Finite Group Databasebeta

← #14532142

Subgroups — #14532142

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×#10587564C212yesmaximal, complemented
3C2×#10587567C211yesmaximal, characteristic
4#11762730C211yesmaximal, characteristic, complemented
5#11762728C211yesmaximal, characteristic, complemented
6#11762721C211yesmaximal, characteristic, complemented
7#11762713C212yesmaximal, complemented
8#11762685C212yesmaximal
9#11762683C212yesmaximal, complemented
10#11762668C211yesmaximal, characteristic, complemented
11C17×#393C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C22×C22⋊D17C2211yescharacteristic
14C23×Dic17C2211yescharacteristic
15C2×Q8×D17—24no—
16C2×Q8×D17—28no—
17C22×C4×D17C2212yes—
18C22×C4×D17C2211yescharacteristic
19C22×Dic34C2211yescharacteristic
20C22×Dic34C2212yes—
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊C4×C17C2212yes—
23C2×C22⋊Dic17C2212yes—
24C2×#63227C2211yescharacteristic
25C2×#63227C2212yes—
26C2×C4⋊Dic17C2212yes—
27C2×#63225C2212yes—
28C2×#63225C2211yescharacteristic
29C2×#63225C2212yes—
30C2×C4×Dic17C2211yescharacteristic
31#10587567—24no—
32#10587564—28no—
33C17×#95C2212yes—
34C17×#95C2211yescharacteristic
35#10587502C2212yes—
36#10587502C2212yes—
37#10587502C2212yes—
38#10587502C2211yescharacteristic
39C24×C17C2311yescharacteristic
40C23×D17C2311yescharacteristic
41C2×Q8×C17—22no—
42C2×Q8×C17—24no—
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2312yes—
45C22×C4×C17C2311yescharacteristic
46C22×C4×C17C2312yes—
47C2×C22⋊D17—22no—
48C2×C22⋊D17—22no—
49C2×C22⋊D17Q814yescomplemented
50C22×Dic17—24no—
51C22×Dic17—22no—
52C22×Dic17C2311yescharacteristic
53C22×Dic17C2312yes—
54C22×Dic17C2312yes—
55C22×Dic17C2311yescharacteristic
56C22×Dic17C2311yescharacteristic
57Q8×D17—232no—
58C2×C4×D17—24no—
59C2×C4×D17—24no—
60C2×C4×D17—24no—
61C2×C4×D17—24no—
62C2×C4×D17D414yes—
63C2×Dic34—22no—
64C2×Dic34—24no—
65C2×Dic34—24no—
66C2×Dic34—24no—
67C2×Dic34D418yescomplemented
68C22⋊C4×C17—24no—
69C22⋊C4×C17—22no—
70C22⋊Dic17—24no—
71#63227—22no—
72#63227—24no—
73C4⋊Dic17—24no—
74#63225—24no—
75#63225—22no—
76#63225—24no—
77C4×Dic17—22no—
78C23×C17—22no—
79C23×C17—21no—
80C23×C17—21no—
81C23×C17—22no—
82C23×C17C2×Q812yescomplemented
83C23×C17C2411yescharacteristic
84C22×D17—22no—
85C22×D17—22no—
86C22×D17C2×D412yes—
87C22×D17C2×Q812yes—
88C22×D17#4112yes—
89Q8×C17—28no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—21no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17C2×D412yes—
99C2×C4×C17C2×D414yes—
100C22⋊D17—28no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—21no—
104C2×Dic17—21no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17—21no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—44no—
111C2×Dic17C2×D414yes—
112C2×Dic17C2×D412yes—
113C2×Dic17C2×D414yes—
114C2×Dic17C2×Q812yescomplemented
115C4×D17—216no—
116C4×D17—28no—
117Dic34—28no—
118Dic34—216no—
119#2409—171nomaximal, Sylow
120C22×C17—22no—
121C22×C17—24no—
122C22×C17—41no—
123C22×C17—41no—
124C22×C17—42no—
125C22×C17C22×D411yescharacteristic
126C22×C17C22×D412yes—
127C22×C17C22×Q811yescharacteristic
128C22×C17C2×#4111yescharacteristic
129C22×C172−1+412yes—
130C2×D17—22no—
131C2×D17—22no—
132C2×D17—24no—
133C2×D17—24no—
134C2×D17C22⋊Q814yescomplemented
135C4×C17—22no—
