Finite Group Databasebeta

← #16932131

Subgroups — #16932131

Order 4352.

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

normal subgroups only

This group has 734 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#14533058C211yesmaximal, characteristic
2#14531137C211yesmaximal, characteristic
3#14528881C212yesmaximal, complemented
4#14526638C212yesmaximal, complemented
5C17×#1462C211yesmaximal, characteristic
6C2×#10587612C2211yescharacteristic
7C22×C22⋊Dic17—21no—
8#11762745C412yescomplemented
9#11762734C412yescomplemented
10#11762697—22no—
11#11762682—21no—
12#11762672C2211yescharacteristic
13#11762665—21no—
14C22⋊C4×Dic17—24no—
15C4×C22⋊Dic17—21no—
16C4×C22⋊Dic17C2212yes—
17#11762570—24no—
18C22×C22⋊C4×C17C2211yescharacteristic
19C17×#407C2212yes—
20C17×#323—22no—
21C25×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17C2311yescharacteristic
24C23×Dic17—22no—
25C2×M16×C17—22no—
26C2×C22⋊C4×C17—22no—
27C2×C22⋊C4×C17—22no—
28C2×C22⋊C4×C17C2×C412yes—
29C2×C22⋊C4×C17D412yes—
30C2×C22⋊Dic17—28no—
31C2×C22⋊Dic17—22no—
32C2×C22⋊Dic17—22no—
33C2×C22⋊Dic17—22no—
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—22no—
37#10587612—22no—
38#10587612—22no—
39C2×C4⋊Dic17C2×C411yescharacteristic, complemented
40C2×C4⋊Dic17C2×C411yescharacteristic, complemented
41C2×#63225—22no—
42C2×C4×Dic17—22no—
43C2×C4×Dic17—21no—
44C2×C4×Dic17D414yes—
45C17×C22⋊C8—22no—
46C17×C22⋊C8—22no—
47#10587502—22no—
48#10587502—22no—
49#10587502—21no—
50#10587502—22no—
51#10587502C2×C412yescomplemented
52C24×C17—21no—
53C24×C17—24no—
54C24×C17—21no—
55C24×C17—42no—
56C24×C17—81no—
57C24×C17C22⋊C412yes—
58C24×C17C2×D411yescharacteristic
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—42no—
62C22×C4×C17—81no—
63C22×C4×C17C22⋊C412yes—
64C22×C4×C17C22×C411yescharacteristic
65C22×C4×C17C2×D412yes—
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17#4112yes—
68C22×Dic17—24no—
69C22×Dic17—28no—
70C22×Dic17—22no—
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—82no—
74C22×Dic17—82no—
75M16×C17—44no—
76C2×C8×C17—42no—
77C22⋊C4×C17—24no—
78C22⋊C4×C17—24no—
79C22⋊Dic17—28no—
80C22⋊Dic17—28no—
81C22⋊Dic17—42no—
82C22⋊Dic17—44no—
83C22⋊Dic17—41no—
84C22⋊Dic17—41no—
85C22⋊Dic17—82no—
86C4⋊Dic17—22no—
87C4⋊Dic17—22no—
88#63225—82no—
89C4×Dic17—42no—
90C4×Dic17—44no—
91C4×Dic17—82no—
92#9477—171nomaximal, Sylow
93C23×C17—22no—
94C23×C17—22no—
95C23×C17—24no—
96C23×C17—22no—
97C23×C17—22no—
98C23×C17—22no—
99C23×C17—22no—
100C23×C17—41no—
101C23×C17—42no—
102C23×C17—42no—
103C23×C17—42no—
104C23×C17—42no—
105C23×C17—81no—
106C23×C17—81no—
107C23×C17—88no—
108C23×C17#10414yes—
109C23×C17C2×C22⋊C411yescharacteristic
110C23×C17#12011yescharacteristic
111C23×C17#12311yescharacteristic
112C2×C4×C17—24no—
113C2×C4×C17—22no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—81no—
120C2×C4×C17C4×D412yes—
121C2×C4×C17#12112yes—
122C2×Dic17—28no—
123C2×Dic17—44no—
124C2×Dic17—44no—
125C2×Dic17—42no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—42no—
131C2×Dic17—42no—
132C2×Dic17—88no—
133C8×C17—82no—
134#2482—171no—
135#2273—171no—
136#1826—172no—
137#1462D1711yescharacteristic
138#1284—172no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—24no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—44no—
148C22×C17—44no—
149C22×C17—44no—
150C22×C17—44no—
151C22×C17—42no—
152C22×C17—42no—
153C22×C17—82no—
154C22×C17—84no—
155C22×C17#38611yescharacteristic
156C22×C17C2×#10412yes—
157C22×C17#45812yes—
158C4×C17—22no—
159C4×C17—22no—
160C4×C17—42no—
161Dic17—48no—
162Dic17—82no—
