Finite Group Databasebeta

← #14535011

Subgroups — #14535011

Order 2176.

796 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 141 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×#10587571C212yesmaximal, complemented
2C2×#10587623C212yesmaximal, complemented
3#11762725C212yesmaximal, complemented
4#11762721C211yesmaximal, characteristic, complemented
5#11762717C212yesmaximal, complemented
6#11762712C212yesmaximal, complemented
7#11762701C212yesmaximal, complemented
8#11762664C211yesmaximal, characteristic, complemented
9C17×#392C211yesmaximal, characteristic, complemented
10C24×D17C2211yescharacteristic
11C22×D4×C17C2212yes—
12C22×D4×C17C2211yescharacteristic, complemented
13C22×C22⋊D17C2212yes—
14C22×C22⋊D17C2212yes—
15C22×Dic34C2211yescharacteristic, complemented
16C2×C22⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2212yescomplemented
18C2×C22⋊Dic17C2212yes—
19C2×C22⋊Dic17C2211yescharacteristic
20C2×C22⋊Dic17C2212yescomplemented
21C2×C22⋊Dic17C2212yescomplemented
22#10587623—28no—
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×#63227C2212yes—
27C2×C4⋊Dic17C2212yescomplemented
28C2×C4×Dic17C2212yescomplemented
29#10587571—28no—
30C17×#95C2211yescharacteristic, complemented
31#10587502C2212yescomplemented
32#10587502C2211yescharacteristic, complemented
33C24×C17C2312yes—
34C24×C17C2311yescharacteristic
35C23×D17—22no—
36C23×D17—24no—
37C23×D17—41no—
38C23×D17—41no—
39C23×D17—42no—
40C23×D17C2312yes—
41C2×D4×C17—24no—
42C2×D4×C17—24no—
43C2×D4×C17—42no—
44C2×D4×C17—41no—
45C2×D4×C17—42no—
46C22×C4×C17C2311yescharacteristic
47C22×C4×C17C2311yescharacteristic
48C22×C4×C17C2312yes—
49C2×C22⋊D17—24no—
50C2×C22⋊D17—24no—
51C2×C22⋊D17—24no—
52C2×C22⋊D17—42no—
53C2×C22⋊D17—42no—
54C2×C22⋊D17—42no—
55C22×Dic17C2312yes—
56C22×Dic17C2312yes—
57C22×Dic17C2312yes—
58C2×Dic34—24no—
59C2×Dic34—41no—
60C22⋊C4×C17—24no—
61C22⋊C4×C17—41no—
62C22⋊Dic17—24no—
63C22⋊Dic17—24no—
64C22⋊Dic17—41no—
65C22⋊Dic17—42no—
66#63227—24no—
67#63227—24no—
68#63227—42no—
69#63227D418yescomplemented
70C4⋊Dic17—42no—
71C4×Dic17—24no—
72C23×C17—22no—
73C23×C17—21no—
74C23×C17—21no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17—22no—
78C23×C17—22no—
79C23×C17—22no—
80C23×C17—22no—
81C23×C17—22no—
82C23×C17—22no—
83C23×C17#4112yes—
84C23×C17C2411yescharacteristic
85C22×D17—22no—
86C22×D17—22no—
87C22×D17—22no—
88C22×D17—22no—
89C22×D17—22no—
90C22×D17—22no—
91C22×D17—44no—
92C22×D17—48no—
93C22×D17—48no—
94C22×D17—44no—
95C22×D17—44no—
96C22×D17C2×D414yes—
97D4×C17—48no—
98D4×C17—44no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—22no—
105C2×C4×C17C2×D414yes—
106C22⋊D17—48no—
107C22⋊D17—48no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—22no—
115C2×Dic17—22no—
116C2×Dic17C2×D414yes—
117Dic34—44no—
118#2670—171nomaximal, Sylow
119C22×C17—22no—
120C22×C17—24no—
121C22×C17—42no—
122C22×C17—41no—
123C22×C17—41no—
124C22×C17—42no—
125C22×C17—42no—
126C22×C17—42no—
127C22×C17—42no—
128C22×C17—42no—
129C22×C17—42no—
130C22×C17—42no—
131C22×C17C22×D412yes—
132C22×C17C2×#4111yescharacteristic
133C22×C17H(5,𝔽2)12yes—
134C22×C17H(5,𝔽2)11yescharacteristic
135C22×C17H(5,𝔽2)11yescharacteristic
136C2×D17—24no—
137C2×D17—24no—
138C2×D17—24no—
139C2×D17—416no—
140C2×D17—42no—
141C2×D17—42no—
142C2×D17—42no—
143C2×D17—42no—
144C4×C17—24no—
145C4×C17—41no—
146C4×C17—41no—
147Dic17—24no—
148Dic17—42no—
149Dic17—42no—
150C2×#124—172no—
151C2×#120—172no—
152#394—172no—
153#393—171no—
154#392—172no—
155#392D1711yescharacteristic, complemented
156#390—172no—
157#388—172no—
158#379—171no—
159C2×C17—42no—
160C2×C17—82no—
