Finite Group Databasebeta

← D17×#342 #14528512

Subgroups — D17×#342 #14528512

Order 2176.

680 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 145 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×C22⋊C4×D17C212yesmaximal
2C2×C22⋊C4×D17C211yesmaximal, characteristic
3#11762576C211yesmaximal, characteristic, complemented
4#11762575C212yesmaximal, complemented
5C17×#342C211yesmaximal, characteristic, complemented
6C24×D17C2211yescharacteristic
7C22×C4×D17—22no—
8C22×C4×D17C412yescomplemented
9C2×C22⋊C4×C17C2211yescharacteristic
10C2×C22⋊C4×C17C2212yes—
11C2×C22⋊Dic17C2211yescharacteristic
12C2×C22⋊Dic17C2212yes—
13C2×#63227—22no—
14C2×#63227C412yescomplemented
15C22⋊C4×D17—24no—
16C22⋊C4×D17—22no—
17C22⋊C4×D17—22no—
18C22⋊C4×D17C418yescomplemented
19C24×C17C2311yescharacteristic
20C23×D17—44no—
21C23×D17C2×C411yescharacteristic
22C23×D17C2×C414yes—
23C23×D17C2×C411yescharacteristic
24C23×D17D412yes—
25C23×D17D412yes—
26C23×D17D412yes—
27C23×D17Q812yes—
28C22×C4×C17—22no—
29C22×C4×C17C2×C412yescomplemented
30C22×Dic17—22no—
31C22×Dic17C2×C412yescomplemented
32C2×C4×D17—28no—
33C2×C4×D17—22no—
34C2×C4×D17—22no—
35C2×C4×D17—48no—
36C2×C4×D17—44no—
37C22⋊C4×C17—22no—
38C22⋊C4×C17—22no—
39C22⋊C4×C17C2×C414yescomplemented
40C22⋊Dic17—22no—
41C22⋊Dic17—22no—
42C22⋊Dic17C2×C414yescomplemented
43#63227—24no—
44#63227—28no—
45C23×C17—42no—
46C23×C17C22×C411yescharacteristic
47C23×C17C22×C412yes—
48C23×C17C2×D412yes—
49C23×C17C2×D411yescharacteristic
50C23×C17C2×Q811yescharacteristic
51C22×D17—24no—
52C22×D17—28no—
53C22×D17—48no—
54C22×D17—44no—
55C22×D17—44no—
56C22×D17—42no—
57C22×D17—44no—
58C22×D17C4212yes—
59C22×D17C4212yes—
60C22×D17C22⋊C414yes—
61C22×D17C22⋊C411yescharacteristic
62C22×D17C22⋊C412yes—
63C22×D17C22⋊C414yes—
64C22×D17C22⋊C411yescharacteristic
65C22×D17C4⋊C412yes—
66C22×D17C4⋊C414yes—
67C22×D17C4⋊C414yes—
68C22×D17C4⋊C412yes—
69C2×C4×C17—22no—
70C2×C4×C17—24no—
71C2×C4×C17—44no—
72C2×C4×C17—42no—
73C2×Dic17—22no—
74C2×Dic17—24no—
75C2×Dic17—44no—
76C2×Dic17—42no—
77C4×D17—416no—
78C4×D17—44no—
79C4×D17—44no—
80C2×#342—171nomaximal, Sylow
81C22×C17—22no—
82C22×C17—24no—
83C22×C17—42no—
84C22×C17—41no—
85C22×C17—41no—
86C22×C17C2×C4211yescharacteristic
87C22×C17C2×C22⋊C412yes—
88C22×C17C2×C22⋊C411yescharacteristic
89C22×C17C2×C4⋊C411yescharacteristic
90C22×C17C2×C4⋊C412yes—
91C2×D17—22no—
92C2×D17—24no—
93C2×D17—28no—
94C2×D17—24no—
95C2×D17—24no—
96C2×D17—28no—
97C2×D17—42no—
98C2×D17—44no—
99C2×D17—44no—
100C2×D17—42no—
101C2×D17#9512yes—
102C2×D17#9514yes—
103C2×D17#9512yes—
104C2×D17#9914yes—
105C4×C17—44no—
106C4×C17—42no—
107Dic17—44no—
108Dic17—42no—
109C22×C22⋊C4—172no—
110C22×C22⋊C4—171no—
111#342—172no—
112#342—171no—
113#342D1711yescharacteristic, direct product factor
114C2×C17—22no—
115C2×C17—21no—
116C2×C17—21no—
117C2×C17—22no—
