Finite Group Databasebeta

← #14538293

Subgroups — #14538293

Order 2176.

494 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#10587618C211yesmaximal, characteristic
2#11763383C211yesmaximal, characteristic, complemented
3#11763284C211yesmaximal, characteristic, complemented
4#11763026C214yesmaximal, complemented
5#11763012C214yesmaximal, complemented
6#11762835C211yesmaximal, characteristic, complemented
7#11762818C211yesmaximal, characteristic, complemented
8#11762806C211yesmaximal, characteristic, complemented
9C22×SD16×C17C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C22×D4×C17C2211yescharacteristic
12C2×#63252C2211yescharacteristic
13C2×SD16×C17—24no—
14C2×SD16×C17C2214yes—
15C22×C8×C17C2211yescharacteristic
16C2×C22⋊Dic17—22no—
17#10587633C2212yescomplemented
18#10587632—22no—
19#10587630C2212yescomplemented
20D4×Dic17—22no—
21D4×Dic17C2214yes—
22C2×#63229C2212yes—
23#10587618—24no—
24C2×#63228C2212yes—
25#10587615—22no—
26#10587612C2211yescharacteristic
27#10587612C2211yescharacteristic
28C2×#63223C2211yescharacteristic
29#10587506C2214yescomplemented
30#10587492C2212yescomplemented
31#10587487C2212yescomplemented
32#10587485C2214yescomplemented
33C24×C17—21no—
34C2×Q8×C17—21no—
35C2×Q8×C17—22no—
36C2×Q8×C17—22no—
37C2×Q8×C17C2312yes—
38C2×D4×C17—21no—
39C2×D4×C17—22no—
40C2×D4×C17D414yes—
41C2×D4×C17C2312yes—
42C22×C4×C17—21no—
43C22×C4×C17C2311yescharacteristic
44C2×C22⋊D17—21no—
45C22×Dic17—24no—
46#63252—22no—
47#63252—22no—
48C2×D68C2311yescharacteristic
49C2×C4×D17—21no—
50C2×Dic34C2311yescharacteristic
51SD16×C17—28no—
52C2×C8×C17—22no—
53C2×C8×C17C2312yes—
54C22⋊Dic17—21no—
55C22⋊Dic17—24no—
56C22⋊Dic17D412yes—
57#63229—24no—
58#63228—24no—
59#63227—24no—
60C4⋊Dic17D412yes—
61C4⋊Dic17C2312yes—
62#63225—24no—
63C4×Dic17—21no—
64C4×Dic17C2312yes—
65#63223—22no—
66C2×#3622C2312yes—
67C23×C17—22no—
68C23×C17—24no—
69C23×C17—82no—
70C23×C17C2×D411yescharacteristic
71C22×D17—21no—
72Q8×C17—22no—
73Q8×C17—22no—
74Q8×C17—42no—
75D4×C17—24no—
76D4×C17—22no—
77D4×C17C2×D414yes—
78C2×C4×C17—22no—
79C2×C4×C17—42no—
80C2×C4×C17C2×D411yescharacteristic
81C2×C4×C17C2×D412yes—
82C2×C4×C17#4112yes—
83C2×C4×C17C2411yescharacteristic
84C22⋊D17—42no—
85C2×Dic17—22no—
86C2×Dic17—21no—
87C2×Dic17—84no—
88C2×Dic17C2×D414yes—
89D68—21no—
90D68—21no—
91C4×D17—42no—
92Dic34—21no—
93Dic34—21no—
94C8×C17—22no—
95#3622—22no—
96#3042—171nomaximal, Sylow
97C22×C17—22no—
98C22×C17—24no—
99C22×C17—44no—
100C22×C17—84no—
101C22×C17#12112yes—
102C22×C17C22×D411yescharacteristic
103C2×D17—41no—
104C2×D17—42no—
105C4×C17—42no—
106C4×C17#12112yes—
107C4×C17C22×D411yescharacteristic
108C4×C17C2×#4111yescharacteristic
109Dic17—24no—
110Dic17—41no—
111Dic17—42no—
112C2×AGL(1,ℤ/8)—171no—
113C22×SD16D1711yescharacteristic, complemented
114#536—171no—
115#518—171no—
116#463—174no—
117#461—174no—
118#424—171no—
119#419—171no—
