Finite Group Databasebeta

← #14538404

Subgroups — #14538404

Order 2176.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×M16×D17C211yesmaximal, characteristic, complemented
3#11763321C211yesmaximal, characteristic, complemented
4C2×#10587581C211yesmaximal, characteristic, complemented
5#11763063C212yesmaximal, complemented
6#11763056C212yesmaximal, complemented
7#11763046C212yesmaximal, complemented
8#11763039C212yesmaximal, complemented
9#11762813C211yesmaximal, characteristic, complemented
10#11762808C211yesmaximal, characteristic, complemented
11C17×#428C211yesmaximal, characteristic, complemented
12C2×#63252C2211yescharacteristic
13C22×D68—21no—
14C22×D68C2211yescharacteristic
15C22×C4×D17C2211yescharacteristic
16C2×M16×C17C2211yescharacteristic
17C17×#117C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×#63227—21no—
20C2×#63223C2211yescharacteristic
21#10587594—24no—
22M16×D17—24no—
23C2×D136C2212yescomplemented
24C2×#63219C2212yescomplemented
25C2×#63218C2212yescomplemented
26C2×C8×D17C2212yescomplemented
27#10587583C2211yescharacteristic, complemented
28#10587583C2211yescharacteristic, complemented
29#10587581—21no—
30#10587581—21no—
31#10587581C2212yes—
32#10587581C2212yes—
33#10587581C2211yescharacteristic, complemented
34#10587581C2211yescharacteristic, complemented
35#10587572—21no—
36#10587568—21no—
37C4×D68—21no—
38C4×D68—21no—
39C17×#107C2212yescomplemented
40C17×#106C2212yescomplemented
41#10587481C2212yescomplemented
42#10587480C2212yescomplemented
43#10587480C2212yescomplemented
44#10587480C2212yescomplemented
45C23×D17—21no—
46C23×D17D411yescharacteristic
47C22×C4×C17—21no—
48C22×C4×C17C2311yescharacteristic
49C2×C22⋊D17—21no—
50C22×Dic17D411yescharacteristic
51#63252—22no—
52#63252—22no—
53C2×D68—22no—
54C2×D68—22no—
55C2×D68—22no—
56C2×D68—22no—
57C2×D68—21no—
58C2×D68—21no—
59C2×D68—21no—
60C2×D68—44no—
61C2×D68C2312yes—
62C2×D68C2311yescharacteristic
63C2×C4×D17—22no—
64C2×C4×D17—22no—
65C2×C4×D17—21no—
66C2×C4×D17D411yescharacteristic
67C2×C4×D17D411yescharacteristic
68C2×C4×D17C2311yescharacteristic
69C2×C4×D17C2311yescharacteristic
70C2×Dic34C2311yescharacteristic
71M16×C17D414yescomplemented
72C2×C8×C17C2312yes—
73C4⋊C4×C17—21no—
74C4⋊C4×C17C2311yescharacteristic
75C4⋊C4×C17C2311yescharacteristic
76C4⋊C4×C17C2312yes—
77C22⋊C4×C17—21no—
78C42×C17—21no—
79#63227—21no—
80#63227—21no—
81#63227—21no—
82#63227—21no—
83#63227—22no—
84#63227—22no—
85C4⋊Dic17—21no—
86C4⋊Dic17—21no—
87#63223—22no—
88C2×#3622C2312yes—
89D136—24no—
90#63219—24no—
91#63218—24no—
92C8×D17—24no—
93C23×C17C2×D411yescharacteristic
94C22×D17—22no—
95C22×D17—22no—
96C22×D17—21no—
97C22×D17—22no—
98C22×D17—41no—
99C22×D17—42no—
100C22×D17—42no—
101C22×D17—42no—
102C22×D17C2×D411yescharacteristic
103C22×D17C2×D411yescharacteristic
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—22no—
