Finite Group Databasebeta

← #16931777

Subgroups — #16931777

Order 4352.

606 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 99 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#14533942C211yesmaximal, characteristic
2#14531382C211yesmaximal, characteristic
3#14527305C212yesmaximal, complemented
4#14526669C212yesmaximal, complemented
5C17×#1462C211yesmaximal, characteristic
6C2×#10587610C2211yescharacteristic
7#11762734C412yescomplemented
8#11762721C412yescomplemented
9#11762703—22no—
10#11762702—24no—
11#11762672—21no—
12#11762669—21no—
13#11762665—21no—
14#11762665—21no—
15C4×C22⋊Dic17C2211yescharacteristic
16C4×C22⋊Dic17C2212yes—
17#11762547—24no—
18C22×C22⋊C4×C17C2211yescharacteristic
19C17×#407C2212yes—
20C17×#323—22no—
21C25×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17C2311yescharacteristic
24C22×Dic34C2×C411yescharacteristic
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—22no—
31C2×C22⋊Dic17—21no—
32C2×C22⋊Dic17—22no—
33C2×C22⋊Dic17—21no—
34C2×C22⋊Dic17—22no—
35C2×C22⋊Dic17—22no—
36C2×C22⋊Dic17—22no—
37C2×C4⋊Dic17—22no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39#10587610—22no—
40#10587610—22no—
41C2×#63225—21no—
42C2×C4×Dic17—22no—
43C2×C4×Dic17—21no—
44C17×C22⋊C8—22no—
45C17×C22⋊C8—22no—
46#10587502—22no—
47#10587502—22no—
48#10587502—22no—
49#10587502C2×C412yes—
50#10587502D414yes—
51C24×C17—21no—
52C24×C17—21no—
53C24×C17—42no—
54C24×C17—42no—
55C24×C17—81no—
56C24×C17C22⋊C412yes—
57C24×C17C2×D411yescharacteristic
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—42no—
61C22×C4×C17—81no—
62C22×C4×C17C22⋊C412yes—
63C22×C4×C17C22×C411yescharacteristic
64C22×C4×C17C2×D412yes—
65C22×C4×C17C2×D411yescharacteristic
66C22×C4×C17#4112yes—
67C22×Dic17—21no—
68C22×Dic17—22no—
69C22×Dic17—21no—
70C22×Dic17—24no—
71C2×Dic34—22no—
72C2×Dic34—81no—
73M16×C17—82no—
74C2×C8×C17—42no—
75C22⋊C4×C17—24no—
76C22⋊C4×C17—24no—
77C22⋊Dic17—44no—
78C22⋊Dic17—42no—
79C22⋊Dic17—44no—
80C22⋊Dic17—41no—
81C22⋊Dic17—41no—
82C22⋊Dic17—82no—
83C22⋊Dic17—82no—
84C22⋊Dic17—82no—
85C4⋊Dic17—22no—
86C4⋊Dic17—82no—
87#63225—42no—
88C4×Dic17—82no—
89C4×Dic17—81no—
90#9411—171nomaximal, Sylow
91C23×C17—22no—
92C23×C17—22no—
93C23×C17—22no—
94C23×C17—22no—
95C23×C17—22no—
96C23×C17—41no—
97C23×C17—42no—
98C23×C17—42no—
99C23×C17—42no—
100C23×C17—42no—
101C23×C17—42no—
102C23×C17—42no—
103C23×C17—81no—
104C23×C17—81no—
105C23×C17—88no—
106C23×C17C2×C22⋊C411yescharacteristic
107C23×C17#12011yescharacteristic
108C23×C17#12311yescharacteristic
109C2×C4×C17—22no—
110C2×C4×C17—42no—
111C2×C4×C17—41no—
112C2×C4×C17—42no—
113C2×C4×C17—41no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—81no—
117C2×C4×C17C4×D412yes—
118C2×C4×C17#12112yes—
119C2×Dic17—42no—
120C2×Dic17—44no—
121C2×Dic17—42no—
122C2×Dic17—42no—
123C2×Dic17—44no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—44no—
129Dic34—82no—
130C8×C17—82no—
131#2561—171no—
132#2312—171no—
133#1508—172no—
134#1462D1711yescharacteristic
135#1297—172no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17—21no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—44no—
144C22×C17—44no—
145C22×C17—42no—
146C22×C17—42no—
147C22×C17—42no—
148C22×C17—82no—
149C22×C17—84no—
150C22×C17—84no—
151C22×C17#38611yescharacteristic
152C22×C17#42112yes—
153C22×C17#45512yes—
154C4×C17—22no—
