Finite Group Databasebeta

← #14538016

Subgroups — #14538016

Order 2176.

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

normal subgroups only

This group has 738 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
1C22×Q8×D17C211yesmaximal, characteristic, complemented
2C2×#10587637C211yesmaximal, characteristic, complemented
3#11763316C211yesmaximal, characteristic
4#11762963C214yesmaximal, complemented
5#11762946C214yesmaximal, complemented
6#11762819C211yesmaximal, characteristic, complemented
7#11762818C211yesmaximal, characteristic, complemented
8#11762772C211yesmaximal, characteristic, complemented
9C2×C17×#137C211yesmaximal, characteristic, complemented
10C2×#41×C17C2211yescharacteristic
11C22×Q8×C17C2211yescharacteristic
12C2×Q8×D17—24no—
13C2×Q8×D17—22no—
14C2×Q8×D17—24no—
15C2×Q8×D17C2214yes—
16C22×C4×D17—21no—
17C22×C4×D17C2211yescharacteristic
18C22×Dic34—21no—
19C22×Dic34C2211yescharacteristic
20C17×#137—24no—
21C2×Q16×C17C2212yescomplemented
22C2×SD16×C17C2212yescomplemented
23C2×M16×C17C2211yescharacteristic
24#10587637—24no—
25#10587632C2212yes—
26C2×#63231C2212yescomplemented
27#10587624C2212yes—
28#10587623—21no—
29#10587621—21no—
30C2×#63229C2212yescomplemented
31C4×C22⋊D17—22no—
32#10587612C2211yescharacteristic
33C2×#63223C2211yescharacteristic
34#10587506C2214yescomplemented
35#10587491C2214yes—
36#10587487C2214yescomplemented
37C23×D17D411yescharacteristic
38#41×C17—22no—
39#41×C17—22no—
40C2×Q8×C17—22no—
41C2×Q8×C17—21no—
42C2×Q8×C17D414yescomplemented
43C2×Q8×C17C2311yescharacteristic
44C2×Q8×C17C2312yes—
45C2×D4×C17—21no—
46C2×D4×C17C2311yescharacteristic
47C22×C4×C17—21no—
48C22×C4×C17—21no—
49C22×C4×C17C2311yescharacteristic
50C2×C22⋊D17—22no—
51C22×Dic17—21no—
52C22×Dic17D411yescharacteristic
53Q8×D17—216no—
54Q8×D17—28no—
55Q8×D17—28no—
56C2×C4×D17—28no—
57C2×C4×D17—24no—
58C2×C4×D17—22no—
59C2×C4×D17—22no—
60C2×C4×D17D412yes—
61C2×C4×D17C2312yes—
62C2×Dic34—28no—
63C2×Dic34—21no—
64C2×Dic34—22no—
65C2×Dic34—24no—
66C2×Dic34D414yescomplemented
67C2×Dic34C2312yes—
68Q16×C17—24no—
69SD16×C17—24no—
70M16×C17—22no—
71C2×C8×C17C2312yes—
72C22⋊Dic17—21no—
73C22⋊Dic17—22no—
74#63231—24no—
75#63229—24no—
76#63227—22no—
77C4⋊Dic17C2312yes—
78#63225—22no—
79C4×Dic17—21no—
80#63223—22no—
81C2×#3622C2312yes—
82C23×C17—21no—
83C23×C17C2×D411yescharacteristic
84C22×D17—22no—
85C22×D17—24no—
86C22×D17C2×D412yes—
87Q8×C17—24no—
88Q8×C17—22no—
89Q8×C17—21no—
90Q8×C17—21no—
91Q8×C17C2×D414yes—
92D4×C17—21no—
93D4×C17—21no—
94D4×C17—42no—
95C2×C4×C17—24no—
96C2×C4×C17—22no—
97C2×C4×C17—21no—
98C2×C4×C17—42no—
99C2×C4×C17C2×D411yescharacteristic
100C2×C4×C17C2×D412yes—
101C2×C4×C17C2×D412yes—
102C2×C4×C17C2411yescharacteristic
103C22⋊D17—44no—
104C2×Dic17—22no—
105C2×Dic17—24no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17C2×D412yes—
109C4×D17—216no—
110C4×D17—24no—
111C4×D17—22no—
112C4×D17—22no—
113Dic34—216no—
114Dic34—24no—
115Dic34—22no—
116Dic34C2×D414yes—
117C8×C17—22no—
118#3622—22no—
119#3004—171nomaximal, Sylow
120C22×C17—22no—
121C22×C17—41no—
122C22×C17—42no—
123C22×C17#12012yes—
124C22×C17C22×D411yescharacteristic
125C2×D17—28no—
126C2×D17—22no—
127C2×D17—22no—
128C2×D17#13714yescomplemented
129C4×C17—24no—
130C4×C17—41no—
131C4×C17#12012yes—
132C4×C17C22×D411yescharacteristic
133C4×C17C22×D411yescharacteristic
134Dic17—22no—
135Dic17—24no—
136Dic17—42no—
137C23×Q8—171no—
138C2×#137—171no—
139C2×#137D1711yescharacteristic, complemented
140#525—171no—
141#451—174no—
