Finite Group Databasebeta

← #14532134

Subgroups — #14532134

Order 2176.

1232 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 205 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×#10587624C212yesmaximal, complemented
2C2×#10587639C212yesmaximal, complemented
3C2×C4×C22⋊D17C211yesmaximal, characteristic, complemented
4#11762723C212yesmaximal, complemented
5#11762717C212yesmaximal, complemented
6#11762712C212yesmaximal, complemented
7#11762669C211yesmaximal, characteristic, complemented
8#11762665C212yesmaximal, complemented
9C23×D4×C17C211yesmaximal, characteristic, complemented
10C25×C17C2212yes—
11C22×D4×C17—28no—
12C22×D4×C17—24no—
13C22×D4×C17C2212yes—
14C22×D4×C17C2212yes—
15C23×C4×C17C2211yescharacteristic
16C22×C22⋊D17C2211yescharacteristic
17C22×C22⋊D17C2214yescomplemented
18C22×C4×D17C2211yescharacteristic, complemented
19#10587639—28no—
20C2×C22⋊Dic17C2211yescharacteristic
21C2×C22⋊Dic17C2212yes—
22C2×C22⋊Dic17C2214yescomplemented
23C2×C22⋊Dic17C2214yescomplemented
24C2×C22⋊Dic17C2214yescomplemented
25#10587624—28no—
26C4×C22⋊D17—24no—
27C2×#63227C2211yescharacteristic, complemented
28C2×C4⋊Dic17C2212yescomplemented
29C2×#63225C2211yescharacteristic, complemented
30C2×C4×Dic17C2211yescharacteristic, complemented
31#10587502C2212yescomplemented
32C24×C17—28no—
33C24×C17—24no—
34C24×C17—24no—
35C24×C17—24no—
36C24×C17—44no—
37C24×C17C2314yes—
38C24×C17C2311yescharacteristic
39C23×D17C2311yescharacteristic
40C2×D4×C17—232no—
41C2×D4×C17—28no—
42C2×D4×C17—24no—
43C2×D4×C17—24no—
44C2×D4×C17—24no—
45C2×D4×C17D418yescomplemented
46C22×C4×C17—24no—
47C22×C4×C17D414yescomplemented
48C22×C4×C17C2311yescharacteristic
49C22×C4×C17C2311yescharacteristic
50C2×C22⋊D17—22no—
51C2×C22⋊D17—22no—
52C2×C22⋊D17—28no—
53C2×C22⋊D17—28no—
54C2×C22⋊D17—44no—
55C2×C22⋊D17D414yescomplemented
56C22×Dic17C2311yescharacteristic
57C22×Dic17C2314yes—
58C22×Dic17C2311yescharacteristic
59C22×Dic17C2311yescharacteristic
60C2×C4×D17—22no—
61C2×C4×D17—24no—
62C22⋊Dic17—22no—
63C22⋊Dic17—28no—
64C22⋊Dic17—24no—
65C22⋊Dic17—28no—
66C22⋊Dic17—44no—
67#63227—22no—
68C4⋊Dic17—24no—
69#63225—22no—
70C4×Dic17—22no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17—216no—
74C23×C17—24no—
75C23×C17—24no—
76C23×C17—24no—
77C23×C17—24no—
78C23×C17—24no—
79C23×C17—24no—
80C23×C17—416no—
81C23×C17—416no—
82C23×C17—48no—
83C23×C17—48no—
84C23×C17C2×D412yes—
85C23×C17C2×D418yes—
86C23×C17C2×D412yes—
87C23×C17#4112yes—
88C23×C17C2411yescharacteristic
89C22×D17—22no—
90C22×D17—21no—
91C22×D17—21no—
92C22×D17—22no—
93C22×D17C2×D412yescomplemented
94D4×C17—216no—
