Finite Group Databasebeta

← #14533715

Subgroups — #14533715

Order 2176.

900 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 173 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×#10587615C212yesmaximal, complemented
2C2×#10587616C211yesmaximal, characteristic, complemented
3C2×#10587639C212yesmaximal, complemented
4#11762745C212yesmaximal, complemented
5#11762718C212yesmaximal, complemented
6#11762709C214yesmaximal, complemented
7C4×C22⋊Dic17C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C17C211yesmaximal, characteristic, complemented
9C25×C17C2211yescharacteristic
10C23×C4×C17C2211yescharacteristic
11C23×C4×C17C2211yescharacteristic
12C22×C22⋊D17C2212yes—
13C22×C22⋊D17C2214yes—
14C22×D68C2211yescharacteristic, complemented
15C2×C22⋊C4×C17—24no—
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊C4×C17C2212yes—
18#10587639—28no—
19C2×C22⋊Dic17C2214yes—
20C2×C22⋊Dic17C2212yes—
21#10587616—24no—
22#10587615—28no—
23C2×#63227C2212yescomplemented
24C2×#63227C2214yescomplemented
25C2×C4⋊Dic17C2211yescharacteristic, complemented
26C2×#63225C2214yescomplemented
27C2×C4×Dic17C2212yescomplemented
28#10587502C2212yescomplemented
29C24×C17—22no—
30C24×C17—24no—
31C24×C17—24no—
32C24×C17—42no—
33C24×C17C2312yes—
34C24×C17C2311yescharacteristic
35C23×D17C2312yes—
36C22×C4×C17—24no—
37C22×C4×C17—22no—
38C22×C4×C17—42no—
39C22×C4×C17D414yescomplemented
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C2×C22⋊D17—24no—
44C2×C22⋊D17—24no—
45C2×C22⋊D17—24no—
46C2×C22⋊D17—28no—
47C2×C22⋊D17—28no—
48C2×C22⋊D17—44no—
49C22×Dic17C2312yes—
50C22×Dic17C2314yes—
51C2×D68—22no—
52C2×D68—42no—
53C22⋊C4×C17—24no—
54C22⋊C4×C17—24no—
55C22⋊Dic17—28no—
56C22⋊Dic17D418yescomplemented
57#63227—24no—
58#63227—28no—
59C4⋊Dic17—22no—
60#63225—28no—
61C4×Dic17—42no—
62C23×C17—22no—
63C23×C17—22no—
64C23×C17—28no—
65C23×C17—24no—
66C23×C17—22no—
67C23×C17—22no—
68C23×C17—24no—
69C23×C17—416no—
70C23×C17—44no—
71C23×C17—44no—
72C23×C17C2×D414yes—
73C23×C17C2×D412yes—
74C23×C17#4114yes—
75C23×C17C2411yescharacteristic
76C22×D17—22no—
77C22×D17—22no—
78C22×D17—22no—
79C22×D17—24no—
80C22×D17—24no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—24no—
85C2×C4×C17—44no—
86C2×C4×C17—44no—
87C2×C4×C17C2×D412yes—
88C2×C4×C17C2×D412yes—
89C22⋊D17—416no—
90C22⋊D17—48no—
91C2×Dic17—24no—
92C2×Dic17—24no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17C2×D418yes—
97D68—44no—
98#2539—171nomaximal, Sylow
99C22×C17—22no—
100C22×C17—22no—
101C22×C17—28no—
102C22×C17—24no—
103C22×C17—28no—
104C22×C17—416no—
105C22×C17—42no—
106C22×C17—42no—
107C22×C17—44no—
108C22×C17#12714yes—
109C22×C17C22×D412yes—
110C22×C17C22×D411yescharacteristic
111C22×C17C2×#4112yes—
112C22×C17H(5,𝔽2)12yes—
113C2×D17—42no—
114C2×D17—42no—
115C2×D17—42no—
116C2×D17—44no—
117C2×D17—44no—
118C4×C17—22no—
119C4×C17—22no—
120C4×C17—42no—
121Dic17—28no—
122Dic17—42no—
123C2×#123—172no—
124C2×#121—171no—
125C2×#120—172no—
126C22×C22⋊C4D1711yescharacteristic, complemented
127#399—172no—
128#392—172no—
129#390—174no—
130C4×C22⋊C4—171no—
131C2×C17—24no—
132C2×C17—44no—
