Finite Group Databasebeta

← #14532167

Subgroups — #14532167

Order 2176.

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

normal subgroups only

This group has 786 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×#10587572C211yesmaximal, characteristic, complemented
2C2×#10587568C211yesmaximal, characteristic
3C2×#10587563C211yesmaximal, characteristic, complemented
4C22×#63227C211yesmaximal, characteristic, complemented
5#11762731C212yesmaximal, complemented
6#11762727C212yesmaximal, complemented
7#11762710C212yesmaximal, complemented
8#11762701C212yesmaximal, complemented
9#11762688C211yesmaximal, characteristic, complemented
10#11762664C211yesmaximal, characteristic, complemented
11C17×#385C211yesmaximal, characteristic, complemented
12C24×D17C2211yescharacteristic
13C23×C4×C17C2211yescharacteristic
14C22×C22⋊D17C2211yescharacteristic
15C23×Dic17C2211yescharacteristic
16C22×D68C2212yes—
17C2×C4⋊C4×C17C2212yescomplemented
18C2×C22⋊C4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C22⋊Dic17C2211yescharacteristic
21C2×#63227—24no—
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×#63227C2212yes—
27C2×#63227C2211yescharacteristic, complemented
28C2×#63227C2211yescharacteristic, complemented
29C2×#63227C2212yescomplemented
30C2×C4⋊Dic17C2211yescharacteristic, complemented
31C2×C4⋊Dic17C2211yescharacteristic, complemented
32C2×#63225C2212yescomplemented
33C2×C4×Dic17C2212yescomplemented
34#10587572—24no—
35#10587568—24no—
36#10587563—24no—
37C17×#95C2212yescomplemented
38#10587502C2212yescomplemented
39C24×C17C2311yescharacteristic
40C23×D17—22no—
41C23×D17—24no—
42C23×D17—22no—
43C23×D17—22no—
44C23×D17—42no—
45C23×D17C2311yescharacteristic
46C23×D17C2312yes—
47C22×C4×C17—22no—
48C22×C4×C17—42no—
49C22×C4×C17C2312yes—
50C22×C4×C17C2311yescharacteristic
51C22×C4×C17C2311yescharacteristic
52C22×C4×C17C2312yes—
53C2×C22⋊D17—22no—
54C2×C22⋊D17—22no—
55C2×C22⋊D17—22no—
56C22×Dic17—22no—
57C22×Dic17—22no—
58C22×Dic17D414yescomplemented
59C22×Dic17C2311yescharacteristic
60C22×Dic17C2312yes—
61C22×Dic17C2311yescharacteristic
62C22×Dic17C2311yescharacteristic
63C2×D68—24no—
64C2×D68—24no—
65C2×D68—42no—
66C4⋊C4×C17—42no—
67C22⋊C4×C17—22no—
68C22⋊C4×C17D414yes—
69C22⋊Dic17—22no—
70#63227—24no—
71#63227—24no—
72#63227—22no—
73#63227—22no—
74#63227—24no—
75#63227—24no—
76#63227—42no—
77#63227—42no—
78C4⋊Dic17—22no—
79C4⋊Dic17—22no—
80#63225—24no—
81C4×Dic17—24no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17C2×D412yes—
85C23×C17C2×D412yes—
86C23×C17#4112yes—
87C23×C17C2411yescharacteristic
88C22×D17—28no—
89C22×D17—22no—
90C22×D17—22no—
91C22×D17—22no—
92C22×D17—22no—
93C22×D17—22no—
94C22×D17—22no—
95C22×D17—21no—
96C22×D17—21no—
97C22×D17—22no—
98C22×D17—21no—
99C22×D17—22no—
100C22×D17—48no—
101C22×D17—48no—
102C22×D17—44no—
103C22×D17—44no—
104C22×D17#4114yes—
105C2×C4×C17—22no—
106C2×C4×C17—22no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
111C2×C4×C17—22no—
112C2×C4×C17—22no—
113C2×C4×C17—22no—
114C2×C4×C17—44no—
115C2×C4×C17C2×D414yes—
116C22⋊D17—44no—
117C2×Dic17—28no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—22no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—22no—
127C2×Dic17—22no—
128C2×Dic17C2×D412yes—
129C2×Dic17C2×D412yes—
130D68—48no—
131#2411—171nomaximal, Sylow
132C22×C17—22no—
133C22×C17—22no—
134C22×C17—24no—
135C22×C17—24no—
136C22×C17#12114yes—
137C22×C17C22×D411yescharacteristic
138C22×C17C22×D411yescharacteristic
139C22×C17C2×#4111yescharacteristic
140C22×C17C2×#4112yes—
141C22×C17H(5,𝔽2)12yes—
142C2×D17—24no—
143C2×D17—24no—
144C2×D17—24no—
145C2×D17—416no—
146C2×D17—42no—
147C2×D17—42no—
148C2×D17—42no—
149C2×D17—42no—
150C2×D17—41no—
151C2×D17—41no—
152C2×D17—41no—
153C2×D17—42no—
154C2×D17—42no—
155C4×C17—24no—
156C4×C17—41no—
157C4×C17—41no—
158C4×C17—42no—
159Dic17—22no—
160Dic17—22no—
161Dic17—41no—
162Dic17—42no—
163C2×#123—171no—
164C2×#120—171no—
165C2×#117—171no—
166C22×C22⋊C4—171no—
167#396—172no—
168#394—172no—
169#390—172no—
170#388—172no—
171#386—171no—
