Finite Group Databasebeta

← #16950317

Subgroups — #16950317

Order 4352.

524 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 101 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#14531497C211yesmaximal, characteristic
2C2×#11762604C211yesmaximal, characteristic
3#14528770C212yesmaximal
4#14526908C212yesmaximal
5C2×C17×#354C211yesmaximal, characteristic
6C2×Q8×Dic17C411yescharacteristic, complemented
7C2×D4×Dic17C2211yescharacteristic
8C2×#10587505C2212yes—
9#11762716C411yescharacteristic, complemented
10#11762712C2211yescharacteristic
11#11762702—22no—
12#11762684—22no—
13#11762679C411yescharacteristic, complemented
14#11762667—22no—
15#11762655C411yescharacteristic, complemented
16#11762604—22no—
17#11762575—22no—
18C2×C17×#124C2211yescharacteristic
19C2×C17×#99C2212yes—
20C17×#354—22no—
21C22×Q8×C17C2×C411yescharacteristic
22C22×D4×C17C2311yescharacteristic
23C23×Dic17D412yes—
24C17×#124—22no—
25C17×#124D412yes—
26C17×#124Q812yes—
27C2×C22⋊C4×C17—22no—
28C2×C22⋊C4×C17—22no—
29C2×C42×C17C2×C411yescharacteristic
30C2×C22⋊Dic17—22no—
31C2×C22⋊Dic17—22no—
32C2×C22⋊Dic17D412yes—
33Q8×Dic17—42no—
34D4×Dic17—42no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—22no—
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×#63225—42no—
40C2×C4×Dic17—21no—
41C2×C4×Dic17C2×C411yescharacteristic
42C2×C4×Dic17C2×C411yescharacteristic
43C17×#99—22no—
44C17×#99—22no—
45#10587505—22no—
46#10587505—22no—
47#10587502—21no—
48#10587502—22no—
49#10587502—21no—
50#10587502—42no—
51C24×C17C2×D412yes—
52C2×Q8×C17—21no—
53C2×Q8×C17—41no—
54C2×Q8×C17—41no—
55C2×Q8×C17C4⋊C412yescomplemented
56C2×D4×C17—41no—
57C2×D4×C17—41no—
58C2×D4×C17C2×D411yescharacteristic
59C2×D4×C17C2×Q811yescharacteristic
60C2×D4×C17#4112yes—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—42no—
64C22×C4×C17C22×C411yescharacteristic
65C22×Dic17—22no—
66C22×Dic17—22no—
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17—42no—
70C22×Dic17—82no—
71C22×Dic17C22⋊C412yes—
72C22×Dic17C22⋊C411yescharacteristic
73C22×Dic17C22⋊C411yescharacteristic
74C22⋊C4×C17—24no—
75C22⋊C4×C17—24no—
76C22⋊C4×C17—42no—
77C22⋊C4×C17—42no—
78C42×C17—21no—
79C42×C17C4⋊C412yescomplemented
80C22⋊Dic17—42no—
81C22⋊Dic17—42no—
82C22⋊Dic17—42no—
83C22⋊Dic17—82no—
84C4⋊Dic17—41no—
85C4⋊Dic17—42no—
86C4⋊Dic17—41no—
87C4⋊Dic17—82no—
88#63225—84no—
89C4×Dic17—22no—
90C4×Dic17—42no—
91C4×Dic17—41no—
92#12800—171nomaximal, Sylow
93C23×C17—22no—
94C23×C17—22no—
95C23×C17—42no—
96C23×C17—42no—
97C23×C17C2×C22⋊C411yescharacteristic
98C23×C17#12112yes—
99C23×C17C22⋊Q812yes—
100Q8×C17—42no—
101Q8×C17—42no—
102D4×C17—41no—
103D4×C17—41no—
104D4×C17—42no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—22no—
108C2×C4×C17—22no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—82no—
114C2×C4×C17—82no—
115C2×C4×C17—82no—
116C2×C4×C17C2×C4⋊C411yescharacteristic
117C2×C4×C17#11711yescharacteristic
118C2×Dic17—21no—
119C2×Dic17—22no—
120C2×Dic17—41no—
121C2×Dic17—44no—
122C2×Dic17—42no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—42no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—82no—
130C2×Dic17—82no—
131C2×Dic17—82no—
132C2×Dic17—82no—
133C2×Dic17#9912yes—
134#2326—171no—
135C2×#354—171no—
136C2×#354D1711yescharacteristic
137#1799—172no—
138#1393—172no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—22no—
142C22×C17—22no—
143C22×C17—42no—
144C22×C17—42no—
145C22×C17—42no—
146C22×C17—42no—
147C22×C17—82no—
148C22×C17#38011yescharacteristic
149C22×C17C2×#9911yescharacteristic
150C22×C17#41011yescharacteristic
151C4×C17—21no—
152C4×C17—21no—
153C4×C17—42no—
154C4×C17—42no—
155C4×C17—162no—
156Dic17—22no—
