Finite Group Databasebeta

← D17×#328 #14527263

Subgroups — D17×#328 #14527263

Order 2176.

336 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C22⋊Q8×D17C211yesmaximal, characteristic
2C22⋊C8×D17C211yesmaximal, characteristic
3#95×D17C211yesmaximal, characteristic
4#11762547C211yesmaximal, characteristic, complemented
5#11762546C211yesmaximal, characteristic, complemented
6#11762545C211yesmaximal, characteristic, complemented
7C17×#328C211yesmaximal, characteristic, complemented
8C2×Q8×D17C411yescharacteristic, complemented
9C22×C4×D17—21no—
10C22×C4×D17C2211yescharacteristic
11C17×C22⋊Q8C2211yescharacteristic
12#10587632C411yescharacteristic, complemented
13#10587610C2211yescharacteristic
14C2×C8×D17—21no—
15#10587583C411yescharacteristic, complemented
16#10587582—21no—
17C4⋊C4×D17—21no—
18C4⋊C4×D17C411yescharacteristic, complemented
19C22⋊C4×D17—21no—
20#10587564—21no—
21C17×C22⋊C8C2211yescharacteristic
22C17×#95C2211yescharacteristic
23#10587502—21no—
24#10587502C2211yescharacteristic
25C22⋊#3622C2211yescharacteristic
26#10587491—21no—
27C23×D17D411yescharacteristic
28C2×Q8×C17C2×C411yescharacteristic, complemented
29C22×C4×C17—21no—
30C22×C4×C17C2311yescharacteristic
31C22×Dic17—21no—
32C22×Dic17D411yescharacteristic
33Q8×D17—42no—
34C2×C4×D17—24no—
35C2×C4×D17—21no—
36C2×C4×D17—21no—
37C2×C4×D17—42no—
38C2×C4×D17—42no—
39C2×C4×D17—42no—
40C2×C4×D17C2×C411yescharacteristic
41C2×C4×D17C2×C411yescharacteristic
42C2×C4×D17D411yescharacteristic
43C2×C4×D17D411yescharacteristic
44C2×Dic34—21no—
45C2×Dic34C2×C411yescharacteristic, complemented
46C2×C8×C17—21no—
47C4⋊C4×C17—21no—
48C4⋊C4×C17C2×C411yescharacteristic, complemented
49C22⋊C4×C17—21no—
50C22⋊Dic17—21no—
51#63227—21no—
52#63227—21no—
53C4⋊Dic17—21no—
54C4⋊Dic17C2×C411yescharacteristic, complemented
55#63225—21no—
56#63225—21no—
57#63225—21no—
58C2×#3622—21no—
59C8×D17—42no—
60C23×C17C2×D411yescharacteristic
61C22×D17—22no—
62C22×D17—22no—
63C22×D17C22⋊C411yescharacteristic
64C22×D17C22⋊C411yescharacteristic
65C22×D17SD1612yes—
66C22×D17Q1612yes—
67Q8×C17—41no—
68C2×C4×C17—22no—
69C2×C4×C17—21no—
70C2×C4×C17—21no—
71C2×C4×C17—41no—
72C2×C4×C17—41no—
73C2×C4×C17—41no—
74C2×C4×C17C22×C411yescharacteristic
75C2×C4×C17C2×D411yescharacteristic
76C2×Dic17—22no—
77C2×Dic17—21no—
78C2×Dic17—21no—
79C2×Dic17—41no—
80C2×Dic17—41no—
81C2×Dic17—41no—
82C2×Dic17C22⋊C411yescharacteristic
83C2×Dic17C22⋊C411yescharacteristic
84C4×D17—22no—
85C4×D17—22no—
86C4×D17—41no—
87C4×D17—41no—
88C4×D17—48no—
89C4×D17—42no—
90C4×D17—42no—
91Dic34—42no—
92Dic34—41no—
93C8×C17—41no—
94#3622—41no—
95C2×#328—171nomaximal, Sylow
96C22×C17—21no—
97C22×C17—21no—
98C22×C17C2×C22⋊C411yescharacteristic
99C22×C17C2×SD1611yescharacteristic
100C22×C17C2×Q1611yescharacteristic
101C2×D17—21no—
102C2×D17—21no—
103C2×D17—24no—
104C2×D17—24no—
105C2×D17#9912yes—
106C2×D17#10312yes—
107C2×D17#10312yes—
108C2×D17#10412yes—
109C4×C17—21no—
110C4×C17—21no—
111C4×C17—42no—
112C4×C17—41no—
113C4×C17—41no—
114Dic17—21no—
115Dic17—21no—
116Dic17—42no—
117Dic17—41no—
118Dic17—41no—
119C2×C22⋊Q8—171no—
120C2×C22⋊C8—171no—
121C2×#95—171no—
122#328—171no—
123#328—171no—
124#328—171no—
