Finite Group Databasebeta

← #14540810

Subgroups — #14540810

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#11763500C211yesmaximal, characteristic, complemented
2D4×D68C211yesmaximal, characteristic
3#11763174C211yesmaximal, characteristic, complemented
4#11763136C211yesmaximal, characteristic, complemented
5#11763124C211yesmaximal, characteristic, complemented
6#11763109C211yesmaximal, characteristic, complemented
7#11763058C211yesmaximal, characteristic, complemented
8#11763041C211yesmaximal, characteristic, complemented
9#11763009C211yesmaximal, characteristic, complemented
10#11762958C211yesmaximal, characteristic, complemented
11#11762958C211yesmaximal, characteristic, complemented
12#11762882C211yesmaximal, characteristic, complemented
13#11762812C211yesmaximal, characteristic, complemented
14#102×D17C211yesmaximal, characteristic, complemented
15C17×#440C211yesmaximal, characteristic, complemented
16C2×D4×D17C2211yescharacteristic
17C2×D4×D17C2211yescharacteristic
18C22×D68—21no—
19C2×SD16×C17C2211yescharacteristic
20C4×Q8×C17C2211yescharacteristic, complemented
21C4×D4×C17C2211yescharacteristic
22#10587632C2211yescharacteristic, complemented
23#10587632C2211yescharacteristic, complemented
24C2×#63230C2211yescharacteristic, complemented
25C2×#63230C2211yescharacteristic, complemented
26#10587616—21no—
27C2×#63218C2211yescharacteristic, complemented
28C2×C8×D17C2211yescharacteristic, complemented
29#10587581C2211yescharacteristic
30#10587581C2211yescharacteristic
31#10587580—21no—
32#10587578C2211yescharacteristic, complemented
33C4⋊C4×D17C2211yescharacteristic, complemented
34#10587575—21no—
35#10587568—21no—
36#10587568—21no—
37#10587562—21no—
38#10587560C2211yescharacteristic
39C4×D68C2211yescharacteristic
40#10587558—21no—
41#10587556C2211yescharacteristic, complemented
42C17×#107C2211yescharacteristic, complemented
43C17×#103C2211yescharacteristic, complemented
44C17×#102C2211yescharacteristic
45C17×#97C2211yescharacteristic, complemented
46#10587503C2211yescharacteristic
47#10587503C2211yescharacteristic
48#10587492C2211yescharacteristic, complemented
49#10587492C2211yescharacteristic, complemented
50#10587492C2211yescharacteristic, complemented
51#10587492C2211yescharacteristic, complemented
52#10587491C2211yescharacteristic, complemented
53#10587491C2211yescharacteristic, complemented
54#10587480C2211yescharacteristic, complemented
55#10587480C2211yescharacteristic, complemented
56#10587479C2211yescharacteristic, complemented
57#10587478C2211yescharacteristic, complemented
58C4⋊#3622C2211yescharacteristic, complemented
59C23×D17—21no—
60C23×D17—21no—
61C2×Q8×C17C2311yescharacteristic
62C2×D4×C17C2311yescharacteristic
63C22×C4×C17—21no—
64C2×C22⋊D17—21no—
65C2×C22⋊D17—21no—
66D4×D17—22no—
67D4×D17—22no—
68D4×D17—41no—
69D4×D17—41no—
70C2×D68—21no—
71C2×D68—21no—
72C2×D68—21no—
73C2×D68—21no—
74C2×D68—44no—
75C2×D68D411yescharacteristic
76C2×D68C2311yescharacteristic
77C2×D68C2311yescharacteristic
78C2×C4×D17—21no—
79C2×C4×D17C2311yescharacteristic
80C2×C4×D17C2311yescharacteristic
81SD16×C17—22no—
82C2×C8×C17C2311yescharacteristic
83C2×C8×C17C2311yescharacteristic
84C4⋊C4×C17—21no—
85C4⋊C4×C17C2311yescharacteristic
86C4⋊C4×C17C2311yescharacteristic
87C22⋊C4×C17—21no—
88C42×C17—21no—
