Finite Group Databasebeta

← #14538697

Subgroups — #14538697

Order 2176.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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
1#11763320C211yesmaximal, characteristic
2#11763150C212yesmaximal, complemented
3#11763144C211yesmaximal, characteristic, complemented
4#11763139C211yesmaximal, characteristic, complemented
5#11763133C212yesmaximal, complemented
6#11763127C211yesmaximal, characteristic, complemented
7#11763122C211yesmaximal, characteristic, complemented
8#11762845C212yesmaximal, complemented
9#11762803C212yesmaximal, complemented
10C2×C22⋊#3622C211yesmaximal, characteristic, complemented
11C2×C17×#117C211yesmaximal, characteristic, complemented
12C23×C4×C17C2211yescharacteristic
13C2×#63252C2211yescharacteristic
14C17×#117—21no—
15C17×#117—21no—
16C17×#117C2212yes—
17C17×#117C2212yes—
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×C4⋊C4×C17C2211yescharacteristic
20C2×C22⋊C4×C17—21no—
21C2×C42×C17—21no—
22#10587639—21no—
23#10587616C2211yescharacteristic
24#10587616C2211yescharacteristic
25#10587615—21no—
26C4×C22⋊D17—21no—
27C4×C22⋊D17—21no—
28#10587612C2211yescharacteristic
29#10587610C2211yescharacteristic
30#10587610C2211yescharacteristic
31C22×#3622C2212yes—
32C22⋊#3622C2211yescharacteristic
33C22⋊#3622C2211yescharacteristic
34C22⋊#3622C2212yes—
35#10587481C2212yescomplemented
36#10587481C2212yescomplemented
37#10587480C2212yescomplemented
38#10587480C2212yescomplemented
39#10587479C2212yescomplemented
40#10587479C2212yescomplemented
41#10587478C2212yescomplemented
42#10587478C2212yescomplemented
43C24×C17D411yescharacteristic
44C22×C4×C17—24no—
45C22×C4×C17—21no—
46C22×C4×C17—21no—
47C22×C4×C17D411yescharacteristic
48C22×C4×C17D411yescharacteristic
49C22×C4×C17D411yescharacteristic
50C22×C4×C17C2311yescharacteristic
51C22×C4×C17C2311yescharacteristic
52C22×C4×C17C2311yescharacteristic
53C2×C22⋊D17—21no—
54C2×C22⋊D17—21no—
55C2×C22⋊D17—21no—
56#63252—24no—
57C2×D68C2311yescharacteristic
58C2×C4×D17—21no—
59C2×Dic34C2311yescharacteristic
60C4⋊C4×C17—21no—
61C4⋊C4×C17—21no—
62C4⋊C4×C17C2312yes—
63C4⋊C4×C17C2312yes—
64C22⋊C4×C17—22no—
65C22⋊C4×C17—22no—
66C42×C17—22no—
67C42×C17—22no—
68C22⋊Dic17—21no—
69C22⋊Dic17—21no—
70C22⋊Dic17—21no—
71#63227—21no—
72#63227—21no—
73C4⋊Dic17C2311yescharacteristic
74C4⋊Dic17C2311yescharacteristic
75#63225—21no—
76#63225—21no—
77C4×Dic17—21no—
78C2×#3622—24no—
79C2×#3622C2312yes—
80C2×#3622C2312yes—
81C23×C17—22no—
82C23×C17—22no—
83C23×C17—41no—
84C23×C17C2×D411yescharacteristic
85C23×C17C2×D411yescharacteristic
86C23×C17C2×D411yescharacteristic
87C22×D17—21no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—42no—
96C2×C4×C17—42no—
97C2×C4×C17C2×D411yescharacteristic
98C2×C4×C17C2×D411yescharacteristic
99C2×C4×C17C2×D411yescharacteristic
100C2×C4×C17#4114yes—
101C2×C4×C17C2411yescharacteristic
102C22⋊D17—42no—
103C22⋊D17—42no—
104C22⋊D17—42no—
105C2×Dic17—21no—
106C2×Dic17—21no—
107C2×Dic17—21no—
108D68—22no—
109C4×D17—42no—
110Dic34—22no—
111#3622—22no—
112#3622—22no—
113#3091—171nomaximal, Sylow
114C22×C17—21no—
115C22×C17—21no—
116C22×C17—21no—
117C22×C17—21no—
118C22×C17—21no—
119C22×C17—21no—
120C22×C17—21no—
121C22×C17—44no—
122C22×C17#12312yes—
123C22×C17#13512yes—
124C22×C17C22×D411yescharacteristic
125C2×D17—41no—
126C2×D17—41no—
127C2×D17—41no—
128C4×C17—21no—
129C4×C17—21no—
130C4×C17—42no—
131C4×C17—42no—
132C4×C17#12312yes—
133C4×C17C2×#4112yes—
134Dic17—41no—
135Dic17—41no—
136Dic17—41no—
137#525—171no—
138C2×#117D1711yescharacteristic, complemented
139#485—172no—
140#484—171no—
141#483—171no—
