Finite Group Databasebeta

← #14539531

Subgroups — #14539531

Order 2176.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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#121×D17C212yesmaximal, complemented
2#11763124C212yesmaximal, complemented
3#11763121C212yesmaximal, complemented
4#11763045C211yesmaximal, characteristic, complemented
5#11763043C212yesmaximal, complemented
6#11762944C212yesmaximal, complemented
7#11762941C212yesmaximal, complemented
8C2×#10587491C211yesmaximal, characteristic, complemented
9C17×#466C211yesmaximal, characteristic, complemented
10C2×D4×D17—22no—
11C2×D4×D17C2212yescomplemented
12C22×C4×D17C2211yescharacteristic
13C2×D8×C17C2211yescharacteristic, complemented
14C22×C8×C17C2211yescharacteristic, complemented
15C17×#121C2212yescomplemented
16#10587625—22no—
17#10587624C2212yes—
18C2×#63228C2212yescomplemented
19#10587616C2212yescomplemented
20C22×#3622C2211yescharacteristic, complemented
21C2×D136C2211yescharacteristic, complemented
22#10587581C2212yes—
23C4⋊C4×D17C2212yescomplemented
24#10587570—22no—
25C22⋊C4×D17—22no—
26C17×#107C2211yescharacteristic, complemented
27C17×#102C2212yescomplemented
28#10587503C2212yescomplemented
29#10587492C2212yescomplemented
30#10587491C2212yescomplemented
31#10587491C2212yescomplemented
32#10587489C2211yescharacteristic, complemented
33#10587480C2212yescomplemented
34#10587478C2212yescomplemented
35C23×D17—22no—
36C23×D17D411yescharacteristic
37C2×D4×C17—22no—
38C2×D4×C17C2312yes—
39C22×C4×C17C2311yescharacteristic
40C2×C22⋊D17—22no—
41C2×C22⋊D17—22no—
42C2×C22⋊D17—22no—
43C22×Dic17D411yescharacteristic
44D4×D17—42no—
45D4×D17—42no—
46D4×D17—48no—
47C2×D68—22no—
48C2×D68C2312yes—
49C2×C4×D17—22no—
50C2×C4×D17—41no—
51C2×C4×D17—41no—
52C2×C4×D17D412yes—
53C2×C4×D17C2312yes—
54D8×C17—41no—
55C2×C8×C17—41no—
56C2×C8×C17C2311yescharacteristic
57C2×C8×C17C2311yescharacteristic
58C4⋊C4×C17C2312yes—
59C22⋊C4×C17—22no—
60C22⋊Dic17—22no—
61#63228—42no—
62#63227—22no—
63#63227—22no—
64C4⋊Dic17C2312yes—
65#63225—22no—
66C2×#3622—41no—
67C2×#3622C2312yes—
68D136—41no—
69C23×C17—22no—
70C23×C17C2×D411yescharacteristic
71C22×D17—22no—
72C22×D17—41no—
73C22×D17—41no—
74C22×D17—44no—
75C22×D17—44no—
76C22×D17—44no—
77C22×D17D814yescomplemented
78C22×D17C2×D412yes—
79D4×C17—22no—
80D4×C17—22no—
81D4×C17—44no—
82C2×C4×C17—21no—
83C2×C4×C17—21no—
84C2×C4×C17—22no—
85C2×C4×C17C2×D411yescharacteristic
86C2×C4×C17C2411yescharacteristic
87C22⋊D17—44no—
88C22⋊D17—44no—
89C22⋊D17—44no—
90C2×Dic17—22no—
91C2×Dic17—41no—
92C2×Dic17C2×D412yes—
93D68—22no—
94D68—22no—
95D68—44no—
96C4×D17—22no—
97C4×D17—22no—
98C4×D17—42no—
99C4×D17—44no—
100C8×C17—21no—
101C8×C17—21no—
102#3622—22no—
103#3186—171nomaximal, Sylow
104C22×C17—21no—
105C22×C17—21no—
106C22×C17—42no—
107C22×C17—42no—
108C22×C17—42no—
109C22×C17C2×D812yes—
110C22×C17C22×D411yescharacteristic
111C2×D17—22no—
112C2×D17—22no—
113C2×D17—44no—
114C2×D17—42no—
115C2×D17—42no—
116C2×D17—42no—
117C2×D17—88no—
118C2×D17C2×D814yes—
119C4×C17—21no—
120C4×C17—42no—
121C4×C17H(5,𝔽2)11yescharacteristic
122C4×C17H(5,𝔽2)11yescharacteristic
123Dic17—22no—
124Dic17—42no—
125C2×#121—172no—
126#480—172no—
127#480—172no—
128#466—171no—
129#466—172no—
130#466D1711yescharacteristic, complemented
131#447—172no—
132#447—172no—
133C2×C22⋊C8—171no—
134C2×C17—22no—
135C2×C17—82no—
136C2×C17#53411yescharacteristic
137C2×C17C22×D811yescharacteristic
138C2×C17#57511yescharacteristic
139D17—24no—
140D17—82no—
141C22×D4—172no—
