Finite Group Databasebeta

← #14533250

Subgroups — #14533250

Order 2176.

708 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 161 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×#10587580C212yesmaximal, complemented
2C2×#10587569C212yesmaximal, complemented
3C2×#10587570C211yesmaximal, characteristic
4C2×C4⋊C4×D17C211yesmaximal, characteristic, complemented
5#11762746C211yesmaximal, characteristic, complemented
6#11762745C211yesmaximal, characteristic, complemented
7#11762744C211yesmaximal, characteristic, complemented
8#11762709C212yesmaximal, complemented
9#11762678C212yesmaximal, complemented
10#11762656C211yesmaximal, characteristic, complemented
11C17×#399C211yesmaximal, characteristic, complemented
12C22×C22⋊D17C2211yescharacteristic
13C22×C22⋊D17C2211yescharacteristic
14C22×D68C2211yescharacteristic
15C22×C4×D17C2212yes—
16C22×C4×D17C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C4⋊C4×C17C2212yes—
19C2×C22⋊C4×C17C2212yes—
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊Dic17C2212yes—
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×#63227C2211yescharacteristic
25C2×#63227C2212yes—
26C2×C4⋊Dic17C2211yescharacteristic
27C2×C4⋊Dic17C2212yes—
28C2×#63225C2212yes—
29C2×#63225C2212yes—
30C2×#63225C2212yes—
31C2×#63225C2211yescharacteristic
32C2×C4×Dic17C2212yes—
33#10587580—28no—
34C4⋊C4×D17—24no—
35#10587570—24no—
36#10587569—28no—
37C17×#95C2211yescharacteristic
38#10587502C2211yescharacteristic
39C24×C17C2311yescharacteristic
40C23×D17C2311yescharacteristic
41C23×D17C2311yescharacteristic
42C22×C4×C17C2312yes—
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C22×C4×C17C2312yes—
46C2×C22⋊D17—24no—
47C2×C22⋊D17—22no—
48C2×C22⋊D17—22no—
49C2×C22⋊D17—24no—
50C22×Dic17C2312yes—
51C22×Dic17C2311yescharacteristic
52C22×Dic17C2312yes—
53C22×Dic17C2311yescharacteristic
54C2×D68—22no—
55C2×D68—24no—
56C2×C4×D17—24no—
57C2×C4×D17—24no—
58C2×C4×D17—24no—
59C2×C4×D17—24no—
60C2×C4×D17D414yes—
61C4⋊C4×C17—22no—
62C4⋊C4×C17—24no—
63C22⋊C4×C17—24no—
64C22⋊C4×C17—22no—
65C22⋊Dic17—24no—
66#63227—22no—
67#63227—24no—
68#63227—24no—
69#63227D418yescomplemented
70C4⋊Dic17—24no—
71C4⋊Dic17—22no—
72#63225—24no—
73#63225—22no—
74#63225—24no—
75#63225—24no—
76C4×Dic17—24no—
77C23×C17—22no—
78C23×C17—21no—
79C23×C17—22no—
80C23×C17—21no—
81C23×C17—21no—
82C23×C17C2411yescharacteristic
83C22×D17—22no—
84C22×D17—21no—
85C22×D17—21no—
86C22×D17—21no—
87C22×D17—22no—
88C22×D17—22no—
89C22×D17—21no—
90C22×D17—21no—
91C22×D17C2×D412yes—
92C22×D17C2×D414yes—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—21no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17C2×D412yes—
102C2×C4×C17C2×D414yes—
103C22⋊D17—44no—
104C22⋊D17—44no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—21no—
112C2×Dic17C2×D414yes—
113C2×Dic17C2×D412yes—
114C2×Dic17#4114yes—
115D68—44no—
116C4×D17—28no—
117C4×D17—48no—
118#2501—171nomaximal, Sylow
119C22×C17—41no—
120C22×C17—41no—
121C22×C17—41no—
122C22×C17—42no—
123C22×C17—42no—
124C22×C17C22×D411yescharacteristic
125C22×C17C22×D412yes—
126C22×C17C2×#4112yes—
127C22×C172−1+411yescharacteristic
128C22×C172−1+411yescharacteristic
129C2×D17—22no—
130C2×D17—22no—
131C2×D17—24no—
132C2×D17—24no—
133C2×D17—41no—
134C2×D17—41no—
135C2×D17—41no—
136C2×D17—42no—
137C2×D17—42no—
138C2×D17#12714yes—
139C4×C17—22no—
140C4×C17—24no—
141C4×C17—42no—
142C4×C17—41no—
143Dic17—24no—
144Dic17—42no—
145Dic17—41no—
146Dic17#12314yescomplemented
147C2×#123—172no—
148C2×#123—172no—
149C2×#121—171no—
150C22×C4⋊C4—171no—
151#399—171no—
