Finite Group Databasebeta

← #14533709

Subgroups — #14533709

Order 2176.

900 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 159 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×#10587569C212yesmaximal, complemented
2C2×#10587624C211yesmaximal, characteristic, complemented
3C2×#10587623C212yesmaximal, complemented
4C2×C22⋊C4×D17C211yesmaximal, characteristic, complemented
5#11762746C211yesmaximal, characteristic, complemented
6#11762745C211yesmaximal, characteristic, complemented
7#11762717C211yesmaximal, characteristic, complemented
8#11762712C212yesmaximal, complemented
9#11762709C212yesmaximal, complemented
10#11762656C211yesmaximal, characteristic, complemented
11C17×#392C211yesmaximal, characteristic, complemented
12C24×D17C2211yescharacteristic
13C22×D4×C17C2212yes—
14C22×D4×C17C2211yescharacteristic, complemented
15C22×C22⋊D17C2212yes—
16C22×C22⋊D17C2212yes—
17C22×C4×D17C2211yescharacteristic
18C22×C4×D17C2211yescharacteristic
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C22⋊C4×C17C2212yescomplemented
21C2×C22⋊Dic17C2211yescharacteristic
22C2×C22⋊Dic17C2212yes—
23C2×C22⋊Dic17C2212yescomplemented
24C2×C22⋊Dic17C2212yescomplemented
25#10587624—24no—
26#10587623—28no—
27C2×#63227C2211yescharacteristic
28C2×#63227C2212yes—
29C2×#63227C2211yescharacteristic
30C2×#63227C2212yes—
31C2×C4⋊Dic17C2212yescomplemented
32C2×C4⋊Dic17C2211yescharacteristic, complemented
33C2×#63225C2212yescomplemented
34C2×C4×Dic17C2212yescomplemented
35#10587569—28no—
36C22⋊C4×D17—24no—
37C17×#95C2211yescharacteristic, complemented
38#10587502C2211yescharacteristic, complemented
39C24×C17C2312yes—
40C24×C17C2311yescharacteristic
41C23×D17—22no—
42C23×D17—22no—
43C23×D17—22no—
44C23×D17—24no—
45C23×D17—42no—
46C23×D17C2311yescharacteristic
47C23×D17C2311yescharacteristic
48C2×D4×C17—24no—
49C2×D4×C17—24no—
50C2×D4×C17—22no—
51C2×D4×C17—42no—
52C2×D4×C17—42no—
53C22×C4×C17C2311yescharacteristic
54C22×C4×C17C2311yescharacteristic
55C22×C4×C17C2312yes—
56C2×C22⋊D17—24no—
57C2×C22⋊D17—24no—
58C2×C22⋊D17—24no—
59C2×C22⋊D17—24no—
60C2×C22⋊D17—24no—
61C2×C22⋊D17—42no—
62C22×Dic17C2311yescharacteristic
63C22×Dic17C2312yes—
64C22×Dic17C2311yescharacteristic
65C22×Dic17C2312yes—
66C2×C4×D17—22no—
67C2×C4×D17—24no—
68C2×C4×D17—42no—
69C2×C4×D17D414yescomplemented
70C22⋊C4×C17—22no—
71C22⋊C4×C17—24no—
72C22⋊Dic17—22no—
73C22⋊Dic17—24no—
74C22⋊Dic17—24no—
75C22⋊Dic17—24no—
76#63227—22no—
77#63227—24no—
78#63227—22no—
79#63227D418yescomplemented
80C4⋊Dic17—24no—
81C4⋊Dic17—22no—
82#63225—24no—
83C4×Dic17—42no—
84C23×C17—21no—
85C23×C17—22no—
86C23×C17—21no—
87C23×C17—22no—
88C23×C17—22no—
89C23×C17—22no—
90C23×C17—22no—
91C23×C17—22no—
92C23×C17—22no—
93C23×C17—22no—
94C23×C17—22no—
95C23×C17#4112yes—
96C23×C17C2411yescharacteristic
97C22×D17—21no—
98C22×D17—21no—
