Finite Group Databasebeta

← #14538415

Subgroups — #14538415

Order 2176.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×#10587595C211yesmaximal, characteristic, complemented
2C2×M16×D17C211yesmaximal, characteristic, complemented
3#11763321C211yesmaximal, characteristic
4C2×#10587583C211yesmaximal, characteristic, complemented
5#11763068C212yesmaximal, complemented
6#11763056C212yesmaximal, complemented
7#11763051C212yesmaximal, complemented
8#11763039C212yesmaximal, complemented
9#11762817C211yesmaximal, characteristic, complemented
10#11762808C211yesmaximal, characteristic, complemented
11C17×#428C211yesmaximal, characteristic, complemented
12C2×#63252C2211yescharacteristic
13C22×C4×D17C2211yescharacteristic
14C22×Dic34C2211yescharacteristic
15C2×M16×C17C2211yescharacteristic
16C17×#117C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×#63227—21no—
19C2×C4⋊Dic17—21no—
20C2×#63223C2211yescharacteristic
21#10587595—24no—
22M16×D17—24no—
23C2×Dic68C2212yescomplemented
24C2×#63219C2212yescomplemented
25C2×#63218C2212yes—
26C2×C8×D17C2212yes—
27#10587583—21no—
28#10587583—21no—
29#10587583C2212yes—
30#10587583C2211yescharacteristic
31#10587583C2212yes—
32#10587583C2211yescharacteristic
33#10587581C2211yescharacteristic
34#10587581C2211yescharacteristic
35#10587572—21no—
36#10587568—21no—
37C4×D68—21no—
38C4×D68—21no—
39C17×#107C2212yescomplemented
40C17×#106C2212yescomplemented
41#10587481C2212yescomplemented
42#10587481C2212yescomplemented
43#10587481C2212yescomplemented
44#10587480C2212yescomplemented
45C23×D17D411yescharacteristic
46C22×C4×C17—21no—
47C22×C4×C17C2311yescharacteristic
48C2×C22⋊D17—21no—
49C22×Dic17—21no—
50C22×Dic17D411yescharacteristic
51#63252—22no—
52#63252—22no—
53C2×D68—21no—
54C2×D68—21no—
55C2×D68C2311yescharacteristic
56C2×C4×D17—22no—
57C2×C4×D17—22no—
58C2×C4×D17—21no—
59C2×C4×D17D411yescharacteristic
60C2×C4×D17D411yescharacteristic
61C2×C4×D17C2311yescharacteristic
62C2×C4×D17C2311yescharacteristic
63C2×Dic34—22no—
64C2×Dic34—22no—
65C2×Dic34—21no—
66C2×Dic34C2312yes—
67C2×Dic34C2311yescharacteristic
68M16×C17D414yescomplemented
69C2×C8×C17C2312yes—
70C4⋊C4×C17—21no—
71C4⋊C4×C17C2312yes—
72C4⋊C4×C17C2311yescharacteristic
73C4⋊C4×C17C2311yescharacteristic
74C22⋊C4×C17—21no—
75C42×C17—21no—
76#63227—21no—
77#63227—21no—
78#63227—22no—
79#63227—22no—
80#63227—21no—
81#63227—21no—
82C4⋊Dic17—21no—
83C4⋊Dic17—21no—
84C4⋊Dic17—22no—
85C4⋊Dic17—22no—
86#63223—22no—
87C2×#3622C2312yes—
88Dic68—24no—
89#63219—24no—
90#63218—24no—
91C8×D17—24no—
92C23×C17C2×D411yescharacteristic
93C22×D17—22no—
94C22×D17—22no—
95C22×D17—21no—
96C22×D17—41no—
97C22×D17C2×D411yescharacteristic
98C22×D17C2×D411yescharacteristic
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—22no—
102C2×C4×C17—42no—
103C2×C4×C17C2×D412yes—
104C2×C4×C17C2×D411yescharacteristic
105C2×C4×C17#4112yes—
106C2×C4×C17C2411yescharacteristic
107C22⋊D17—42no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—21no—
111C2×Dic17—42no—
112C2×Dic17C2×D411yescharacteristic
113C2×Dic17C2×D411yescharacteristic
114D68—21no—
115D68—21no—
116D68—42no—
117D68—42no—
118C4×D17—22no—
119C4×D17—22no—
120C4×D17—21no—
121C4×D17—21no—
122C4×D17—21no—
123C4×D17—21no—
124C4×D17—42no—
125Dic34—22no—
126Dic34—21no—
127Dic34—22no—
128Dic34—21no—
129Dic34—42no—
130C8×C17C2×D414yes—
131#3622—22no—
132#3056—171nomaximal, Sylow
133C22×C17—21no—
134C22×C17—21no—
135C22×C17#12112yes—
136C22×C17C22×D411yescharacteristic
137C2×D17—21no—
138C2×D17—21no—
139C2×D17—21no—
140C2×D17—21no—
141C2×D17—21no—
142C2×D17—44no—
143C2×D17—41no—
144C2×D17—41no—
145C2×D17—41no—
146C2×D17#13712yescomplemented
147C4×C17—41no—
148C4×C17—41no—
149C4×C17—42no—
150C4×C17#12112yes—
151C4×C17C22×D411yescharacteristic
152C4×C17C2×#4111yescharacteristic
153Dic17—21no—
154Dic17—21no—
155Dic17—42no—
156Dic17—41no—
157C2×#137—171no—
158C22×M16—171no—
159#525—171no—
160C2×C22⋊Q8—171no—
161#470—172no—
162#468—172no—
163#467—172no—
164#465—172no—
