Finite Group Databasebeta

← #14540405

Subgroups — #14540405

Order 2176.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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×#10587600C212yesmaximal, complemented
2#11763481C211yesmaximal, characteristic, complemented
3D4×D68C211yesmaximal, characteristic, complemented
4#11763186C211yesmaximal, characteristic, complemented
5#11763063C212yesmaximal, complemented
6#11763009C211yesmaximal, characteristic, complemented
7#11763004C211yesmaximal, characteristic, complemented
8#11762957C212yesmaximal, complemented
9#11762944C212yesmaximal, complemented
10#11762888C211yesmaximal, characteristic, complemented
11C4×SD16×C17C211yesmaximal, characteristic, complemented
12C2×#63256C2212yescomplemented
13C2×D4×D17C2212yes—
14C22×D68—21no—
15C2×SD16×C17C2211yescharacteristic
16C4×Q8×C17C2211yescharacteristic, complemented
17C4×D4×C17C2211yescharacteristic
18C2×#63230C2212yescomplemented
19C2×#63228C2212yes—
20#10587616—21no—
21#10587600—28no—
22C2×D136C2212yescomplemented
23C2×D136C2212yescomplemented
24C2×#63218C2212yescomplemented
25#10587581—22no—
26#10587581C2212yes—
27#10587581C2212yescomplemented
28#10587568—22no—
29#10587560—21no—
30#10587560C2211yescharacteristic
31#10587560C2211yescharacteristic, complemented
32C4×D68—21no—
33C4×D68C2211yescharacteristic
34C17×#106C2211yescharacteristic, complemented
35C17×#103C2211yescharacteristic, complemented
36C17×#102C2211yescharacteristic
37C4×C8×C17C2211yescharacteristic, complemented
38#10587491C2212yescomplemented
39#10587480C2212yescomplemented
40#10587480C2212yescomplemented
41C4⋊#3622C2211yescharacteristic, complemented
42C23×D17—22no—
43C2×Q8×C17C2311yescharacteristic
44C2×D4×C17C2311yescharacteristic
45C22×C4×C17—21no—
46C2×C22⋊D17—22no—
47#63256—24no—
48#63256—42no—
49D4×D17—24no—
50D4×D17—42no—
51C2×D68—22no—
52C2×D68—22no—
53C2×D68—21no—
54C2×D68—21no—
55C2×D68—22no—
56C2×D68—21no—
57C2×D68—21no—
58C2×D68—44no—
59C2×D68D411yescharacteristic
60C2×D68C2312yes—
61C2×D68C2312yes—
62C2×C4×D17—22no—
63C2×C4×D17C2312yes—
64SD16×C17D414yescomplemented
65C2×C8×C17C2311yescharacteristic
66C2×C8×C17C2311yescharacteristic
67C4⋊C4×C17—21no—
68C4⋊C4×C17C2311yescharacteristic
69C4⋊C4×C17C2311yescharacteristic
70C22⋊C4×C17—21no—
71C42×C17—21no—
72C42×C17C2311yescharacteristic
73#63230—24no—
74#63228—24no—
75#63227—22no—
76#63227—22no—
77#63227D412yes—
78C4⋊Dic17D411yescharacteristic
79C2×#3622C2312yes—
80D136—24no—
81D136—42no—
82#63218—24no—
83C23×C17—21no—
84C22×D17—22no—
85C22×D17—22no—
86C22×D17—44no—
87C22×D17—44no—
88C22×D17—82no—
89C22×D17C2×D412yes—
90Q8×C17—21no—
91Q8×C17C2×D412yescomplemented
92D4×C17—21no—
93D4×C17C2×D412yes—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—42no—
98C2×C4×C17C2×D411yescharacteristic
99C2×C4×C17C2×D411yescharacteristic
100C2×C4×C17C2411yescharacteristic
101C22⋊D17—44no—
102C2×Dic17C2×D412yes—
103D68—22no—
104D68—22no—
105D68—22no—
106D68—22no—
107D68—22no—
108D68—44no—
109D68—44no—
110D68—42no—
111D68—42no—
112D68—81no—
113D68—81no—
114D68—82no—
115C4×D17—22no—
116C4×D17—22no—
117C4×D17—44no—
118C8×C17—21no—
119C8×C17C2×D412yescomplemented
120#3622—22no—
121#3274—171nomaximal, Sylow
122C22×C17—22no—
123C22×C17—41no—
124C22×C17—41no—
125C22×C17C22×D411yescharacteristic
126C2×D17—22no—
127C2×D17—22no—
128C2×D17—22no—
129C2×D17—42no—
130C2×D17—42no—
131C2×D17—42no—
132C2×D17—42no—
133C2×D17—42no—
134C2×D17—42no—
135C2×D17—88no—
136C4×C17—22no—
137C4×C17—21no—
138C4×C17—41no—
139C4×C17—41no—
140C4×C17AGL(1,ℤ/8)12yescomplemented
141C4×C17C22×D411yescharacteristic
142C4×C17H(5,𝔽2)11yescharacteristic
143Dic17—22no—
144C2×AGL(1,ℤ/8)—172no—
145#550—171no—
146D42—171no—
147#493—171no—
148#469—172no—
149#460—171no—
150#459—171no—
151#449—172no—
152#447—172no—
153C4×SD16D1711yescharacteristic, complemented
154#436—171no—
155C2×C17—42no—
156C2×C17D4211yescharacteristic
157C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
158C2×C17#57611yescharacteristic
159D17—42no—
160D17—82no—
161D17—82no—
162C2×#41—172no—
163C22×D4—172no—