125#328D1711yescharacteristic, direct product factor
126C2×C17—21no—
127C2×C17—21no—
128C2×C17C2×#9911yescharacteristic
129C2×C17C2×#10311yescharacteristic
130C2×C17C2×#10411yescharacteristic
131D17—22no—
132D17#32814yesdirect product factor
133C22×Q8—171no—
134C23×C4—171no—
135C23×C4—341no—
136C22×C8—341no—
137C22⋊Q8—171no—
138C22⋊Q8—171no—
139C22⋊Q8—171no—
140C22⋊Q8—341no—
141C22⋊Q8—341no—
142C22⋊Q8C2×D1711yescharacteristic
143C2×C4⋊C4—171no—
144C2×C4⋊C4—341no—
145C2×C22⋊C4—341no—
146C22⋊C8—171no—
147C22⋊C8—341no—
148C22⋊C8C2×D1711yescharacteristic
149#95—171no—
150#95—341no—
151#95C2×D1711yescharacteristic
152C17C2×#32811yescharacteristic, complemented, Sylow
153C24—171no—
154C2×Q8—171no—
155C2×Q8—341no—
156C2×Q8—682no—
157C2×Q8C4×D1711yescharacteristic, complemented
158C22×C4—21no—
159C22×C4—171no—
160C22×C4—171no—
161C22×C4—171no—
162C22×C4—171no—
163C22×C4—171no—
164C22×C4—341no—
165C22×C4—341no—
166C22×C4—344no—
167C22×C4—341no—
168C22×C4—682no—
169C22×C4—682no—
170C22×C4—682no—
171C22×C4C22×D1711yescharacteristic
172C2×C8—21no—
173C2×C8—341no—
174C2×C8—682no—
175C4⋊C4—21no—
176C4⋊C4—171no—
177C4⋊C4—341no—
178C4⋊C4—341no—
179C4⋊C4—341no—
180C4⋊C4—341no—
181C4⋊C4C4×D1711yescharacteristic, complemented
182C22⋊C4—21no—
183C22⋊C4—341no—
184C22⋊C4—341no—
185C22⋊C4—341no—
186C23—171no—
187C23—171no—
188C23—172no—
189C23—172no—
190C23—342no—
191C23—342no—
192C23D4×D1711yescharacteristic
193Q8—41no—
194Q8—681no—
195Q8—682no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—171no—
203C2×C4—171no—
204C2×C4—342no—
205C2×C4—342no—
206C2×C4—342no—
207C2×C4—341no—
208C2×C4—341no—
209C2×C4—681no—
210C2×C4—681no—
211C2×C4—682no—
212C2×C4—682no—
213C2×C4—688no—
214C2×C4—681no—
215C2×C4—681no—
216C2×C4—681no—
217C2×C4C2×C4×D1711yescharacteristic
218C2×C4D4×D1711yescharacteristic
219C8—41no—
220C8—681no—
221C22—21no—
222C22—21no—
223C22—172no—
224C22—172no—
225C22—172no—
226C22—172no—
227C22—344no—
228C22—344no—
229C22—341no—
230C22—341no—
231C22C22⋊C4×D1711yescharacteristic
232C22SD16×D1711yescharacteristic
233C22Q16×D1711yescharacteristic
234C4—21no—
235C4—21no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—341no—
240C4—341no—
241C4—682no—
242C4—681no—
243C4—681no—
244C2—21no—
245C2—21no—
246C2—174no—
247C2—342no—
248C2#99×D1711yescharacteristic
249C2#103×D1711yescharacteristic
250C2#104×D1711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D17
commutatorC2×C4×C17C22×C4
FrattiniC22×C4C22×D17
FittingC17×#328C2
socleC22×C17C2×C22⋊C4
prime radicalC23×D17D4
cosocleC22×C4×C17C23
hypercenter#328D17
hypocenterC17C2×#328
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #11762547C2semidirect
5. #11762546C2semidirect
6. #11762545C2semidirect
7. C17×#328C2semidirect
8. C2×Q8×D17C4semidirect
12. #10587632C4semidirect
15. #10587583C4semidirect
18. C4⋊C4×D17C4semidirect
28. C2×Q8×C17C2×C4semidirect
45. C2×Dic34C2×C4semidirect
48. C4⋊C4×C17C2×C4semidirect
54. C4⋊Dic17C2×C4semidirect
125. #328D17direct (N × H)
132. D17#328direct (N × H)
152. C17C2×#328semidirect
157. C2×Q8C4×D17semidirect
181. C4⋊C4C4×D17semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D172
C22⋊C8×D172
#95×D172
#117625472
#117625462
#117625452
C17×#3282
C2×#32817