Finite Group Databasebeta

← #14535160

Subgroups — #14535160

Order 2176.

688 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 137 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×#10587623C211yesmaximal, characteristic
2#11762737C212yesmaximal, complemented
3#11762724C212yesmaximal, complemented
4#11762720C212yesmaximal, complemented
5#11762700C212yesmaximal, complemented
6#11762689C211yesmaximal, characteristic, complemented
7#11762686C212yesmaximal, complemented
8#11762664C212yesmaximal, complemented
9C2×C17×#124C211yesmaximal, characteristic, complemented
10C24×D17C2211yescharacteristic
11C22×Q8×C17C2211yescharacteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C22×C22⋊D17C2212yes—
14C23×Dic17C2211yescharacteristic, complemented
15C17×#124—24no—
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊C4×C17C2212yescomplemented
18C2×C42×C17C2211yescharacteristic, complemented
19C2×C22⋊Dic17C2211yescharacteristic
20C2×C22⋊Dic17C2212yes—
21C2×C22⋊Dic17C2211yescharacteristic, complemented
22#10587623—24no—
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×#63227C2212yes—
27C2×#63227C2212yes—
28C2×#63225C2212yescomplemented
29#10587502C2212yescomplemented
30#10587502C2212yescomplemented
31#10587502C2212yescomplemented
32#10587502C2212yescomplemented
33C24×C17C2311yescharacteristic
34C24×C17C2311yescharacteristic
35C23×D17—22no—
36C23×D17—24no—
37C23×D17—42no—
38C23×D17—41no—
39C23×D17—41no—
40C23×D17C2312yes—
41C2×Q8×C17—22no—
42C2×Q8×C17—41no—
43C2×Q8×C17—41no—
44C2×D4×C17—41no—
45C2×D4×C17—41no—
46C2×D4×C17D414yes—
47C22×C4×C17C2312yes—
48C22×C4×C17C2312yes—
49C22×C4×C17C2311yescharacteristic
50C2×C22⋊D17—24no—
51C2×C22⋊D17—24no—
52C2×C22⋊D17—42no—
53C22×Dic17—42no—
54C22×Dic17—41no—
55C22×Dic17C2312yes—
56C22×Dic17C2312yes—
57C22×Dic17C2312yes—
58C22⋊C4×C17—24no—
59C22⋊C4×C17—24no—
60C42×C17—22no—
61C22⋊Dic17—22no—
62C22⋊Dic17—42no—
63C22⋊Dic17—41no—
64#63227—24no—
65#63227—24no—
66#63227—42no—
67#63227—42no—
68#63227—42no—
69#63225—42no—
70C23×C17—22no—
71C23×C17—22no—
72C23×C17—21no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—21no—
76C23×C17—21no—
77C23×C17—21no—
78C23×C17C2×D412yes—
79C23×C17C2×D412yes—
80C23×C17C2411yescharacteristic
81C22×D17—22no—
82C22×D17—22no—
83C22×D17—22no—
84C22×D17—22no—
85C22×D17—22no—
86C22×D17—22no—
87C22×D17—44no—
88C22×D17—48no—
89C22×D17—48no—
90C22×D17—44no—
91C22×D17—44no—
92C22×D17#4114yes—
93Q8×C17—44no—
94D4×C17—44no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—22no—
103C2×C4×C17C2×D412yes—
104C22⋊D17—48no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—44no—
115#2684—171nomaximal, Sylow
116C22×C17—22no—
117C22×C17—22no—
118C22×C17—24no—
119C22×C17—24no—
120C22×C17—42no—
121C22×C17—42no—
122C22×C17—41no—
123C22×C17—41no—
124C22×C17—41no—
125C22×C17—41no—
126C22×C17C22×D411yescharacteristic
127C22×C17C2×#4112yes—
128C22×C17H(5,𝔽2)11yescharacteristic
129C22×C17H(5,𝔽2)11yescharacteristic
130C22×C17H(5,𝔽2)12yes—
131C2×D17—24no—
132C2×D17—24no—
133C2×D17—24no—
134C2×D17—416no—
135C2×D17—42no—
136C2×D17—42no—
137C2×D17—42no—
138C2×D17—42no—
139C4×C17—22no—
140C4×C17—42no—
141C4×C17—42no—
142Dic17—42no—
143Dic17—42no—
144Dic17—42no—
145C2×#124D1711yescharacteristic, complemented
146C2×#120—171no—
147#397—172no—
148#394—172no—
149#393—172no—
150#388—172no—
151#386—171no—
152#386—172no—
153#379—172no—
154C2×C17—42no—
155C2×C17—42no—
156C2×C17#53411yescharacteristic
157C2×C17#53812yes—
158C2×C17#54612yes—