99C22×D17—22no—
100C22×D17—28no—
101C22×D17—22no—
102C22×D17—21no—
103C22×D17—21no—
104C22×D17—22no—
105C22×D17—48no—
106C22×D17—48no—
107C22×D17—44no—
108C22×D17—44no—
109C22×D17C2×D412yes—
110C22×D17C2×D414yes—
111C22×D17#4112yes—
112D4×C17—48no—
113D4×C17—44no—
114C2×C4×C17—22no—
115C2×C4×C17—22no—
116C2×C4×C17—22no—
117C2×C4×C17—22no—
118C2×C4×C17—21no—
119C2×C4×C17C2×D412yes—
120C2×C4×C17C2×D414yes—
121C22⋊D17—48no—
122C22⋊D17—48no—
123C2×Dic17—22no—
124C2×Dic17—22no—
125C2×Dic17—22no—
126C2×Dic17—22no—
127C2×Dic17—21no—
128C2×Dic17—22no—
129C2×Dic17—22no—
130C2×Dic17—22no—
131C2×Dic17C2×D412yes—
132C2×Dic17C2×D414yes—
133C4×D17—44no—
134C4×D17—44no—
135#2539—171nomaximal, Sylow
136C22×C17—22no—
137C22×C17—24no—
138C22×C17—41no—
139C22×C17—41no—
140C22×C17—42no—
141C22×C17—42no—
142C22×C17—42no—
143C22×C17—42no—
144C22×C17—42no—
145C22×C17—42no—
146C22×C17—42no—
147C22×C17—42no—
148C22×C17C22×D411yescharacteristic
149C22×C17C22×D412yes—
150C22×C17C2×#4111yescharacteristic
151C22×C17C2×#4111yescharacteristic
152C22×C17H(5,𝔽2)12yes—
153C2×D17—22no—
154C2×D17—22no—
155C2×D17—24no—
156C2×D17—24no—
157C2×D17—22no—
158C2×D17—24no—
159C2×D17—416no—
160C2×D17—41no—
161C2×D17—41no—
162C2×D17—42no—
163C2×D17#12714yes—
164C4×C17—22no—
165C4×C17—24no—
166C4×C17—41no—
167Dic17—22no—
168Dic17—24no—
169Dic17—41no—
170Dic17—42no—
171C2×#123—172no—
172C2×#121—171no—
173C2×#120—172no—
174C22×C22⋊C4—171no—
175#399—171no—
176#399—171no—
177#392—171no—
178#392D1711yescharacteristic, complemented
179#390—172no—
180#390—172no—
181C4×C22⋊C4—171no—
182C2×C17—42no—
183C2×C17—82no—
184C2×C17#52511yescharacteristic
185C2×C17C2×#12711yescharacteristic
186C2×C17#53411yescharacteristic
187C2×C17#54612yes—
188C2×C17#54612yes—
189D17—24no—
190D17—42no—
191C25—171no—
192C22×D4—172no—
193C22×D4—172no—
194C22×D4C2×D1712yes—
195C22×D4C2×D1711yescharacteristic, complemented
196C23×C4—171no—
197C23×C4—171no—
198#123—348no—
199#121—344no—
200#120—348no—
201C2×C4⋊C4—171no—
202C2×C4⋊C4—172no—
203C2×C4⋊C4—172no—
204C2×C22⋊C4—172no—
205C2×C22⋊C4—171no—
206C2×C22⋊C4—172no—
207C2×C22⋊C4—172no—
208C2×C22⋊C4—172no—
209C2×C22⋊C4—171no—
210C2×C22⋊C4—171no—
211C2×C22⋊C4—172no—
212C2×C22⋊C4—344no—
213C2×C22⋊C4C2×D1711yescharacteristic
214C2×C22⋊C4C2×D1712yescomplemented
215C2×C42—172no—
216#95—171no—
217#95C2×D1711yescharacteristic, complemented
218C17#253911yescharacteristic, complemented, Sylow
219C24—171no—
220C24—171no—
221C24—342no—
222C24—342no—
223C24—344no—
224C24—342no—
225C24—682no—
226C24C22×D1712yes—
227C24C22×D1711yescharacteristic
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—344no—
234C2×D4—344no—
235C2×D4—344no—
236C2×D4—344no—
237C2×D4—344no—
238C2×D4—682no—
239C22×C4—172no—
240C22×C4—171no—
241C22×C4—174no—
242C22×C4—171no—
243C22×C4—172no—
244C22×C4—342no—
245C22×C4—344no—
