Finite Group Databasebeta

← #14536778

Subgroups — #14536778

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×#10587623C212yesmaximal, complemented
2#11762738C212yesmaximal, complemented
3#11762734C211yesmaximal, characteristic, complemented
4#11762725C212yesmaximal, complemented
5#11762724C212yesmaximal, complemented
6#11762723C212yesmaximal, complemented
7#11762717C212yesmaximal, complemented
8#11762664C211yesmaximal, characteristic, complemented
9C17×#392C211yesmaximal, characteristic, complemented
10C24×D17C2211yescharacteristic
11C22×D4×C17C2212yes—
12C22×D4×C17C2211yescharacteristic, complemented
13C22×C22⋊D17C2212yes—
14C22×C22⋊D17C2212yes—
15C2×C22⋊C4×C17C2211yescharacteristic
16C2×C22⋊C4×C17C2212yescomplemented
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2211yescharacteristic
19C2×C22⋊Dic17C2212yescomplemented
20C2×C22⋊Dic17C2212yescomplemented
21#10587623—28no—
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×C4⋊Dic17C2211yescharacteristic, complemented
27C2×#63225C2212yescomplemented
28C17×#95C2211yescharacteristic, complemented
29#10587502C2212yescomplemented
30#10587502C2212yescomplemented
31#10587502C2211yescharacteristic, complemented
32C24×C17C2312yes—
33C24×C17C2311yescharacteristic
34C23×D17—41no—
35C23×D17—41no—
36C23×D17—42no—
37C23×D17—42no—
38C23×D17D414yescomplemented
39C23×D17C2312yes—
40C2×D4×C17—24no—
41C2×D4×C17—42no—
42C2×D4×C17—42no—
43C2×D4×C17—41no—
44C2×D4×C17—42no—
45C22×C4×C17C2312yes—
46C22×C4×C17C2311yescharacteristic
47C22×C4×C17C2311yescharacteristic
48C2×C22⋊D17—24no—
49C2×C22⋊D17—24no—
50C2×C22⋊D17—42no—
51C2×C22⋊D17—42no—
52C2×C22⋊D17—42no—
53C2×C22⋊D17—42no—
54C22×Dic17C2312yes—
55C22×Dic17C2312yes—
56C22×Dic17C2312yes—
57C22⋊C4×C17—41no—
58C22⋊C4×C17—42no—
59C22⋊Dic17—24no—
60C22⋊Dic17—41no—
61C22⋊Dic17—42no—
62C22⋊Dic17—42no—
63#63227—24no—
64#63227—24no—
65#63227—42no—
66#63227—42no—
67C4⋊Dic17—41no—
68#63225—42no—
69C23×C17—21no—
70C23×C17—21no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17—22no—
78C23×C17—22no—
79C23×C17—22no—
80C23×C17C2×D412yes—
81C23×C17C2411yescharacteristic
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—
94D4×C17—48no—
95D4×C17—44no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—22no—
103C22⋊D17—48no—
104C22⋊D17—48no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114#2812—171nomaximal, Sylow
115C22×C17—24no—
116C22×C17—22no—
117C22×C17—41no—
118C22×C17—42no—
119C22×C17—41no—
120C22×C17—42no—
121C22×C17—42no—
122C22×C17—42no—
123C22×C17—42no—
124C22×C17—42no—
125C22×C17—42no—
126C22×C17—42no—
127C22×C17C22×D411yescharacteristic
128C22×C17H(5,𝔽2)12yes—
129C22×C17H(5,𝔽2)11yescharacteristic
130C22×C17H(5,𝔽2)12yes—
131C22×C17H(5,𝔽2)11yescharacteristic
132C2×D17—24no—
133C2×D17—24no—
134C2×D17—24no—
135C2×D17—416no—
136C2×D17—42no—
137C2×D17—42no—
138C2×D17—42no—
139C2×D17—42no—
140C4×C17—42no—
141C4×C17—41no—
142C4×C17—41no—
143Dic17—42no—
144Dic17—42no—
145Dic17—42no—
146C2×#120—172no—
147#397—171no—
148#397—172no—
149#394—172no—
150#394—172no—
151#394—172no—
152#392—172no—
153#392D1711yescharacteristic, complemented
154#379—171no—
155C2×C17—42no—
