Finite Group Databasebeta

← #14535002

Subgroups — #14535002

Order 2176.

932 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 932 conjugacy classes of subgroups — too many to draw. Only its 157 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×#10587626C211yesmaximal, characteristic, complemented
2C2×#10587616C212yesmaximal, complemented
3C2×#10587568C212yesmaximal, complemented
4C2×#10587623C211yesmaximal, characteristic, complemented
5#11762725C212yesmaximal, complemented
6#11762719C212yesmaximal, complemented
7#11762718C212yesmaximal, complemented
8#11762701C212yesmaximal, complemented
9C17×#386C211yesmaximal, characteristic, complemented
10C24×D17C2211yescharacteristic
11C22×D4×C17C2211yescharacteristic
12C23×C4×C17C2211yescharacteristic, complemented
13C22×C22⋊D17C2212yes—
14C22×C22⋊D17C2212yes—
15C22×C22⋊D17C2212yescomplemented
16C22×D68C2212yes—
17C22×D68C2212yes—
18C22×D68C2211yescharacteristic, complemented
19C2×C22⋊C4×C17C2212yes—
20C2×C22⋊C4×C17C2211yescharacteristic, complemented
21C2×C22⋊Dic17C2211yescharacteristic
22#10587626—24no—
23#10587623—24no—
24#10587616—28no—
25C2×#63227C2212yes—
26C2×#63227C2212yes—
27C2×#63227C2212yes—
28C2×#63227C2212yescomplemented
29C2×#63227C2212yescomplemented
30C2×C4⋊Dic17C2212yescomplemented
31C2×C4×Dic17C2211yescharacteristic, complemented
32#10587568—28no—
33C17×#95C2212yescomplemented
34#10587502C2212yescomplemented
35C24×C17C2311yescharacteristic
36C24×C17C2311yescharacteristic
37C23×D17—22no—
38C23×D17—24no—
39C23×D17—41no—
40C23×D17—41no—
41C23×D17—42no—
42C23×D17C2312yes—
43C23×D17C2312yes—
44C2×D4×C17—41no—
45C2×D4×C17—41no—
46C2×D4×C17D414yescomplemented
47C22×C4×C17—24no—
48C22×C4×C17—41no—
49C22×C4×C17C2312yes—
50C22×C4×C17C2312yes—
51C22×C4×C17C2311yescharacteristic
52C2×C22⋊D17—24no—
53C2×C22⋊D17—24no—
54C2×C22⋊D17—24no—
55C2×C22⋊D17—24no—
56C2×C22⋊D17—24no—
57C2×C22⋊D17—24no—
58C2×C22⋊D17—42no—
59C2×C22⋊D17—42no—
60C2×C22⋊D17—42no—
61C22×Dic17C2312yes—
62C22×Dic17C2312yes—
63C2×D68—24no—
64C2×D68—24no—
65C2×D68—24no—
66C2×D68—22no—
67C2×D68—42no—
68C2×D68—42no—
69C2×D68—42no—
70C2×D68—41no—
71C2×D68—41no—
72C22⋊C4×C17—24no—
73C22⋊C4×C17—41no—
74C22⋊Dic17—22no—
75#63227—24no—
76#63227—24no—
77#63227—24no—
78#63227—42no—
79#63227D418yescomplemented
80C4⋊Dic17—24no—
81C4×Dic17—22no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17—21no—
85C23×C17—22no—
86C23×C17—22no—
87C23×C17—21no—
88C23×C17—21no—
89C23×C17—21no—
90C23×C17C2×D412yes—
91C23×C17C2×D412yes—
92C23×C17C2411yescharacteristic
93C22×D17—22no—
94C22×D17—22no—
95C22×D17—22no—
96C22×D17—22no—
97C22×D17—22no—
98C22×D17—22no—
99C22×D17—22no—
100C22×D17—22no—
101C22×D17—22no—
102C22×D17—22no—
103C22×D17—22no—
104C22×D17—22no—
105C22×D17—22no—
106C22×D17—44no—
