Finite Group Databasebeta

← #14539617

Subgroups — #14539617

Order 2176.

414 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 113 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#11763322C211yesmaximal, characteristic
2C2×#10587632C211yesmaximal, characteristic, complemented
3#11763148C212yesmaximal, complemented
4#11763142C211yesmaximal, characteristic, complemented
5#11763128C211yesmaximal, characteristic, complemented
6#11763064C212yesmaximal, complemented
7#11763058C212yesmaximal, complemented
8#11762970C212yesmaximal, complemented
9#11762948C211yesmaximal, characteristic, complemented
10#11762771C211yesmaximal, characteristic, complemented
11C17×#451C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C2×#63256C2211yescharacteristic
14C22×C4×D17C2211yescharacteristic
15C2×Q16×C17C2212yescomplemented
16C17×C22⋊Q8C2211yescharacteristic
17#10587632—21no—
18#10587632—21no—
19#10587632C2212yes—
20#10587632C2212yes—
21#10587632C2211yescharacteristic
22C2×#63230C2212yescomplemented
23#10587616C2211yescharacteristic
24C2×#63227—21no—
25C2×C4⋊Dic17C2211yescharacteristic
26C2×#63225—21no—
27C2×#63218C2212yes—
28#10587581C2211yescharacteristic
29#10587580—21no—
30#10587579—21no—
31#10587578C2211yescharacteristic
32#10587568—21no—
33#10587567—21no—
34C17×#103C2212yescomplemented
35C17×C22⋊C8C2211yescharacteristic
36#10587506C2212yescomplemented
37#10587506C2212yescomplemented
38C22⋊#3622C2211yescharacteristic
39#10587492C2212yescomplemented
40#10587491C2212yes—
41#10587488C2212yescomplemented
42#10587478C2212yescomplemented
43C23×D17D411yescharacteristic
44C2×Q8×C17—21no—
45C2×Q8×C17—42no—
46C2×Q8×C17C2311yescharacteristic
47C2×Q8×C17C2312yes—
48C22×C4×C17—21no—
49C22×C4×C17C2311yescharacteristic
50C2×C22⋊D17—21no—
51C22×Dic17—21no—
52C22×Dic17D411yescharacteristic
53#63256—42no—
54C2×D68—21no—
55C2×D68—21no—
56C2×D68C2311yescharacteristic
57C2×C4×D17—21no—
58C2×C4×D17—42no—
59C2×C4×D17D411yescharacteristic
60C2×C4×D17D411yescharacteristic
61C2×C4×D17C2311yescharacteristic
62C2×C4×D17C2311yescharacteristic
63Q16×C17—42no—
64C2×C8×C17C2312yes—
65C4⋊C4×C17—21no—
66C4⋊C4×C17C2311yescharacteristic
67C22⋊C4×C17—21no—
68#63230—42no—
69#63227—21no—
70#63227—21no—
71#63227—22no—
72#63227—22no—
73#63227—21no—
74#63227—21no—
75C4⋊Dic17—21no—
76C4⋊Dic17C2312yes—
77C4⋊Dic17C2311yescharacteristic
78#63225—21no—
79#63225—22no—
80#63225—22no—
81C4×Dic17—21no—
82C2×#3622C2312yes—
83#63218—42no—
84C23×C17C2×D411yescharacteristic
85C22×D17—22no—
86C22×D17—21no—
87C22×D17—41no—
88C22×D17—41no—
89C22×D17C2×D411yescharacteristic
90C22×D17C2×D411yescharacteristic
91Q8×C17—22no—
92Q8×C17—22no—
93Q8×C17—22no—
94Q8×C17—42no—
95C2×C4×C17—22no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—22no—
99C2×C4×C17—42no—
100C2×C4×C17C2×D411yescharacteristic
101C2×C4×C17C2411yescharacteristic
102C22⋊D17—42no—
103C2×Dic17—22no—
104C2×Dic17—21no—
105C2×Dic17—41no—
106C2×Dic17—42no—
107C2×Dic17C2×D411yescharacteristic
108C2×Dic17C2×D411yescharacteristic
109D68—22no—
110D68—42no—
111D68—42no—
112C4×D17—22no—
113C4×D17—22no—
114C4×D17—41no—
115C4×D17—41no—
116C4×D17—42no—
117C8×C17—22no—
118#3622—22no—
119#3196—171nomaximal, Sylow
120C22×C17—21no—
121C22×C17—21no—
122C22×C17#13712yes—
123C22×C17C22×D411yescharacteristic
124C2×D17—21no—
125C2×D17—21no—
126C2×D17—21no—
127C2×D17—21no—
128C2×D17—21no—
129C2×D17—21no—
130C2×D17—44no—
131C2×D17—41no—
132C2×D17—41no—
133C2×D17—41no—
134C4×C17—21no—
135C4×C17—41no—
136C4×C17—42no—
137C4×C17—41no—
138C4×C17H(5,𝔽2)12yes—
139Dic17—21no—
140Dic17—21no—
141Dic17—41no—
142Dic17—42no—
143#525—171no—
144C2×C22⋊Q8—171no—
145#485—172no—
146#484—171no—
147#481—171no—
148#470—172no—
