Finite Group Databasebeta

← #14535346

Subgroups — #14535346

Order 2176.

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

normal subgroups only

This group has 680 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×#10587621C211yesmaximal, characteristic, complemented
2C2×#10587623C211yesmaximal, characteristic
3#11762734C212yesmaximal, complemented
4#11762731C212yesmaximal, complemented
5#11762725C212yesmaximal, complemented
6#11762700C212yesmaximal, complemented
7#11762686C212yesmaximal, complemented
8#11762664C212yesmaximal, complemented
9C17×#390C211yesmaximal, characteristic, complemented
10C24×D17C2211yescharacteristic
11C22×D4×C17C2211yescharacteristic
12C22×C22⋊D17C2212yes—
13C23×Dic17C2211yescharacteristic, complemented
14C2×C4⋊C4×C17C2211yescharacteristic, complemented
15C2×C22⋊C4×C17C2212yes—
16C2×C22⋊C4×C17C2212yescomplemented
17C2×C42×C17C2211yescharacteristic, complemented
18C2×C22⋊Dic17C2211yescharacteristic
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2211yescharacteristic, complemented
21#10587623—24no—
22#10587621—24no—
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×#63227C2212yes—
27C2×#63227C2212yes—
28C2×C4⋊Dic17C2212yescomplemented
29C2×#63225C2212yescomplemented
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×D4×C17—41no—
42C2×D4×C17—41no—
43C2×D4×C17D414yes—
44C22×C4×C17C2312yes—
45C22×C4×C17C2312yes—
46C22×C4×C17C2311yescharacteristic
47C2×C22⋊D17—24no—
48C2×C22⋊D17—24no—
49C2×C22⋊D17—42no—
50C22×Dic17—22no—
51C22×Dic17—41no—
52C22×Dic17—41no—
53C22×Dic17C2312yes—
54C22×Dic17C2312yes—
55C22×Dic17C2312yes—
56C4⋊C4×C17—41no—
57C22⋊C4×C17—42no—
58C22⋊C4×C17—42no—
59C42×C17—41no—
60C22⋊Dic17—24no—
61C22⋊Dic17—22no—
62C22⋊Dic17—22no—
63#63227—24no—
64#63227—24no—
65#63227—42no—
66#63227—42no—
67#63227—42no—
68C4⋊Dic17—42no—
69#63225—24no—
70C23×C17—22no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—21no—
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—
93D4×C17—44no—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
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×C17C2×D412yes—
103C22⋊D17—48no—
104C2×Dic17—22no—
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—44no—
114#2697—171nomaximal, Sylow
115C22×C17—22no—
116C22×C17—22no—
117C22×C17—24no—
118C22×C17—24no—
119C22×C17—42no—
120C22×C17—42no—
121C22×C17—41no—
122C22×C17—41no—
123C22×C17—41no—
124C22×C17—41no—
125C22×C17C22×D411yescharacteristic
126C22×C17C2×#4112yes—
127C22×C17H(5,𝔽2)11yescharacteristic
128C22×C17H(5,𝔽2)11yescharacteristic
129C22×C17H(5,𝔽2)12yes—
130C2×D17—24no—
131C2×D17—24no—
132C2×D17—24no—
133C2×D17—416no—
134C2×D17—42no—
135C2×D17—42no—
136C2×D17—42no—
137C2×D17—42no—
138C4×C17—22no—
139C4×C17—42no—
140C4×C17—42no—
141Dic17—42no—
142Dic17—42no—
143Dic17—42no—
144C2×#123—171no—
145C2×#120—171no—
146#397—172no—
147#396—172no—
148#394—172no—
149#390D1711yescharacteristic, complemented
150#388—172no—
151#386—172no—
152#379—172no—
153C2×C17—42no—
154C2×C17—42no—
155C2×C17#53411yescharacteristic
156C2×C17#53812yes—
157C2×C17#54612yes—
158C2×C17#55111yescharacteristic
159C2×C17#56011yescharacteristic
160D17—44no—
161C25—171no—
162C22×D4—172no—
163C22×D4C2×D1711yescharacteristic
