Finite Group Databasebeta

← #16931775

Subgroups — #16931775

Order 4352.

644 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 115 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#14531141C211yesmaximal, characteristic, complemented
2#14527300C212yesmaximal
3#14527114C211yesmaximal, characteristic, complemented
4#14526670C212yesmaximal, complemented
5C17×#2591C211yesmaximal, characteristic
6C2×#10587612—21no—
7#11762773C2212yes—
8C4⋊C4×Dic17—24no—
9C4×C22⋊Dic17—21no—
10C4×C22⋊Dic17C2212yes—
11#11762543—24no—
12#11762535—22no—
13C2×C17×#121C2211yescharacteristic
14C22×C4⋊C4×C17C2211yescharacteristic
15C17×#400C412yescomplemented
16C17×#394C412yescomplemented
17C17×#388—22no—
18C17×#386—21no—
19C17×#386—21no—
20C4×C22⋊C4×C17C2211yescharacteristic
21C22×D4×C17—21no—
22C22×D4×C17C2×C411yescharacteristic
23C23×C4×C17D411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17C2311yescharacteristic
26C17×#121—22no—
27C17×#121—22no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C4⋊C4×C17C2×C412yes—
32C2×C4⋊C4×C17D412yes—
33C2×C22⋊C4×C17—21no—
34C2×C22⋊C4×C17—22no—
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—22no—
37C2×C42×C17—21no—
38C2×C22⋊Dic17—22no—
39#10587612—28no—
40C2×C4⋊Dic17—22no—
41C2×C4×Dic17—22no—
42C2×C4×Dic17—22no—
43C2×C4×Dic17—22no—
44C2×#63223—22no—
45C17×#95—22no—
46C17×#95—22no—
47C17×#95C2×C412yes—
48#10587502—22no—
49#10587502D414yes—
50C22⋊#3622—22no—
51C22⋊#3622—22no—
52C24×C17—21no—
53C24×C17C2×D411yescharacteristic
54C2×D4×C17—22no—
55C2×D4×C17—44no—
56C2×D4×C17—41no—
57C2×D4×C17—41no—
58C2×D4×C17—81no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—24no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—22no—
66C22×C4×C17—81no—
67C22×C4×C17—81no—
68C22×C4×C17C22⋊C412yes—
69C22×C4×C17C22⋊C412yes—
70C22×C4×C17C22×C411yescharacteristic
71C22×C4×C17C2×D412yes—
72C22×C4×C17C2×D411yescharacteristic
73C22×C4×C17#4112yes—
74C22×Dic17—24no—
75C4⋊C4×C17—24no—
76C4⋊C4×C17—24no—
77C4⋊C4×C17—22no—
78C22⋊C4×C17—42no—
79C22⋊C4×C17—44no—
80C22⋊C4×C17—42no—
81C22⋊C4×C17—82no—
82C42×C17—81no—
83C22⋊Dic17—28no—
84C4⋊Dic17—28no—
85C4×Dic17—28no—
86C4×Dic17—82no—
87C4×Dic17—82no—
88#63223—82no—
89C2×#3622—42no—
90#9410—171nomaximal, Sylow
91C23×C17—22no—
92C23×C17—22no—
93C23×C17—41no—
94C23×C17—42no—
95C23×C17—42no—
96C23×C17—41no—
97C23×C17—42no—
98C23×C17C2×C22⋊C411yescharacteristic
99C23×C17#12011yescharacteristic
100C23×C17#12311yescharacteristic
101D4×C17—82no—
102D4×C17—82no—
103D4×C17—82no—
104C2×C4×C17—24no—
105C2×C4×C17—22no—
106C2×C4×C17—22no—
107C2×C4×C17—22no—
108C2×C4×C17—22no—
109C2×C4×C17—22no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—41no—
120C2×C4×C17—42no—
121C2×C4×C17—81no—
122C2×C4×C17#10414yes—
123C2×C4×C17C4×D412yes—
124C2×C4×C17#12112yes—
125C2×Dic17—28no—
126C2×Dic17—44no—
127C2×Dic17—44no—
128#3622—82no—
129#2591D1711yescharacteristic
130#2274—171no—
131#1507—172no—
132#1463—171no—
133#1297—172no—
134C22×C17—21no—
135C22×C17—21no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17—21no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—44no—
142C22×C17—84no—
143C22×C17—81no—
144C22×C17—81no—
145C22×C17—81no—
146C22×C17#38611yescharacteristic
147C22×C17C2×#10412yes—
148C22×C17#45412yes—
149C4×C17—24no—
150C4×C17—22no—
151C4×C17—22no—
152C4×C17—42no—
153C4×C17—82no—