155C4×C17—22no—
156C4×C17—42no—
157Dic17—84no—
158Dic17—82no—
159Dic17—81no—
160Dic17—81no—
161C2×C22⋊Q8—171no—
162C22×C22⋊C4C2×D1711yescharacteristic
163#407C2×D1712yes—
164#397—172no—
165#393—172no—
166#388—344no—
167#388—342no—
168#381—341no—
169#380—341no—
170#379—341no—
171#379—341no—
172C4×C22⋊C4—171no—
173C4×C22⋊C4—172no—
174#328—344no—
175#323—22no—
176C2×C17—22no—
177C2×C17—22no—
178C2×C17—41no—
179C2×C17—82no—
180C2×C17#188511yescharacteristic
181C2×C17#189711yescharacteristic
182C2×C17#191311yescharacteristic
183C25D6811yescharacteristic
184C22×Q8—171no—
185C23×C4D6811yescharacteristic
186C23×C4C22×D1711yescharacteristic
187C2×M16—22no—
188C22⋊Q8—342no—
189C22⋊Q8—342no—
190C2×C4⋊C4—171no—
191C2×C4⋊C4—341no—
192C2×C4⋊C4—342no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—341no—
196C2×C22⋊C4—342no—
197C2×C22⋊C4—341no—
198C2×C22⋊C4—342no—
199C2×C22⋊C4—342no—
200C2×C22⋊C4—342no—
201C2×C22⋊C4—342no—
202C2×C22⋊C4C4×D1712yes—
203C2×C22⋊C4C22⋊D1712yes—
204C2×C42—342no—
205C2×C42—341no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208#95—172no—
209#95—174no—
210#95—342no—
211#95—342no—
212#95—342no—
213C17#941111yescharacteristic, complemented, Sylow
214C24—21no—
215C24—21no—
216C24—42no—
217C24—42no—
218C24—81no—
219C24#6322712yes—
220C24C2×D6811yescharacteristic
221C2×Q8—342no—
222C2×Q8—1361no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—81no—
227C22×C4—344no—
228C22×C4—342no—
229C22×C4—341no—
230C22×C4—341no—
231C22×C4#6322712yes—
232C22×C4C2×C4×D1711yescharacteristic
233C22×C4D4×D1712yes—
234C22×C4#6325412yes—
235C22×C4C2×C22⋊D1711yescharacteristic
236M16—82no—
237C2×C8—42no—
238C4⋊C4—342no—
239C4⋊C4—682no—
240C4⋊C4—1362no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—682no—
244C22⋊C4—684no—
245C22⋊C4—684no—
246C22⋊C4—681no—
247C22⋊C4—681no—
248C22⋊C4—1362no—
249C22⋊C4—1362no—
250C22⋊C4—1362no—
251C42—1362no—
252C42—1361no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—41no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—81no—
266C23—81no—
267C23—88no—
268C23#1058756811yescharacteristic
269C23#1058757211yescharacteristic
270C23C2×#6322711yescharacteristic
271Q8—1362no—
272C2×C4—22no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—81no—
280C2×C4—684no—
281C2×C4—682no—
282C2×C4—682no—
283C2×C4—684no—
284C2×C4—682no—
285C2×C4—684no—
286C2×C4—682no—
287C2×C4—682no—
288C2×C4—681no—
289C2×C4—681no—
290C2×C4#1058756712yes—
291C2×C4#1058762512yes—
292C8—82no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—44no—
301C22—44no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22—84no—
307C22—84no—
308C22#1176268611yescharacteristic
309C22#1176282512yes—
310C22#1176298912yes—
311C4—22no—
312C4—22no—
313C4—42no—
314C4—1364no—
315C4—1362no—
316C4—1361no—
317C4—1361no—
318C2—22no—
319C2—22no—
320C2—41no—
321C2—82no—
322C2#1452917011yescharacteristic
323C2#1452923011yescharacteristic
324C2#1452932411yescharacteristic
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#9411
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. #14527305C2semidirect
4. #14526669C2semidirect
7. #11762734C4semidirect
8. #11762721C4semidirect
213. C17#9411semidirect

Maximal subgroups

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

subgroupindex
#145339422
#145313822
#145273052
#145266692
C17×#14622
#941117