Finite Group Databasebeta

← #14533941

Subgroups — #14533941

Order 2176.

672 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 145 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×#10587564C211yesmaximal, characteristic, complemented
2C22×#63227C211yesmaximal, characteristic
3#11762738C212yesmaximal, complemented
4#11762722C212yesmaximal, complemented
5#11762701C212yesmaximal, complemented
6#11762698C212yesmaximal, complemented
7#11762672C211yesmaximal, characteristic, complemented
8#11762664C211yesmaximal, characteristic
9#11762664C211yesmaximal, characteristic
10C22⋊C4×Dic17C211yesmaximal, characteristic, complemented
11C17×#381C211yesmaximal, characteristic, complemented
12C24×D17C2211yescharacteristic
13C23×C4×C17C2211yescharacteristic
14C23×Dic17C2211yescharacteristic
15C22×Dic34C2211yescharacteristic, complemented
16C2×C22⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊Dic17C2211yescharacteristic
19C2×C22⋊Dic17C2211yescharacteristic
20C2×#63227—24no—
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×#63227C2212yes—
27C2×C4⋊Dic17C2211yescharacteristic, complemented
28C2×#63225C2211yescharacteristic, complemented
29C2×#63225C2211yescharacteristic, complemented
30C2×C4×Dic17C2212yescomplemented
31#10587564—24no—
32C17×#95C2212yescomplemented
33C17×#95C2212yescomplemented
34#10587502C2212yescomplemented
35#10587502C2212yescomplemented
36#10587502C2212yescomplemented
37C24×C17C2311yescharacteristic
38C23×D17—22no—
39C23×D17—24no—
40C23×D17—22no—
41C23×D17—22no—
42C23×D17—42no—
43C23×D17C2312yes—
44C22×C4×C17—22no—
45C22×C4×C17—22no—
46C22×C4×C17—22no—
47C22×C4×C17C2311yescharacteristic
48C22×C4×C17C2312yes—
49C22×C4×C17C2312yes—
50C22×C4×C17C2311yescharacteristic
51C22×Dic17—42no—
52C22×Dic17D414yes—
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2312yes—
55C22×Dic17C2311yescharacteristic
56C22×Dic17C2311yescharacteristic
57C22×Dic17C2311yescharacteristic
58C2×Dic34—22no—
59C2×Dic34—42no—
60C22⋊C4×C17—22no—
61C22⋊C4×C17—22no—
62C22⋊Dic17—22no—
63C22⋊Dic17—22no—
64#63227—24no—
65#63227—24no—
66#63227—24no—
67#63227—24no—
68#63227—42no—
69#63227—42no—
70C4⋊Dic17—22no—
71#63225—22no—
72#63225—22no—
73C4×Dic17—42no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17C2×D412yes—
77C23×C17#4112yes—
78C23×C17#4112yes—
79C23×C17C2411yescharacteristic
80C22×D17—28no—
81C22×D17—22no—
82C22×D17—22no—
83C22×D17—22no—
84C22×D17—22no—
85C22×D17—22no—
86C22×D17—22no—
87C22×D17—48no—
88C22×D17—48no—
89C22×D17—44no—
90C22×D17—44no—
91C22×D17#4114yes—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—44no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—21no—
106C2×Dic17—21no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—44no—
113C2×Dic17C2×D412yes—
114C2×Dic17C2×D412yes—
115Dic34—44no—
116#2561—171nomaximal, Sylow
117C22×C17—22no—
118C22×C17—22no—
119C22×C17—24no—
120C22×C17—24no—
121C22×C17#12414yes—
122C22×C17C22×D411yescharacteristic
123C22×C17C2×#4111yescharacteristic
124C22×C17C2×#4111yescharacteristic
125C22×C17C2×#4112yes—
126C22×C17H(5,𝔽2)12yes—
127C2×D17—24no—
128C2×D17—24no—
129C2×D17—24no—
130C2×D17—416no—
131C2×D17—42no—
132C2×D17—42no—
133C2×D17—42no—
134C2×D17—42no—
135C4×C17—41no—
136C4×C17—42no—
137C4×C17—41no—
138C4×C17—42no—
139Dic17—22no—
140Dic17—22no—
141Dic17—41no—
142Dic17—41no—
143Dic17—42no—
144C2×C22⋊Q8—171no—
145C22×C22⋊C4—171no—
146#397—172no—
147#393—172no—
148#388—172no—
149#388—172no—
150#381—171no—
151#381D1711yescharacteristic, complemented
152#379—171no—
153#379—171no—
154C4×C22⋊C4—171no—
155C2×C17—24no—
