Finite Group Databasebeta

← #16944613

Subgroups — #16944613

Order 4352.

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

normal subgroups only

This group has 448 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#14528517C211yesmaximal, characteristic
2#14527098C211yesmaximal, characteristic
3#14526805C214yesmaximal
4C17×#2628C211yesmaximal, characteristic
5C2×#10587612—21no—
6#11762862—21no—
7#11762576—21no—
8#11762576C2212yes—
9#11762535—21no—
10#11762533C2212yes—
11C2×C17×C4⋊Q8C411yescharacteristic, complemented
12C2×C17×#117C2211yescharacteristic
13C17×#400—21no—
14C17×#399—21no—
15C17×#395C412yescomplemented
16C17×#393C2212yes—
17C17×#389—21no—
18C17×#382C411yescharacteristic, complemented
19C17×#380—21no—
20C22×Q8×C17C2×C412yes—
21C23×C4×C17D411yescharacteristic
22C17×C4⋊Q8—41no—
23C17×C4⋊Q8—41no—
24C17×#117—41no—
25C17×#117—41no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C22⋊C4×C17—22no—
33C2×C22⋊C4×C17D411yescharacteristic
34C2×C22⋊C4×C17C2311yescharacteristic
35C2×C42×C17C2×C411yescharacteristic
36C2×C42×C17D411yescharacteristic
37C2×C22⋊Dic17—22no—
38#10587612—24no—
39#10587612—42no—
40C2×C4⋊Dic17—41no—
41C2×C4×Dic17—41no—
42C17×#95—22no—
43C17×#95—21no—
44C17×#95—22no—
45C17×#95C2×C412yes—
46C22⋊#3622—22no—
47C4⋊#3622—41no—
48C4×#3622—41no—
49C24×C17C2×D411yescharacteristic
50C2×Q8×C17—42no—
51C2×Q8×C17—82no—
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—22no—
55C22×C4×C17—22no—
56C22×C4×C17—22no—
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17SD1612yes—
62C22×C4×C17Q1612yes—
63C22×C4×C17C22×C411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C22×Dic17—42no—
66C4⋊C4×C17—41no—
67C4⋊C4×C17—41no—
68C4⋊C4×C17—42no—
69C4⋊C4×C17—41no—
70C4⋊C4×C17—41no—
71C4⋊C4×C17—42no—
72C4⋊C4×C17—82no—
73C22⋊C4×C17—41no—
74C22⋊C4×C17—41no—
75C22⋊C4×C17—82no—
76C42×C17—21no—
77C42×C17—21no—
78C42×C17—21no—
79C42×C17—21no—
80C22⋊Dic17—24no—
81C22⋊Dic17—42no—
82C4⋊Dic17—44no—
83C4×Dic17—44no—
84C2×#3622—42no—
85#11717—171nomaximal, Sylow
86C23×C17—21no—
87C23×C17—21no—
88C23×C17—41no—
89C23×C17—41no—
90C23×C17C2×C22⋊C411yescharacteristic
91C23×C17C2×SD1611yescharacteristic
92C23×C17C2×Q1611yescharacteristic
93Q8×C17—84no—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—41no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17#9911yescharacteristic
115C2×C4×C17#9911yescharacteristic
116C2×C4×C17#10312yes—
117C2×C4×C17#10312yes—
118C2×C4×C17#10412yes—
119C2×Dic17—44no—
120C2×Dic17—82no—
121C2×Dic17—82no—
122#3622—82no—
123#2628D1711yescharacteristic
124#1740—171no—
125#1458—171no—
126#1351—174no—
127C22×C17—21no—
128C22×C17—21no—
129C22×C17—21no—
130C22×C17—21no—
131C22×C17—41no—
132C22×C17—41no—
133C22×C17—41no—
134C22×C17—41no—
135C22×C17—81no—
136C22×C17C2×#9911yescharacteristic
137C22×C17C2×#10311yescharacteristic
138C22×C17C2×#10411yescharacteristic
139C4×C17—22no—
140C4×C17—41no—
141C4×C17—41no—
142C4×C17—41no—
143C4×C17—41no—
144C4×C17—82no—
145C4×C17—82no—
146C4×C17#32812yes—
147C4×C17#32812yes—
148Dic17—84no—
149C2×C4⋊Q8Dic1711yescharacteristic, complemented
150C2×#117—341no—
151C2×#117C2×D1711yescharacteristic
