Finite Group Databasebeta

← #14532169

Subgroups — #14532169

Order 2176.

786 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 165 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×#10587615C211yesmaximal, characteristic, complemented
2C2×#10587639C211yesmaximal, characteristic
3C2×#10587612C211yesmaximal, characteristic, complemented
4#11762732C212yesmaximal, complemented
5#11762725C212yesmaximal, complemented
6#11762709C212yesmaximal, complemented
7#11762703C212yesmaximal, complemented
8#11762688C211yesmaximal, characteristic, complemented
9#11762682C211yesmaximal, characteristic, complemented
10#11762665C211yesmaximal, characteristic, complemented
11C22×C22⋊C4×C17C211yesmaximal, characteristic, complemented
12C25×C17C2211yescharacteristic
13C23×C4×C17C2211yescharacteristic
14C23×C4×C17C2211yescharacteristic
15C22×C22⋊D17C2211yescharacteristic
16C22×C22⋊D17C2212yes—
17C2×C22⋊C4×C17—24no—
18C2×C22⋊C4×C17C2212yes—
19C2×C22⋊C4×C17C2212yes—
20#10587639—24no—
21C2×C22⋊Dic17C2211yescharacteristic
22C2×C22⋊Dic17C2212yes—
23C2×C22⋊Dic17C2212yes—
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2212yes—
26C2×C22⋊Dic17C2211yescharacteristic
27#10587615—24no—
28C2×#63227C2211yescharacteristic, complemented
29C2×#63227C2211yescharacteristic, complemented
30C2×#63227C2212yescomplemented
31#10587612—24no—
32C2×C4⋊Dic17C2212yescomplemented
33C2×#63225C2211yescharacteristic, complemented
34C2×#63225C2211yescharacteristic, complemented
35C2×#63225C2212yescomplemented
36C2×C4×Dic17C2212yescomplemented
37#10587502C2212yescomplemented
38#10587502C2212yescomplemented
39C24×C17—22no—
40C24×C17—24no—
41C24×C17—22no—
42C24×C17—22no—
43C24×C17—42no—
44C24×C17C2312yes—
45C24×C17C2311yescharacteristic
46C23×D17C2311yescharacteristic
47C22×C4×C17—22no—
48C22×C4×C17—22no—
49C22×C4×C17—22no—
50C22×C4×C17—42no—
51C22×C4×C17D414yescomplemented
52C22×C4×C17C2312yes—
53C22×C4×C17C2311yescharacteristic
54C22×C4×C17C2311yescharacteristic
55C2×C22⋊D17—22no—
56C2×C22⋊D17—22no—
57C2×C22⋊D17—24no—
58C2×C22⋊D17—22no—
59C2×C22⋊D17—24no—
60C2×C22⋊D17—42no—
61C22×Dic17C2311yescharacteristic
62C22×Dic17C2311yescharacteristic
63C22×Dic17C2312yes—
64C22×Dic17C2311yescharacteristic
65C22×Dic17C2312yes—
66C22⋊C4×C17—24no—
67C22⋊C4×C17—24no—
68C22⋊Dic17—22no—
69C22⋊Dic17—22no—
70C22⋊Dic17—24no—
71C22⋊Dic17—24no—
72C22⋊Dic17—42no—
73C22⋊Dic17D414yes—
74#63227—22no—
75#63227—22no—
76#63227—42no—
77C4⋊Dic17—24no—
78#63225—22no—
79#63225—22no—
80#63225—42no—
81C4×Dic17—24no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17—28no—
85C23×C17—22no—
86C23×C17—22no—
87C23×C17—22no—
88C23×C17—22no—
89C23×C17—22no—
90C23×C17—22no—
91C23×C17—48no—
92C23×C17—48no—
93C23×C17—44no—
94C23×C17—44no—
95C23×C17C2×D412yes—
96C23×C17C2×D412yes—
97C23×C17#4114yes—
98C23×C17#4112yes—
99C23×C17C2411yescharacteristic
100C22×D17—21no—
101C22×D17—21no—
102C22×D17—22no—
103C22×D17—21no—
104C22×D17—22no—
105C2×C4×C17—28no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—22no—
111C2×C4×C17—22no—
112C2×C4×C17—22no—
113C2×C4×C17—44no—
114C2×C4×C17C2×D412yes—
115C2×C4×C17C2×D412yes—
116C22⋊D17—48no—
117C22⋊D17—44no—
118C2×Dic17—22no—
119C2×Dic17—22no—
120C2×Dic17—21no—
121C2×Dic17—22no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—22no—
126C2×Dic17—22no—
127C2×Dic17—21no—
128C2×Dic17—22no—
129C2×Dic17—22no—
130C2×Dic17C2×D414yes—
131#2411—171nomaximal, Sylow
132C22×C17—22no—
133C22×C17—22no—
134C22×C17—24no—
135C22×C17—24no—
136C22×C17—24no—
137C22×C17—24no—
138C22×C17—24no—
139C22×C17—416no—
140C22×C17—42no—
141C22×C17—42no—
142C22×C17—42no—
143C22×C17—42no—
144C22×C17#12114yes—
145C22×C17C22×D411yescharacteristic
146C22×C17C22×D411yescharacteristic
147C22×C17C2×#4112yes—
148C22×C17C2×#4111yescharacteristic
149C22×C17H(5,𝔽2)12yes—
150C2×D17—41no—
151C2×D17—41no—
152C2×D17—41no—
153C2×D17—42no—
154C2×D17—42no—
155C4×C17—22no—
156C4×C17—22no—
157C4×C17—42no—
158Dic17—24no—
159Dic17—41no—
160Dic17—41no—
161Dic17—41no—
162Dic17—42no—
163C2×#123—171no—
164C2×#120—171no—
165C2×#117—171no—
