Finite Group Databasebeta

← #16931675

Subgroups — #16931675

Order 4352.

446 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 103 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#14537422C211yesmaximal, characteristic
2#14527386C212yesmaximal, complemented
3#14527118C211yesmaximal, characteristic
4#14526677C212yesmaximal, complemented
5C17×#2611C211yesmaximal, characteristic
6#11762870—24no—
7#11762866C2212yes—
8#11762833—21no—
9#11762773—21no—
10C4×#63223C2212yes—
11#11762551—24no—
12#11762535—21no—
13#11762533—21no—
14C2×C17×#125C2211yescharacteristic
15C2×C17×#117C2211yescharacteristic
16C17×#400C411yescharacteristic
17C17×#399C411yescharacteristic
18C17×#398C411yescharacteristic
19C17×#397C411yescharacteristic
20C17×#389—21no—
21C17×#388—21no—
22C17×#387—21no—
23C17×#385—21no—
24C17×#376C2211yescharacteristic
25C23×C4×C17D411yescharacteristic
26C17×#125—22no—
27C17×#125—22no—
28C17×#117—22no—
29C17×#117—22no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C4⋊C4×C17C2×C411yescharacteristic
37C2×C4⋊C4×C17D411yescharacteristic
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17C2×C411yescharacteristic
41C2×C22⋊C4×C17D411yescharacteristic
42C2×C42×C17—21no—
43C2×C42×C17D411yescharacteristic
44C2×C42×C17C2311yescharacteristic
45C2×#63223—22no—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95C2×C411yescharacteristic
52C17×#95C2×C411yescharacteristic
53C22⋊#3622—24no—
54C22⋊#3622—22no—
55C4⋊#3622—24no—
56C4⋊#3622—22no—
57#10587474—22no—
58#10587474D414yes—
59C4×#3622—22no—
60C24×C17C22⋊C411yescharacteristic
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—41no—
68C22×C4×C17—41no—
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22×C411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C4⋊C4×C17—22no—
76C4⋊C4×C17—22no—
77C4⋊C4×C17—22no—
78C4⋊C4×C17—22no—
79C4⋊C4×C17—42no—
80C4⋊C4×C17—42no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—42no—
83C4⋊C4×C17—41no—
84C22⋊C4×C17—22no—
85C22⋊C4×C17—22no—
86C22⋊C4×C17—42no—
87C22⋊C4×C17—42no—
88C42×C17—21no—
89C42×C17—21no—
90C42×C17—42no—
91C42×C17C2×D412yes—
92C42×C17#4112yes—
93#63223—42no—
94#63223—42no—
95C2×#3622—24no—
96C2×#3622—42no—
97#9390—171nomaximal, Sylow
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—41no—
102C23×C17—41no—
103C23×C17C2×C22⋊C411yescharacteristic
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—22no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17C4×D412yes—
130C2×C4×C17#12011yescharacteristic
131C2×C4×C17#12112yes—
132C2×C4×C17#12311yescharacteristic
133#3622—44no—
134#3622—82no—
135#2921—171no—
136#2611D1711yescharacteristic
137#1528—172no—
138#1464—171no—
139#1300—172no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17—42no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—41no—
146C22×C17—81no—
147C22×C17#38611yescharacteristic
148C22×C17#42111yescharacteristic
149C22×C17#42111yescharacteristic
150C4×C17—22no—
151C4×C17—21no—
152C4×C17—21no—
153C4×C17—22no—
154C4×C17—42no—
155C4×C17—41no—
156C4×C17—41no—
157C4×C17—41no—
158C4×C17—81no—
159C4×C17—81no—
160C4×C17—81no—
161C2×#125C2×D1711yescharacteristic
162C2×#117C2×D1711yescharacteristic
163#433—344no—
164#432—172no—
165#423—341no—
166#407—341no—
167C4×M16—172no—
168#400Dic1711yescharacteristic
169#399Dic1711yescharacteristic
170#398Dic1711yescharacteristic
171#397Dic1711yescharacteristic
172#389—21no—
173#388—21no—
174#387—21no—
175#385—21no—
176#376C2×D1711yescharacteristic
177#330—344no—
178#324—341no—
179#323—341no—
180C2×C17—21no—
181C2×C17—21no—
182C2×C17—81no—
183C2×C17#188411yescharacteristic
184C2×C17#189711yescharacteristic
185C2×C17#191511yescharacteristic
186C23×C4C22⋊D1711yescharacteristic
187C2×M16—342no—
188#125—22no—
189#125—22no—
190#117—22no—
191#117—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C2×Dic1711yescharacteristic
197C2×C4⋊C4C2×Dic1711yescharacteristic
198C2×C4⋊C4C2×Dic1711yescharacteristic
199C2×C4⋊C4C22⋊D1711yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4C2×Dic1711yescharacteristic
203C2×C22⋊C4C22⋊D1711yescharacteristic
204C2×C42—21no—
205C2×C42C22⋊D1711yescharacteristic
206C2×C42C22×D1711yescharacteristic
207C4⋊C8—344no—
208C4⋊C8—342no—
209C22⋊C8—344no—
210C22⋊C8—342no—
211#97—174no—
212#97—342no—
213C4×C8—342no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95C2×Dic1711yescharacteristic
220#95C2×Dic1711yescharacteristic
221C17#939011yescharacteristic, complemented, Sylow
222C24C22⋊Dic1711yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4C22⋊Dic1711yescharacteristic
232C22×C4C22⋊Dic1711yescharacteristic
233C22×C4C22⋊Dic1711yescharacteristic
234C22×C4C22×Dic1711yescharacteristic
235C22×C4C2×C22⋊D1711yescharacteristic
236C22×C4C2×C22⋊D1711yescharacteristic
237M16—682no—
238M16—682no—
239C2×C8—344no—
240C2×C8—682no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C42—21no—
255C42—21no—
256C42—42no—
257C42D4×D1712yes—
258C42#6325412yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×C22⋊Dic1711yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4D4×Dic1712yes—
291C2×C4#1058762111yescharacteristic
292C2×C4#1058762311yescharacteristic
293C2×C4#1058762512yes—
294C8—684no—
295C8—1362no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#1176268911yescharacteristic
304C22#1176282611yescharacteristic
305C22#1176282611yescharacteristic
306C4—22no—
307C4—22no—
308C4—21no—
309C4—21no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—81no—
315C4—81no—
316C4—81no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#1452916811yescharacteristic
321C2#1452923311yescharacteristic
322C2#1452933711yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2611C2
socleC22×C17#386
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#2611D17
hypocenterC17#9390
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2611

Complements

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

NH ≅ G/Ntype
2. #14527386C2semidirect
4. #14526677C2semidirect
221. C17#9390semidirect

Maximal subgroups

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

subgroupindex
#145374222
#145273862
#145271182
#145266772
C17×#26112
#939017