Finite Group Databasebeta

← #14538024

Subgroups — #14538024

Order 2176.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#63252C211yesmaximal, characteristic, complemented
2C2×#10587637C211yesmaximal, characteristic, complemented
3C2×#10587635C211yesmaximal, characteristic, complemented
4#11762969C214yesmaximal, complemented
5#11762952C214yesmaximal, complemented
6#11762817C211yesmaximal, characteristic, complemented
7#11762813C211yesmaximal, characteristic, complemented
8#11762773C211yesmaximal, characteristic, complemented
9C17×#525C211yesmaximal, characteristic, complemented
10C2×#41×C17C2211yescharacteristic
11C23×C4×C17C2211yescharacteristic
12C22×C22⋊D17—21no—
13C2×#63252—22no—
14C2×#63252—24no—
15C2×#63252—24no—
16C2×#63252C2214yes—
17C22×D68C2211yescharacteristic
18C22×C4×D17—21no—
19C22×Dic34C2211yescharacteristic
20C17×#123—21no—
21C17×C22⋊Q8C2212yescomplemented
22C17×#121C2212yescomplemented
23C17×#120—21no—
24C4×D4×C17—22no—
25C17×#117C2211yescharacteristic
26#10587637—24no—
27#10587635—24no—
28C2×#63231C2212yescomplemented
29C2×#63230C2212yescomplemented
30C2×#63229C2212yescomplemented
31C2×#63228C2212yescomplemented
32C2×#63223C2211yescharacteristic
33C2×#63223C2211yescharacteristic
34C22⋊#3622C2214yescomplemented
35#10587481C2214yescomplemented
36#10587480C2214yescomplemented
37C24×C17D411yescharacteristic
38C23×D17—21no—
39#41×C17—22no—
40#41×C17—22no—
41C2×Q8×C17C2311yescharacteristic
42C2×D4×C17—21no—
43C2×D4×C17—22no—
44C2×D4×C17C2311yescharacteristic
45C22×C4×C17—22no—
46C22×C4×C17—22no—
47C22×C4×C17—21no—
48C22×C4×C17D411yescharacteristic
49C22×C4×C17D412yes—
50C22×C4×C17C2312yes—
51C22×C4×C17C2311yescharacteristic
52C2×C22⋊D17—28no—
53C2×C22⋊D17—24no—
54C22×Dic17—21no—
55#63252—216no—
56#63252—28no—
57#63252—28no—
58C2×D68—22no—
59C2×D68—21no—
60C2×D68D414yescomplemented
61C2×D68C2312yes—
62C2×C4×D17—28no—
63C2×C4×D17—24no—
64C2×Dic34—22no—
65C2×Dic34—21no—
66C2×Dic34D414yescomplemented
67C2×Dic34C2312yes—
68C4⋊C4×C17—22no—
69C4⋊C4×C17C2312yes—
70C22⋊C4×C17—22no—
71C22⋊C4×C17—21no—
72C22⋊C4×C17—22no—
73C42×C17—21no—
74#63231—24no—
75#63230—24no—
76#63229—24no—
77#63228—24no—
78#63223—22no—
79#63223—22no—
80C2×#3622C2312yes—
81C2×#3622C2312yes—
82C23×C17—22no—
83C23×C17—21no—
84C23×C17—42no—
85C23×C17C2×D412yes—
86C23×C17C2×D411yescharacteristic
87C22×D17—24no—
88C22×D17—22no—
89C22×D17—44no—
90Q8×C17—21no—
91Q8×C17—21no—
92D4×C17—21no—
93D4×C17—21no—
94D4×C17—44no—
95D4×C17—42no—
96C2×C4×C17—22no—
97C2×C4×C17—24no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—21no—
102C2×C4×C17—42no—
103C2×C4×C17C2×D412yes—
104C2×C4×C17C2×D412yes—
105C2×C4×C17C2×D412yes—
106C2×C4×C17C2×D411yescharacteristic
107C2×C4×C17C2411yescharacteristic
108C22⋊D17—216no—
109C2×Dic17—24no—
110C2×Dic17—22no—
111D68—24no—
112D68—22no—
113D68C2×D414yes—
114C4×D17—216no—
115Dic34—24no—
116Dic34—22no—
117Dic34C2×D414yes—
118#3622—22no—
119#3622—22no—
120#3005—171nomaximal, Sylow
121C22×C17—22no—
122C22×C17—22no—
123C22×C17—22no—
124C22×C17—22no—
125C22×C17—44no—
126C22×C17—41no—
127C22×C17—42no—
128C22×C17#12012yes—
129C22×C17C22×D411yescharacteristic
130C2×D17—24no—
131C2×D17—48no—
132C2×D17—44no—
133C4×C17—22no—
134C4×C17—41no—
135C4×C17—42no—
136C4×C17#12012yes—
137C4×C17C22×D411yescharacteristic
138C4×C17C22×D411yescharacteristic
139Dic17—24no—
140C22×#41—171no—
141C2×#137—171no—
142C2×AGL(1,ℤ/8)—171no—
143#525D1711yescharacteristic, complemented
144#452—174no—
145#449—174no—
