Finite Group Databasebeta

← #14538726

Subgroups — #14538726

Order 2176.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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×#10587581C211yesmaximal, characteristic
2#117×D17C211yesmaximal, characteristic, complemented
3#11763136C212yesmaximal, complemented
4#11763135C212yesmaximal, complemented
5#11763129C212yesmaximal, complemented
6#11763124C212yesmaximal, complemented
7#11762849C211yesmaximal, characteristic, complemented
8#11762815C211yesmaximal, characteristic, complemented
9#11762813C211yesmaximal, characteristic, complemented
10#11762772C211yesmaximal, characteristic, complemented
11C17×#428C211yesmaximal, characteristic, complemented
12C22×D68—21no—
13C22×D68C2211yescharacteristic
14C22×C4×D17C2211yescharacteristic
15C2×M16×C17C2211yescharacteristic
16C17×#117C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×#63227—21no—
19#10587612C2211yescharacteristic
20C2×#63223C2211yescharacteristic
21#10587581—21no—
22#10587581—21no—
23#10587581C2212yes—
24#10587581C2212yes—
25#10587578C2211yescharacteristic, complemented
26#10587578C2211yescharacteristic, complemented
27C4⋊C4×D17C2211yescharacteristic, complemented
28C4⋊C4×D17C2211yescharacteristic, complemented
29C22⋊C4×D17—21no—
30#10587563—21no—
31#10587558—21no—
32C42×D17—21no—
33C17×#107C2212yescomplemented
34C17×#106C2212yescomplemented
35#10587492C2212yescomplemented
36#10587492C2212yescomplemented
37#10587491C2212yescomplemented
38#10587491C2212yescomplemented
39#10587480C2212yescomplemented
40#10587480C2212yescomplemented
41#10587479C2212yescomplemented
42#10587478C2212yescomplemented
43C23×D17—21no—
44C23×D17D411yescharacteristic
45C22×C4×C17—21no—
46C22×C4×C17C2311yescharacteristic
47C22×Dic17D411yescharacteristic
48C2×D68—22no—
49C2×D68—22no—
50C2×D68—22no—
51C2×D68—22no—
52C2×D68—21no—
53C2×D68—44no—
54C2×D68C2312yes—
55C2×C4×D17—22no—
56C2×C4×D17—22no—
57C2×C4×D17—21no—
58C2×C4×D17—21no—
59C2×C4×D17D411yescharacteristic
60C2×C4×D17D411yescharacteristic
61C2×C4×D17C2311yescharacteristic
62C2×C4×D17C2311yescharacteristic
63M16×C17—22no—
64C2×C8×C17C2312yes—
65C4⋊C4×C17—21no—
66C4⋊C4×C17C2312yes—
67C4⋊C4×C17C2311yescharacteristic
68C4⋊C4×C17C2311yescharacteristic
69C22⋊C4×C17—21no—
70C42×C17—21no—
71C22⋊Dic17—21no—
72#63227—21no—
73#63227—21no—
74#63227—22no—
75#63227—22no—
76C4⋊Dic17C2311yescharacteristic
77C4⋊Dic17C2311yescharacteristic
78#63225—21no—
79#63225—21no—
80C4×Dic17—21no—
81C4×Dic17—21no—
82C4×Dic17—21no—
83#63223—22no—
84C2×#3622C2312yes—
85C23×C17C2×D411yescharacteristic
86C22×D17—22no—
87C22×D17—22no—
88C22×D17—22no—
89C22×D17—41no—
90C22×D17—42no—
91C22×D17—42no—
92C22×D17—42no—
93C22×D17C2×D411yescharacteristic
94C22×D17C2×D411yescharacteristic
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—22no—
98C2×C4×C17—42no—
99C2×C4×C17C2×D411yescharacteristic
100C2×C4×C17#4112yes—
101C2×C4×C17#4112yes—
102C2×C4×C17C2411yescharacteristic
103C2×Dic17—22no—
104C2×Dic17—21no—
105C2×Dic17—21no—
106C2×Dic17C2×D411yescharacteristic
107C2×Dic17C2×D411yescharacteristic
108D68—22no—
109D68—22no—
110D68—44no—
111D68—44no—
112D68—42no—
113C4×D17—22no—
114C4×D17—22no—
115C4×D17—21no—
116C4×D17—21no—
117C4×D17—21no—
118C4×D17—21no—
119C4×D17—42no—
120C4×D17—42no—
121C8×C17—22no—
122#3622—22no—
123#3094—171nomaximal, Sylow
124C22×C17—21no—
125C22×C17—21no—
126C22×C17#12312yes—
127C22×C17C22×D411yescharacteristic
128C2×D17—21no—
129C2×D17—21no—
130C2×D17—21no—
131C2×D17—21no—
132C2×D17—21no—
133C2×D17—44no—
134C2×D17—42no—
135C2×D17—42no—
136C2×D17—42no—
137C2×D17—84no—
138C2×D17AGL(1,ℤ/8)12yes—
139C4×C17—41no—
140C4×C17—42no—
141C4×C17—41no—
142C4×C17#12312yes—
143C4×C17C2×#4111yescharacteristic
144C4×C17C2×#4111yescharacteristic
145Dic17—21no—
146Dic17—21no—
147Dic17—41no—
148Dic17—41no—
149C2×#121—171no—
150C2×#117—171no—
151#482—172no—
152#482—172no—
153#481—172no—
154#480—172no—
155#428—171no—
156#428D1711yescharacteristic, complemented