89C42×C17C2311yescharacteristic
90#63230—41no—
91#63230—41no—
92#63227—21no—
93#63227—21no—
94#63227—21no—
95#63227—21no—
96#63227—21no—
97#63227D411yescharacteristic
98#63227D411yescharacteristic
99C4⋊Dic17D411yescharacteristic
100C4⋊Dic17C2311yescharacteristic
101C4⋊Dic17C2311yescharacteristic
102#63225—21no—
103#63225—21no—
104#63225—21no—
105#63225—21no—
106#63225—21no—
107C4×Dic17—21no—
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110#63218—41no—
111C8×D17—41no—
112C23×C17—21no—
113C22×D17—21no—
114C22×D17—21no—
115C22×D17—42no—
116C22×D17—42no—
117C22×D17—42no—
118C22×D17—42no—
119C22×D17—81no—
120C22×D17—81no—
121C22×D17C2×D411yescharacteristic
122C22×D17C2×D411yescharacteristic
123Q8×C17—21no—
124Q8×C17—21no—
125D4×C17—21no—
126D4×C17#4112yes—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—42no—
131C2×C4×C17C2×D411yescharacteristic
132C2×C4×C17C2×D411yescharacteristic
133C2×C4×C17C2411yescharacteristic
134C22⋊D17—42no—
135C22⋊D17—42no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17C2×D411yescharacteristic
139C2×Dic17C2×D411yescharacteristic
140D68—22no—
141D68—21no—
142D68—21no—
143D68—21no—
144D68—21no—
145D68—42no—
146D68—42no—
147D68—42no—
148D68—81no—
149C4×D17—21no—
150C4×D17—21no—
151C4×D17—21no—
152C4×D17—21no—
153C4×D17—42no—
154C8×C17—21no—
155C8×C17—21no—
156#3622—21no—
157#3622—21no—
158#3310—171nomaximal, Sylow
159C22×C17—22no—
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—41no—
172C2×D17—41no—
173C2×D17—41no—
174C2×D17—84no—
175C2×D17—84no—
176C2×D17AGL(1,ℤ/8)12yes—
177C4×C17—21no—
178C4×C17—21no—
179C4×C17—41no—
180C4×C17—41no—
181C4×C17—41no—
182C4×C17C2×#4111yescharacteristic
183C4×C17H(5,𝔽2)11yescharacteristic
184Dic17—21no—
185Dic17—21no—
186Dic17—41no—
187Dic17—41no—
188#552—171no—
189D42—171no—
190#490—171no—
191#482—171no—
192#480—171no—
193#478—171no—
194#468—171no—
195#465—171no—
196#460—171no—
197#450—171no—
198#450—171no—
199#440D1711yescharacteristic, complemented
200#435—171no—
201#418—171no—
202C2×#102—171no—
203C2×C17—42no—
204C2×C17#54611yescharacteristic
205C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
206C2×C17#57711yescharacteristic
207D17—22no—
208D17—41no—
209D17—81no—
210D17—81no—
211C22×D4—171no—
212C22×D4—171no—
213C22×D4—341no—
214C2×SD16—171no—
215C2×SD16—171no—
216C2×SD16C2×D1711yescharacteristic
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—171no—
228#121—341no—
229#120—341no—
230#120—341no—
231C4×Q8C2×D1711yescharacteristic, complemented
232C4×D4—171no—
233C4×D4C2×D1711yescharacteristic
234#117—171no—
235#117—341no—
236C2×C4⋊C4—171no—
237#107—171no—
238#107C2×D1711yescharacteristic, complemented
239#106—171no—
240C4⋊C8—171no—
241#103C2×D1711yescharacteristic, complemented
242#102—171no—
243#102—171no—
244#102—171no—
245#102—171no—
246#102—171no—
247#102—171no—
248#102—171no—
249#102—171no—
250#102C2×D1711yescharacteristic
251C22⋊C8—171no—
252C22⋊C8—171no—
253#97C2×D1711yescharacteristic, complemented
254C17#331011yescharacteristic, complemented, Sylow
255C24—341no—
256C24—341no—
257C2×Q8C22×D1711yescharacteristic
258C2×D4—171no—
259C2×D4—171no—
260C2×D4—171no—
261C2×D4—341no—
262C2×D4—341no—
263C2×D4—341no—
264C2×D4—341no—
265C2×D4—342no—
266C2×D4—342no—
267C2×D4—341no—
268C2×D4—341no—
269C2×D4—684no—
270C2×D4—681no—