164C23×C4—171no—
165#123—344no—
166#120—344no—
167C2×C4⋊C4—172no—
168C2×C4⋊C4—172no—
169C2×C4⋊C4C2×D1711yescharacteristic, complemented
170C2×C22⋊C4—171no—
171C2×C22⋊C4—172no—
172C2×C22⋊C4—171no—
173C2×C22⋊C4—172no—
174C2×C22⋊C4—172no—
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—
184C17#269711yescharacteristic, complemented, Sylow
185C24—172no—
186C24—344no—
187C24—342no—
188C24—682no—
189C24—681no—
190C24—681no—
191C24C22×D1711yescharacteristic
192C24C22×D1711yescharacteristic
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—344no—
196C2×D4—344no—
197C2×D4—682no—
198C2×D4D6814yes—
199C22×C4—172no—
200C22×C4—172no—
201C22×C4—172no—
202C22×C4—342no—
203C22×C4—681no—
204C22×C4—681no—
205C22×C4C22×D1712yes—
206C22×C4C22×D1712yes—
207C22×C4C22×D1711yescharacteristic
208C4⋊C4—41no—
209C4⋊C4—344no—
210C4⋊C4—682no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—344no—
214C22⋊C4—342no—
215C22⋊C4—342no—
216C22⋊C4—344no—
217C22⋊C4—344no—
218C22⋊C4—682no—
219C22⋊C4—682no—
220C22⋊C4—682no—
221C42—41no—
222C23—22no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—22no—
229C23—22no—
230C23—174no—
231C23—342no—
232C23—342no—
233C23—342no—
234C23—342no—
235C23—342no—
236C23—342no—
237C23—688no—
238C23—684no—
239C23—684no—
240C23—688no—
241C23—684no—
242C23C2×D6812yes—
243C23C2×D6812yes—
244C23C23×D1711yescharacteristic
245D4—44no—
246D4—688no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—342no—
256C2×C4—342no—
257C2×C4—342no—
258C2×C4—342no—
259C2×C4—342no—
260C2×C4—342no—
261C2×C4—342no—
262C2×C4—342no—
263C2×C4—342no—
264C2×C4—684no—
265C2×C4C2×D6812yes—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—24no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—344no—
277C22—344no—
278C22—344no—
279C22—6816no—
280C22—682no—
281C22—682no—
282C22—682no—
283C22—682no—
284C22C22×D6811yescharacteristic
285C22#1058769111yescharacteristic
286C22#1058769112yes—
287C22#1058769111yescharacteristic
288C22#41×D1712yes—
289C4—22no—
290C4—42no—
291C4—42no—
292C4—682no—
293C4—682no—
294C4—682no—
295C2—42no—
296C2—42no—
297C2—684no—
298C2#1176337111yescharacteristic
299C2#1176339112yes—
300C2#1176345412yes—
301C2#1176348911yescharacteristic
302C2#1176354611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#390C2
prime radicalC2×#10587623C2
hypercenter#390D17
hypocenterC17#2697
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#10587621C2semidirect
3. #11762734C2semidirect
4. #11762731C2semidirect
5. #11762725C2semidirect
6. #11762700C2semidirect
7. #11762686C2semidirect
8. #11762664C2semidirect
9. C17×#390C2semidirect
13. C23×Dic17C22semidirect
14. C2×C4⋊C4×C17C22semidirect
16. C2×C22⋊C4×C17C22semidirect
17. C2×C42×C17C22semidirect
20. C2×C22⋊Dic17C22semidirect
28. C2×C4⋊Dic17C22semidirect
29. C2×#63225C22semidirect
30. #10587502C22semidirect
31. #10587502C22semidirect
32. #10587502C22semidirect
149. #390D17semidirect
169. C2×C4⋊C4C2×D17semidirect
179. C2×C22⋊C4C2×D17semidirect
180. C2×C42C2×D17semidirect
184. C17#2697semidirect

Maximal subgroups

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

subgroupindex
C2×#105876212
C2×#105876232
#117627342
#117627312
#117627252
#117627002
#117626862
#117626642
C17×#3902
#269717