Finite Group Databasebeta

← #14540805

Subgroups — #14540805

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#11763499C211yesmaximal, characteristic, complemented
2#11763173C211yesmaximal, characteristic, complemented
3#11763130C211yesmaximal, characteristic, complemented
4#11763120C211yesmaximal, characteristic, complemented
5#11763109C211yesmaximal, characteristic, complemented
6#11763053C211yesmaximal, characteristic, complemented
7#11763036C211yesmaximal, characteristic, complemented
8#11763006C211yesmaximal, characteristic, complemented
9#11762960C211yesmaximal, characteristic, complemented
10#11762960C211yesmaximal, characteristic, complemented
11#11762909C211yesmaximal, characteristic, complemented
12#11762881C211yesmaximal, characteristic, complemented
13#11762814C211yesmaximal, characteristic, complemented
14C2×#10587503C211yesmaximal, characteristic, complemented
15D42×C17C211yesmaximal, characteristic
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#10587621—21no—
28C2×#63229C2211yescharacteristic
29C2×#63229C2211yescharacteristic, complemented
30C2×#63229C2211yescharacteristic, complemented
31#10587612C2211yescharacteristic, complemented
32C2×C4⋊Dic17C2211yescharacteristic, complemented
33#10587610C2211yescharacteristic, complemented
34#10587610C2211yescharacteristic, complemented
35C2×#63223C2211yescharacteristic, complemented
36C22×#3622C2211yescharacteristic, complemented
37#10587575—21no—
38#10587565—21no—
39#10587563—21no—
40#10587556C2211yescharacteristic, complemented
41C4×Dic34C2211yescharacteristic, 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#10587481C2211yescharacteristic, complemented
54#10587479C2211yescharacteristic, complemented
55#10587478C2211yescharacteristic, 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
79C22×Dic17—21no—
80C2×Dic34C2311yescharacteristic
81C4⋊C4×C17D411yescharacteristic
82C4⋊C4×C17C2311yescharacteristic
83C22⋊C4×C17—21no—
84C22⋊C4×C17—21no—
85C22⋊C4×C17—21no—
86C22⋊C4×C17D411yescharacteristic
87C22⋊C4×C17D411yescharacteristic
88C42×C17C2311yescharacteristic
89C22⋊Dic17—21no—
90C22⋊Dic17—21no—
91C22⋊Dic17—21no—
92#63229—22no—
93#63229—41no—
94#63229—41no—
95C4⋊Dic17—21no—
96C4⋊Dic17C2311yescharacteristic
97C4⋊Dic17C2311yescharacteristic
98C4⋊Dic17C2311yescharacteristic
99#63225—21no—
100#63225—21no—
101#63225—21no—
102#63225—21no—
103#63225—21no—
104C4×Dic17—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
123D4×C17—22no—
124D4×C17—21no—
125D4×C17—21no—
126D4×C17—21no—
127D4×C17—21no—
128D4×C17—21no—
129D4×C17—42no—
130D4×C17—42no—
131D4×C17—42no—
132D4×C17—42no—
133D4×C17—42no—
134D4×C17—81no—
135D4×C17#4112yes—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—42no—
142C2×C4×C17C2×D411yescharacteristic
143C2×C4×C17C2×D411yescharacteristic
144C2×C4×C17C2×D411yescharacteristic
145C2×C4×C17C2×D411yescharacteristic
146C2×C4×C17C2411yescharacteristic
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—42no—
152Dic34—21no—
153Dic34—21no—
154#3622—21no—
155#3622—21no—
156#3622—21no—
157#3622—21no—
158#3310—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—41no—
171C22×C17—41no—
172C22×C17—41no—
173C22×C17—84no—
174C22×C17—84no—
175C22×C17AGL(1,ℤ/8)12yes—
176C22×C17C22×D411yescharacteristic
177C4×C17—21no—
178C4×C17—21no—
179C4×C17—21no—
180C4×C17—21no—
181C4×C17—41no—
182C4×C17C2×#4111yescharacteristic
183C4×C17H(5,𝔽2)11yescharacteristic
184Dic17—41no—
185Dic17—41no—
186Dic17—41no—
187Dic17—41no—
188#552—171no—
189D42D1711yescharacteristic
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#440—171no—
200#435—171no—
201#418—171no—
202C2×#102—171no—
203C2×C17—22no—
204C2×C17—41no—
205C2×C17—42no—
206C2×C17—81no—
207C2×C17—81no—
208C2×C17#54611yescharacteristic
209C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
210C2×C17#57711yescharacteristic
211C22×D4—21no—
212C22×D4C2×D1711yescharacteristic
213C22×D4C2×D1711yescharacteristic
214C2×SD16—171no—
215C2×SD16—171no—
216C2×SD16—171no—
217C2×M16—171no—
218C22×C8—171no—
