Finite Group Databasebeta

← #16932299

Subgroups — #16932299

Order 4352.

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

normal subgroups only

This group has 522 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×#2487C211yesmaximal, characteristic
6#11762887—24no—
7#11762862—21no—
8#11762773—21no—
9#11762773C2212yes—
10#11762773C2212yes—
11#11762578—24no—
12#11762535—21no—
13#11762533—21no—
14C2×C4×D4×C17C2211yescharacteristic
15C22×C4⋊C4×C17C2211yescharacteristic
16C17×#400—21no—
17C17×#399—21no—
18C17×#389—21no—
19C17×#386C412yes—
20C17×#385C412yes—
21C17×#380—21no—
22C17×#380C2211yescharacteristic
23C22×D4×C17C2×C411yescharacteristic
24C23×C4×C17C2×C412yes—
25C23×C4×C17D411yescharacteristic
26C23×C4×C17C2311yescharacteristic
27C4×D4×C17—22no—
28C4×D4×C17—22no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—22no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—22no—
34C2×C4⋊C4×C17—22no—
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C22⋊C4×C17—22no—
37C2×C22⋊C4×C17C2×C411yescharacteristic
38C2×C22⋊C4×C17C2×C412yes—
39C2×C22⋊C4×C17D411yescharacteristic
40C2×C42×C17D411yescharacteristic
41C2×#63223—22no—
42C2×#63223—22no—
43C2×#63223D414yes—
44C17×#95—22no—
45C17×#95—22no—
46C17×#95—21no—
47C22⋊#3622—22no—
48C22⋊#3622—22no—
49C22⋊#3622—22no—
50C22⋊#3622—22no—
51C22⋊#3622—22no—
52C4⋊#3622—22no—
53C4×#3622—22no—
54C24×C17C22⋊C411yescharacteristic
55C24×C17C22×C411yescharacteristic
56C2×D4×C17—22no—
57C2×D4×C17—22no—
58C2×D4×C17—41no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—22no—
62C22×C4×C17—22no—
63C22×C4×C17—24no—
64C22×C4×C17—42no—
65C22×C4×C17—42no—
66C22×C4×C17—82no—
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C2×D412yes—
71C22×C4×C17C2×D411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×C4×C17#4112yes—
74C4⋊C4×C17—22no—
75C4⋊C4×C17—42no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—42no—
78C4⋊C4×C17—41no—
79C4⋊C4×C17—82no—
80C22⋊C4×C17—22no—
81C22⋊C4×C17—22no—
82C22⋊C4×C17—42no—
83C22⋊C4×C17—82no—
84C42×C17—21no—
85C42×C17—21no—
86#63223—44no—
87#63223—44no—
88#63223—82no—
89C2×#3622—24no—
90C2×#3622—42no—
91#9510—171nomaximal, Sylow
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17—41no—
100C23×C17—41no—
101C23×C17C2×C22⋊C411yescharacteristic
102C23×C17C4×D412yes—
103D4×C17—44no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—22no—
107C2×C4×C17—24no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—41no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17—42no—
121C2×C4×C17—81no—
122C2×C4×C17—84no—
123C2×C4×C17#10112yes—
124C2×C4×C17#12011yescharacteristic
125C2×C4×C17#12112yes—
126C2×C4×C17#12311yescharacteristic
127#3622—44no—
128#3622—82no—
129#2975—171no—
130#2487D1711yescharacteristic
131#1821—172no—
132#1465—171no—
133#1305—172no—
134C22×C17—21no—
135C22×C17—21no—
136C22×C17—22no—
137C22×C17—22no—
138C22×C17—22no—
139C22×C17—42no—
140C22×C17—42no—
141C22×C17—42no—
142C22×C17—41no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—81no—
146C22×C17#38611yescharacteristic
147C22×C17C2×#10111yescharacteristic
148C22×C17#41311yescharacteristic
149C4×C17—22no—
150C4×C17—21no—
151C4×C17—22no—
152C4×C17—41no—
153C4×C17—41no—
154C4×C17—82no—
155C4×C17—82no—
156C4×C17#45812yes—
157C2×C4×D4C2×D1711yescharacteristic
158C22×C4⋊C4C2×D1711yescharacteristic
159#436—344no—
160#431—341no—
161#407—172no—
162#407—172no—
163#407—341no—
164#400—21no—
165#399—21no—
166#389—21no—
167#386Dic1712yes—
168#385Dic1712yes—
169#380—21no—
170#380C2×D1711yescharacteristic
171#343—344no—
172#324—341no—
173#323—341no—
174C2×C17—21no—
175C2×C17—21no—
176C2×C17—42no—
177C2×C17—81no—
178C2×C17#189011yescharacteristic
179C2×C17#190111yescharacteristic
180C2×C17#191411yescharacteristic
181C22×D4C2×Dic1711yescharacteristic
182C23×C4C2×Dic1712yes—
183C23×C4C22⋊D1711yescharacteristic
184C23×C4C22×D1711yescharacteristic
185C2×M16—174no—
186C2×M16—342no—
187C2×M16—342no—
188C4×D4—22no—
189C4×D4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4C22⋊D1711yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4C2×Dic1711yescharacteristic
199C2×C22⋊C4C2×Dic1712yes—
200C2×C22⋊C4C22⋊D1711yescharacteristic
201C2×C42C22⋊D1711yescharacteristic
202C4⋊C8—342no—
203C22⋊C8—342no—
204C22⋊C8—342no—
205C22⋊C8—342no—
206C22⋊C8—342no—
207C22⋊C8—342no—
208C4×C8—342no—
209#95—22no—
210#95—22no—
211#95—21no—
212C17#951011yescharacteristic, complemented, Sylow
213C24C22⋊Dic1711yescharacteristic
214C24C22×Dic1711yescharacteristic
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—41no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—82no—
226C22×C4C22⋊Dic1711yescharacteristic
227C22×C4C22⋊Dic1711yescharacteristic
228C22×C4C22⋊Dic1711yescharacteristic
229C22×C4D4×D1712yes—
230C22×C4#6325412yes—
231C22×C4C2×C22⋊D1711yescharacteristic
232C22×C4C2×C22⋊D1711yescharacteristic
233M16—684no—
234M16—684no—
235M16—1362no—
236C2×C8—344no—
237C2×C8—682no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—82no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—42no—
247C22⋊C4—82no—
248C42—21no—
249C42—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23D4×Dic1712yes—
260C23C2×C22⋊Dic1711yescharacteristic
261D4—44no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—81no—
280C2×C4—84no—
281C2×C4#1058750712yes—
282C2×C4#1058762111yescharacteristic
283C2×C4#1058762311yescharacteristic
284C2×C4#1058762512yes—
285C8—684no—
286C8—1362no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22#1176268911yescharacteristic
300C22C2×#1058750711yescharacteristic
301C22#1176279311yescharacteristic
302C4—22no—
303C4—21no—
304C4—22no—
305C4—41no—
306C4—41no—
307C4—82no—
308C4—82no—
309C4#1176299812yes—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—81no—
314C2#1452919811yescharacteristic
315C2#1452925311yescharacteristic
316C2#1452933211yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC24×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2487C2
socleC22×C17#386
prime radicalC22×D4×C17C2×C4
cosocleC23×C4×C17C23
hypercenter#2487D17
hypocenterC17#9510
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. #14528864C2semidirect
4. #14526690C2semidirect
212. C17#9510semidirect

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×#24872
#951017