163Dic17—81no—
164Dic17—81no—
165C2×#117—171no—
166C22×C22⋊C4—341no—
167C22×C22⋊C4C2×D1711yescharacteristic
168#407C2×D1712yes—
169#399—172no—
170#397—172no—
171#388—342no—
172#385—341no—
173#381—171no—
174#379—341no—
175C4×C22⋊C4—172no—
176C4×C22⋊C4—344no—
177C4×C22⋊C4—341no—
178#339—344no—
179#323—22no—
180C2×C17—22no—
181C2×C17—22no—
182C2×C17—41no—
183C2×C17—44no—
184C2×C17#189011yescharacteristic
185C2×C17#189611yescharacteristic
186C2×C17#191111yescharacteristic
187C25D6811yescharacteristic
188C23×C4—342no—
189C23×C4D6811yescharacteristic
190C23×C4C22×D1711yescharacteristic
191C2×M16—22no—
192#117—342no—
193#117—342no—
194C2×C4⋊C4—171no—
195C2×C4⋊C4—171no—
196C2×C4⋊C4—342no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—342no—
200C2×C22⋊C4—348no—
201C2×C22⋊C4—342no—
202C2×C22⋊C4—341no—
203C2×C22⋊C4—341no—
204C2×C22⋊C4—342no—
205C2×C22⋊C4—342no—
206C2×C22⋊C4C4×D1712yes—
207C2×C22⋊C4C22⋊D1712yes—
208C2×C42—174no—
209C2×C42—341no—
210C2×C42—342no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213#95—172no—
214#95—342no—
215#95—342no—
216#95—341no—
217#95—342no—
218C17#947711yescharacteristic, complemented, Sylow
219C24—21no—
220C24—24no—
221C24—21no—
222C24—42no—
223C24—81no—
224C24#6322712yes—
225C24C2×D6811yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—42no—
229C22×C4—81no—
230C22×C4—344no—
231C22×C4—348no—
232C22×C4—342no—
233C22×C4—341no—
234C22×C4—341no—
235C22×C4—1362no—
236C22×C4—1362no—
237C22×C4#6322712yes—
238C22×C4C2×C4×D1711yescharacteristic
239C22×C4D4×D1712yes—
240C22×C4#6325412yes—
241C22×C4C2×C22⋊D1711yescharacteristic
242M16—44no—
243C2×C8—42no—
244C4⋊C4—342no—
245C4⋊C4—342no—
246C4⋊C4—1362no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—348no—
250C22⋊C4—348no—
251C22⋊C4—682no—
252C22⋊C4—684no—
253C22⋊C4—681no—
254C22⋊C4—681no—
255C22⋊C4—1362no—
256C42—684no—
257C42—682no—
258C42—1362no—
259C23—22no—
260C23—22no—
261C23—24no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—41no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—81no—
272C23—81no—
273C23—88no—
274C23#1058749614yes—
275C23#1058756811yescharacteristic
276C23#1058757211yescharacteristic
277C23C2×#6322711yescharacteristic
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—81no—
286C2×C4—348no—
287C2×C4—684no—
288C2×C4—684no—
289C2×C4—682no—
290C2×C4—682no—
291C2×C4—682no—
292C2×C4—682no—
293C2×C4—682no—
294C2×C4—681no—
295C2×C4—681no—
296C2×C4—1368no—
297C2×C4#1058756712yes—
298C2×C4#1058762512yes—
299C8—82no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—24no—
307C22—21no—
308C22—44no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—42no—
313C22—42no—
314C22—82no—
315C22—84no—
316C22#1176268611yescharacteristic
317C22C2×#1058749612yes—
318C22#1176300012yes—
319C4—22no—
320C4—22no—
321C4—42no—
322C4—688no—
323C4—1362no—
324C4—1361no—
325C4—1361no—
326C2—22no—
327C2—22no—
328C2—41no—
329C2—44no—
330C2#1452919511yescharacteristic
331C2#1452922511yescharacteristic
332C2#1452931411yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1462C2
socleC22×C17#386
prime radicalC25×C17D4
cosocleC23×C4×C17C23
hypercenter#1462D17
hypocenterC17#9477
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
3. #14528881C2semidirect
4. #14526638C2semidirect
8. #11762745C4semidirect
9. #11762734C4semidirect
39. C2×C4⋊Dic17C2×C4semidirect
40. C2×C4⋊Dic17C2×C4semidirect
51. #10587502C2×C4semidirect
218. C17#9477semidirect

Maximal subgroups

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

subgroupindex
#145330582
#145311372
#145288812
#145266382
C17×#14622
#947717