Finite Group Databasebeta

← #16950319

Subgroups — #16950319

Order 4352.

524 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 119 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×#11762604C212yesmaximal
2#14528772C212yesmaximal
3#14526909C212yesmaximal
4C17×#2326C211yesmaximal, characteristic
5C2×#10587505C2214yes—
6#11762669—22no—
7#11762604—24no—
8#11762576—22no—
9C2×C17×#124C2211yescharacteristic
10C2×C4×Q8×C17C411yescharacteristic, complemented
11C2×C4×D4×C17C2211yescharacteristic
12C17×#391C411yescharacteristic, complemented
13C17×#390C2211yescharacteristic
14C17×#388—22no—
15C17×#385—22no—
16C17×#384C411yescharacteristic, complemented
17C17×#376C411yescharacteristic, complemented
18C22×Q8×C17C2×C411yescharacteristic
19C22×D4×C17C2311yescharacteristic
20C23×C4×C17D412yes—
21C17×#124—22no—
22C17×#124D412yes—
23C17×#124Q812yes—
24C4×Q8×C17—42no—
25C4×D4×C17—42no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C22⋊C4×C17—22no—
31C2×C22⋊C4×C17—22no—
32C2×C22⋊C4×C17D412yes—
33C2×C42×C17—21no—
34C2×C42×C17C2×C411yescharacteristic
35C2×C42×C17C2×C411yescharacteristic
36C2×C42×C17C2×C411yescharacteristic
37C2×C22⋊Dic17—24no—
38C2×C4⋊Dic17—42no—
39C17×#95—21no—
40C17×#95—21no—
41C17×#95—22no—
42#10587505—24no—
43#10587505—24no—
44#10587502—42no—
45C24×C17C2×D412yes—
46C2×Q8×C17—21no—
47C2×Q8×C17—41no—
48C2×Q8×C17—41no—
49C2×Q8×C17C4⋊C412yescomplemented
50C2×D4×C17—41no—
51C2×D4×C17—41no—
52C2×D4×C17C2×D411yescharacteristic
53C2×D4×C17C2×Q811yescharacteristic
54C2×D4×C17#4112yes—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—22no—
58C22×C4×C17—22no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—82no—
62C22×C4×C17C22⋊C412yes—
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22×C411yescharacteristic
66C22×Dic17—44no—
67C4⋊C4×C17—41no—
68C4⋊C4×C17—41no—
69C4⋊C4×C17—42no—
70C4⋊C4×C17—82no—
71C22⋊C4×C17—24no—
72C22⋊C4×C17—24no—
73C22⋊C4×C17—42no—
74C22⋊C4×C17—82no—
75C42×C17—22no—
76C42×C17—21no—
77C42×C17—41no—
78C42×C17—42no—
79C42×C17C4⋊C412yescomplemented
80C22⋊Dic17—44no—
81C22⋊Dic17—44no—
82C4⋊Dic17—84no—
83#12800—171nomaximal, Sylow
84C23×C17—22no—
85C23×C17—22no—
86C23×C17—42no—
87C23×C17—42no—
88C23×C17C2×C22⋊C411yescharacteristic
89C23×C17#12112yes—
90C23×C17C22⋊Q812yes—
91Q8×C17—42no—
92Q8×C17—42no—
93D4×C17—41no—
94D4×C17—41no—
95D4×C17—42no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—21no—
101C2×C4×C17—22no—
102C2×C4×C17—41no—
103C2×C4×C17—44no—
104C2×C4×C17—41no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17—42no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—42no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—82no—
116C2×C4×C17#9912yes—
117C2×C4×C17C2×C4⋊C411yescharacteristic
118C2×C4×C17#11711yescharacteristic
119C2×Dic17—84no—
120C2×Dic17—84no—
121C2×Dic17—84no—
122#2326D1711yescharacteristic
123C2×#354—172no—
124#1799—172no—
125#1393—172no—
126C22×C17—21no—
127C22×C17—21no—
128C22×C17—22no—
129C22×C17—22no—
130C22×C17—42no—
131C22×C17—42no—
132C22×C17—42no—
133C22×C17—42no—
134C22×C17—82no—
135C22×C17#38011yescharacteristic
136C22×C17C2×#9911yescharacteristic
137C22×C17#41011yescharacteristic
138C4×C17—21no—
139C4×C17—21no—
140C4×C17—22no—
141C4×C17—42no—
142C4×C17—42no—
143C4×C17—42no—
144C4×C17—41no—
145C4×C17—81no—
146C4×C17—81no—
147Dic17—164no—
148C2×#124C2×D1711yescharacteristic
149C2×C4×Q8Dic1711yescharacteristic, complemented
150C2×C4×D4C2×D1711yescharacteristic
151C2×#99—174no—
152#391Dic1711yescharacteristic, complemented
