Finite Group Databasebeta

← #14532433

Subgroups — #14532433

Order 2176.

1088 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1088 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×C22⋊D17C211yesmaximal, characteristic
2C2×#10587571C212yesmaximal, complemented
3C2×#10587564C211yesmaximal, characteristic, complemented
4#11762725C212yesmaximal, complemented
5#11762724C212yesmaximal, complemented
6#11762709C212yesmaximal, complemented
7#11762665C211yesmaximal, characteristic, complemented
8#11762664C211yesmaximal, characteristic, complemented
9#11762664C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic17C211yesmaximal, characteristic, complemented
11C17×#379C211yesmaximal, characteristic, complemented
12C25×C17C2211yescharacteristic
13C24×D17C2211yescharacteristic
14C22×C22⋊D17—28no—
15C22×C22⋊D17—24no—
16C22×C22⋊D17C2212yes—
17C22×C22⋊D17C2212yes—
18C23×Dic17C2211yescharacteristic
19C22×Dic34C2211yescharacteristic, complemented
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊C4×C17C2212yescomplemented
23C2×C22⋊C4×C17C2212yescomplemented
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2211yescharacteristic
26C2×C22⋊Dic17C2212yescomplemented
27C2×C22⋊Dic17C2212yescomplemented
28C2×#63227C2212yescomplemented
29C2×#63227C2212yescomplemented
30C2×#63227C2212yescomplemented
31C2×#63227C2212yescomplemented
32C2×C4⋊Dic17C2211yescharacteristic, complemented
33C2×#63225C2211yescharacteristic, complemented
34C2×#63225C2211yescharacteristic, complemented
35C2×C4×Dic17C2212yescomplemented
36#10587571—28no—
37#10587564—24no—
38#10587502C2212yescomplemented
39C24×C17—24no—
40C24×C17—22no—
41C24×C17—22no—
42C24×C17—22no—
43C24×C17—42no—
44C24×C17C2312yes—
45C24×C17C2311yescharacteristic
46C23×D17—22no—
47C23×D17—24no—
48C23×D17—22no—
49C23×D17—22no—
50C23×D17—42no—
51C23×D17C2312yes—
52C22×C4×C17C2311yescharacteristic
53C22×C4×C17C2312yes—
54C22×C4×C17C2311yescharacteristic
55C2×C22⋊D17—232no—
56C2×C22⋊D17—24no—
57C2×C22⋊D17—24no—
58C2×C22⋊D17—24no—
59C2×C22⋊D17—24no—
60C2×C22⋊D17—24no—
61C2×C22⋊D17D418yes—
62C22×Dic17—24no—
63C22×Dic17D414yes—
64C22×Dic17C2311yescharacteristic
65C22×Dic17C2312yes—
66C22×Dic17C2311yescharacteristic
67C22×Dic17C2311yescharacteristic
68C22×Dic17C2311yescharacteristic
69C2×Dic34—22no—
70C2×Dic34—24no—
71C22⋊C4×C17—22no—
72C22⋊C4×C17—24no—
73C22⋊C4×C17—22no—
74C22⋊C4×C17—42no—
75C22⋊Dic17—22no—
76C22⋊Dic17—24no—
77C22⋊Dic17—22no—
78C22⋊Dic17—42no—
79#63227—24no—
80#63227—24no—
81#63227—42no—
82#63227—42no—
83C4⋊Dic17—22no—
84#63225—22no—
85#63225—22no—
86C4×Dic17—24no—
87C23×C17—22no—
88C23×C17—22no—
89C23×C17—28no—
90C23×C17—22no—
91C23×C17—22no—
92C23×C17—22no—
93C23×C17—22no—
94C23×C17—22no—
95C23×C17—22no—
96C23×C17—48no—
97C23×C17—48no—
98C23×C17—44no—
99C23×C17—44no—
100C23×C17C2×D414yes—
101C23×C17C2×D412yes—
102C23×C17#4112yes—
103C23×C17#4112yes—
104C23×C17C2411yescharacteristic
