Finite Group Databasebeta

← #14537633

Subgroups — #14537633

Order 2176.

528 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 175 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×D8×D17C211yesmaximal, characteristic
2#11763281C212yesmaximal, complemented
3#11762922C211yesmaximal, characteristic, complemented
4#11762921C211yesmaximal, characteristic, complemented
5#11762920C211yesmaximal, characteristic, complemented
6C4×#63228C212yesmaximal, complemented
7#11762889C212yesmaximal, complemented
8#11762844C211yesmaximal, characteristic, complemented
9#11762812C212yesmaximal, complemented
10#11762766C211yesmaximal, characteristic, complemented
11C17×#442C211yesmaximal, characteristic, complemented
12C2×D4×D17C2212yes—
13C2×D8×C17C2211yescharacteristic
14C4×D4×C17C2212yescomplemented
15D4×Dic17C2212yescomplemented
16C2×#63228C2212yes—
17C4×C22⋊D17—22no—
18C2×#63227—22no—
19D8×D17C418yescomplemented
20C2×D136C2211yescharacteristic
21C2×#63218C2211yescharacteristic
22C2×C8×D17C2211yescharacteristic
23#10587579C2212yescomplemented
24#10587578C2212yes—
25#10587567—22no—
26C22⋊C4×D17—22no—
27C4×D68C2212yescomplemented
28#10587558C2211yescharacteristic
29C17×#106C2211yescharacteristic, complemented
30C17×#102C2212yescomplemented
31C17×#97C2211yescharacteristic, complemented
32#10587503C2212yescomplemented
33#10587492C2212yescomplemented
34#10587488C2211yescharacteristic, complemented
35#10587486C2211yescharacteristic, complemented
36C8×Dic17C2211yescharacteristic, complemented
37#10587480C2212yescomplemented
38#10587478C2212yescomplemented
39C4×#3622C2211yescharacteristic, complemented
40C23×D17—22no—
41C2×D4×C17C2312yes—
42C22×C4×C17—22no—
43C2×C22⋊D17—22no—
44C22×Dic17—22no—
45D4×D17—24no—
46D4×D17C2×C418yescomplemented
47C2×D68C2312yes—
48C2×C4×D17—22no—
49C2×C4×D17—22no—
50C2×C4×D17C2311yescharacteristic
51D8×C17C2×C414yescomplemented
52C2×C8×C17C2311yescharacteristic
53C2×C8×C17C2311yescharacteristic
54C4⋊C4×C17C2312yes—
55C22⋊C4×C17—22no—
56C42×C17C2311yescharacteristic
57C22⋊Dic17—22no—
58#63228C2×C418yescomplemented
59#63227—22no—
60#63227—22no—
61#63227—22no—
62#63227D412yes—
63C4⋊Dic17C2312yes—
64#63225D412yes—
65C4×Dic17—22no—
66C4×Dic17C2311yescharacteristic
67C2×#3622C2311yescharacteristic
68C2×#3622C2311yescharacteristic
69D136C2×C414yescomplemented
70#63218—22no—
71C8×D17C2×C414yescomplemented
72C23×C17—22no—
73C22×D17—22no—
74C22×D17—28no—
75C22×D17—82no—
76C22×D17—82no—
77C22×D17C2×D411yescharacteristic
78D4×C17—22no—
79D4×C17C22×C414yes—
80C2×C4×C17—22no—
81C2×C4×C17—82no—
82C2×C4×C17C2×D412yes—
83C2×C4×C17C2411yescharacteristic
84C22⋊D17—28no—
85C2×Dic17—22no—
86C2×Dic17—82no—
87C2×Dic17C2×D412yes—
88C2×Dic17C2×D411yescharacteristic
89D68—22no—
90D68C22×C414yes—
91C4×D17—82no—
92C4×D17—82no—
93C4×D17C22×C412yes—
94C4×D17#4112yes—
95C8×C17—21no—
96C8×C17C22×C412yes—
97#3622—21no—
98#3622C22×C412yes—
99#2948—171nomaximal, Sylow
100C22×C17—24no—
101C22×C17—42no—
102C22×C17—42no—
103C22×C17C22×D411yescharacteristic
104C2×D17—22no—
105C2×D17—24no—
106C2×D17—42no—
107C2×D17—42no—
108C2×D17—88no—
109C2×D17C4×D412yes—
110C4×C17—22no—
111C4×C17—42no—
112C4×C17C23×C411yescharacteristic
113C4×C17C2×#4111yescharacteristic
114Dic17—22no—
115Dic17—42no—
116Dic17C4×D412yes—
117C22×D8—171no—
118#518—172no—
119#442—171no—
120#442—171no—
121#442—171no—
122#442D1711yescharacteristic, complemented
123C4×D8—172no—
124C4×D8—172no—
125#427—171no—
126#418—172no—
127#405—171no—
128C2×C17—44no—