142#482—172no—
143#481—171no—
144#480—171no—
145#427—172no—
146#416—172no—
147C2×C22⋊C8—171no—
148C2×C17—22no—
149C2×C17—22no—
150C2×C17—41no—
151C2×C17C2×#12311yescharacteristic
152C2×C17C2×#13511yescharacteristic
153C2×C17#57511yescharacteristic
154D17—81no—
155C2×#41—171no—
156C23×C4C2×D1711yescharacteristic
157C22×C8—172no—
158#123—341no—
159C22⋊Q8—171no—
160C22⋊Q8—171no—
161#121—171no—
162#121—171no—
163#120—341no—
164C4×D4—341no—
165C4×D4—341no—
166#117—21no—
167#117—21no—
168#117—171no—
169#117C2×D1712yes—
170#117C2×D1712yes—
171C2×C4⋊C4C2×D1711yescharacteristic
172C2×C4⋊C4C2×D1711yescharacteristic
173C2×C22⋊C4—21no—
174C2×C42—21no—
175#107—172no—
176#107—172no—
177#106—172no—
178#106—172no—
179#103—172no—
180#103—172no—
181#102—172no—
182#102—172no—
183C22⋊C8—171no—
184C22⋊C8—171no—
185C22⋊C8—172no—
186C17#309111yescharacteristic, complemented, Sylow
187C24C22⋊D1711yescharacteristic
188#41—344no—
189C2×Q8—171no—
190C2×D4—171no—
191C2×D4—341no—
192C2×D4—341no—
193C2×D4—341no—
194C22×C4—24no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—341no—
198C22×C4C22⋊D1711yescharacteristic
199C22×C4C22⋊D1711yescharacteristic
200C22×C4C22⋊D1711yescharacteristic
201C22×C4C22×D1711yescharacteristic
202C22×C4C22×D1711yescharacteristic
203C22×C4C22×D1711yescharacteristic
204C2×C8—172no—
205C2×C8—172no—
206C2×C8—344no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—171no—
210C4⋊C4—171no—
211C4⋊C4—341no—
212C4⋊C4—341no—
213C4⋊C4C22×D1712yes—
214C4⋊C4C22×D1712yes—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—341no—
218C22⋊C4—341no—
219C22⋊C4—341no—
220C22⋊C4—341no—
221C22⋊C4—341no—
222C42—22no—
223C42—22no—
224C42—341no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—341no—
229C23C2×C22⋊D1711yescharacteristic
230C23C2×C22⋊D1711yescharacteristic
231C23C2×C22⋊D1711yescharacteristic
232Q8—342no—
233D4—342no—
234D4—682no—
235D4—682no—
236D4—682no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—341no—
247C2×C4—341no—
248C2×C4—341no—
249C2×C4—682no—
250C2×C4#6325214yes—
251C2×C4C2×C22⋊D1711yescharacteristic
252C2×C4C2×C22⋊D1711yescharacteristic
253C2×C4C2×C22⋊D1711yescharacteristic
254C2×C4C23×D1711yescharacteristic
255C8—342no—
256C8—342no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—681no—
266C22—681no—
267C22—681no—
268C22#1058761512yes—
269C22#1058763612yes—
270C22C22×C22⋊D1711yescharacteristic
271C4—21no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—681no—
276C4—681no—
277C4—681no—
278C4#1058761512yes—
279C4C2×#6325212yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—1361no—
284C2C2×#1058761511yescharacteristic
285C2C2×#1058763611yescharacteristic
286C2#1176360711yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC2×C17×#117C2
socleC22×C17C22×D4
prime radical#11763320C2
hypercenterC2×#117D17
hypocenterC17#3091
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #11763150C2semidirect
3. #11763144C2semidirect
4. #11763139C2semidirect
5. #11763133C2semidirect
6. #11763127C2semidirect
7. #11763122C2semidirect
8. #11762845C2semidirect
9. #11762803C2semidirect
10. C2×C22⋊#3622C2semidirect
11. C2×C17×#117C2semidirect
35. #10587481C22semidirect
36. #10587481C22semidirect
37. #10587480C22semidirect
38. #10587480C22semidirect
39. #10587479C22semidirect
40. #10587479C22semidirect
41. #10587478C22semidirect
42. #10587478C22semidirect
138. C2×#117D17semidirect
186. C17#3091semidirect

Maximal subgroups

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

subgroupindex
#117633202
#117631502
#117631442
#117631392
#117631332
#117631272
#117631222
#117628452
#117628032
C2×C22⋊#36222
C2×C17×#1172
#309117