Finite Group Databasebeta

← #14532113

Subgroups — #14532113

Order 2176.

1116 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×D68C211yesmaximal, characteristic
2C2×#10587580C212yesmaximal, complemented
3C2×C22⋊C4×D17C211yesmaximal, characteristic, complemented
4#11762727C212yesmaximal, complemented
5#11762726C212yesmaximal, complemented
6#11762687C212yesmaximal, complemented
7#11762682C211yesmaximal, characteristic, complemented
8#11762664C211yesmaximal, characteristic, complemented
9#11762664C211yesmaximal, characteristic, complemented
10#11762664C211yesmaximal, characteristic, complemented
11C17×#385C211yesmaximal, characteristic, complemented
12C24×D17C2211yescharacteristic
13C24×D17C2211yescharacteristic
14C23×C4×C17C2211yescharacteristic
15C22×D68—24no—
16C22×D68—28no—
17C22×D68C2212yes—
18C22×D68C2212yes—
19C22×C4×D17C2211yescharacteristic, complemented
20C22×C4×D17C2211yescharacteristic, complemented
21C2×C4⋊C4×C17C2212yescomplemented
22C2×C22⋊C4×C17C2211yescharacteristic
23C2×C22⋊C4×C17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2211yescharacteristic
26C2×#63227C2211yescharacteristic, complemented
27C2×#63227C2212yescomplemented
28C2×#63227C2211yescharacteristic, complemented
29C2×#63227C2212yescomplemented
30C2×#63227C2212yescomplemented
31C2×#63227C2212yescomplemented
32C2×#63227C2212yescomplemented
33C2×#63227C2212yescomplemented
34C2×#63225C2212yescomplemented
35#10587580—28no—
36C22⋊C4×D17—24no—
37C17×#95C2212yescomplemented
38#10587502C2212yescomplemented
39C24×C17C2311yescharacteristic
40C23×D17—22no—
41C23×D17—22no—
42C23×D17—22no—
43C23×D17—24no—
44C23×D17—24no—
45C23×D17—22no—
46C23×D17—22no—
47C23×D17—42no—
48C23×D17—42no—
49C23×D17D414yes—
50C23×D17C2311yescharacteristic
51C23×D17C2312yes—
52C23×D17C2311yescharacteristic
53C22×C4×C17—22no—
54C22×C4×C17—24no—
55C22×C4×C17C2311yescharacteristic
56C22×C4×C17C2311yescharacteristic
57C22×C4×C17C2312yes—
58C22×C4×C17C2312yes—
59C22×Dic17C2311yescharacteristic
60C22×Dic17C2311yescharacteristic
61C22×Dic17C2312yes—
62C2×D68—232no—
63C2×D68—24no—
64C2×D68—24no—
65C2×D68—24no—
66C2×D68—24no—
67C2×D68—24no—
68C2×D68D418yes—
69C2×C4×D17—22no—
70C2×C4×D17—22no—
71C2×C4×D17—24no—
72C2×C4×D17—42no—
73C4⋊C4×C17—24no—
74C22⋊C4×C17—22no—
75C22⋊C4×C17—22no—
76C22⋊Dic17—22no—
77C22⋊Dic17—22no—
78#63227—22no—
79#63227—22no—
80#63227—24no—
81#63227—24no—
82#63227—24no—
83#63227—42no—
84#63227—42no—
85#63227—42no—
86#63225—24no—
87C23×C17—22no—
88C23×C17—22no—
89C23×C17C2×D412yes—
90C23×C17#4112yes—
91C23×C17#4112yes—
92C23×C17C2411yescharacteristic
93C22×D17—21no—
94C22×D17—21no—
95C22×D17—22no—
96C22×D17—28no—
97C22×D17—28no—
98C22×D17—22no—
99C22×D17—22no—
100C22×D17—22no—
101C22×D17—21no—
102C22×D17—22no—
103C22×D17—21no—
104C22×D17—22no—
105C22×D17—22no—
106C22×D17—22no—
107C22×D17—22no—
108C22×D17—28no—
109C22×D17—44no—
