Finite Group Databasebeta

← D17⋊#327 #14544150

Subgroups — D17⋊#327 #14544150

Order 2176.

388 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 74 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#11763777C211yesmaximal, characteristic, complemented
2C2×(C2×#595)⋊C4C211yesmaximal, characteristic, complemented
3#11763661C212yesmaximal, complemented
4#11763660C212yesmaximal, complemented
5#121×D17C211yesmaximal, characteristic
6C23×#595—21no—
7C2×D4×D17—21no—
8C2×D4×D17C411yescharacteristic, complemented
9C22×C4×D17C2211yescharacteristic
10C2×#63242—21no—
11C17×#121C411yescharacteristic, complemented
12#10587625—21no—
13#10587624C2211yescharacteristic
14#10587616C411yescharacteristic, complemented
15#10587581C2211yescharacteristic
16C4⋊C4×D17C411yescharacteristic, complemented
17#10587570—21no—
18C22⋊C4×D17—21no—
19C22⋊#3624—21no—
20(C2×#595)⋊C4—21no—
21(C2×#595)⋊C4C2212yescomplemented
22#10587543C2212yescomplemented
23C23×D17—21no—
24C23×D17D411yescharacteristic
25C22×#595—24no—
26C22×#595—22no—
27C22×#595—42no—
28C22×#595—42no—
29C2×D4×C17—21no—
30C2×D4×C17C2×C411yescharacteristic, complemented
31C22×C4×C17C2×C411yescharacteristic
32C2×C22⋊D17—21no—
33C2×C22⋊D17—21no—
34C2×C22⋊D17—21no—
35C22×Dic17D411yescharacteristic
36D4×D17—42no—
37D4×D17—42no—
38D4×D17—42no—
39C2×D68—21no—
40C2×D68C2×C411yescharacteristic, complemented
41C2×C4×D17—21no—
42C2×C4×D17—42no—
43C2×C4×D17C2×C411yescharacteristic
44C2×C4×D17D411yescharacteristic
45C2×C4×D17D411yescharacteristic
46C2×C4×D17C2311yescharacteristic
47C2×#3624—22no—
48#63242—42no—
49C4⋊C4×C17C2×C411yescharacteristic, complemented
50C22⋊C4×C17—21no—
51C22⋊Dic17—21no—
52#63227—21no—
53#63227—21no—
54C4⋊Dic17C2×C411yescharacteristic, complemented
55#63225—21no—
56C23×C17—21no—
57C23×C17C22⋊C411yescharacteristic
58C22×D17—22no—
59C22×D17—22no—
60C22×D17—21no—
61C22×D17—42no—
62C22×D17—42no—
63C22×D17—42no—
64C22×D17C22⋊C411yescharacteristic
65C22×D17D812yes—
66C22×D17SD1612yes—
67C22×D17C2×D411yescharacteristic
68C2×#595—24no—
69C2×#595—42no—
70C2×#595—48no—
71C2×#595—42no—
72D4×C17—41no—
73D4×C17—42no—
74C2×C4×C17—21no—
75C2×C4×C17—41no—
76C2×C4×C17C22⋊C411yescharacteristic
77C2×C4×C17C22×C411yescharacteristic
78C22⋊D17—42no—
79C22⋊D17—42no—
80C22⋊D17—41no—
81C22⋊D17—41no—
82C2×Dic17—21no—
83C2×Dic17—41no—
84C2×Dic17C22⋊C411yescharacteristic
85C2×Dic17C2×D411yescharacteristic
86D68—41no—
87D68—41no—
88D68—41no—
89C4×D17—22no—
90C4×D17—22no—
91C4×D17—41no—
92C4×D17—41no—
93C4×D17—42no—
94#3624—42no—
95C2×#327—171nomaximal, Sylow
96C22×C17—21no—
97C22×C17—21no—
98C22×C17—41no—
99C22×C17—41no—
100C22×C17—41no—
101C22×C17#10212yes—
102C22×C17C2×C22⋊C411yescharacteristic
103C2×D17—21no—
104C2×D17—21no—
105C2×D17—24no—
106C2×D17—24no—
107C2×D17—41no—
108C2×D17—41no—
109C2×D17—41no—
110C2×D17—84no—
111C2×D17#9912yes—
112C2×D17#10212yes—
113C2×D17#10412yes—
114C2×D17C2×D811yescharacteristic
115C2×D17C2×SD1611yescharacteristic
116#595—44no—
117C4×C17—21no—
118C4×C17—21no—
119C4×C17—41no—
120Dic17—21no—
121Dic17—21no—
122Dic17—41no—
123C2×#121—171no—
124C2×C22⋊C8—171no—
125C2×#95—171no—
126#327—172no—
127#327—172no—
128C2×C17—22no—
129C2×C17—81no—
130C2×C17C2×#9911yescharacteristic