133C2×C17#52511yescharacteristic
134C2×C17C2×#12711yescharacteristic
135C2×C17#53411yescharacteristic
136C2×C17#54614yes—
137D17—82no—
138C25C2×D1711yescharacteristic
139C22×D4—172no—
140C22×D4—174no—
141C22×D4—171no—
142C23×C4C2×D1711yescharacteristic
143C23×C4C2×D1711yescharacteristic
144#123—348no—
145#121—344no—
146#120—348no—
147C2×C4⋊C4—171no—
148C2×C4⋊C4—174no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—172no—
151C2×C22⋊C4—174no—
152C2×C22⋊C4—172no—
153C2×C22⋊C4—174no—
154C2×C22⋊C4C2×D1712yes—
155C2×C22⋊C4C2×D1712yes—
156C2×C42—172no—
157#95—172no—
158C17#253911yescharacteristic, complemented, Sylow
159C24—24no—
160C24—22no—
161C24—24no—
162C24—42no—
163C24—172no—
164C24C22×D1712yes—
165C24C22×D1711yescharacteristic
166C2×D4—348no—
167C2×D4—348no—
168C2×D4—344no—
169C2×D4—344no—
170C2×D4—344no—
171C2×D4—342no—
172C2×D4—682no—
173C2×D4—684no—
174C22×C4—24no—
175C22×C4—22no—
176C22×C4—42no—
177C22×C4—174no—
178C22×C4—172no—
179C22×C4C22⋊D1714yescomplemented
180C22×C4C22×D1712yes—
181C22×C4C22×D1711yescharacteristic
182C22×C4C22×D1711yescharacteristic
183C4⋊C4—348no—
184C4⋊C4—342no—
185C22⋊C4—24no—
186C22⋊C4—24no—
187C22⋊C4—178no—
188C22⋊C4—348no—
189C22⋊C4—348no—
190C22⋊C4—344no—
191C42—682no—
192C23—22no—
193C23—22no—
194C23—28no—
195C23—22no—
196C23—24no—
197C23—24no—
198C23—22no—
199C23—416no—
200C23—44no—
201C23—44no—
202C23—344no—
203C23—342no—
204C23—342no—
205C23—344no—
206C23—342no—
207C23#6325214yes—
208C23D4×D1714yes—
209C23C2×C22⋊D1712yes—
210C23C23×D1711yescharacteristic
211D4—684no—
212D4—6816no—
213D4—688no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—178no—
221C2×C4—344no—
222C2×C4—344no—
223C2×C4—342no—
224C2×C4—342no—
225C2×C4—342no—
226C2×C4C2×C22⋊D1712yes—
227C2×C4C2×C22⋊D1712yes—
228C22—22no—
229C22—22no—
230C22—28no—
231C22—24no—
232C22—28no—
233C22—416no—
234C22—42no—
235C22—42no—
236C22—44no—
237C22—682no—
238C22—682no—
239C22—682no—
240C22—684no—
241C22—684no—
242C22#1058762614yes—
243C22C2×#6325212yes—
244C22C2×D4×D1712yes—
245C22#1058769112yes—
246C22C22×C22⋊D1711yescharacteristic
247C4—22no—
248C4—22no—
249C4—42no—
250C4—348no—
251C4—682no—
252C2—24no—
253C2—44no—
254C2—1362no—
255C2#1176332011yescharacteristic
256C2C2×#1058762611yescharacteristic
257C2#1176336811yescharacteristic
258C2#1176344914yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC22×C22⋊C4×C17C2
hypercenterC22×C22⋊C4D17
hypocenterC17#2539
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#10587615C2semidirect
2. C2×#10587616C2semidirect
3. C2×#10587639C2semidirect
4. #11762745C2semidirect
5. #11762718C2semidirect
6. #11762709C2semidirect
7. C4×C22⋊Dic17C2semidirect
8. C22×C22⋊C4×C17C2semidirect
14. C22×D68C22semidirect
23. C2×#63227C22semidirect
24. C2×#63227C22semidirect
25. C2×C4⋊Dic17C22semidirect
26. C2×#63225C22semidirect
27. C2×C4×Dic17C22semidirect
28. #10587502C22semidirect
39. C22×C4×C17D4semidirect
56. C22⋊Dic17D4semidirect
126. C22×C22⋊C4D17semidirect
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×#105876152
C2×#105876162
C2×#105876392
#117627452
#117627182
#117627092
C4×C22⋊Dic172
C22×C22⋊C4×C172
#253917