Finite Group Databasebeta

← #14540404

Subgroups — #14540404

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×#10587635C212yesmaximal, complemented
2D4×D68C211yesmaximal, characteristic, complemented
3#11763185C211yesmaximal, characteristic, complemented
4#11763060C212yesmaximal, complemented
5#11763009C211yesmaximal, characteristic, complemented
6#11763001C211yesmaximal, characteristic, complemented
7#11762954C212yesmaximal, complemented
8#11762942C212yesmaximal, complemented
9C4×#63230C211yesmaximal, characteristic, complemented
10#11762887C211yesmaximal, characteristic, complemented
11C17×#550C211yesmaximal, characteristic, complemented
12C2×#41×C17C2212yescomplemented
13C2×D4×D17—21no—
14C22×D68C2212yes—
15C17×#127—21no—
16C17×#127C2211yescharacteristic, complemented
17C17×#121—22no—
18C17×#121C2212yescomplemented
19C4×Q8×C17C2211yescharacteristic, complemented
20C4×D4×C17—21no—
21C4×D4×C17C2211yescharacteristic
22#10587635—28no—
23C2×#63230C2211yescharacteristic
24C2×#63230C2212yescomplemented
25C2×#63228C2212yes—
26C2×#63228C2212yescomplemented
27C2×#63228C2212yescomplemented
28#10587616C2212yes—
29C2×#63223C2212yescomplemented
30#10587581—21no—
31#10587568—22no—
32#10587560C2211yescharacteristic
33C4×D68C2211yescharacteristic
34#10587506C2211yescharacteristic, complemented
35#10587503C2212yescomplemented
36C22⋊#3622C2212yescomplemented
37#10587480C2211yescharacteristic
38#10587480C2212yescomplemented
39#10587479C2211yescharacteristic, complemented
40C4⋊#3622C2211yescharacteristic, complemented
41C4×#3622C2211yescharacteristic, complemented
42C23×D17—22no—
43#41×C17—24no—
44#41×C17—42no—
45C2×Q8×C17C2311yescharacteristic
46C2×D4×C17—22no—
47C2×D4×C17—22no—
48C2×D4×C17—21no—
49C2×D4×C17—21no—
50C2×D4×C17D411yescharacteristic
51C2×D4×C17C2312yes—
52C22×C4×C17—22no—
53C22×C4×C17C2312yes—
54C2×C22⋊D17—22no—
55D4×D17—44no—
56C2×D68—24no—
57C2×D68—22no—
58C2×D68—21no—
59C2×D68—21no—
60C2×D68—42no—
61C2×D68C2312yes—
62C2×D68C2311yescharacteristic
63C2×C4×D17—21no—
64C4⋊C4×C17—21no—
65C4⋊C4×C17D411yescharacteristic
66C4⋊C4×C17C2311yescharacteristic
67C22⋊C4×C17—22no—
68C22⋊C4×C17D412yes—
69C42×C17—21no—
70C42×C17C2311yescharacteristic
71#63230—24no—
72#63230D414yescomplemented
73#63228—24no—
74#63228—24no—
75#63228—42no—
76#63227—21no—
77#63227—22no—
78C4⋊Dic17C2311yescharacteristic
79#63223—24no—
80C2×#3622C2311yescharacteristic
81C2×#3622C2312yes—
82C2×#3622C2311yescharacteristic
83C23×C17—22no—
84C23×C17C2×D412yes—
85C22×D17—21no—
86C22×D17—22no—
87C22×D17—44no—
88C22×D17—44no—
89C22×D17—82no—
90Q8×C17—21no—
91Q8×C17C2×D412yescomplemented
92D4×C17—22no—
93D4×C17—22no—
94D4×C17—22no—
95D4×C17—44no—
96D4×C17—42no—
97D4×C17—81no—
98D4×C17—82no—
99C2×C4×C17—21no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—21no—
103C2×C4×C17—44no—
104C2×C4×C17C2×D411yescharacteristic
105C2×C4×C17C2×D412yes—
106C2×C4×C17C2×D411yescharacteristic
107C2×C4×C17C2411yescharacteristic
108C22⋊D17—44no—
109C2×Dic17—21no—
110D68—22no—
111D68—21no—
112D68—22no—
113D68—44no—
114D68—42no—
115D68—81no—
116D68C2×D412yes—
117C4×D17—42no—
118#3622—22no—
119#3622—21no—
120#3622C2×D412yescomplemented
121#3274—171nomaximal, Sylow
122C22×C17—22no—
123C22×C17—22no—
124C22×C17—22no—
125C22×C17—42no—
126C22×C17—42no—
127C22×C17—42no—
128C22×C17C22×D411yescharacteristic
129C2×D17—22no—
130C2×D17—41no—
131C2×D17—41no—
132C2×D17—42no—
133C2×D17—42no—
134C2×D17—42no—
135C2×D17—88no—
136C4×C17—22no—
137C4×C17—22no—
138C4×C17—21no—
139C4×C17—41no—
140C4×C17AGL(1,ℤ/8)12yescomplemented
141C4×C17C22×D411yescharacteristic
142C4×C17H(5,𝔽2)11yescharacteristic
143Dic17—41no—
144C2×AGL(1,ℤ/8)—172no—
145#550D1711yescharacteristic, complemented
146D42—171no—
147#493—171no—
148#469—172no—
149#460—171no—
150#459—171no—
151#449—172no—
152#447—172no—
153C4×SD16—171no—
154#436—171no—
155C2×C17—42no—
156C2×C17—82no—
157C2×C17D4211yescharacteristic
