Finite Group Databasebeta

← #14539552

Subgroups — #14539552

Order 2176.

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

normal subgroups only

This group has 472 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
1#11763322C211yesmaximal, characteristic, complemented
2C2×#10587624C211yesmaximal, characteristic, complemented
3#11763142C211yesmaximal, characteristic, complemented
4#11763132C212yesmaximal, complemented
5#11763128C211yesmaximal, characteristic, complemented
6#11763044C212yesmaximal, complemented
7#11763035C212yesmaximal, complemented
8#11762953C212yesmaximal, complemented
9#11762943C211yesmaximal, characteristic, complemented
10C22⋊C8×D17C211yesmaximal, characteristic, complemented
11C17×#450C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C22×C22⋊D17—21no—
14C2×#63256C2211yescharacteristic, complemented
15C22×C4×D17C2211yescharacteristic
16C2×SD16×C17C2212yescomplemented
17C17×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Dic17—21no—
19#10587632C2211yescharacteristic, complemented
20#10587624—21no—
21#10587624—21no—
22#10587624C2212yes—
23#10587624C2212yes—
24C2×#63228C2212yescomplemented
25#10587616C2211yescharacteristic
26C2×C4⋊Dic17C2211yescharacteristic
27C2×C8×D17C2212yescomplemented
28#10587581C2211yescharacteristic, complemented
29#10587580—21no—
30#10587579—21no—
31#10587578C2211yescharacteristic, complemented
32#10587568—21no—
33#10587567—21no—
34C17×#102C2212yescomplemented
35C17×C22⋊C8C2211yescharacteristic
36#10587506C2212yescomplemented
37#10587503C2212yescomplemented
38C22⋊#3622C2211yescharacteristic
39#10587492C2212yescomplemented
40#10587491C2212yescomplemented
41#10587488C2212yescomplemented
42#10587478C2212yescomplemented
43C24×C17—21no—
44C23×D17D411yescharacteristic
45C2×Q8×C17C2311yescharacteristic
46C2×D4×C17—21no—
47C2×D4×C17—42no—
48C2×D4×C17C2312yes—
49C22×C4×C17C2311yescharacteristic
50C2×C22⋊D17—21no—
51C2×C22⋊D17—22no—
52C2×C22⋊D17—22no—
53C2×C22⋊D17—44no—
54C22×Dic17—21no—
55C22×Dic17D411yescharacteristic
56#63256—42no—
57C2×D68—21no—
58C2×D68—21no—
59C2×D68C2311yescharacteristic
60C2×C4×D17—21no—
61C2×C4×D17—42no—
62C2×C4×D17D411yescharacteristic
63C2×C4×D17D411yescharacteristic
64C2×C4×D17C2311yescharacteristic
65C2×C4×D17C2311yescharacteristic
66SD16×C17—42no—
67C2×C8×C17C2312yes—
68C4⋊C4×C17—21no—
69C4⋊C4×C17C2311yescharacteristic
70C22⋊C4×C17—21no—
71C22⋊Dic17—22no—
72C22⋊Dic17—22no—
73#63228—42no—
74#63227—21no—
75#63227—21no—
76#63227—21no—
77#63227—21no—
78C4⋊Dic17—21no—
79C4⋊Dic17C2312yes—
80C4⋊Dic17C2311yescharacteristic
81#63225—21no—
82C4×Dic17—21no—
83C2×#3622C2312yes—
84C8×D17—42no—
85C23×C17—22no—
86C23×C17—42no—
87C23×C17—42no—
88C23×C17—42no—
89C23×C17C2×D411yescharacteristic
90C22×D17—22no—
91C22×D17—22no—
92C22×D17—21no—
93C22×D17—41no—
94C22×D17C2×D411yescharacteristic
95C22×D17C2×D411yescharacteristic
96Q8×C17—22no—
97D4×C17—22no—
98D4×C17—22no—
99D4×C17—42no—
100C2×C4×C17—22no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17C2×D411yescharacteristic
104C2×C4×C17C2411yescharacteristic
105C22⋊D17—44no—
106C22⋊D17—42no—
107C22⋊D17—44no—
108C2×Dic17—22no—
109C2×Dic17—21no—
110C2×Dic17—41no—
111C2×Dic17—42no—
112C2×Dic17C2×D411yescharacteristic
113C2×Dic17C2×D411yescharacteristic
114D68—22no—
115D68—42no—
116D68—42no—
117C4×D17—22no—
118C4×D17—22no—
119C4×D17—41no—
120C4×D17—41no—
121C4×D17—42no—
122C8×C17—22no—
123#3622—22no—
124#3188—171nomaximal, Sylow
125C22×C17—21no—
126C22×C17—21no—
127C22×C17—42no—
128C22×C17—42no—
129C22×C17—42no—
130C22×C17—84no—
131C22×C17#13512yes—
132C22×C17C22×D411yescharacteristic
133C2×D17—21no—
134C2×D17—21no—
135C2×D17—21no—
136C2×D17—21no—
137C2×D17—21no—
138C2×D17—44no—
139C2×D17—41no—
140C2×D17—41no—
141C2×D17—41no—
142C2×D17AGL(1,ℤ/8)12yes—
143C4×C17—21no—
144C4×C17—41no—
145C4×C17—41no—
146C4×C17H(5,𝔽2)12yes—
147Dic17—21no—
148Dic17—21no—
149Dic17—41no—
150Dic17—42no—
151#525—171no—
152C2×#121—171no—
153#484—171no—
154#482—172no—
155#481—171no—
156#466—172no—
157#465—172no—
