Finite Group Databasebeta

← #14532141

Subgroups — #14532141

Order 2176.

1232 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 189 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
1C23×D68C211yesmaximal, characteristic, complemented
2C2×#10587581C212yesmaximal, complemented
3C2×#10587568C212yesmaximal, complemented
4C2×C4×C22⋊D17C211yesmaximal, characteristic, complemented
5#11762727C212yesmaximal, complemented
6#11762719C212yesmaximal, complemented
7#11762710C212yesmaximal, complemented
8#11762664C212yesmaximal, complemented
9C17×#380C211yesmaximal, characteristic, complemented
10C24×D17C2212yes—
11C23×C4×C17C2211yescharacteristic
12C22×C22⋊D17C2211yescharacteristic
13C22×D68—24no—
14C22×D68—28no—
15C22×D68C2212yes—
16C22×D68C2212yes—
17C22×D68C2214yescomplemented
18C22×C4×D17C2211yescharacteristic, complemented
19C2×C4⋊C4×C17C2212yescomplemented
20C2×C22⋊C4×C17C2212yes—
21C2×C22⋊Dic17C2211yescharacteristic
22C4×C22⋊D17—24no—
23C2×#63227C2214yescomplemented
24C2×#63227C2211yescharacteristic, complemented
25C2×#63227C2214yescomplemented
26C2×#63227C2214yescomplemented
27C2×#63225C2211yescharacteristic, complemented
28C2×C4×Dic17C2211yescharacteristic, complemented
29#10587581—28no—
30#10587568—28no—
31C17×#95C2212yescomplemented
32C24×C17C2311yescharacteristic
33C23×D17—24no—
34C23×D17—28no—
35C23×D17—24no—
36C23×D17—24no—
37C23×D17—44no—
38C23×D17C2314yes—
39C23×D17C2311yescharacteristic
40C22×C4×C17—24no—
41C22×C4×C17D414yescomplemented
42C22×C4×C17C2314yes—
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C2×C22⋊D17—22no—
46C2×C22⋊D17—22no—
47C2×C22⋊D17D414yescomplemented
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2311yescharacteristic
50C22×Dic17C2311yescharacteristic
51C2×D68—232no—
52C2×D68—28no—
53C2×D68—24no—
54C2×D68—24no—
55C2×D68—24no—
56C2×D68—28no—
57C2×D68—28no—
58C2×D68—44no—
59C2×D68D418yescomplemented
60C2×C4×D17—24no—
61C2×C4×D17—22no—
62C4⋊C4×C17—24no—
63C22⋊C4×C17—24no—
64C22⋊Dic17—22no—
65#63227—28no—
66#63227—22no—
67#63227—28no—
68#63227—44no—
69#63225—22no—
70C4×Dic17—22no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17C2×D412yes—
74C23×C17C2×D412yes—
75C23×C17#4112yes—
76C23×C17C2411yescharacteristic
77C22×D17—216no—
78C22×D17—22no—
79C22×D17—21no—
80C22×D17—24no—
81C22×D17—24no—
82C22×D17—24no—
83C22×D17—21no—
84C22×D17—24no—
85C22×D17—22no—
86C22×D17—24no—
87C22×D17—24no—
88C22×D17—416no—
89C22×D17—416no—
90C22×D17—48no—
91C22×D17—48no—
92C22×D17C2×D418yes—
93C22×D17C2×D412yescomplemented
94C2×C4×C17—22no—
95C2×C4×C17—28no—
96C2×C4×C17—24no—
97C2×C4×C17—24no—
98C2×C4×C17—24no—
99C2×C4×C17C2×D412yes—
100C2×C4×C17C2×D412yes—
101C2×C4×C17C2×D414yes—
102C22⋊D17—28no—
103C2×Dic17—21no—
104C2×Dic17—21no—
105C2×Dic17—21no—
106C2×Dic17—21no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17C2×D412yescomplemented
110D68—216no—
111D68—216no—
112D68—416no—
113C4×D17—44no—
114#2408—171nomaximal, Sylow
115C22×C17—22no—
116C22×C17—22no—
117C22×C17—24no—
118C22×C17—24no—
119C22×C17#12114yescomplemented
120C22×C17C22×D411yescharacteristic
121C22×C17C22×D412yes—
122C22×C17C22×D411yescharacteristic
123C22×C17C2×#4111yescharacteristic
124C22×C17H(5,𝔽2)12yes—
125C2×D17—28no—
126C2×D17—22no—
127C2×D17—24no—
128C2×D17—28no—
129C2×D17—28no—
130C2×D17—432no—
131C2×D17—41no—
132C2×D17—44no—
133C2×D17—41no—
134C2×D17—44no—
135C2×D17—44no—
136C2×D17—42no—
137C2×D17—44no—
138C4×C17—22no—
139C4×C17—44no—
140C4×C17#12014yescomplemented
141Dic17—22no—
142Dic17—41no—
143Dic17—41no—
144C23×D4—171no—
145C2×#121—172no—
146C2×#120—172no—
147C2×C4×D4—171no—
148#394—172no—
149#392—172no—
150#390—172no—
151#380D1711yescharacteristic, complemented
152#379—172no—
153C2×C17—24no—
154C2×C17C2×#12011yescharacteristic
155C2×C17C2×#12111yescharacteristic
156C2×C17#53511yescharacteristic
157C2×C17D4212yes—
158C2×C17#54612yes—
