Finite Group Databasebeta

← #14538247

Subgroups — #14538247

Order 2176.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#10587635C211yesmaximal, characteristic, complemented
3#11763355C211yesmaximal, characteristic, complemented
4C2×#10587579C211yesmaximal, characteristic, complemented
5#11763027C212yesmaximal, complemented
6#11763023C212yesmaximal, complemented
7#11763013C212yesmaximal, complemented
8#11763003C212yesmaximal, complemented
9#11762832C211yesmaximal, characteristic, complemented
10#11762808C211yesmaximal, characteristic, complemented
11C17×#417C211yesmaximal, characteristic, complemented
12C2×#41×C17C2211yescharacteristic
13C2×#63252C2211yescharacteristic
14C22×D68C2211yescharacteristic
15C22×C4×D17—21no—
16C2×M16×C17C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18#10587635—24no—
19#10587633C2211yescharacteristic, complemented
20#10587630C2211yescharacteristic, complemented
21C2×#63230C2212yes—
22#10587626C2211yescharacteristic, complemented
23#10587625—21no—
24#10587625—21no—
25#10587622C2211yescharacteristic, complemented
26C2×#63228C2212yes—
27C4×C22⋊D17—21no—
28C4×C22⋊D17—21no—
29C2×#63227—21no—
30C2×C4×Dic17C2211yescharacteristic
31C2×#63223C2211yescharacteristic
32#10587594—24no—
33C2×D136C2212yes—
34C2×#63219C2212yes—
35#10587579—21no—
36#10587579—21no—
37#10587579C2212yes—
38#10587579C2212yes—
39C17×#103C2212yescomplemented
40C17×#102C2212yescomplemented
41#10587485C2212yescomplemented
42#10587485C2212yescomplemented
43#10587481C2212yescomplemented
44#10587480C2212yescomplemented
45C23×D17—21no—
46#41×C17—22no—
47#41×C17—22no—
48C2×Q8×C17C2311yescharacteristic
49C2×D4×C17—21no—
50C2×D4×C17C2311yescharacteristic
51C22×C4×C17—21no—
52C22×C4×C17—21no—
53C22×C4×C17C2311yescharacteristic
54C2×C22⋊D17—21no—
55C2×C22⋊D17—21no—
56C2×C22⋊D17—21no—
57C22×Dic17D411yescharacteristic
58C22×Dic17D411yescharacteristic
59#63252—22no—
60#63252—22no—
61C2×D68—21no—
62C2×D68—22no—
63C2×D68D414yescomplemented
64C2×D68C2311yescharacteristic
65C2×D68C2312yes—
66C2×C4×D17—21no—
67C2×C4×D17—22no—
68C2×C4×D17—22no—
69C2×C4×D17—44no—
70C2×Dic34C2311yescharacteristic
71M16×C17—22no—
72C2×C8×C17C2312yes—
73C4⋊C4×C17—21no—
74C4⋊C4×C17C2312yes—
75C22⋊Dic17—21no—
76C22⋊Dic17—21no—
77#63230—24no—
78#63228—24no—
79#63227—22no—
80#63227—22no—
81#63227—21no—
82#63227—21no—
83#63225—21no—
84#63225—21no—
85C4×Dic17D411yescharacteristic
86C4×Dic17D411yescharacteristic
87C4×Dic17C2311yescharacteristic
88C4×Dic17C2311yescharacteristic
89#63223—22no—
90C2×#3622C2312yes—
91D136—24no—
92#63219—24no—
93C23×C17—21no—
94C23×C17C2×D411yescharacteristic
95C22×D17—24no—
96C22×D17—22no—
97C22×D17—21no—
98C22×D17—42no—
99C22×D17—42no—
100Q8×C17—21no—
101Q8×C17—21no—
102D4×C17—21no—
103D4×C17—21no—
104D4×C17—42no—
105C2×C4×C17—22no—
106C2×C4×C17—21no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17C2×D411yescharacteristic
110C2×C4×C17C2×D412yes—
111C2×C4×C17#4112yes—
112C2×C4×C17C2411yescharacteristic
113C22⋊D17—42no—
114C22⋊D17—42no—
115C22⋊D17—42no—
116C2×Dic17—22no—
117C2×Dic17—21no—
118C2×Dic17—41no—
119C2×Dic17C2×D411yescharacteristic
120C2×Dic17C2×D411yescharacteristic
121C2×Dic17C2×D411yescharacteristic
122C2×Dic17C2×D411yescharacteristic
123D68—24no—
124D68—21no—
125D68—22no—
126D68—21no—
127D68C2×D414yes—
128C4×D17—44no—
129C4×D17—42no—
130C4×D17—44no—
131Dic34—21no—
132Dic34—21no—
133C8×C17—22no—
134#3622—22no—
135#3036—171nomaximal, Sylow
136C22×C17—21no—
137C22×C17—21no—
138C22×C17—41no—
139C22×C17—41no—
140C22×C17—41no—
141C22×C17#12112yes—
142C22×C17C22×D411yescharacteristic
143C2×D17—24no—
144C2×D17—48no—
145C2×D17—42no—
146C2×D17—42no—
147C2×D17—41no—
148C2×D17—41no—
149C2×D17—41no—
150C4×C17—41no—
151C4×C17—42no—
152C4×C17#12112yes—
153C4×C17C22×D411yescharacteristic
154C4×C17C2×#4111yescharacteristic
155Dic17—21no—
156Dic17—21no—
157Dic17—21no—
158Dic17—41no—
159Dic17AGL(1,ℤ/8)12yescomplemented
160C2×AGL(1,ℤ/8)—171no—
161C2×AGL(1,ℤ/8)—171no—
162#532—171no—
163C2×C4×D4—171no—
164#463—172no—
165#463—172no—
166#461—172no—
167#459—172no—
168#423—171no—