172#385D1711yescharacteristic, complemented
173#379—171no—
174C2×C17—24no—
175C2×C17C2×#12111yescharacteristic
176C2×C17#52511yescharacteristic
177C2×C17#53511yescharacteristic
178C2×C17#54612yes—
179C2×C17#55112yes—
180D17—44no—
181D17—81no—
182C25—171no—
183C22×D4—171no—
184C22×D4—172no—
185C23×C4—171no—
186C23×C4C2×D1711yescharacteristic
187#123—344no—
188#120—344no—
189#117—344no—
190C2×C4⋊C4—172no—
191C2×C4⋊C4—171no—
192C2×C4⋊C4—171no—
193C2×C4⋊C4C2×D1712yescomplemented
194C2×C22⋊C4—171no—
195C2×C22⋊C4—172no—
196C2×C22⋊C4—172no—
197C2×C22⋊C4—172no—
198C2×C22⋊C4—171no—
199C2×C22⋊C4—171no—
200C2×C22⋊C4—172no—
201C2×C22⋊C4—172no—
202C2×C22⋊C4—172no—
203C2×C22⋊C4—344no—
204C2×C22⋊C4C2×D1711yescharacteristic
205C2×C22⋊C4C2×D1711yescharacteristic
206C2×C42—172no—
207#95—172no—
208#95C2×D1712yescomplemented
209C17#241111yescharacteristic, complemented, Sylow
210C24—172no—
211C24—171no—
212C24—344no—
213C24—342no—
214C24—342no—
215C24—342no—
216C24—682no—
217C24C22×D1711yescharacteristic
218C2×D4—344no—
219C2×D4—342no—
220C2×D4—342no—
221C2×D4—342no—
222C2×D4—344no—
223C2×D4—682no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—174no—
227C22×C4—171no—
228C22×C4—171no—
229C22×C4—171no—
230C22×C4—172no—
231C22×C4—342no—
232C22×C4—342no—
233C22×C4C22×D1712yes—
234C22×C4C22×D1711yescharacteristic
235C22×C4C22×D1711yescharacteristic
236C22×C4C22×D1712yes—
237C4⋊C4—42no—
238C4⋊C4—342no—
239C4⋊C4—342no—
240C4⋊C4—344no—
241C22⋊C4—22no—
242C22⋊C4—344no—
243C22⋊C4—344no—
244C22⋊C4—342no—
245C22⋊C4—342no—
246C22⋊C4—344no—
247C22⋊C4—342no—
248C22⋊C4—344no—
249C22⋊C4—682no—
250C22⋊C4—682no—
251C22⋊C4D6814yes—
252C42—344no—
253C23—22no—
254C23—22no—
255C23—174no—
256C23—348no—
257C23—342no—
258C23—342no—
259C23—342no—
260C23—342no—
261C23—342no—
262C23—342no—
263C23—341no—
264C23—341no—
265C23—342no—
266C23—341no—
267C23—342no—
268C23—688no—
269C23—688no—
270C23—684no—
271C23—684no—
272C23C2×D6812yes—
273C23D4×D1712yes—
274C23#6325612yes—
275C23C23×D1711yescharacteristic
276D4—688no—
277D4—684no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—172no—
289C2×C4—172no—
290C2×C4—341no—
291C2×C4—341no—
292C2×C4—341no—
293C2×C4—341no—
294C2×C4—348no—
295C2×C4—342no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—342no—
299C2×C4—341no—
300C2×C4—342no—
301C2×C4C2×D6814yes—
302C22—24no—
303C22—22no—
304C22—22no—
305C22—24no—
306C22—344no—
307C22—344no—
308C22—344no—
309C22—6816no—
310C22—682no—
311C22—682no—
312C22—682no—
313C22—682no—
314C22—682no—
315C22—681no—
316C22—681no—
317C22—681no—
318C22—682no—
319C22#1058758114yes—
320C22C22×D6811yescharacteristic
321C22C2×D4×D1711yescharacteristic
322C22#1058769112yes—
323C22C2×#6325611yescharacteristic
324C22#41×D1712yes—
325C4—24no—
326C4—41no—
327C4—42no—
328C4—41no—
329C4—342no—
330C4—342no—
331C4—681no—
332C4—682no—
333C2—24no—
334C2—684no—
335C2—1361no—
336C2C2×#1058758111yescharacteristic
337C2#1176331711yescharacteristic
338C2#1176337411yescharacteristic
339C2#1176345412yes—
340C2#1176348612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#385C2
prime radicalC2×#10587568C2
hypercenter#385D17
hypocenterC17#2411
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#10587572C2semidirect
3. C2×#10587563C2semidirect
4. C22×#63227C2semidirect
5. #11762731C2semidirect
6. #11762727C2semidirect
7. #11762710C2semidirect
8. #11762701C2semidirect
9. #11762688C2semidirect
10. #11762664C2semidirect
11. C17×#385C2semidirect
17. C2×C4⋊C4×C17C22semidirect
27. C2×#63227C22semidirect
28. C2×#63227C22semidirect
29. C2×#63227C22semidirect
30. C2×C4⋊Dic17C22semidirect
31. C2×C4⋊Dic17C22semidirect
32. C2×#63225C22semidirect
33. C2×C4×Dic17C22semidirect
37. C17×#95C22semidirect
38. #10587502C22semidirect
58. C22×Dic17D4semidirect
172. #385D17semidirect
193. C2×C4⋊C4C2×D17semidirect
208. #95C2×D17semidirect
209. C17#2411semidirect

Maximal subgroups

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

subgroupindex
C2×#105875722
C2×#105875682
C2×#105875632
C22×#632272
#117627312
#117627272
#117627102
#117627012
#117626882
#117626642
C17×#3852
#241117