Finite Group Databasebeta

← #16944880

Subgroups — #16944880

Order 4352.

482 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#10587505C211yesmaximal, characteristic
2#14527125C211yesmaximal, characteristic
3#14526784C214yesmaximal, complemented
4C17×#2486C211yesmaximal, characteristic
5#11762866—21no—
6C2×#10587505—21no—
7C4×C22⋊Dic17—21no—
8#11762576C2212yes—
9#11762533—21no—
10#11762533C2212yes—
11C2×C17×#123—21no—
12C2×C4×D4×C17C2211yescharacteristic
13C22×C4⋊C4×C17C411yescharacteristic, complemented
14C17×#398C412yescomplemented
15C17×#397C2212yes—
16C17×#389—21no—
17C17×#381—21no—
18C17×#380—21no—
19C17×#380C411yescharacteristic, complemented
20C22×D4×C17D411yescharacteristic
21C23×C4×C17—21no—
22C23×C4×C17C2×C411yescharacteristic
23C23×C4×C17D411yescharacteristic
24C17×#123—44no—
25C4×D4×C17—41no—
26C4×D4×C17—41no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—41no—
31C2×C4⋊C4×C17—41no—
32C2×C4⋊C4×C17C2×C412yes—
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C22⋊C4×C17—21no—
35C2×C22⋊C4×C17—22no—
36C2×C22⋊C4×C17D411yescharacteristic
37C2×C22⋊C4×C17C2311yescharacteristic
38C2×C42×C17D411yescharacteristic
39C2×C22⋊Dic17—22no—
40C2×C4×Dic17—41no—
41C17×#95—22no—
42C17×#95—22no—
43C17×#95C2×C412yes—
44#10587505—24no—
45#10587502—41no—
46C22⋊#3622—22no—
47C4⋊#3622—41no—
48#10587474—41no—
49C24×C17C22⋊C411yescharacteristic
50C24×C17C2×D411yescharacteristic
51C2×D4×C17—22no—
52C2×D4×C17—22no—
53C2×D4×C17—41no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—22no—
59C22×C4×C17—22no—
60C22×C4×C17—42no—
61C22×C4×C17—42no—
62C22×C4×C17—42no—
63C22×C4×C17—42no—
64C22×C4×C17—42no—
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22×C411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×Dic17—42no—
71C4⋊C4×C17—41no—
72C4⋊C4×C17—42no—
73C4⋊C4×C17—44no—
74C4⋊C4×C17—42no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C22⋊C4×C17—41no—
78C22⋊C4×C17—41no—
79C22⋊C4×C17—42no—
80C22⋊C4×C17—44no—
81C42×C17—21no—
82C42×C17—21no—
83C22⋊Dic17—24no—
84C22⋊Dic17—42no—
85C4×Dic17—82no—
86C2×#3622—42no—
87#11773—171nomaximal, Sylow
88C23×C17—21no—
89C23×C17—21no—
90C23×C17—21no—
91C23×C17—21no—
92C23×C17—41no—
93C23×C17—41no—
94C23×C17—41no—
95C23×C17—41no—
96C23×C17#10412yes—
97C23×C17C2×C22⋊C411yescharacteristic
98C23×C17AGL(1,ℤ/8)12yes—
99D4×C17—44no—
100C2×C4×C17—22no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—81no—
120C2×C4×C17—84no—
121C2×C4×C17#9911yescharacteristic
122C2×C4×C17#9911yescharacteristic
123C2×Dic17—44no—
124C2×Dic17—82no—
125C2×Dic17—82no—
126#3622—82no—
127#2486D1711yescharacteristic
128C4×#99—171no—
129#1466—171no—
130#1344—174no—
131C22×C17—21no—
132C22×C17—21no—
133C22×C17—22no—
134C22×C17—22no—
135C22×C17—42no—
136C22×C17—42no—
137C22×C17—42no—
138C22×C17—41no—
139C22×C17—41no—
140C22×C17—41no—
141C22×C17—41no—
142C22×C17—81no—
143C22×C17C2×#9911yescharacteristic
144C22×C17#41811yescharacteristic
145C22×C17C2×#10411yescharacteristic
146C4×C17—21no—
147C4×C17—41no—
148C4×C17—41no—
149C4×C17—41no—
150C4×C17—41no—
151C4×C17—82no—
