Finite Group Databasebeta

← #14539613

Subgroups — #14539613

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#11763319C211yesmaximal, characteristic
2C2×#10587583C211yesmaximal, characteristic, complemented
3#11763153C212yesmaximal, complemented
4#11763144C211yesmaximal, characteristic, complemented
5#11763068C212yesmaximal, complemented
6#11763054C212yesmaximal, complemented
7#11762972C212yesmaximal, complemented
8#11762967C211yesmaximal, characteristic, complemented
9#11762947C211yesmaximal, characteristic, complemented
10#11762771C211yesmaximal, characteristic, complemented
11C17×#481C211yesmaximal, characteristic, complemented
12C2×#63254C2211yescharacteristic
13C22×C4×D17C2211yescharacteristic
14C22×Dic34C2211yescharacteristic
15C17×#121C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17#10587624C2211yescharacteristic
18#10587623—21no—
19D4×Dic17—21no—
20C2×#63229C2212yescomplemented
21C2×#63227—21no—
22C2×C4⋊Dic17—21no—
23#10587610C2211yescharacteristic
24C2×Dic68C2212yescomplemented
25C2×#63218C2212yes—
26#10587583—21no—
27#10587583—21no—
28#10587583C2212yes—
29#10587583C2212yes—
30#10587583C2211yescharacteristic
31#10587578C2211yescharacteristic
32#10587569—21no—
33#10587567—21no—
34C17×#106C2212yescomplemented
35C17×#102C2212yescomplemented
36C17×C22⋊C8C2211yescharacteristic
37C22⋊#3622C2211yescharacteristic
38#10587491C2212yes—
39#10587487C2212yescomplemented
40#10587481C2212yescomplemented
41#10587481C2212yescomplemented
42#10587478C2212yescomplemented
43C23×D17D411yescharacteristic
44C2×D4×C17—21no—
45C2×D4×C17C2311yescharacteristic
46C22×C4×C17—21no—
47C22×C4×C17C2311yescharacteristic
48C2×C22⋊D17—21no—
49C2×C22⋊D17—21no—
50C22×Dic17—21no—
51C22×Dic17—21no—
52C22×Dic17D411yescharacteristic
53#63254—42no—
54C2×C4×D17—42no—
55C2×C4×D17D411yescharacteristic
56C2×C4×D17D411yescharacteristic
57C2×C4×D17C2311yescharacteristic
58C2×C4×D17C2311yescharacteristic
59C2×Dic34—21no—
60C2×Dic34—42no—
61C2×Dic34C2312yes—
62C2×Dic34C2311yescharacteristic
63C2×C8×C17C2312yes—
64C4⋊C4×C17—21no—
65C4⋊C4×C17C2312yes—
66C4⋊C4×C17C2311yescharacteristic
67C22⋊C4×C17—21no—
68C22⋊Dic17—21no—
69C22⋊Dic17—21no—
70#63229—42no—
71#63227—21no—
72#63227—21no—
73#63227—22no—
74#63227—22no—
75C4⋊Dic17—21no—
76C4⋊Dic17—22no—
77C4⋊Dic17—22no—
78C4⋊Dic17C2311yescharacteristic
79#63225—21no—
80C4×Dic17—21no—
81C2×#3622C2312yes—
82Dic68—42no—
83#63218—42no—
84C23×C17—21no—
85C23×C17C2×D411yescharacteristic
86C22×D17—22no—
87C22×D17—41no—
88C22×D17—41no—
89C22×D17C2×D411yescharacteristic
90C22×D17C2×D411yescharacteristic
91D4×C17—22no—
92D4×C17—42no—
93C2×C4×C17—22no—
94C2×C4×C17—21no—
95C2×C4×C17—22no—
96C2×C4×C17—42no—
97C2×C4×C17C2×D411yescharacteristic
98C2×C4×C17C2411yescharacteristic
99C22⋊D17—42no—
100C22⋊D17—42no—
101C2×Dic17—21no—
102C2×Dic17—21no—
103C2×Dic17—22no—
104C2×Dic17—41no—
105C2×Dic17—42no—
106C2×Dic17—42no—
107C2×Dic17C2×D411yescharacteristic
108C2×Dic17C2×D411yescharacteristic
109C4×D17—22no—
110C4×D17—22no—
111C4×D17—41no—
112C4×D17—41no—
113Dic34—22no—
114Dic34—22no—
115Dic34—22no—
116Dic34—42no—
117C8×C17—22no—
118#3622—22no—
119#3196—171nomaximal, Sylow
120C22×C17—21no—
121C22×C17—21no—
122C22×C17—41no—
123C22×C17—41no—
124C22×C17—41no—
125C22×C17#13712yes—
126C22×C17C22×D411yescharacteristic
127C2×D17—21no—
128C2×D17—21no—
129C2×D17—21no—
130C2×D17—21no—
131C2×D17—21no—
132C2×D17—21no—
133C2×D17—44no—
134C4×C17—21no—
135C4×C17—41no—
136C4×C17—42no—
137C4×C17H(5,𝔽2)12yes—
138Dic17—21no—
139Dic17—21no—
140Dic17—42no—
141Dic17—41no—
142Dic17—41no—
143#525—171no—
144C2×C22⋊Q8—171no—
