Finite Group Databasebeta

← #17010444

Subgroups — #17010444

Order 4352.

982 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 982 conjugacy classes of subgroups — too many to draw. Only its 179 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#14539158C211yesmaximal, characteristic
2#14538914C212yesmaximal, complemented
3#14527970C218yesmaximal, complemented
4#14527498C211yesmaximal, characteristic
5#14527231C212yesmaximal, complemented
6C2×C17×#559C211yesmaximal, characteristic
7#11763366C2211yescharacteristic
8#11763173—22no—
9#11763155C2214yes—
10#11763139—22no—
11#11763094C2214yescomplemented
12#11763019C2214yescomplemented
13#11762965—22no—
14#11762914—24no—
15#11762866C2212yes—
16#11762814—22no—
17C4×#63223C2211yescharacteristic
18#11762553C2214yes—
19Q8⋊#3622C2218yescomplemented
20C17×#559—24no—
21C17×#559—22no—
22C17×#559C2214yes—
23C2×C17×#127C2211yescharacteristic
24C2×C17×#124—22no—
25C2×C17×#123—22no—
26C2×C4×Q8×C17C2212yes—
27C2×C4×D4×C17—22no—
28C22×Q8×C17D412yes—
29C22×D4×C17—22no—
30C22×D4×C17D412yes—
31C23×C4×C17—22no—
32C17×#127—22no—
33C17×#127D412yes—
34C17×#127C2312yes—
35C17×#124—28no—
36C17×#124—28no—
37C17×#123—28no—
38C17×#123—28no—
39C4×Q8×C17—22no—
40C4×Q8×C17—22no—
41C4×Q8×C17D414yes—
42C4×Q8×C17C2314yes—
43C4×D4×C17—28no—
44C4×D4×C17—28no—
45C2×C4⋊C4×C17—22no—
46C2×C4⋊C4×C17D412yes—
47C2×C22⋊C4×C17—24no—
48C2×C22⋊C4×C17—22no—
49C2×C42×C17—22no—
50C2×C42×C17C2311yescharacteristic
51C2×#63231—24no—
52#10587612—21no—
53#10587610—22no—
54C2×#63223—21no—
55#10587556—21no—
56#10587555C2312yes—
57C4×Dic34—22no—
58#10587506—24no—
59#10587503—28no—
60C22⋊#3622—22no—
61#10587481—24no—
62#10587479—24no—
63C4⋊#3622C2314yes—
64#10587474—21no—
65#10587474—22no—
66C4×#3622C2312yes—
67C24×C17—22no—
68C2×Q8×C17—22no—
69C2×Q8×C17—44no—
70C2×Q8×C17C2×D414yes—
71C2×D4×C17—24no—
72C2×D4×C17—22no—
73C2×D4×C17—24no—
74C2×D4×C17—48no—
75C2×D4×C17—44no—
76C2×D4×C17C2×D414yes—
77C22×C4×C17—22no—
78C22×C4×C17—22no—
79C22×C4×C17—28no—
80C22×C4×C17—84no—
81C22×C4×C17C2×D412yes—
82C22×C4×C17C2×D411yescharacteristic
83C2×Dic34—22no—
84C4⋊C4×C17—24no—
85C4⋊C4×C17—22no—
86C4⋊C4×C17—24no—
87C4⋊C4×C17C2×D414yes—
88C22⋊C4×C17—28no—
89C22⋊C4×C17—216no—
90C42×C17—24no—
91C42×C17—24no—
92C42×C17C2×D412yes—
93C42×C17C2×D411yescharacteristic
94C42×C17C2411yescharacteristic
95C22⋊Dic17—41no—
96#63231—48no—
97C4⋊Dic17—22no—
98C4⋊Dic17—22no—
99#63225—42no—
100C4×Dic17—41no—
101#63223—42no—
102C2×#3622—22no—
103C2×#3622—24no—
104#18531—171nomaximal, Sylow
105C23×C17—24no—
106C23×C17—48no—
107C23×C17—44no—
108C23×C17#12011yescharacteristic
109Q8×C17—24no—
110Q8×C17—24no—
111Q8×C17—44no—
112D4×C17—44no—
113D4×C17—48no—
114D4×C17—48no—
115D4×C17—82no—
116C2×C4×C17—22no—
117C2×C4×C17—24no—
118C2×C4×C17—24no—
119C2×C4×C17—24no—
120C2×C4×C17—44no—
121C2×C4×C17—44no—
122C2×C4×C17—88no—
123C2×C4×C17#12012yes—
124C2×C4×C17#12011yescharacteristic
125C2×C4×C17C22×D411yescharacteristic
126C2×C4×C17C22×D412yes—
127C2×Dic17—42no—
128Dic34—44no—
129#3622—42no—
130#3622—44no—
131C2×#559D1711yescharacteristic
132#3138—171no—
133#3113—172no—
134#1640—178no—
135#1555—171no—
136#1491—172no—
137C22×C17—22no—
138C22×C17—48no—
139C22×C17—44no—
140C22×C17—88no—
141C22×C17#45512yes—
142C22×C17C2×#12011yescharacteristic
143C4×C17—22no—
