Finite Group Databasebeta

← #14533716

Subgroups — #14533716

Order 2176.

900 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 169 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×#10587621C212yesmaximal, complemented
2C2×#10587623C212yesmaximal, complemented
3C22×#63227C211yesmaximal, characteristic, complemented
4#11762744C212yesmaximal, complemented
5#11762717C212yesmaximal, complemented
6#11762709C214yesmaximal, complemented
7C4×C22⋊Dic17C211yesmaximal, characteristic, complemented
8C2×C17×#121C211yesmaximal, characteristic, complemented
9C24×D17C2211yescharacteristic
10C22×D4×C17C2212yes—
11C22×D4×C17C2211yescharacteristic, complemented
12C23×C4×C17C2211yescharacteristic
13C22×C22⋊D17C2214yes—
14C23×Dic17C2211yescharacteristic
15C17×#121—24no—
16C2×C4⋊C4×C17C2211yescharacteristic, complemented
17C2×C22⋊C4×C17C2212yescomplemented
18C2×C22⋊Dic17C2212yes—
19C2×C22⋊Dic17C2214yescomplemented
20#10587623—28no—
21#10587621—28no—
22C2×#63227—24no—
23C2×#63227C2212yes—
24C2×#63227C2214yes—
25C2×#63227C2212yes—
26C2×#63225C2214yescomplemented
27C2×C4×Dic17C2212yescomplemented
28#10587502C2212yescomplemented
29C24×C17C2312yes—
30C24×C17C2311yescharacteristic
31C23×D17—22no—
32C23×D17—24no—
33C23×D17—24no—
34C23×D17—42no—
35C23×D17C2312yes—
36C2×D4×C17—24no—
37C2×D4×C17—24no—
38C2×D4×C17—24no—
39C2×D4×C17—22no—
40C2×D4×C17—42no—
41C22×C4×C17—42no—
42C22×C4×C17D414yescomplemented
43C22×C4×C17C2312yes—
44C22×C4×C17C2311yescharacteristic
45C22×C4×C17C2311yescharacteristic
46C2×C22⋊D17—28no—
47C2×C22⋊D17—28no—
48C2×C22⋊D17—44no—
49C22×Dic17—22no—
50C22×Dic17—24no—
51C22×Dic17C2312yes—
52C22×Dic17C2314yes—
53C4⋊C4×C17—22no—
54C22⋊C4×C17—24no—
55C22⋊Dic17—28no—
56C22⋊Dic17D418yescomplemented
57#63227—24no—
58#63227—24no—
59#63227—28no—
60#63225—28no—
61C4×Dic17—42no—
62C23×C17—22no—
63C23×C17—22no—
64C23×C17—22no—
65C23×C17—22no—
66C23×C17—22no—
67C23×C17—24no—
68C23×C17—24no—
69C23×C17C2×D414yes—
70C23×C17C2×D412yes—
71C23×C17C2411yescharacteristic
72C22×D17—28no—
73C22×D17—24no—
74C22×D17—22no—
75C22×D17—22no—
76C22×D17—24no—
77C22×D17—416no—
78C22×D17—44no—
79C22×D17—44no—
80C22×D17#4114yes—
81D4×C17—44no—
82D4×C17—48no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—44no—
89C2×C4×C17C2×D412yes—
90C2×C4×C17C2×D412yes—
91C22⋊D17—416no—
92C2×Dic17—24no—
93C2×Dic17—24no—
94C2×Dic17—24no—
95C2×Dic17—22no—
96C2×Dic17—44no—
97C2×Dic17C2×D418yes—
98#2539—171nomaximal, Sylow
99C22×C17—22no—
100C22×C17—22no—
101C22×C17—28no—
102C22×C17—42no—
103C22×C17—42no—
104C22×C17—42no—
105C22×C17—44no—
106C22×C17—44no—
107C22×C17#12714yes—
108C22×C17C22×D412yes—
109C22×C17C22×D411yescharacteristic
110C22×C17C2×#4112yes—
111C22×C17H(5,𝔽2)12yes—
112C2×D17—24no—
113C2×D17—28no—
114C2×D17—416no—
115C2×D17—42no—
116C2×D17—42no—
117C2×D17—44no—
118C4×C17—22no—
119C4×C17—22no—
120C4×C17—42no—
121Dic17—28no—
122Dic17—42no—
123C2×#123—172no—
124C2×#121D1711yescharacteristic, complemented
125C2×#120—172no—
126C22×C22⋊C4—171no—
127#399—172no—
128#392—172no—
129#390—174no—
130C4×C22⋊C4—171no—
131C2×C17—24no—
132C2×C17—82no—
133C2×C17#52511yescharacteristic
134C2×C17C2×#12711yescharacteristic
135C2×C17#53411yescharacteristic
