Finite Group Databasebeta

← #16950311

Subgroups — #16950311

Order 4352.

544 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×#11762604C212yesmaximal
2#14528772C212yesmaximal
3#14526909C212yesmaximal
4C4×C17×#124C211yesmaximal, characteristic
5C2×#10587505C2214yes—
6#11762669—22no—
7#11762604—24no—
8#11762576—22no—
9C2×C17×#124C2211yescharacteristic
10C2×C4×Q8×C17C411yescharacteristic, complemented
11C2×C4×D4×C17C2211yescharacteristic
12C17×#391C411yescharacteristic, complemented
13C17×#390C2211yescharacteristic
14C17×#388—22no—
15C17×#382C411yescharacteristic, complemented
16C4×C22⋊C4×C17—22no—
17C43×C17C411yescharacteristic, complemented
18C22×Q8×C17C2×C411yescharacteristic
19C22×D4×C17C2311yescharacteristic
20C23×C4×C17D412yes—
21C17×#124—22no—
22C17×#124D412yes—
23C17×#124Q812yes—
24C4×Q8×C17—41no—
25C4×Q8×C17—41no—
26C4×D4×C17—41no—
27C4×D4×C17—41no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C22⋊C4×C17—22no—
32C2×C22⋊C4×C17—22no—
33C2×C22⋊C4×C17D412yes—
34C2×C42×C17—21no—
35C2×C42×C17—21no—
36C2×C42×C17—21no—
37C2×C42×C17C2×C411yescharacteristic
38C2×C42×C17C2×C411yescharacteristic
39C2×C42×C17C2×C411yescharacteristic
40C2×C22⋊Dic17—24no—
41C2×C4⋊Dic17—42no—
42C17×#95—22no—
43C17×#95—22no—
44#10587505—24no—
45#10587505—24no—
46#10587502—42no—
47C24×C17C2×D412yes—
48C2×Q8×C17—21no—
49C2×Q8×C17—41no—
50C2×Q8×C17—41no—
51C2×Q8×C17C4⋊C412yescomplemented
52C2×D4×C17—41no—
53C2×D4×C17—41no—
54C2×D4×C17C2×D411yescharacteristic
55C2×D4×C17C2×Q811yescharacteristic
56C2×D4×C17#4112yes—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—22no—
61C22×C4×C17—22no—
62C22×C4×C17—21no—
63C22×C4×C17—44no—
64C22×C4×C17C22⋊C412yes—
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22×C411yescharacteristic
68C22×Dic17—44no—
69C4⋊C4×C17—41no—
70C4⋊C4×C17—41no—
71C4⋊C4×C17—42no—
72C22⋊C4×C17—24no—
73C22⋊C4×C17—24no—
74C22⋊C4×C17—42no—
75C22⋊C4×C17—44no—
76C42×C17—21no—
77C42×C17—21no—
78C42×C17—21no—
79C42×C17—41no—
80C42×C17—42no—
81C42×C17—42no—
82C42×C17—41no—
83C42×C17—41no—
84C42×C17C4⋊C412yescomplemented
85C22⋊Dic17—44no—
86C22⋊Dic17—44no—
87C4⋊Dic17—84no—
88#12798—171nomaximal, Sylow
89C23×C17—22no—
90C23×C17—22no—
91C23×C17—42no—
92C23×C17—42no—
93C23×C17C2×C22⋊C411yescharacteristic
94C23×C17#12112yes—
95C23×C17C22⋊Q812yes—
96Q8×C17—42no—
97Q8×C17—42no—
98D4×C17—41no—
99D4×C17—41no—
100D4×C17—42no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—22no—
104C2×C4×C17—22no—
105C2×C4×C17—22no—
106C2×C4×C17—22no—
107C2×C4×C17—21no—
108C2×C4×C17—22no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—44no—
116C2×C4×C17—41no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—41no—
120C2×C4×C17—82no—
121C2×C4×C17#9911yescharacteristic
122C2×C4×C17#9911yescharacteristic
123C2×C4×C17C2×C4⋊C411yescharacteristic
124C2×C4×C17#11711yescharacteristic
125C2×Dic17—84no—
126C2×Dic17—84no—
127C2×Dic17—84no—
128C4×#124D1711yescharacteristic
129C2×#354—172no—
130#1799—172no—
131#1393—172no—
132C22×C17—21no—
133C22×C17—21no—
134C22×C17—22no—
135C22×C17—22no—
136C22×C17—42no—
137C22×C17—42no—
138C22×C17—42no—
139C22×C17—42no—
140C22×C17—82no—
141C22×C17#38011yescharacteristic
142C22×C17C2×#9911yescharacteristic
143C22×C17#41011yescharacteristic
144C4×C17—21no—
145C4×C17—21no—
146C4×C17—21no—
147C4×C17—42no—
148C4×C17—42no—
149C4×C17—42no—
150C4×C17—42no—
151C4×C17—41no—
152C4×C17—81no—
153C4×C17#35412yescomplemented
154Dic17—164no—
155C2×#124C2×D1711yescharacteristic
156C2×C4×Q8Dic1711yescharacteristic, complemented
157C2×C4×D4C2×D1711yescharacteristic
158C2×#99—174no—
159#391Dic1711yescharacteristic, complemented