157#418—171no—
158#418—171no—
159#407—171no—
160C2×C17—22no—
161C2×C17C2×#12311yescharacteristic
162C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
163C2×C17#57511yescharacteristic
164D17—22no—
165D17—41no—
166D17—82no—
167C22×D4—171no—
168C22×D4—341no—
169C23×C4—171no—
170C2×M16—171no—
171C2×M16C2×D1711yescharacteristic
172#121—172no—
173#121—172no—
174#121—341no—
175#121—341no—
176#117—171no—
177#117—171no—
178#117—171no—
179#117—341no—
180#117—341no—
181#117C2×D1711yescharacteristic
182C2×C4⋊C4—171no—
183C2×C4⋊C4—171no—
184C2×C4⋊C4C2×D1711yescharacteristic
185C2×C22⋊C4—341no—
186C2×C22⋊C4—341no—
187C2×C42—341no—
188#107—172no—
189#107C2×D1712yescomplemented
190#106—172no—
191#106C2×D1712yescomplemented
192#102—172no—
193#102—172no—
194#102—172no—
195#102—172no—
196C22⋊C8—172no—
197C22⋊C8—172no—
198C17#309411yescharacteristic, complemented, Sylow
199C24—171no—
200C24—341no—
201C2×D4—172no—
202C2×D4—342no—
203C2×D4—342no—
204C2×D4—342no—
205C2×D4—341no—
206C2×D4—342no—
207C2×D4—684no—
208C22×C4—21no—
209C22×C4—171no—
210C22×C4—171no—
211C22×C4—171no—
212C22×C4—171no—
213C22×C4—171no—
214C22×C4—341no—
215C22×C4—341no—
216C22×C4—342no—
217C22×C4—342no—
218C22×C4C22×D1711yescharacteristic
219M16—22no—
220M16—342no—
221C2×C8—172no—
222C2×C8C22×D1712yes—
223C4⋊C4—21no—
224C4⋊C4—171no—
225C4⋊C4—171no—
226C4⋊C4—341no—
227C4⋊C4—341no—
228C4⋊C4C22×D1711yescharacteristic
229C4⋊C4C22×D1711yescharacteristic
230C4⋊C4C22×D1712yes—
231C22⋊C4—21no—
232C22⋊C4—342no—
233C22⋊C4—342no—
234C22⋊C4—341no—
235C22⋊C4—341no—
236C22⋊C4—341no—
237C42—21no—
238C42—341no—
239C42—341no—
240C42—341no—
241C23—171no—
242C23—171no—
243C23—342no—
244C23—342no—
245C23—342no—
246C23—681no—
247C23—682no—
248C23—682no—
249C23—682no—
250C23D4×D1711yescharacteristic
251D4—342no—
252D4—342no—
253D4—684no—
254D4—684no—
255D4—682no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—171no—
261C2×C4—171no—
262C2×C4—342no—
263C2×C4—341no—
264C2×C4—341no—
265C2×C4—341no—
266C2×C4—341no—
267C2×C4—342no—
268C2×C4—342no—
269C2×C4—341no—
270C2×C4—341no—
271C2×C4—682no—
272C2×C4—682no—
273C2×C4#6325212yes—
274C2×C4D4×D1711yescharacteristic
275C2×C4#6325612yes—
276C2×C4C23×D1711yescharacteristic
277C8—22no—
278C8—342no—
279C22—21no—
280C22—21no—
281C22—172no—
282C22—341no—
283C22—341no—
284C22—341no—
285C22—341no—
286C22—341no—
287C22—684no—
288C22—682no—
289C22—682no—
290C22—682no—
291C22—1364no—
292C22#1058758012yes—
293C22C2×D4×D1711yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—341no—
298C4—341no—
299C4—681no—
300C4—681no—
301C4#1058758012yes—
302C4C2×#6325211yescharacteristic
303C4C2×#6325611yescharacteristic
304C2—22no—
305C2—342no—
306C2—681no—
307C2—1362no—
308C2C2×#1058758011yescharacteristic
309C2AGL(1,ℤ/8)×D1711yescharacteristic
310C2#1176360811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#428C2
socleC22×C17C22×D4
prime radicalC2×#10587581C2
hypercenter#428D17
hypocenterC17#3094
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
2. #117×D17C2semidirect
3. #11763136C2semidirect
4. #11763135C2semidirect
5. #11763129C2semidirect
6. #11763124C2semidirect
7. #11762849C2semidirect
8. #11762815C2semidirect
9. #11762813C2semidirect
10. #11762772C2semidirect
11. C17×#428C2semidirect
25. #10587578C22semidirect
26. #10587578C22semidirect
27. C4⋊C4×D17C22semidirect
28. C4⋊C4×D17C22semidirect
33. C17×#107C22semidirect
34. C17×#106C22semidirect
35. #10587492C22semidirect
36. #10587492C22semidirect
37. #10587491C22semidirect
38. #10587491C22semidirect
39. #10587480C22semidirect
40. #10587480C22semidirect
41. #10587479C22semidirect
42. #10587478C22semidirect
156. #428D17semidirect
189. #107C2×D17semidirect
191. #106C2×D17semidirect
198. C17#3094semidirect

Maximal subgroups

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

subgroupindex
C2×#105875812
#117×D172
#117631362
#117631352
#117631292
#117631242
#117628492
#117628152
#117628132
#117627722
C17×#4282
#309417