95D4×C17—216no—
96C2×C4×C17—22no—
97C2×C4×C17—28no—
98C2×C4×C17C2×D412yes—
99C2×C4×C17C2×D414yes—
100C2×C4×C17C2×D412yes—
101C22⋊D17—28no—
102C22⋊D17—416no—
103C2×Dic17—21no—
104C2×Dic17—21no—
105C2×Dic17—22no—
106C2×Dic17—24no—
107C2×Dic17—21no—
108C2×Dic17—24no—
109C2×Dic17—21no—
110C2×Dic17—22no—
111C2×Dic17—24no—
112C2×Dic17C2×D412yescomplemented
113C4×D17—44no—
114#2408—171nomaximal, Sylow
115C22×C17—22no—
116C22×C17—22no—
117C22×C17—24no—
118C22×C17—24no—
119C22×C17—28no—
120C22×C17—28no—
121C22×C17—28no—
122C22×C17—432no—
123C22×C17—44no—
124C22×C17—44no—
125C22×C17—44no—
126C22×C17—44no—
127C22×C17#12114yescomplemented
128C22×C17C22×D411yescharacteristic
129C22×C17C22×D411yescharacteristic
130C22×C17C22×D412yes—
131C22×C17C2×#4111yescharacteristic
132C22×C17H(5,𝔽2)12yes—
133C2×D17—22no—
134C2×D17—24no—
135C2×D17—41no—
136C2×D17—41no—
137C2×D17—42no—
138C4×C17—22no—
139C4×C17#12014yescomplemented
140Dic17—22no—
141Dic17—41no—
142Dic17—41no—
143Dic17—44no—
144C23×D4D1711yescharacteristic, complemented
145C2×#121—172no—
146C2×#120—172no—
147C2×C4×D4—171no—
148#394—172no—
149#392—172no—
150#390—172no—
151#380—171no—
152#379—172no—
153C2×C17—24no—
154C2×C17—48no—
155C2×C17C2×#12011yescharacteristic
156C2×C17C2×#12111yescharacteristic
157C2×C17#53511yescharacteristic
158C2×C17D4212yes—
159C2×C17#54612yes—
160D17—42no—
161C25C2×D1712yes—
162C22×D4—28no—
163C22×D4—24no—
164C22×D4—171no—
165C22×D4—174no—
166C22×D4C2×D1712yes—
167C22×D4C2×D1712yes—
168C23×C4—171no—
169C23×C4C2×D1711yescharacteristic
170#121—348no—
171#120—348no—
172C4×D4—344no—
173C2×C4⋊C4—171no—
174C2×C4⋊C4—172no—
175C2×C22⋊C4—171no—
176C2×C22⋊C4—172no—
177C2×C22⋊C4—174no—
178C2×C22⋊C4—174no—
179C2×C22⋊C4—171no—
180C2×C22⋊C4—174no—
181C2×C42—171no—
182#95—172no—
183C17#240811yescharacteristic, complemented, Sylow
184C24—28no—
185C24—24no—
186C24—24no—
187C24—24no—
188C24—44no—
189C24—171no—
190C24C22×D1714yes—
191C24C22×D1711yescharacteristic
192C2×D4—232no—
193C2×D4—28no—
194C2×D4—24no—
195C2×D4—24no—
196C2×D4—24no—
197C2×D4—174no—
198C2×D4—342no—
199C2×D4—342no—
200C2×D4—348no—
201C2×D4—348no—
202C2×D4—684no—
203C2×D4C22⋊D1718yescomplemented
204C22×C4—24no—
205C22×C4—171no—
206C22×C4—174no—
207C22×C4—171no—
208C22×C4—171no—
209C22×C4—342no—
210C22×C4—344no—
211C22×C4C22⋊D1714yescomplemented
212C22×C4C22×D1711yescharacteristic
213C22×C4C22×D1711yescharacteristic
214C4⋊C4—342no—
215C4⋊C4—344no—
216C22⋊C4—348no—
217C22⋊C4—344no—
218C22⋊C4—342no—
219C22⋊C4—348no—
220C22⋊C4—342no—
221C22⋊C4—684no—
222C42—342no—
223C23—22no—
224C23—22no—
225C23—216no—
226C23—24no—