131C2×C17C2×#10211yescharacteristic
132C2×C17C2×#10411yescharacteristic
133D17—21no—
134D17—21no—
135D17—81no—
136D17#32714yescomplemented
137C22×D4—171no—
138C22×D4—341no—
139C23×C4—171no—
140C23×C4—341no—
141C22×C8—341no—
142#121—171no—
143#121—171no—
144#121—171no—
145#121—341no—
146#121—341no—
147#121#59511yescharacteristic, complemented
148C2×C4⋊C4—171no—
149C2×C22⋊C4—341no—
150C22⋊C8—172no—
151C22⋊C8—341no—
152#95—172no—
153#95—341no—
154C17C2×#32711yescharacteristic, complemented, Sylow
155C24—171no—
156C24—341no—
157C2×D4—21no—
158C2×D4—171no—
159C2×D4—341no—
160C2×D4—341no—
161C2×D4—341no—
162C2×D4—341no—
163C2×D4—682no—
164C2×D4—682no—
165C2×D4—682no—
166C2×D4C2×#59511yescharacteristic, complemented
167C22×C4—171no—
168C22×C4—171no—
169C22×C4—171no—
170C22×C4—171no—
171C22×C4—171no—
172C22×C4—341no—
173C22×C4—342no—
174C22×C4—344no—
175C22×C4—682no—
176C22×C4—682no—
177C22×C4—682no—
178C22×C4C2×#59511yescharacteristic
179C2×C8—342no—
180C2×C8—682no—
181C4⋊C4—171no—
182C4⋊C4—341no—
183C4⋊C4C2×#59511yescharacteristic, complemented
184C22⋊C4—21no—
185C22⋊C4—341no—
186C22⋊C4—341no—
187C22⋊C4—341no—
188C23—21no—
189C23—171no—
190C23—171no—
191C23—172no—
192C23—172no—
193C23—342no—
194C23—342no—
195C23—341no—
196C23—682no—
197C23—682no—
198C23—682no—
199C23C22⋊#59511yescharacteristic
200D4—41no—
201D4—42no—
202D4—681no—
203D4—681no—
204D4—681no—
205D4—682no—
206D4—681no—
207D4—681no—
208D4—682no—
209C2×C4—21no—
210C2×C4—41no—
211C2×C4—171no—
212C2×C4—171no—
213C2×C4—342no—
214C2×C4—342no—
215C2×C4—344no—
216C2×C4—341no—
217C2×C4—681no—
218C2×C4—681no—
219C2×C4—682no—
220C2×C4—681no—
221C2×C4—682no—
222C2×C4—688no—
223C2×C4—682no—
224C2×C4C22⋊#59511yescharacteristic
225C2×C4C22×#59511yescharacteristic
226C8—682no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—172no—
233C22—172no—
234C22—172no—
235C22—171no—
236C22—171no—
237C22—344no—
238C22—344no—
239C22—341no—
240C22—341no—
241C22—681no—
242C22—681no—
243C22—681no—
244C22—1364no—
245C22#1058754712yes—
246C22C2×C22⋊#59511yescharacteristic
247C4—21no—
248C4—21no—
249C4—41no—
250C4—341no—
251C4—341no—
252C4—684no—
253C4—681no—
254C2—22no—
255C2—81no—
256C2—174no—
257C2—341no—
258C2—341no—
259C2—1361no—
260C2D17⋊#9911yescharacteristic
261C2C2×#1058754711yescharacteristic
262C2D17⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#121C4
socleC22×C17C2×C22⋊C4
prime radical#121×D17C2
cosocleC2×C4×D17C23
hypercenter#121#595
hypocenterC17C2×#327
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #11763777C2semidirect
2. C2×(C2×#595)⋊C4C2semidirect
3. #11763661C2semidirect
4. #11763660C2semidirect
8. C2×D4×D17C4semidirect
11. C17×#121C4semidirect
14. #10587616C4semidirect
16. C4⋊C4×D17C4semidirect
21. (C2×#595)⋊C4C22semidirect
22. #10587543C22semidirect
30. C2×D4×C17C2×C4semidirect
40. C2×D68C2×C4semidirect
49. C4⋊C4×C17C2×C4semidirect
54. C4⋊Dic17C2×C4semidirect
136. D17#327semidirect
147. #121#595semidirect
154. C17C2×#327semidirect
166. C2×D4C2×#595semidirect
183. C4⋊C4C2×#595semidirect

Maximal subgroups

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

subgroupindex
#117637772
C2×(C2×#595)⋊C42
#117636612
#117636602
#121×D172
C2×#32717