107C22×D17—48no—
108C22×D17—48no—
109C22×D17—44no—
110C22×D17—44no—
111C22×D17C2×D414yes—
112D4×C17—44no—
113C2×C4×C17—22no—
114C2×C4×C17—22no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—22no—
118C2×C4×C17—22no—
119C2×C4×C17—22no—
120C2×C4×C17—44no—
121C2×C4×C17C2×D412yes—
122C2×C4×C17C2×D414yes—
123C22⋊D17—48no—
124C22⋊D17—48no—
125C22⋊D17—48no—
126C2×Dic17—22no—
127C2×Dic17—22no—
128C2×Dic17—22no—
129C2×Dic17—22no—
130C2×Dic17—22no—
131C2×Dic17C2×D414yes—
132D68—48no—
133D68—44no—
134D68—48no—
135#2669—171nomaximal, Sylow
136C22×C17—22no—
137C22×C17—22no—
138C22×C17—24no—
139C22×C17—24no—
140C22×C17—42no—
141C22×C17—42no—
142C22×C17—41no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—41no—
146C22×C17C22×D411yescharacteristic
147C22×C17C22×D412yes—
148C22×C17H(5,𝔽2)11yescharacteristic
149C22×C17H(5,𝔽2)11yescharacteristic
150C22×C17H(5,𝔽2)12yes—
151C2×D17—24no—
152C2×D17—24no—
153C2×D17—24no—
154C2×D17—416no—
155C2×D17—42no—
156C2×D17—42no—
157C2×D17—42no—
158C2×D17—42no—
159C2×D17—42no—
160C2×D17—42no—
161C2×D17—42no—
162C2×D17—42no—
163C2×D17—42no—
164C2×D17—42no—
165C2×D17—42no—
166C4×C17—22no—
167C4×C17—24no—
168C4×C17—42no—
169Dic17—24no—
170Dic17—42no—
171C2×#127—171no—
172C2×#121—172no—
173C2×#120—171no—
174C2×#120—172no—
175#394—172no—
176#392—172no—
177#392—172no—
178#388—172no—
179#386D1711yescharacteristic, complemented
180C2×C17—42no—
181C2×C17—42no—
182C2×C17#53411yescharacteristic
183C2×C17#53512yes—
184C2×C17D4211yescharacteristic
185C2×C17D4212yes—
186C2×C17#56011yescharacteristic
187D17—44no—
188D17—82no—
189C25—171no—
190C22×D4—172no—
191C22×D4—172no—
192C22×D4—172no—
193C22×D4—172no—
194C22×D4—172no—
195C22×D4—171no—
196C22×D4C2×D1711yescharacteristic
197C23×C4C2×D1711yescharacteristic, complemented
198#127—344no—
199#121—348no—
200#120—348no—
201#120—344no—
202C2×C4⋊C4—172no—
203C2×C22⋊C4—172no—
204C2×C22⋊C4—171no—
205C2×C22⋊C4—172no—
206C2×C22⋊C4—172no—
207C2×C22⋊C4—172no—
208C2×C22⋊C4—172no—
209C2×C22⋊C4C2×D1712yes—
210C2×C22⋊C4C2×D1711yescharacteristic, complemented
211C2×C42—171no—
212#95—172no—
213#95C2×D1712yescomplemented
214C17#266911yescharacteristic, complemented, Sylow
215C24—172no—
216C24—172no—
217C24—344no—
218C24—342no—
219C24—681no—
220C24—681no—
221C24—682no—
222C24C22×D1711yescharacteristic
223C24C22×D1711yescharacteristic
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—344no—
227C2×D4—344no—
228C2×D4—344no—
229C2×D4—344no—
230C2×D4—344no—
231C2×D4—344no—
232C2×D4—344no—
233C2×D4—344no—
234C2×D4—342no—
235C2×D4—344no—
236C2×D4—682no—
237C2×D4—681no—
238C2×D4—681no—
239C2×D4—682no—
240C2×D4—682no—
241C2×D4—682no—
242C2×D4—682no—
243C2×D4—682no—
244C2×D4D6814yescomplemented
245C22×C4—24no—
246C22×C4—41no—
247C22×C4—172no—
