Finite Group Databasebeta

← #14539548

Subgroups — #14539548

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#11763319C211yesmaximal, characteristic, complemented
2C2×#10587581C211yesmaximal, characteristic, complemented
3#11763144C211yesmaximal, characteristic, complemented
4#11763135C212yesmaximal, complemented
5#11763044C212yesmaximal, complemented
6#11763039C212yesmaximal, complemented
7#11762962C211yesmaximal, characteristic, complemented
8#11762955C212yesmaximal, complemented
9#11762942C211yesmaximal, characteristic, complemented
10C22⋊C8×D17C211yesmaximal, characteristic, complemented
11C17×#481C211yesmaximal, characteristic, complemented
12C2×#63254C2211yescharacteristic, complemented
13C22×D68—21no—
14C22×D68C2211yescharacteristic
15C22×C4×D17C2211yescharacteristic
16C17×#121C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18#10587624C2211yescharacteristic, complemented
19#10587623—21no—
20D4×Dic17—21no—
21C2×#63228C2212yescomplemented
22C2×#63227—21no—
23#10587610C2211yescharacteristic
24C2×#63219C2212yescomplemented
25C2×C8×D17C2212yescomplemented
26#10587583C2211yescharacteristic, complemented
27#10587581—21no—
28#10587581—21no—
29#10587581C2212yes—
30#10587581C2212yes—
31#10587578C2211yescharacteristic, complemented
32#10587569—21no—
33#10587567—21no—
34C17×#106C2212yescomplemented
35C17×#102C2212yescomplemented
36C17×C22⋊C8C2211yescharacteristic
37C22⋊#3622C2211yescharacteristic
38#10587492C2212yescomplemented
39#10587491C2212yescomplemented
40#10587481C2212yescomplemented
41#10587480C2212yescomplemented
42#10587478C2212yescomplemented
43C23×D17—21no—
44C23×D17D411yescharacteristic
45C2×D4×C17—21no—
46C2×D4×C17C2311yescharacteristic
47C22×C4×C17—21no—
48C22×C4×C17C2311yescharacteristic
49C2×C22⋊D17—21no—
50C2×C22⋊D17—21no—
51C22×Dic17—21no—
52C22×Dic17D411yescharacteristic
53#63254—42no—
54C2×D68—22no—
55C2×D68—22no—
56C2×D68—21no—
57C2×D68—42no—
58C2×D68—44no—
59C2×D68C2312yes—
60C2×C4×D17—42no—
61C2×C4×D17D411yescharacteristic
62C2×C4×D17D411yescharacteristic
63C2×C4×D17C2311yescharacteristic
64C2×C4×D17C2311yescharacteristic
65C2×Dic34C2311yescharacteristic
66C2×C8×C17C2312yes—
67C4⋊C4×C17—21no—
68C4⋊C4×C17C2312yes—
69C4⋊C4×C17C2311yescharacteristic
70C22⋊C4×C17—21no—
71C22⋊Dic17—21no—
72C22⋊Dic17—21no—
73#63228—42no—
74#63227—21no—
75#63227—21no—
76#63227—22no—
77#63227—22no—
78C4⋊Dic17—21no—
79C4⋊Dic17C2311yescharacteristic
80#63225—21no—
81C4×Dic17—21no—
82C2×#3622C2312yes—
83#63219—42no—
84C8×D17—42no—
85C23×C17—21no—
86C23×C17C2×D411yescharacteristic
87C22×D17—22no—
88C22×D17—22no—
89C22×D17—22no—
90C22×D17—41no—
91C22×D17—42no—
92C22×D17—42no—
93C22×D17—42no—
94C22×D17C2×D411yescharacteristic
95C22×D17C2×D411yescharacteristic
96D4×C17—22no—
97D4×C17—42no—
98C2×C4×C17—22no—
99C2×C4×C17—21no—
100C2×C4×C17—22no—
101C2×C4×C17—42no—
102C2×C4×C17C2×D411yescharacteristic
103C2×C4×C17C2411yescharacteristic
104C22⋊D17—42no—
105C22⋊D17—42no—
106C2×Dic17—21no—
107C2×Dic17—21no—
108C2×Dic17—41no—
109C2×Dic17—42no—
110C2×Dic17C2×D411yescharacteristic
111C2×Dic17C2×D411yescharacteristic
112D68—22no—
113D68—22no—
114D68—42no—
115D68—44no—
116D68—44no—
117C4×D17—22no—
118C4×D17—22no—
119C4×D17—41no—
120C4×D17—41no—
121Dic34—22no—
122C8×C17—22no—
123#3622—22no—
124#3188—171nomaximal, Sylow
125C22×C17—21no—
126C22×C17—21no—
127C22×C17—41no—
128C22×C17—41no—
129C22×C17—41no—
130C22×C17#13512yes—
131C22×C17C22×D411yescharacteristic
132C2×D17—21no—
133C2×D17—21no—
134C2×D17—21no—
135C2×D17—21no—
136C2×D17—21no—
137C2×D17—44no—
138C2×D17—42no—
139C2×D17—42no—
140C2×D17—42no—
141C2×D17—84no—
142C2×D17AGL(1,ℤ/8)12yes—
143C4×C17—21no—
144C4×C17—41no—
145C4×C17—42no—
146C4×C17H(5,𝔽2)12yes—
147Dic17—21no—
148Dic17—21no—
149Dic17—41no—
150Dic17—41no—
151#525—171no—
152C2×#121—171no—
153#484—171no—
