Finite Group Databasebeta

← #14532172

Subgroups — #14532172

Order 2176.

786 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 153 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×#10587623C211yesmaximal, characteristic
3C2×#10587563C211yesmaximal, characteristic, complemented
4C22×#63227C211yesmaximal, characteristic, complemented
5#11762733C212yesmaximal, complemented
6#11762724C212yesmaximal, complemented
7#11762709C212yesmaximal, complemented
8#11762698C212yesmaximal, complemented
9#11762682C211yesmaximal, characteristic, complemented
10#11762664C211yesmaximal, characteristic, complemented
11C17×#386C211yesmaximal, characteristic, complemented
12C24×D17C2211yescharacteristic
13C22×D4×C17C2211yescharacteristic
14C23×C4×C17C2211yescharacteristic
15C22×C22⋊D17C2212yes—
16C23×Dic17C2211yescharacteristic
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊C4×C17C2212yescomplemented
19C2×C22⋊Dic17C2211yescharacteristic
20C2×C22⋊Dic17C2211yescharacteristic
21C2×C22⋊Dic17C2211yescharacteristic, complemented
22C2×C22⋊Dic17C2211yescharacteristic, complemented
23#10587623—24no—
24#10587621—24no—
25C2×#63227—24no—
26C2×#63227C2212yes—
27C2×#63227C2212yes—
28C2×#63227C2212yes—
29C2×#63227C2212yes—
30C2×#63227C2212yes—
31C2×#63225C2211yescharacteristic, complemented
32C2×#63225C2212yescomplemented
33C2×#63225C2211yescharacteristic, complemented
34C2×#63225C2212yescomplemented
35C2×C4×Dic17C2212yescomplemented
36#10587563—24no—
37C17×#95C2212yescomplemented
38#10587502C2212yescomplemented
39C24×C17C2311yescharacteristic
40C24×C17C2311yescharacteristic
41C23×D17—22no—
42C23×D17—24no—
43C23×D17—22no—
44C23×D17—22no—
45C23×D17—42no—
46C23×D17C2312yes—
47C2×D4×C17—22no—
48C2×D4×C17—22no—
49C2×D4×C17—22no—
50C22×C4×C17—22no—
51C22×C4×C17—42no—
52C22×C4×C17C2312yes—
53C22×C4×C17C2311yescharacteristic
54C22×C4×C17C2312yes—
55C2×C22⋊D17—24no—
56C2×C22⋊D17—24no—
57C2×C22⋊D17—42no—
58C22×Dic17—22no—
59C22×Dic17—22no—
60C22×Dic17D414yescomplemented
61C22×Dic17C2311yescharacteristic
62C22×Dic17C2311yescharacteristic
63C22×Dic17C2311yescharacteristic
64C22×Dic17C2311yescharacteristic
65C22×Dic17C2312yes—
66C22⋊C4×C17—22no—
67C22⋊C4×C17—42no—
68C22⋊Dic17—22no—
69C22⋊Dic17—22no—
70C22⋊Dic17—22no—
71C22⋊Dic17D414yes—
72#63227—24no—
73#63227—24no—
74#63227—24no—
75#63227—24no—
76#63227—42no—
77#63225—24no—
78#63225—22no—
79#63225—22no—
80#63225—42no—
81C4×Dic17—24no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17—21no—
85C23×C17—21no—
86C23×C17—22no—
87C23×C17—21no—
88C23×C17—22no—
89C23×C17C2×D412yes—
90C23×C17C2×D412yes—
91C23×C17#4112yes—
92C23×C17C2411yescharacteristic
93C22×D17—28no—
94C22×D17—22no—
95C22×D17—22no—
96C22×D17—22no—
97C22×D17—22no—
98C22×D17—22no—
99C22×D17—22no—
100C22×D17—48no—
101C22×D17—48no—
102C22×D17—44no—
103C22×D17—44no—
104C22×D17#4114yes—
105D4×C17—44no—
106C2×C4×C17—22no—
107C2×C4×C17—22no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—22no—
112C2×C4×C17—22no—
113C2×C4×C17—22no—
114C2×C4×C17—22no—
115C2×C4×C17—44no—
116C22⋊D17—48no—
117C2×Dic17—28no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—22no—
123C2×Dic17—22no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—22no—
127C2×Dic17—22no—
128C2×Dic17C2×D412yes—
129C2×Dic17C2×D412yes—
130C2×Dic17C2×D414yes—
131#2411—171nomaximal, Sylow
132C22×C17—22no—
133C22×C17—22no—
134C22×C17—24no—
135C22×C17—24no—
136C22×C17—41no—
137C22×C17—41no—
138C22×C17—41no—
139C22×C17—42no—
140C22×C17—42no—
141C22×C17#12114yes—
142C22×C17C22×D411yescharacteristic
143C22×C17C22×D411yescharacteristic
144C22×C17C2×#4111yescharacteristic
145C22×C17C2×#4112yes—
146C22×C17H(5,𝔽2)12yes—
147C2×D17—24no—
148C2×D17—24no—
149C2×D17—24no—
150C2×D17—416no—
151C2×D17—42no—
152C2×D17—42no—
153C2×D17—42no—
154C2×D17—42no—
155C4×C17—41no—
156C4×C17—42no—
157C4×C17—42no—
158Dic17—22no—
159Dic17—22no—
160Dic17—24no—
161Dic17—41no—
162Dic17—41no—
163C2×#123—171no—
164C2×#120—171no—