246C22×C4—682no—
247C22×C4C22×D1711yescharacteristic
248C22×C4C22×D1711yescharacteristic
249C22×C4C22×D1712yes—
250C4⋊C4—344no—
251C4⋊C4—342no—
252C4⋊C4—344no—
253C22⋊C4—22no—
254C22⋊C4—24no—
255C22⋊C4—178no—
256C22⋊C4—342no—
257C22⋊C4—342no—
258C22⋊C4—344no—
259C22⋊C4—344no—
260C22⋊C4—344no—
261C22⋊C4—342no—
262C22⋊C4—344no—
263C42—682no—
264C23—21no—
265C23—22no—
266C23—22no—
267C23—21no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—174no—
276C23—172no—
277C23—172no—
278C23—348no—
279C23—341no—
280C23—342no—
281C23—341no—
282C23—341no—
283C23—342no—
284C23—341no—
285C23—342no—
286C23—688no—
287C23—684no—
288C23—688no—
289C23—684no—
290C23#6325412yes—
291C23C23×D1711yescharacteristic
292D4—48no—
293D4—44no—
294D4—688no—
295D4—688no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—174no—
302C2×C4—172no—
303C2×C4—342no—
304C2×C4—342no—
305C2×C4—342no—
306C2×C4—341no—
307C2×C4—342no—
308C2×C4—342no—
309C2×C4—342no—
310C2×C4—342no—
311C2×C4—684no—
312C2×C4—684no—
313C2×C4D4×D1714yes—
314C2×C4D4×D1712yes—
315C22—22no—
316C22—24no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—174no—
328C22—344no—
329C22—342no—
330C22—342no—
331C22—344no—
332C22—344no—
333C22—342no—
334C22—6816no—
335C22—681no—
336C22—681no—
337C22—682no—
338C22C2×D4×D1712yes—
339C22C2×D4×D1711yescharacteristic
340C22C2×#6325411yescharacteristic
341C22#1058769112yes—
342C22#41×D1711yescharacteristic
343C4—22no—
344C4—24no—
345C4—41no—
346C4—344no—
347C4—342no—
348C4—681no—
349C4—682no—
350C2—42no—
351C2—82no—
352C2—344no—
353C2—682no—
354C2#1176331911yescharacteristic
355C2#127×D1711yescharacteristic
356C2#1176336911yescharacteristic
357C2#1176344612yes—
358C2#1176345612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#392C2
hypercenter#392D17
hypocenterC17#2539
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#10587569C2semidirect
2. C2×#10587624C2semidirect
3. C2×#10587623C2semidirect
4. C2×C22⋊C4×D17C2semidirect
5. #11762746C2semidirect
6. #11762745C2semidirect
7. #11762717C2semidirect
8. #11762712C2semidirect
9. #11762709C2semidirect
10. #11762656C2semidirect
11. C17×#392C2semidirect
14. C22×D4×C17C22semidirect
20. C2×C22⋊C4×C17C22semidirect
23. C2×C22⋊Dic17C22semidirect
24. C2×C22⋊Dic17C22semidirect
31. C2×C4⋊Dic17C22semidirect
32. C2×C4⋊Dic17C22semidirect
33. C2×#63225C22semidirect
34. C2×C4×Dic17C22semidirect
37. C17×#95C22semidirect
38. #10587502C22semidirect
69. C2×C4×D17D4semidirect
79. #63227D4semidirect
178. #392D17semidirect
195. C22×D4C2×D17semidirect
214. C2×C22⋊C4C2×D17semidirect
217. #95C2×D17semidirect
218. C17#2539semidirect

Maximal subgroups

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

subgroupindex
C2×#105875692
C2×#105876242
C2×#105876232
C2×C22⋊C4×D172
#117627462
#117627452
#117627172
#117627122
#117627092
#117626562
C17×#3922
#253917