Finite Group Databasebeta

← #14532176

Subgroups — #14532176

Order 2176.

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

normal subgroups only

This group has 814 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×#10587623C211yesmaximal, characteristic
2C22×#63227C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D17C211yesmaximal, characteristic, complemented
4#11762735C212yesmaximal, complemented
5#11762725C212yesmaximal, complemented
6#11762689C211yesmaximal, characteristic, complemented
7#11762685C212yesmaximal, complemented
8#11762684C211yesmaximal, characteristic, complemented
9#11762672C211yesmaximal, characteristic, complemented
10#11762666C212yesmaximal, complemented
11C2×C17×#123C211yesmaximal, characteristic, complemented
12C24×D17C2211yescharacteristic
13C22×D4×C17C2211yescharacteristic
14C23×C4×C17C2211yescharacteristic
15C22×C22⋊D17C2211yescharacteristic
16C22×C22⋊D17C2211yescharacteristic
17C23×Dic17C2211yescharacteristic
18C22×C4×D17C2212yes—
19C17×#123—24no—
20C2×C4⋊C4×C17C2212yescomplemented
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊C4×C17C2212yescomplemented
23C2×C22⋊Dic17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2212yescomplemented
26#10587623—24no—
27C2×#63227—24no—
28C2×#63227C2212yes—
29C2×#63227C2212yes—
30C2×#63227C2211yescharacteristic
31C2×#63227C2212yes—
32C2×#63227C2211yescharacteristic
33C2×C4⋊Dic17C2212yescomplemented
34C22⋊C4×D17—24no—
35#10587502C2212yescomplemented
36#10587502C2212yescomplemented
37#10587502C2212yescomplemented
38#10587502C2212yescomplemented
39C24×C17C2311yescharacteristic
40C24×C17C2311yescharacteristic
41C23×D17—22no—
42C23×D17—22no—
43C23×D17—24no—
44C23×D17—42no—
45C23×D17D414yes—
46C23×D17C2311yescharacteristic
47C23×D17C2311yescharacteristic
48C2×D4×C17—22no—
49C2×D4×C17—22no—
50C2×D4×C17—22no—
51C22×C4×C17—22no—
52C22×C4×C17—22no—
53C22×C4×C17—22no—
54C22×C4×C17C2312yes—
55C22×C4×C17C2311yescharacteristic
56C22×C4×C17C2312yes—
57C2×C22⋊D17—22no—
58C2×C22⋊D17—22no—
59C2×C22⋊D17—24no—
60C2×C22⋊D17—42no—
61C22×Dic17—22no—
62C22×Dic17—42no—
63C22×Dic17C2312yes—
64C22×Dic17C2312yes—
65C22×Dic17C2311yescharacteristic
66C22×Dic17C2311yescharacteristic
67C2×C4×D17—24no—
68C2×C4×D17—24no—
69C2×C4×D17—42no—
70C4⋊C4×C17—24no—
71C22⋊C4×C17—24no—
72C22⋊C4×C17D414yescomplemented
73C22⋊Dic17—22no—
74C22⋊Dic17—22no—
75C22⋊Dic17—42no—
76#63227—24no—
77#63227—24no—
78#63227—22no—
79#63227—22no—
80#63227—24no—
81C4⋊Dic17—42no—
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—21no—
94C22×D17—21no—
95C22×D17—22no—
96C22×D17—28no—
97C22×D17—21no—
98C22×D17—21no—
99C22×D17—22no—
100C22×D17—22no—
101C22×D17—28no—
102C22×D17—44no—
103C22×D17—48no—
104C22×D17—48no—
105C22×D17C2×D412yes—
106C22×D17C2×D412yes—
107C22×D17#4114yes—
108D4×C17—44no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
111C2×C4×C17—22no—
112C2×C4×C17—21no—
113C2×C4×C17—22no—
114C2×C4×C17—21no—
115C2×C4×C17—22no—
116C2×C4×C17—21no—
117C2×C4×C17—44no—
118C2×C4×C17C2×D414yes—
119C22⋊D17—44no—
120C22⋊D17—44no—
121C2×Dic17—22no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—22no—
125C2×Dic17—22no—
126C2×Dic17—22no—
127C2×Dic17—22no—
128C2×Dic17—22no—
129C2×Dic17—22no—
130C2×Dic17—22no—
131C2×Dic17—44no—
132C4×D17—48no—
133#2412—171nomaximal, Sylow
134C22×C17—22no—
135C22×C17—22no—
136C22×C17—24no—
137C22×C17—24no—
138C22×C17—41no—
139C22×C17—41no—
140C22×C17—41no—
141C22×C17—42no—
142C22×C17—42no—
143C22×C17#12114yes—
144C22×C17C22×D411yescharacteristic
145C22×C17C22×D411yescharacteristic
146C22×C17C2×#4111yescharacteristic
147C22×C17C2×#4112yes—
148C22×C17H(5,𝔽2)12yes—
149C2×D17—22no—
150C2×D17—22no—
151C2×D17—24no—
152C2×D17—24no—
153C2×D17—22no—
154C2×D17—22no—
155C2×D17—22no—
156C2×D17—416no—
157C2×D17—42no—
158C2×D17—42no—
159C2×D17#12314yescomplemented
160C4×C17—24no—
161C4×C17—42no—
162C4×C17—41no—
163Dic17—41no—
164Dic17—42no—
165Dic17—41no—
166Dic17—42no—
167C2×#123D1711yescharacteristic, complemented
168C2×#120—171no—
169C22×C22⋊C4—171no—
170C22×C22⋊C4—171no—
171#397—172no—
172#394—172no—
173#386—171no—
174#386—172no—
175#385—171no—