118C2×C17C2×#9511yescharacteristic
119C2×C17C2×#9912yes—
120D17—22no—
121D17—24no—
122D17#34214yesdirect product factor
123C25—171no—
124C23×C4—172no—
125C23×C4—342no—
126C2×C22⋊C4—172no—
127C2×C22⋊C4—178no—
128C2×C22⋊C4—171no—
129C2×C22⋊C4—172no—
130C2×C22⋊C4—344no—
131C2×C22⋊C4—342no—
132C2×C22⋊C4—342no—
133C2×C22⋊C4—342no—
134C2×C22⋊C4C2×D1711yescharacteristic
135C2×C22⋊C4C2×D1712yes—
136C17C2×#34211yescharacteristic, complemented, Sylow
137C24—171no—
138C24—172no—
139C24—174no—
140C24—171no—
141C24—172no—
142C24—172no—
143C24—172no—
144C24—684no—
145C24C22×D1711yescharacteristic
146C22×C4—22no—
147C22×C4—172no—
148C22×C4—348no—
149C22×C4—342no—
150C22×C4—342no—
151C22×C4—342no—
152C22×C4—688no—
153C22×C4—684no—
154C22×C4C4×D1712yescomplemented
155C22⋊C4—22no—
156C22⋊C4—22no—
157C22⋊C4—174no—
158C22⋊C4—348no—
159C22⋊C4—344no—
160C22⋊C4—342no—
161C22⋊C4—342no—
162C22⋊C4C4×D1714yescomplemented
163C23—42no—
164C23—171no—
165C23—172no—
166C23—174no—
167C23—172no—
168C23—171no—
169C23—172no—
170C23—174no—
171C23—172no—
172C23—174no—
173C23—174no—
174C23—172no—
175C23—344no—
176C23—348no—
177C23—682no—
178C23—684no—
179C23—688no—
180C23—684no—
181C23—684no—
182C23C2×C4×D1711yescharacteristic
183C23C2×C4×D1712yes—
184C23D4×D1712yes—
185C23D4×D1711yescharacteristic
186C23Q8×D1711yescharacteristic
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—344no—
192C2×C4—342no—
193C2×C4—684no—
194C2×C4—6816no—
195C2×C4—684no—
196C2×C4—684no—
197C2×C4—682no—
198C22—22no—
199C22—24no—
200C22—42no—
201C22—41no—
202C22—41no—
203C22—174no—
204C22—172no—
205C22—172no—
206C22—174no—
207C22—348no—
208C22—344no—
209C22—344no—
210C22—348no—
211C22—342no—
212C22—344no—
213C22—684no—
214C22—682no—
215C22—682no—
216C22—684no—
217C22C42×D1711yescharacteristic
218C22C22⋊C4×D1712yes—
219C22C22⋊C4×D1711yescharacteristic
220C22C4⋊C4×D1711yescharacteristic
221C22C4⋊C4×D1712yes—
222C4—44no—
223C4—42no—
224C4—684no—
225C4—682no—
226C2—22no—
227C2—21no—
228C2—21no—
229C2—22no—
230C2—174no—
231C2—344no—
232C2—342no—
233C2#95×D1711yescharacteristic
234C2#99×D1712yes—
subgroupG/H
centerC22C22⋊C4×D17
commutatorC22×C17C2×C42
FrattiniC24C22×D17
FittingC17×#342C2
socleC22×C17C2×C22⋊C4
prime radicalC24×D17C22
cosocleC24×C17C23
hypercenter#342D17
hypocenterC17C2×#342
LCSG ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
3. #11762576C2semidirect
4. #11762575C2semidirect
5. C17×#342C2semidirect
8. C22×C4×D17C4semidirect
14. C2×#63227C4semidirect
18. C22⋊C4×D17C4semidirect
29. C22×C4×C17C2×C4semidirect
31. C22×Dic17C2×C4semidirect
39. C22⋊C4×C17C2×C4semidirect
42. C22⋊Dic17C2×C4semidirect
113. #342D17direct (N × H)
122. D17#342direct (N × H)
136. C17C2×#342semidirect
154. C22×C4C4×D17semidirect
162. C22⋊C4C4×D17semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D172
#117625762
#117625752
C17×#3422
C2×#34217