Finite Group Databasebeta

← #14539232

Subgroups — #14539232

Order 2176.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#10587618C211yesmaximal, characteristic, complemented
3C2×#10587635C211yesmaximal, characteristic, complemented
4#11763355C211yesmaximal, characteristic, complemented
5C2×#10587626C211yesmaximal, characteristic, complemented
6#11763197C212yesmaximal, complemented
7#11763195C212yesmaximal, complemented
8#11763184C212yesmaximal, complemented
9#11763180C212yesmaximal, complemented
10M16×Dic17C211yesmaximal, characteristic, complemented
11C2×C17×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
12C2×#41×C17C2211yescharacteristic
13C22×D4×C17C2211yescharacteristic
14C22×C22⋊D17—22no—
15C2×#63252C2211yescharacteristic
16C22×D68C2211yescharacteristic
17C17×AGL(1,ℤ/8)—24no—
18C2×SD16×C17C2212yescomplemented
19C2×D8×C17C2212yescomplemented
20C2×M16×C17C2211yescharacteristic
21#10587635—24no—
22#10587633C2211yescharacteristic, complemented
23#10587630C2211yescharacteristic, complemented
24C2×#63230C2212yescomplemented
25#10587626—22no—
26#10587626C2212yes—
27#10587626C2212yes—
28#10587626C2211yescharacteristic, complemented
29#10587625—22no—
30#10587622C2211yescharacteristic, complemented
31C2×#63229C2212yescomplemented
32#10587618—24no—
33C2×#63228C2212yescomplemented
34C2×#63228C2212yescomplemented
35C4×C22⋊D17—22no—
36C2×C4×Dic17C2211yescharacteristic
37C2×#63223C2211yescharacteristic
38#10587594—24no—
39C2×D136C2212yescomplemented
40C2×#63219C2212yescomplemented
41#10587486C2212yescomplemented
42C8×Dic17C2212yescomplemented
43C24×C17—21no—
44C23×D17—21no—
45#41×C17—22no—
46#41×C17—22no—
47C2×Q8×C17C2311yescharacteristic
48C2×D4×C17—21no—
49C2×D4×C17—22no—
50C2×D4×C17—22no—
51C2×D4×C17—21no—
52C2×D4×C17C2311yescharacteristic
53C2×D4×C17C2312yes—
54C22×C4×C17—21no—
55C22×C4×C17C2311yescharacteristic
56C2×C22⋊D17—21no—
57C2×C22⋊D17—24no—
58C2×C22⋊D17—24no—
59C2×C22⋊D17—22no—
60C2×C22⋊D17—48no—
61C22×Dic17D412yes—
62#63252—22no—
63#63252—22no—
64C2×D68—22no—
65C2×D68—22no—
66C2×D68—21no—
67C2×D68C2312yes—
68C2×D68C2311yescharacteristic
69C2×C4×D17—21no—
70C2×Dic34C2311yescharacteristic
71SD16×C17—24no—
72D8×C17—24no—
73M16×C17D414yescomplemented
74C2×C8×C17C2312yes—
75C22⋊Dic17—22no—
76#63230—24no—
77#63229—24no—
78#63228—24no—
79#63228—24no—
80#63227—22no—
81#63225—22no—
82C4×Dic17D412yes—
83C4×Dic17C2311yescharacteristic
84C4×Dic17C2311yescharacteristic
85#63223D414yescomplemented
86C2×#3622C2312yes—
87D136—24no—
88#63219—24no—
89C23×C17—21no—
90C23×C17—22no—
91C23×C17—44no—
92C23×C17—42no—
93C23×C17C2×D411yescharacteristic
94C22×D17—21no—
95C22×D17—22no—
96C22×D17—44no—
97C22×D17—42no—
98Q8×C17—21no—
99Q8×C17—21no—
100D4×C17—21no—
101D4×C17—22no—
102D4×C17—22no—
103D4×C17—21no—
104D4×C17—42no—
105D4×C17—42no—
106C2×C4×C17—21no—
107C2×C4×C17—42no—
108C2×C4×C17C2×D411yescharacteristic
109C2×C4×C17C2×D412yes—
110C2×C4×C17C2×D412yes—
111C2×C4×C17C2411yescharacteristic
112C22⋊D17—48no—
113C22⋊D17—42no—
114C22⋊D17—44no—
115C22⋊D17—48no—
116C2×Dic17—24no—
117C2×Dic17—21no—
118C2×Dic17C2×D412yes—
119C2×Dic17C2×D412yes—
120D68—22no—
121D68—22no—
122D68—21no—
123D68—21no—
124D68—42no—
125C4×D17—42no—
126Dic34—21no—
127Dic34—21no—
128C8×C17C2×D414yescomplemented
129#3622C2×D414yescomplemented
130#3151—171nomaximal, Sylow
131C22×C17—22no—
132C22×C17—41no—
133C22×C17—42no—
134C22×C17—44no—
135C22×C17—42no—
136C22×C17—84no—
137C22×C17#12712yes—
138C22×C17C22×D411yescharacteristic
139C2×D17—41no—
140C2×D17—42no—
141C2×D17—44no—
142C2×D17—42no—
143C2×D17—84no—
144C4×C17—41no—
145C4×C17#12712yes—
146C4×C17C22×D411yescharacteristic
147C4×C17C22×D411yescharacteristic
148Dic17—22no—
149Dic17—41no—
150Dic17AGL(1,ℤ/8)14yescomplemented
151C2×AGL(1,ℤ/8)—171no—
152C2×AGL(1,ℤ/8)—171no—
153C2×AGL(1,ℤ/8)—171no—
154C2×AGL(1,ℤ/8)D1711yescharacteristic, complemented
155#532—171no—
156C2×#127—171no—
157#496—172no—
158#496—172no—
159#493—172no—
160#492—172no—
161C4×M16—171no—
