Finite Group Databasebeta

← #14537073

Subgroups — #14537073

Order 2176.

1102 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C23×C4×D17C211yesmaximal, characteristic, complemented
2C2×#10587621C211yesmaximal, characteristic, complemented
3C2×#10587623C211yesmaximal, characteristic
4#11762685C214yesmaximal, complemented
5#11762680C214yesmaximal, complemented
6#11762672C211yesmaximal, characteristic, complemented
7#11762670C211yesmaximal, characteristic, complemented
8#11762664C211yesmaximal, characteristic, complemented
9C2×C17×#123C211yesmaximal, characteristic, complemented
10C24×D17C2211yescharacteristic
11C22×D4×C17C2211yescharacteristic
12C23×C4×C17C2211yescharacteristic
13C22×C22⋊D17C2212yes—
14C23×Dic17C2211yescharacteristic
15C22×C4×D17—24no—
16C22×C4×D17—24no—
17C22×C4×D17—24no—
18C22×C4×D17C2214yes—
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—
27#10587621—24no—
28C2×#63227C2212yes—
29C2×#63227C2214yes—
30C2×#63225C2212yescomplemented
31#10587502C2214yescomplemented
32#10587502C2214yescomplemented
33C24×C17C2311yescharacteristic
34C24×C17C2311yescharacteristic
35C23×D17—22no—
36C23×D17—22no—
37C23×D17—28no—
38C23×D17D414yes—
39C23×D17C2312yes—
40C2×D4×C17—22no—
41C2×D4×C17—22no—
42C2×D4×C17—22no—
43C22×C4×C17—22no—
44C22×C4×C17—22no—
45C22×C4×C17D414yescomplemented
46C22×C4×C17C2311yescharacteristic
47C22×C4×C17C2312yes—
48C22×C4×C17C2312yes—
49C2×C22⋊D17—24no—
50C2×C22⋊D17—28no—
51C22×Dic17—22no—
52C22×Dic17—22no—
53C22×Dic17D414yescomplemented
54C22×Dic17C2312yes—
55C22×Dic17C2312yes—
56C22×Dic17C2312yes—
57C2×C4×D17—232no—
58C2×C4×D17—28no—
59C2×C4×D17—28no—
60C2×C4×D17—28no—
61C4⋊C4×C17—24no—
62C22⋊C4×C17—22no—
63C22⋊C4×C17—24no—
64C22⋊Dic17—22no—
65C22⋊Dic17—22no—
66C22⋊Dic17—24no—
67#63227—24no—
68#63227—28no—
69#63225—24no—
70C23×C17—24no—
71C23×C17—21no—
72C23×C17—21no—
73C23×C17—21no—
74C23×C17—24no—
75C23×C17C2×D412yes—
76C23×C17C2×D412yes—
77C23×C17C2×D412yes—
78C23×C17C2411yescharacteristic
79C22×D17—216no—
80C22×D17—216no—
81C22×D17—28no—
82C22×D17—22no—
83C22×D17—22no—
84C22×D17—24no—
85C22×D17—216no—
86C22×D17C2×D414yes—
87C22×D17#4118yes—
88D4×C17—44no—
89C2×C4×C17—28no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—21no—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
96C2×C4×C17—24no—
97C2×C4×C17C2×D414yes—
98C22⋊D17—48no—
99C2×Dic17—28no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—24no—
105C2×Dic17—24no—
106C2×Dic17C2×D414yes—
107C4×D17—232no—
108#2849—171nomaximal, Sylow
109C22×C17—24no—
110C22×C17—24no—
111C22×C17—24no—
112C22×C17—41no—
113C22×C17—42no—
114C22×C17—42no—
115C22×C17—42no—
116C22×C17#12014yes—
117C22×C17C22×D411yescharacteristic
118C22×C17C22×D411yescharacteristic
119C22×C17C22×D411yescharacteristic
120C22×C17C2×#4114yes—
121C2×D17—232no—
122C2×D17—24no—
123C2×D17—24no—
124C2×D17—28no—
125C2×D17—28no—
126C2×D17#12318yescomplemented
127C4×C17—24no—
128C4×C17—41no—
129C4×C17—42no—
130Dic17—24no—
131Dic17—42no—
132Dic17—42no—
133C24×C4—171no—
134C2×#123—171no—
135C2×#123D1711yescharacteristic, complemented
136C2×#120—171no—
137#386—174no—
138#385—174no—
139#381—171no—
140#381—171no—
141#379—171no—
142C2×C17—24no—
143C2×C17—81no—
144C2×C17C2×#12011yescharacteristic
145C2×C17C2×#12312yes—
146C2×C17#52512yes—
147C2×C17#52512yes—