149#468—172no—
150#452—172no—
151#451D1711yescharacteristic, complemented
152#448—171no—
153#407—171no—
154C2×C17—22no—
155C2×C17#53411yescharacteristic
156C2×C17C2×#13711yescharacteristic
157C2×C17#57511yescharacteristic
158D17—41no—
159D17—41no—
160D17—81no—
161C2×#41—171no—
162C22×Q8C2×D1711yescharacteristic
163C23×C4—171no—
164C2×Q16C2×D1712yescomplemented
165C2×SD16—172no—
166C2×M16—172no—
167#123—341no—
168C22⋊Q8—172no—
169C22⋊Q8—172no—
170C22⋊Q8—171no—
171C22⋊Q8—341no—
172C22⋊Q8—341no—
173C22⋊Q8C2×D1711yescharacteristic
174#121—171no—
175#121—171no—
176#120—341no—
177C4×D4—341no—
178C4×D4—341no—
179#117—171no—
180C2×C4⋊C4—171no—
181C2×C4⋊C4—341no—
182C2×C22⋊C4—341no—
183#107—172no—
184#106—172no—
185#103—172no—
186#103—172no—
187#103C2×D1712yescomplemented
188#102—172no—
189C22⋊C8—171no—
190C22⋊C8—172no—
191C22⋊C8C2×D1711yescharacteristic
192C17#319611yescharacteristic, complemented, Sylow
193C24—171no—
194#41—682no—
195C2×Q8—21no—
196C2×Q8—42no—
197C2×Q8C22×D1711yescharacteristic
198C2×Q8C22×D1712yes—
199C2×D4—171no—
200C2×D4—341no—
201C2×D4—341no—
202C2×D4—341no—
203C22×C4—21no—
204C22×C4—171no—
205C22×C4—171no—
206C22×C4—171no—
207C22×C4—171no—
208C22×C4—171no—
209C22×C4—341no—
210C22×C4—341no—
211C22×C4—682no—
212C22×C4C22×D1711yescharacteristic
213Q16—42no—
214SD16—682no—
215M16—682no—
216C2×C8—172no—
217C2×C8C22×D1712yes—
218C4⋊C4—21no—
219C4⋊C4—172no—
220C4⋊C4—171no—
221C4⋊C4—342no—
222C4⋊C4—342no—
223C4⋊C4—341no—
224C4⋊C4—341no—
225C4⋊C4C22×D1711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—341no—
228C22⋊C4—342no—
229C22⋊C4—342no—
230C22⋊C4—341no—
231C22⋊C4—341no—
232C22⋊C4—341no—
233C42—341no—
234C23—171no—
235C23—171no—
236C23—342no—
237C23—341no—
238C23—681no—
239C23—681no—
240C23D4×D1711yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—22no—
244Q8—42no—
245D4—342no—
246D4—682no—
247D4—682no—
248D4—682no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—171no—
255C2×C4—171no—
256C2×C4—342no—
257C2×C4—342no—
258C2×C4—341no—
259C2×C4—342no—
260C2×C4—681no—
261C2×C4—681no—
262C2×C4—682no—
263C2×C4—681no—
264C2×C4—682no—
265C2×C4D4×D1711yescharacteristic
266C2×C4C23×D1711yescharacteristic
267C8—22no—
268C8—342no—
269C22—21no—
270C22—21no—
271C22—341no—
272C22—341no—
273C22—341no—
274C22—341no—
275C22—341no—
276C22—341no—
277C22—684no—
278C22—681no—
279C22—681no—
280C22—681no—
281C22#1058760412yes—
282C22C2×D4×D1711yescharacteristic
283C4—21no—
284C4—41no—
285C4—42no—
286C4—41no—
287C4—341no—
288C4—341no—
289C4—681no—
290C4—682no—
291C4#1058769112yes—
292C2—22no—
293C2—681no—
294C2—681no—
295C2—1361no—
296C2#1176336911yescharacteristic
297C2C2×#1058760411yescharacteristic
298C2#1176360811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#451C2
socleC22×C17C22×D4
prime radical#11763322C2
hypercenter#451D17
hypocenterC17#3196
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×#10587632C2semidirect
3. #11763148C2semidirect
4. #11763142C2semidirect
5. #11763128C2semidirect
6. #11763064C2semidirect
7. #11763058C2semidirect
8. #11762970C2semidirect
9. #11762948C2semidirect
10. #11762771C2semidirect
11. C17×#451C2semidirect
15. C2×Q16×C17C22semidirect
22. C2×#63230C22semidirect
34. C17×#103C22semidirect
36. #10587506C22semidirect
37. #10587506C22semidirect
39. #10587492C22semidirect
41. #10587488C22semidirect
42. #10587478C22semidirect
151. #451D17semidirect
164. C2×Q16C2×D17semidirect
187. #103C2×D17semidirect
192. C17#3196semidirect

Maximal subgroups

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

subgroupindex
#117633222
C2×#105876322
#117631482
#117631422
#117631282
#117630642
#117630582
#117629702
#117629482
#117627712
C17×#4512
#319617