Finite Group Databasebeta

← #16932014

Subgroups — #16932014

Order 4352.

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

normal subgroups only

This group has 498 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#14537813C211yesmaximal, characteristic
2#14527501C212yesmaximal, complemented
3#14527125C211yesmaximal, characteristic
4#14526640C212yesmaximal, complemented
5C17×#2563C211yesmaximal, characteristic
6#11762881—24no—
7#11762866C2212yes—
8#11762866C2212yes—
9#11762773—21no—
10C2×C22⋊#3622—21no—
11#11762555—24no—
12#11762533—21no—
13#11762533—21no—
14C2×C17×C4⋊Q8C2211yescharacteristic
15C2×C4×D4×C17C2211yescharacteristic
16C17×#393—21no—
17C17×#393—21no—
18C17×#390C412yes—
19C17×#382C412yes—
20C17×#380—21no—
21C17×#380—21no—
22C4×C4⋊C4×C17C2211yescharacteristic
23C22×Q8×C17—21no—
24C22×D4×C17C2×C411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C17×C4⋊Q8—22no—
28C17×C4⋊Q8—22no—
29C4×D4×C17—22no—
30C4×D4×C17—22no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17C2×C412yes—
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C22⋊C4×C17—22no—
36C2×C22⋊C4×C17—22no—
37C2×C22⋊C4×C17D411yescharacteristic
38C2×C22⋊C4×C17D411yescharacteristic
39C2×C42×C17C2×C412yes—
40C2×C42×C17C2311yescharacteristic
41C2×#63223—22no—
42C22×#3622—22no—
43C17×#95—22no—
44C17×#95—22no—
45C17×#95—21no—
46C22⋊#3622—22no—
47C22⋊#3622—22no—
48C22⋊#3622—22no—
49C22⋊#3622—22no—
50C4⋊#3622—22no—
51C4⋊#3622D414yes—
52#10587474—22no—
53#10587474—22no—
54C24×C17C22⋊C411yescharacteristic
55C24×C17C22⋊C411yescharacteristic
56C2×Q8×C17—42no—
57C2×Q8×C17—82no—
58C2×D4×C17—22no—
59C2×D4×C17—22no—
60C2×D4×C17—41no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—22no—
66C22×C4×C17—22no—
67C22×C4×C17—42no—
68C22×C4×C17—42no—
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22×C411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C4⋊C4×C17—22no—
75C4⋊C4×C17—24no—
76C4⋊C4×C17—42no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—41no—
79C22⋊C4×C17—22no—
80C22⋊C4×C17—22no—
81C22⋊C4×C17—82no—
82C22⋊C4×C17—82no—
83C42×C17—42no—
84C42×C17C2×D412yes—
85C42×C17#4112yes—
86#63223—82no—
87C2×#3622—24no—
88C2×#3622—42no—
89C2×#3622—82no—
90#9453—171nomaximal, Sylow
91C23×C17—21no—
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—41no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17—41no—
100C23×C17#10112yes—
101C23×C17C2×C22⋊C411yescharacteristic
102Q8×C17—84no—
103D4×C17—44no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—22no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—41no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17—81no—
121C2×C4×C17—81no—
122C2×C4×C17C4×D412yes—
123C2×C4×C17#12011yescharacteristic
124C2×C4×C17#12112yes—
125C2×C4×C17#12311yescharacteristic
126#3622—44no—
127#3622—82no—
128#2973—171no—
129#2563D1711yescharacteristic
130#1556—172no—
131#1466—171no—
132#1285—172no—
133C22×C17—21no—
134C22×C17—21no—
135C22×C17—22no—
136C22×C17—42no—
137C22×C17—42no—
138C22×C17—41no—
139C22×C17—41no—
140C22×C17—41no—
141C22×C17—41no—
142C22×C17—41no—
143C22×C17—81no—
144C22×C17—81no—
145C22×C17#38611yescharacteristic
146C22×C17C2×#10111yescharacteristic
147C22×C17#41311yescharacteristic