164C22×D4—341no—
165AGL(1,ℤ/8)—348no—
166C2×SD16—172no—
167C2×SD16C2×D1711yescharacteristic
168C2×D8—172no—
169C2×D8—172no—
170C2×D8—172no—
171C2×M16—172no—
172#127—171no—
173#127—171no—
174#127—341no—
175#121—172no—
176#121—172no—
177#121—342no—
178#121—341no—
179#120—342no—
180C4×Q8C2×D1711yescharacteristic, complemented
181C4×D4—171no—
182C4×D4—341no—
183C4×D4C2×D1711yescharacteristic
184#106C2×D1711yescharacteristic, complemented
185C4⋊C8—171no—
186#103C2×D1711yescharacteristic, complemented
187#102—172no—
188#102—172no—
189#102C2×D1711yescharacteristic
190C22⋊C8—172no—
191C4×C8C2×D1711yescharacteristic, complemented
192C17#327411yescharacteristic, complemented, Sylow
193C24—342no—
194#41—344no—
195#41—682no—
196C2×Q8C22×D1711yescharacteristic
197C2×D4—171no—
198C2×D4—172no—
199C2×D4—172no—
200C2×D4—342no—
201C2×D4—342no—
202C2×D4—342no—
203C2×D4—342no—
204C2×D4—344no—
205C2×D4—341no—
206C2×D4—341no—
207C2×D4—341no—
208C2×D4—341no—
209C2×D4—684no—
210C2×D4—682no—
211C2×D4C22×D1711yescharacteristic
212C22×C4—21no—
213C22×C4—172no—
214C22×C4—342no—
215SD16—344no—
216SD16D6814yescomplemented
217D8—344no—
218D8—344no—
219D8—682no—
220M16—344no—
221C2×C8—172no—
222C2×C8C22×D1711yescharacteristic
223C2×C8C22×D1711yescharacteristic
224C4⋊C4—21no—
225C4⋊C4—171no—
226C4⋊C4C22×D1711yescharacteristic
227C4⋊C4C22×D1711yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—172no—
230C22⋊C4—342no—
231C22⋊C4—342no—
232C42—21no—
233C42C22×D1711yescharacteristic
234C23—21no—
235C23—172no—
236C23—342no—
237C23—342no—
238C23—684no—
239C23—684no—
240C23—1362no—
241Q8—21no—
242Q8C2×D6812yescomplemented
243D4—21no—
244D4—342no—
245D4—342no—
246D4—342no—
247D4—342no—
248D4—342no—
249D4—684no—
250D4—684no—
251D4—684no—
252D4—682no—
253D4—682no—
254D4—1361no—
255D4—1361no—
256D4—1362no—
257D4C2×D6812yes—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—172no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—684no—
266C2×C4D4×D1711yescharacteristic
267C2×C4D4×D1711yescharacteristic
268C2×C4C23×D1711yescharacteristic
269C8—21no—
270C8—342no—
271C8C2×D6812yescomplemented
272C22—22no—
273C22—41no—
274C22—41no—
275C22—342no—
276C22—342no—
277C22—342no—
278C22—682no—
279C22—682no—
280C22—682no—
281C22—682no—
282C22—682no—
283C22—682no—
284C22—1368no—
285C22C2×D4×D1711yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—41no—
290C4—342no—
291C4#1058760012yescomplemented
292C4C22×D6811yescharacteristic
293C4#1058769611yescharacteristic
294C2—42no—
295C2—682no—
296C2—1362no—
297C2—1362no—
298C2D4×D6811yescharacteristic
299C2C2×#1058760011yescharacteristic
300C2#1176361111yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC4×SD16×C17C2
socleC22×C17C22×D4
hypercenterC4×SD16D17
hypocenterC17#3274
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#10587600C2semidirect
2. #11763481C2semidirect
3. D4×D68C2semidirect
4. #11763186C2semidirect
5. #11763063C2semidirect
6. #11763009C2semidirect
7. #11763004C2semidirect
8. #11762957C2semidirect
9. #11762944C2semidirect
10. #11762888C2semidirect
11. C4×SD16×C17C2semidirect
12. C2×#63256C22semidirect
16. C4×Q8×C17C22semidirect
18. C2×#63230C22semidirect
22. C2×D136C22semidirect
23. C2×D136C22semidirect
24. C2×#63218C22semidirect
27. #10587581C22semidirect
31. #10587560C22semidirect
34. C17×#106C22semidirect
35. C17×#103C22semidirect
37. C4×C8×C17C22semidirect
38. #10587491C22semidirect
39. #10587480C22semidirect
40. #10587480C22semidirect
41. C4⋊#3622C22semidirect
64. SD16×C17D4semidirect
91. Q8×C17C2×D4semidirect
119. C8×C17C2×D4semidirect
140. C4×C17AGL(1,ℤ/8)semidirect
153. C4×SD16D17semidirect
180. C4×Q8C2×D17semidirect
184. #106C2×D17semidirect
186. #103C2×D17semidirect
191. C4×C8C2×D17semidirect
192. C17#3274semidirect
216. SD16D68semidirect
242. Q8C2×D68semidirect
271. C8C2×D68semidirect
291. C4#10587600semidirect

Maximal subgroups

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

subgroupindex
C2×#105876002
#117634812
D4×D682
#117631862
#117630632
#117630092
#117630042
#117629572
#117629442
#117628882
C4×SD16×C172
#327417