145#485—172no—
146#484—171no—
147#481D1711yescharacteristic, complemented
148#470—172no—
149#468—172no—
150#452—172no—
151#451—171no—
152#448—171no—
153#407—171no—
154C2×C17—22no—
155C2×C17—81no—
156C2×C17#53411yescharacteristic
157C2×C17C2×#13711yescharacteristic
158C2×C17#57511yescharacteristic
159D17—41no—
160D17—41no—
161C2×#41—171no—
162C22×Q8—171no—
163C23×C4—171no—
164C2×Q16—172no—
165C2×SD16—172no—
166C2×M16—172no—
167#123—341no—
168C22⋊Q8—171no—
169C22⋊Q8—172no—
170C22⋊Q8—172no—
171C22⋊Q8—171no—
172C22⋊Q8—341no—
173C22⋊Q8—341no—
174#121—171no—
175#121C2×D1711yescharacteristic
176#120—341no—
177C4×D4—341no—
178C4×D4—341no—
179#117—171no—
180C2×C4⋊C4—341no—
181C2×C4⋊C4C2×D1711yescharacteristic
182C2×C22⋊C4—341no—
183#107—172no—
184#106C2×D1712yescomplemented
185#103—172no—
186#103—172no—
187#103—172no—
188#102C2×D1712yescomplemented
189C22⋊C8—171no—
190C22⋊C8—172no—
191C22⋊C8C2×D1711yescharacteristic
192C17#319611yescharacteristic, complemented, Sylow
193C24—171no—
194#41—682no—
195C2×Q8—172no—
196C2×Q8—171no—
197C2×Q8—341no—
198C2×Q8—682no—
199C2×D4—21no—
200C2×D4—341no—
201C2×D4—341no—
202C2×D4C22×D1711yescharacteristic
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—682no—
214SD16—682no—
215M16—682no—
216C2×C8—172no—
217C2×C8C22×D1712yes—
218C4⋊C4—21no—
219C4⋊C4—171no—
220C4⋊C4—342no—
221C4⋊C4—342no—
222C4⋊C4—341no—
223C4⋊C4—341no—
224C4⋊C4C22×D1712yes—
225C4⋊C4C22×D1711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—342no—
228C22⋊C4—342no—
229C22⋊C4—341no—
230C22⋊C4—341no—
231C22⋊C4—341no—
232C22⋊C4—341no—
233C42—341no—
234C23—21no—
235C23—171no—
236C23—171no—
237C23—342no—
238C23—681no—
239C23—681no—
240C23D4×D1711yescharacteristic
241Q8—342no—
242Q8—342no—
243Q8—342no—
244Q8—682no—
245D4—22no—
246D4—42no—
247D4—682no—
248D4—682no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—171no—
254C2×C4—171no—
255C2×C4—342no—
256C2×C4—342no—
257C2×C4—341no—
258C2×C4—341no—
259C2×C4—342no—
260C2×C4—681no—
261C2×C4—681no—
262C2×C4—681no—
263C2×C4—682no—
264C2×C4—682no—
265C2×C4D4×D1711yescharacteristic
266C2×C4C23×D1711yescharacteristic
267C8—22no—
268C8—342no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—341no—
275C22—341no—
276C22—341no—
277C22—341no—
278C22—341no—
279C22—341no—
280C22—684no—
281C22#1058760112yes—
282C22C2×D4×D1711yescharacteristic
283C4—21no—
284C4—41no—
285C4—42no—
286C4—341no—
287C4—341no—
288C4—681no—
289C4—681no—
290C4—682no—
291C4#1058769612yes—
292C2—22no—
293C2—81no—
294C2—681no—
295C2—681no—
296C2#1176337011yescharacteristic
297C2C2×#1058760111yescharacteristic
298C2#1176360511yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#481C2
socleC22×C17C22×D4
prime radical#11763319C2
hypercenter#481D17
hypocenterC17#3196
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×#10587583C2semidirect
3. #11763153C2semidirect
4. #11763144C2semidirect
5. #11763068C2semidirect
6. #11763054C2semidirect
7. #11762972C2semidirect
8. #11762967C2semidirect
9. #11762947C2semidirect
10. #11762771C2semidirect
11. C17×#481C2semidirect
20. C2×#63229C22semidirect
24. C2×Dic68C22semidirect
34. C17×#106C22semidirect
35. C17×#102C22semidirect
39. #10587487C22semidirect
40. #10587481C22semidirect
41. #10587481C22semidirect
42. #10587478C22semidirect
147. #481D17semidirect
184. #106C2×D17semidirect
188. #102C2×D17semidirect
192. C17#3196semidirect

Maximal subgroups

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

subgroupindex
#117633192
C2×#105875832
#117631532
#117631442
#117630682
#117630542
#117629722
#117629672
#117629472
#117627712
C17×#4812
#319617