156C2×C17—82no—
157C2×C17#53411yescharacteristic
158C2×C17#53412yes—
159C2×C17#56011yescharacteristic
160C2×C17#56012yes—
161C2×C17H(3,𝔽4)11yescharacteristic
162D17—44no—
163C25—171no—
164C22×D4—172no—
165C22×D4—172no—
166C22×D4C2×D1712yes—
167C22×D4C2×D1711yescharacteristic, complemented
168#120—348no—
169C2×C4⋊C4—172no—
170C2×C4⋊C4—171no—
171C2×C22⋊C4—172no—
172C2×C22⋊C4—172no—
173C2×C22⋊C4—172no—
174C2×C22⋊C4—171no—
175C2×C22⋊C4—172no—
176C2×C22⋊C4—172no—
177C2×C22⋊C4—172no—
178C2×C22⋊C4—172no—
179C2×C22⋊C4C2×D1711yescharacteristic
180C2×C22⋊C4C2×D1712yescomplemented
181#95—172no—
182#95—171no—
183#95—172no—
184#95C2×D1711yescharacteristic, complemented
185C17#281211yescharacteristic, complemented, Sylow
186C24—172no—
187C24—174no—
188C24—681no—
189C24—682no—
190C24—682no—
191C24—681no—
192C24C22×D1712yes—
193C24C22×D1711yescharacteristic
194C2×D4—24no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—41no—
198C2×D4—42no—
199C2×D4—344no—
200C2×D4—344no—
201C2×D4—682no—
202C2×D4—682no—
203C2×D4—682no—
204C2×D4—682no—
205C22×C4—172no—
206C22×C4—172no—
207C22×C4—172no—
208C22×C4C22×D1711yescharacteristic
209C22×C4C22×D1712yes—
210C22×C4C22×D1711yescharacteristic
211C4⋊C4—682no—
212C4⋊C4—681no—
213C22⋊C4—41no—
214C22⋊C4—42no—
215C22⋊C4—344no—
216C22⋊C4—344no—
217C22⋊C4—344no—
218C22⋊C4—682no—
219C22⋊C4—682no—
220C22⋊C4—682no—
221C22⋊C4—681no—
222C22⋊C4—682no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—174no—
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—48no—
249D4—44no—
250D4—688no—
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—342no—
260C2×C4—342no—
261C2×C4—342no—
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—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—344no—
281C22—344no—
282C22—344no—
283C22—6816no—
284C22—682no—
285C22—682no—
286C22—682no—
287C22—682no—
288C22C2×D4×D1711yescharacteristic
289C22#1058769112yes—
290C22#1058769111yescharacteristic
291C22#1058769112yes—
292C22#1058769611yescharacteristic
293C4—41no—
294C4—41no—
295C4—42no—
296C4—682no—
297C4—682no—
298C4—682no—
299C2—42no—
300C2—82no—
301C2—684no—
302C2#1176336711yescharacteristic
303C2#1176336912yes—
304C2#1176354611yescharacteristic
305C2#1176354712yes—
306C2#1176355211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#392C2
hypercenter#392D17
hypocenterC17#2812
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#10587623C2semidirect
2. #11762738C2semidirect
3. #11762734C2semidirect
4. #11762725C2semidirect
5. #11762724C2semidirect
6. #11762723C2semidirect
7. #11762717C2semidirect
8. #11762664C2semidirect
9. C17×#392C2semidirect
12. C22×D4×C17C22semidirect
16. C2×C22⋊C4×C17C22semidirect
19. C2×C22⋊Dic17C22semidirect
20. C2×C22⋊Dic17C22semidirect
26. C2×C4⋊Dic17C22semidirect
27. C2×#63225C22semidirect
28. C17×#95C22semidirect
29. #10587502C22semidirect
30. #10587502C22semidirect
31. #10587502C22semidirect
38. C23×D17D4semidirect
153. #392D17semidirect
167. C22×D4C2×D17semidirect
180. C2×C22⋊C4C2×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×#105876232
#117627382
#117627342
#117627252
#117627242
#117627232
#117627172
#117626642
C17×#3922
#281217