Finite Group Databasebeta

← #16945217

Subgroups — #16945217

Order 4352.

382 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 87 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#14528861C211yesmaximal, characteristic
2#14527098C211yesmaximal, characteristic
3#14526808C212yesmaximal
4#14526784C212yesmaximal
5C17×#2611C211yesmaximal, characteristic
6#11762862—21no—
7#11762672—21no—
8#11762576—21no—
9#11762576C2212yes—
10#11762535—21no—
11#11762533C2212yes—
12C2×C17×#125C411yescharacteristic, complemented
13C2×C17×#117C2211yescharacteristic
14C17×#400C411yescharacteristic, complemented
15C17×#399C2211yescharacteristic
16C17×#398C411yescharacteristic, complemented
17C17×#397C2211yescharacteristic
18C17×#389—21no—
19C17×#388—21no—
20C17×#387—21no—
21C17×#385—21no—
22C17×#376C411yescharacteristic, complemented
23C23×C4×C17D411yescharacteristic
24C17×#125—41no—
25C17×#125—41no—
26C17×#117—41no—
27C17×#117—41no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17D411yescharacteristic
39C2×C22⋊C4×C17C2311yescharacteristic
40C2×C42×C17—21no—
41C2×C42×C17C2×C411yescharacteristic
42C2×C42×C17D411yescharacteristic
43C2×C22⋊Dic17—22no—
44C17×#95—21no—
45C17×#95—21no—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95C2×C411yescharacteristic
50C17×#95C2×C411yescharacteristic
51#10587502—41no—
52#10587502—41no—
53C22⋊#3622—22no—
54C4⋊#3622—41no—
55C4×#3622—41no—
56C24×C17C2×D411yescharacteristic
57C22×C4×C17—21no—
58C22×C4×C17—22no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—41no—
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22×C411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×Dic17—42no—
72C4⋊C4×C17—41no—
73C4⋊C4×C17—41no—
74C4⋊C4×C17—41no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—42no—
77C4⋊C4×C17—42no—
78C4⋊C4×C17—42no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C22⋊C4×C17—41no—
82C22⋊C4×C17—41no—
83C22⋊C4×C17—42no—
84C22⋊C4×C17—42no—
85C42×C17—21no—
86C42×C17—21no—
87C42×C17—21no—
88C42×C17—21no—
89C42×C17—42no—
90C22⋊Dic17—42no—
91C22⋊Dic17—42no—
92C2×#3622—42no—
93#11846—171nomaximal, Sylow
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—41no—
97C23×C17—41no—
98C23×C17C2×C22⋊C411yescharacteristic
99C23×C17#13512yes—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—22no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
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#9911yescharacteristic
128C2×C4×C17#9911yescharacteristic
129C2×Dic17—82no—
130C2×Dic17—82no—
131C2×Dic17—82no—
132#3622—82no—
133#2611D1711yescharacteristic
134#1820—171no—
135#1458—171no—
136#1352—172no—
137#1344—172no—
138C22×C17—21no—
139C22×C17—21no—
140C22×C17—22no—
141C22×C17—42no—
142C22×C17—41no—
143C22×C17—41no—
144C22×C17—81no—
145C22×C17C2×#9911yescharacteristic
146C22×C17#41611yescharacteristic
147C22×C17#42111yescharacteristic
148C4×C17—21no—
149C4×C17—21no—
150C4×C17—42no—
151C4×C17—41no—
152C4×C17—41no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—81no—
157C4×C17—81no—
158C4×C17—81no—
159Dic17—162no—
160C2×#125Dic1711yescharacteristic, complemented
161C2×#117C2×D1711yescharacteristic
162#431—341no—
163#400Dic1711yescharacteristic, complemented
164#399C2×D1711yescharacteristic
165#398Dic1711yescharacteristic, complemented
166#397C2×D1711yescharacteristic
167#389—21no—
168#388—21no—
169#387—21no—
170#385—21no—
171#381—341no—
172#376Dic1711yescharacteristic, complemented
173#342—172no—
174#342—341no—
175#324—341no—
176#323—172no—
177C2×C17—21no—
178C2×C17—21no—
179C2×C17—42no—
180C2×C17#151611yescharacteristic
181C2×C17#212311yescharacteristic
182C2×C17#212611yescharacteristic
183C23×C4C22⋊D1711yescharacteristic
184#125—41no—
185#125—41no—
186#117—41no—
187#117—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C2×Dic1711yescharacteristic
193C2×C4⋊C4C2×Dic1711yescharacteristic
194C2×C4⋊C4C2×Dic1711yescharacteristic
195C2×C4⋊C4C22⋊D1711yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—342no—
199C2×C22⋊C4C22⋊D1711yescharacteristic
200C2×C22⋊C4C22×D1711yescharacteristic
201C2×C42—21no—
202C2×C42C2×Dic1711yescharacteristic
203C2×C42C22⋊D1711yescharacteristic
204C4⋊C8—681no—
205C22⋊C8—342no—
206C4×C8—681no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—681no—
213#95—681no—
214#95C2×Dic1711yescharacteristic
215#95C2×Dic1711yescharacteristic
216C17#1184611yescharacteristic, complemented, Sylow
217C24C2×C22⋊D1711yescharacteristic
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—682no—
227C22×C4C22⋊Dic1711yescharacteristic
228C22×C4C22⋊Dic1711yescharacteristic
229C22×C4C22⋊Dic1711yescharacteristic
230C22×C4C22⋊Dic1711yescharacteristic
231C22×C4C22×Dic1711yescharacteristic
232C22×C4C2×C22⋊D1711yescharacteristic
233C2×C8—682no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—682no—
248C22⋊C4—682no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—42no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23#1058763612yes—
259C23C2×C22⋊Dic1711yescharacteristic
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
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—1362no—
288C2×C4—1362no—
289C2×C4—1362no—
290C2×C4#1058750511yescharacteristic
291C2×C4#1058750511yescharacteristic
292C8—1362no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22C2×#1058750511yescharacteristic
301C22#1176280511yescharacteristic
302C22#1176282611yescharacteristic
303C4—21no—
304C4—21no—
305C4—42no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—2722no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#1452734011yescharacteristic
319C2#1453043811yescharacteristic
320C2#1453044911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2611C2
socleC22×C17C2×#99
prime radicalC24×C17C2×D4
cosocleC2×C22⋊C4×C17C23
hypercenter#2611D17
hypocenterC17#11846
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2611

Complements

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

NH ≅ G/Ntype
12. C2×C17×#125C4semidirect
14. C17×#400C4semidirect
16. C17×#398C4semidirect
22. C17×#376C4semidirect
160. C2×#125Dic17semidirect
163. #400Dic17semidirect
165. #398Dic17semidirect
172. #376Dic17semidirect
216. C17#11846semidirect

Maximal subgroups

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

subgroupindex
#145288612
#145270982
#145268082
#145267842
C17×#26112
#1184617