10587481C22semidirect
56. #10587480C22semidirect
57. #10587480C22semidirect
58. #10587479C22semidirect
192. #480D17semidirect
236. C2×C4⋊C4C2×D17semidirect
237. #107C2×D17semidirect
238. #107C2×D17semidirect
249. #102C2×D17semidirect
250. #102C2×D17semidirect
252. C22⋊C8C2×D17semidirect
254. C17#3310semidirect

Maximal subgroups

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

subgroupindex
#117634942
#117634432
#117631722
#117631362
#117631122
#117630562
#117630362
#117630082
#117629622
#117629612
#117629082
#117628772
#117628122
C2×#105874802
C17×#4802
#331017