136C2×C17#54614yes—
137D17—44no—
138C25—171no—
139C22×D4—174no—
140C22×D4C2×D1712yes—
141C22×D4C2×D1711yescharacteristic, complemented
142C23×C4—171no—
143C23×C4C2×D1711yescharacteristic
144#123—348no—
145#121—24no—
146#120—348no—
147C2×C4⋊C4—174no—
148C2×C4⋊C4C2×D1711yescharacteristic, complemented
149C2×C22⋊C4—172no—
150C2×C22⋊C4—174no—
151C2×C22⋊C4—172no—
152C2×C22⋊C4—172no—
153C2×C22⋊C4—174no—
154C2×C22⋊C4—344no—
155C2×C22⋊C4C2×D1712yescomplemented
156C2×C42—172no—
157#95—172no—
158C17#253911yescharacteristic, complemented, Sylow
159C24—172no—
160C24—344no—
161C24—342no—
162C24—344no—
163C24—682no—
164C24C22×D1712yes—
165C24C22×D1711yescharacteristic
166C2×D4—24no—
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—22no—
170C2×D4—42no—
171C2×D4—348no—
172C2×D4—348no—
173C2×D4—684no—
174C22×C4—42no—
175C22×C4—174no—
176C22×C4—172no—
177C22×C4—344no—
178C22×C4—342no—
179C22×C4C22⋊D1714yescomplemented
180C22×C4C22×D1712yes—
181C22×C4C22×D1711yescharacteristic
182C22×C4C22×D1711yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—348no—
185C22⋊C4—24no—
186C22⋊C4—178no—
187C22⋊C4—348no—
188C22⋊C4—348no—
189C22⋊C4—344no—
190C22⋊C4—344no—
191C42—682no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—24no—
197C23—24no—
198C23—22no—
199C23—174no—
200C23—348no—
201C23—344no—
202C23—344no—
203C23—342no—
204C23—342no—
205C23—6816no—
206C23—684no—
207C23—684no—
208C23D4×D1714yes—
209C23C2×C22⋊D1712yes—
210C23C23×D1711yescharacteristic
211D4—44no—
212D4—48no—
213D4—6816no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—44no—
220C2×C4—178no—
221C2×C4—344no—
222C2×C4—344no—
223C2×C4—344no—
224C2×C4—342no—
225C2×C4—684no—
226C2×C4C2×C22⋊D1712yes—
227C2×C4C2×C22⋊D1712yes—
228C22—22no—
229C22—22no—
230C22—28no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—44no—
235C22—44no—
236C22—344no—
237C22—348no—
238C22—6816no—
239C22—682no—
240C22—682no—
241C22—684no—
242C22#1058762614yes—
243C22C2×D4×D1712yes—
244C22#1058769112yes—
245C22#41×D1712yes—
246C22C22×C22⋊D1711yescharacteristic
247C4—22no—
248C4—22no—
249C4—42no—
250C4—348no—
251C4—682no—
252C2—24no—
253C2—82no—
254C2—684no—
255C2#1176331611yescharacteristic
256C2C2×#1058762611yescharacteristic
257C2#1176336811yescharacteristic
258C2#1176344814yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#121C2
hypercenterC2×#121D17
hypocenterC17#2539
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#10587621C2semidirect
2. C2×#10587623C2semidirect
3. C22×#63227C2semidirect
4. #11762744C2semidirect
5. #11762717C2semidirect
6. #11762709C2semidirect
7. C4×C22⋊Dic17C2semidirect
8. C2×C17×#121C2semidirect
11. C22×D4×C17C22semidirect
16. C2×C4⋊C4×C17C22semidirect
17. C2×C22⋊C4×C17C22semidirect
19. C2×C22⋊Dic17C22semidirect
26. C2×#63225C22semidirect
27. C2×C4×Dic17C22semidirect
28. #10587502C22semidirect
42. C22×C4×C17D4semidirect
56. C22⋊Dic17D4semidirect
124. C2×#121D17semidirect
141. C22×D4C2×D17semidirect
148. C2×C4⋊C4C2×D17semidirect
155. C2×C22⋊C4C2×D17semidirect
158. C17#2539semidirect
179. C22×C4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876212
C2×#105876232
C22×#632272
#117627442
#117627172
#117627092
C4×C22⋊Dic172
C2×C17×#1212
#253917