Finite Group Databasebeta

← #16944850

Subgroups — #16944850

Order 4352.

420 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 91 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#14528523C211yesmaximal, characteristic
2#14527118C211yesmaximal, characteristic
3#14526811C212yesmaximal, complemented
4#14526790C212yesmaximal, complemented
5C17×#2611C211yesmaximal, characteristic
6C22×#63223—21no—
7#11762866—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—24no—
44C2×#63223—22no—
45C2×#63223—41no—
46C2×#63223—42no—
47C22×#3622—41no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95—21no—
52C17×#95—21no—
53C17×#95C2×C411yescharacteristic
54C17×#95C2×C411yescharacteristic
55C22⋊#3622—22no—
56C4⋊#3622—41no—
57#10587474—41no—
58C24×C17C22⋊C411yescharacteristic
59C22×C4×C17—22no—
60C22×C4×C17—21no—
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—21no—
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22×C411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C4⋊C4×C17—41no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—42no—
79C4⋊C4×C17—42no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—41no—
84C22⋊C4×C17—41no—
85C22⋊C4×C17—41no—
86C22⋊C4×C17—42no—
87C22⋊C4×C17—42no—
88C42×C17—21no—
89C42×C17—21no—
90C42×C17—21no—
91C42×C17—21no—
92C42×C17—42no—
93#63223—24no—
94#63223—44no—
95#63223—42no—
96C2×#3622—44no—
97C2×#3622—42no—
98C2×#3622—42no—
99#11767—171nomaximal, Sylow
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17#10412yes—
105C23×C17C2×C22⋊C411yescharacteristic
106C2×C4×C17—22no—
107C2×C4×C17—22no—
108C2×C4×C17—22no—
109C2×C4×C17—22no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—22no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
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×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17#9911yescharacteristic
135C2×C4×C17#9911yescharacteristic
136C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C17#13711yescharacteristic
138#3622—44no—
139#3622—82no—
140#2611D1711yescharacteristic
141#1742—171no—
142#1464—171no—
143#1353—172no—
144#1346—172no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—22no—
148C22×C17—42no—
149C22×C17—41no—
150C22×C17—41no—
151C22×C17—81no—
152C22×C17C2×#9911yescharacteristic
153C22×C17#41711yescharacteristic
154C22×C17C2×#10411yescharacteristic
155C4×C17—21no—
156C4×C17—21no—
157C4×C17—21no—
158C4×C17—21no—
159C4×C17—42no—
160C4×C17—41no—
161C4×C17—41no—
162C4×C17—41no—
163C4×C17—41no—
164C4×C17—81no—
165C4×C17—81no—
166C4×C17—81no—
167C22×M16—341no—
168C2×#125Dic1711yescharacteristic
169C2×#117C2×D1711yescharacteristic
170#432—341no—
171#400C2×D1711yescharacteristic
172#399Dic1711yescharacteristic
173#398C2×D1711yescharacteristic
174#397Dic1711yescharacteristic
175#389—21no—
176#388—21no—
177#387—21no—
178#385—21no—
179#376Dic1711yescharacteristic
180#343—172no—
181#343—341no—
182#324—172no—
183#323—341no—
184C2×C17—21no—
185C2×C17—21no—
186C2×C17—42no—
187C2×C17#150411yescharacteristic
188C2×C17#212311yescharacteristic
189C2×C17#213211yescharacteristic
190C23×C4C22⋊D1711yescharacteristic
191C2×M16—344no—
192C2×M16—342no—
193C2×M16—681no—
194C2×M16—682no—
195C22×C8—681no—
196#125—41no—
197#125—41no—
198#117—41no—
199#117—41no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4C2×Dic1711yescharacteristic
205C2×C4⋊C4C2×Dic1711yescharacteristic
206C2×C4⋊C4C22⋊D1711yescharacteristic
207C2×C4⋊C4C22×D1711yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4C2×Dic1711yescharacteristic
211C2×C22⋊C4C22⋊D1711yescharacteristic
212C2×C42—21no—
213C2×C42C2×Dic1711yescharacteristic
214C2×C42C22⋊D1711yescharacteristic
215C4⋊C8—681no—
216C22⋊C8—342no—
217#97—681no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95C2×Dic1711yescharacteristic
224#95C2×Dic1711yescharacteristic
225C17#1176711yescharacteristic, complemented, Sylow
226C24C22⋊Dic1711yescharacteristic
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4C22⋊Dic1711yescharacteristic
237C22×C4C22⋊Dic1711yescharacteristic
238C22×C4C22⋊Dic1711yescharacteristic
239C22×C4C22×Dic1711yescharacteristic
240C22×C4C2×C22⋊D1711yescharacteristic
241C22×C4C2×C22⋊D1711yescharacteristic
242M16—344no—
243M16—684no—
244M16—682no—
245C2×C8—684no—
246C2×C8—682no—
247C2×C8—682no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—42no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23#1058750812yes—
272C23C2×C22⋊Dic1711yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4#1058750511yescharacteristic
302C2×C4#1058750511yescharacteristic
303C2×C4#1058763511yescharacteristic
304C2×C4#1058763711yescharacteristic
305C8—684no—
306C8—1362no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—81no—
314C22C2×#1058750511yescharacteristic
315C22#1176281011yescharacteristic
316C22C2×#1058750811yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—42no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—81no—
327C4—81no—
328C4—81no—
329C2—21no—
330C2—21no—
331C2—42no—
332C2#1452728811yescharacteristic
333C2#1453043811yescharacteristic
334C2#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#11767
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
225. C17#11767semidirect

Maximal subgroups

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

subgroupindex
#145285232
#145271182
#145268112
#145267902
C17×#26112
#1176717