169#417—171no—
170#417D1711yescharacteristic, complemented
171C2×C17—22no—
172C2×C17—81no—
173C2×C17C2×#12111yescharacteristic
174C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
175C2×C17#57511yescharacteristic
176D17—44no—
177D17—81no—
178C2×#41—171no—
179C2×#41C2×D1711yescharacteristic
180C22×D4—171no—
181C23×C4—341no—
182AGL(1,ℤ/8)—344no—
183AGL(1,ℤ/8)—344no—
184C2×SD16—172no—
185C2×SD16—172no—
186C2×D8—172no—
187C2×D8—172no—
188C2×M16—171no—
189C2×M16C2×D1711yescharacteristic
190C4⋊Q8—171no—
191#127—171no—
192#124—171no—
193#124—171no—
194#121—341no—
195#121—341no—
196C4×D4—172no—
197C4×D4—172no—
198C4×D4—341no—
199C4×D4—341no—
200C4×D4—341no—
201C4×D4—341no—
202C2×C4⋊C4C2×D1711yescharacteristic
203C2×C22⋊C4—341no—
204C2×C42—171no—
205C4⋊C8—172no—
206C4⋊C8—172no—
207#103—172no—
208#103C2×D1712yescomplemented
209#102—172no—
210#102C2×D1712yescomplemented
211C17#303611yescharacteristic, complemented, Sylow
212C24—341no—
213#41—22no—
214#41—22no—
215#41—342no—
216#41—342no—
217C2×Q8—171no—
218C2×Q8C22×D1711yescharacteristic
219C2×D4—21no—
220C2×D4—174no—
221C2×D4—172no—
222C2×D4—171no—
223C2×D4—341no—
224C2×D4—341no—
225C2×D4—341no—
226C2×D4—342no—
227C2×D4—341no—
228C2×D4C22×D1711yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—171no—
232C22×C4—171no—
233C22×C4—342no—
234C22×C4—342no—
235C22×C4—341no—
236C22×C4—684no—
237C22×C4C22×D1711yescharacteristic
238SD16—344no—
239SD16—344no—
240D8—344no—
241D8—344no—
242M16—22no—
243M16—342no—
244C2×C8—172no—
245C2×C8C22×D1712yes—
246C4⋊C4—21no—
247C4⋊C4—341no—
248C4⋊C4—341no—
249C4⋊C4C22×D1712yes—
250C22⋊C4—342no—
251C22⋊C4—342no—
252C22⋊C4—341no—
253C22⋊C4—341no—
254C22⋊C4—341no—
255C22⋊C4—341no—
256C42—171no—
257C42—171no—
258C42—171no—
259C42—171no—
260C23—21no—
261C23—342no—
262C23—344no—
263C23—341no—
264C23—682no—
265C23—682no—
266C23D4×D1711yescharacteristic
267Q8—21no—
268Q8—21no—
269Q8—341no—
270Q8—341no—
271D4—21no—
272D4—21no—
273D4—42no—
274D4—174no—
275D4—342no—
276D4—341no—
277D4—341no—
278D4—344no—
279D4—682no—
280D4—682no—
281D4—682no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—171no—
287C2×C4—171no—
288C2×C4—171no—
289C2×C4—171no—
290C2×C4—342no—
291C2×C4—341no—
292C2×C4—684no—
293C2×C4—684no—
294C2×C4—681no—
295C2×C4—682no—
296C2×C4#6325212yes—
297C2×C4D4×D1712yes—
298C2×C4D4×D1711yescharacteristic
299C2×C4C23×D1711yescharacteristic
300C8—22no—
301C8—342no—
302C22—21no—
303C22—21no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—344no—
308C22—688no—
309C22—682no—
310C22—682no—
311C22—681no—
312C22—681no—
313C22—681no—
314C22#1058757012yes—
315C22C2×D4×D1711yescharacteristic
316C4—41no—
317C4—42no—
318C4—172no—
319C4—341no—
320C4—341no—
321C4—341no—
322C4—681no—
323C4#1058757012yes—
324C4C2×#6325211yescharacteristic
325C4C2×D4×D1711yescharacteristic
326C2—22no—
327C2—81no—
328C2—684no—
329C2—1361no—
330C2C2×#1058757011yescharacteristic
331C2AGL(1,ℤ/8)×D1711yescharacteristic
332C2#1176360811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#417C2
socleC22×C17C22×D4
hypercenter#417D17
hypocenterC17#3036
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#10587594C2semidirect
2. C2×#10587635C2semidirect
3. #11763355C2semidirect
4. C2×#10587579C2semidirect
5. #11763027C2semidirect
6. #11763023C2semidirect
7. #11763013C2semidirect
8. #11763003C2semidirect
9. #11762832C2semidirect
10. #11762808C2semidirect
11. C17×#417C2semidirect
19. #10587633C22semidirect
20. #10587630C22semidirect
22. #10587626C22semidirect
25. #10587622C22semidirect
39. C17×#103C22semidirect
40. C17×#102C22semidirect
41. #10587485C22semidirect
42. #10587485C22semidirect
43. #10587481C22semidirect
44. #10587480C22semidirect
63. C2×D68D4semidirect
159. Dic17AGL(1,ℤ/8)semidirect
170. #417D17semidirect
208. #103C2×D17semidirect
210. #102C2×D17semidirect
211. C17#3036semidirect

Maximal subgroups

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

subgroupindex
C2×#105875942
C2×#105876352
#117633552
C2×#105875792
#117630272
#117630232
#117630132
#117630032
#117628322
#117628082
C17×#4172
#303617