227C23—24no—
228C23—24no—
229C23—24no—
230C23—24no—
231C23—24no—
232C23—416no—
233C23—416no—
234C23—48no—
235C23—48no—
236C23—172no—
237C23—341no—
238C23—342no—
239C23—341no—
240C23—342no—
241C23D4×D1712yes—
242C23#6325412yes—
243C23C2×C22⋊D1718yes—
244C23C2×C22⋊D1712yes—
245C23C23×D1711yescharacteristic
246D4—216no—
247D4—216no—
248D4—348no—
249D4—6816no—
250C2×C4—22no—
251C2×C4—28no—
252C2×C4—172no—
253C2×C4—341no—
254C2×C4—341no—
255C2×C4—344no—
256C2×C4—342no—
257C2×C4—344no—
258C2×C4—341no—
259C2×C4—342no—
260C2×C4—341no—
261C2×C4—344no—
262C2×C4—684no—
263C2×C4C2×C22⋊D1712yes—
264C2×C4C2×C22⋊D1712yes—
265C2×C4C2×C22⋊D1714yes—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—24no—
270C22—28no—
271C22—28no—
272C22—28no—
273C22—432no—
274C22—44no—
275C22—44no—
276C22—44no—
277C22—44no—
278C22—342no—
279C22—344no—
280C22—681no—
281C22—681no—
282C22—682no—
283C22#1058762414yescomplemented
284C22C2×D4×D1711yescharacteristic
285C22C2×#6325411yescharacteristic
286C22#1058769112yes—
287C22C22×C22⋊D1711yescharacteristic
288C22C22×C22⋊D1712yes—
289C4—22no—
290C4—342no—
291C4—681no—
292C4—681no—
293C4—684no—
294C4#1058763914yescomplemented
295C2—24no—
296C2—48no—
297C2—682no—
298C2C2×#1058763911yescharacteristic
299C2C2×#1058762411yescharacteristic
300C2#1176337611yescharacteristic
301C2D4×C22⋊D1712yes—
302C2#1176345312yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC23×D4×C17C2
hypercenterC23×D4D17
hypocenterC17#2408
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#10587624C2semidirect
2. C2×#10587639C2semidirect
3. C2×C4×C22⋊D17C2semidirect
4. #11762723C2semidirect
5. #11762717C2semidirect
6. #11762712C2semidirect
7. #11762669C2semidirect
8. #11762665C2semidirect
9. C23×D4×C17C2semidirect
17. C22×C22⋊D17C22semidirect
18. C22×C4×D17C22semidirect
22. C2×C22⋊Dic17C22semidirect
23. C2×C22⋊Dic17C22semidirect
24. C2×C22⋊Dic17C22semidirect
27. C2×#63227C22semidirect
28. C2×C4⋊Dic17C22semidirect
29. C2×#63225C22semidirect
30. C2×C4×Dic17C22semidirect
31. #10587502C22semidirect
45. C2×D4×C17D4semidirect
47. C22×C4×C17D4semidirect
55. C2×C22⋊D17D4semidirect
93. C22×D17C2×D4semidirect
112. C2×Dic17C2×D4semidirect
127. C22×C17#121semidirect
139. C4×C17#120semidirect
144. C23×D4D17semidirect
183. C17#2408semidirect
203. C2×D4C22⋊D17semidirect
211. C22×C4C22⋊D17semidirect
283. C22#10587624semidirect
294. C4#10587639semidirect

Maximal subgroups

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

subgroupindex
C2×#105876242
C2×#105876392
C2×C4×C22⋊D172
#117627232
#117627172
#117627122
#117626692
#117626652
C23×D4×C172
#240817