156C2×C17#52511yescharacteristic
157C2×C17C2×#12411yescharacteristic
158C2×C17#53411yescharacteristic
159C2×C17#55112yes—
160C2×C17#55112yes—
161D17—44no—
162C25—171no—
163C22×Q8—171no—
164C23×C4—171no—
165C23×C4C2×D1711yescharacteristic
166C22⋊Q8—344no—
167C2×C4⋊C4—171no—
168C2×C4⋊C4—171no—
169C2×C4⋊C4—171no—
170C2×C22⋊C4—171no—
171C2×C22⋊C4—171no—
172C2×C22⋊C4—172no—
173C2×C22⋊C4—172no—
174C2×C22⋊C4—172no—
175C2×C22⋊C4—172no—
176C2×C22⋊C4—172no—
177C2×C22⋊C4—172no—
178C2×C22⋊C4—344no—
179C2×C22⋊C4C2×D1711yescharacteristic
180C2×C22⋊C4C2×D1711yescharacteristic
181C2×C42—172no—
182#95—172no—
183#95—172no—
184#95—172no—
185#95C2×D1712yescomplemented
186#95C2×D1712yescomplemented
187C17#256111yescharacteristic, complemented, Sylow
188C24—172no—
189C24—344no—
190C24—342no—
191C24—342no—
192C24—342no—
193C24—682no—
194C24C22×D1711yescharacteristic
195C2×Q8—342no—
196C2×Q8—682no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—171no—
201C22×C4—171no—
202C22×C4—174no—
203C22×C4—172no—
204C22×C4—171no—
205C22×C4—171no—
206C22×C4—682no—
207C22×C4C22×D1712yes—
208C22×C4C22×D1712yes—
209C22×C4C22×D1711yescharacteristic
210C22×C4C22×D1711yescharacteristic
211C4⋊C4—342no—
212C4⋊C4—342no—
213C4⋊C4—342no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—344no—
217C22⋊C4—344no—
218C22⋊C4—344no—
219C22⋊C4—344no—
220C22⋊C4—342no—
221C22⋊C4—342no—
222C22⋊C4—682no—
223C22⋊C4—682no—
224C42—682no—
225C23—22no—
226C23—22no—
227C23—174no—
228C23—348no—
229C23—342no—
230C23—342no—
231C23—342no—
232C23—342no—
233C23—342no—
234C23—342no—
235C23—688no—
236C23—688no—
237C23—684no—
238C23—684no—
239C23#6325212yes—
240C23D4×D1712yes—
241C23#6325412yes—
242C23C23×D1711yescharacteristic
243Q8—684no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—44no—
255C2×C4—172no—
256C2×C4—172no—
257C2×C4—342no—
258C2×C4—342no—
259C2×C4—342no—
260C2×C4—341no—
261C2×C4—341no—
262C2×C4—342no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—342no—
266C2×C4—684no—
267C22—24no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—344no—
272C22—344no—
273C22—344no—
274C22—6816no—
275C22—682no—
276C22—682no—
277C22—682no—
278C22—682no—
279C22#1058757114yes—
280C22C2×#6325211yescharacteristic
281C22C2×D4×D1711yescharacteristic
282C22C2×#6325411yescharacteristic
283C22#41×D1712yes—
284C22#1058769612yes—
285C4—42no—
286C4—41no—
287C4—42no—
288C4—41no—
289C4—342no—
290C4—342no—
291C4—682no—
292C4—681no—
293C4—681no—
294C2—24no—
295C2—684no—
296C2#1176331711yescharacteristic
297C2C2×#1058757111yescharacteristic
298C2#1176337011yescharacteristic
299C2#1176348212yes—
300C2#1176348312yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#381C2
prime radicalC24×D17C22
hypercenter#381D17
hypocenterC17#2561
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C2×#10587564C2semidirect
3. #11762738C2semidirect
4. #11762722C2semidirect
5. #11762701C2semidirect
6. #11762698C2semidirect
7. #11762672C2semidirect
10. C22⋊C4×Dic17C2semidirect
11. C17×#381C2semidirect
15. C22×Dic34C22semidirect
27. C2×C4⋊Dic17C22semidirect
28. C2×#63225C22semidirect
29. C2×#63225C22semidirect
30. C2×C4×Dic17C22semidirect
32. C17×#95C22semidirect
33. C17×#95C22semidirect
34. #10587502C22semidirect
35. #10587502C22semidirect
36. #10587502C22semidirect
151. #381D17semidirect
185. #95C2×D17semidirect
186. #95C2×D17semidirect
187. C17#2561semidirect

Maximal subgroups

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

subgroupindex
C2×#105875642
C22×#632272
#117627382
#117627222
#117627012
#117626982
#117626722
#117626642
C22⋊C4×Dic172
C17×#3812
#256117