Finite Group Databasebeta

← #14536782

Subgroups — #14536782

Order 2176.

712 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 125 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×#10587568C212yesmaximal, complemented
2#11762738C212yesmaximal, complemented
3#11762727C212yesmaximal, complemented
4#11762726C212yesmaximal, complemented
5#11762724C212yesmaximal, complemented
6#11762719C212yesmaximal, complemented
7#11762718C211yesmaximal, characteristic, complemented
8#11762664C211yesmaximal, characteristic, complemented
9C17×#397C211yesmaximal, characteristic, complemented
10C24×D17C2211yescharacteristic
11C22×C22⋊D17C2212yes—
12C22×D68C2212yes—
13C22×D68C2212yes—
14C22×D68C2211yescharacteristic, complemented
15C2×C4⋊C4×C17C2211yescharacteristic, complemented
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊Dic17C2211yescharacteristic
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×#63227C2212yescomplemented
24C2×#63227C2212yescomplemented
25C2×#63227C2212yescomplemented
26C2×#63225C2212yescomplemented
27#10587568—28no—
28C17×#95C2212yescomplemented
29C17×#95C2211yescharacteristic, complemented
30#10587502C2211yescharacteristic, complemented
31#10587502C2212yescomplemented
32C24×C17C2311yescharacteristic
33C23×D17—41no—
34C23×D17—41no—
35C23×D17—42no—
36C23×D17—42no—
37C23×D17D414yescomplemented
38C23×D17C2312yes—
39C23×D17C2312yes—
40C22×C4×C17C2312yes—
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C2×C22⋊D17—24no—
45C2×C22⋊D17—42no—
46C2×C22⋊D17—42no—
47C22×Dic17C2312yes—
48C22×Dic17C2312yes—
49C2×D68—24no—
50C2×D68—24no—
51C2×D68—42no—
52C2×D68—42no—
53C2×D68—42no—
54C2×D68—42no—
55C2×D68—41no—
56C2×D68—42no—
57C4⋊C4×C17—41no—
58C22⋊C4×C17—24no—
59C22⋊C4×C17—41no—
60C22⋊Dic17—41no—
61#63227—24no—
62#63227—24no—
63#63227—42no—
64#63227—42no—
65#63227—42no—
66#63227—42no—
67#63227—42no—
68#63225—42no—
69C23×C17—21no—
70C23×C17—21no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17C2×D412yes—
74C23×C17C2411yescharacteristic
75C22×D17—22no—
76C22×D17—22no—
77C22×D17—22no—
78C22×D17—22no—
79C22×D17—22no—
80C22×D17—22no—
81C22×D17—22no—
82C22×D17—22no—
83C22×D17—22no—
84C22×D17—22no—
85C22×D17—22no—
86C22×D17—22no—
87C22×D17—22no—
88C22×D17—44no—
89C22×D17—48no—
90C22×D17—44no—
91C22×D17—48no—
92C22×D17—44no—
93C22×D17C2×D414yes—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—22no—
103C2×C4×C17—22no—
104C22⋊D17—48no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111D68—48no—
112D68—48no—
113D68—44no—
114#2812—171nomaximal, Sylow
115C22×C17—24no—
116C22×C17—22no—
117C22×C17—41no—
118C22×C17—42no—
119C22×C17—41no—
120C22×C17C22×D411yescharacteristic
121C22×C17H(5,𝔽2)11yescharacteristic
122C22×C17H(5,𝔽2)12yes—
123C22×C17H(5,𝔽2)11yescharacteristic
124C22×C17H(5,𝔽2)12yes—
125C2×D17—24no—
126C2×D17—24no—
127C2×D17—24no—
128C2×D17—416no—
129C2×D17—42no—
130C2×D17—42no—
131C2×D17—42no—
132C2×D17—42no—
133C2×D17—42no—
134C2×D17—42no—
135C2×D17—42no—
136C2×D17—42no—
137C2×D17—42no—
138C2×D17—42no—
139C2×D17—42no—
140C4×C17—42no—
141C4×C17—42no—
142C4×C17—41no—
143C4×C17—41no—
144Dic17—42no—
145Dic17—42no—
146C2×#120—172no—
147#397—172no—
148#397D1711yescharacteristic, complemented
149#394—172no—
150#394—172no—
151#394—172no—
152#392—171no—
153#392—172no—
154#379—171no—
155C2×C17—42no—
156C2×C17#53411yescharacteristic
157C2×C17#53412yes—
158C2×C17#56011yescharacteristic
159C2×C17#56012yes—
160C2×C17H(3,𝔽4)11yescharacteristic
161D17—44no—