154#482—172no—
155#481D1711yescharacteristic, complemented
156#466—172no—
157#465—172no—
158#450—171no—
159#449—172no—
160#447—171no—
161C2×C22⋊C8—171no—
162C2×C17—22no—
163C2×C17—81no—
164C2×C17#53411yescharacteristic
165C2×C17C2×#13511yescharacteristic
166C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
167D17—22no—
168D17—41no—
169D17—82no—
170C2×#41—171no—
171C22×D4—171no—
172C22×D4—341no—
173C23×C4—171no—
174C2×SD16—172no—
175C2×D8—172no—
176C22×C8—172no—
177#123—341no—
178C22⋊Q8—171no—
179C22⋊Q8—171no—
180#121—171no—
181#121—172no—
182#121—172no—
183#121—341no—
184#121—341no—
185#121C2×D1711yescharacteristic
186#120—341no—
187C4×D4—341no—
188C4×D4—341no—
189#117—171no—
190C2×C4⋊C4C2×D1711yescharacteristic
191C2×C22⋊C4—341no—
192#107—172no—
193#106C2×D1712yescomplemented
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—171no—
203C24—341no—
204#41—682no—
205C2×Q8—171no—
206C2×D4—21no—
207C2×D4—172no—
208C2×D4—342no—
209C2×D4—342no—
210C2×D4—341no—
211C2×D4—341no—
212C2×D4—341no—
213C2×D4—682no—
214C2×D4—684no—
215C2×D4C22×D1711yescharacteristic
216C22×C4—21no—
217C22×C4—171no—
218C22×C4—171no—
219C22×C4—171no—
220C22×C4—171no—
221C22×C4—171no—
222C22×C4—341no—
223C22×C4—682no—
224C22×C4C22×D1711yescharacteristic
225SD16—682no—
226D8—682no—
227C2×C8—172no—
228C2×C8—682no—
229C2×C8C22×D1712yes—
230C4⋊C4—21no—
231C4⋊C4—171no—
232C4⋊C4—341no—
233C4⋊C4—341no—
234C4⋊C4C22×D1711yescharacteristic
235C4⋊C4C22×D1712yes—
236C22⋊C4—21no—
237C22⋊C4—342no—
238C22⋊C4—342no—
239C22⋊C4—341no—
240C22⋊C4—341no—
241C22⋊C4—341no—
242C22⋊C4—341no—
243C42—341no—
244C23—21no—
245C23—171no—
246C23—171no—
247C23—342no—
248C23—342no—
249C23—342no—
250C23—681no—
251C23—682no—
252C23—682no—
253C23—682no—
254C23D4×D1711yescharacteristic
255Q8—342no—
256D4—22no—
257D4—42no—
258D4—342no—
259D4—342no—
260D4—682no—
261D4—684no—
262D4—684no—
263D4—682no—
264D4—682no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—171no—
270C2×C4—171no—
271C2×C4—342no—
272C2×C4—342no—
273C2×C4—341no—
274C2×C4—341no—
275C2×C4—681no—
276C2×C4—681no—
277C2×C4—681no—
278C2×C4—682no—
279C2×C4D4×D1711yescharacteristic
280C2×C4C23×D1711yescharacteristic
281C8—22no—
282C8—342no—
283C22—21no—
284C22—21no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—172no—
289C22—341no—
290C22—341no—
291C22—341no—
292C22—341no—
293C22—341no—
294C22—684no—
295C22—682no—
296C22—682no—
297C22—682no—
298C22—1364no—
299C22#1058760212yes—
300C22C2×D4×D1711yescharacteristic
301C4—21no—
302C4—41no—
303C4—42no—
304C4—341no—
305C4—341no—
306C4—681no—
307C4—681no—
308C4#1058769612yes—
309C2—22no—
310C2—81no—
311C2—342no—
312C2—681no—
313C2—1362no—
314C2#1176337011yescharacteristic
315C2C2×#1058760211yescharacteristic
316C2AGL(1,ℤ/8)×D1711yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#481C2
socleC22×C17C22×D4
hypercenter#481D17
hypocenterC17#3188
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #11763319C2semidirect
2. C2×#10587581C2semidirect
3. #11763144C2semidirect
4. #11763135C2semidirect
5. #11763044C2semidirect
6. #11763039C2semidirect
7. #11762962C2semidirect
8. #11762955C2semidirect
9. #11762942C2semidirect
10. C22⋊C8×D17C2semidirect
11. C17×#481C2semidirect
12. C2×#63254C22semidirect
18. #10587624C22semidirect
21. C2×#63228C22semidirect
24. C2×#63219C22semidirect
25. C2×C8×D17C22semidirect
26. #10587583C22semidirect
31. #10587578C22semidirect
34. C17×#106C22semidirect
35. C17×#102C22semidirect
38. #10587492C22semidirect
39. #10587491C22semidirect
40. #10587481C22semidirect
41. #10587480C22semidirect
42. #10587478C22semidirect
155. #481D17semidirect
193. #106C2×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
#117633192
C2×#105875812
#117631442
#117631352
#117630442
#117630392
#117629622
#117629552
#117629422
C22⋊C8×D172
C17×#4812
#318817