Finite Group Databasebeta

← #14539261

Subgroups — #14539261

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×#10587637C211yesmaximal, characteristic, complemented
2C2×#10587635C211yesmaximal, characteristic, complemented
3C22×#63219C211yesmaximal, characteristic, complemented
4#11763380C211yesmaximal, characteristic, complemented
5#11763373C211yesmaximal, characteristic, complemented
6#11763202C212yesmaximal, complemented
7#11763197C212yesmaximal, complemented
8#11763190C212yesmaximal, complemented
9#11763180C212yesmaximal, complemented
10#11762764C211yesmaximal, characteristic, complemented
11C2×C17×#135C211yesmaximal, characteristic, complemented
12C2×#41×C17C2211yescharacteristic
13C2×#41×C17C2211yescharacteristic
14C22×D68C2211yescharacteristic
15C22×Dic34C2211yescharacteristic
16C17×#135—24no—
17C2×Q16×C17C2211yescharacteristic, complemented
18C2×SD16×C17C2211yescharacteristic, complemented
19C2×SD16×C17C2211yescharacteristic, complemented
20C2×D8×C17C2211yescharacteristic, complemented
21C22×C8×C17C2211yescharacteristic
22#10587637—24no—
23#10587635—24no—
24#10587633C2212yes—
25#10587630C2212yescomplemented
26C2×#63231C2212yescomplemented
27C2×#63230C2212yes—
28#10587626C2212yes—
29#10587623—22no—
30#10587622C2212yescomplemented
31#10587621—22no—
32C2×#63229C2212yescomplemented
33C2×#63228C2212yes—
34#10587616—22no—
35#10587612C2211yescharacteristic
36#10587610—22no—
37C2×#63223C2211yescharacteristic
38C2×#63219—24no—
39C2×#63219C2212yes—
40C2×#63219C2212yes—
41#10587486C2212yescomplemented
42C8×Dic17C2212yescomplemented
43C23×D17—21no—
44#41×C17—22no—
45#41×C17—22no—
46#41×C17—22no—
47#41×C17—22no—
48C2×Q8×C17C2311yescharacteristic
49C2×Q8×C17C2311yescharacteristic
50C2×D4×C17—21no—
51C2×D4×C17—21no—
52C2×D4×C17C2311yescharacteristic
53C2×D4×C17C2311yescharacteristic
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17C2311yescharacteristic
57C2×C22⋊D17—24no—
58C22×Dic17—21no—
59C2×D68—22no—
60C2×D68—22no—
61C2×D68—21no—
62C2×D68C2312yes—
63C2×Dic34—22no—
64C2×Dic34—22no—
65C2×Dic34—21no—
66C2×Dic34C2312yes—
67Q16×C17—22no—
68SD16×C17—22no—
69SD16×C17—22no—
70D8×C17—22no—
71C2×C8×C17D414yescomplemented
72C2×C8×C17C2311yescharacteristic
73C2×C8×C17C2311yescharacteristic
74C22⋊Dic17—24no—
75C22⋊Dic17D412yes—
76#63231—24no—
77#63230—24no—
78#63229—24no—
79#63228—24no—
80#63227—24no—
81C4⋊Dic17D412yes—
82#63225—24no—
83C4×Dic17C2312yes—
84#63223D414yescomplemented
85C2×#3622C2312yes—
86#63219—24no—
87#63219—24no—
88C23×C17—21no—
89C23×C17—21no—
90C23×C17C2×D411yescharacteristic
91C22×D17—22no—
92C22×D17—42no—
93C22×D17—82no—
94Q8×C17—21no—
95Q8×C17—21no—
96Q8×C17—21no—
97Q8×C17—21no—
98D4×C17—21no—
99D4×C17—21no—
100D4×C17—21no—
101D4×C17—21no—
102D4×C17—42no—
103D4×C17—42no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17C2×D411yescharacteristic
109C2×C4×C17C2×D412yes—
110C2×C4×C17C2×D412yes—
111C2×C4×C17C2411yescharacteristic
112C22⋊D17—84no—
113C2×Dic17—22no—
114C2×Dic17—42no—
115C2×Dic17C2×D414yes—
116D68—22no—
117D68—22no—
118D68—42no—
119Dic34—22no—
120Dic34—22no—
121Dic34—42no—
122C8×C17C2×D412yes—
123C8×C17C2×D412yes—
124#3622C2×D414yes—
125#3154—171nomaximal, Sylow
126C22×C17—22no—
127C22×C17—41no—
128C22×C17—41no—
129C22×C17—42no—
130C22×C17—42no—
131C22×C17#12712yes—
132C22×C17C22×D411yescharacteristic
133C2×D17—42no—
134C2×D17—44no—
135C2×D17—84no—
136C4×C17—41no—
137C4×C17—41no—
138C4×C17#12712yes—
139C4×C17C22×D411yescharacteristic
140C4×C17C22×D411yescharacteristic
141Dic17—24no—
142Dic17—42no—
143C2×#137—171no—
144C2×AGL(1,ℤ/8)—171no—
145C2×#135D1711yescharacteristic, complemented
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×C17—81no—