165C2×#117—171no—
166C22×C22⋊C4—171no—
167#396—172no—
168#394—172no—
169#390—172no—
170#388—172no—
171#386D1711yescharacteristic, complemented
172#385—171no—
173#379—171no—
174C2×C17—24no—
175C2×C17—81no—
176C2×C17C2×#12111yescharacteristic
177C2×C17#52511yescharacteristic
178C2×C17#53511yescharacteristic
179C2×C17#54612yes—
180C2×C17#55112yes—
181D17—44no—
182C25—171no—
183C22×D4—172no—
184C22×D4C2×D1711yescharacteristic
185C23×C4—171no—
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—171no—
195C2×C22⋊C4—171no—
196C2×C22⋊C4—171no—
197C2×C22⋊C4—171no—
198C2×C22⋊C4—172no—
199C2×C22⋊C4—172no—
200C2×C22⋊C4—172no—
201C2×C22⋊C4—172no—
202C2×C22⋊C4—172no—
203C2×C22⋊C4—344no—
204C2×C22⋊C4C2×D1711yescharacteristic
205C2×C22⋊C4C2×D1712yescomplemented
206C2×C42—172no—
207#95—172no—
208#95C2×D1712yescomplemented
209C17#241111yescharacteristic, complemented, Sylow
210C24—172no—
211C24—344no—
212C24—342no—
213C24—342no—
214C24—342no—
215C24—682no—
216C24C22×D1711yescharacteristic
217C24C22×D1711yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—344no—
222C2×D4—344no—
223C2×D4—682no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—174no—
227C22×C4—172no—
228C22×C4—171no—
229C22×C4—171no—
230C22×C4—171no—
231C22×C4—171no—
232C22×C4—342no—
233C22×C4—342no—
234C22×C4C22×D1712yes—
235C22×C4C22×D1711yescharacteristic
236C22×C4C22×D1712yes—
237C4⋊C4—344no—
238C4⋊C4—342no—
239C4⋊C4—342no—
240C4⋊C4—682no—
241C22⋊C4—22no—
242C22⋊C4—42no—
243C22⋊C4—174no—
244C22⋊C4—344no—
245C22⋊C4—344no—
246C22⋊C4—344no—
247C22⋊C4—344no—
248C22⋊C4—342no—
249C22⋊C4—342no—
250C22⋊C4—342no—
251C22⋊C4—682no—
252C42—344no—
253C23—22no—
254C23—22no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23—174no—
261C23—348no—
262C23—342no—
263C23—342no—
264C23—342no—
265C23—342no—
266C23—342no—
267C23—342no—
268C23—688no—
269C23—688no—
270C23—684no—
271C23—684no—
272C23#6325212yes—
273C23D4×D1712yes—
274C23D4×D1712yes—
275C23C23×D1711yescharacteristic
276D4—44no—
277D4—688no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—174no—
289C2×C4—172no—
290C2×C4—172no—
291C2×C4—341no—
292C2×C4—341no—
293C2×C4—341no—
294C2×C4—341no—
295C2×C4—342no—
296C2×C4—342no—
297C2×C4—348no—
298C2×C4—341no—
299C2×C4—341no—
300C2×C4—342no—
301C2×C4—342no—
302C22—24no—
303C22—22no—
304C22—22no—
305C22—24no—
306C22—42no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—42no—
311C22—344no—
312C22—344no—
313C22—344no—
314C22—6816no—
315C22—682no—
316C22—682no—
317C22—682no—
318C22—682no—
319C22#1058757014yes—
320C22C2×#6325211yescharacteristic
321C22C2×D4×D1711yescharacteristic
322C22C2×D4×D1711yescharacteristic
323C22#1058769112yes—
324C22#41×D1712yes—
325C4—41no—
326C4—42no—
327C4—42no—
328C4—342no—
329C4—342no—
330C4—344no—
331C4—681no—
332C4—681no—
333C2—24no—
334C2—81no—
335C2—684no—
336C2C2×#1058757011yescharacteristic
337C2#1176331711yescharacteristic
338C2#1176337411yescharacteristic
339C2#1176344812yes—
340C2#1176348812yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#386C2
prime radicalC2×#10587623C2
hypercenter#386D17
hypocenterC17#2411
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#10587621C2semidirect
3. C2×#10587563C2semidirect
4. C22×#63227C2semidirect
5. #11762733C2semidirect
6. #11762724C2semidirect
7. #11762709C2semidirect
8. #11762698C2semidirect
9. #11762682C2semidirect
10. #11762664C2semidirect
11. C17×#386C2semidirect
18. C2×C22⋊C4×C17C22semidirect
21. C2×C22⋊Dic17C22semidirect
22. C2×C22⋊Dic17C22semidirect
31. C2×#63225C22semidirect
32. C2×#63225C22semidirect
33. C2×#63225C22semidirect
34. C2×#63225C22semidirect
35. C2×C4×Dic17C22semidirect
37. C17×#95C22semidirect
38. #10587502C22semidirect
60. C22×Dic17D4semidirect
171. #386D17semidirect
205. C2×C22⋊C4C2×D17semidirect
208. #95C2×D17semidirect
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×#105876232
C2×#105875632
C22×#632272
#117627332
#117627242
#117627092
#117626982
#117626822
#117626642
C17×#3862
#241117