159C2×C17#55111yescharacteristic
160C2×C17H(3,𝔽4)11yescharacteristic
161D17—44no—
162C25—171no—
163C22×Q8C2×D1711yescharacteristic, complemented
164C22×D4—172no—
165C22×D4C2×D1711yescharacteristic
166C23×C4—171no—
167#124—24no—
168#120—344no—
169C2×C4⋊C4—172no—
170C2×C22⋊C4—171no—
171C2×C22⋊C4—172no—
172C2×C22⋊C4—172no—
173C2×C22⋊C4—172no—
174C2×C22⋊C4—171no—
175C2×C22⋊C4—172no—
176C2×C22⋊C4—172no—
177C2×C22⋊C4—172no—
178C2×C22⋊C4C2×D1712yes—
179C2×C22⋊C4C2×D1712yescomplemented
180C2×C42C2×D1711yescharacteristic, complemented
181#95—172no—
182#95—172no—
183#95—172no—
184#95—172no—
185C17#268411yescharacteristic, complemented, Sylow
186C24—172no—
187C24—344no—
188C24—342no—
189C24—681no—
190C24—681no—
191C24—682no—
192C24C22×D1711yescharacteristic
193C24C22×D1711yescharacteristic
194C2×Q8—22no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—344no—
200C2×D4—344no—
201C2×D4—682no—
202C2×D4C22⋊D1714yes—
203C22×C4—172no—
204C22×C4—172no—
205C22×C4—172no—
206C22×C4—681no—
207C22×C4—682no—
208C22×C4C22×D1712yes—
209C22×C4C22×D1712yes—
210C22×C4C22×D1711yescharacteristic
211C4⋊C4—682no—
212C22⋊C4—24no—
213C22⋊C4—24no—
214C22⋊C4—344no—
215C22⋊C4—342no—
216C22⋊C4—344no—
217C22⋊C4—682no—
218C22⋊C4—682no—
219C22⋊C4—682no—
220C22⋊C4—681no—
221C22⋊C4—682no—
222C42—22no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—174no—
232C23—342no—
233C23—342no—
234C23—342no—
235C23—342no—
236C23—342no—
237C23—342no—
238C23—688no—
239C23—684no—
240C23—684no—
241C23—688no—
242C23—684no—
243C23C2×C22⋊D1712yes—
244C23C2×C22⋊D1712yes—
245C23C23×D1711yescharacteristic
246Q8—44no—
247D4—44no—
248D4—688no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—342no—
258C2×C4—342no—
259C2×C4—342no—
260C2×C4—342no—
261C2×C4—342no—
262C2×C4—342no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—342no—
266C2×C4—684no—
267C2×C4C2×C22⋊D1712yes—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—24no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—42no—
278C22—344no—
279C22—344no—
280C22—344no—
281C22—6816no—
282C22—682no—
283C22—682no—
284C22—682no—
285C22—682no—
286C22#1058769111yescharacteristic
287C22#1058769111yescharacteristic
288C22#41×D1712yes—
289C22#1058769612yes—
290C22C22×C22⋊D1711yescharacteristic
291C4—22no—
292C4—42no—
293C4—42no—
294C4—682no—
295C4—682no—
296C4—682no—
297C2—42no—
298C2—42no—
299C2—684no—
300C2#1176336811yescharacteristic
301C2#1176339312yes—
302C2#1176344712yes—
303C2#1176348911yescharacteristic
304C2#1176355111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#124C2
prime radicalC2×#10587623C2
hypercenterC2×#124D17
hypocenterC17#2684
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. #11762737C2semidirect
3. #11762724C2semidirect
4. #11762720C2semidirect
5. #11762700C2semidirect
6. #11762689C2semidirect
7. #11762686C2semidirect
8. #11762664C2semidirect
9. C2×C17×#124C2semidirect
11. C22×Q8×C17C22semidirect
14. C23×Dic17C22semidirect
17. C2×C22⋊C4×C17C22semidirect
18. C2×C42×C17C22semidirect
21. C2×C22⋊Dic17C22semidirect
28. C2×#63225C22semidirect
29. #10587502C22semidirect
30. #10587502C22semidirect
31. #10587502C22semidirect
32. #10587502C22semidirect
145. C2×#124D17semidirect
163. C22×Q8C2×D17semidirect
179. C2×C22⋊C4C2×D17semidirect
180. C2×C42C2×D17semidirect
185. C17#2684semidirect

Maximal subgroups

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

subgroupindex
C2×#105876232
#117627372
#117627242
#117627202
#117627002
#117626892
#117626862
#117626642
C2×C17×#1242
#268417