Finite Group Databasebeta

← #16932289

Subgroups — #16932289

Order 4352.

542 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 111 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#14537825C211yesmaximal, characteristic
2#14528864C212yesmaximal, complemented
3#14527122C211yesmaximal, characteristic
4#14526690C212yesmaximal, complemented
5C17×#2486C211yesmaximal, characteristic
6#11762887—24no—
7#11762862—21no—
8#11762773—21no—
9#11762773C2212yes—
10#11762773C2212yes—
11#11762578—24no—
12#11762535—21no—
13#11762533—21no—
14C2×C17×#123C412yes—
15C2×C4×D4×C17C2211yescharacteristic
16C22×C4⋊C4×C17C2211yescharacteristic
17C17×#398—21no—
18C17×#397—21no—
19C17×#389—21no—
20C17×#381C412yes—
21C17×#380—21no—
22C17×#380C2211yescharacteristic
23C22×D4×C17C2×C411yescharacteristic
24C23×C4×C17C2×C412yes—
25C23×C4×C17D411yescharacteristic
26C23×C4×C17C2311yescharacteristic
27C17×#123—44no—
28C4×D4×C17—22no—
29C4×D4×C17—22no—
30C2×C4⋊C4×C17—22no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—22no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17—22no—
36C2×C4⋊C4×C17D411yescharacteristic
37C2×C22⋊C4×C17—22no—
38C2×C22⋊C4×C17C2×C411yescharacteristic
39C2×C22⋊C4×C17C2×C412yes—
40C2×C22⋊C4×C17D411yescharacteristic
41C2×C42×C17D411yescharacteristic
42C2×#63223—22no—
43C2×#63223—22no—
44C2×#63223D414yes—
45C17×#95—22no—
46C17×#95—22no—
47C17×#95—21no—
48C22⋊#3622—22no—
49C22⋊#3622—22no—
50C22⋊#3622—22no—
51C22⋊#3622—22no—
52C22⋊#3622—22no—
53C4⋊#3622—22no—
54C4×#3622—22no—
55C24×C17C22⋊C411yescharacteristic
56C24×C17C22×C411yescharacteristic
57C2×D4×C17—22no—
58C2×D4×C17—22no—
59C2×D4×C17—41no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—22no—
63C22×C4×C17—22no—
64C22×C4×C17—24no—
65C22×C4×C17—42no—
66C22×C4×C17—42no—
67C22×C4×C17—42no—
68C22×C4×C17—42no—
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C2×D412yes—
73C22×C4×C17C2×D411yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C22×C4×C17#4112yes—
76C4⋊C4×C17—22no—
77C4⋊C4×C17—42no—
78C4⋊C4×C17—44no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—42no—
81C4⋊C4×C17—41no—
82C22⋊C4×C17—22no—
83C22⋊C4×C17—22no—
84C22⋊C4×C17—44no—
85C22⋊C4×C17—42no—
86C42×C17—21no—
87C42×C17—21no—
88#63223—44no—
89#63223—44no—
90#63223—82no—
91C2×#3622—24no—
92C2×#3622—42no—
93#9508—171nomaximal, Sylow
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—41no—
100C23×C17—41no—
101C23×C17—41no—
102C23×C17—41no—
103C23×C17C2×C22⋊C411yescharacteristic
104C23×C17C4×D412yes—
105D4×C17—44no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—22no—
109C2×C4×C17—24no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—41no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17—42no—
121C2×C4×C17—42no—
122C2×C4×C17—42no—
123C2×C4×C17—81no—
124C2×C4×C17—84no—
125C2×C4×C17#10012yes—
126C2×C4×C17#12011yescharacteristic
127C2×C4×C17#12112yes—
128C2×C4×C17#12311yescharacteristic
129#3622—44no—
130#3622—82no—
131#2975—171no—
132#2486D1711yescharacteristic
133#1821—172no—
134#1465—171no—
135#1305—172no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17—22no—
139C22×C17—22no—
140C22×C17—22no—
141C22×C17—42no—
142C22×C17—42no—
143C22×C17—42no—
144C22×C17—41no—
145C22×C17—41no—
146C22×C17—41no—
147C22×C17—81no—
148C22×C17#38611yescharacteristic
149C22×C17C2×#10011yescharacteristic