162C2×C17—22no—
163C2×C17—81no—
164C2×C17—82no—
165C2×C17C2×#12711yescharacteristic
166C2×C17C2×AGL(1,ℤ/8)12yes—
167D17—81no—
168D17—82no—
169C2×#41—171no—
170C2×#41C2×D1711yescharacteristic
171C22×D4—171no—
172C22×D4—342no—
173C22×D4C2×D1711yescharacteristic
174AGL(1,ℤ/8)—24no—
175AGL(1,ℤ/8)—344no—
176AGL(1,ℤ/8)—344no—
177AGL(1,ℤ/8)—344no—
178C2×SD16—172no—
179C2×SD16—172no—
180C2×SD16—172no—
181C2×SD16C2×D1712yescomplemented
182C2×D8—172no—
183C2×D8—172no—
184C2×D8—172no—
185C2×D8C2×D1712yescomplemented
186C2×M16—171no—
187C2×M16C2×D1711yescharacteristic
188C4⋊Q8—171no—
189#127—172no—
190#127—172no—
191#127—171no—
192#127—342no—
193#124—171no—
194#124—171no—
195#121—342no—
196C4×D4—342no—
197C2×C42—171no—
198#97—172no—
199C4×C8—172no—
200C17#315111yescharacteristic, complemented, Sylow
201C24—21no—
202C24—341no—
203#41—22no—
204#41—22no—
205#41—342no—
206#41—342no—
207C2×Q8—171no—
208C2×Q8C22×D1711yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—171no—
214C2×D4—172no—
215C2×D4—342no—
216C2×D4—344no—
217C2×D4—344no—
218C2×D4—341no—
219C2×D4—342no—
220C2×D4—341no—
221C2×D4—342no—
222C2×D4—688no—
223C2×D4C22×D1711yescharacteristic
224C2×D4C22×D1712yes—
225C22×C4—21no—
226C22×C4—172no—
227C22×C4—341no—
228C22×C4C22×D1711yescharacteristic
229SD16—24no—
230SD16—344no—
231SD16—344no—
232SD16—344no—
233D8—24no—
234D8—344no—
235D8—344no—
236D8—344no—
237M16—174no—
238M16C22⋊D1714yescomplemented
239C2×C8—172no—
240C2×C8C22×D1712yes—
241C4⋊C4—342no—
242C22⋊C4—342no—
243C22⋊C4—342no—
244C42—172no—
245C42—171no—
246C42—171no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—341no—
252C23—342no—
253C23—684no—
254C23—682no—
255C23D4×D1711yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—341no—
259Q8—341no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—341no—
267D4—342no—
268D4—341no—
269D4—342no—
270D4—688no—
271D4—684no—
272D4—688no—
273D4—682no—
274D4—682no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—172no—
278C2×C4—172no—
279C2×C4—344no—
280C2×C4—341no—
281C2×C4—682no—
282C2×C4D4×D1712yes—
283C2×C4D4×D1711yescharacteristic
284C2×C4C2×C22⋊D1712yes—
285C2×C4C23×D1711yescharacteristic
286C8—174no—
287C8C2×C22⋊D1714yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—684no—
295C22—682no—
296C22—681no—
297C22—682no—
298C22—1364no—
299C22#1058762612yes—
300C22C2×D4×D1711yescharacteristic
301C4—41no—
302C4—174no—
303C4—342no—
304C4—681no—
305C4#1058762612yes—
306C4C2×D4×D1711yescharacteristic
307C4C22×C22⋊D1711yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—1362no—
312C2—1361no—
313C2C2×#1058762611yescharacteristic
314C2AGL(1,ℤ/8)×D1712yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC2×C17×AGL(1,ℤ/8)C2
socleC22×C17C22×D4
hypercenterC2×AGL(1,ℤ/8)D17
hypocenterC17#3151
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#10587594C2semidirect
2. C2×#10587618C2semidirect
3. C2×#10587635C2semidirect
4. #11763355C2semidirect
5. C2×#10587626C2semidirect
6. #11763197C2semidirect
7. #11763195C2semidirect
8. #11763184C2semidirect
9. #11763180C2semidirect
10. M16×Dic17C2semidirect
11. C2×C17×AGL(1,ℤ/8)C2semidirect
18. C2×SD16×C17C22semidirect
19. C2×D8×C17C22semidirect
22. #10587633C22semidirect
23. #10587630C22semidirect
24. C2×#63230C22semidirect
28. #10587626C22semidirect
30. #10587622C22semidirect
31. C2×#63229C22semidirect
33. C2×#63228C22semidirect
34. C2×#63228C22semidirect
39. C2×D136C22semidirect
40. C2×#63219C22semidirect
41. #10587486C22semidirect
42. C8×Dic17C22semidirect
73. M16×C17D4semidirect
85. #63223D4semidirect
128. C8×C17C2×D4semidirect
129. #3622C2×D4semidirect
150. Dic17AGL(1,ℤ/8)semidirect
154. C2×AGL(1,ℤ/8)D17semidirect
181. C2×SD16C2×D17semidirect
185. C2×D8C2×D17semidirect
200. C17#3151semidirect
238. M16C22⋊D17semidirect
287. C8C2×C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105875942
C2×#105876182
C2×#105876352
#117633552
C2×#105876262
#117631972
#117631952
#117631842
#117631802
M16×Dic172
C2×C17×AGL(1,ℤ/8)2
#315117