157Dic17—42no—
158Dic17—41no—
159Dic17—81no—
160Dic17—81no—
161Dic17—162no—
162C2×#124C2×D1711yescharacteristic
163C2×C4×Q8—171no—
164C2×C4×D4—171no—
165C2×#99—172no—
166C2×#99C2×D1712yes—
167#391—171no—
168#390—171no—
169#388—342no—
170#385—342no—
171#384—171no—
172#380—342no—
173#376—171no—
174#354—22no—
175#354—342no—
176#342—342no—
177C2×C17—21no—
178C2×C17—21no—
179C2×C17—42no—
180C2×C17—42no—
181C2×C17#179911yescharacteristic
182C2×C17#212711yescharacteristic
183C2×C17#212911yescharacteristic
184C22×Q8C4×D1711yescharacteristic
185C22×D4C22×D1711yescharacteristic
186C23×C4—172no—
187#124—22no—
188#124Dic3412yes—
189#124C22⋊D1712yes—
190C4×Q8—682no—
191C4×D4—682no—
192C2×C4⋊C4—171no—
193C2×C4⋊C4—171no—
194C2×C4⋊C4—341no—
195C2×C4⋊C4—342no—
196C2×C4⋊C4—682no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—172no—
200C2×C22⋊C4—342no—
201C2×C22⋊C4—342no—
202C2×C42—171no—
203C2×C42—171no—
204C2×C42—341no—
205C2×C42C4×D1711yescharacteristic
206#99—22no—
207#99—22no—
208#99—342no—
209#99—342no—
210#95—341no—
211#95—341no—
212#95—342no—
213#95—682no—
214C17#1280011yescharacteristic, complemented, Sylow
215C24D4×D1712yes—
216C2×Q8—21no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8#6322512yescomplemented
220C2×D4—41no—
221C2×D4—41no—
222C2×D4C2×Dic3411yescharacteristic
223C2×D4#6325412yes—
224C2×D4C2×C22⋊D1711yescharacteristic
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—171no—
229C22×C4—171no—
230C22×C4—172no—
231C22×C4—342no—
232C22×C4—342no—
233C22×C4—341no—
234C22×C4—341no—
235C22×C4—682no—
236C22×C4—1362no—
237C22×C4C2×C4×D1711yescharacteristic
238C4⋊C4—681no—
239C4⋊C4—681no—
240C4⋊C4—682no—
241C4⋊C4—1362no—
242C4⋊C4—1364no—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—682no—
248C22⋊C4—682no—
249C22⋊C4—682no—
250C22⋊C4—1362no—
251C42—21no—
252C42—342no—
253C42—681no—
254C42—682no—
255C42#6322512yescomplemented
256C23—22no—
257C23—22no—
258C23—42no—
259C23—42no—
260C23#1058756412yes—
261C23C22⋊C4×D1711yescharacteristic
262C23#1058762512yes—
263Q8—42no—
264Q8—42no—
265D4—41no—
266D4—42no—
267D4—41no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—172no—
280C2×C4—341no—
281C2×C4—342no—
282C2×C4—681no—
283C2×C4—684no—
284C2×C4—682no—
285C2×C4—681no—
286C2×C4—681no—
287C2×C4—682no—
288C2×C4—681no—
289C2×C4—681no—
290C2×C4—681no—
291C2×C4—1362no—
292C2×C4—1362no—
293C2×C4—1362no—
294C2×C4—1362no—
295C2×C4#1058756311yescharacteristic
296C2×C4C2×#6322511yescharacteristic
297C22—21no—
298C22—21no—
299C22—22no—
300C22—22no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22#1176266711yescharacteristic
307C22#99×D1711yescharacteristic
308C22#1176278111yescharacteristic
309C4—21no—
310C4—21no—
311C4—42no—
312C4—42no—
313C4—162no—
314C4—2722no—
315C4—342no—
316C4—682no—
317C4—681no—
318C4—1361no—
319C4—1361no—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—42no—
324C2#1452877011yescharacteristic
325C2#1453045211yescharacteristic
326C2#1453045711yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC2×C17×#354C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenterC2×#354D17
hypocenterC17#12800
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#354

Complements

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

NH ≅ G/Ntype
6. C2×Q8×Dic17C4semidirect
9. #11762716C4semidirect
13. #11762679C4semidirect
15. #11762655C4semidirect
55. C2×Q8×C17C4⋊C4semidirect
79. C42×C17C4⋊C4semidirect
214. C17#12800semidirect
219. C2×Q8#63225semidirect
255. C42#63225semidirect

Maximal subgroups

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

subgroupindex
#145314972
C2×#117626042
#145287702
#145269082
C2×C17×#3542
#1280017