110C22×D17—48no—
111C22×D17—48no—
112C22×D17—48no—
113C22×D17—48no—
114C22×D17—44no—
115C22×D17—44no—
116C22×D17C2×D412yes—
117C22×D17C2×D414yes—
118C22×D17C2×D412yes—
119C22×D17C2×D414yes—
120C2×C4×C17—28no—
121C2×C4×C17—22no—
122C2×C4×C17—22no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—22no—
126C2×C4×C17—22no—
127C2×C4×C17—22no—
128C2×C4×C17—22no—
129C2×C4×C17—22no—
130C2×C4×C17C2×D414yes—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—22no—
134C2×Dic17—22no—
135C2×Dic17—22no—
136C2×Dic17—22no—
137C2×Dic17—22no—
138D68—216no—
139D68—216no—
140C4×D17—44no—
141C4×D17—44no—
142#2405—171nomaximal, Sylow
143C22×C17—22no—
144C22×C17—22no—
145C22×C17—24no—
146C22×C17—24no—
147C22×C17#12314yescomplemented
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—22no—
160C2×D17—22no—
161C2×D17—24no—
162C2×D17—24no—
163C2×D17—416no—
164C2×D17—416no—
165C2×D17—42no—
166C2×D17—42no—
167C2×D17—42no—
168C2×D17—42no—
169C2×D17—42no—
170C2×D17—42no—
171C2×D17#12014yes—
172C4×C17—24no—
173C4×C17—41no—
174C4×C17—41no—
175C4×C17—42no—
176Dic17—41no—
177Dic17—41no—
178Dic17—42no—
179C23×D4—171no—
180C2×#123—172no—
181C22×C22⋊C4—171no—
182#394—172no—
183#394—172no—
184#386—172no—
185#385—171no—
186#385D1711yescharacteristic, complemented
187#379—171no—
188#379—171no—
189#379—171no—
190C2×C17—24no—
191C2×C17C2×#12011yescharacteristic
192C2×C17C2×#12311yescharacteristic
193C2×C17#53411yescharacteristic
194C2×C17#54612yes—
195C2×C17#54612yes—
196D17—24no—
197D17—42no—
198D17—44no—
199C25—171no—
200C25—171no—
201C22×D4—172no—
202C22×D4—172no—
203C22×D4—348no—
204C22×D4—344no—
205C23×C4—171no—
206C23×C4—171no—
207C23×C4C2×D1711yescharacteristic
208#123—348no—
209C2×C4⋊C4—172no—
210C2×C4⋊C4C2×D1712yescomplemented
211C2×C22⋊C4—171no—
212C2×C22⋊C4—171no—
213C2×C22⋊C4—171no—
214C2×C22⋊C4—171no—
215C2×C22⋊C4—172no—
216C2×C22⋊C4—172no—
217C2×C22⋊C4—172no—
218C2×C22⋊C4—172no—
219C2×C22⋊C4—172no—
220C2×C22⋊C4—172no—
221C2×C22⋊C4—344no—
222C2×C22⋊C4C2×D1711yescharacteristic
223C2×C22⋊C4C2×D1711yescharacteristic
224#95—172no—
225#95C2×D1712yescomplemented
226C17#240511yescharacteristic, complemented, Sylow
227C24—171no—
228C24—172no—
229C24—171no—
230C24—174no—
231C24—344no—
232C24—344no—
233C24—342no—
234C24—342no—
235C24—342no—
236C24—342no—
237C24—342no—
238C24—682no—
239C24—682no—
240C24C22×D1711yescharacteristic
241C2×D4—178no—
242C2×D4—344no—
243C2×D4—344no—
244C2×D4—344no—
245C2×D4—344no—
246C2×D4—3432no—
247C2×D4—344no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—171no—
251C22×C4—172no—
252C22×C4—171no—
253C22×C4—342no—
254C22×C4—342no—
255C22×C4—344no—
256C22×C4—682no—
257C22×C4C22×D1711yescharacteristic
258C22×C4C22×D1712yes—
259C22×C4C22×D1711yescharacteristic
260C22×C4C22×D1712yes—
261C4⋊C4—24no—
262C4⋊C4—344no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—342no—