129C2×C17C2×C4×D411yescharacteristic
130C2×C17#57612yes—
131D17—22no—
132D17—44no—
133C22×D4—172no—
134C2×D8—172no—
135C2×D8—178no—
136C2×D8—171no—
137C2×D8C2×D1711yescharacteristic
138C2×M16—171no—
139C22×C8—171no—
140C4×D4—172no—
141C4×D4—172no—
142C4×D4—172no—
143C4×D4—342no—
144C4×D4—342no—
145C4×D4C2×D1712yescomplemented
146#117—171no—
147#117—172no—
148C2×C22⋊C4—342no—
149C2×C22⋊C4—342no—
150#107—172no—
151#106—171no—
152#106C2×D1711yescharacteristic, complemented
153#102—172no—
154#102—172no—
155#102—172no—
156#102C2×D1712yescomplemented
157#97—171no—
158#97C2×D1711yescharacteristic, complemented
159C4×C8—171no—
160C4×C8—171no—
161C17#294811yescharacteristic, complemented, Sylow
162C24—342no—
163C2×D4—178no—
164C2×D4—172no—
165C2×D4—344no—
166C2×D4—342no—
167C2×D4C22×D1712yes—
168C22×C4—22no—
169C22×C4—171no—
170C22×C4—342no—
171C22×C4—342no—
172C22×C4—342no—
173D8—178no—
174D8—174no—
175D8C4×D1714yescomplemented
176M16—342no—
177C2×C8—171no—
178C2×C8—174no—
179C2×C8—171no—
180C2×C8C22×D1711yescharacteristic
181C2×C8C22×D1711yescharacteristic
182C4⋊C4—172no—
183C4⋊C4—172no—
184C4⋊C4C22×D1712yes—
185C22⋊C4—22no—
186C22⋊C4—172no—
187C22⋊C4—342no—
188C22⋊C4—342no—
189C22⋊C4—342no—
190C22⋊C4—342no—
191C42—171no—
192C42—342no—
193C42C22×D1711yescharacteristic
194C23—22no—
195C23—171no—
196C23—348no—
197C23—342no—
198C23—1362no—
199C23—1362no—
200D4—22no—
201D4—174no—
202D4—348no—
203D4—342no—
204D4C2×C4×D1714yes—
205C2×C4—22no—
206C2×C4—82no—
207C2×C4—171no—
208C2×C4—172no—
209C2×C4—172no—
210C2×C4—172no—
211C2×C4—342no—
212C2×C4—1362no—
213C2×C4—1362no—
214C2×C4—1362no—
215C2×C4D4×D1712yes—
216C2×C4C23×D1711yescharacteristic
217C8—21no—
218C8—172no—
219C8—341no—
220C8C2×C4×D1712yes—
221C22—24no—
222C22—42no—
223C22—42no—
224C22—172no—
225C22—342no—
226C22—344no—
227C22—682no—
228C22—682no—
229C22—1368no—
230C22C2×D4×D1711yescharacteristic
231C4—22no—
232C4—42no—
233C4—172no—
234C4—342no—
235C4—682no—
236C4C22×C4×D1711yescharacteristic
237C4#41×D1711yescharacteristic
238C2—44no—
239C2—342no—
240C2—684no—
241C2C4×D4×D1711yescharacteristic
242C2#1176361312yes—
subgroupG/H
centerC22C2×D4×D17
commutatorC4×C17C23×C4
FrattiniC2×C4C23×D17
FittingC17×#442C2
socleC22×C17C22×D4
prime radicalC2×D8×D17C2
cosocleC2×C4×C17C24
hypercenter#442D17
hypocenterC17#2948
LCSG ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #11763281C2semidirect
3. #11762922C2semidirect
4. #11762921C2semidirect
5. #11762920C2semidirect
6. C4×#63228C2semidirect
7. #11762889C2semidirect
8. #11762844C2semidirect
9. #11762812C2semidirect
10. #11762766C2semidirect
11. C17×#442C2semidirect
14. C4×D4×C17C22semidirect
15. D4×Dic17C22semidirect
19. D8×D17C4semidirect
23. #10587579C22semidirect
27. C4×D68C22semidirect
29. C17×#106C22semidirect
30. C17×#102C22semidirect
31. C17×#97C22semidirect
32. #10587503C22semidirect
33. #10587492C22semidirect
34. #10587488C22semidirect
35. #10587486C22semidirect
36. C8×Dic17C22semidirect
37. #10587480C22semidirect
38. #10587478C22semidirect
39. C4×#3622C22semidirect
46. D4×D17C2×C4semidirect
51. D8×C17C2×C4semidirect
58. #63228C2×C4semidirect
69. D136C2×C4semidirect
71. C8×D17C2×C4semidirect
122. #442D17semidirect
145. C4×D4C2×D17semidirect
152. #106C2×D17semidirect
156. #102C2×D17semidirect
158. #97C2×D17semidirect
161. C17#2948semidirect
175. D8C4×D17semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D172
#117632812
#117629222
#117629212
#117629202
C4×#632282
#117628892
#117628442
#117628122
#117627662
C17×#4422
#294817