Finite Group Databasebeta

← #16932049

Subgroups — #16932049

Order 4352.

512 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 107 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
1#14531387C211yesmaximal, characteristic
2#14528864C211yesmaximal, characteristic
3#14527317C212yesmaximal
4#14526637C212yesmaximal
5C17×#2503C211yesmaximal, characteristic
6#11762773C2212yes—
7#11762695—24no—
8#11762672C2212yes—
9#11762669—21no—
10C4×C22⋊Dic17—21no—
11#11762578—22no—
12#11762546—24no—
13C2×C17×C22⋊Q8C2211yescharacteristic
14C22×C4⋊C4×C17C2211yescharacteristic
15C17×#400—22no—
16C17×#391C412yescomplemented
17C17×#382C412yescomplemented
18C17×#381—21no—
19C17×#381—21no—
20C4×C22⋊C4×C17C2211yescharacteristic
21C22×Q8×C17C2×C411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17D411yescharacteristic
24C23×C4×C17C2311yescharacteristic
25C17×C22⋊Q8—22no—
26C17×C22⋊Q8—22no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C4⋊C4×C17C2×C412yes—
32C2×C4⋊C4×C17D412yes—
33C2×C22⋊C4×C17—21no—
34C2×C22⋊C4×C17—21no—
35C2×C42×C17C2×C412yes—
36C2×C22⋊Dic17—22no—
37C2×C4⋊Dic17—22no—
38C2×C4×Dic17—22no—
39C2×#63223—22no—
40C17×#95—22no—
41C17×#95—22no—
42C17×#95—22no—
43C17×#95—22no—
44C17×#95—21no—
45#10587502—22no—
46#10587502—22no—
47#10587502—22no—
48#10587502D414yes—
49C22⋊#3622—22no—
50C22⋊#3622—22no—
51C24×C17C2×D411yescharacteristic
52C2×Q8×C17—22no—
53C2×Q8×C17—42no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—22no—
60C22×C4×C17—21no—
61C22×C4×C17—42no—
62C22×C4×C17—42no—
63C22×C4×C17—42no—
64C22×C4×C17C22⋊C412yes—
65C22×C4×C17C22⋊C412yes—
66C22×C4×C17C22×C411yescharacteristic
67C22×C4×C17C2×D412yes—
68C22×C4×C17C2×D411yescharacteristic
69C22×C4×C17#4112yes—
70C22×Dic17—24no—
71C4⋊C4×C17—24no—
72C4⋊C4×C17—24no—
73C4⋊C4×C17—22no—
74C4⋊C4×C17—42no—
75C22⋊C4×C17—42no—
76C22⋊C4×C17—41no—
77C22⋊C4×C17—41no—
78C42×C17—42no—
79C22⋊Dic17—44no—
80C4⋊Dic17—82no—
81C4×Dic17—82no—
82#63223—44no—
83C2×#3622—42no—
84#9460—171nomaximal, Sylow
85C23×C17—22no—
86C23×C17—22no—
87C23×C17—41no—
88C23×C17C2×C22⋊C411yescharacteristic
89C23×C17#12011yescharacteristic
90C23×C17#12311yescharacteristic
91Q8×C17—82no—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—42no—
97C2×C4×C17—42no—
98C2×C4×C17—42no—
99C2×C4×C17—42no—
100C2×C4×C17—42no—
101C2×C4×C17—42no—
102C2×C4×C17—42no—
103C2×C4×C17—42no—
104C2×C4×C17—42no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—81no—
110C2×C4×C17—81no—
111C2×C4×C17C4×D412yes—
112C2×C4×C17#12112yes—
113C2×Dic17—44no—
114C2×Dic17—44no—
115C2×Dic17—44no—
116#3622—82no—
117#2503D1711yescharacteristic
118#2313—171no—
119#1821—171no—
120#1511—172no—
121#1284—172no—
122C22×C17—21no—
123C22×C17—21no—
124C22×C17—21no—
125C22×C17—21no—
126C22×C17—21no—
127C22×C17—21no—
128C22×C17—21no—
129C22×C17—44no—
130C22×C17#38611yescharacteristic
131C22×C17#41012yes—
132C22×C17#45512yes—
133C4×C17—22no—
134C4×C17—22no—
135C4×C17—42no—
136C4×C17—42no—
137C4×C17—82no—
138C4×C17—81no—
139C4×C17—81no—
140Dic17—84no—
141C2×C22⋊Q8C2×D1711yescharacteristic
142C22×C4⋊C4C2×D1711yescharacteristic
143#407—172no—
144#400—22no—
145#391Dic1712yescomplemented
146#387—344no—
147#382Dic1712yescomplemented
148#381—21no—
149#381—21no—
150#381—172no—
151#380—341no—
152C4×C22⋊C4—341no—
153C4×C22⋊C4C2×D1711yescharacteristic
154#343—342no—
155#328—344no—
156C2×C17—22no—
157C2×C17—22no—
158C2×C17—41no—
159C2×C17#188511yescharacteristic
160C2×C17#189911yescharacteristic
161C2×C17#191211yescharacteristic
162C22×Q8C2×Dic1711yescharacteristic
163C23×C4C22⋊D1711yescharacteristic
164C23×C4C22⋊D1711yescharacteristic
165C23×C4C22×D1711yescharacteristic
166C2×M16—342no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—342no—
173C2×C4⋊C4C2×Dic1711yescharacteristic
174C2×C4⋊C4C2×Dic1712yes—
175C2×C4⋊C4C22⋊D1712yes—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—342no—
179C2×C42—342no—
180C2×C42C2×Dic1712yes—
181C22⋊C8—342no—
182C22⋊C8—342no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—21no—
188#95—174no—
189#95—342no—
190#95—342no—
191#95—342no—
192C17#946011yescharacteristic, complemented, Sylow
193C24C2×C22⋊D1711yescharacteristic
194C2×Q8—22no—
195C2×Q8—42no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—344no—
207C22×C4C22⋊Dic1712yes—
208C22×C4C22⋊Dic1712yes—
209C22×C4D4×D1712yes—
210C22×C4#6325412yes—
211C22×C4C22×Dic1711yescharacteristic
212C22×C4C2×C22⋊D1711yescharacteristic
213M16—684no—
214C2×C8—682no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—22no—
218C4⋊C4—42no—
219C4⋊C4—1362no—
220C22⋊C4—42no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—684no—
224C42—42no—
225C42—1362no—
226C23—22no—
227C23—22no—
228C23—41no—
229C23#1058762111yescharacteristic
230C23#1058762311yescharacteristic
231C23C2×C22⋊Dic1711yescharacteristic
232Q8—82no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—684no—
253C2×C4—684no—
254C2×C4—684no—
255C2×C4D4×Dic1712yes—
256C2×C4#1058762512yes—
257C8—1362no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—44no—
266C22#1176268911yescharacteristic
267C22#1176278312yes—
268C22#1176298612yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—82no—
274C4—81no—
275C4—81no—
276C4—1364no—
277C2—22no—
278C2—22no—
279C2—41no—
280C2#1452917311yescharacteristic
281C2#1452924311yescharacteristic
282C2#1452932211yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2503C2
socleC22×C17#386
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenter#2503D17
hypocenterC17#9460
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2503

Complements

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

NH ≅ G/Ntype
16. C17×#391C4semidirect
17. C17×#382C4semidirect
145. #391Dic17semidirect
147. #382Dic17semidirect
192. C17#9460semidirect

Maximal subgroups

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

subgroupindex
#145313872
#145288642
#145273172
#145266372
C17×#25032
#946017