Finite Group Databasebeta

← #16944941

Subgroups — #16944941

Order 4352.

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

normal subgroups only

This group has 460 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#14527122C211yesmaximal, characteristic
3#14526805C214yesmaximal, complemented
4C17×#2563C211yesmaximal, characteristic
5C2×#10587505—21no—
6#11762773—21no—
7C4×C22⋊Dic17—21no—
8#11762576C2212yes—
9#11762535—21no—
10#11762533C2212yes—
11C2×C17×C4⋊Q8C411yescharacteristic, complemented
12C2×C4×D4×C17C2211yescharacteristic
13C17×#393C412yescomplemented
14C17×#393C2212yes—
15C17×#390—21no—
16C17×#382—21no—
17C17×#380—21no—
18C17×#380—21no—
19C4×C4⋊C4×C17C411yescharacteristic, complemented
20C22×Q8×C17C2×C412yes—
21C22×D4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17D411yescharacteristic
24C17×C4⋊Q8—41no—
25C17×C4⋊Q8—41no—
26C4×D4×C17—41no—
27C4×D4×C17—41no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C22⋊C4×C17—22no—
33C2×C22⋊C4×C17—22no—
34C2×C22⋊C4×C17C2×C411yescharacteristic
35C2×C22⋊C4×C17C2311yescharacteristic
36C2×C42×C17—21no—
37C2×C42×C17C2×C411yescharacteristic
38C2×C22⋊Dic17—22no—
39C2×C4×Dic17—41no—
40C2×#63223—41no—
41C17×#95—22no—
42C17×#95—22no—
43C17×#95C2×C412yes—
44#10587505—24no—
45#10587502—41no—
46C22⋊#3622—22no—
47C22⋊#3622—41no—
48C24×C17C22⋊C411yescharacteristic
49C24×C17C2×D411yescharacteristic
50C2×Q8×C17—42no—
51C2×Q8×C17—82no—
52C2×D4×C17—22no—
53C2×D4×C17—22no—
54C2×D4×C17—41no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—22no—
60C22×C4×C17—22no—
61C22×C4×C17—42no—
62C22×C4×C17—42no—
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22×C411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×Dic17—42no—
69C4⋊C4×C17—41no—
70C4⋊C4×C17—42no—
71C4⋊C4×C17—42no—
72C4⋊C4×C17—41no—
73C4⋊C4×C17—41no—
74C22⋊C4×C17—41no—
75C22⋊C4×C17—41no—
76C22⋊C4×C17—82no—
77C22⋊C4×C17—82no—
78C42×C17—21no—
79C42×C17—21no—
80C42×C17—42no—
81C22⋊Dic17—24no—
82C22⋊Dic17—42no—
83C4×Dic17—82no—
84#63223—82no—
85C2×#3622—42no—
86#11787—171nomaximal, Sylow
87C23×C17—21no—
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×C17C2×C22⋊C411yescharacteristic
97C23×C17#13712yes—
98Q8×C17—84no—
99D4×C17—44no—
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—42no—
107C2×C4×C17—42no—
108C2×C4×C17—41no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—81no—
118C2×C4×C17—81no—
119C2×C4×C17#9911yescharacteristic
120C2×C4×C17#9911yescharacteristic
121C2×C4×C17#10412yes—
122C2×Dic17—44no—
123C2×Dic17—82no—
124C2×Dic17—82no—
125#3622—82no—
126#2563D1711yescharacteristic
127C4×#99—171no—
128#1465—171no—
129#1351—174no—
130C22×C17—21no—
131C22×C17—21no—
132C22×C17—22no—
133C22×C17—42no—
134C22×C17—42no—
135C22×C17—41no—
136C22×C17—41no—
137C22×C17—41no—
138C22×C17—41no—
139C22×C17—41no—
140C22×C17—81no—
141C22×C17—81no—
142C22×C17C2×#9911yescharacteristic
143C22×C17#41911yescharacteristic
144C22×C17C2×#10411yescharacteristic
145C4×C17—21no—
146C4×C17—22no—
147C4×C17—41no—
148C4×C17—41no—
149C4×C17—41no—
150C4×C17—82no—
151C4×C17—82no—
152C4×C17#35412yes—
153Dic17—84no—