148D17—28no—
149C25—171no—
150C22×D4—172no—
151C22×D4C2×D1711yescharacteristic
152C23×C4—171no—
153C23×C4—174no—
154C23×C4—344no—
155C23×C4—344no—
156C23×C4—344no—
157C23×C4C2×D1711yescharacteristic
158#123—24no—
159#123—344no—
160#120—344no—
161C2×C4⋊C4—172no—
162C2×C4⋊C4C2×D1712yescomplemented
163C2×C22⋊C4—171no—
164C2×C22⋊C4—171no—
165C2×C22⋊C4—172no—
166C2×C22⋊C4—172no—
167C2×C22⋊C4—174no—
168C2×C22⋊C4C2×D1711yescharacteristic
169C2×C22⋊C4C2×D1712yescomplemented
170#95—174no—
171#95—174no—
172C17#284911yescharacteristic, complemented, Sylow
173C24—172no—
174C24—174no—
175C24—348no—
176C24—342no—
177C24—342no—
178C24C22×D1711yescharacteristic
179C24C22×D1711yescharacteristic
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—344no—
184C2×D4—348no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—174no—
188C22×C4—172no—
189C22×C4—172no—
190C22×C4—172no—
191C22×C4—348no—
192C22×C4—348no—
193C22×C4—3432no—
194C22×C4—342no—
195C22×C4—342no—
196C22×C4—348no—
197C22×C4C22⋊D1714yescomplemented
198C22×C4C22×D1712yes—
199C22×C4C22×D1711yescharacteristic
200C22×C4C22×D1712yes—
201C4⋊C4—24no—
202C4⋊C4—344no—
203C22⋊C4—22no—
204C22⋊C4—24no—
205C22⋊C4—344no—
206C22⋊C4—348no—
207C22⋊C4—344no—
208C22⋊C4—342no—
209C22⋊C4—342no—
210C23—24no—
211C23—21no—
212C23—21no—
213C23—24no—
214C23—21no—
215C23—174no—
216C23—178no—
217C23—3416no—
218C23—3416no—
219C23—3416no—
220C23—342no—
221C23—344no—
222C23—342no—
223C23—348no—
224C23D4×D1712yes—
225C23D4×D1712yes—
226C23C2×C22⋊D1712yes—
227C23C23×D1711yescharacteristic
228D4—44no—
229D4—688no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—28no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—174no—
239C2×C4—3432no—
240C2×C4—342no—
241C2×C4—342no—
242C2×C4—348no—
243C2×C4—342no—
244C2×C4—342no—
245C2×C4—344no—
246C2×C4—344no—
247C2×C4C2×C22⋊D1714yes—
248C22—24no—
249C22—24no—
250C22—24no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—178no—
256C22—3432no—
257C22—344no—
258C22—344no—
259C22—348no—
260C22—348no—
261C22#1058762314yes—
262C22C2×D4×D1711yescharacteristic
263C22C2×D4×D1711yescharacteristic
264C22#41×D1714yes—
265C22C22×C22⋊D1711yescharacteristic
266C4—24no—
267C4—41no—
268C4—42no—
269C4—344no—
270C4—682no—
271C4—682no—
272C2—24no—
273C2—81no—
274C2—348no—
275C2C2×#1058762311yescharacteristic
276C2#123×D1712yes—
277C2#1176331612yes—
278C2#1176331712yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#123C2
prime radicalC2×#10587623C2
hypercenterC2×#123D17
hypocenterC17#2849
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C23×C4×D17C2semidirect
2. C2×#10587621C2semidirect
4. #11762685C2semidirect
5. #11762680C2semidirect
6. #11762672C2semidirect
7. #11762670C2semidirect
8. #11762664C2semidirect
9. C2×C17×#123C2semidirect
20. C2×C4⋊C4×C17C22semidirect
22. C2×C22⋊C4×C17C22semidirect
25. C2×C22⋊Dic17C22semidirect
30. C2×#63225C22semidirect
31. #10587502C22semidirect
32. #10587502C22semidirect
45. C22×C4×C17D4semidirect
53. C22×Dic17D4semidirect
126. C2×D17#123semidirect
135. C2×#123D17semidirect
162. C2×C4⋊C4C2×D17semidirect
169. C2×C22⋊C4C2×D17semidirect
172. C17#2849semidirect
197. C22×C4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D172
C2×#105876212
C2×#105876232
#117626852
#117626802
#117626722
#117626702
#117626642
C2×C17×#1232
#284917