Finite Group Databasebeta

← #14533264

Subgroups — #14533264

Order 2176.

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

normal subgroups only

This group has 512 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×#10587582C211yesmaximal, characteristic
2C2×C4⋊C4×D17C211yesmaximal, characteristic
3#11762749C211yesmaximal, characteristic, complemented
4#11762748C211yesmaximal, characteristic, complemented
5#11762747C211yesmaximal, characteristic, complemented
6#11762713C212yesmaximal, complemented
7#11762673C212yesmaximal, complemented
8#11762671C212yesmaximal
9#11762671C212yesmaximal
10#11762656C211yesmaximal, characteristic
11C17×#400C211yesmaximal, characteristic, complemented
12C22×C4×D17C2212yes—
13C22×C4×D17C2211yescharacteristic
14C22×Dic34C2211yescharacteristic
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×#63227C2211yescharacteristic
18C2×#63227C2211yescharacteristic
19C2×#63227C2212yes—
20C2×C4⋊Dic17C2211yescharacteristic
21C2×#63225C2212yes—
22C2×#63225C2211yescharacteristic
23C2×#63225C2211yescharacteristic
24C2×C4×Dic17C2212yes—
25#10587582—24no—
26C4⋊C4×D17—24no—
27C17×#95C2212yes—
28C17×#95C2211yescharacteristic
29C17×#95C2212yes—
30#10587502C2212yes—
31#10587502C2212yes—
32#10587502C2212yes—
33#10587502C2211yescharacteristic
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C23×D17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2312yes—
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×Dic17C2311yescharacteristic
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2311yescharacteristic
48C2×C4×D17—24no—
49C2×C4×D17—24no—
50C2×C4×D17—28no—
51C2×C4×D17D414yes—
52C2×Dic34—22no—
53C2×Dic34—24no—
54C4⋊C4×C17—22no—
55C4⋊C4×C17—22no—
56#63227—22no—
57#63227—22no—
58#63227—24no—
59C4⋊Dic17—22no—
60#63225—22no—
61#63225—24no—
62#63225—22no—
63C4×Dic17—24no—
64C23×C17C2411yescharacteristic
65C22×D17—22no—
66C22×D17—21no—
67C22×D17—21no—
68C22×D17C2×D412yes—
69C22×D17#4114yes—
70C2×C4×C17—22no—
71C2×C4×C17—21no—
72C2×C4×C17—22no—
73C2×C4×C17—21no—
74C2×C4×C17—22no—
75C2×C4×C17—24no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—21no—
79C2×C4×C17—21no—
80C2×C4×C17—22no—
81C2×C4×C17C2×D412yes—
82C2×Dic17—21no—
83C2×Dic17—22no—
84C2×Dic17—21no—
85C2×Dic17—22no—
86C2×Dic17—24no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—21no—
90C2×Dic17—22no—
91C2×Dic17—21no—
92C2×Dic17C2×D412yes—
93C2×Dic17#4114yes—
94C4×D17—28no—
95C4×D17—48no—
96Dic34—44no—
97#2503—171nomaximal, Sylow
98C22×C17C22×D411yescharacteristic
99C22×C17C2×#4112yes—
100C22×C17C2×#4112yes—
101C22×C172−1+411yescharacteristic
102C22×C172−1+411yescharacteristic
103C2×D17—22no—
104C2×D17—22no—
105C2×D17—24no—
106C2×D17—24no—
107C2×D17#12414yes—
108C4×C17—22no—
109C4×C17—41no—
110C4×C17—42no—
111C4×C17—41no—
112C4×C17—42no—
113Dic17—41no—
114Dic17—42no—
115Dic17—41no—
116Dic17—42no—
117Dic17#12314yes—
118C2×C22⋊Q8—171no—