166C22×C22⋊C4D1711yescharacteristic, complemented
167#396—172no—
168#394—172no—
169#390—172no—
170#388—172no—
171#386—171no—
172#385—171no—
173#379—171no—
174C2×C17—24no—
175C2×C17—44no—
176C2×C17C2×#12111yescharacteristic
177C2×C17#52511yescharacteristic
178C2×C17#53511yescharacteristic
179C2×C17#54612yes—
180C2×C17#55112yes—
181D17—81no—
182C25C2×D1711yescharacteristic
183C22×D4—171no—
184C22×D4—172no—
185C23×C4C2×D1711yescharacteristic
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⋊C4—172no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—171no—
196C2×C22⋊C4—171no—
197C2×C22⋊C4—172no—
198C2×C22⋊C4—171no—
199C2×C22⋊C4—172no—
200C2×C22⋊C4—171no—
201C2×C22⋊C4—171no—
202C2×C22⋊C4—172no—
203C2×C22⋊C4—172no—
204C2×C22⋊C4C2×D1712yes—
205C2×C22⋊C4C2×D1712yes—
206C2×C42—172no—
207#95—172no—
208#95—172no—
209C17#241111yescharacteristic, complemented, Sylow
210C24—24no—
211C24—22no—
212C24—22no—
213C24—22no—
214C24—42no—
215C24—171no—
216C24C22×D1712yes—
217C24C22×D1711yescharacteristic
218C2×D4—344no—
219C2×D4—344no—
220C2×D4—342no—
221C2×D4—342no—
222C2×D4—342no—
223C2×D4—682no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—172no—
229C22×C4—171no—
230C22×C4—171no—
231C22×C4—172no—
232C22×C4—171no—
233C22×C4C22⋊D1714yescomplemented
234C22×C4C22×D1712yes—
235C22×C4C22×D1711yescharacteristic
236C22×C4C22×D1711yescharacteristic
237C4⋊C4—344no—
238C4⋊C4—342no—
239C4⋊C4—342no—
240C4⋊C4—682no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—174no—
244C22⋊C4—344no—
245C22⋊C4—344no—
246C22⋊C4—342no—
247C22⋊C4—342no—
248C22⋊C4—342no—
249C22⋊C4—342no—
250C22⋊C4—682no—
251C22⋊C4—682no—
252C42—344no—
253C23—22no—
254C23—22no—
255C23—28no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—48no—
263C23—48no—
264C23—44no—
265C23—44no—
266C23—342no—
267C23—341no—
268C23—341no—
269C23—342no—
270C23—341no—
271C23#6325214yes—
272C23D4×D1712yes—
273C23#6325412yes—
274C23C2×C22⋊D1712yes—
275C23C23×D1711yescharacteristic
276D4—688no—
277D4—684no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—28no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—44no—
287C2×C4—174no—
288C2×C4—342no—
289C2×C4—342no—
290C2×C4—342no—
291C2×C4—342no—
292C2×C4—342no—
293C2×C4—341no—
294C2×C4—341no—
295C2×C4—342no—
296C2×C4—342no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—341no—
300C2×C4C2×C22⋊D1712yes—
301C2×C4C2×C22⋊D1712yes—
302C22—24no—
303C22—22no—
304C22—22no—
305C22—24no—
306C22—24no—
307C22—24no—
308C22—24no—
309C22—416no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—682no—
315C22—681no—
316C22—681no—
317C22—681no—
318C22—682no—
319C22#1058762514yes—
320C22C2×#6325212yes—
321C22C2×D4×D1711yescharacteristic
322C22C2×#6325411yescharacteristic
323C22#1058769112yes—
324C22C22×C22⋊D1711yescharacteristic
325C4—22no—
326C4—22no—
327C4—42no—
328C4—344no—
329C4—681no—
330C4—681no—
331C4—681no—
332C4—682no—
333C2—24no—
334C2—44no—
335C2—1361no—
336C2C2×#1058762511yescharacteristic
337C2#1176332011yescharacteristic
338C2#1176337611yescharacteristic
339C2#1176344912yes—
340C2#1176348512yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC22×C22⋊C4×C17C2
prime radicalC2×#10587639C2
hypercenterC22×C22⋊C4D17
hypocenterC17#2411
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#10587615C2semidirect
3. C2×#10587612C2semidirect
4. #11762732C2semidirect
5. #11762725C2semidirect
6. #11762709C2semidirect
7. #11762703C2semidirect
8. #11762688C2semidirect
9. #11762682C2semidirect
10. #11762665C2semidirect
11. C22×C22⋊C4×C17C2semidirect
28. C2×#63227C22semidirect
29. C2×#63227C22semidirect
30. C2×#63227C22semidirect
32. C2×C4⋊Dic17C22semidirect
33. C2×#63225C22semidirect
34. C2×#63225C22semidirect
35. C2×#63225C22semidirect
36. C2×C4×Dic17C22semidirect
37. #10587502C22semidirect
38. #10587502C22semidirect
51. C22×C4×C17D4semidirect
166. C22×C22⋊C4D17semidirect
209. C17#2411semidirect
233. C22×C4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876152
C2×#105876392
C2×#105876122
#117627322
#117627252
#117627092
#117627032
#117626882
#117626822
#117626652
C22×C22⋊C4×C172
#241117