161C2×C17#53512yes—
162C2×C17#53811yescharacteristic
163C2×C17D4211yescharacteristic
164C2×C17#54612yes—
165C2×C17H(3,𝔽4)11yescharacteristic
166D17—44no—
167C25—171no—
168C22×Q8—171no—
169C22×D4—172no—
170C22×D4—172no—
171C22×D4C2×D1712yes—
172C22×D4C2×D1711yescharacteristic, complemented
173#124—348no—
174#120—348no—
175C2×C4⋊C4—172no—
176C2×C22⋊C4—172no—
177C2×C22⋊C4—172no—
178C2×C22⋊C4—172no—
179C2×C22⋊C4—172no—
180C2×C22⋊C4—172no—
181C2×C22⋊C4—171no—
182C2×C22⋊C4—172no—
183C2×C22⋊C4—172no—
184C2×C22⋊C4C2×D1711yescharacteristic
185C2×C22⋊C4C2×D1712yescomplemented
186C2×C42—172no—
187#95—172no—
188#95—171no—
189#95C2×D1711yescharacteristic, complemented
190C17#267011yescharacteristic, complemented, Sylow
191C24—172no—
192C24—344no—
193C24—342no—
194C24—681no—
195C24—682no—
196C24—681no—
197C24C22×D1712yes—
198C24C22×D1711yescharacteristic
199C2×Q8—344no—
200C2×Q8—681no—
201C2×D4—24no—
202C2×D4—24no—
203C2×D4—42no—
204C2×D4—41no—
205C2×D4—42no—
206C2×D4—344no—
207C2×D4—344no—
208C2×D4—344no—
209C2×D4—682no—
210C2×D4—682no—
211C2×D4—682no—
212C22×C4—172no—
213C22×C4—172no—
214C22×C4—172no—
215C22×C4C22×D1711yescharacteristic
216C22×C4C22×D1711yescharacteristic
217C22×C4C22×D1712yes—
218C4⋊C4—682no—
219C22⋊C4—24no—
220C22⋊C4—41no—
221C22⋊C4—178no—
222C22⋊C4—344no—
223C22⋊C4—344no—
224C22⋊C4—344no—
225C22⋊C4—344no—
226C22⋊C4—682no—
227C22⋊C4—681no—
228C22⋊C4—682no—
229C42—344no—
230C23—22no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—174no—
242C23—342no—
243C23—342no—
244C23—342no—
245C23—342no—
246C23—342no—
247C23—342no—
248C23—688no—
249C23—684no—
250C23—684no—
251C23—688no—
252C23—684no—
253C23#6325412yes—
254C23C23×D1711yescharacteristic
255Q8—684no—
256D4—48no—
257D4—44no—
258D4—688no—
259D4—688no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—174no—
267C2×C4—342no—
268C2×C4—342no—
269C2×C4—342no—
270C2×C4—342no—
271C2×C4—342no—
272C2×C4—342no—
273C2×C4—342no—
274C2×C4—342no—
275C2×C4D4×D1714yes—
276C22—22no—
277C22—24no—
278C22—41no—
279C22—42no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—344no—
289C22—344no—
290C22—344no—
291C22—6816no—
292C22—682no—
293C22—682no—
294C22—682no—
295C22—682no—
296C22C2×D4×D1712yes—
297C22C2×#6325411yescharacteristic
298C22#1058769112yes—
299C22#1058769111yescharacteristic
300C22#1058769611yescharacteristic
301C4—24no—
302C4—41no—
303C4—41no—
304C4—344no—
305C4—682no—
306C4—682no—
307C2—42no—
308C2—82no—
309C2—684no—
310C2#1176337512yes—
311C2#1176339611yescharacteristic
312C2#1176344411yescharacteristic
313C2#1176344612yes—
314C2#1176355211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#392C2
hypercenter#392D17
hypocenterC17#2670
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#10587571C2semidirect
2. C2×#10587623C2semidirect
3. #11762725C2semidirect
4. #11762721C2semidirect
5. #11762717C2semidirect
6. #11762712C2semidirect
7. #11762701C2semidirect
8. #11762664C2semidirect
9. C17×#392C2semidirect
12. C22×D4×C17C22semidirect
15. C22×Dic34C22semidirect
17. C2×C22⋊C4×C17C22semidirect
20. C2×C22⋊Dic17C22semidirect
21. C2×C22⋊Dic17C22semidirect
27. C2×C4⋊Dic17C22semidirect
28. C2×C4×Dic17C22semidirect
30. C17×#95C22semidirect
31. #10587502C22semidirect
32. #10587502C22semidirect
69. #63227D4semidirect
155. #392D17semidirect
172. C22×D4C2×D17semidirect
185. C2×C22⋊C4C2×D17semidirect
189. #95C2×D17semidirect
190. C17#2670semidirect

Maximal subgroups

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

subgroupindex
C2×#105875712
C2×#105876232
#117627252
#117627212
#117627172
#117627122
#117627012
#117626642
C17×#3922
#267017