158#450D1711yescharacteristic, complemented
159#449—172no—
160#447—171no—
161C2×C22⋊C8—171no—
162C2×C17—22no—
163C2×C17—82no—
164C2×C17#53411yescharacteristic
165C2×C17C2×#13511yescharacteristic
166C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
167D17—22no—
168D17—41no—
169D17—81no—
170C2×#41—171no—
171C22×D4—341no—
172C22×D4C2×D1711yescharacteristic
173C23×C4—171no—
174C2×SD16C2×D1712yescomplemented
175C2×D8—172no—
176C22×C8—172no—
177#123—341no—
178C22⋊Q8—171no—
179C22⋊Q8C2×D1711yescharacteristic
180#121—172no—
181#121—172no—
182#121—171no—
183#121—171no—
184#121—341no—
185#121—341no—
186#120—341no—
187C4×D4—341no—
188C4×D4—341no—
189#117—171no—
190C2×C4⋊C4—171no—
191C2×C22⋊C4—341no—
192#107—172no—
193#106—172no—
194#103—172no—
195#102—172no—
196#102—172no—
197#102C2×D1712yescomplemented
198C22⋊C8—171no—
199C22⋊C8—172no—
200C22⋊C8C2×D1711yescharacteristic
201C17#318811yescharacteristic, complemented, Sylow
202C24—21no—
203C24—171no—
204#41—682no—
205C2×Q8C22×D1711yescharacteristic
206C2×D4—21no—
207C2×D4—42no—
208C2×D4—171no—
209C2×D4—341no—
210C2×D4—341no—
211C2×D4—342no—
212C2×D4—342no—
213C2×D4—341no—
214C2×D4—684no—
215C2×D4C22×D1712yes—
216C22×C4—171no—
217C22×C4—171no—
218C22×C4—171no—
219C22×C4—171no—
220C22×C4—171no—
221C22×C4—341no—
222C22×C4—341no—
223C22×C4—682no—
224C22×C4C22×D1711yescharacteristic
225SD16—42no—
226D8—682no—
227C2×C8—172no—
228C2×C8—682no—
229C2×C8C22×D1712yes—
230C4⋊C4—21no—
231C4⋊C4—171no—
232C4⋊C4—172no—
233C4⋊C4—341no—
234C4⋊C4—341no—
235C4⋊C4C22×D1711yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—341no—
238C22⋊C4—341no—
239C22⋊C4—341no—
240C22⋊C4—342no—
241C22⋊C4—342no—
242C22⋊C4—341no—
243C42—341no—
244C23—22no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—171no—
249C23—171no—
250C23—342no—
251C23—342no—
252C23—341no—
253C23—681no—
254C23D4×D1711yescharacteristic
255Q8—22no—
256D4—22no—
257D4—22no—
258D4—42no—
259D4—342no—
260D4—682no—
261D4—682no—
262D4—684no—
263D4—684no—
264D4—682no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—171no—
269C2×C4—171no—
270C2×C4—342no—
271C2×C4—342no—
272C2×C4—341no—
273C2×C4—342no—
274C2×C4—681no—
275C2×C4—681no—
276C2×C4—682no—
277C2×C4—681no—
278C2×C4—682no—
279C2×C4D4×D1711yescharacteristic
280C2×C4C23×D1711yescharacteristic
281C8—22no—
282C8—342no—
283C22—21no—
284C22—21no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—172no—
290C22—341no—
291C22—341no—
292C22—341no—
293C22—341no—
294C22—341no—
295C22—684no—
296C22—681no—
297C22—681no—
298C22—681no—
299C22#1058760212yes—
300C22C2×D4×D1711yescharacteristic
301C4—21no—
302C4—41no—
303C4—41no—
304C4—341no—
305C4—341no—
306C4—681no—
307C4—682no—
308C4#1058769112yes—
309C2—22no—
310C2—82no—
311C2—342no—
312C2—681no—
313C2—1361no—
314C2#1176336911yescharacteristic
315C2C2×#1058760211yescharacteristic
316C2AGL(1,ℤ/8)×D1711yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#450C2
socleC22×C17C22×D4
hypercenter#450D17
hypocenterC17#3188
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #11763322C2semidirect
2. C2×#10587624C2semidirect
3. #11763142C2semidirect
4. #11763132C2semidirect
5. #11763128C2semidirect
6. #11763044C2semidirect
7. #11763035C2semidirect
8. #11762953C2semidirect
9. #11762943C2semidirect
10. C22⋊C8×D17C2semidirect
11. C17×#450C2semidirect
14. C2×#63256C22semidirect
16. C2×SD16×C17C22semidirect
19. #10587632C22semidirect
24. C2×#63228C22semidirect
27. C2×C8×D17C22semidirect
28. #10587581C22semidirect
31. #10587578C22semidirect
34. C17×#102C22semidirect
36. #10587506C22semidirect
37. #10587503C22semidirect
39. #10587492C22semidirect
40. #10587491C22semidirect
41. #10587488C22semidirect
42. #10587478C22semidirect
158. #450D17semidirect
174. C2×SD16C2×D17semidirect
197. #102C2×D17semidirect
201. C17#3188semidirect

Maximal subgroups

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

subgroupindex
#117633222
C2×#105876242
#117631422
#117631322
#117631282
#117630442
#117630352
#117629532
#117629432
C22⋊C8×D172
C17×#4502
#318817