271C2×D4—681no—
272C2×D4C22×D1711yescharacteristic
273C22×C4—21no—
274C22×C4—171no—
275C22×C4—171no—
276C22×C4—341no—
277SD16—22no—
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—171no—
290C4⋊C4—341no—
291C4⋊C4—341no—
292C4⋊C4—341no—
293C4⋊C4—341no—
294C4⋊C4—341no—
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—21no—
306C42—341no—
307C42C22×D1711yescharacteristic
308C23—21no—
309C23—171no—
310C23—171no—
311C23—341no—
312C23—341no—
313C23—682no—
314C23—682no—
315C23—682no—
316C23—682no—
317C23—1361no—
318C23—1361no—
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—342no—
323D4—341no—
324D4—341no—
325D4—341no—
326D4—341no—
327D4—682no—
328D4—682no—
329D4—682no—
330D4—682no—
331D4—682no—
332D4—1361no—
333D4#6325212yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—171no—
339C2×C4—171no—
340C2×C4—341no—
341C2×C4—341no—
342C2×C4—341no—
343C2×C4—341no—
344C2×C4—341no—
345C2×C4—341no—
346C2×C4—682no—
347C2×C4D4×D1711yescharacteristic
348C2×C4D4×D1711yescharacteristic
349C2×C4C23×D1711yescharacteristic
350C8—21no—
351C8—21no—
352C8—341no—
353C8—341no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—172no—
358C22—341no—
359C22—341no—
360C22—341no—
361C22—341no—
362C22—341no—
363C22—681no—
364C22—681no—
365C22—681no—
366C22—681no—
367C22—681no—
368C22—681no—
369C22—1364no—
370C22—1364no—
371C22C2×D4×D1711yescharacteristic
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—341no—
378C4—341no—
379C4—681no—
380C4—681no—
381C4C2×#6325211yescharacteristic
382C4#1058769611yescharacteristic
383C2—42no—
384C2—342no—
385C2—681no—
386C2—1361no—
387C2—1361no—
388C2#1176345511yescharacteristic
389C2AGL(1,ℤ/8)×D1711yescharacteristic
390C2#1176361811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#440C2
socleC22×C17C22×D4
prime radicalD4×D68C2
hypercenter#440D17
hypocenterC17#3310
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #11763500C2semidirect
3. #11763174C2semidirect
4. #11763136C2semidirect
5. #11763124C2semidirect
6. #11763109C2semidirect
7. #11763058C2semidirect
8. #11763041C2semidirect
9. #11763009C2semidirect
10. #11762958C2semidirect
11. #11762958C2semidirect
12. #11762882C2semidirect
13. #11762812C2semidirect
14. #102×D17C2semidirect
15. C17×#440C2semidirect
20. C4×Q8×C17C22semidirect
22. #10587632C22semidirect
23. #10587632C22semidirect
24. C2×#63230C22semidirect
25. C2×#63230C22semidirect
27. C2×#63218C22semidirect
28. C2×C8×D17C22semidirect
32. #10587578C22semidirect
33. C4⋊C4×D17C22semidirect
41. #10587556C22semidirect
42. C17×#107C22semidirect
43. C17×#103C22semidirect
45. C17×#97C22semidirect
48. #10587492C22semidirect
49. #10587492C22semidirect
50. #10587492C22semidirect
51. #10587492C22semidirect
52. #10587491C22semidirect
53. #10587491C22semidirect
54. #10587480C22semidirect
55. #10587480C22semidirect
56. #10587479C22semidirect
57. #10587478C22semidirect
58. C4⋊#3622C22semidirect
199. #440D17semidirect
231. C4×Q8C2×D17semidirect
238. #107C2×D17semidirect
241. #103C2×D17semidirect
253. #97C2×D17semidirect
254. C17#3310semidirect

Maximal subgroups

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

subgroupindex
#117635002
D4×D682
#117631742
#117631362
#117631242
#117631092
#117630582
#117630412
#117630092
#117629582
#117628822
#117628122
#102×D172
C17×#4402
#331017