176#381—171no—
177#380—172no—
178C2×C17—24no—
179C2×C17—81no—
180C2×C17C2×#12111yescharacteristic
181C2×C17C2×#12311yescharacteristic
182C2×C17#53411yescharacteristic
183C2×C17#54612yes—
184C2×C17#55112yes—
185D17—24no—
186D17—42no—
187C25—171no—
188C22×D4—171no—
189C22×D4—171no—
190C22×D4C2×D1711yescharacteristic
191C23×C4—171no—
192C23×C4—172no—
193C23×C4C2×D1711yescharacteristic
194#123—24no—
195#120—344no—
196C2×C4⋊C4—172no—
197C2×C4⋊C4C2×D1712yescomplemented
198C2×C22⋊C4—171no—
199C2×C22⋊C4—171no—
200C2×C22⋊C4—172no—
201C2×C22⋊C4—172no—
202C2×C22⋊C4—172no—
203C2×C22⋊C4—171no—
204C2×C22⋊C4—171no—
205C2×C22⋊C4—172no—
206C2×C22⋊C4—344no—
207C2×C22⋊C4—344no—
208C2×C22⋊C4C2×D1711yescharacteristic
209C2×C22⋊C4C2×D1712yescomplemented
210#95—172no—
211#95—172no—
212#95—172no—
213#95—172no—
214C17#241211yescharacteristic, complemented, Sylow
215C24—171no—
216C24—171no—
217C24—174no—
218C24—344no—
219C24—342no—
220C24—342no—
221C24—682no—
222C24C22×D1711yescharacteristic
223C24C22×D1711yescharacteristic
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—342no—
228C2×D4—342no—
229C2×D4—344no—
230C2×D4—682no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—171no—
235C22×C4—172no—
236C22×C4—172no—
237C22×C4—171no—
238C22×C4—344no—
239C22×C4—342no—
240C22×C4—344no—
241C22×C4—682no—
242C22×C4—682no—
243C22×C4C22×D1712yes—
244C22×C4C22×D1711yescharacteristic
245C22×C4C22×D1712yes—
246C4⋊C4—24no—
247C4⋊C4—682no—
248C22⋊C4—24no—
249C22⋊C4—344no—
250C22⋊C4—344no—
251C22⋊C4—342no—
252C22⋊C4—342no—
253C22⋊C4—342no—
254C22⋊C4—342no—
255C22⋊C4—344no—
256C22⋊C4—682no—
257C22⋊C4C22⋊D1714yescomplemented
258C23—22no—
259C23—22no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—22no—
264C23—21no—
265C23—172no—
266C23—174no—
267C23—172no—
268C23—348no—
269C23—341no—
270C23—342no—
271C23—341no—
272C23—341no—
273C23—342no—
274C23—342no—
275C23—341no—
276C23—348no—
277C23—684no—
278C23—688no—
279C23—688no—
280C23D4×D1712yes—
281C23#6325412yes—
282C23C2×C22⋊D1712yes—
283C23C23×D1711yescharacteristic
284D4—44no—
285D4—684no—
286D4—684no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—44no—
296C2×C4—342no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—342no—
300C2×C4—342no—
301C2×C4—342no—
302C2×C4—342no—
303C2×C4—342no—
304C2×C4—342no—
305C2×C4—342no—
306C2×C4—688no—
307C2×C4—684no—
308C2×C4C2×C22⋊D1714yes—
309C22—24no—
310C22—22no—
311C22—22no—
312C22—24no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—174no—
319C22—344no—
320C22—342no—
321C22—342no—
322C22—344no—
323C22—342no—
324C22—342no—
325C22—342no—
326C22—6816no—
327C22—682no—
328C22—682no—
329C22#1058762414yes—
330C22C2×D4×D1711yescharacteristic
331C22C2×#6325411yescharacteristic
332C22#41×D1712yes—
333C22#1058769612yes—
334C22C22×C22⋊D1711yescharacteristic
335C4—24no—
336C4—42no—
337C4—41no—
338C4—681no—
339C4—682no—
340C4—681no—
341C4—682no—
342C2—24no—
343C2—81no—
344C2—344no—
345C2—682no—
346C2C2×#1058762411yescharacteristic
347C2#123×D1711yescharacteristic
348C2#1176337011yescharacteristic
349C2#1176344712yes—
350C2#1176348212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#123C2
prime radicalC2×#10587623C2
hypercenterC2×#123D17
hypocenterC17#2412
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C22×#63227C2semidirect
3. C2×C22⋊C4×D17C2semidirect
4. #11762735C2semidirect
5. #11762725C2semidirect
6. #11762689C2semidirect
7. #11762685C2semidirect
8. #11762684C2semidirect
9. #11762672C2semidirect
10. #11762666C2semidirect
11. C2×C17×#123C2semidirect
20. C2×C4⋊C4×C17C22semidirect
22. C2×C22⋊C4×C17C22semidirect
25. C2×C22⋊Dic17C22semidirect
33. C2×C4⋊Dic17C22semidirect
35. #10587502C22semidirect
36. #10587502C22semidirect
37. #10587502C22semidirect
38. #10587502C22semidirect
72. C22⋊C4×C17D4semidirect
159. C2×D17#123semidirect
167. C2×#123D17semidirect
197. C2×C4⋊C4C2×D17semidirect
209. C2×C22⋊C4C2×D17semidirect
214. C17#2412semidirect
257. C22⋊C4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876232
C22×#632272
C2×C22⋊C4×D172
#117627352
#117627252
#117626892
#117626852
#117626842
#117626722
#117626662
C2×C17×#1232
#241217