Finite Group Databasebeta

← #14532144

Subgroups — #14532144

Order 2176.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C22×Q8×D17C211yesmaximal, characteristic, complemented
2C2×#10587582C212yesmaximal, complemented
3C2×#10587579C211yesmaximal, characteristic
4#11762729C211yesmaximal, characteristic, complemented
5#11762722C211yesmaximal, characteristic, complemented
6#11762720C211yesmaximal, characteristic, complemented
7#11762713C212yesmaximal, complemented
8#11762687C212yesmaximal
9#11762680C212yesmaximal, complemented
10#11762668C211yesmaximal, characteristic, complemented
11C17×#395C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C2×Q8×D17—24no—
14C2×Q8×D17—28no—
15C22×D68C2211yescharacteristic
16C22×C4×D17C2211yescharacteristic
17C22×C4×D17C2212yes—
18C22×C4×D17C2211yescharacteristic
19C22×Dic34C2211yescharacteristic
20C22×Dic34C2212yes—
21C2×C4⋊C4×C17C2211yescharacteristic
22C2×C4⋊C4×C17C2212yes—
23C2×#63227C2211yescharacteristic
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×#63227C2211yescharacteristic
27C2×#63227C2212yes—
28C2×#63225C2212yes—
29C2×#63225C2212yes—
30C2×C4×Dic17C2211yescharacteristic
31#10587582—28no—
32#10587579—24no—
33C17×#95C2212yes—
34C17×#95C2211yescharacteristic
35#10587502C2212yes—
36#10587502C2212yes—
37#10587502C2212yes—
38#10587502C2211yescharacteristic
39C23×D17C2311yescharacteristic
40C23×D17C2311yescharacteristic
41C2×Q8×C17—22no—
42C2×Q8×C17—24no—
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2312yes—
45C22×C4×C17C2311yescharacteristic
46C22×C4×C17C2312yes—
47C22×C4×C17C2311yescharacteristic
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2312yes—
50C22×Dic17C2312yes—
51C22×Dic17C2311yescharacteristic
52Q8×D17—232no—
53C2×D68—22no—
54C2×D68—22no—
55C2×D68Q814yescomplemented
56C2×C4×D17—24no—
57C2×C4×D17—24no—
58C2×C4×D17—24no—
59C2×C4×D17—24no—
60C2×C4×D17—22no—
61C2×C4×D17—24no—
62C2×C4×D17D414yes—
63C2×Dic34—22no—
64C2×Dic34—24no—
65C2×Dic34—24no—
66C2×Dic34—24no—
67C2×Dic34D418yescomplemented
68C4⋊C4×C17—22no—
69C4⋊C4×C17—24no—
70#63227—22no—
71#63227—24no—
72#63227—22no—
73#63227—24no—
74#63227—24no—
75#63225—24no—
76#63225—24no—
77C4×Dic17—22no—
78C23×C17C2411yescharacteristic
79C22×D17—22no—
80C22×D17—22no—
81C22×D17—21no—
82C22×D17—22no—
83C22×D17—21no—
84C22×D17—22no—
85C22×D17C2×D412yes—
86C22×D17C2×Q812yes—
87C22×D17C2×Q812yescomplemented
88C22×D17#4112yes—
89Q8×C17—28no—
90C2×C4×C17—21no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17—21no—
95C2×C4×C17—22no—
96C2×C4×C17—21no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17C2×D412yes—
101C2×C4×C17C2×D414yes—
102C2×C4×C17C2×Q812yescomplemented
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—21no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17C2×D414yes—
111C2×Dic17C2×D412yes—
112C2×Dic17C2×D414yes—
113D68—28no—
114C4×D17—216no—
115C4×D17—28no—
116C4×D17—44no—
117Dic34—28no—
118Dic34—216no—
119#2409—171nomaximal, Sylow
120C22×C17C22×D411yescharacteristic
121C22×C17C22×D412yes—
122C22×C17C22×Q811yescharacteristic
123C22×C17C2×#4111yescharacteristic
124C22×C172−1+412yes—
125C2×D17—22no—
126C2×D17—22no—
127C2×D17—24no—
128C2×D17—24no—
129C2×D17—22no—
130C2×D17—24no—
131C2×D17—41no—
132C2×D17—41no—
133C2×D17—42no—
134C2×D17C22⋊Q814yescomplemented
135C4×C17—22no—
136C4×C17—24no—
137C4×C17—22no—
138C4×C17—41no—
139C4×C17—42no—
140Dic17—24no—
141Dic17—41no—
142Dic17—42no—
143Dic17#12014yes—
144C23×Q8—171no—
145C2×C22⋊Q8—172no—
146C2×C4×D4—171no—
147#395—171no—