157C2×C17C2×#12711yescharacteristic
158C2×C17#57712yes—
159D17—82no—
160C2×#41C2×D1711yescharacteristic
161C2×#41C2×D1711yescharacteristic
162C22×Q8—171no—
163C22×D4—171no—
164#137—344no—
165AGL(1,ℤ/8)—344no—
166#135—24no—
167C2×Q16—172no—
168C2×Q16C2×D1711yescharacteristic, complemented
169C2×SD16—172no—
170C2×SD16—172no—
171C2×SD16—172no—
172C2×SD16—172no—
173C2×SD16—344no—
174C2×SD16C2×D1711yescharacteristic, complemented
175C2×SD16C2×D1711yescharacteristic, complemented
176C2×D8—172no—
177C2×D8C2×D1711yescharacteristic, complemented
178C2×M16—171no—
179C22×C8C2×D1711yescharacteristic
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—22no—
196#41—22no—
197C2×Q8—172no—
198C2×Q8—341no—
199C2×Q8—342no—
200C2×Q8—342no—
201C2×Q8C22×D1711yescharacteristic
202C2×Q8C22×D1711yescharacteristic
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—172no—
206C2×D4—344no—
207C2×D4—342no—
208C2×D4—341no—
209C2×D4—342no—
210C2×D4C22×D1711yescharacteristic
211C2×D4C22×D1711yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—341no—
215C22×C4C22×D1711yescharacteristic
216Q16—22no—
217Q16—344no—
218SD16—22no—
219SD16—22no—
220SD16—344no—
221SD16—344no—
222SD16—344no—
223SD16—344no—
224D8—22no—
225D8—344no—
226M16—174no—
227C2×C8—172no—
228C2×C8C22⋊D1714yescomplemented
229C2×C8C22×D1711yescharacteristic
230C2×C8C22×D1711yescharacteristic
231C4⋊C4—172no—
232C4⋊C4—344no—
233C22⋊C4—172no—
234C22⋊C4—344no—
235C22⋊C4—344no—
236C42—172no—
237C23—21no—
238C23—21no—
239C23—342no—
240C23—682no—
241C23—1362no—
242C23D4×D1711yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—21no—
246Q8—21no—
247Q8—342no—
248Q8—342no—
249Q8—682no—
250D4—21no—
251D4—21no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—42no—
256D4—342no—
257D4—342no—
258D4—682no—
259D4—1364no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—174no—
265C2×C4—342no—
266C2×C4—682no—
267C2×C4D4×D1712yes—
268C2×C4D4×D1711yescharacteristic
269C2×C4C2×C22⋊D1712yes—
270C2×C4C23×D1711yescharacteristic
271C8—174no—
272C8C2×C22⋊D1712yes—
273C8C2×C22⋊D1712yes—
274C22—22no—
275C22—41no—
276C22—41no—
277C22—42no—
278C22—42no—
279C22—682no—
280C22—684no—
281C22—1364no—
282C22#1058762612yes—
283C22C2×D4×D1711yescharacteristic
284C4—41no—
285C4—41no—
286C4—344no—
287C4—682no—
288C4#1058762612yes—
289C4C2×D4×D1711yescharacteristic
290C4C22×C22⋊D1711yescharacteristic
291C2—22no—
292C2—81no—
293C2—81no—
294C2—1362no—
295C2C2×#1058762611yescharacteristic
296C2#1176361612yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC2×C17×#135C2
socleC22×C17C22×D4
hypercenterC2×#135D17
hypocenterC17#3154
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#10587637C2semidirect
2. C2×#10587635C2semidirect
3. C22×#63219C2semidirect
4. #11763380C2semidirect
5. #11763373C2semidirect
6. #11763202C2semidirect
7. #11763197C2semidirect
8. #11763190C2semidirect
9. #11763180C2semidirect
10. #11762764C2semidirect
11. C2×C17×#135C2semidirect
17. C2×Q16×C17C22semidirect
18. C2×SD16×C17C22semidirect
19. C2×SD16×C17C22semidirect
20. C2×D8×C17C22semidirect
25. #10587630C22semidirect
26. C2×#63231C22semidirect
30. #10587622C22semidirect
32. C2×#63229C22semidirect
41. #10587486C22semidirect
42. C8×Dic17C22semidirect
71. C2×C8×C17D4semidirect
84. #63223D4semidirect
145. C2×#135D17semidirect
168. C2×Q16C2×D17semidirect
174. C2×SD16C2×D17semidirect
175. C2×SD16C2×D17semidirect
177. C2×D8C2×D17semidirect
191. C17#3154semidirect
228. C2×C8C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876372
C2×#105876352
C22×#632192
#117633802
#117633732
#117632022
#117631972
#117631902
#117631802
#117627642
C2×C17×#1352
#315417