219#127C2×D1711yescharacteristic
220#126—341no—
221#125—171no—
222#125—341no—
223#123—341no—
224C22⋊Q8—171no—
225C22⋊Q8—171no—
226#121—21no—
227#121C2×D1711yescharacteristic
228#121C2×D1711yescharacteristic
229#120—21no—
230#120—21no—
231C4×Q8—171no—
232C4×D4C2×D1711yescharacteristic
233C4×D4C2×D1711yescharacteristic
234#117—171no—
235#117—341no—
236C2×C4⋊C4—171no—
237#107—171no—
238#107—171no—
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#102—171no—
250#102—171no—
251C22⋊C8—171no—
252C22⋊C8—171no—
253#97—171no—
254C17#331011yescharacteristic, complemented, Sylow
255C24—21no—
256C24—21no—
257C2×Q8—171no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—44no—
269C2×D4C22⋊D1711yescharacteristic
270C2×D4C22×D1711yescharacteristic
271C2×D4C22×D1711yescharacteristic
272C2×D4C22×D1711yescharacteristic
273C22×C4—21no—
274C22×C4—341no—
275C22×C4C22×D1711yescharacteristic
276C22×C4C22×D1711yescharacteristic
277SD16—342no—
278SD16—681no—
279SD16—681no—
280M16—681no—
281C2×C8—171no—
282C2×C8—171no—
283C2×C8—171no—
284C2×C8—171no—
285C2×C8—681no—
286C4⋊C4—171no—
287C4⋊C4—171no—
288C4⋊C4—171no—
289C4⋊C4—341no—
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—21no—
299C22⋊C4—21no—
300C22⋊C4—341no—
301C22⋊C4—341no—
302C22⋊C4—341no—
303C22⋊C4C22⋊D1711yescharacteristic
304C22⋊C4C22⋊D1711yescharacteristic
305C42—341no—
306C42—341no—
307C42C22×D1711yescharacteristic
308C23—21no—
309C23—21no—
310C23—21no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23C2×C22⋊D1711yescharacteristic
318C23C2×C22⋊D1711yescharacteristic
319Q8—341no—
320Q8—341no—
321D4—22no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—42no—
329D4—42no—
330D4—42no—
331D4—42no—
332D4—81no—
333D4#6325412yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—341no—
341C2×C4—341no—
342C2×C4—341no—
343C2×C4—341no—
344C2×C4—682no—
345C2×C4C2×C22⋊D1711yescharacteristic
346C2×C4C2×C22⋊D1711yescharacteristic
347C2×C4C2×C22⋊D1711yescharacteristic
348C2×C4C2×C22⋊D1711yescharacteristic
349C2×C4C23×D1711yescharacteristic
350C8—341no—
351C8—341no—
352C8—341no—
353C8—341no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—84no—
369C22—84no—
370C22#1058761812yes—
371C22C22×C22⋊D1711yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—681no—
378C4—681no—
379C4—681no—
380C4—681no—
381C4C2×#6325411yescharacteristic
382C4#1058769111yescharacteristic
383C2—22no—
384C2—41no—
385C2—42no—
386C2—81no—
387C2—81no—
388C2#1176345311yescharacteristic
389C2C2×#1058761811yescharacteristic
390C2#1176362011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
Fitting, prime radicalD42×C17C2
socleC22×C17C22×D4
hypercenterD42D17
hypocenterC17#3310
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. #11763499C2semidirect
2. #11763173C2semidirect
3. #11763130C2semidirect
4. #11763120C2semidirect
5. #11763109C2semidirect
6. #11763053C2semidirect
7. #11763036C2semidirect
8. #11763006C2semidirect
9. #11762960C2semidirect
10. #11762960C2semidirect
11. #11762909C2semidirect
12. #11762881C2semidirect
13. #11762814C2semidirect
14. C2×#10587503C2semidirect
29. C2×#63229C22semidirect
30. C2×#63229C22semidirect
31. #10587612C22semidirect
32. C2×C4⋊Dic17C22semidirect
33. #10587610C22semidirect
34. #10587610C22semidirect
35. C2×#63223C22semidirect
36. C22×#3622C22semidirect
40. #10587556C22semidirect
41. C4×Dic34C22semidirect
45. #10587503C22semidirect
46. #10587503C22semidirect
47. #10587503C22semidirect
48. #10587503C22semidirect
49. #10587503C22semidirect
50. #10587503C22semidirect
51. C22⋊#3622C22semidirect
52. C22⋊#3622C22semidirect
53. #10587481C22semidirect
54. #10587479C22semidirect
55. #10587478C22semidirect
56. #10587478C22semidirect
57. C4⋊#3622C22semidirect
58. #10587474C22semidirect
254. C17#3310semidirect

Maximal subgroups

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

subgroupindex
#117634992
#117631732
#117631302
#117631202
#117631092
#117630532
#117630362
#117630062
#117629602
#117629092
#117628812
#117628142
C2×#105875032
D42×C172
#331017