152C4×C17—82no—
153C4×C17#35212yes—
154Dic17—84no—
155C2×#123—21no—
156C2×C4×D4C2×D1711yescharacteristic
157C22×C4⋊C4Dic1711yescharacteristic, complemented
158#432—341no—
159C2×#99—341no—
160#398Dic1712yescomplemented
161#397C2×D1712yes—
162#389—21no—
163#381—21no—
164#380—21no—
165#380Dic1711yescharacteristic, complemented
166C4×C22⋊C4—341no—
167#342—172no—
168#323—172no—
169#323—341no—
170C2×C17—21no—
171C2×C17—21no—
172C2×C17—42no—
173C2×C17—42no—
174C2×C17#150711yescharacteristic
175C2×C17C2×#35211yescharacteristic
176C2×C17#212911yescharacteristic
177C22×D4C22⋊D1711yescharacteristic
178C23×C4—21no—
179C23×C4C2×Dic1711yescharacteristic
180C23×C4C22⋊D1711yescharacteristic
181#123—44no—
182C4×D4—41no—
183C4×D4—41no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4C2×Dic1712yes—
190C2×C4⋊C4C2×Dic1711yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—342no—
194C2×C22⋊C4C22⋊D1711yescharacteristic
195C2×C22⋊C4C22×D1711yescharacteristic
196C2×C42—681no—
197C2×C42C22⋊D1711yescharacteristic
198C4⋊C8—681no—
199#99—344no—
200C22⋊C8—342no—
201#97—681no—
202#95—22no—
203#95—22no—
204#95—681no—
205#95C2×Dic1712yes—
206C17#1177311yescharacteristic, complemented, Sylow
207C24C22⋊Dic1711yescharacteristic
208C24C2×C22⋊D1711yescharacteristic
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—41no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—682no—
224C22×C4C22⋊Dic1711yescharacteristic
225C22×C4C22⋊Dic1711yescharacteristic
226C22×C4C22⋊Dic1711yescharacteristic
227C22×C4C22×Dic1711yescharacteristic
228C22×C4C2×C22⋊D1711yescharacteristic
229C2×C8—682no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—44no—
240C22⋊C4—344no—
241C22⋊C4—682no—
242C42—21no—
243C42—21no—
244C42—1362no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23#1058750812yes—
254C23#1058761812yes—
255C23C2×C22⋊Dic1711yescharacteristic
256D4—44no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—81no—
277C2×C4—84no—
278C2×C4—684no—
279C2×C4—1362no—
280C2×C4—1362no—
281C2×C4#1058750511yescharacteristic
282C2×C4#1058750511yescharacteristic
283C8—1362no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22C2×#1058750511yescharacteristic
297C22#1176281411yescharacteristic
298C22C2×#1058750811yescharacteristic
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—82no—
305C4—82no—
306C4—1364no—
307C4#1176259912yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#1452730011yescharacteristic
313C2C2×#1176259911yescharacteristic
314C2#1453045911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2486C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenter#2486D17
hypocenterC17#11773
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
3. #14526784C2semidirect
13. C22×C4⋊C4×C17C4semidirect
14. C17×#398C4semidirect
19. C17×#380C4semidirect
157. C22×C4⋊C4Dic17semidirect
160. #398Dic17semidirect
165. #380Dic17semidirect
206. C17#11773semidirect

Maximal subgroups

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

subgroupindex
C4×#105875052
#145271252
#145267842
C17×#24862
#1177317