Finite Group Databasebeta

← #14538465

Subgroups — #14538465

Order 2176.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#10587594C211yesmaximal, characteristic, complemented
2C2×#10587593C211yesmaximal, characteristic, complemented
3#11763390C211yesmaximal, characteristic, complemented
4#11763378C211yesmaximal, characteristic, complemented
5#11763063C212yesmaximal, complemented
6#11763056C212yesmaximal, complemented
7#11763046C212yesmaximal, complemented
8#11763039C212yesmaximal, complemented
9#11762803C211yesmaximal, characteristic, complemented
10C2×#10587480C211yesmaximal, characteristic, complemented
11C17×#428C211yesmaximal, characteristic, complemented
12C2×#63252C2211yescharacteristic
13C2×#63252C2211yescharacteristic
14C22×D68C2211yescharacteristic
15C2×M16×C17C2211yescharacteristic
16C17×#117C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18#10587616—22no—
19C22×#3622C2211yescharacteristic
20#10587594—24no—
21#10587593—24no—
22C2×D136C2212yes—
23C2×#63219C2212yes—
24C2×#63218C2212yescomplemented
25C2×C8×D17C2212yescomplemented
26#10587583—21no—
27#10587583C2212yescomplemented
28#10587581—21no—
29#10587581C2212yes—
30#10587581C2212yes—
31#10587581C2212yescomplemented
32#10587568—22no—
33#10587561—21no—
34#10587560—21no—
35C17×#107C2212yescomplemented
36C17×#106C2212yescomplemented
37#10587481C2211yescharacteristic, complemented
38#10587481C2211yescharacteristic, complemented
39#10587480C2212yes—
40#10587480C2212yes—
41#10587480C2211yescharacteristic, complemented
42#10587480C2211yescharacteristic, complemented
43C23×D17—21no—
44C22×C4×C17—21no—
45C22×C4×C17C2311yescharacteristic
46C2×C22⋊D17—21no—
47C2×C22⋊D17D412yes—
48#63252—22no—
49#63252—22no—
50#63252—22no—
51#63252—22no—
52C2×D68—22no—
53C2×D68—22no—
54C2×D68—21no—
55C2×D68—22no—
56C2×D68—22no—
57C2×D68—22no—
58C2×D68D411yescharacteristic
59C2×D68C2312yes—
60C2×D68C2311yescharacteristic
61C2×C4×D17—21no—
62C2×C4×D17C2312yes—
63C2×Dic34D411yescharacteristic
64C2×Dic34C2311yescharacteristic
65M16×C17D414yescomplemented
66C2×C8×C17C2312yes—
67C4⋊C4×C17—21no—
68C4⋊C4×C17C2312yes—
69C4⋊C4×C17C2311yescharacteristic
70C4⋊C4×C17C2311yescharacteristic
71C22⋊C4×C17—21no—
72C42×C17—21no—
73#63227—22no—
74#63227—22no—
75#63227—22no—
76#63227—22no—
77C4⋊Dic17—22no—
78C2×#3622—22no—
79C2×#3622C2311yescharacteristic
80C2×#3622C2311yescharacteristic
81D136—24no—
82#63219—24no—
83#63218—24no—
84C8×D17—24no—
85C23×C17C2×D411yescharacteristic
86C22×D17—21no—
87C22×D17—22no—
88C22×D17—42no—
89C22×D17—81no—
90C22×D17—81no—
91C22×D17C2×D412yes—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—22no—
95C2×C4×C17—81no—
96C2×C4×C17C2×D412yes—
97C2×C4×C17C2×D411yescharacteristic
98C2×C4×C17#4112yes—
99C2×C4×C17C2411yescharacteristic
100C22⋊D17—24no—
101C22⋊D17—42no—
102C2×Dic17—21no—
103C2×Dic17C2×D412yes—
104D68—22no—
105D68—22no—
106D68—21no—
107D68—22no—
108D68—21no—
109D68—42no—
110D68—82no—
111D68—82no—
112D68—82no—
113C4×D17—22no—
114C4×D17—22no—
115C4×D17—42no—
116Dic34—22no—
117Dic34—21no—
118Dic34—21no—
119C8×C17C2×D414yes—
120#3622—21no—
121#3622—21no—
122#3062—171nomaximal, Sylow
123C22×C17—21no—
124C22×C17—21no—
125C22×C17#12112yes—
126C22×C17C22×D411yescharacteristic
127C2×D17—22no—
128C2×D17—22no—
129C2×D17—22no—
130C2×D17—41no—
131C2×D17—42no—
132C2×D17—41no—
133C2×D17—42no—
134C2×D17—42no—
135C2×D17—41no—
136C2×D17—84no—
137C4×C17—41no—
138C4×C17—42no—
139C4×C17—41no—
140C4×C17#12112yes—
141C4×C17C22×D411yescharacteristic
142C4×C17C2×#4111yescharacteristic
143Dic17—22no—
144Dic17—41no—
145C2×AGL(1,ℤ/8)—171no—
146C2×#131—171no—
147#537—171no—
148#535—171no—
149#469—172no—
150#468—172no—
151#466—172no—
152#465—172no—
153#428D1711yescharacteristic, complemented
154#416—171no—
155C2×#102—171no—
156C2×C17—22no—
