Finite Group Databasebeta

← #14539800

Subgroups — #14539800

Order 2176.

490 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587626C211yesmaximal, characteristic, complemented
2#11763340C211yesmaximal, characteristic, complemented
3#11763197C212yesmaximal, complemented
4#11763172C212yesmaximal, complemented
5#11763008C212yesmaximal, complemented
6#11763002C212yesmaximal, complemented
7#11762962C211yesmaximal, characteristic, complemented
8#11762960C211yesmaximal, characteristic, complemented
9#11762942C211yesmaximal, characteristic, complemented
10#11762863C211yesmaximal, characteristic, complemented
11C17×#447C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C22×C22⋊D17—21no—
14C22×C22⋊D17—21no—
15C22×D68C2211yescharacteristic
16C2×D8×C17C2212yescomplemented
17C17×#121C2211yescharacteristic
18#10587626—21no—
19#10587626—21no—
20#10587626C2212yes—
21#10587626C2212yes—
22#10587622C2211yescharacteristic, complemented
23#10587621—21no—
24D4×Dic17—21no—
25D4×Dic17C2211yescharacteristic, complemented
26C2×#63229C2212yescomplemented
27C2×#63228C2212yescomplemented
28#10587610C2211yescharacteristic
29C2×C4×Dic17C2211yescharacteristic
30C2×#63219C2212yescomplemented
31#10587576C2211yescharacteristic, complemented
32#10587573C2211yescharacteristic, complemented
33#10587565—21no—
34#10587563—21no—
35C17×#102C2212yescomplemented
36C17×C22⋊C8C2211yescharacteristic
37#10587503C2212yescomplemented
38C22⋊#3622C2211yescharacteristic
39#10587492C2212yescomplemented
40#10587486C2212yescomplemented
41#10587485C2212yescomplemented
42#10587480C2212yescomplemented
43C24×C17—21no—
44C23×D17—21no—
45C2×D4×C17—21no—
46C2×D4×C17—21no—
47C2×D4×C17—42no—
48C2×D4×C17C2311yescharacteristic
49C2×D4×C17C2312yes—
50C22×C4×C17C2311yescharacteristic
51C2×C22⋊D17—22no—
52C2×C22⋊D17—22no—
53C2×C22⋊D17—22no—
54C2×C22⋊D17—22no—
55C2×C22⋊D17—44no—
56C2×C22⋊D17—44no—
57C22×Dic17—21no—
58C22×Dic17D411yescharacteristic
59C22×Dic17D411yescharacteristic
60C2×D68—21no—
61C2×D68—42no—
62C2×D68C2312yes—
63C2×Dic34C2311yescharacteristic
64D8×C17—42no—
65C2×C8×C17C2312yes—
66C4⋊C4×C17C2311yescharacteristic
67C22⋊C4×C17—21no—
68C22⋊Dic17—21no—
69C22⋊Dic17—21no—
70C22⋊Dic17—21no—
71C22⋊Dic17—21no—
72#63229—42no—
73#63228—42no—
74C4⋊Dic17—21no—
75C4⋊Dic17C2311yescharacteristic
76#63225—21no—
77#63225—21no—
78#63225—21no—
79C4×Dic17—21no—
80C4×Dic17D411yescharacteristic
81C4×Dic17D411yescharacteristic
82C4×Dic17C2311yescharacteristic
83C4×Dic17C2311yescharacteristic
84C2×#3622C2312yes—
85#63219—42no—
86C23×C17—21no—
87C23×C17—22no—
88C23×C17—42no—
89C23×C17—42no—
90C23×C17—42no—
91C23×C17C2×D411yescharacteristic
92C22×D17—22no—
93C22×D17—42no—
94C22×D17—42no—
95C22×D17—42no—
96D4×C17—22no—
97D4×C17—22no—
98D4×C17—22no—
99D4×C17—42no—
100D4×C17—42no—
101C2×C4×C17—22no—
102C2×C4×C17—21no—
103C2×C4×C17C2×D411yescharacteristic
104C2×C4×C17C2411yescharacteristic
105C22⋊D17—44no—
106C22⋊D17—44no—
107C22⋊D17—44no—
108C22⋊D17—44no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—21no—
112C2×Dic17—21no—
113C2×Dic17—42no—
114C2×Dic17C2×D411yescharacteristic
115C2×Dic17C2×D411yescharacteristic
116C2×Dic17C2×D411yescharacteristic
117C2×Dic17C2×D411yescharacteristic
118D68—22no—
119D68—22no—
120D68—42no—
121Dic34—22no—
122C8×C17—22no—
123#3622—22no—
124#3217—171nomaximal, Sylow
125C22×C17—21no—
126C22×C17—21no—
127C22×C17—42no—
128C22×C17—41no—
129C22×C17—41no—
130C22×C17—42no—
131C22×C17—42no—
132C22×C17—41no—
133C22×C17—84no—
134C22×C17AGL(1,ℤ/8)12yes—
135C22×C17C22×D411yescharacteristic
136C2×D17—42no—
137C2×D17—42no—
138C2×D17—42no—
139C2×D17—84no—
140C4×C17—21no—
141C4×C17—41no—
142C4×C17H(5,𝔽2)12yes—
143Dic17—21no—
144Dic17—21no—
145Dic17—21no—
146Dic17—41no—
147Dic17—41no—
148Dic17AGL(1,ℤ/8)12yes—
149C2×#127—171no—
150#529—171no—
151#496—172no—
152#490—172no—
153#460—172no—
154#459—172no—
155#450—171no—
156#450—171no—
157#447—171no—