152#399—171no—
153#399—171no—
154#399D1711yescharacteristic, complemented
155#390—172no—
156#384—172no—
157C4×C22⋊C4—171no—
158C2×C17—81no—
159C2×C17C2×#12311yescharacteristic
160C2×C17C2×#12711yescharacteristic
161C2×C17#53611yescharacteristic
162C2×C17#54712yes—
163C2×C17#54712yes—
164D17—24no—
165D17—81no—
166C22×D4—171no—
167C22×D4—171no—
168C22×D4—171no—
169C23×C4—171no—
170C23×C4—172no—
171#123—348no—
172#123—348no—
173#121—344no—
174C2×C4⋊C4—172no—
175C2×C4⋊C4—171no—
176C2×C4⋊C4—171no—
177C2×C4⋊C4—172no—
178C2×C4⋊C4—172no—
179C2×C4⋊C4—172no—
180C2×C4⋊C4—344no—
181C2×C4⋊C4C2×D1711yescharacteristic
182C2×C4⋊C4C2×D1712yes—
183C2×C22⋊C4—172no—
184C2×C22⋊C4—172no—
185C2×C22⋊C4—171no—
186C2×C22⋊C4—172no—
187C2×C22⋊C4—172no—
188C2×C22⋊C4C2×D1711yescharacteristic
189C2×C22⋊C4C2×D1712yes—
190C2×C42—172no—
191#95—171no—
192#95C2×D1711yescharacteristic
193C17#250111yescharacteristic, complemented, Sylow
194C24—171no—
195C24—171no—
196C24C22×D1711yescharacteristic
197C2×D4—342no—
198C2×D4—344no—
199C2×D4—342no—
200C2×D4—344no—
201C2×D4—342no—
202C2×D4—344no—
203C22×C4—171no—
204C22×C4—172no—
205C22×C4—174no—
206C22×C4—172no—
207C22×C4—171no—
208C22×C4—344no—
209C22×C4—344no—
210C22×C4—344no—
211C22×C4—344no—
212C22×C4C22×D1712yes—
213C22×C4C22×D1711yescharacteristic
214C22×C4C22×D1711yescharacteristic
215C22×C4C22×D1712yes—
216C4⋊C4—22no—
217C4⋊C4—24no—
218C4⋊C4—344no—
219C4⋊C4—342no—
220C4⋊C4—344no—
221C4⋊C4—344no—
222C4⋊C4—342no—
223C4⋊C4—344no—
224C22⋊C4—22no—
225C22⋊C4—24no—
226C22⋊C4—178no—
227C22⋊C4—344no—
228C22⋊C4—344no—
229C22⋊C4—342no—
230C22⋊C4—344no—
231C42—344no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—21no—
237C23—174no—
238C23—172no—
239C23—341no—
240C23—342no—
241C23—341no—
242C23—342no—
243C23—341no—
244C23—341no—
245C23—342no—
246C23—341no—
247C23C23×D1711yescharacteristic
248D4—684no—
249D4—684no—
250D4—684no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—172no—
260C2×C4—174no—
261C2×C4—174no—
262C2×C4—348no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—342no—
266C2×C4—342no—
267C2×C4—341no—
268C2×C4—342no—
269C2×C4—342no—
270C2×C4—688no—
271C2×C4D4×D1714yes—
272C2×C4D4×D1712yes—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—174no—
279C22—344no—
280C22—344no—
281C22—342no—
282C22—342no—
283C22—682no—
284C22—682no—
285C22—681no—
286C22—681no—
287C22—681no—
288C22C2×D4×D1712yes—
289C22C2×D4×D1711yescharacteristic
290C22#1058769411yescharacteristic
291C22#41×D1712yes—
292C22#1058769711yescharacteristic
293C4—24no—
294C4—22no—
295C4—42no—
296C4—41no—
297C4—174no—
298C4—344no—
299C4—682no—
300C4—681no—
301C2—81no—
302C2—344no—
303C2—1361no—
304C2#123×D1711yescharacteristic
305C2#127×D1711yescharacteristic
306C2#1176338211yescharacteristic
307C2#1176345712yes—
308C2#1176346112yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#399C2
prime radicalC2×#10587570C2
hypercenter#399D17
hypocenterC17#2501
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#10587580C2semidirect
2. C2×#10587569C2semidirect
4. C2×C4⋊C4×D17C2semidirect
5. #11762746C2semidirect
6. #11762745C2semidirect
7. #11762744C2semidirect
8. #11762709C2semidirect
9. #11762678C2semidirect
10. #11762656C2semidirect
11. C17×#399C2semidirect
69. #63227D4semidirect
146. Dic17#123semidirect
154. #399D17semidirect
193. C17#2501semidirect

Maximal subgroups

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

subgroupindex
C2×#105875802
C2×#105875692
C2×#105875702
C2×C4⋊C4×D172
#117627462
#117627452
#117627442
#117627092
#117626782
#117626562
C17×#3992
#250117