154C4×C17—81no—
155Dic17—48no—
156C2×#121C2×D1711yescharacteristic
157C2×#117—341no—
158C22×C4⋊C4C2×D1711yescharacteristic
159#407—172no—
160#400Dic1712yescomplemented
161#394Dic1712yescomplemented
162#388—22no—
163#386—21no—
164#386—21no—
165C4×C4⋊C4—344no—
166C4×C22⋊C4—172no—
167C4×C22⋊C4—341no—
168C4×C22⋊C4C2×D1711yescharacteristic
169#327—344no—
170#324—342no—
171C2×C17—22no—
172C2×C17—22no—
173C2×C17—41no—
174C2×C17—161no—
175C2×C17#188811yescharacteristic
176C2×C17#189611yescharacteristic
177C2×C17#191311yescharacteristic
178C22×D4—21no—
179C22×D4C2×Dic1711yescharacteristic
180C23×C4C22⋊D1711yescharacteristic
181C23×C4C22⋊D1711yescharacteristic
182C23×C4C22×D1711yescharacteristic
183C2×M16—342no—
184#121—22no—
185#121—22no—
186#117—348no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—342no—
190C2×C4⋊C4C2×Dic1711yescharacteristic
191C2×C4⋊C4C2×Dic1712yes—
192C2×C4⋊C4C22⋊D1712yes—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—342no—
198C2×C42—21no—
199C2×C42—342no—
200C2×C42—342no—
201C2×C42—342no—
202C22⋊C8—342no—
203C22⋊C8—342no—
204#95—22no—
205#95—22no—
206#95—174no—
207#95—342no—
208#95C2×Dic1712yes—
209C17#941011yescharacteristic, complemented, Sylow
210C24—21no—
211C24C2×C22⋊D1711yescharacteristic
212C2×D4—22no—
213C2×D4—44no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—81no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—344no—
227C22×C4C22⋊Dic1712yes—
228C22×C4C22⋊Dic1712yes—
229C22×C4D4×D1712yes—
230C22×C4#6325412yes—
231C22×C4C22×Dic1711yescharacteristic
232C22×C4C2×C22⋊D1711yescharacteristic
233M16—1362no—
234C2×C8—682no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—348no—
239C22⋊C4—42no—
240C22⋊C4—44no—
241C22⋊C4—42no—
242C22⋊C4—82no—
243C22⋊C4—348no—
244C42—81no—
245C42—348no—
246C42—1362no—
247C42—1362no—
248C23—22no—
249C23—22no—
250C23—41no—
251C23—42no—
252C23—42no—
253C23—41no—
254C23—42no—
255C23#1058762111yescharacteristic
256C23#1058762311yescharacteristic
257C23C2×C22⋊Dic1711yescharacteristic
258D4—82no—
259D4—82no—
260D4—82no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—81no—
279C2×C4—348no—
280C2×C4—684no—
281C2×C4—684no—
282C2×C4#1058750814yes—
283C2×C4D4×Dic1712yes—
284C2×C4#1058762512yes—
285C8—1362no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—44no—
294C22—84no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22#1176268911yescharacteristic
299C22C2×#1058750812yes—
300C22#1176298112yes—
301C4—22no—
302C4—22no—
303C4—24no—
304C4—42no—
305C4—82no—
306C4—81no—
307C4—688no—
308C2—22no—
309C2—22no—
310C2—41no—
311C2—161no—
312C2#1452918811yescharacteristic
313C2#1452922811yescharacteristic
314C2#1452932711yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2591C2
socleC22×C17#386
prime radicalC2×C17×#121C22
cosocleC23×C4×C17C23
hypercenter#2591D17
hypocenterC17#9410
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2591

Complements

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

NH ≅ G/Ntype
1. #14531141C2semidirect
3. #14527114C2semidirect
4. #14526670C2semidirect
15. C17×#400C4semidirect
16. C17×#394C4semidirect
160. #400Dic17semidirect
161. #394Dic17semidirect
209. C17#9410semidirect

Maximal subgroups

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

subgroupindex
#145311412
#145273002
#145271142
#145266702
C17×#25912
#941017