107C2×C4×C17—42no—
108C2×C4×C17C2×D412yes—
109C2×C4×C17C2×D411yescharacteristic
110C2×C4×C17#4112yes—
111C2×C4×C17C2411yescharacteristic
112C22⋊D17—42no—
113C2×Dic17—22no—
114C2×Dic17—21no—
115C2×Dic17C2×D411yescharacteristic
116C2×Dic17C2×D411yescharacteristic
117D68—22no—
118D68—21no—
119D68—22no—
120D68—21no—
121D68—42no—
122D68—44no—
123D68—42no—
124D68—44no—
125D68—42no—
126C4×D17—22no—
127C4×D17—22no—
128C4×D17—21no—
129C4×D17—21no—
130C4×D17—21no—
131C4×D17—21no—
132C4×D17—42no—
133Dic34—21no—
134Dic34—21no—
135C8×C17C2×D414yescomplemented
136#3622—22no—
137#3055—171nomaximal, Sylow
138C22×C17—21no—
139C22×C17—21no—
140C22×C17#12112yes—
141C22×C17C22×D411yescharacteristic
142C2×D17—21no—
143C2×D17—21no—
144C2×D17—21no—
145C2×D17—21no—
146C2×D17—21no—
147C2×D17—44no—
148C2×D17—41no—
149C2×D17—42no—
150C2×D17—42no—
151C2×D17—41no—
152C2×D17—41no—
153C2×D17—42no—
154C2×D17—84no—
155C2×D17AGL(1,ℤ/8)12yescomplemented
156C4×C17—42no—
157C4×C17—41no—
158C4×C17—41no—
159C4×C17#12112yes—
160C4×C17C22×D411yescharacteristic
161C4×C17C2×#4111yescharacteristic
162Dic17—21no—
163Dic17—21no—
164Dic17—41no—
165C2×AGL(1,ℤ/8)—171no—
166C22×M16—171no—
167#525—171no—
168C2×#121—171no—
169#469—172no—
170#468—172no—
171#466—172no—
172#465—172no—
173#428D1711yescharacteristic, complemented
174#418—171no—
175#417—171no—
176C2×C17—22no—
177C2×C17C2×#12111yescharacteristic
178C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
179C2×C17#57511yescharacteristic
180D17—22no—
181D17—41no—
182D17—81no—
183D17—82no—
184C2×#41—171no—
185C22×D4—171no—
186C22×D4—341no—
187C23×C4—171no—
188AGL(1,ℤ/8)—344no—
189C2×SD16—172no—
190C2×D8—172no—
191C2×M16—171no—
192C2×M16—172no—
193C2×M16—344no—
194C2×M16C2×D1711yescharacteristic
195C22×C8—172no—
196#123—341no—
197C22⋊Q8—171no—
198C22⋊Q8—171no—
199#121—172no—
200#121—172no—
201#121—171no—
202#121—171no—
203#121—341no—
204#121—341no—
205#120—341no—
206C4×D4—341no—
207C4×D4—341no—
208#117C2×D1711yescharacteristic
209C2×C4⋊C4C2×D1711yescharacteristic
210C2×C22⋊C4—341no—
211#107C2×D1712yescomplemented
212#106C2×D1712yescomplemented
213#103—172no—
214#102—172no—
215#102—172no—
216#102—172no—
217C17#305511yescharacteristic, complemented, Sylow
218C24—171no—
219C24—341no—
220#41—342no—
221#41—342no—
222C2×Q8—171no—
223C2×D4—171no—
224C2×D4—172no—
225C2×D4—341no—
226C2×D4—341no—
227C2×D4—342no—
228C2×D4—342no—
229C2×D4—341no—
230C2×D4—342no—
231C2×D4—341no—
232C2×D4—342no—
233C2×D4—684no—
234C22×C4—21no—
235C22×C4—171no—
236C22×C4—171no—
237C22×C4—171no—
238C22×C4—171no—
239C22×C4—171no—
240C22×C4—342no—
241C22×C4—342no—
242C22×C4—341no—
243C22×C4C22×D1711yescharacteristic
244SD16—344no—
245D8—344no—
246M16—342no—
247M16—344no—
248M16D6814yescomplemented
249C2×C8—172no—
250C2×C8—344no—
251C2×C8C22×D1712yes—
252C4⋊C4—21no—
