Finite Group Databasebeta

← #14532165

Subgroups — #14532165

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×#10587621C211yesmaximal, characteristic, complemented
2C2×#10587563C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic17C211yesmaximal, characteristic, complemented
4#11762732C212yesmaximal, complemented
5#11762723C212yesmaximal, complemented
6#11762712C212yesmaximal, complemented
7#11762697C212yesmaximal, complemented
8#11762689C211yesmaximal, characteristic, complemented
9#11762684C211yesmaximal, characteristic, complemented
10#11762665C211yesmaximal, characteristic, complemented
11C2×C17×#120C211yesmaximal, characteristic
12C25×C17C2211yescharacteristic
13C22×D4×C17C2211yescharacteristic
14C22×D4×C17C2212yes—
15C23×Dic17C2211yescharacteristic
16C23×Dic17C2211yescharacteristic
17C17×#120—24no—
18C2×C22⋊C4×C17C2212yes—
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C22⋊Dic17—24no—
21C2×C22⋊Dic17C2211yescharacteristic
22C2×C22⋊Dic17C2212yes—
23C2×C22⋊Dic17C2212yes—
24C2×C22⋊Dic17C2212yes—
25C2×C22⋊Dic17C2211yescharacteristic
26C2×C22⋊Dic17C2212yes—
27C2×C22⋊Dic17C2211yescharacteristic, complemented
28C2×C22⋊Dic17C2211yescharacteristic, complemented
29C2×C22⋊Dic17C2212yescomplemented
30#10587621—24no—
31C2×C4⋊Dic17C2212yescomplemented
32C2×#63225C2211yescharacteristic, complemented
33C2×#63225C2212yescomplemented
34C2×#63225C2211yescharacteristic, complemented
35C2×C4×Dic17C2212yescomplemented
36#10587563—24no—
37#10587502C2212yescomplemented
38#10587502C2212yescomplemented
39C24×C17—22no—
40C24×C17—24no—
41C24×C17—22no—
42C24×C17—22no—
43C24×C17—42no—
44C24×C17C2311yescharacteristic
45C24×C17C2312yes—
46C24×C17C2311yescharacteristic
47C2×D4×C17—22no—
48C2×D4×C17—22no—
49C2×D4×C17—24no—
50C2×D4×C17—22no—
51C2×D4×C17—24no—
52C2×D4×C17—42no—
53C22×C4×C17C2311yescharacteristic
54C22×C4×C17C2312yes—
55C22×Dic17—22no—
56C22×Dic17—22no—
57C22×Dic17—22no—
58C22×Dic17—42no—
59C22×Dic17D414yescomplemented
60C22×Dic17C2312yes—
61C22×Dic17C2311yescharacteristic
62C22×Dic17C2312yes—
63C22×Dic17C2311yescharacteristic
64C22×Dic17C2311yescharacteristic
65C22×Dic17C2311yescharacteristic
66C22⋊C4×C17—24no—
67C22⋊C4×C17D414yes—
68C22⋊Dic17—22no—
69C22⋊Dic17—22no—
70C22⋊Dic17—24no—
71C22⋊Dic17—22no—
72C22⋊Dic17—22no—
73C22⋊Dic17—24no—
74C22⋊Dic17—24no—
75C22⋊Dic17—42no—
76C22⋊Dic17—42no—
77C4⋊Dic17—42no—
78#63225—24no—
79#63225—22no—
80#63225—22no—
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—21no—
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—22no—
95C23×C17—22no—
96C23×C17—48no—
97C23×C17—48no—
98C23×C17—44no—
99C23×C17—44no—
100C23×C17C2×D412yes—
101C23×C17C2×D412yes—
102C23×C17#4114yes—
103C23×C17#4112yes—
104C23×C17C2411yescharacteristic
105D4×C17—48no—
106D4×C17—44no—
107C2×C4×C17—22no—
108C2×C4×C17—22no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17C2×D414yes—
113C2×Dic17—28no—
114C2×Dic17—21no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—22no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—22no—
122C2×Dic17—22no—
123C2×Dic17—22no—
124C2×Dic17—22no—
125C2×Dic17—22no—
126C2×Dic17—22no—
127C2×Dic17—22no—
128C2×Dic17—44no—
129C2×Dic17C2×D412yes—
130C2×Dic17C2×D412yes—
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—41no—
145C22×C17—41no—
146C22×C17—41no—
147C22×C17—42no—
148C22×C17—42no—
149C22×C17#12114yes—
150C22×C17C22×D411yescharacteristic
151C22×C17C22×D411yescharacteristic
152C22×C17C2×#4111yescharacteristic
153C22×C17C2×#4112yes—
154C22×C17H(5,𝔽2)12yes—
155C4×C17—24no—
156C4×C17—41no—
157Dic17—22no—
158Dic17—22no—
159Dic17—41no—
160Dic17—41no—
161Dic17—42no—
162Dic17—42no—
163C2×#123—171no—
164C2×#120D1711yescharacteristic
165C2×#117—171no—
166C22×C22⋊C4—171no—
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×C17—81no—