150C22×C17#41311yescharacteristic
151C4×C17—22no—
152C4×C17—21no—
153C4×C17—22no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—82no—
157C4×C17—82no—
158C4×C17#45812yes—
159C2×#123Dic1712yes—
160C2×C4×D4C2×D1711yescharacteristic
161C22×C4⋊C4C2×D1711yescharacteristic
162#436—344no—
163#431—341no—
164#407—172no—
165#407—172no—
166#407—341no—
167#398—21no—
168#397—21no—
169#389—21no—
170#381Dic1712yes—
171#380—21no—
172#380C2×D1711yescharacteristic
173#343—344no—
174#324—341no—
175#323—341no—
176C2×C17—21no—
177C2×C17—21no—
178C2×C17—42no—
179C2×C17—81no—
180C2×C17#189011yescharacteristic
181C2×C17#190011yescharacteristic
182C2×C17#191511yescharacteristic
183C22×D4C2×Dic1711yescharacteristic
184C23×C4C2×Dic1712yes—
185C23×C4C22⋊D1711yescharacteristic
186C23×C4C22×D1711yescharacteristic
187C2×M16—174no—
188C2×M16—342no—
189C2×M16—342no—
190#123—44no—
191C4×D4—22no—
192C4×D4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C22⋊D1711yescharacteristic
200C2×C22⋊C4—22no—
201C2×C22⋊C4C2×Dic1711yescharacteristic
202C2×C22⋊C4C2×Dic1712yes—
203C2×C22⋊C4C22⋊D1711yescharacteristic
204C2×C42C22⋊D1711yescharacteristic
205C4⋊C8—342no—
206C22⋊C8—342no—
207C22⋊C8—342no—
208C22⋊C8—342no—
209C22⋊C8—342no—
210C22⋊C8—342no—
211C4×C8—342no—
212#95—22no—
213#95—22no—
214#95—21no—
215C17#950811yescharacteristic, complemented, Sylow
216C24C22⋊Dic1711yescharacteristic
217C24C22×Dic1711yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4C22⋊Dic1711yescharacteristic
231C22×C4C22⋊Dic1711yescharacteristic
232C22×C4C22⋊Dic1711yescharacteristic
233C22×C4D4×D1712yes—
234C22×C4#6325412yes—
235C22×C4C2×C22⋊D1711yescharacteristic
236C22×C4C2×C22⋊D1711yescharacteristic
237M16—684no—
238M16—684no—
239M16—1362no—
240C2×C8—344no—
241C2×C8—682no—
242C4⋊C4—22no—
243C4⋊C4—42no—
244C4⋊C4—44no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—44no—
251C22⋊C4—42no—
252C42—21no—
253C42—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23D4×Dic1712yes—
264C23C2×C22⋊Dic1711yescharacteristic
265D4—44no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—81no—
284C2×C4—84no—
285C2×C4#1058750412yes—
286C2×C4#1058762111yescharacteristic
287C2×C4#1058762311yescharacteristic
288C2×C4#1058762512yes—
289C8—684no—
290C8—1362no—
291C22—21no—
292C22—21no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#1176268911yescharacteristic
304C22C2×#1058750411yescharacteristic
305C22#1176279311yescharacteristic
306C4—22no—
307C4—21no—
308C4—22no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—82no—
313C4#1176299812yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—81no—
318C2#1452919811yescharacteristic
319C2#1452924811yescharacteristic
320C2#1452933711yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC24×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2486C2
socleC22×C17#386
prime radicalC22×D4×C17C2×C4
cosocleC23×C4×C17C23
hypercenter#2486D17
hypocenterC17#9508
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2486

Complements

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

NH ≅ G/Ntype
2. #14528864C2semidirect
4. #14526690C2semidirect
215. C17#9508semidirect

Maximal subgroups

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

subgroupindex
#145378252
#145288642
#145271222
#145266902
C17×#24862
#950817