120#416—171no—
121C2×C17—22no—
122C2×C17—44no—
123C2×C17C2×#12111yescharacteristic
124C2×C17#57712yes—
125D17—81no—
126C2×#41—171no—
127C22×Q8C2×D1711yescharacteristic
128C22×D4C2×D1711yescharacteristic
129AGL(1,ℤ/8)—344no—
130C2×SD16—24no—
131C2×SD16—172no—
132C2×SD16C2×D1714yes—
133C2×D8—172no—
134C2×M16—171no—
135C22×C8C2×D1711yescharacteristic
136C4⋊Q8—172no—
137#124—172no—
138#123—342no—
139C22⋊Q8—342no—
140C4×D4—174no—
141C4×D4—342no—
142#117—171no—
143#117—171no—
144C2×C22⋊C4—342no—
145C4⋊C8—174no—
146#103—174no—
147#103—172no—
148#102—172no—
149C17#304211yescharacteristic, complemented, Sylow
150C24—21no—
151#41—342no—
152#41—342no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—171no—
157C2×Q8C22×D1712yes—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—171no—
161C2×D4—341no—
162C2×D4C22⋊D1714yes—
163C2×D4C22×D1712yes—
164C22×C4—21no—
165C22×C4—344no—
166C22×C4—341no—
167C22×C4C22×D1711yescharacteristic
168SD16—28no—
169SD16—344no—
170D8—344no—
171M16—342no—
172C2×C8—22no—
173C2×C8—172no—
174C2×C8C22×D1712yes—
175C4⋊C4—172no—
176C4⋊C4—172no—
177C4⋊C4—344no—
178C22⋊C4—172no—
179C22⋊C4—344no—
180C22⋊C4—344no—
181C22⋊C4—341no—
182C42—172no—
183C42—341no—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—341no—
188C23D4×D1711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—341no—
193Q8—341no—
194D4—22no—
195D4—24no—
196D4—341no—
197D4—341no—
198D4—682no—
199D4C2×C22⋊D1714yes—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—174no—
203C2×C4—342no—
204C2×C4—341no—
205C2×C4—682no—
206C2×C4—1364no—
207C2×C4D4×D1711yescharacteristic
208C2×C4#6325412yes—
209C2×C4C2×C22⋊D1712yes—
210C2×C4C23×D1711yescharacteristic
211C8—22no—
212C8—342no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—681no—
218C22—682no—
219C22#1058762512yes—
220C22C2×D4×D1711yescharacteristic
221C4—42no—
222C4—344no—
223C4—681no—
224C4—682no—
225C4#1058762512yes—
226C4C2×#6325411yescharacteristic
227C4C22×C22⋊D1711yescharacteristic
228C2—22no—
229C2—44no—
230C2—1361no—
231C2C2×#1058762511yescharacteristic
232C2#1176361512yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC22×SD16×C17C2
socleC22×C17C22×D4
prime radicalC2×#10587618C2
hypercenterC22×SD16D17
hypocenterC17#3042
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. #11763383C2semidirect
3. #11763284C2semidirect
4. #11763026C2semidirect
5. #11763012C2semidirect
6. #11762835C2semidirect
7. #11762818C2semidirect
8. #11762806C2semidirect
9. C22×SD16×C17C2semidirect
17. #10587633C22semidirect
19. #10587630C22semidirect
29. #10587506C22semidirect
30. #10587492C22semidirect
31. #10587487C22semidirect
32. #10587485C22semidirect
113. C22×SD16D17semidirect
149. C17#3042semidirect

Maximal subgroups

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

subgroupindex
C2×#105876182
#117633832
#117632842
#117630262
#117630122
#117628352
#117628182
#117628062
C22×SD16×C172
#304217