Finite Group Databasebeta

← #14539263

Subgroups — #14539263

Order 2176.

564 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10587636C211yesmaximal, characteristic, complemented
3C22×#63230C211yesmaximal, characteristic, complemented
4#11763383C211yesmaximal, characteristic, complemented
5#11763373C211yesmaximal, characteristic, complemented
6#11763202C212yesmaximal, complemented
7#11763195C212yesmaximal, complemented
8#11763194C212yesmaximal, complemented
9#11763180C212yesmaximal, complemented
10#11762767C211yesmaximal, characteristic, complemented
11C2×C17×#137C211yesmaximal, characteristic, complemented
12C2×#41×C17C2211yescharacteristic
13C22×Q8×C17C2211yescharacteristic
14C2×#63252C2211yescharacteristic
15C22×D68C2211yescharacteristic
16C17×#137—24no—
17C2×Q16×C17C2212yescomplemented
18C2×SD16×C17C2212yescomplemented
19C2×M16×C17C2211yescharacteristic
20#10587636—24no—
21#10587633C2212yes—
22#10587633C2212yescomplemented
23#10587632—22no—
24#10587630C2212yescomplemented
25C2×#63231C2211yescharacteristic, complemented
26C2×#63230—24no—
27C2×#63230C2212yes—
28C2×#63230C2212yes—
29C2×#63230C2211yescharacteristic, complemented
30#10587626C2212yes—
31#10587623—22no—
32C2×#63229C2211yescharacteristic, complemented
33C2×#63228C2211yescharacteristic, complemented
34#10587616—22no—
35#10587615—22no—
36#10587612C2211yescharacteristic
37C22×#3622C2211yescharacteristic
38#10587594—24no—
39C2×D136C2212yes—
40C2×#63219C2212yes—
41#10587486C2212yescomplemented
42C8×Dic17C2212yescomplemented
43C23×D17—21no—
44#41×C17—22no—
45#41×C17—22no—
46C2×Q8×C17—21no—
47C2×Q8×C17—22no—
48C2×Q8×C17—22no—
49C2×Q8×C17C2312yes—
50C2×Q8×C17C2311yescharacteristic
51C2×D4×C17—21no—
52C2×D4×C17C2311yescharacteristic
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17C2311yescharacteristic
56C2×C22⋊D17—21no—
57C2×C22⋊D17—24no—
58#63252—22no—
59#63252—22no—
60C2×D68—22no—
61C2×D68—22no—
62C2×D68—21no—
63C2×D68C2312yes—
64C2×D68C2311yescharacteristic
65C2×C4×D17—21no—
66C2×Dic34C2311yescharacteristic
67Q16×C17—24no—
68SD16×C17—24no—
69M16×C17D414yescomplemented
70C2×C8×C17C2312yes—
71C22⋊Dic17D412yes—
72#63231—22no—
73#63230—24no—
74#63230—24no—
75#63230—22no—
76#63229—22no—
77#63228—22no—
78#63227—24no—
79#63227—24no—
80C4⋊Dic17D412yes—
81#63225—24no—
82C4×Dic17C2312yes—
83C2×#3622D414yescomplemented
84C2×#3622C2311yescharacteristic
85C2×#3622C2311yescharacteristic
86D136—24no—
87#63219—24no—
88C23×C17—21no—
89C23×C17C2×D411yescharacteristic
90C22×D17—21no—
91C22×D17—22no—
92C22×D17—42no—
93C22×D17—82no—
94Q8×C17—22no—
95Q8×C17—21no—
96Q8×C17—22no—
97Q8×C17—21no—
98Q8×C17—42no—
99D4×C17—21no—
100D4×C17—21no—
101D4×C17—42no—
102C2×C4×C17—21no—
103C2×C4×C17—22no—
104C2×C4×C17—42no—
105C2×C4×C17—42no—
106C2×C4×C17C2×D411yescharacteristic
107C2×C4×C17C2×D412yes—
108C2×C4×C17C2×D412yes—
109C2×C4×C17C2411yescharacteristic
110C22⋊D17—42no—
111C22⋊D17—84no—
112C2×Dic17—21no—
113C2×Dic17C2×D414yes—
114D68—22no—
115D68—22no—
116D68—21no—
117D68—21no—
118D68—42no—
119C4×D17—42no—
120Dic34—21no—
121Dic34—21no—
122C8×C17C2×D414yes—
123#3622C2×D412yes—
124#3622C2×D412yes—
125#3154—171nomaximal, Sylow
126C22×C17—22no—
127C22×C17—41no—
128C22×C17—42no—
129C22×C17#12712yes—
130C22×C17C22×D411yescharacteristic
131C2×D17—41no—
132C2×D17—42no—
133C2×D17—44no—
134C2×D17—42no—
135C2×D17—84no—
136C4×C17—41no—
137C4×C17—42no—
138C4×C17#12712yes—
139C4×C17C22×D411yescharacteristic
140C4×C17C22×D411yescharacteristic
141Dic17—24no—
142Dic17—41no—
143C2×#137D1711yescharacteristic, complemented
144C2×AGL(1,ℤ/8)—171no—
145C2×#135—171no—
146C22×SD16—171no—
147#536—171no—
148#535—171no—
149#497—172no—
150#496—172no—
