Finite Group Databasebeta

← #14537243

Subgroups — #14537243

Order 2176.

800 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 209 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×#10587696C211yesmaximal, characteristic
2#11763283C212yesmaximal, complemented
3#11762783C211yesmaximal, characteristic, complemented
4#11762782C212yesmaximal, complemented
5#99×D17C218yesmaximal, complemented
6C17×#410C211yesmaximal, characteristic, complemented
7C2×#41×C17C2211yescharacteristic
8#10587696—24no—
9#10587696—24no—
10C2×#63256C412yescomplemented
11C2×D4×D17C412yescomplemented
12C2×D4×D17C2214yes—
13C2×#63252C412yescomplemented
14C2×#63252C2211yescharacteristic
15C22×D68C412yescomplemented
16C22×D68C2211yescharacteristic
17C17×#117C2212yescomplemented
18#10587612C2212yescomplemented
19#10587579—24no—
20C22⋊C4×D17C2218yescomplemented
21C4×D68—24no—
22C17×#99C2214yescomplemented
23#10587505C2214yescomplemented
24#10587477C2218yescomplemented
25C23×D17C2×C414yescomplemented
26C23×D17C2312yes—
27#41×C17—22no—
28#41×C17—22no—
29C2×Q8×C17C2×C411yescharacteristic, complemented
30C2×D4×C17C2×C411yescharacteristic, complemented
31C2×D4×C17C2312yes—
32C22×C4×C17C2×C412yescomplemented
33C22×C4×C17C2311yescharacteristic
34C2×C22⋊D17C2×C414yescomplemented
35C2×C22⋊D17C2312yes—
36#63256—28no—
37D4×D17—216no—
38D4×D17—28no—
39#63252—22no—
40#63252—22no—
41#63252—24no—
42#63252—24no—
43C2×D68—24no—
44C2×D68—24no—
45C2×D68C2×C412yes—
46C2×D68C2×C412yes—
47C2×D68C2×C411yescharacteristic, complemented
48C2×D68C2×C414yescomplemented
49C2×D68D414yes—
50C2×D68D414yes—
51C2×C4×D17—28no—
52C2×C4×D17C2×C414yescomplemented
53C2×C4×D17C2×C412yescomplemented
54C2×Dic34C2×C411yescharacteristic, complemented
55C4⋊C4×C17—22no—
56C22⋊C4×C17C2314yes—
57C42×C17—22no—
58C22⋊Dic17C2314yes—
59#63227—28no—
60C4⋊Dic17—22no—
61C4×Dic17—22no—
62C23×C17C22×C412yes—
63C23×C17C2411yescharacteristic
64C22×D17—28no—
65C22×D17—82no—
66C22×D17—84no—
67C22×D17C22×C412yes—
68C22×D17C22×C414yes—
69C22×D17C2×D418yes—
70Q8×C17—22no—
71D4×C17—24no—
72D4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—24no—
76C2×C4×C17C22×C412yes—
77C2×C4×C17C22×C411yescharacteristic
78C2×C4×C17C22×C411yescharacteristic
79C2×C4×C17C2×D412yes—
80C2×C4×C17C2×D412yes—
81C22⋊D17—24no—
82C22⋊D17—28no—
83C2×Dic17—24no—
84C2×Dic17C22×C412yes—
85D68—28no—
86D68—21no—
87D68—21no—
88D68C22⋊C414yes—
89D68C22⋊C414yes—
90D68C22⋊C418yes—
91C4×D17—28no—
92C4×D17—24no—
93C4×D17—88no—
94Dic34—21no—
95Dic34—21no—
96#2883—171nomaximal, Sylow
97C22×C17—22no—
98C22×C17—41no—
99C22×C17—42no—
100C22×C17C23×C411yescharacteristic
101C22×C17C22×D412yes—
102C2×D17—22no—
103C2×D17—48no—
104C2×D17—44no—
105C2×D17—42no—
106C2×D17—88no—
107C2×D17C2×C22⋊C418yes—
108C4×C17—22no—
109C4×C17—44no—
110C4×C17C2×C22⋊C411yescharacteristic
111C4×C17C2×C22⋊C411yescharacteristic
112C4×C17C2×C22⋊C412yes—
113Dic17—22no—
114Dic17—44no—
115C2×H(5,𝔽2)—171no—
116#518—172no—
117#410—171no—
118#410—172no—
119#410D1711yescharacteristic, complemented
120C2×#99—178no—
121C2×C17—22no—
122C2×C17—21no—
123C2×C17—42no—
124C2×C17C22×C22⋊C411yescharacteristic
125D17—28no—
126D17—42no—
127H(5,𝔽2)—344no—
128H(5,𝔽2)—344no—
129C2×#41—172no—
130C2×#41—172no—
131C2×#41—171no—