144C4×C17—21no—
145C4×C17—21no—
146C4×C17—44no—
147C4×C17—44no—
148C4×C17#57514yes—
149Dic17—82no—
150#559—22no—
151#559—24no—
152#559C2×D1714yes—
153#533—171no—
154C2×#127C2×D1711yescharacteristic
155C2×#124—22no—
156C2×#123—22no—
157C2×C4×Q8C2×D1712yes—
158C2×C4×D4—22no—
159#490—342no—
160#486—174no—
161#483—342no—
162#475—174no—
163#462—174no—
164#451—342no—
165#441—344no—
166#432—172no—
167#418—342no—
168C4×M16—171no—
169#331—174no—
170Q8⋊C8—178no—
171C2×C17—22no—
172C2×C17—84no—
173C2×C17#300512yes—
174C2×C17C2×#45511yescharacteristic
175C22×Q8C22⋊D1712yes—
176C22×D4—22no—
177C22×D4C22⋊D1712yes—
178C23×C4—22no—
179C2×Q16—344no—
180C2×M16—341no—
181C4⋊Q8—172no—
182#127—22no—
183#127C22⋊D1712yes—
184#127C22×D1712yes—
185#125—341no—
186#124—28no—
187#124—28no—
188#123—28no—
189#123—28no—
190C22⋊Q8—342no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—342no—
194C4×Q8C22⋊D1714yes—
195C4×Q8C22×D1714yes—
196C4×D4—28no—
197C4×D4—28no—
198#117—341no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4C22⋊D1712yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—22no—
203C2×C42—22no—
204C2×C42C22×D1711yescharacteristic
205#106—344no—
206C4⋊C8—174no—
207#103—344no—
208#103—344no—
209#102—348no—
210C22⋊C8—342no—
211#97—341no—
212#97—342no—
213C4×C8—172no—
214C17#1853111yescharacteristic, complemented, Sylow
215C24—22no—
216C2×Q8—22no—
217C2×Q8—44no—
218C2×Q8—342no—
219C2×Q8C2×C22⋊D1714yes—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—22no—
223C2×D4—48no—
224C2×D4—44no—
225C2×D4C2×C22⋊D1714yes—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—28no—
229C22×C4—84no—
230C22×C4C2×C22⋊D1712yes—
231C22×C4C2×C22⋊D1711yescharacteristic
232Q16—688no—
233M16—682no—
234C2×C8—342no—
235C2×C8—344no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—22no—
239C4⋊C4—342no—
240C4⋊C4—342no—
241C4⋊C4—682no—
242C4⋊C4C2×C22⋊D1714yes—
243C22⋊C4—216no—
244C22⋊C4—28no—
245C22⋊C4—681no—
246C42—24no—
247C42—24no—
248C42—681no—
249C42C2×C22⋊D1711yescharacteristic
250C42C2×C22⋊D1712yes—
251C42C23×D1711yescharacteristic
252C23—24no—
253C23—48no—
254C23—44no—
255C23#1058763911yescharacteristic
256Q8—24no—
257Q8—24no—
258Q8—44no—
259Q8—684no—
260D4—44no—
261D4—48no—
262D4—48no—
263D4—82no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—88no—
271C2×C4—682no—
272C2×C4#1058763912yes—
273C2×C4#1058763911yescharacteristic
274C2×C4C22×C22⋊D1711yescharacteristic
275C2×C4C22×C22⋊D1712yes—
276C8—682no—
277C8—684no—
278C22—22no—
279C22—48no—
280C22—44no—
281C22—88no—
282C22#1176298712yes—
283C22C2×#1058763911yescharacteristic
284C4—22no—
285C4—21no—
286C4—21no—
287C4—44no—
288C4—44no—
289C4—1362no—
290C4#1176360714yes—
291C2—22no—
292C2—84no—
293C2#1453802612yes—
294C2C2×#1176298711yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC2×C17×#559C2
socleC22×C17C2×#120
prime radicalC2×C17×#127C22
hypercenterC2×#559D17
hypocenterC17#18531
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. #14538914C2semidirect
3. #14527970C2semidirect
5. #14527231C2semidirect
11. #11763094C22semidirect
12. #11763019C22semidirect
19. Q8⋊#3622C22semidirect
214. C17#18531semidirect

Maximal subgroups

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

subgroupindex
#145391582
#145389142
#145279702
#145274982
#145272312
C2×C17×#5592
#1853117