158C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C17#57611yescharacteristic
160D17—42no—
161D17—82no—
162C2×#41C2×D1712yescomplemented
163C22×D4—172no—
164C22×D4—341no—
165AGL(1,ℤ/8)—348no—
166C2×SD16—171no—
167C2×SD16—172no—
168C2×D8—172no—
169C2×D8—172no—
170C2×D8—172no—
171C2×M16—172no—
172#127—21no—
173#127—171no—
174#127C2×D1711yescharacteristic, complemented
175#121—22no—
176#121—172no—
177#121—341no—
178#121C2×D1712yescomplemented
179#120—342no—
180C4×Q8C2×D1711yescharacteristic, complemented
181C4×D4—21no—
182C4×D4—171no—
183C4×D4C2×D1711yescharacteristic
184#106—171no—
185C4⋊C8—171no—
186#103—171no—
187#102—172no—
188#102—171no—
189#102—172no—
190C22⋊C8—172no—
191C4×C8—171no—
192C17#327411yescharacteristic, complemented, Sylow
193C24—342no—
194#41—24no—
195#41—42no—
196C2×Q8C22×D1711yescharacteristic
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—171no—
202C2×D4—172no—
203C2×D4—344no—
204C2×D4—342no—
205C2×D4—342no—
206C2×D4—341no—
207C2×D4—341no—
208C2×D4—682no—
209C2×D4—684no—
210C2×D4C22⋊D1711yescharacteristic
211C2×D4C22×D1712yes—
212C22×C4—22no—
213C22×C4—341no—
214C22×C4C22×D1712yes—
215SD16—174no—
216SD16—344no—
217D8—344no—
218D8—344no—
219D8—682no—
220M16—344no—
221C2×C8—171no—
222C2×C8—171no—
223C2×C8—172no—
224C4⋊C4—21no—
225C4⋊C4—171no—
226C4⋊C4C22⋊D1711yescharacteristic
227C4⋊C4C22×D1711yescharacteristic
228C22⋊C4—22no—
229C22⋊C4—341no—
230C22⋊C4—342no—
231C22⋊C4C22⋊D1712yes—
232C42—21no—
233C42C22×D1711yescharacteristic
234C23—22no—
235C23—341no—
236C23—342no—
237C23—684no—
238C23—684no—
239C23—1362no—
240C23C2×C22⋊D1712yes—
241Q8—21no—
242Q8D4×D1712yescomplemented
243D4—22no—
244D4—22no—
245D4—22no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—82no—
250D4—172no—
251D4—342no—
252D4—341no—
253D4—342no—
254D4—684no—
255D4—684no—
256D4—682no—
257D4—1361no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—44no—
263C2×C4—341no—
264C2×C4—682no—
265C2×C4C2×C22⋊D1711yescharacteristic
266C2×C4C2×C22⋊D1712yes—
267C2×C4C2×C22⋊D1711yescharacteristic
268C2×C4C23×D1711yescharacteristic
269C8—172no—
270C8—342no—
271C8—341no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—342no—
279C22—681no—
280C22—681no—
281C22—682no—
282C22—682no—
283C22—682no—
284C22—1368no—
285C22C22×C22⋊D1711yescharacteristic
286C4—21no—
287C4—22no—
288C4—22no—
289C4—41no—
290C4—681no—
291C4#1058763512yescomplemented
292C4C2×D4×D1711yescharacteristic
293C4#1058769111yescharacteristic
294C2—42no—
295C2—82no—
296C2—682no—
297C2—1362no—
298C2D4×C22⋊D1711yescharacteristic
299C2C2×#1058763511yescharacteristic
300C2#1176361211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#550C2
socleC22×C17C22×D4
hypercenter#550D17
hypocenterC17#3274
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#10587635C2semidirect
2. D4×D68C2semidirect
3. #11763185C2semidirect
4. #11763060C2semidirect
5. #11763009C2semidirect
6. #11763001C2semidirect
7. #11762954C2semidirect
8. #11762942C2semidirect
9. C4×#63230C2semidirect
10. #11762887C2semidirect
11. C17×#550C2semidirect
12. C2×#41×C17C22semidirect
16. C17×#127C22semidirect
18. C17×#121C22semidirect
19. C4×Q8×C17C22semidirect
24. C2×#63230C22semidirect
26. C2×#63228C22semidirect
27. C2×#63228C22semidirect
29. C2×#63223C22semidirect
34. #10587506C22semidirect
35. #10587503C22semidirect
36. C22⋊#3622C22semidirect
38. #10587480C22semidirect
39. #10587479C22semidirect
40. C4⋊#3622C22semidirect
41. C4×#3622C22semidirect
72. #63230D4semidirect
91. Q8×C17C2×D4semidirect
120. #3622C2×D4semidirect
140. C4×C17AGL(1,ℤ/8)semidirect
145. #550D17semidirect
162. C2×#41C2×D17semidirect
174. #127C2×D17semidirect
178. #121C2×D17semidirect
180. C4×Q8C2×D17semidirect
192. C17#3274semidirect
242. Q8D4×D17semidirect
291. C4#10587635semidirect

Maximal subgroups

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

subgroupindex
C2×#105876352
D4×D682
#117631852
#117630602
#117630092
#117630012
#117629542
#117629422
C4×#632302
#117628872
C17×#5502
#327417