Finite Group Databasebeta

← #14533061

Subgroups — #14533061

Order 2176.

700 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 141 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×C22⋊C4×D17C212yesmaximal
2#11762737C212yesmaximal, complemented
3#11762734C211yesmaximal, characteristic, complemented
4#11762683C212yesmaximal, complemented
5#11762680C212yesmaximal, complemented
6#11762671C212yesmaximal, complemented
7#11762671C212yesmaximal, complemented
8#11762664C211yesmaximal, characteristic
9C17×#397C211yesmaximal, characteristic, complemented
10C24×D17C2211yescharacteristic
11C22×C4×D17C2212yes—
12C22×C4×D17C2212yes—
13C2×C4⋊C4×C17C2211yescharacteristic, complemented
14C2×C22⋊C4×C17C2212yes—
15C2×C22⋊C4×C17C2211yescharacteristic
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2211yescharacteristic
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×C4⋊Dic17C2211yescharacteristic, complemented
23C2×#63225C2212yescomplemented
24C22⋊C4×D17—28no—
25C17×#95C2211yescharacteristic, complemented
26C17×#95C2212yescomplemented
27#10587502C2211yescharacteristic, complemented
28#10587502C2212yescomplemented
29#10587502C2212yescomplemented
30#10587502C2212yescomplemented
31#10587502C2212yescomplemented
32C24×C17C2311yescharacteristic
33C23×D17—24no—
34C23×D17—24no—
35C23×D17—41no—
36C23×D17—41no—
37C23×D17D414yes—
38C23×D17C2312yes—
39C22×C4×C17C2312yes—
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C2×C4×D17—24no—
47C2×C4×D17—24no—
48C2×C4×D17—24no—
49C2×C4×D17—24no—
50C2×C4×D17—42no—
51C2×C4×D17—42no—
52C4⋊C4×C17—41no—
53C22⋊C4×C17—24no—
54C22⋊C4×C17—41no—
55C22⋊Dic17—24no—
56C22⋊Dic17—41no—
57#63227—24no—
58#63227—24no—
59#63227—24no—
60#63227—24no—
61C4⋊Dic17—41no—
62#63225—42no—
63C23×C17—22no—
64C23×C17—22no—
65C23×C17—21no—
66C23×C17—21no—
67C23×C17C2×D412yes—
68C23×C17C2411yescharacteristic
69C22×D17—22no—
70C22×D17—22no—
71C22×D17—22no—
72C22×D17—22no—
73C22×D17—216no—
74C22×D17—44no—
75C22×D17—48no—
76C22×D17—44no—
77C22×D17—44no—
78C22×D17C2×D414yes—
79C22×D17#4114yes—
80C22×D17#4114yes—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—21no—
88C2×C4×C17—21no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C4×D17—48no—
101C4×D17—48no—
102#2483—171nomaximal, Sylow
103C22×C17—24no—
104C22×C17—22no—
105C22×C17—42no—
106C22×C17—41no—
107C22×C17—41no—
108C22×C17C22×D411yescharacteristic
109C22×C17C2×#4112yes—
110C22×C17C2×#4112yes—
111C22×C17H(5,𝔽2)11yescharacteristic
112C22×C17H(5,𝔽2)11yescharacteristic
113C2×D17—24no—
114C2×D17—24no—
115C2×D17—24no—
116C2×D17—24no—
117C2×D17—24no—
118C2×D17—24no—
119C2×D17—416no—
120C2×D17#12318yes—
121C4×C17—42no—
122C4×C17—42no—
123C4×C17—41no—
124C4×C17—41no—
125Dic17—42no—
126Dic17—42no—
127Dic17—42no—
128C22×C22⋊C4—172no—
129#397—171no—
130#397—172no—
131#397D1711yescharacteristic, complemented
132#385—172no—
133#385—172no—
134#381—172no—
135#381—172no—
136#379—171no—
137C2×C17—42no—
138C2×C17C2×#12312yes—
139C2×C17#53411yescharacteristic
140C2×C17#55112yes—
141C2×C17#55111yescharacteristic
142C2×C17#55111yescharacteristic