154C2×C4⋊Q8Dic1711yescharacteristic, complemented
155C2×C4×D4C2×D1711yescharacteristic
156C2×#99—341no—
157#407—341no—
158#393Dic1712yescomplemented
159#393C2×D1712yes—
160#390—21no—
161#382—21no—
162#380—21no—
163#380—21no—
164C4×C4⋊C4Dic1711yescharacteristic, complemented
165C4×C22⋊C4—341no—
166#342—172no—
167#324—341no—
168#323—172no—
169C2×C17—21no—
170C2×C17—21no—
171C2×C17—42no—
172C2×C17—81no—
173C2×C17#150811yescharacteristic
174C2×C17#212211yescharacteristic
175C2×C17C2×#35411yescharacteristic
176C22×Q8C2×Dic1712yes—
177C22×D4C22⋊D1711yescharacteristic
178C23×C4C22⋊D1711yescharacteristic
179C23×C4C22⋊D1711yescharacteristic
180C2×M16—681no—
181C4⋊Q8—41no—
182C4⋊Q8—41no—
183C4×D4—41no—
184C4×D4—41no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4C22⋊D1711yescharacteristic
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—342no—
192C2×C22⋊C4C2×Dic1711yescharacteristic
193C2×C22⋊C4C22×D1711yescharacteristic
194C2×C42—21no—
195C2×C42—681no—
196C2×C42C2×Dic1711yescharacteristic
197#99—344no—
198C22⋊C8—342no—
199C22⋊C8—681no—
200#95—22no—
201#95—22no—
202#95—681no—
203#95C2×Dic1712yes—
204C17#1178711yescharacteristic, complemented, Sylow
205C24C22⋊Dic1711yescharacteristic
206C24C2×C22⋊D1711yescharacteristic
207C2×Q8—42no—
208C2×Q8—82no—
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—682no—
221C22×C4C22⋊Dic1711yescharacteristic
222C22×C4C22⋊Dic1711yescharacteristic
223C22×C4C22⋊Dic1711yescharacteristic
224C22×C4C22×Dic1711yescharacteristic
225C22×C4C2×C22⋊D1711yescharacteristic
226M16—1362no—
227C2×C8—682no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C22⋊C4—344no—
238C22⋊C4—682no—
239C42—21no—
240C42—21no—
241C42—42no—
242C42—1362no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23#1058762812yes—
253C23C2×C22⋊Dic1711yescharacteristic
254Q8—84no—
255D4—44no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—684no—
276C2×C4—1362no—
277C2×C4—1362no—
278C2×C4#1058750511yescharacteristic
279C2×C4#1058750511yescharacteristic
280C2×C4#1058750812yes—
281C8—1362no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22—81no—
294C22C2×#1058750511yescharacteristic
295C22#1176281811yescharacteristic
296C22C2×#1058750811yescharacteristic
297C4—22no—
298C4—21no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4—82no—
303C4—82no—
304C4—1364no—
305C4#1176260412yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2#1452730411yescharacteristic
311C2#1453043411yescharacteristic
312C2C2×#1176260411yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2563C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenter#2563D17
hypocenterC17#11787
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2563

Complements

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

NH ≅ G/Ntype
3. #14526805C2semidirect
11. C2×C17×C4⋊Q8C4semidirect
13. C17×#393C4semidirect
19. C4×C4⋊C4×C17C4semidirect
154. C2×C4⋊Q8Dic17semidirect
158. #393Dic17semidirect
164. C4×C4⋊C4Dic17semidirect
204. C17#11787semidirect

Maximal subgroups

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

subgroupindex
C4×#105875052
#145271222
#145268052
C17×#25632
#1178717