153#390C2×D1711yescharacteristic
154#388—22no—
155#385—22no—
156#384Dic1711yescharacteristic, complemented
157#380—342no—
158#376Dic1711yescharacteristic, complemented
159#354—344no—
160#342—342no—
161C2×C17—21no—
162C2×C17—21no—
163C2×C17—42no—
164C2×C17—42no—
165C2×C17#179911yescharacteristic
166C2×C17#212711yescharacteristic
167C2×C17#212911yescharacteristic
168C22×Q8C2×Dic1711yescharacteristic
169C22×D4C22×D1711yescharacteristic
170C23×C4C22⋊D1712yes—
171#124—22no—
172#124Dic3412yes—
173#124D6812yes—
174C4×Q8—42no—
175C4×D4—42no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—682no—
179C2×C4⋊C4C2×Dic1711yescharacteristic
180C2×C4⋊C4C2×Dic1711yescharacteristic
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—344no—
184C2×C22⋊C4C22⋊D1712yes—
185C2×C42—21no—
186C2×C42C2×Dic1711yescharacteristic
187C2×C42C2×Dic1711yescharacteristic
188C2×C42C2×Dic1711yescharacteristic
189#99—344no—
190#99—344no—
191#95—21no—
192#95—22no—
193#95—21no—
194#95—682no—
195C17#1280011yescharacteristic, complemented, Sylow
196C24C2×C22⋊D1712yes—
197C2×Q8—21no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8C4⋊Dic1712yescomplemented
201C2×D4—41no—
202C2×D4—41no—
203C2×D4C2×Dic3411yescharacteristic
204C2×D4C2×D6811yescharacteristic
205C2×D4#6325212yes—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—82no—
213C22×C4—684no—
214C22×C4C22⋊Dic1711yescharacteristic
215C22×C4C22⋊Dic1712yes—
216C22×C4C22⋊Dic1711yescharacteristic
217C22×C4C22×Dic1711yescharacteristic
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C4⋊C4—1364no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—42no—
226C22⋊C4—82no—
227C22⋊C4—684no—
228C22⋊C4—684no—
229C42—22no—
230C42—21no—
231C42—41no—
232C42—42no—
233C42C4⋊Dic1712yescomplemented
234C23—22no—
235C23—22no—
236C23—42no—
237C23—42no—
238C23#1058761012yes—
239C23#1058761612yes—
240C23C2×C22⋊Dic1711yescharacteristic
241Q8—42no—
242Q8—42no—
243D4—41no—
244D4—41no—
245D4—42no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—44no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—82no—
266C2×C4—1364no—
267C2×C4—1364no—
268C2×C4—1364no—
269C2×C4#1058750512yes—
270C2×C4C2×C4⋊Dic1711yescharacteristic
271C2×C4#1058761211yescharacteristic
272C22—21no—
273C22—21no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—82no—
281C22#1176266911yescharacteristic
282C22C2×#1058750511yescharacteristic
283C22#1176278311yescharacteristic
284C4—21no—
285C4—21no—
286C4—22no—
287C4—42no—
288C4—42no—
289C4—42no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4—2724no—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—42no—
298C2#1452877211yescharacteristic
299C2#1453045311yescharacteristic
300C2#1453045911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#2326C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenter#2326D17
hypocenterC17#12800
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2326

Complements

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

NH ≅ G/Ntype
10. C2×C4×Q8×C17C4semidirect
12. C17×#391C4semidirect
16. C17×#384C4semidirect
17. C17×#376C4semidirect
49. C2×Q8×C17C4⋊C4semidirect
79. C42×C17C4⋊C4semidirect
149. C2×C4×Q8Dic17semidirect
152. #391Dic17semidirect
156. #384Dic17semidirect
158. #376Dic17semidirect
195. C17#12800semidirect
200. C2×Q8C4⋊Dic17semidirect
233. C42C4⋊Dic17semidirect

Maximal subgroups

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

subgroupindex
C2×#117626042
#145287722
#145269092
C17×#23262
#1280017