Finite Group Databasebeta

← #14538705

Subgroups — #14538705

Order 2176.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#11763316C211yesmaximal, characteristic
2C2×C4⋊C4×D17C211yesmaximal, characteristic, complemented
3#11763143C212yesmaximal, complemented
4#11763138C212yesmaximal, complemented
5#11763126C212yesmaximal, complemented
6#11763121C212yesmaximal, complemented
7#11762851C211yesmaximal, characteristic, complemented
8#11762849C211yesmaximal, characteristic, complemented
9#11762810C211yesmaximal, characteristic, complemented
10#11762772C211yesmaximal, characteristic, complemented
11C17×#417C211yesmaximal, characteristic, complemented
12C2×#41×C17C2211yescharacteristic
13C22×C4×D17—21no—
14C22×C4×D17C2211yescharacteristic
15C2×M16×C17C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17#10587632C2211yescharacteristic
18#10587632C2211yescharacteristic
19#10587624C2211yescharacteristic
20#10587624C2211yescharacteristic
21#10587623—21no—
22#10587621—21no—
23C4×C22⋊D17—21no—
24C4×C22⋊D17—21no—
25#10587612C2211yescharacteristic
26C2×C4⋊Dic17C2211yescharacteristic
27C2×#63225—21no—
28C2×#63223C2211yescharacteristic
29C4⋊C4×D17—21no—
30C4⋊C4×D17—21no—
31C4⋊C4×D17C2212yes—
32C4⋊C4×D17C2212yes—
33C17×#103C2212yescomplemented
34C17×#102C2212yescomplemented
35#10587506C2212yescomplemented
36#10587503C2212yescomplemented
37#10587491C2212yes—
38#10587491C2212yes—
39#10587489C2212yescomplemented
40#10587488C2212yescomplemented
41#10587479C2212yescomplemented
42#10587478C2212yescomplemented
43C23×D17D411yescharacteristic
44#41×C17—22no—
45#41×C17—22no—
46C2×Q8×C17C2311yescharacteristic
47C2×D4×C17—21no—
48C2×D4×C17C2311yescharacteristic
49C22×C4×C17—21no—
50C22×C4×C17—21no—
51C22×C4×C17C2311yescharacteristic
52C2×C22⋊D17—21no—
53C2×C22⋊D17—21no—
54C22×Dic17—21no—
55C22×Dic17D411yescharacteristic
56C2×C4×D17—22no—
57C2×C4×D17—22no—
58C2×C4×D17—22no—
59C2×C4×D17—22no—
60C2×C4×D17—44no—
61C2×C4×D17D411yescharacteristic
62C2×C4×D17D411yescharacteristic
63C2×C4×D17C2311yescharacteristic
64C2×C4×D17C2311yescharacteristic
65M16×C17—22no—
66C2×C8×C17C2312yes—
67C4⋊C4×C17—21no—
68C4⋊C4×C17C2312yes—
69C22⋊Dic17—21no—
70C22⋊Dic17—21no—
71C22⋊Dic17—21no—
72#63227—21no—
73#63227—21no—
74C4⋊Dic17—21no—
75C4⋊Dic17C2311yescharacteristic
76C4⋊Dic17C2311yescharacteristic
77C4⋊Dic17C2312yes—
78#63225—22no—
79#63225—22no—
80#63225—21no—
81#63225—21no—
82C4×Dic17—21no—
83#63223—22no—
84C2×#3622C2312yes—
85C23×C17—21no—
86C23×C17C2×D411yescharacteristic
87C22×D17—22no—
88C22×D17—22no—
89C22×D17—22no—
90C22×D17C2×D411yescharacteristic
91C22×D17C2×D411yescharacteristic
92Q8×C17—21no—
93Q8×C17—21no—
94D4×C17—21no—
95D4×C17—21no—
96D4×C17—42no—
97C2×C4×C17—22no—
98C2×C4×C17—21no—
99C2×C4×C17—42no—
100C2×C4×C17—42no—
101C2×C4×C17C2×D411yescharacteristic
102C2×C4×C17#4112yes—
103C2×C4×C17#4112yes—
104C2×C4×C17C2411yescharacteristic
105C22⋊D17—42no—
106C22⋊D17—42no—
107C2×Dic17—22no—
108C2×Dic17—21no—
109C2×Dic17—21no—
110C2×Dic17—22no—
111C2×Dic17—42no—
112C2×Dic17C2×D411yescharacteristic
113C2×Dic17C2×D411yescharacteristic
114C4×D17—22no—
115C4×D17—22no—
116C4×D17—21no—
117C4×D17—21no—
118C4×D17—21no—
119C4×D17—21no—
120C4×D17—44no—
121C4×D17—44no—
122C8×C17—22no—
123#3622—22no—
124#3092—171nomaximal, Sylow
125C22×C17—21no—
126C22×C17—21no—
127C22×C17—41no—
128C22×C17—41no—
129C22×C17—41no—
130C22×C17#12312yes—
131C22×C17C22×D411yescharacteristic
132C2×D17—28no—
133C2×D17—21no—
134C2×D17—21no—
135C2×D17—21no—
136C2×D17—21no—
137C2×D17AGL(1,ℤ/8)12yes—
138C2×D17#13712yes—
139C4×C17—41no—
140C4×C17—42no—
141C4×C17#12312yes—
142C4×C17C2×#4111yescharacteristic
143C4×C17C2×#4111yescharacteristic
144Dic17—21no—
145Dic17—21no—
146Dic17—42no—
147Dic17—41no—
148Dic17—41no—
149#525—171no—
150C22×C4⋊C4—171no—
151#484—172no—
152#483—172no—
153#481—172no—
154#480—172no—