148#395D1711yescharacteristic, complemented
149#393—171no—
150#393—171no—
151#391—172no—
152#386—172no—
153#385—172no—
154#380—171no—
155C2×C17C2×#12011yescharacteristic
156C2×C17C2×C22⋊Q811yescharacteristic
157C2×C17#53611yescharacteristic
158C2×C17#54812yes—
159C2×C17D4×Q812yes—
160D17—24no—
161D17—42no—
162C22×Q8—172no—
163C22×Q8—171no—
164C22×Q8—344no—
165C22×Q8—348no—
166C22×Q8C2×D1711yescharacteristic
167C22×D4—171no—
168C23×C4—171no—
169C23×C4—171no—
170C23×C4—172no—
171C22⋊Q8—348no—
172C4×D4—344no—
173C2×C4⋊C4—172no—
174C2×C4⋊C4—172no—
175C2×C4⋊C4C2×D1711yescharacteristic
176C2×C4⋊C4C2×D1712yes—
177C2×C22⋊C4—171no—
178C2×C22⋊C4—171no—
179C2×C22⋊C4—172no—
180C2×C22⋊C4—172no—
181C2×C22⋊C4—172no—
182C2×C42—171no—
183#95—171no—
184#95—172no—
185#95—172no—
186#95—172no—
187#95C2×D1711yescharacteristic
188#95C2×D1712yes—
189C17#240911yescharacteristic, complemented, Sylow
190C24—171no—
191C24—171no—
192C2×Q8—22no—
193C2×Q8—24no—
194C2×Q8—178no—
195C2×Q8—344no—
196C2×Q8—344no—
197C2×Q8—344no—
198C2×Q8—3432no—
199C2×Q8—342no—
200C2×D4—174no—
201C2×D4—342no—
202C2×D4—342no—
203C22×C4—171no—
204C22×C4—172no—
205C22×C4—174no—
206C22×C4—171no—
207C22×C4—172no—
208C22×C4—344no—
209C22×C4—344no—
210C22×C4—342no—
211C22×C4—344no—
212C22×C4—344no—
213C22×C4—344no—
214C22×C4C22×D1712yes—
215C22×C4C22×D1711yescharacteristic
216C22×C4C22×D1711yescharacteristic
217C22×C4C22×D1712yes—
218C22×C4C22×D1711yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—24no—
221C4⋊C4—344no—
222C4⋊C4—344no—
223C22⋊C4—344no—
224C22⋊C4—342no—
225C22⋊C4—344no—
226C22⋊C4—342no—
227C22⋊C4—344no—
228C42—342no—
229C23—172no—
230C23—172no—
231C23—172no—
232C23—172no—
233C23—342no—
234C23—342no—
235C23—342no—
236C23—341no—
237C23—342no—
238C23—341no—
239C23C23×D1711yescharacteristic
240Q8—28no—
241Q8—3416no—
242Q8—348no—
243D4—348no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—172no—
255C2×C4—174no—
256C2×C4—174no—
257C2×C4—348no—
258C2×C4—3416no—
259C2×C4—342no—
260C2×C4—342no—
261C2×C4—341no—
262C2×C4—342no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—342no—
266C2×C4—684no—
267C2×C4D4×D1712yes—
268C2×C4D4×D1714yes—
269C2×C4Q8×D1712yescomplemented
270C22—174no—
271C22—344no—
272C22—344no—
273C22—342no—
274C22—342no—
275C22—344no—
276C22—342no—
277C22—682no—
278C22—681no—
279C22—681no—
280C22C2×D4×D1712yes—
281C22C2×D4×D1711yescharacteristic
282C22C2×Q8×D1711yescharacteristic
283C22#1058769412yes—
284C22#41×D1711yescharacteristic
285C4—22no—
286C4—24no—
287C4—22no—
288C4—42no—
289C4—41no—
290C4—174no—
291C4—344no—
292C4—682no—
293C4—681no—
294C2—344no—
295C2—682no—
296C2#120×D1711yescharacteristic
297C2C22⋊Q8×D1711yescharacteristic
298C2#1176338411yescharacteristic
299C2#1176346912yes—
300C2#1176347412yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#395C2
prime radicalC22×D68C22
hypercenter#395D17
hypocenterC17#2409
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D17C2semidirect
2. C2×#10587582C2semidirect
4. #11762729C2semidirect
5. #11762722C2semidirect
6. #11762720C2semidirect
7. #11762713C2semidirect
9. #11762680C2semidirect
10. #11762668C2semidirect
11. C17×#395C2semidirect
55. C2×D68Q8semidirect
67. C2×Dic34D4semidirect
87. C22×D17C2×Q8semidirect
102. C2×C4×C17C2×Q8semidirect
134. C2×D17C22⋊Q8semidirect
148. #395D17semidirect
189. C17#2409semidirect
269. C2×C4Q8×D17semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D172
C2×#105875822
C2×#105875792
#117627292
#117627222
#117627202
#117627132
#117626872
#117626802
#117626682
C17×#3952
#240917