Finite Group Databasebeta

← #14540925

Subgroups — #14540925

Order 2176.

462 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763505C211yesmaximal, characteristic, complemented
2#11763173C211yesmaximal, characteristic, complemented
3#11763130C211yesmaximal, characteristic, complemented
4#11763125C211yesmaximal, characteristic, complemented
5#11763109C211yesmaximal, characteristic, complemented
6#11763060C211yesmaximal, characteristic, complemented
7#11763043C211yesmaximal, characteristic, complemented
8#11763001C211yesmaximal, characteristic, complemented
9#11762943C211yesmaximal, characteristic, complemented
10#11762943C211yesmaximal, characteristic, complemented
11#11762919C211yesmaximal, characteristic, complemented
12#11762881C211yesmaximal, characteristic, complemented
13#11762814C211yesmaximal, characteristic, complemented
14C2×#10587503C211yesmaximal, characteristic, complemented
15D42×C17C211yesmaximal, characteristic, complemented
16C22×D4×C17—21no—
17C22×D4×C17C2211yescharacteristic
18C22×D4×C17C2211yescharacteristic
19C17×#127C2211yescharacteristic
20C17×#121—21no—
21C17×#121C2211yescharacteristic
22C17×#121C2211yescharacteristic
23C17×#120—21no—
24C17×#120—21no—
25C4×D4×C17C2211yescharacteristic
26C4×D4×C17C2211yescharacteristic
27#10587625—21no—
28C2×#63228C2211yescharacteristic
29C2×#63228C2211yescharacteristic, complemented
30C2×#63228C2211yescharacteristic, complemented
31#10587616C2211yescharacteristic, complemented
32#10587616C2211yescharacteristic, complemented
33#10587612C2211yescharacteristic, complemented
34C2×C4⋊Dic17C2211yescharacteristic, complemented
35C2×#63223C2211yescharacteristic, complemented
36C22×#3622C2211yescharacteristic, complemented
37#10587584—21no—
38#10587569—21no—
39#10587567—21no—
40C4×D68C2211yescharacteristic, complemented
41#10587556C2211yescharacteristic, complemented
42#10587503C2211yescharacteristic
43#10587503C2211yescharacteristic
44#10587503C2211yescharacteristic
45#10587503C2211yescharacteristic, complemented
46#10587503C2211yescharacteristic, complemented
47#10587503C2211yescharacteristic, complemented
48#10587503C2211yescharacteristic, complemented
49#10587503C2211yescharacteristic, complemented
50#10587503C2211yescharacteristic, complemented
51C22⋊#3622C2211yescharacteristic, complemented
52C22⋊#3622C2211yescharacteristic, complemented
53#10587480C2211yescharacteristic, complemented
54#10587479C2211yescharacteristic, complemented
55#10587479C2211yescharacteristic, complemented
56#10587478C2211yescharacteristic, complemented
57C4⋊#3622C2211yescharacteristic, complemented
58#10587474C2211yescharacteristic, complemented
59C24×C17—21no—
60C24×C17—21no—
61C2×D4×C17—22no—
62C2×D4×C17—22no—
63C2×D4×C17—21no—
64C2×D4×C17—21no—
65C2×D4×C17—21no—
66C2×D4×C17—21no—
67C2×D4×C17—21no—
68C2×D4×C17—21no—
69C2×D4×C17—41no—
70C2×D4×C17—41no—
71C2×D4×C17—44no—
72C2×D4×C17D411yescharacteristic
73C2×D4×C17C2311yescharacteristic
74C2×D4×C17C2311yescharacteristic
75C2×D4×C17C2311yescharacteristic
76C22×C4×C17—21no—
77C22×C4×C17C2311yescharacteristic
78C22×C4×C17C2311yescharacteristic
79C2×C22⋊D17—21no—
80C2×C22⋊D17—21no—
81C22×Dic17—21no—
82C2×D68C2311yescharacteristic
83C2×C4×D17—21no—
84C4⋊C4×C17D411yescharacteristic
85C4⋊C4×C17C2311yescharacteristic
86C22⋊C4×C17—21no—
87C22⋊C4×C17—21no—
88C22⋊C4×C17—21no—
89C22⋊C4×C17D411yescharacteristic
90C22⋊C4×C17D411yescharacteristic