253C4⋊C4—341no—
254C4⋊C4—341no—
255C4⋊C4C22×D1711yescharacteristic
256C4⋊C4C22×D1711yescharacteristic
257C4⋊C4C22×D1712yes—
258C22⋊C4—21no—
259C22⋊C4—342no—
260C22⋊C4—342no—
261C22⋊C4—341no—
262C22⋊C4—341no—
263C22⋊C4—341no—
264C22⋊C4—341no—
265C42—21no—
266C23—171no—
267C23—171no—
268C23—342no—
269C23—342no—
270C23—341no—
271C23—342no—
272C23—681no—
273C23—682no—
274C23—682no—
275C23—682no—
276C23D4×D1711yescharacteristic
277Q8—341no—
278Q8—341no—
279D4—341no—
280D4—342no—
281D4—342no—
282D4—341no—
283D4—684no—
284D4—682no—
285D4—684no—
286D4—682no—
287D4—682no—
288D4—682no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—171no—
294C2×C4—171no—
295C2×C4—342no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—341no—
300C2×C4—342no—
301C2×C4—342no—
302C2×C4—341no—
303C2×C4—682no—
304C2×C4C2×D6812yes—
305C2×C4D4×D1711yescharacteristic
306C2×C4#6325612yes—
307C2×C4C23×D1711yescharacteristic
308C8—342no—
309C8C2×D6814yescomplemented
310C22—21no—
311C22—21no—
312C22—172no—
313C22—341no—
314C22—341no—
315C22—341no—
316C22—341no—
317C22—341no—
318C22—684no—
319C22—681no—
320C22—682no—
321C22—682no—
322C22—681no—
323C22—681no—
324C22—682no—
325C22—1364no—
326C22#1058758112yes—
327C22C2×D4×D1711yescharacteristic
328C4—42no—
329C4—41no—
330C4—41no—
331C4—341no—
332C4—341no—
333C4—681no—
334C4#1058758112yes—
335C4C22×D6811yescharacteristic
336C4C2×#6325611yescharacteristic
337C2—22no—
338C2—342no—
339C2—681no—
340C2—1362no—
341C2—1361no—
342C2C2×#1058758111yescharacteristic
343C2AGL(1,ℤ/8)×D1711yescharacteristic
344C2#1176360811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#428C2
socleC22×C17C22×D4
hypercenter#428D17
hypocenterC17#3055
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#10587594C2semidirect
2. C2×M16×D17C2semidirect
3. #11763321C2semidirect
4. C2×#10587581C2semidirect
5. #11763063C2semidirect
6. #11763056C2semidirect
7. #11763046C2semidirect
8. #11763039C2semidirect
9. #11762813C2semidirect
10. #11762808C2semidirect
11. C17×#428C2semidirect
23. C2×D136C22semidirect
24. C2×#63219C22semidirect
25. C2×#63218C22semidirect
26. C2×C8×D17C22semidirect
27. #10587583C22semidirect
28. #10587583C22semidirect
33. #10587581C22semidirect
34. #10587581C22semidirect
39. C17×#107C22semidirect
40. C17×#106C22semidirect
41. #10587481C22semidirect
42. #10587480C22semidirect
43. #10587480C22semidirect
44. #10587480C22semidirect
71. M16×C17D4semidirect
135. C8×C17C2×D4semidirect
155. C2×D17AGL(1,ℤ/8)semidirect
173. #428D17semidirect
211. #107C2×D17semidirect
212. #106C2×D17semidirect
217. C17#3055semidirect
248. M16D68semidirect
309. C8C2×D68semidirect

Maximal subgroups

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

subgroupindex
C2×#105875942
C2×M16×D172
#117633212
C2×#105875812
#117630632
#117630562
#117630462
#117630392
#117628132
#117628082
C17×#4282
#305517