157C2×C17C2×#12111yescharacteristic
158C2×C17#57611yescharacteristic
159C2×C17#57711yescharacteristic
160D17—42no—
161D17—81no—
162D17—82no—
163C2×#41—171no—
164C2×#41—171no—
165C22×D4—171no—
166AGL(1,ℤ/8)—344no—
167C2×SD16—172no—
168C2×D8—172no—
169#131—344no—
170C2×M16—172no—
171C2×M16C2×D1711yescharacteristic
172C22×C8—171no—
173C22×C8—172no—
174#127—341no—
175#124—341no—
176C22⋊Q8—172no—
177C22⋊Q8—341no—
178#121—172no—
179#121—172no—
180#121—172no—
181#121—341no—
182#121—342no—
183#120—342no—
184#117C2×D1711yescharacteristic
185C2×C4⋊C4C2×D1711yescharacteristic
186#107C2×D1712yescomplemented
187#106C2×D1712yescomplemented
188#103—171no—
189#103—171no—
190#102—172no—
191#102—172no—
192#102—171no—
193#102—171no—
194C17#306211yescharacteristic, complemented, Sylow
195C24—341no—
196#41—342no—
197#41—342no—
198#41—342no—
199#41—342no—
200C2×Q8—171no—
201C2×Q8—171no—
202C2×D4—172no—
203C2×D4—171no—
204C2×D4—171no—
205C2×D4—172no—
206C2×D4—342no—
207C2×D4—342no—
208C2×D4—342no—
209C2×D4—341no—
210C2×D4—342no—
211C2×D4—341no—
212C2×D4—342no—
213C22×C4—21no—
214C22×C4—172no—
215C22×C4—341no—
216C22×C4C22×D1711yescharacteristic
217SD16—344no—
218D8—344no—
219M16—344no—
220M16D6814yescomplemented
221C2×C8—171no—
222C2×C8—171no—
223C2×C8—342no—
224C2×C8—344no—
225C2×C8C22×D1712yes—
226C4⋊C4—21no—
227C4⋊C4—342no—
228C4⋊C4C22×D1712yes—
229C4⋊C4C22×D1711yescharacteristic
230C4⋊C4C22×D1711yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—342no—
233C22⋊C4—342no—
234C22⋊C4—342no—
235C22⋊C4—342no—
236C42—21no—
237C23—172no—
238C23—341no—
239C23—342no—
240C23—682no—
241C23—1361no—
242C23—1361no—
243C23D4×D1711yescharacteristic
244Q8—342no—
245Q8—341no—
246Q8—341no—
247D4—344no—
248D4—341no—
249D4—342no—
250D4—342no—
251D4—341no—
252D4—342no—
253D4—682no—
254D4—682no—
255D4—1362no—
256D4—1362no—
257D4—1362no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—81no—
262C2×C4—172no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—341no—
266C2×C4—682no—
267C2×C4C2×D6812yes—
268C2×C4D4×D1711yescharacteristic
269C2×C4#6325612yes—
270C2×C4C23×D1711yescharacteristic
271C8—341no—
272C8—341no—
273C8C2×D6814yes—
274C22—21no—
275C22—21no—
276C22—342no—
277C22—342no—
278C22—342no—
279C22—681no—
280C22—682no—
281C22—681no—
282C22—682no—
283C22—682no—
284C22—681no—
285C22—1364no—
286C22#1058758112yes—
287C22C2×D4×D1711yescharacteristic
288C4—42no—
289C4—41no—
290C4—41no—
291C4—342no—
292C4—681no—
293C4#1058758112yes—
294C4C22×D6811yescharacteristic
295C4C2×#6325611yescharacteristic
296C2—22no—
297C2—682no—
298C2—1362no—
299C2—1361no—
300C2C2×#1058758111yescharacteristic
301C2#1176361311yescharacteristic
302C2#1176361811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#428C2
socleC22×C17C22×D4
hypercenter#428D17
hypocenterC17#3062
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×#10587594C2semidirect
2. C2×#10587593C2semidirect
3. #11763390C2semidirect
4. #11763378C2semidirect
5. #11763063C2semidirect
6. #11763056C2semidirect
7. #11763046C2semidirect
8. #11763039C2semidirect
9. #11762803C2semidirect
10. C2×#10587480C2semidirect
11. C17×#428C2semidirect
24. C2×#63218C22semidirect
25. C2×C8×D17C22semidirect
27. #10587583C22semidirect
31. #10587581C22semidirect
35. C17×#107C22semidirect
36. C17×#106C22semidirect
37. #10587481C22semidirect
38. #10587481C22semidirect
41. #10587480C22semidirect
42. #10587480C22semidirect
65. M16×C17D4semidirect
153. #428D17semidirect
186. #107C2×D17semidirect
187. #106C2×D17semidirect
194. C17#3062semidirect
220. M16D68semidirect

Maximal subgroups

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

subgroupindex
C2×#105875942
C2×#105875932
#117633902
#117633782
#117630632
#117630562
#117630462
#117630392
#117628032
C2×#105874802
C17×#4282
#306217