91C42×C17C2311yescharacteristic
92C22⋊Dic17—21no—
93#63228—22no—
94#63228—41no—
95#63228—41no—
96#63227—21no—
97#63227—21no—
98#63227—21no—
99C4⋊Dic17—21no—
100C4⋊Dic17C2311yescharacteristic
101C4⋊Dic17C2311yescharacteristic
102C4⋊Dic17C2311yescharacteristic
103#63225—21no—
104#63225—21no—
105C4×Dic17—21no—
106#63223—41no—
107C2×#3622—41no—
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—42no—
116C23×C17—42no—
117C23×C17—42no—
118C23×C17—42no—
119C23×C17—81no—
120C23×C17—81no—
121C23×C17C2×D411yescharacteristic
122C23×C17C2×D411yescharacteristic
123C22×D17—21no—
124D4×C17—22no—
125D4×C17—21no—
126D4×C17—21no—
127D4×C17—21no—
128D4×C17—21no—
129D4×C17—21no—
130D4×C17—42no—
131D4×C17—42no—
132D4×C17—42no—
133D4×C17—42no—
134D4×C17—42no—
135D4×C17—81no—
136D4×C17#4112yes—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—42no—
143C2×C4×C17C2×D411yescharacteristic
144C2×C4×C17C2×D411yescharacteristic
145C2×C4×C17C2×D411yescharacteristic
146C2×C4×C17C2×D411yescharacteristic
147C2×C4×C17C2411yescharacteristic
148C22⋊D17—42no—
149C22⋊D17—81no—
150C2×Dic17—21no—
151C2×Dic17—21no—
152C2×Dic17—21no—
153C2×Dic17—81no—
154D68—21no—
155D68—21no—
156C4×D17—81no—
157#3622—21no—
158#3622—21no—
159#3622—21no—
160#3622—21no—
161#3318—171nomaximal, Sylow
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—22no—
166C22×C17—21no—
167C22×C17—21no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17—41no—
172C22×C17—41no—
173C22×C17—41no—
174C22×C17—41no—
175C22×C17—41no—
176C22×C17—84no—
177C22×C17—84no—
178C22×C17AGL(1,ℤ/8)12yes—
179C22×C17C22×D411yescharacteristic
180C2×D17—41no—
181C2×D17—41no—
182C2×D17—41no—
183C4×C17—21no—
184C4×C17—21no—
185C4×C17—21no—
186C4×C17—21no—
187C4×C17—41no—
188C4×C17C2×#4111yescharacteristic
189C4×C17H(5,𝔽2)11yescharacteristic
190Dic17—41no—
191Dic17—41no—
192Dic17—41no—
193#553—171no—
194D42D1711yescharacteristic, complemented
195#490—171no—
196#482—171no—
197#481—171no—
198#478—171no—
199#469—171no—
200#466—171no—
201#459—171no—
202#447—171no—
203#447—171no—
204#442—171no—
205#435—171no—
206#418—171no—
207C2×#102—171no—
208C2×C17—22no—
209C2×C17—41no—
210C2×C17—42no—
211C2×C17—81no—
212C2×C17—81no—
213C2×C17#54611yescharacteristic
214C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C17#57611yescharacteristic
216D17—81no—
217C22×D4—21no—
218C22×D4C2×D1711yescharacteristic
219C22×D4C2×D1711yescharacteristic
220C2×D8—171no—
221C2×D8—171no—
222C2×D8—171no—
223C2×M16—171no—
224C22×C8—171no—
225#127C2×D1711yescharacteristic
226#126—341no—
227#125—171no—
228#123—341no—
229#121—21no—
230#121—171no—
231#121—171no—
232#121—341no—
233#121C2×D1711yescharacteristic
234#121C2×D1711yescharacteristic
235#120—21no—
236#120—21no—
237C4×D4—171no—
238C4×D4—341no—
239C4×D4C2×D1711yescharacteristic
240C4×D4C2×D1711yescharacteristic
241#117—171no—
242C2×C4⋊C4—171no—
243#107—171no—
244#106—171no—
245#106—171no—
246C4⋊C8—171no—
247#102—171no—
248#102—171no—
249#102—171no—
250#102—171no—
251#102—171no—
252#102—171no—
253#102—171no—
254#102—171no—
255#102—171no—
256#102—171no—
257C22⋊C8—171no—
258C22⋊C8—171no—
259#97—171no—
260C17#331811yescharacteristic, complemented, Sylow
261C24—21no—
262C24—21no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—44no—
274C2×D4—171no—
275C2×D4—341no—
276C2×D4—341no—
277C2×D4C22⋊D1711yescharacteristic
278C2×D4C22×D1711yescharacteristic
279C2×D4C22×D1711yescharacteristic
280C2×D4C22×D1711yescharacteristic