151#495—172no—
152#492—172no—
153#405—171no—
154C2×C17—22no—
155C2×C17—81no—
156C2×C17C2×#12711yescharacteristic
157C2×C17#57712yes—
158D17—81no—
159D17—82no—
160C2×#41—171no—
161C2×#41C2×D1711yescharacteristic
162C22×Q8C2×D1711yescharacteristic
163C22×D4—171no—
164#137—24no—
165AGL(1,ℤ/8)—344no—
166#135—344no—
167C2×Q16—171no—
168C2×Q16C2×D1712yescomplemented
169C2×SD16—172no—
170C2×SD16—172no—
171C2×SD16—171no—
172C2×SD16—171no—
173C2×SD16—172no—
174C2×SD16—344no—
175C2×SD16C2×D1712yescomplemented
176C2×D8—171no—
177C2×D8—172no—
178C2×M16C2×D1711yescharacteristic
179C22×C8—171no—
180C4⋊Q8—172no—
181#127—172no—
182#124—172no—
183#124—172no—
184#123—342no—
185C22⋊Q8—342no—
186#121—342no—
187#120—342no—
188#117—171no—
189#97—172no—
190C4×C8—172no—
191C17#315411yescharacteristic, complemented, Sylow
192C24—341no—
193#41—22no—
194#41—22no—
195#41—342no—
196#41—342no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—171no—
201C2×Q8C22×D1711yescharacteristic
202C2×Q8C22×D1712yes—
203C2×D4—21no—
204C2×D4—171no—
205C2×D4—172no—
206C2×D4—344no—
207C2×D4—341no—
208C2×D4—342no—
209C2×D4—341no—
210C2×D4—342no—
211C2×D4C22×D1711yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—341no—
215C22×C4C22×D1711yescharacteristic
216Q16—24no—
217Q16—342no—
218SD16—24no—
219SD16—342no—
220SD16—344no—
221SD16—344no—
222SD16—342no—
223SD16—344no—
224D8—342no—
225D8—344no—
226M16C22⋊D1714yescomplemented
227C2×C8—174no—
228C2×C8—171no—
229C2×C8—171no—
230C2×C8C22×D1712yes—
231C4⋊C4—172no—
232C4⋊C4—344no—
233C22⋊C4—172no—
234C22⋊C4—344no—
235C22⋊C4—344no—
236C42—172no—
237C23—21no—
238C23—341no—
239C23—342no—
240C23—682no—
241C23—1362no—
242C23D4×D1711yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—341no—
249Q8—341no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—341no—
254D4—342no—
255D4—341no—
256D4—342no—
257D4—682no—
258D4—682no—
259D4—1364no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—174no—
265C2×C4—341no—
266C2×C4—682no—
267C2×C4D4×D1712yes—
268C2×C4D4×D1711yescharacteristic
269C2×C4C2×C22⋊D1712yes—
270C2×C4C23×D1711yescharacteristic
271C8—172no—
272C8—172no—
273C8C2×C22⋊D1714yes—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—681no—
278C22—682no—
279C22—684no—
280C22—682no—
281C22—1364no—
282C22#1058762612yes—
283C22C2×D4×D1711yescharacteristic
284C4—41no—
285C4—42no—
286C4—344no—
287C4—681no—
288C4#1058762612yes—
289C4C2×D4×D1711yescharacteristic
290C4C22×C22⋊D1711yescharacteristic
291C2—22no—
292C2—81no—
293C2—1362no—
294C2—1361no—
295C2C2×#1058762611yescharacteristic
296C2#1176361812yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC2×C17×#137C2
socleC22×C17C22×D4
hypercenterC2×#137D17
hypocenterC17#3154
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#10587594C2semidirect
2. C2×#10587636C2semidirect
3. C22×#63230C2semidirect
4. #11763383C2semidirect
5. #11763373C2semidirect
6. #11763202C2semidirect
7. #11763195C2semidirect
8. #11763194C2semidirect
9. #11763180C2semidirect
10. #11762767C2semidirect
11. C2×C17×#137C2semidirect
17. C2×Q16×C17C22semidirect
18. C2×SD16×C17C22semidirect
22. #10587633C22semidirect
24. #10587630C22semidirect
25. C2×#63231C22semidirect
29. C2×#63230C22semidirect
32. C2×#63229C22semidirect
33. C2×#63228C22semidirect
41. #10587486C22semidirect
42. C8×Dic17C22semidirect
69. M16×C17D4semidirect
83. C2×#3622D4semidirect
143. C2×#137D17semidirect
168. C2×Q16C2×D17semidirect
175. C2×SD16C2×D17semidirect
191. C17#3154semidirect
226. M16C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105875942
C2×#105876362
C22×#632302
#117633832
#117633732
#117632022
#117631952
#117631942
#117631802
#117627672
C2×C17×#1372
#315417