148C4×C17—21no—
149C4×C17—22no—
150C4×C17—22no—
151C4×C17—41no—
152C4×C17—41no—
153C4×C17—82no—
154C4×C17—82no—
155C4×C17#45512yes—
156C2×C4⋊Q8C2×D1711yescharacteristic
157C2×C4×D4C2×D1711yescharacteristic
158#435—344no—
159#432—172no—
160#432—172no—
161#407—341no—
162C2×C22⋊C8—341no—
163#393—21no—
164#393—21no—
165#390Dic1712yes—
166#382Dic1712yes—
167#380—21no—
168#380—21no—
169C4×C4⋊C4C2×D1711yescharacteristic
170#332—344no—
171#323—341no—
172#323—341no—
173C2×C17—21no—
174C2×C17—21no—
175C2×C17—42no—
176C2×C17—81no—
177C2×C17#188511yescharacteristic
178C2×C17#190111yescharacteristic
179C2×C17#191011yescharacteristic
180C22×Q8—21no—
181C22×D4C2×Dic1711yescharacteristic
182C23×C4C22⋊D1711yescharacteristic
183C23×C4C22⋊D1711yescharacteristic
184C2×M16—342no—
185C22×C8—342no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4×D4—22no—
189C4×D4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C2×Dic1712yes—
193C2×C4⋊C4C2×Dic1711yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4C22⋊D1711yescharacteristic
197C2×C22⋊C4C22⋊D1711yescharacteristic
198C2×C42C2×Dic1712yes—
199C2×C42C22×D1711yescharacteristic
200C4⋊C8—174no—
201C4⋊C8—342no—
202C22⋊C8—342no—
203C22⋊C8—342no—
204C22⋊C8—342no—
205C22⋊C8—342no—
206#97—342no—
207#97—342no—
208#95—22no—
209#95—22no—
210#95—21no—
211C17#945311yescharacteristic, complemented, Sylow
212C24C22⋊Dic1711yescharacteristic
213C24C22⋊Dic1711yescharacteristic
214C2×Q8—42no—
215C2×Q8—82no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4C22⋊Dic1711yescharacteristic
228C22×C4C22⋊Dic1711yescharacteristic
229C22×C4C22×Dic1711yescharacteristic
230C22×C4C2×C22⋊D1711yescharacteristic
231C22×C4C2×C22⋊D1711yescharacteristic
232M16—1362no—
233C2×C8—344no—
234C2×C8—682no—
235C2×C8—1362no—
236C4⋊C4—22no—
237C4⋊C4—24no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—82no—
244C22⋊C4—82no—
245C42—42no—
246C42D4×D1712yes—
247C42#6325412yes—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23#1058750712yes—
258C23C2×C22⋊Dic1711yescharacteristic
259Q8—84no—
260D4—44no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4D4×Dic1712yes—
280C2×C4#1058762111yescharacteristic
281C2×C4#1058762311yescharacteristic
282C2×C4#1058762512yes—
283C8—684no—
284C8—1362no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#1176268911yescharacteristic
298C22C2×#1058750711yescharacteristic
299C22#1176279311yescharacteristic
300C4—22no—
301C4—21no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—82no—
306C4—82no—
307C4#1176298612yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—81no—
312C2#1452917311yescharacteristic
313C2#1452925311yescharacteristic
314C2#1452931211yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2563C2
socleC22×C17#386
prime radicalC22×D4×C17C2×C4
cosocleC2×C42×C17C23
hypercenter#2563D17
hypocenterC17#9453
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2563

Complements

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

NH ≅ G/Ntype
2. #14527501C2semidirect
4. #14526640C2semidirect
211. C17#9453semidirect

Maximal subgroups

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

subgroupindex
#145378132
#145275012
#145271252
#145266402
C17×#25632
#945317