160#390C2×D1711yescharacteristic
161#388—22no—
162#382Dic1711yescharacteristic, complemented
163#380—342no—
164C4×C22⋊C4—22no—
165C43Dic1711yescharacteristic, complemented
166#354—344no—
167#342—342no—
168C2×C17—21no—
169C2×C17—21no—
170C2×C17—42no—
171C2×C17—42no—
172C2×C17#179911yescharacteristic
173C2×C17C2×#35411yescharacteristic
174C2×C17#212611yescharacteristic
175C22×Q8C2×Dic1711yescharacteristic
176C22×D4C22×D1711yescharacteristic
177C23×C4C22⋊D1712yes—
178#124—22no—
179#124Dic3412yes—
180#124D6812yes—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×D4—41no—
184C4×D4—41no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—682no—
187C2×C4⋊C4C2×Dic1711yescharacteristic
188C2×C4⋊C4C2×Dic1711yescharacteristic
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—344no—
192C2×C22⋊C4C22⋊D1712yes—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42C2×Dic1711yescharacteristic
197C2×C42C2×Dic1711yescharacteristic
198C2×C42C2×Dic1711yescharacteristic
199#99—344no—
200#99—344no—
201#95—22no—
202#95—22no—
203#95—682no—
204C17#1279811yescharacteristic, complemented, Sylow
205C24C2×C22⋊D1712yes—
206C2×Q8—21no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8C4⋊Dic1712yescomplemented
210C2×D4—41no—
211C2×D4—41no—
212C2×D4C2×Dic3411yescharacteristic
213C2×D4C2×D6811yescharacteristic
214C2×D4#6325212yes—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—44no—
222C22×C4—684no—
223C22×C4C22⋊Dic1712yes—
224C22×C4C22⋊Dic1711yescharacteristic
225C22×C4C22⋊Dic1711yescharacteristic
226C22×C4C22×Dic1711yescharacteristic
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—1364no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C22⋊C4—42no—
234C22⋊C4—44no—
235C22⋊C4—684no—
236C22⋊C4—684no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—41no—
241C42—42no—
242C42—42no—
243C42—41no—
244C42—41no—
245C42C4⋊Dic1712yescomplemented
246C23—22no—
247C23—22no—
248C23—42no—
249C23—42no—
250C23#1058761012yes—
251C23#1058761612yes—
252C23C2×C22⋊Dic1711yescharacteristic
253Q8—42no—
254Q8—42no—
255D4—41no—
256D4—41no—
257D4—42no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—44no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—82no—
278C2×C4—1364no—
279C2×C4—1364no—
280C2×C4—1364no—
281C2×C4#1058750511yescharacteristic
282C2×C4#1058750511yescharacteristic
283C2×C4C2×C4⋊Dic1711yescharacteristic
284C2×C4#1058761211yescharacteristic
285C22—21no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—82no—
294C22#1176266911yescharacteristic
295C22C2×#1058750511yescharacteristic
296C22#1176278311yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—42no—
301C4—42no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—81no—
306C4—2724no—
307C4#1176260412yescomplemented
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#1452877211yescharacteristic
313C2C2×#1176260411yescharacteristic
314C2#1453044911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC4×C17×#124C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenterC4×#124D17
hypocenterC17#12798
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
10. C2×C4×Q8×C17C4semidirect
12. C17×#391C4semidirect
15. C17×#382C4semidirect
17. C43×C17C4semidirect
51. C2×Q8×C17C4⋊C4semidirect
84. C42×C17C4⋊C4semidirect
153. C4×C17#354semidirect
156. C2×C4×Q8Dic17semidirect
159. #391Dic17semidirect
162. #382Dic17semidirect
165. C43Dic17semidirect
204. C17#12798semidirect
209. C2×Q8C4⋊Dic17semidirect
245. C42C4⋊Dic17semidirect
307. C4#11762604semidirect

Maximal subgroups

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

subgroupindex
C2×#117626042
#145287722
#145269092
C4×C17×#1242
#1279817