Finite Group Databasebeta

← #14532111

Subgroups — #14532111

Order 2176.

1116 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 181 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×C22⋊D17C211yesmaximal, characteristic
2C2×#10587615C212yesmaximal, complemented
3#11762724C214yesmaximal, complemented
4#11762688C212yesmaximal, complemented
5#11762681C212yesmaximal, complemented
6#11762665C211yesmaximal, characteristic, complemented
7#11762664C212yesmaximal, complemented
8C22×C22⋊C4×C17C211yesmaximal, characteristic, complemented
9C25×C17C2211yescharacteristic
10C24×D17C2211yescharacteristic
11C23×C4×C17C2211yescharacteristic, complemented
12C23×C4×C17C2211yescharacteristic, complemented
13C22×C22⋊D17—24no—
14C22×C22⋊D17—28no—
15C22×C22⋊D17C2212yes—
16C22×C22⋊D17C2212yes—
17C23×Dic17C2211yescharacteristic
18C2×C22⋊C4×C17—24no—
19C2×C22⋊C4×C17C2212yes—
20C2×C22⋊C4×C17C2212yescomplemented
21C2×C22⋊Dic17C2212yes—
22C2×C22⋊Dic17C2212yescomplemented
23C2×C22⋊Dic17C2212yescomplemented
24#10587615—28no—
25C2×#63227C2214yescomplemented
26C2×#63227C2214yescomplemented
27C2×#63225C2214yescomplemented
28#10587502C2214yescomplemented
29C24×C17—24no—
30C24×C17—24no—
31C24×C17—42no—
32C24×C17D414yes—
33C24×C17C2311yescharacteristic
34C24×C17C2311yescharacteristic
35C24×C17C2311yescharacteristic
36C23×D17—22no—
37C23×D17—24no—
38C23×D17—24no—
39C23×D17—42no—
40C23×D17C2312yes—
41C22×C4×C17—24no—
42C22×C4×C17—22no—
43C22×C4×C17—22no—
44C22×C4×C17—42no—
45C22×C4×C17C2312yes—
46C22×C4×C17C2312yes—
47C2×C22⋊D17—232no—
48C2×C22⋊D17—24no—
49C2×C22⋊D17—24no—
50C2×C22⋊D17—24no—
51C2×C22⋊D17—24no—
52C2×C22⋊D17—24no—
53C2×C22⋊D17D418yes—
54C22×Dic17—22no—
55C22×Dic17—24no—
56C22×Dic17C2314yes—
57C22×Dic17C2312yes—
58C22⋊C4×C17—24no—
59C22⋊C4×C17—24no—
60C22⋊Dic17—24no—
61C22⋊Dic17—24no—
62C22⋊Dic17—42no—
63#63227—28no—
64#63227—44no—
65#63225—28no—
66C23×C17—22no—
67C23×C17—22no—
68C23×C17—22no—
69C23×C17—22no—
70C23×C17—28no—
71C23×C17—28no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—44no—
76C23×C17—416no—
77C23×C17C2×D414yes—
78C23×C17C2×D412yes—
79C23×C17C2×D412yes—
80C23×C17C2×D412yes—
81C23×C17#4114yes—
82C23×C17C2411yescharacteristic
83C22×D17—28no—
84C22×D17—22no—
85C22×D17—22no—
86C22×D17—24no—
87C22×D17—24no—
88C22×D17—416no—
89C22×D17—44no—
90C22×D17—44no—
91C22×D17C2×D414yes—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17—24no—
95C2×C4×C17—24no—
96C2×C4×C17—44no—
97C2×C4×C17—44no—
98C22⋊D17—216no—
99C22⋊D17—216no—
100C2×Dic17—28no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—24no—
104C2×Dic17—24no—
105C2×Dic17—24no—
106C2×Dic17C2×D414yes—
107#2405—171nomaximal, Sylow
108C22×C17—22no—
109C22×C17—22no—
110C22×C17—22no—
111C22×C17—22no—
112C22×C17—28no—
113C22×C17—28no—
114C22×C17—22no—
115C22×C17—22no—
116C22×C17—22no—
117C22×C17—416no—
118C22×C17—44no—
119C22×C17#12014yes—
120C22×C17#12314yescomplemented
121C22×C17C22×D412yes—
122C22×C17C22×D411yescharacteristic
123C22×C17C2×#4112yes—
124C22×C17H(5,𝔽2)12yes—
125C2×D17—24no—
126C2×D17—28no—
127C2×D17—416no—
128C2×D17—42no—
129C2×D17—42no—
130C2×D17—44no—
131C4×C17—42no—
132C4×C17—42no—
133Dic17—24no—
134Dic17—44no—
135C23×D4—171no—
136C2×#123—172no—
137C22×C22⋊C4D1711yescharacteristic, complemented
138#394—174no—
139#386—172no—
140#385—172no—
141#379—171no—
142#379—172no—
143C2×C17—24no—
144C2×C17—24no—
145C2×C17—42no—
146C2×C17C2×#12011yescharacteristic