162D17—82no—
163C25—171no—
164C22×D4—172no—
165C22×D4—172no—
166C22×D4—172no—
167C22×D4—171no—
168#120—348no—
169C2×C4⋊C4—172no—
170C2×C4⋊C4C2×D1711yescharacteristic, complemented
171C2×C22⋊C4—172no—
172C2×C22⋊C4—172no—
173C2×C22⋊C4—171no—
174C2×C22⋊C4—172no—
175C2×C22⋊C4—172no—
176C2×C22⋊C4—172no—
177C2×C22⋊C4—172no—
178C2×C22⋊C4—172no—
179C2×C22⋊C4C2×D1712yes—
180C2×C22⋊C4C2×D1711yescharacteristic
181#95—171no—
182#95—172no—
183#95C2×D1712yescomplemented
184#95C2×D1711yescharacteristic, complemented
185C17#281211yescharacteristic, complemented, Sylow
186C24—172no—
187C24—174no—
188C24—172no—
189C24—681no—
190C24—681no—
191C24—682no—
192C24—682no—
193C24C22×D1711yescharacteristic
194C2×D4—344no—
195C2×D4—344no—
196C2×D4—344no—
197C2×D4—681no—
198C2×D4—682no—
199C2×D4—682no—
200C2×D4—682no—
201C2×D4—682no—
202C2×D4—682no—
203C2×D4—682no—
204C2×D4—682no—
205C22×C4—172no—
206C22×C4—172no—
207C22×C4C22×D1711yescharacteristic
208C22×C4C22×D1712yes—
209C22×C4C22×D1712yes—
210C22×C4C22×D1711yescharacteristic
211C4⋊C4—41no—
212C4⋊C4—682no—
213C22⋊C4—24no—
214C22⋊C4—41no—
215C22⋊C4—344no—
216C22⋊C4—344no—
217C22⋊C4—682no—
218C22⋊C4—682no—
219C22⋊C4—682no—
220C22⋊C4—682no—
221C22⋊C4—682no—
222C22⋊C4—681no—
223C23—21no—
224C23—21no—
225C23—22no—
226C23—22no—
227C23—174no—
228C23—342no—
229C23—342no—
230C23—342no—
231C23—342no—
232C23—342no—
233C23—342no—
234C23—342no—
235C23—342no—
236C23—342no—
237C23—342no—
238C23—342no—
239C23—342no—
240C23—342no—
241C23—684no—
242C23—688no—
243C23—684no—
244C23—688no—
245C23—684no—
246C23D4×D1712yes—
247C23C23×D1711yescharacteristic
248D4—688no—
249D4—688no—
250D4—684no—
251D4—688no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—342no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—342no—
266C2×C4—342no—
267C2×C4—342no—
268C22—22no—
269C22—24no—
270C22—41no—
271C22—42no—
272C22—41no—
273C22—344no—
274C22—344no—
275C22—344no—
276C22—6816no—
277C22—682no—
278C22—682no—
279C22—682no—
280C22—682no—
281C22—682no—
282C22—682no—
283C22—682no—
284C22—682no—
285C22—682no—
286C22—682no—
287C22—682no—
288C22C2×D4×D1711yescharacteristic
289C22#1058769111yescharacteristic
290C22#1058769612yes—
291C22#1058769611yescharacteristic
292C22#1058769612yes—
293C4—41no—
294C4—41no—
295C4—42no—
296C4—42no—
297C4—682no—
298C4—682no—
299C2—42no—
300C2—684no—
301C2—1362no—
302C2#1176336711yescharacteristic
303C2#1176337012yes—
304C2#1176354911yescharacteristic
305C2#1176355012yes—
306C2#1176355111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#397C2
hypercenter#397D17
hypocenterC17#2812
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#10587568C2semidirect
2. #11762738C2semidirect
3. #11762727C2semidirect
4. #11762726C2semidirect
5. #11762724C2semidirect
6. #11762719C2semidirect
7. #11762718C2semidirect
8. #11762664C2semidirect
9. C17×#397C2semidirect
14. C22×D68C22semidirect
15. C2×C4⋊C4×C17C22semidirect
23. C2×#63227C22semidirect
24. C2×#63227C22semidirect
25. C2×#63227C22semidirect
26. C2×#63225C22semidirect
28. C17×#95C22semidirect
29. C17×#95C22semidirect
30. #10587502C22semidirect
31. #10587502C22semidirect
37. C23×D17D4semidirect
148. #397D17semidirect
170. C2×C4⋊C4C2×D17semidirect
183. #95C2×D17semidirect
184. #95C2×D17semidirect
185. C17#2812semidirect

Maximal subgroups

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

subgroupindex
C2×#105875682
#117627382
#117627272
#117627262
#117627242
#117627192
#117627182
#117626642
C17×#3972
#281217