248C22×C4—172no—
249C22×C4C22×D1712yes—
250C22×C4C22×D1711yescharacteristic
251C22×C4C22×D1712yes—
252C4⋊C4—344no—
253C22⋊C4—24no—
254C22⋊C4—41no—
255C22⋊C4—178no—
256C22⋊C4—344no—
257C22⋊C4—342no—
258C22⋊C4—344no—
259C22⋊C4—344no—
260C22⋊C4—682no—
261C42—342no—
262C23—22no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—22no—
269C23—22no—
270C23—174no—
271C23—342no—
272C23—342no—
273C23—342no—
274C23—342no—
275C23—342no—
276C23—342no—
277C23—342no—
278C23—342no—
279C23—342no—
280C23—342no—
281C23—342no—
282C23—342no—
283C23—342no—
284C23—688no—
285C23—684no—
286C23—684no—
287C23—688no—
288C23—684no—
289C23C2×D6812yes—
290C23C2×D6812yes—
291C23C23×D1711yescharacteristic
292D4—44no—
293D4—688no—
294D4—684no—
295D4—688no—
296D4—688no—
297D4—688no—
298D4—688no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—44no—
307C2×C4—174no—
308C2×C4—342no—
309C2×C4—342no—
310C2×C4—342no—
311C2×C4—342no—
312C2×C4—342no—
313C2×C4C2×D6812yes—
314C2×C4D4×D1714yes—
315C22—22no—
316C22—22no—
317C22—24no—
318C22—24no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—344no—
326C22—344no—
327C22—344no—
328C22—6816no—
329C22—682no—
330C22—682no—
331C22—682no—
332C22—682no—
333C22—682no—
334C22—682no—
335C22—682no—
336C22—682no—
337C22—682no—
338C22—682no—
339C22—682no—
340C22C22×D6811yescharacteristic
341C22C2×D4×D1712yes—
342C22#1058769111yescharacteristic
343C22#1058769111yescharacteristic
344C22#1058769612yes—
345C4—22no—
346C4—24no—
347C4—42no—
348C4—344no—
349C4—682no—
350C2—42no—
351C2—42no—
352C2—684no—
353C2—1362no—
354C2#1176337111yescharacteristic
355C2#1176337512yes—
356C2#1176344111yescharacteristic
357C2D4×D6812yes—
358C2#1176354911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#386C2
hypercenter#386D17
hypocenterC17#2669
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#10587626C2semidirect
2. C2×#10587616C2semidirect
3. C2×#10587568C2semidirect
4. C2×#10587623C2semidirect
5. #11762725C2semidirect
6. #11762719C2semidirect
7. #11762718C2semidirect
8. #11762701C2semidirect
9. C17×#386C2semidirect
12. C23×C4×C17C22semidirect
15. C22×C22⋊D17C22semidirect
18. C22×D68C22semidirect
20. C2×C22⋊C4×C17C22semidirect
28. C2×#63227C22semidirect
29. C2×#63227C22semidirect
30. C2×C4⋊Dic17C22semidirect
31. C2×C4×Dic17C22semidirect
33. C17×#95C22semidirect
34. #10587502C22semidirect
46. C2×D4×C17D4semidirect
79. #63227D4semidirect
179. #386D17semidirect
197. C23×C4C2×D17semidirect
210. C2×C22⋊C4C2×D17semidirect
213. #95C2×D17semidirect
214. C17#2669semidirect
244. C2×D4D68semidirect

Maximal subgroups

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

subgroupindex
C2×#105876262
C2×#105876162
C2×#105875682
C2×#105876232
#117627252
#117627192
#117627182
#117627012
C17×#3862
#266917