266C22⋊C4—342no—
267C22⋊C4—344no—
268C22⋊C4—344no—
269C22⋊C4—344no—
270C22⋊C4—342no—
271C22⋊C4—342no—
272C22⋊C4—682no—
273C22⋊C4—682no—
274C22⋊C4—682no—
275C23—22no—
276C23—22no—
277C23—174no—
278C23—172no—
279C23—174no—
280C23—172no—
281C23—348no—
282C23—348no—
283C23—341no—
284C23—341no—
285C23—342no—
286C23—341no—
287C23—342no—
288C23—342no—
289C23—342no—
290C23—342no—
291C23—341no—
292C23—342no—
293C23—342no—
294C23—342no—
295C23—342no—
296C23—348no—
297C23—684no—
298C23—688no—
299C23—688no—
300C23—688no—
301C23—688no—
302C23—684no—
303C23—684no—
304C23#6325212yes—
305C23D4×D1712yes—
306C23#6325612yes—
307C23C23×D1711yescharacteristic
308D4—3416no—
309D4—3416no—
310C2×C4—28no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—342no—
321C2×C4—341no—
322C2×C4—341no—
323C2×C4—342no—
324C2×C4—342no—
325C2×C4—342no—
326C2×C4—342no—
327C2×C4—684no—
328C2×C4—684no—
329C2×C4D4×D1714yes—
330C22—22no—
331C22—22no—
332C22—24no—
333C22—24no—
334C22—174no—
335C22—342no—
336C22—342no—
337C22—344no—
338C22—344no—
339C22—344no—
340C22—342no—
341C22—342no—
342C22—342no—
343C22—344no—
344C22—344no—
345C22—6816no—
346C22—6816no—
347C22—682no—
348C22—682no—
349C22—682no—
350C22—682no—
351C22—682no—
352C22—682no—
353C22#1058758014yescomplemented
354C22C2×#6325211yescharacteristic
355C22C2×D4×D1711yescharacteristic
356C22C2×D4×D1712yes—
357C22C2×#6325611yescharacteristic
358C22#1058769612yes—
359C4—24no—
360C4—41no—
361C4—41no—
362C4—42no—
363C4—681no—
364C4—681no—
365C4—682no—
366C2—24no—
367C2—344no—
368C2—682no—
369C2—684no—
370C2#120×D1711yescharacteristic
371C2C2×#1058758011yescharacteristic
372C2#1176337011yescharacteristic
373C2#1176345012yes—
374C2#1176345512yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#385C2
prime radicalC23×D68C2
hypercenter#385D17
hypocenterC17#2405
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#10587580C2semidirect
3. C2×C22⋊C4×D17C2semidirect
4. #11762727C2semidirect
5. #11762726C2semidirect
6. #11762687C2semidirect
7. #11762682C2semidirect
8. #11762664C2semidirect
9. #11762664C2semidirect
10. #11762664C2semidirect
11. C17×#385C2semidirect
19. C22×C4×D17C22semidirect
20. C22×C4×D17C22semidirect
21. C2×C4⋊C4×C17C22semidirect
26. C2×#63227C22semidirect
27. C2×#63227C22semidirect
28. C2×#63227C22semidirect
29. C2×#63227C22semidirect
30. C2×#63227C22semidirect
31. C2×#63227C22semidirect
32. C2×#63227C22semidirect
33. C2×#63227C22semidirect
34. C2×#63225C22semidirect
37. C17×#95C22semidirect
38. #10587502C22semidirect
147. C22×C17#123semidirect
186. #385D17semidirect
210. C2×C4⋊C4C2×D17semidirect
225. #95C2×D17semidirect
226. C17#2405semidirect
353. C22#10587580semidirect

Maximal subgroups

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

subgroupindex
C23×D682
C2×#105875802
C2×C22⋊C4×D172
#117627272
#117627262
#117626872
#117626822
#117626642
C17×#3852
#240517