165#428D1711yescharacteristic, complemented
166#419—171no—
167#417—171no—
168C2×C17—22no—
169C2×C17C2×#12111yescharacteristic
170C2×C17C2×#13711yescharacteristic
171C2×C17#57511yescharacteristic
172D17—22no—
173D17—41no—
174D17—81no—
175C2×#41—171no—
176C22×Q8—171no—
177C23×C4—171no—
178#137—344no—
179C2×Q16—172no—
180C2×SD16—172no—
181C2×M16—171no—
182C2×M16—172no—
183C2×M16—344no—
184C2×M16C2×D1711yescharacteristic
185C22×C8—172no—
186#123—341no—
187C22⋊Q8—172no—
188C22⋊Q8—172no—
189C22⋊Q8—171no—
190C22⋊Q8—171no—
191C22⋊Q8—341no—
192C22⋊Q8—341no—
193#121—171no—
194#121—171no—
195#120—341no—
196C4×D4—341no—
197C4×D4—341no—
198#117C2×D1711yescharacteristic
199C2×C4⋊C4—341no—
200C2×C4⋊C4C2×D1711yescharacteristic
201C2×C22⋊C4—341no—
202#107C2×D1712yescomplemented
203#106C2×D1712yescomplemented
204#103—172no—
205#103—172no—
206#103—172no—
207#102—172no—
208C17#305611yescharacteristic, complemented, Sylow
209C24—171no—
210#41—342no—
211#41—342no—
212C2×Q8—172no—
213C2×Q8—171no—
214C2×Q8—341no—
215C2×Q8—342no—
216C2×Q8—342no—
217C2×D4—171no—
218C2×D4—341no—
219C2×D4—341no—
220C2×D4—341no—
221C22×C4—21no—
222C22×C4—171no—
223C22×C4—171no—
224C22×C4—171no—
225C22×C4—171no—
226C22×C4—171no—
227C22×C4—342no—
228C22×C4—342no—
229C22×C4—341no—
230C22×C4—341no—
231C22×C4C22×D1711yescharacteristic
232Q16—344no—
233SD16—344no—
234M16—342no—
235M16—344no—
236M16D6814yescomplemented
237C2×C8—172no—
238C2×C8—344no—
239C2×C8C22×D1712yes—
240C4⋊C4—21no—
241C4⋊C4—342no—
242C4⋊C4—342no—
243C4⋊C4—341no—
244C4⋊C4—341no—
245C4⋊C4C22×D1712yes—
246C4⋊C4C22×D1711yescharacteristic
247C4⋊C4C22×D1711yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—341no—
250C22⋊C4—341no—
251C22⋊C4—342no—
252C22⋊C4—342no—
253C22⋊C4—341no—
254C22⋊C4—341no—
255C42—21no—
256C23—171no—
257C23—171no—
258C23—342no—
259C23—342no—
260C23—341no—
261C23—681no—
262C23D4×D1711yescharacteristic
263Q8—342no—
264Q8—341no—
265Q8—342no—
266Q8—341no—
267Q8—682no—
268D4—341no—
269D4—341no—
270D4—682no—
271D4—682no—
272D4—682no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—171no—
278C2×C4—171no—
279C2×C4—342no—
280C2×C4—341no—
281C2×C4—341no—
282C2×C4—341no—
283C2×C4—341no—
284C2×C4—342no—
285C2×C4—342no—
286C2×C4—341no—
287C2×C4—342no—
288C2×C4—682no—
289C2×C4—682no—
290C2×C4C2×D6812yes—
291C2×C4D4×D1711yescharacteristic
292C2×C4#6325612yes—
293C2×C4C23×D1711yescharacteristic
294C8—342no—
295C8C2×D6814yes—
296C22—21no—
297C22—21no—
298C22—172no—
299C22—341no—
300C22—341no—
301C22—341no—
302C22—341no—
303C22—341no—
304C22—684no—
305C22—681no—
306C22—681no—
307C22—681no—
308C22#1058758112yes—
309C22C2×D4×D1711yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—341no—
314C4—341no—
315C4—681no—
316C4—682no—
317C4#1058758112yes—
318C4C22×D6811yescharacteristic
319C4C2×#6325611yescharacteristic
320C2—22no—
321C2—342no—
322C2—681no—
323C2—1361no—
324C2C2×#1058758111yescharacteristic
325C2#137×D1711yescharacteristic
326C2#1176360511yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#428C2
socleC22×C17C22×D4
prime radical#11763321C2
hypercenter#428D17
hypocenterC17#3056
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#10587595C2semidirect
2. C2×M16×D17C2semidirect
4. C2×#10587583C2semidirect
5. #11763068C2semidirect
6. #11763056C2semidirect
7. #11763051C2semidirect
8. #11763039C2semidirect
9. #11762817C2semidirect
10. #11762808C2semidirect
11. C17×#428C2semidirect
23. C2×Dic68C22semidirect
24. C2×#63219C22semidirect
39. C17×#107C22semidirect
40. C17×#106C22semidirect
41. #10587481C22semidirect
42. #10587481C22semidirect
43. #10587481C22semidirect
44. #10587480C22semidirect
68. M16×C17D4semidirect
146. C2×D17#137semidirect
165. #428D17semidirect
202. #107C2×D17semidirect
203. #106C2×D17semidirect
208. C17#3056semidirect
236. M16D68semidirect

Maximal subgroups

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

subgroupindex
C2×#105875952
C2×M16×D172
#117633212
C2×#105875832
#117630682
#117630562
#117630512
#117630392
#117628172
#117628082
C17×#4282
#305617