142#448—174no—
143#419—171no—
144#419—171no—
145#407—171no—
146C2×C17—22no—
147C2×C17—81no—
148C2×C17C2×#12011yescharacteristic
149C2×C17C2×#13712yes—
150D17—24no—
151C2×#41C2×D1711yescharacteristic
152C22×Q8—174no—
153C22×Q8—171no—
154C22×Q8—341no—
155C22×Q8—342no—
156C22×Q8—344no—
157C22×Q8—344no—
158C22×Q8C2×D1711yescharacteristic
159C23×C4—171no—
160C23×C4—341no—
161#137—24no—
162#137—344no—
163C2×Q16—172no—
164C2×Q16C2×D1712yescomplemented
165C2×SD16—172no—
166C2×SD16C2×D1712yescomplemented
167C2×M16—171no—
168C2×M16C2×D1711yescharacteristic
169#123—341no—
170C22⋊Q8—172no—
171#121—172no—
172#120—341no—
173C4×D4—342no—
174#117—171no—
175#103—174no—
176#103—174no—
177C22⋊C8—174no—
178C17#300411yescharacteristic, complemented, Sylow
179C24—171no—
180#41—22no—
181#41—22no—
182C2×Q8—21no—
183C2×Q8—22no—
184C2×Q8—174no—
185C2×Q8—172no—
186C2×Q8—344no—
187C2×Q8—348no—
188C2×Q8—348no—
189C2×Q8—3416no—
190C2×Q8—348no—
191C2×Q8—341no—
192C2×Q8—342no—
193C2×Q8C22⋊D1714yescomplemented
194C2×Q8C22×D1711yescharacteristic
195C2×Q8C22×D1712yes—
196C2×D4—21no—
197C2×D4—342no—
198C2×D4C22×D1711yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—171no—
202C22×C4—172no—
203C22×C4—172no—
204C22×C4—344no—
205C22×C4—348no—
206C22×C4—341no—
207C22×C4—342no—
208C22×C4—342no—
209C22×C4C22×D1711yescharacteristic
210Q16—24no—
211Q16—344no—
212SD16—24no—
213SD16—344no—
214M16—22no—
215M16—342no—
216C2×C8—172no—
217C2×C8C22×D1712yes—
218C4⋊C4—172no—
219C4⋊C4—342no—
220C22⋊C4—342no—
221C22⋊C4—342no—
222C22⋊C4—341no—
223C42—341no—
224C23—21no—
225C23—172no—
226C23—344no—
227C23—342no—
228C23D4×D1711yescharacteristic
229Q8—22no—
230Q8—24no—
231Q8—21no—
232Q8—21no—
233Q8—174no—
234Q8—3416no—
235Q8—342no—
236Q8—344no—
237Q8C2×C22⋊D1714yes—
238D4—21no—
239D4—21no—
240D4—42no—
241D4—684no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—21no—
245C2×C4—42no—
246C2×C4—172no—
247C2×C4—3416no—
248C2×C4—342no—
249C2×C4—342no—
250C2×C4—344no—
251C2×C4—342no—
252C2×C4—342no—
253C2×C4—344no—
254C2×C4—342no—
255C2×C4D4×D1712yes—
256C2×C4D4×D1711yescharacteristic
257C2×C4C2×C22⋊D1712yes—
258C2×C4C23×D1711yescharacteristic
259C8—22no—
260C8—342no—
261C22—22no—
262C22—41no—
263C22—42no—
264C22—174no—
265C22—348no—
266C22—342no—
267C22—342no—
268C22#1058762312yes—
269C22C2×D4×D1711yescharacteristic
270C4—24no—
271C4—41no—
272C4—342no—
273C4—344no—
274C4—682no—
275C4#1058762312yes—
276C4C2×D4×D1711yescharacteristic
277C4C22×C22⋊D1711yescharacteristic
278C2—22no—
279C2—81no—
280C2—344no—
281C2C2×#1058762311yescharacteristic
282C2#137×D1712yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC2×C17×#137C2
socleC22×C17C22×D4
prime radical#11763316C2
hypercenterC2×#137D17
hypocenterC17#3004
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D17C2semidirect
2. C2×#10587637C2semidirect
4. #11762963C2semidirect
5. #11762946C2semidirect
6. #11762819C2semidirect
7. #11762818C2semidirect
8. #11762772C2semidirect
9. C2×C17×#137C2semidirect
21. C2×Q16×C17C22semidirect
22. C2×SD16×C17C22semidirect
26. C2×#63231C22semidirect
30. C2×#63229C22semidirect
34. #10587506C22semidirect
36. #10587487C22semidirect
42. C2×Q8×C17D4semidirect
66. C2×Dic34D4semidirect
128. C2×D17#137semidirect
139. C2×#137D17semidirect
164. C2×Q16C2×D17semidirect
166. C2×SD16C2×D17semidirect
178. C17#3004semidirect
193. C2×Q8C22⋊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×#105876372
#117633162
#117629632
#117629462
#117628192
#117628182
#117627722
C2×C17×#1372
#300417