146#419—171no—
147#418—171no—
148#407—171no—
149C2×C17—22no—
150C2×C17—42no—
151C2×C17—81no—
152C2×C17C2×#12011yescharacteristic
153C2×C17#57512yes—
154D17—44no—
155C2×#41—174no—
156C2×#41—344no—
157C2×#41—344no—
158C2×#41—342no—
159C2×#41C2×D1711yescharacteristic
160C22×Q8—171no—
161C22×D4—171no—
162C22×D4—341no—
163C23×C4—341no—
164C23×C4C2×D1711yescharacteristic
165#137—344no—
166AGL(1,ℤ/8)—344no—
167C2×Q16—172no—
168C2×SD16—172no—
169C2×SD16—172no—
170C2×D8—172no—
171C2×M16—171no—
172C2×M16—171no—
173#123—21no—
174C22⋊Q8C2×D1712yescomplemented
175#121C2×D1712yescomplemented
176#120—21no—
177C4×D4—22no—
178#117C2×D1711yescharacteristic
179#103—174no—
180#102—174no—
181C22⋊C8—174no—
182C17#300511yescharacteristic, complemented, Sylow
183C24—341no—
184C24C22⋊D1711yescharacteristic
185#41—22no—
186#41—22no—
187#41—348no—
188#41—348no—
189#41—3416no—
190C2×Q8—174no—
191C2×Q8—172no—
192C2×Q8—342no—
193C2×Q8—341no—
194C2×Q8C22×D1711yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—174no—
198C2×D4—172no—
199C2×D4—348no—
200C2×D4—344no—
201C2×D4—342no—
202C2×D4—341no—
203C2×D4C22×D1711yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—348no—
208C22×C4—341no—
209C22×C4—344no—
210C22×C4C22⋊D1712yes—
211C22×C4C22⋊D1711yescharacteristic
212C22×C4C22×D1712yes—
213C22×C4C22×D1711yescharacteristic
214Q16—344no—
215SD16—344no—
216SD16—344no—
217D8—344no—
218M16—342no—
219M16—342no—
220C2×C8—172no—
221C2×C8—172no—
222C4⋊C4—22no—
223C4⋊C4C22×D1712yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—342no—
232C23—344no—
233C23—684no—
234C23C2×C22⋊D1712yes—
235C23C2×C22⋊D1711yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—174no—
239Q8—342no—
240Q8—344no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—174no—
246D4—3416no—
247D4—342no—
248D4—344no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—3416no—
257C2×C4—342no—
258C2×C4—344no—
259C2×C4D4×D1712yes—
260C2×C4D4×D1712yes—
261C2×C4C2×C22⋊D1712yes—
262C2×C4C2×C22⋊D1711yescharacteristic
263C2×C4C23×D1711yescharacteristic
264C8—342no—
265C8—342no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—41no—
272C22—42no—
273C22—344no—
274C22—688no—
275C22—684no—
276C22#1058762312yes—
277C22C22×C22⋊D1711yescharacteristic
278C4—22no—
279C4—41no—
280C4—42no—
281C4—344no—
282C4#1058762312yes—
283C4C2×D4×D1711yescharacteristic
284C4C2×D4×D1711yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—684no—
289C2C2×#1058762311yescharacteristic
290C2#1176360712yes—
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#525C2
socleC22×C17C22×D4
hypercenter#525D17
hypocenterC17#3005
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#63252C2semidirect
2. C2×#10587637C2semidirect
3. C2×#10587635C2semidirect
4. #11762969C2semidirect
5. #11762952C2semidirect
6. #11762817C2semidirect
7. #11762813C2semidirect
8. #11762773C2semidirect
9. C17×#525C2semidirect
21. C17×C22⋊Q8C22semidirect
22. C17×#121C22semidirect
28. C2×#63231C22semidirect
29. C2×#63230C22semidirect
30. C2×#63229C22semidirect
31. C2×#63228C22semidirect
34. C22⋊#3622C22semidirect
35. #10587481C22semidirect
36. #10587480C22semidirect
60. C2×D68D4semidirect
66. C2×Dic34D4semidirect
143. #525D17semidirect
174. C22⋊Q8C2×D17semidirect
175. #121C2×D17semidirect
182. C17#3005semidirect

Maximal subgroups

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

subgroupindex
C22×#632522
C2×#105876372
C2×#105876352
#117629692
#117629522
#117628172
#117628132
#117627732
C17×#5252
#300517