177C2×C17C2×#12111yescharacteristic
178C2×C17#52511yescharacteristic
179C2×C17#53511yescharacteristic
180C2×C17#54612yes—
181C2×C17#55112yes—
182C25C2×D1711yescharacteristic
183C22×D4C2×D1711yescharacteristic
184C22×D4C2×D1712yes—
185C23×C4—171no—
186C23×C4—171no—
187#123—344no—
188#120—24no—
189#117—344no—
190C2×C4⋊C4—172no—
191C2×C4⋊C4—171no—
192C2×C4⋊C4—171no—
193C2×C4⋊C4—172no—
194C2×C22⋊C4—172no—
195C2×C22⋊C4—172no—
196C2×C22⋊C4—171no—
197C2×C22⋊C4—171no—
198C2×C22⋊C4—171no—
199C2×C22⋊C4—171no—
200C2×C22⋊C4—172no—
201C2×C22⋊C4—172no—
202C2×C22⋊C4—172no—
203C2×C22⋊C4—344no—
204C2×C22⋊C4C2×D1712yes—
205C2×C22⋊C4C2×D1711yescharacteristic
206C2×C42—172no—
207#95—172no—
208#95—172no—
209C17#241111yescharacteristic, complemented, Sylow
210C24—22no—
211C24—24no—
212C24—22no—
213C24—22no—
214C24—42no—
215C24C22×D1711yescharacteristic
216C24C22×D1712yes—
217C24C22×D1711yescharacteristic
218C2×D4—22no—
219C2×D4—24no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—42no—
224C22×C4—174no—
225C22×C4—171no—
226C22×C4—171no—
227C22×C4—171no—
228C22×C4—172no—
229C22×C4—172no—
230C22×C4—171no—
231C22×C4—342no—
232C22×C4—342no—
233C22×C4—342no—
234C22×C4—682no—
235C22×C4C22×D1711yescharacteristic
236C22×C4C22×D1712yes—
237C4⋊C4—342no—
238C4⋊C4—342no—
239C4⋊C4—344no—
240C4⋊C4—682no—
241C22⋊C4—24no—
242C22⋊C4—344no—
243C22⋊C4—344no—
244C22⋊C4—342no—
245C22⋊C4—342no—
246C22⋊C4—342no—
247C22⋊C4—344no—
248C22⋊C4—342no—
249C22⋊C4—682no—
250C22⋊C4—682no—
251C22⋊C4C22⋊D1714yes—
252C42—344no—
253C23—22no—
254C23—22no—
255C23—28no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—21no—
263C23—21no—
264C23—22no—
265C23—22no—
266C23—21no—
267C23—48no—
268C23—48no—
269C23—44no—
270C23—44no—
271C23D4×D1712yes—
272C23#6325414yes—
273C23#6325412yes—
274C23C2×C22⋊D1712yes—
275C23C23×D1711yescharacteristic
276D4—48no—
277D4—44no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—172no—
284C2×C4—172no—
285C2×C4—341no—
286C2×C4—341no—
287C2×C4—341no—
288C2×C4—341no—
289C2×C4—348no—
290C2×C4—341no—
291C2×C4—341no—
292C2×C4—342no—
293C2×C4—342no—
294C2×C4—342no—
295C2×C4—342no—
296C2×C4—342no—
297C2×C4—342no—
298C2×C4—342no—
299C2×C4—342no—
300C2×C4—684no—
301C2×C4C2×C22⋊D1714yes—
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—42no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—42no—
319C22#1058762414yes—
320C22C2×D4×D1711yescharacteristic
321C22C2×#6325411yescharacteristic
322C22C2×#6325412yes—
323C22#1058769112yes—
324C22C22×C22⋊D1711yescharacteristic
325C4—24no—
326C4—41no—
327C4—342no—
328C4—342no—
329C4—681no—
330C4—682no—
331C4—681no—
332C4—682no—
333C2—24no—
334C2—44no—
335C2—81no—
336C2C2×#1058762411yescharacteristic
337C2#1176331511yescharacteristic
338C2#1176337411yescharacteristic
339C2#1176345312yes—
340C2#1176349012yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
Fitting, prime radicalC2×C17×#120C2
hypercenterC2×#120D17
hypocenterC17#2411
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#10587621C2semidirect
2. C2×#10587563C2semidirect
3. C22×C22⋊Dic17C2semidirect
4. #11762732C2semidirect
5. #11762723C2semidirect
6. #11762712C2semidirect
7. #11762697C2semidirect
8. #11762689C2semidirect
9. #11762684C2semidirect
10. #11762665C2semidirect
27. C2×C22⋊Dic17C22semidirect
28. C2×C22⋊Dic17C22semidirect
29. C2×C22⋊Dic17C22semidirect
31. C2×C4⋊Dic17C22semidirect
32. C2×#63225C22semidirect
33. C2×#63225C22semidirect
34. C2×#63225C22semidirect
35. C2×C4×Dic17C22semidirect
37. #10587502C22semidirect
38. #10587502C22semidirect
59. C22×Dic17D4semidirect
209. C17#2411semidirect

Maximal subgroups

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

subgroupindex
C2×#105876212
C2×#105875632
C22×C22⋊Dic172
#117627322
#117627232
#117627122
#117626972
#117626892
#117626842
#117626652
C2×C17×#1202
#241117