136C4×C17—24no—
137C4×C17—41no—
138C4×C17—42no—
139Dic17—24no—
140Dic17—22no—
141Dic17—41no—
142Dic17—42no—
143Dic17#12014yes—
144C23×Q8—171no—
145C2×C22⋊Q8—172no—
146C2×C4×D4—171no—
147#395—171no—
148#395—171no—
149#393—171no—
150#393D1711yescharacteristic, complemented
151#391—172no—
152#386—172no—
153#385—172no—
154#380—171no—
155C2×C17—42no—
156C2×C17C2×#12011yescharacteristic
157C2×C17C2×C22⋊Q811yescharacteristic
158C2×C17#53611yescharacteristic
159C2×C17#54812yes—
160C2×C17D4×Q812yes—
161D17—24no—
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—171no—
174C2×C4⋊C4—172no—
175C2×C4⋊C4—172no—
176C2×C4⋊C4—172no—
177C2×C22⋊C4—172no—
178C2×C22⋊C4—171no—
179C2×C22⋊C4—172no—
180C2×C22⋊C4C2×D1711yescharacteristic
181C2×C22⋊C4C2×D1712yes—
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—
191C24C22×D1711yescharacteristic
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—171no—
205C22×C4—172no—
206C22×C4—174no—
207C22×C4—172no—
208C22×C4—171no—
209C22×C4—344no—
210C22×C4—344no—
211C22×C4—344no—
212C22×C4—342no—
213C22×C4—344no—
214C22×C4—344no—
215C22×C4C22×D1711yescharacteristic
216C22×C4C22×D1712yes—
217C22×C4C22×D1711yescharacteristic
218C22×C4C22×D1712yes—
219C4⋊C4—344no—
220C4⋊C4—344no—
221C4⋊C4—342no—
222C4⋊C4—344no—
223C22⋊C4—22no—
224C22⋊C4—24no—
225C22⋊C4—342no—
226C22⋊C4—344no—
227C22⋊C4—344no—
228C42—342no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—172no—
234C23—172no—
235C23—172no—
236C23—342no—
237C23—342no—
238C23Q8×D1712yescomplemented
239C23C23×D1711yescharacteristic
240Q8—28no—
241Q8—3416no—
242Q8—348no—
243D4—348no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—172no—
253C2×C4—174no—
254C2×C4—174no—
255C2×C4—172no—
256C2×C4—348no—
257C2×C4—3416no—
258C2×C4—341no—
259C2×C4—342no—
260C2×C4—342no—
261C2×C4—341no—
262C2×C4—342no—
263C2×C4—341no—
264C2×C4—342no—
265C2×C4—342no—
266C2×C4—342no—
267C2×C4—684no—
268C2×C4D4×D1712yes—
269C2×C4D4×D1714yes—
270C22—24no—
271C22—22no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—174no—
276C22—344no—
277C22—344no—
278C22—342no—
279C22—342no—
280C22C2×D4×D1712yes—
281C22C2×D4×D1711yescharacteristic
282C22C2×Q8×D1711yescharacteristic
283C22#41×D1711yescharacteristic
284C22#1058769712yes—
285C4—22no—
286C4—24no—
287C4—42no—
288C4—41no—
289C4—174no—
290C4—344no—
291C4—342no—
292C4—682no—
293C4—681no—
294C2—42no—
295C2—344no—
296C2#120×D1711yescharacteristic
297C2C22⋊Q8×D1711yescharacteristic
298C2#1176337911yescharacteristic
299C2#1176346712yes—
300C2#1176347212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#393C2
prime radicalC22×C22⋊D17C22
hypercenter#393D17
hypocenterC17#2409
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D17C2semidirect
2. C2×#10587564C2semidirect
4. #11762730C2semidirect
5. #11762728C2semidirect
6. #11762721C2semidirect
7. #11762713C2semidirect
9. #11762683C2semidirect
10. #11762668C2semidirect
11. C17×#393C2semidirect
49. C2×C22⋊D17Q8semidirect
67. C2×Dic34D4semidirect
82. C23×C17C2×Q8semidirect
114. C2×Dic17C2×Q8semidirect
134. C2×D17C22⋊Q8semidirect
150. #393D17semidirect
189. C17#2409semidirect
238. C23Q8×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×#105875642
C2×#105875672
#117627302
#117627282
#117627212
#117627132
#117626852
#117626832
#117626682
C17×#3932
#240917