281C22×C4—21no—
282C22×C4—341no—
283C22×C4—341no—
284C22×C4C22×D1711yescharacteristic
285C22×C4C22×D1711yescharacteristic
286D8—342no—
287D8—681no—
288D8—681no—
289M16—681no—
290C2×C8—171no—
291C2×C8—171no—
292C2×C8—171no—
293C2×C8—171no—
294C2×C8—681no—
295C4⋊C4—171no—
296C4⋊C4—171no—
297C4⋊C4—171no—
298C4⋊C4—341no—
299C4⋊C4—341no—
300C4⋊C4—341no—
301C4⋊C4C22⋊D1711yescharacteristic
302C4⋊C4C22×D1711yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—341no—
307C22⋊C4—341no—
308C22⋊C4—341no—
309C22⋊C4—341no—
310C22⋊C4C22⋊D1711yescharacteristic
311C22⋊C4C22⋊D1711yescharacteristic
312C42—341no—
313C42C22×D1711yescharacteristic
314C23—21no—
315C23—21no—
316C23—21no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—81no—
322C23—81no—
323C23—341no—
324C23C2×C22⋊D1711yescharacteristic
325C23C2×C22⋊D1711yescharacteristic
326D4—22no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—42no—
333D4—42no—
334D4—42no—
335D4—42no—
336D4—42no—
337D4—81no—
338D4—341no—
339D4—341no—
340D4—682no—
341D4—1361no—
342D4#6325412yes—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—341no—
350C2×C4—341no—
351C2×C4—341no—
352C2×C4—1361no—
353C2×C4—1361no—
354C2×C4C2×C22⋊D1711yescharacteristic
355C2×C4C2×C22⋊D1711yescharacteristic
356C2×C4C2×C22⋊D1711yescharacteristic
357C2×C4C2×C22⋊D1711yescharacteristic
358C2×C4C23×D1711yescharacteristic
359C8—341no—
360C8—341no—
361C8—341no—
362C8—341no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—84no—
378C22—84no—
379C22—681no—
380C22—681no—
381C22—681no—
382C22#1058761812yes—
383C22C22×C22⋊D1711yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—41no—
389C4—681no—
390C4—681no—
391C4—681no—
392C4C2×#6325411yescharacteristic
393C4#1058769111yescharacteristic
394C2—22no—
395C2—41no—
396C2—42no—
397C2—81no—
398C2—81no—
399C2—1361no—
400C2#1176345311yescharacteristic
401C2C2×#1058761811yescharacteristic
402C2#1176361211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingD42×C17C2
socleC22×C17C22×D4
hypercenterD42D17
hypocenterC17#3318
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #11763505C2semidirect
2. #11763173C2semidirect
3. #11763130C2semidirect
4. #11763125C2semidirect
5. #11763109C2semidirect
6. #11763060C2semidirect
7. #11763043C2semidirect
8. #11763001C2semidirect
9. #11762943C2semidirect
10. #11762943C2semidirect
11. #11762919C2semidirect
12. #11762881C2semidirect
13. #11762814C2semidirect
14. C2×#10587503C2semidirect
15. D42×C17C2semidirect
29. C2×#63228C22semidirect
30. C2×#63228C22semidirect
31. #10587616C22semidirect
32. #10587616C22semidirect
33. #10587612C22semidirect
34. C2×C4⋊Dic17C22semidirect
35. C2×#63223C22semidirect
36. C22×#3622C22semidirect
40. C4×D68C22semidirect
41. #10587556C22semidirect
45. #10587503C22semidirect
46. #10587503C22semidirect
47. #10587503C22semidirect
48. #10587503C22semidirect
49. #10587503C22semidirect
50. #10587503C22semidirect
51. C22⋊#3622C22semidirect
52. C22⋊#3622C22semidirect
53. #10587480C22semidirect
54. #10587479C22semidirect
55. #10587479C22semidirect
56. #10587478C22semidirect
57. C4⋊#3622C22semidirect
58. #10587474C22semidirect
194. D42D17semidirect
260. C17#3318semidirect

Maximal subgroups

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

subgroupindex
#117635052
#117631732
#117631302
#117631252
#117631092
#117630602
#117630432
#117630012
#117629432
#117629192
#117628812
#117628142
C2×#105875032
D42×C172
#331817