105C22×D17—28no—
106C22×D17—22no—
107C22×D17—22no—
108C22×D17—22no—
109C22×D17—22no—
110C22×D17—22no—
111C22×D17—22no—
112C22×D17—48no—
113C22×D17—48no—
114C22×D17—44no—
115C22×D17—44no—
116C22×D17C2×D414yes—
117C2×C4×C17—21no—
118C2×C4×C17—22no—
119C2×C4×C17—22no—
120C2×C4×C17—21no—
121C2×C4×C17—22no—
122C2×C4×C17—22no—
123C2×C4×C17—22no—
124C22⋊D17—216no—
125C22⋊D17—216no—
126C2×Dic17—22no—
127C2×Dic17—28no—
128C2×Dic17—21no—
129C2×Dic17—22no—
130C2×Dic17—22no—
131C2×Dic17—21no—
132C2×Dic17—22no—
133C2×Dic17—22no—
134C2×Dic17—22no—
135C2×Dic17C2×D412yes—
136C2×Dic17C2×D414yes—
137C2×Dic17C2×D412yes—
138Dic34—44no—
139#2435—171nomaximal, Sylow
140C22×C17—22no—
141C22×C17—22no—
142C22×C17—24no—
143C22×C17—24no—
144C22×C17—24no—
145C22×C17—24no—
146C22×C17—24no—
147C22×C17—416no—
148C22×C17—42no—
149C22×C17—42no—
150C22×C17—42no—
151C22×C17—42no—
152C22×C17#12414yescomplemented
153C22×C17C22×D411yescharacteristic
154C22×C17C22×D412yes—
155C22×C17C2×#4111yescharacteristic
156C22×C17C2×#4111yescharacteristic
157C22×C17H(5,𝔽2)12yes—
158C2×D17—24no—
159C2×D17—24no—
160C2×D17—24no—
161C2×D17—416no—
162C2×D17—42no—
163C2×D17—42no—
164C2×D17—42no—
165C2×D17—42no—
166C4×C17—41no—
167C4×C17—41no—
168C4×C17—42no—
169Dic17—22no—
170Dic17—41no—
171Dic17—41no—
172Dic17—42no—
173Dic17#12014yes—
174C23×D4—171no—
175C2×#124—172no—
176C2×C22⋊Q8—171no—
177#394—172no—
178#394—172no—
179#390—172no—
180#379—171no—
181#379—171no—
182#379—171no—
183#379D1711yescharacteristic, complemented
184C4×C22⋊C4—171no—
185C2×C17—24no—
186C2×C17—44no—
187C2×C17C2×#12011yescharacteristic
188C2×C17C2×#12411yescharacteristic
189C2×C17#53511yescharacteristic
190C2×C17#54612yes—
191C2×C17#54612yes—
192D17—44no—
193C25—171no—
194C25C2×D1711yescharacteristic
195C22×Q8—171no—
196C22×D4—172no—
197C22×D4—172no—
198C22×D4—344no—
199C22×D4—348no—
200C23×C4—171no—
201#124—348no—
202C22⋊Q8—344no—
203C2×C4⋊C4—171no—
204C2×C4⋊C4—171no—
205C2×C4⋊C4—171no—
206C2×C22⋊C4—171no—
207C2×C22⋊C4—172no—
208C2×C22⋊C4—171no—
209C2×C22⋊C4—172no—
210C2×C22⋊C4—172no—
211C2×C22⋊C4—172no—
212C2×C22⋊C4—172no—
213C2×C22⋊C4—172no—
214C2×C22⋊C4C2×D1711yescharacteristic
215C2×C22⋊C4C2×D1711yescharacteristic
216C2×C22⋊C4C2×D1712yescomplemented
217C2×C22⋊C4C2×D1712yescomplemented
218C2×C42—172no—
219#95—172no—
220C17#243511yescharacteristic, complemented, Sylow
221C24—24no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24—42no—
226C24—172no—
227C24—344no—
228C24—342no—
229C24—342no—
230C24—342no—
231C24—682no—
232C24C22×D1712yes—
233C24C22×D1711yescharacteristic
234C2×Q8—342no—
235C2×Q8—344no—
236C2×D4—178no—
237C2×D4—344no—
238C2×D4—344no—
239C2×D4—344no—
240C2×D4—344no—
241C2×D4—3432no—
242C2×D4—344no—
243C22×C4—171no—
244C22×C4—174no—
245C22×C4—171no—
246C22×C4—172no—
247C22×C4—171no—
248C22×C4—171no—
249C22×C4—344no—
250C22×C4C22×D1712yes—
251C22×C4C22×D1711yescharacteristic