159D17—42no—
160D17—48no—
161C25—172no—
162C22×D4—171no—
163C22×D4—172no—
164C22×D4—172no—
165C22×D4—174no—
166C22×D4—348no—
167C22×D4—344no—
168C23×C4—171no—
169C23×C4C2×D1711yescharacteristic
170#121—348no—
171#120—348no—
172C4×D4—344no—
173C2×C4⋊C4—171no—
174C2×C4⋊C4C2×D1712yescomplemented
175C2×C22⋊C4—171no—
176C2×C22⋊C4—171no—
177C2×C22⋊C4—174no—
178C2×C22⋊C4—174no—
179C2×C22⋊C4—174no—
180C2×C22⋊C4C2×D1712yes—
181C2×C42—171no—
182#95C2×D1712yescomplemented
183C17#240811yescharacteristic, complemented, Sylow
184C24—174no—
185C24—171no—
186C24—348no—
187C24—344no—
188C24—344no—
189C24—344no—
190C24—684no—
191C24C22×D1711yescharacteristic
192C2×D4—174no—
193C2×D4—178no—
194C2×D4—348no—
195C2×D4—3432no—
196C2×D4—348no—
197C2×D4—348no—
198C2×D4—342no—
199C2×D4—342no—
200C2×D4—344no—
201C2×D4—344no—
202C2×D4—344no—
203C2×D4—684no—
204C22×C4—24no—
205C22×C4—171no—
206C22×C4—171no—
207C22×C4—171no—
208C22×C4—342no—
209C22×C4—344no—
210C22×C4D6814yescomplemented
211C22×C4C22×D1714yes—
212C22×C4C22×D1711yescharacteristic
213C22×C4C22×D1711yescharacteristic
214C4⋊C4—24no—
215C4⋊C4—342no—
216C22⋊C4—24no—
217C22⋊C4—348no—
218C22⋊C4—348no—
219C22⋊C4—342no—
220C22⋊C4—342no—
221C22⋊C4—684no—
222C42—342no—
223C23—22no—
224C23—22no—
225C23—178no—
226C23—172no—
227C23—3416no—
228C23—341no—
229C23—344no—
230C23—344no—
231C23—342no—
232C23—344no—
233C23—341no—
234C23—344no—
235C23—344no—
236C23—342no—
237C23—344no—
238C23—6816no—
239C23—6816no—
240C23—688no—
241C23—688no—
242C23C2×D6812yes—
243C23D4×D1712yes—
244C23#6325612yes—
245C23C23×D1711yescharacteristic
246D4—3416no—
247D4—348no—
248D4—3416no—
249D4—6816no—
250C2×C4—22no—
251C2×C4—28no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—172no—
256C2×C4—341no—
257C2×C4—341no—
258C2×C4—341no—
259C2×C4—341no—
260C2×C4—342no—
261C2×C4—342no—
262C2×C4—684no—
263C2×C4C2×D6812yes—
264C2×C4C2×D6812yes—
265C2×C4D4×D1714yes—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—24no—
270C22—348no—
271C22—342no—
272C22—344no—
273C22—348no—
274C22—348no—
275C22—6832no—
276C22—681no—
277C22—684no—
278C22—684no—
279C22—681no—
280C22—684no—
281C22—684no—
282C22—682no—
283C22#1058758114yescomplemented
284C22C22×D6811yescharacteristic
285C22C2×D4×D1712yes—
286C22C2×D4×D1711yescharacteristic
287C22C2×#6325611yescharacteristic
288C22#1058769612yes—
289C4—22no—
290C4—44no—
291C4—342no—
292C4—681no—
293C4—681no—
294C4#1058756814yescomplemented
295C2—24no—
296C2—688no—
297C2—682no—
298C2C2×#1058756811yescharacteristic
299C2C2×#1058758111yescharacteristic
300C2#1176337811yescharacteristic
301C2#1176344312yes—
302C2#1176345012yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#380C2
hypercenter#380D17
hypocenterC17#2408
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C23×D68C2semidirect
2. C2×#10587581C2semidirect
3. C2×#10587568C2semidirect
4. C2×C4×C22⋊D17C2semidirect
5. #11762727C2semidirect
6. #11762719C2semidirect
7. #11762710C2semidirect
8. #11762664C2semidirect
9. C17×#380C2semidirect
17. C22×D68C22semidirect
18. C22×C4×D17C22semidirect
19. C2×C4⋊C4×C17C22semidirect
23. C2×#63227C22semidirect
24. C2×#63227C22semidirect
25. C2×#63227C22semidirect
26. C2×#63227C22semidirect
27. C2×#63225C22semidirect
28. C2×C4×Dic17C22semidirect
31. C17×#95C22semidirect
41. C22×C4×C17D4semidirect
47. C2×C22⋊D17D4semidirect
59. C2×D68D4semidirect
93. C22×D17C2×D4semidirect
109. C2×Dic17C2×D4semidirect
119. C22×C17#121semidirect
140. C4×C17#120semidirect
151. #380D17semidirect
174. C2×C4⋊C4C2×D17semidirect
182. #95C2×D17semidirect
183. C17#2408semidirect
210. C22×C4D68semidirect
283. C22#10587581semidirect
294. C4#10587568semidirect

Maximal subgroups

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

subgroupindex
C23×D682
C2×#105875812
C2×#105875682
C2×C4×C22⋊D172
#117627272
#117627192
#117627102
#117626642
C17×#3802
#240817