152#431—341no—
153#400—21no—
154#399—21no—
155#395Dic1712yescomplemented
156#393C2×D1712yes—
157#389—21no—
158#382Dic1711yescharacteristic, complemented
159#380—21no—
160#342—172no—
161#342—341no—
162#324—341no—
163#323—172no—
164C2×C17—21no—
165C2×C17—21no—
166C2×C17—41no—
167C2×C17—41no—
168C2×C17C2×#32811yescharacteristic
169C2×C17#212211yescharacteristic
170C2×C17#212911yescharacteristic
171C22×Q8C2×Dic1712yes—
172C23×C4C22⋊D1711yescharacteristic
173C4⋊Q8—41no—
174C4⋊Q8—41no—
175#117—41no—
176#117—41no—
177#117—344no—
178#117—682no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—681no—
184C2×C4⋊C4C2×Dic1711yescharacteristic
185C2×C4⋊C4C22⋊D1711yescharacteristic
186C2×C22⋊C4—22no—
187C2×C22⋊C4—342no—
188C2×C22⋊C4C22⋊D1711yescharacteristic
189C2×C22⋊C4C22×D1711yescharacteristic
190C2×C42—681no—
191C2×C42C2×Dic1711yescharacteristic
192C2×C42C22⋊D1711yescharacteristic
193C4⋊C8—681no—
194C22⋊C8—342no—
195C4×C8—681no—
196#95—22no—
197#95—21no—
198#95—22no—
199#95C2×Dic1712yes—
200C17#1171711yescharacteristic, complemented, Sylow
201C24C2×C22⋊D1711yescharacteristic
202C2×Q8—42no—
203C2×Q8—82no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—682no—
210C22×C4#6323012yes—
211C22×C4#6323112yes—
212C22×C4C22⋊Dic1711yescharacteristic
213C22×C4C22⋊Dic1711yescharacteristic
214C22×C4C22⋊Dic1711yescharacteristic
215C22×C4C22⋊Dic1711yescharacteristic
216C22×C4C22×Dic1711yescharacteristic
217C22×C4C2×C22⋊D1711yescharacteristic
218C2×C8—682no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—82no—
226C4⋊C4—684no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—82no—
230C22⋊C4—344no—
231C22⋊C4—682no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—684no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23C2×#6323011yescharacteristic
242C23C2×#6323111yescharacteristic
243C23C2×C22⋊Dic1711yescharacteristic
244Q8—84no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—684no—
266C2×C4—1362no—
267C2×C4—1362no—
268C2×C4#1058750511yescharacteristic
269C2×C4#1058750511yescharacteristic
270C2×C4#1058750612yes—
271C2×C4#1058750612yes—
272C2×C4#1058750812yes—
273C8—1362no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22C2×#1058750511yescharacteristic
284C22C2×#1058750611yescharacteristic
285C22C2×#1058750811yescharacteristic
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—82no—
292C4—82no—
293C4—1364no—
294C4#1176254612yes—
295C4#1176254612yes—
296C2—21no—
297C2—21no—
298C2—41no—
299C2—41no—
300C2C2×#1176254611yescharacteristic
301C2#1453043411yescharacteristic
302C2#1453045911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2628C2
socleC22×C17C2×#99
prime radicalC24×C17C2×D4
cosocleC2×C22⋊C4×C17C23
hypercenter#2628D17
hypocenterC17#11717
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
11. C2×C17×C4⋊Q8C4semidirect
15. C17×#395C4semidirect
18. C17×#382C4semidirect
149. C2×C4⋊Q8Dic17semidirect
155. #395Dic17semidirect
158. #382Dic17semidirect
200. C17#11717semidirect

Maximal subgroups

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

subgroupindex
#145285172
#145270982
#145268052
C17×#26282
#1171717