252C22×C4C22×D1711yescharacteristic
253C4⋊C4—342no—
254C4⋊C4—342no—
255C4⋊C4—342no—
256C22⋊C4—22no—
257C22⋊C4—24no—
258C22⋊C4—22no—
259C22⋊C4—42no—
260C22⋊C4—344no—
261C22⋊C4—344no—
262C22⋊C4—342no—
263C22⋊C4—342no—
264C22⋊C4—344no—
265C22⋊C4—682no—
266C22⋊C4—682no—
267C22⋊C4—682no—
268C42—344no—
269C23—22no—
270C23—22no—
271C23—28no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—48no—
279C23—48no—
280C23—44no—
281C23—44no—
282C23—174no—
283C23—348no—
284C23—342no—
285C23—342no—
286C23—342no—
287C23—342no—
288C23—342no—
289C23—342no—
290C23—688no—
291C23—688no—
292C23—684no—
293C23—684no—
294C23#6325212yes—
295C23D4×D1714yes—
296C23D4×D1712yes—
297C23#6325412yes—
298C23C23×D1711yescharacteristic
299Q8—684no—
300D4—3416no—
301D4—3416no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—174no—
310C2×C4—172no—
311C2×C4—172no—
312C2×C4—348no—
313C2×C4—342no—
314C2×C4—342no—
315C2×C4—342no—
316C2×C4—342no—
317C2×C4—341no—
318C2×C4—341no—
319C2×C4—342no—
320C2×C4—342no—
321C22—22no—
322C22—22no—
323C22—24no—
324C22—24no—
325C22—24no—
326C22—24no—
327C22—24no—
328C22—416no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—344no—
334C22—344no—
335C22—344no—
336C22—6816no—
337C22—682no—
338C22—682no—
339C22—682no—
340C22—682no—
341C22#1058757114yescomplemented
342C22C2×#6325211yescharacteristic
343C22C2×D4×D1711yescharacteristic
344C22C2×D4×D1712yes—
345C22C2×#6325411yescharacteristic
346C22#1058769112yes—
347C4—41no—
348C4—41no—
349C4—42no—
350C4—174no—
351C4—342no—
352C4—681no—
353C4—681no—
354C4—682no—
355C2—24no—
356C2—44no—
357C2—684no—
358C2#120×D1711yescharacteristic
359C2C2×#1058757111yescharacteristic
360C2#1176337411yescharacteristic
361C2#1176344612yes—
362C2#1176344912yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#379C2
prime radicalC23×C22⋊D17C2
hypercenter#379D17
hypocenterC17#2435
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#10587571C2semidirect
3. C2×#10587564C2semidirect
4. #11762725C2semidirect
5. #11762724C2semidirect
6. #11762709C2semidirect
7. #11762665C2semidirect
8. #11762664C2semidirect
9. #11762664C2semidirect
10. C22⋊C4×Dic17C2semidirect
11. C17×#379C2semidirect
19. C22×Dic34C22semidirect
22. C2×C22⋊C4×C17C22semidirect
23. C2×C22⋊C4×C17C22semidirect
26. C2×C22⋊Dic17C22semidirect
27. C2×C22⋊Dic17C22semidirect
28. C2×#63227C22semidirect
29. C2×#63227C22semidirect
30. C2×#63227C22semidirect
31. C2×#63227C22semidirect
32. C2×C4⋊Dic17C22semidirect
33. C2×#63225C22semidirect
34. C2×#63225C22semidirect
35. C2×C4×Dic17C22semidirect
38. #10587502C22semidirect
152. C22×C17#124semidirect
183. #379D17semidirect
216. C2×C22⋊C4C2×D17semidirect
217. C2×C22⋊C4C2×D17semidirect
220. C17#2435semidirect
341. C22#10587571semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D172
C2×#105875712
C2×#105875642
#117627252
#117627242
#117627092
#117626652
#117626642
C22⋊C4×Dic172
C17×#3792
#243517