Finite Group Databasebeta

← #16945252

Subgroups — #16945252

Order 4352.

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

normal subgroups only

This group has 368 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#14528867C211yesmaximal, characteristic
2#14527098C211yesmaximal, characteristic
3#14526811C212yesmaximal, complemented
4#14526790C212yesmaximal, complemented
5C17×#2611C211yesmaximal, characteristic
6#11762862—21no—
7#11762773—21no—
8#11762578—21no—
9#11762578C2212yes—
10#11762535C2212yes—
11#11762533—21no—
12C2×C17×#125C411yescharacteristic
13C2×C17×#117C2211yescharacteristic
14C17×#400C2211yescharacteristic
15C17×#399C411yescharacteristic
16C17×#398C2211yescharacteristic
17C17×#397C411yescharacteristic
18C17×#389—21no—
19C17×#388—21no—
20C17×#387—21no—
21C17×#385—21no—
22C17×#376C411yescharacteristic
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×C17D411yescharacteristic
35C2×C4⋊C4×C17C2311yescharacteristic
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17C2×C411yescharacteristic
39C2×C22⋊C4×C17D411yescharacteristic
40C2×C42×C17—21no—
41C2×C42×C17C2×C411yescharacteristic
42C2×C42×C17D411yescharacteristic
43C2×#63223—22no—
44C17×#95—21no—
45C17×#95—21no—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95C2×C411yescharacteristic
50C17×#95C2×C411yescharacteristic
51C22⋊#3622—22no—
52C22⋊#3622—41no—
53C22⋊#3622—41no—
54C4⋊#3622—41no—
55C4×#3622—41no—
56C24×C17C22⋊C411yescharacteristic
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—22no—
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×C17C2×D411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C4⋊C4×C17—41no—
72C4⋊C4×C17—41no—
73C4⋊C4×C17—41no—
74C4⋊C4×C17—41no—
75C4⋊C4×C17—42no—
76C4⋊C4×C17—42no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—41no—
79C4⋊C4×C17—42no—
80C22⋊C4×C17—41no—
81C22⋊C4×C17—41no—
82C22⋊C4×C17—42no—
83C22⋊C4×C17—42no—
84C42×C17—21no—
85C42×C17—21no—
86C42×C17—21no—
87C42×C17—21no—
88C42×C17—42no—
89#63223—42no—
90#63223—42no—
91C2×#3622—42no—
92C2×#3622—42no—
93#11853—171nomaximal, Sylow
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17C2×C22⋊C411yescharacteristic
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—22no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17—41no—
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#9911yescharacteristic
127C2×C4×C17#9911yescharacteristic
128C2×C4×C17#13512yes—
129#3622—82no—
130#3622—82no—
131#2611D1711yescharacteristic
132#1822—171no—
133#1458—171no—
134#1353—172no—
135#1346—172no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17—42no—
139C22×C17—41no—
140C22×C17—41no—
141C22×C17—41no—
142C22×C17—81no—
143C22×C17C2×#9911yescharacteristic
144C22×C17#41611yescharacteristic
145C22×C17#42111yescharacteristic
146C4×C17—22no—
147C4×C17—21no—
148C4×C17—21no—
149C4×C17—42no—
150C4×C17—41no—
151C4×C17—41no—
152C4×C17—41no—
153C4×C17—41no—
154C4×C17—81no—
155C4×C17—81no—
156C4×C17—81no—
157C2×#125Dic1711yescharacteristic
158C2×#117C2×D1711yescharacteristic
159#431—341no—
160#407—341no—
161#400C2×D1711yescharacteristic
162#399Dic1711yescharacteristic
163#398C2×D1711yescharacteristic
164#397Dic1711yescharacteristic
165#389—21no—
166#388—21no—
167#387—21no—
168#385—21no—
169#376Dic1711yescharacteristic
170#343—172no—
171#343—341no—
172#324—172no—
173#323—341no—
174C2×C17—21no—
175C2×C17—21no—
176C2×C17—81no—
177C2×C17#151611yescharacteristic
178C2×C17#212311yescharacteristic
179C2×C17#213211yescharacteristic
180C23×C4C22⋊D1711yescharacteristic
181C2×M16—342no—
182#125—41no—
183#125—41no—
184#117—41no—
185#117—41no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C2×Dic1711yescharacteristic
191C2×C4⋊C4C2×Dic1711yescharacteristic
192C2×C4⋊C4C22⋊D1711yescharacteristic
193C2×C4⋊C4C22×D1711yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4C2×Dic1711yescharacteristic
197C2×C22⋊C4C22⋊D1711yescharacteristic
198C2×C42—21no—
199C2×C42C2×Dic1711yescharacteristic
200C2×C42C22⋊D1711yescharacteristic
201C4⋊C8—681no—
202C22⋊C8—342no—
203C22⋊C8—681no—
204C22⋊C8—681no—
205C4×C8—681no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95C2×Dic1711yescharacteristic
212#95C2×Dic1711yescharacteristic
213C17#1185311yescharacteristic, complemented, Sylow
214C24C22⋊Dic1711yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—41no—
223C22×C4C22⋊Dic1711yescharacteristic
224C22×C4C22⋊Dic1711yescharacteristic
225C22×C4C22⋊Dic1711yescharacteristic
226C22×C4C22×Dic1711yescharacteristic
227C22×C4C2×C22⋊D1711yescharacteristic
228C22×C4C2×C22⋊D1711yescharacteristic
229M16—682no—
230M16—682no—
231C2×C8—682no—
232C2×C8—682no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—42no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23C2×C22⋊Dic1711yescharacteristic
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
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#1058750511yescharacteristic
284C2×C4#1058750511yescharacteristic
285C2×C4#1058763612yes—
286C8—1362no—
287C8—1362no—
288C22—21no—
289C22—21no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—81no—
295C22C2×#1058750511yescharacteristic
296C22#1176280511yescharacteristic
297C22#1176282611yescharacteristic
298C4—22no—
299C4—21no—
300C4—21no—
301C4—42no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—81no—
307C4—81no—
308C4—81no—
309C2—21no—
310C2—21no—
311C2—81no—
312C2#1452734011yescharacteristic
313C2#1453043811yescharacteristic
314C2#1453046911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2611C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#2611D17
hypocenterC17#11853
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
3. #14526811C2semidirect
4. #14526790C2semidirect
213. C17#11853semidirect

Maximal subgroups

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

subgroupindex
#145288672
#145270982
#145268112
#145267902
C17×#26112
#1185317