119C22×C4⋊C4—171no—
120#400—171no—
121#400—171no—
122#400—171no—
123#400D1711yescharacteristic, complemented
124#391—172no—
125#382—172no—
126#381—172no—
127#381—172no—
128C4×C22⋊C4—171no—
129C2×C17C2×#12311yescharacteristic
130C2×C17C2×#12411yescharacteristic
131C2×C17#53611yescharacteristic
132C2×C17#55212yes—
133C2×C17#55212yes—
134D17—24no—
135C22×Q8—171no—
136C23×C4—171no—
137C23×C4—172no—
138C22⋊Q8—344no—
139C2×C4⋊C4—172no—
140C2×C4⋊C4—171no—
141C2×C4⋊C4—171no—
142C2×C4⋊C4—171no—
143C2×C4⋊C4—344no—
144C2×C4⋊C4C2×D1711yescharacteristic
145C2×C4⋊C4C2×D1711yescharacteristic
146C2×C22⋊C4—171no—
147C2×C22⋊C4—171no—
148C2×C22⋊C4—172no—
149C2×C42—172no—
150#95—171no—
151#95—172no—
152#95—172no—
153#95—172no—
154#95—172no—
155#95—172no—
156#95—172no—
157#95C2×D1711yescharacteristic
158#95C2×D1712yes—
159#95C2×D1712yes—
160C17#250311yescharacteristic, complemented, Sylow
161C24—171no—
162C2×Q8—342no—
163C2×Q8—344no—
164C22×C4—171no—
165C22×C4—174no—
166C22×C4—172no—
167C22×C4—171no—
168C22×C4—171no—
169C22×C4—172no—
170C22×C4—344no—
171C22×C4—344no—
172C22×C4—348no—
173C22×C4C22×D1712yes—
174C22×C4C22×D1711yescharacteristic
175C22×C4C22×D1711yescharacteristic
176C22×C4C22×D1712yes—
177C22×C4C22×D1711yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—344no—
181C4⋊C4—342no—
182C4⋊C4—342no—
183C4⋊C4—342no—
184C22⋊C4—342no—
185C22⋊C4—342no—
186C22⋊C4—344no—
187C42—344no—
188C23—172no—
189C23—174no—
190C23—341no—
191C23—342no—
192C23—341no—
193C23C23×D1711yescharacteristic
194Q8—684no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—172no—
207C2×C4—174no—
208C2×C4—348no—
209C2×C4—341no—
210C2×C4—341no—
211C2×C4—342no—
212C2×C4—342no—
213C2×C4—341no—
214C2×C4—342no—
215C2×C4—342no—
216C2×C4—344no—
217C2×C4—342no—
218C2×C4—341no—
219C2×C4—688no—
220C2×C4D4×D1712yes—
221C22—174no—
222C22—344no—
223C22—344no—
224C22—342no—
225C22—342no—
226C22C2×D4×D1711yescharacteristic
227C22#1058769411yescharacteristic
228C22#41×D1712yes—
229C22#41×D1712yes—
230C22#1058769711yescharacteristic
231C4—22no—
232C4—42no—
233C4—41no—
234C4—42no—
235C4—41no—
236C4—174no—
237C4—682no—
238C4—681no—
239C4—682no—
240C4—681no—
241C2—344no—
242C2#123×D1711yescharacteristic
243C2#124×D1711yescharacteristic
244C2#1176338211yescharacteristic
245C2#1176349112yes—
246C2#1176349512yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#400C2
prime radicalC23×D17C23
hypercenter#400D17
hypocenterC17#2503
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #11762749C2semidirect
4. #11762748C2semidirect
5. #11762747C2semidirect
6. #11762713C2semidirect
7. #11762673C2semidirect
11. C17×#400C2semidirect
123. #400D17semidirect
160. C17#2503semidirect

Maximal subgroups

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

subgroupindex
C2×#105875822
C2×C4⋊C4×D172
#117627492
#117627482
#117627472
#117627132
#117626732
#117626712
#117626562
C17×#4002
#250317