147C2×C17C2×#12311yescharacteristic
148C2×C17#53411yescharacteristic
149C2×C17#54614yes—
150D17—44no—
151C25—171no—
152C25C2×D1711yescharacteristic
153C22×D4—172no—
154C22×D4—172no—
155C22×D4—348no—
156C22×D4—344no—
157C23×C4—171no—
158C23×C4C2×D1711yescharacteristic, complemented
159C23×C4C2×D1711yescharacteristic, complemented
160#123—348no—
161C2×C4⋊C4—174no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—172no—
164C2×C22⋊C4—172no—
165C2×C22⋊C4—174no—
166C2×C22⋊C4—172no—
167C2×C22⋊C4—174no—
168C2×C22⋊C4C2×D1712yes—
169C2×C22⋊C4C2×D1712yescomplemented
170#95—174no—
171C17#240511yescharacteristic, complemented, Sylow
172C24—24no—
173C24—24no—
174C24—42no—
175C24—172no—
176C24—344no—
177C24—342no—
178C24—344no—
179C24—682no—
180C24C22⋊D1714yes—
181C24C22×D1711yescharacteristic
182C24C22×D1711yescharacteristic
183C24C22×D1711yescharacteristic
184C2×D4—178no—
185C2×D4—344no—
186C2×D4—344no—
187C2×D4—3432no—
188C2×D4—344no—
189C2×D4—344no—
190C2×D4—344no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—172no—
196C22×C4—174no—
197C22×C4—344no—
198C22×C4—342no—
199C22×C4C22×D1712yes—
200C22×C4C22×D1712yes—
201C4⋊C4—348no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—348no—
205C22⋊C4—344no—
206C22⋊C4—344no—
207C22⋊C4—684no—
208C22⋊C4—682no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—28no—
217C23—28no—
218C23—44no—
219C23—416no—
220C23—174no—
221C23—348no—
222C23—344no—
223C23—342no—
224C23—344no—
225C23—342no—
226C23—6816no—
227C23—684no—
228C23—684no—
229C23#6325214yes—
230C23D4×D1714yes—
231C23C2×C22⋊D1712yes—
232C23C2×C22⋊D1712yes—
233C23C2×C22⋊D1712yes—
234C23C23×D1711yescharacteristic
235D4—3416no—
236D4—3416no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—174no—
244C2×C4—348no—
245C2×C4—342no—
246C2×C4—342no—
247C2×C4—344no—
248C2×C4—344no—
249C2×C4—344no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—28no—
255C22—28no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—416no—
260C22—44no—
261C22—344no—
262C22—348no—
263C22—6816no—
264C22—682no—
265C22—682no—
266C22—684no—
267C22#1058761514yescomplemented
268C22#1058762314yes—
269C22C2×#6325212yes—
270C22C2×D4×D1712yes—
271C22#1058769112yes—
272C22C22×C22⋊D1711yescharacteristic
273C4—42no—
274C4—42no—
275C4—344no—
276C4—684no—
277C2—24no—
278C2—24no—
279C2—42no—
280C2—684no—
281C2C2×#1058762311yescharacteristic
282C2C2×#1058761511yescharacteristic
283C2#1176336811yescharacteristic
284C2#1176344914yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC22×C22⋊C4×C17C2
prime radicalC23×C22⋊D17C2
hypercenterC22×C22⋊C4D17
hypocenterC17#2405
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#10587615C2semidirect
3. #11762724C2semidirect
4. #11762688C2semidirect
5. #11762681C2semidirect
6. #11762665C2semidirect
7. #11762664C2semidirect
8. C22×C22⋊C4×C17C2semidirect
11. C23×C4×C17C22semidirect
12. C23×C4×C17C22semidirect
20. C2×C22⋊C4×C17C22semidirect
22. C2×C22⋊Dic17C22semidirect
23. C2×C22⋊Dic17C22semidirect
25. C2×#63227C22semidirect
26. C2×#63227C22semidirect
27. C2×#63225C22semidirect
28. #10587502C22semidirect
120. C22×C17#123semidirect
137. C22×C22⋊C4D17semidirect
158. C23×C4C2×D17semidirect
159. C23×C4C2×D17semidirect
169. C2×C22⋊C4C2×D17semidirect
171. C17#2405semidirect
267. C22#10587615semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D172
C2×#105876152
#117627242
#117626882
#117626812
#117626652
#117626642
C22×C22⋊C4×C172
#240517