155#428—171no—
156#428—171no—
157#417—171no—
158#417D1711yescharacteristic, complemented
159#407—171no—
160C2×C17—22no—
161C2×C17—81no—
162C2×C17C2×#12311yescharacteristic
163C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
164C2×C17C2×#13711yescharacteristic
165D17—22no—
166D17—22no—
167C2×#41C2×D1711yescharacteristic
168C23×C4—171no—
169C23×C4—341no—
170C2×M16—171no—
171C2×M16C2×D1711yescharacteristic
172#123—341no—
173C22⋊Q8—171no—
174C22⋊Q8—171no—
175#121—171no—
176#121—171no—
177#120—341no—
178C4×D4—341no—
179C4×D4—341no—
180#117—171no—
181C2×C4⋊C4—171no—
182C2×C4⋊C4—172no—
183C2×C4⋊C4—172no—
184C2×C4⋊C4—341no—
185C2×C4⋊C4—341no—
186C2×C4⋊C4—341no—
187C2×C4⋊C4C2×D1711yescharacteristic
188#107—172no—
189#107—172no—
190#106—172no—
191#106—172no—
192#103—172no—
193#103C2×D1712yescomplemented
194#102—172no—
195#102C2×D1712yescomplemented
196C22⋊C8—172no—
197C22⋊C8—172no—
198C17#309211yescharacteristic, complemented, Sylow
199C24—171no—
200#41—22no—
201#41—22no—
202C2×Q8C22×D1711yescharacteristic
203C2×D4—21no—
204C2×D4—341no—
205C2×D4—341no—
206C2×D4C22×D1711yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—171no—
210C22×C4—171no—
211C22×C4—171no—
212C22×C4—171no—
213C22×C4—171no—
214C22×C4—342no—
215C22×C4—342no—
216C22×C4—342no—
217C22×C4—342no—
218C22×C4—341no—
219C22×C4—684no—
220C22×C4C22×D1711yescharacteristic
221M16—22no—
222M16—342no—
223C2×C8—172no—
224C2×C8C22×D1712yes—
225C4⋊C4—21no—
226C4⋊C4—172no—
227C4⋊C4—171no—
228C4⋊C4—171no—
229C4⋊C4—342no—
230C4⋊C4—342no—
231C4⋊C4—341no—
232C4⋊C4—341no—
233C4⋊C4—341no—
234C4⋊C4C22×D1712yes—
235C22⋊C4—341no—
236C22⋊C4—341no—
237C22⋊C4—341no—
238C22⋊C4—341no—
239C22⋊C4—341no—
240C42—341no—
241C23—21no—
242C23—171no—
243C23—171no—
244C23—342no—
245C23—342no—
246C23—342no—
247C23D4×D1711yescharacteristic
248Q8—21no—
249Q8—21no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—682no—
254D4—682no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—171no—
260C2×C4—171no—
261C2×C4—342no—
262C2×C4—341no—
263C2×C4—341no—
264C2×C4—341no—
265C2×C4—341no—
266C2×C4—342no—
267C2×C4—342no—
268C2×C4—341no—
269C2×C4—341no—
270C2×C4—342no—
271C2×C4—684no—
272C2×C4—684no—
273C2×C4—682no—
274C2×C4#6325212yes—
275C2×C4D4×D1711yescharacteristic
276C2×C4#6325412yes—
277C2×C4C23×D1711yescharacteristic
278C8—22no—
279C8—342no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—172no—
286C22—172no—
287C22—348no—
288C22—341no—
289C22—341no—
290C22—341no—
291C22—341no—
292C22#1058756912yes—
293C22C2×D4×D1711yescharacteristic
294C4—41no—
295C4—42no—
296C4—341no—
297C4—341no—
298C4—682no—
299C4—681no—
300C4—681no—
301C4#1058756912yes—
302C4C2×#6325211yescharacteristic
303C4C2×#6325411yescharacteristic
304C2—22no—
305C2—81no—
306C2—342no—
307C2—342no—
308C2C2×#1058756911yescharacteristic
309C2AGL(1,ℤ/8)×D1711yescharacteristic
310C2#137×D1711yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#417C2
socleC22×C17C22×D4
prime radical#11763316C2
hypercenter#417D17
hypocenterC17#3092
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
2. C2×C4⋊C4×D17C2semidirect
3. #11763143C2semidirect
4. #11763138C2semidirect
5. #11763126C2semidirect
6. #11763121C2semidirect
7. #11762851C2semidirect
8. #11762849C2semidirect
9. #11762810C2semidirect
10. #11762772C2semidirect
11. C17×#417C2semidirect
33. C17×#103C22semidirect
34. C17×#102C22semidirect
35. #10587506C22semidirect
36. #10587503C22semidirect
39. #10587489C22semidirect
40. #10587488C22semidirect
41. #10587479C22semidirect
42. #10587478C22semidirect
158. #417D17semidirect
193. #103C2×D17semidirect
195. #102C2×D17semidirect
198. C17#3092semidirect

Maximal subgroups

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

subgroupindex
#117633162
C2×C4⋊C4×D172
#117631432
#117631382
#117631262
#117631212
#117628512
#117628492
#117628102
#117627722
C17×#4172
#309217