143D17—28no—
144C25—171no—
145C23×C4—172no—
146C23×C4—172no—
147C2×C4⋊C4—172no—
148C2×C4⋊C4—171no—
149C2×C4⋊C4C2×D1711yescharacteristic, complemented
150C2×C22⋊C4—172no—
151C2×C22⋊C4—172no—
152C2×C22⋊C4—172no—
153C2×C22⋊C4—172no—
154C2×C22⋊C4—172no—
155C2×C22⋊C4—171no—
156C2×C22⋊C4—348no—
157C2×C22⋊C4C2×D1712yes—
158C2×C22⋊C4C2×D1711yescharacteristic
159#95—171no—
160#95—172no—
161#95—172no—
162#95—172no—
163#95—172no—
164#95C2×D1711yescharacteristic, complemented
165#95C2×D1712yescomplemented
166C17#248311yescharacteristic, complemented, Sylow
167C24—172no—
168C24—174no—
169C24—344no—
170C24—344no—
171C24—681no—
172C24—681no—
173C24C22×D1711yescharacteristic
174C22×C4—172no—
175C22×C4—172no—
176C22×C4—172no—
177C22×C4—344no—
178C22×C4—344no—
179C22×C4—344no—
180C22×C4—344no—
181C22×C4—682no—
182C22×C4—682no—
183C22×C4C22×D1712yes—
184C22×C4C22×D1711yescharacteristic
185C22×C4C22×D1711yescharacteristic
186C22×C4C22×D1712yes—
187C4⋊C4—41no—
188C4⋊C4—682no—
189C4⋊C4—681no—
190C22⋊C4—24no—
191C22⋊C4—41no—
192C22⋊C4—344no—
193C22⋊C4—344no—
194C22⋊C4—344no—
195C22⋊C4—344no—
196C22⋊C4—344no—
197C22⋊C4—681no—
198C23—21no—
199C23—22no—
200C23—22no—
201C23—21no—
202C23—174no—
203C23—174no—
204C23—174no—
205C23—3416no—
206C23—342no—
207C23—342no—
208C23—342no—
209C23—342no—
210C23—684no—
211C23—688no—
212C23—684no—
213C23—684no—
214C23D4×D1712yes—
215C23C23×D1711yescharacteristic
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—342no—
227C2×C4—342no—
228C2×C4—342no—
229C2×C4—342no—
230C2×C4—342no—
231C2×C4—342no—
232C2×C4—342no—
233C2×C4—342no—
234C2×C4—342no—
235C2×C4—688no—
236C2×C4—688no—
237C22—24no—
238C22—22no—
239C22—42no—
240C22—41no—
241C22—41no—
242C22—178no—
243C22—344no—
244C22—344no—
245C22—344no—
246C22—344no—
247C22—344no—
248C22—344no—
249C22—6816no—
250C22C2×D4×D1711yescharacteristic
251C22#1058769111yescharacteristic
252C22#41×D1712yes—
253C22#41×D1712yes—
254C22#1058769611yescharacteristic
255C4—41no—
256C4—41no—
257C4—42no—
258C4—42no—
259C4—682no—
260C4—682no—
261C4—682no—
262C2—42no—
263C2—348no—
264C2#123×D1712yes—
265C2#1176336711yescharacteristic
266C2#1176348412yes—
267C2#1176348711yescharacteristic
268C2#1176348911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#397C2
prime radicalC24×D17C22
hypercenter#397D17
hypocenterC17#2483
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #11762737C2semidirect
3. #11762734C2semidirect
4. #11762683C2semidirect
5. #11762680C2semidirect
6. #11762671C2semidirect
7. #11762671C2semidirect
9. C17×#397C2semidirect
13. C2×C4⋊C4×C17C22semidirect
22. C2×C4⋊Dic17C22semidirect
23. C2×#63225C22semidirect
25. C17×#95C22semidirect
26. C17×#95C22semidirect
27. #10587502C22semidirect
28. #10587502C22semidirect
29. #10587502C22semidirect
30. #10587502C22semidirect
31. #10587502C22semidirect
131. #397D17semidirect
149. C2×C4⋊C4C2×D17semidirect
164. #95C2×D17semidirect
165. #95C2×D17semidirect
166. C17#2483semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D172
#117627372
#117627342
#117626832
#117626802
#117626712
#117626642
C17×#3972
#248317