132C2×#41C2×D1711yescharacteristic
133C22×D4—172no—
134C22×D4—174no—
135C22×D4—172no—
136C22×D4—171no—
137C4×D4—344no—
138C4×D4—344no—
139#117—172no—
140#117C2×D1712yescomplemented
141C2×C22⋊C4—178no—
142#99—174no—
143#99—178no—
144#99C2×D1714yescomplemented
145C17#288311yescharacteristic, complemented, Sylow
146C24—172no—
147C24—174no—
148#41—22no—
149#41—22no—
150#41—344no—
151#41—344no—
152#41—348no—
153#41—342no—
154#41—342no—
155C2×Q8—171no—
156C2×Q8C4×D1711yescharacteristic, complemented
157C2×D4—174no—
158C2×D4—172no—
159C2×D4—172no—
160C2×D4—172no—
161C2×D4—174no—
162C2×D4—171no—
163C2×D4—174no—
164C2×D4—174no—
165C2×D4—344no—
166C2×D4—344no—
167C2×D4—348no—
168C2×D4—3416no—
169C2×D4C4×D1711yescharacteristic, complemented
170C2×D4C22×D1712yes—
171C22×C4—174no—
172C22×C4—172no—
173C22×C4—348no—
174C22×C4C4×D1712yescomplemented
175C22×C4C22×D1711yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—342no—
178C22⋊C4—174no—
179C22⋊C4—348no—
180C22⋊C4C22×D1714yes—
181C42—22no—
182C42—342no—
183C23—174no—
184C23—178no—
185C23—172no—
186C23—348no—
187C23—1362no—
188C23—1364no—
189C23C2×C4×D1712yes—
190C23C23×D1711yescharacteristic
191Q8—22no—
192Q8—341no—
193Q8—341no—
194D4—24no—
195D4—22no—
196D4—174no—
197D4—174no—
198D4—178no—
199D4—348no—
200D4—348no—
201D4—344no—
202D4—341no—
203D4—341no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—172no—
208C2×C4—348no—
209C2×C4—344no—
210C2×C4—344no—
211C2×C4—1368no—
212C2×C4C2×C4×D1712yes—
213C2×C4C2×C4×D1711yescharacteristic
214C2×C4C2×C4×D1711yescharacteristic
215C2×C4D4×D1712yes—
216C2×C4D4×D1712yes—
217C22—22no—
218C22—41no—
219C22—42no—
220C22—178no—
221C22—342no—
222C22—688no—
223C22—684no—
224C22—682no—
225C22—1368no—
226C22C22×C4×D1711yescharacteristic
227C22C2×D4×D1712yes—
228C4—22no—
229C4—44no—
230C4—342no—
231C4—684no—
232C4C22⋊C4×D1711yescharacteristic
233C4C22⋊C4×D1711yescharacteristic
234C4C22⋊C4×D1712yes—
235C2—22no—
236C2—21no—
237C2—42no—
238C2—348no—
239C2—682no—
240C2C2×C22⋊C4×D1711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D17
commutatorC22×C17C23×C4
FrattiniC23C23×D17
FittingC17×#410C2
socleC2×C17C22×C22⋊C4
prime radicalC2×#10587696C2
cosocleC23×C17C24
hypercenter#410D17
hypocenterC17#2883
LCSG ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #11763283C2semidirect
3. #11762783C2semidirect
4. #11762782C2semidirect
5. #99×D17C2semidirect
6. C17×#410C2semidirect
10. C2×#63256C4semidirect
11. C2×D4×D17C4semidirect
13. C2×#63252C4semidirect
15. C22×D68C4semidirect
17. C17×#117C22semidirect
18. #10587612C22semidirect
20. C22⋊C4×D17C22semidirect
22. C17×#99C22semidirect
23. #10587505C22semidirect
24. #10587477C22semidirect
25. C23×D17C2×C4semidirect
29. C2×Q8×C17C2×C4semidirect
30. C2×D4×C17C2×C4semidirect
32. C22×C4×C17C2×C4semidirect
34. C2×C22⋊D17C2×C4semidirect
47. C2×D68C2×C4semidirect
48. C2×D68C2×C4semidirect
52. C2×C4×D17C2×C4semidirect
53. C2×C4×D17C2×C4semidirect
54. C2×Dic34C2×C4semidirect
119. #410D17semidirect
140. #117C2×D17semidirect
144. #99C2×D17semidirect
145. C17#2883semidirect
156. C2×Q8C4×D17semidirect
169. C2×D4C4×D17semidirect
174. C22×C4C4×D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876962
#117632832
#117627832
#117627822
#99×D172
C17×#4102
#288317