158#447D1711yescharacteristic, complemented
159#432—171no—
160C2×C17—22no—
161C2×C17—81no—
162C2×C17—82no—
163C2×C17#53511yescharacteristic
164C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
165C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
166D17—82no—
167C22×D4—171no—
168C22×D4—341no—
169C22×D4—341no—
170C22×D4C2×D1711yescharacteristic
171C2×SD16—172no—
172C2×SD16—172no—
173C2×D8—172no—
174C2×D8C2×D1712yescomplemented
175#127—172no—
176#127—172no—
177#127—341no—
178#127—341no—
179#126—341no—
180#125—171no—
181#124—171no—
182#123—341no—
183C22⋊Q8—171no—
184#121C2×D1711yescharacteristic
185C4×Q8—171no—
186C4×D4—171no—
187C4×D4—341no—
188#117—341no—
189C2×C42—171no—
190C4⋊C8—172no—
191#102—172no—
192#102—172no—
193#102—172no—
194#102C2×D1712yescomplemented
195C22⋊C8—171no—
196C22⋊C8C2×D1711yescharacteristic
197#97—172no—
198C17#321711yescharacteristic, complemented, Sylow
199C24—21no—
200C24—341no—
201C2×Q8—171no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—42no—
205C2×D4—172no—
206C2×D4—342no—
207C2×D4—342no—
208C2×D4—342no—
209C2×D4—342no—
210C2×D4—341no—
211C2×D4—682no—
212C2×D4—684no—
213C2×D4—684no—
214C2×D4C22×D1712yes—
215C2×D4C22×D1711yescharacteristic
216C22×C4—171no—
217C22×C4—171no—
218C22×C4—341no—
219C22×C4C22×D1711yescharacteristic
220SD16—682no—
221SD16—682no—
222D8—42no—
223D8—682no—
224C2×C8—172no—
225C2×C8C22×D1712yes—
226C4⋊C4—171no—
227C4⋊C4—341no—
228C4⋊C4—341no—
229C4⋊C4—341no—
230C4⋊C4—341no—
231C4⋊C4C22×D1711yescharacteristic
232C22⋊C4—21no—
233C22⋊C4—341no—
234C22⋊C4—341no—
235C22⋊C4—341no—
236C22⋊C4—341no—
237C42—171no—
238C42—171no—
239C42—171no—
240C42—171no—
241C42—341no—
242C23—21no—
243C23—22no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—342no—
248C23—682no—
249C23—682no—
250C23—682no—
251C23D4×D1711yescharacteristic
252Q8—342no—
253D4—22no—
254D4—22no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—342no—
259D4—342no—
260D4—682no—
261D4—684no—
262D4—684no—
263D4—684no—
264D4—684no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—171no—
268C2×C4—171no—
269C2×C4—171no—
270C2×C4—171no—
271C2×C4—342no—
272C2×C4—342no—
273C2×C4—341no—
274C2×C4—341no—
275C2×C4—682no—
276C2×C4D4×D1711yescharacteristic
277C2×C4C23×D1711yescharacteristic
278C8—22no—
279C8—342no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—42no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—84no—
289C22—682no—
290C22—682no—
291C22—682no—
292C22—1364no—
293C22#1058759712yes—
294C22C2×D4×D1711yescharacteristic
295C4—21no—
296C4—41no—
297C4—172no—
298C4—341no—
299C4—341no—
300C4—341no—
301C4—681no—
302C4—681no—
303C4#1058769112yes—
304C2—22no—
305C2—82no—
306C2—81no—
307C2—1362no—
308C2#1176337411yescharacteristic
309C2AGL(1,ℤ/8)×D1711yescharacteristic
310C2C2×#1058759711yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#447C2
socleC22×C17C22×D4
hypercenter#447D17
hypocenterC17#3217
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#10587626C2semidirect
2. #11763340C2semidirect
3. #11763197C2semidirect
4. #11763172C2semidirect
5. #11763008C2semidirect
6. #11763002C2semidirect
7. #11762962C2semidirect
8. #11762960C2semidirect
9. #11762942C2semidirect
10. #11762863C2semidirect
11. C17×#447C2semidirect
16. C2×D8×C17C22semidirect
22. #10587622C22semidirect
25. D4×Dic17C22semidirect
26. C2×#63229C22semidirect
27. C2×#63228C22semidirect
30. C2×#63219C22semidirect
31. #10587576C22semidirect
32. #10587573C22semidirect
35. C17×#102C22semidirect
37. #10587503C22semidirect
39. #10587492C22semidirect
40. #10587486C22semidirect
41. #10587485C22semidirect
42. #10587480C22semidirect
158. #447D17semidirect
174. C2×D8C2×D17semidirect
194. #102C2×D17semidirect
198. C17#3217semidirect

Maximal subgroups

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

subgroupindex
C2×#105876262
#117633402
#117631972
#117631722
#117630082
#117630022
#117629622
#117629602
#117629422
#117628632
C17×#4472
#321717