142C22×D4—342no—
143C23×C4—171no—
144C2×D8—172no—
145C2×D8—171no—
146C2×D8C2×D1711yescharacteristic, complemented
147C22×C8—171no—
148C22×C8C2×D1711yescharacteristic, complemented
149#121—172no—
150#121—172no—
151#121—172no—
152#121—342no—
153#121—342no—
154#121C2×D1712yescomplemented
155C2×C4⋊C4—172no—
156C2×C22⋊C4—342no—
157#107—171no—
158#107—172no—
159#107C2×D1711yescharacteristic, complemented
160#102—172no—
161#102—172no—
162#102—172no—
163#102C2×D1712yescomplemented
164C22⋊C8—172no—
165C22⋊C8—172no—
166C17#318611yescharacteristic, complemented, Sylow
167C24—171no—
168C24—342no—
169C2×D4—22no—
170C2×D4—172no—
171C2×D4—342no—
172C2×D4—342no—
173C2×D4—342no—
174C2×D4—342no—
175C2×D4—682no—
176C2×D4—682no—
177C2×D4—688no—
178C2×D4C22×D1712yes—
179C22×C4—171no—
180C22×C4—172no—
181C22×C4—172no—
182C22×C4—342no—
183C22×C4—681no—
184C22×C4—681no—
185C22×C4C22×D1711yescharacteristic
186D8—41no—
187D8—681no—
188D8—682no—
189C2×C8—41no—
190C2×C8—172no—
191C2×C8—681no—
192C2×C8C22×D1711yescharacteristic
193C2×C8C22×D1711yescharacteristic
194C4⋊C4—172no—
195C4⋊C4—342no—
196C4⋊C4C22×D1712yes—
197C22⋊C4—22no—
198C22⋊C4—342no—
199C22⋊C4—342no—
200C22⋊C4—342no—
201C23—22no—
202C23—172no—
203C23—174no—
204C23—342no—
205C23—681no—
206C23—681no—
207C23—684no—
208C23—684no—
209C23—684no—
210C23D4×D1711yescharacteristic
211D4—22no—
212D4—22no—
213D4—44no—
214D4—342no—
215D4—342no—
216D4—684no—
217D4—684no—
218D4—684no—
219D4—684no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—172no—
224C2×C4—342no—
225C2×C4—342no—
226C2×C4—342no—
227C2×C4—682no—
228C2×C4—684no—
229C2×C4—681no—
230C2×C4D4×D1711yescharacteristic
231C2×C4C23×D1711yescharacteristic
232C8—21no—
233C8—21no—
234C8—342no—
235C22—21no—
236C22—21no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—174no—
241C22—342no—
242C22—342no—
243C22—684no—
244C22—682no—
245C22—682no—
246C22—682no—
247C22—1368no—
248C22D8×D1712yes—
249C22C2×D4×D1711yescharacteristic
250C4—21no—
251C4—42no—
252C4—342no—
253C4—682no—
254C4#1058769111yescharacteristic
255C4#1058769611yescharacteristic
256C2—22no—
257C2—82no—
258C2—344no—
259C2—1362no—
260C2#1176336711yescharacteristic
261C2C2×D8×D1711yescharacteristic
262C2#1176360611yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#466C2
socleC22×C17C22×D4
hypercenter#466D17
hypocenterC17#3186
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #121×D17C2semidirect
2. #11763124C2semidirect
3. #11763121C2semidirect
4. #11763045C2semidirect
5. #11763043C2semidirect
6. #11762944C2semidirect
7. #11762941C2semidirect
8. C2×#10587491C2semidirect
9. C17×#466C2semidirect
11. C2×D4×D17C22semidirect
13. C2×D8×C17C22semidirect
14. C22×C8×C17C22semidirect
15. C17×#121C22semidirect
18. C2×#63228C22semidirect
19. #10587616C22semidirect
20. C22×#3622C22semidirect
21. C2×D136C22semidirect
23. C4⋊C4×D17C22semidirect
26. C17×#107C22semidirect
27. C17×#102C22semidirect
28. #10587503C22semidirect
29. #10587492C22semidirect
30. #10587491C22semidirect
31. #10587491C22semidirect
32. #10587489C22semidirect
33. #10587480C22semidirect
34. #10587478C22semidirect
77. C22×D17D8semidirect
130. #466D17semidirect
146. C2×D8C2×D17semidirect
148. C22×C8C2×D17semidirect
154. #121C2×D17semidirect
159. #107C2×D17semidirect
163. #102C2×D17semidirect
166. C17#3186semidirect

Maximal subgroups

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

subgroupindex
#121×D172
#117631242
#117631212
#117630452
#117630432
#117629442
#117629412
C2×#105874912
C17×#4662
#318617