Finite Group Databasebeta

← D17⋊#328 #14544153

Subgroups — D17⋊#328 #14544153

Order 2176.

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

normal subgroups only

This group has 336 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
2C2×(C2×#595)⋊C4C211yesmaximal, characteristic
3#11763663C212yesmaximal, complemented
4#11763662C212yesmaximal, complemented
5C22⋊Q8×D17C211yesmaximal, characteristic
6C23×#595—21no—
7C2×Q8×D17C411yescharacteristic, complemented
8C22×C4×D17C2211yescharacteristic
9C2×#63242—21no—
10C17×C22⋊Q8C411yescharacteristic, complemented
11#10587632C2211yescharacteristic
12#10587610C411yescharacteristic, complemented
13#10587583C2211yescharacteristic
14#10587582—21no—
15C4⋊C4×D17—21no—
16C4⋊C4×D17C411yescharacteristic, complemented
17C22⋊C4×D17—21no—
18#10587564—21no—
19C22⋊#3624—21no—
20(C2×#595)⋊C4—21no—
21(C2×#595)⋊C4C2212yes—
22#10587543C2212yes—
23C23×D17D411yescharacteristic
24C22×#595—24no—
25C22×#595—22no—
26C22×#595—42no—
27C22×#595—42no—
28C2×Q8×C17C2×C411yescharacteristic, complemented
29C22×C4×C17C2×C411yescharacteristic
30C22×Dic17D411yescharacteristic
31Q8×D17—42no—
32C2×C4×D17—21no—
33C2×C4×D17—21no—
34C2×C4×D17—42no—
35C2×C4×D17C2×C411yescharacteristic
36C2×C4×D17D411yescharacteristic
37C2×C4×D17D411yescharacteristic
38C2×C4×D17C2311yescharacteristic
39C2×Dic34—21no—
40C2×Dic34C2×C411yescharacteristic, complemented
41C2×#3624—22no—
42#63242—42no—
43C4⋊C4×C17—21no—
44C4⋊C4×C17C2×C411yescharacteristic, complemented
45C22⋊C4×C17—21no—
46C22⋊Dic17—21no—
47#63227—21no—
48#63227—21no—
49C4⋊Dic17—21no—
50C4⋊Dic17C2×C411yescharacteristic, complemented
51#63225—21no—
52#63225—21no—
53#63225—21no—
54C23×C17C22⋊C411yescharacteristic
55C22×D17—22no—
56C22×D17—22no—
57C22×D17C22⋊C411yescharacteristic
58C22×D17SD1612yes—
59C22×D17Q1612yes—
60C22×D17C2×D411yescharacteristic
61C2×#595—24no—
62C2×#595—42no—
63C2×#595—48no—
64C2×#595—42no—
65Q8×C17—41no—
66C2×C4×C17—21no—
67C2×C4×C17—21no—
68C2×C4×C17—41no—
69C2×C4×C17C22⋊C411yescharacteristic
70C2×C4×C17C22×C411yescharacteristic
71C2×Dic17—21no—
72C2×Dic17—21no—
73C2×Dic17—41no—
74C2×Dic17C22⋊C411yescharacteristic
75C2×Dic17C2×D411yescharacteristic
76C4×D17—22no—
77C4×D17—22no—
78C4×D17—41no—
79C4×D17—41no—
80C4×D17—42no—
81C4×D17—42no—
82Dic34—42no—
83Dic34—41no—
84#3624—42no—
85C2×#328—171nomaximal, Sylow
86C22×C17—21no—
87C22×C17—21no—
88C22×C17#10312yes—
89C22×C17C2×C22⋊C411yescharacteristic
90C2×D17—21no—
91C2×D17—21no—
92C2×D17—24no—
93C2×D17—24no—
94C2×D17#9912yes—
95C2×D17#10312yes—
96C2×D17#10412yes—
97C2×D17C2×SD1611yescharacteristic
98C2×D17C2×Q1611yescharacteristic
99#595—44no—
100C4×C17—21no—
101C4×C17—21no—
102C4×C17—41no—
103C4×C17—41no—
104Dic17—21no—
105Dic17—21no—
106Dic17—41no—
107Dic17—41no—
108C2×C22⋊Q8—171no—
109C2×C22⋊C8—171no—
110C2×#95—171no—
111#328—172no—
112#328—172no—
113C2×C17—22no—
114C2×C17C2×#9911yescharacteristic
115C2×C17C2×#10311yescharacteristic
116C2×C17C2×#10411yescharacteristic
117D17—21no—
118D17—21no—
119D17#32814yescomplemented
120C22×Q8—171no—
121C23×C4—171no—
122C23×C4—341no—
123C22×C8—341no—
124C22⋊Q8—171no—
125C22⋊Q8—171no—
126C22⋊Q8—171no—
127C22⋊Q8—341no—
128C22⋊Q8—341no—
129C22⋊Q8#59511yescharacteristic, complemented
130C2×C4⋊C4—171no—
131C2×C4⋊C4—341no—
132C2×C22⋊C4—341no—
133C22⋊C8—172no—
134C22⋊C8—341no—
135#95—172no—
136#95—341no—
137C17C2×#32811yescharacteristic, complemented, Sylow
138C24—171no—
139C2×Q8—171no—
140C2×Q8—341no—
141C2×Q8—682no—
142C2×Q8C2×#59511yescharacteristic, complemented
143C22×C4—171no—
144C22×C4—171no—
145C22×C4—171no—
146C22×C4—171no—
147C22×C4—171no—
148C22×C4—341no—
149C22×C4—341no—
150C22×C4—342no—
151C22×C4—344no—
152C22×C4—682no—
153C22×C4—682no—
154C22×C4—682no—
155C22×C4C2×#59511yescharacteristic
156C2×C8—342no—
157C2×C8—682no—
158C4⋊C4—21no—
159C4⋊C4—171no—
160C4⋊C4—341no—
161C4⋊C4—341no—
162C4⋊C4—341no—
163C4⋊C4—341no—
164C4⋊C4C2×#59511yescharacteristic, complemented
165C22⋊C4—21no—
166C22⋊C4—341no—
167C22⋊C4—341no—
168C22⋊C4—341no—
169C23—171no—
170C23—171no—
171C23—172no—
172C23—172no—
173C23—342no—
174C23—342no—
175C23C22⋊#59511yescharacteristic
176Q8—41no—
177Q8—681no—
178Q8—682no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—171no—
183C2×C4—171no—
184C2×C4—342no—
185C2×C4—342no—
186C2×C4—344no—
187C2×C4—341no—
188C2×C4—341no—
189C2×C4—681no—
190C2×C4—681no—
191C2×C4—682no—
192C2×C4—682no—
193C2×C4—681no—
194C2×C4—682no—
195C2×C4—688no—
196C2×C4—682no—
197C2×C4C22⋊#59511yescharacteristic
198C2×C4C22×#59511yescharacteristic
199C8—682no—
200C22—21no—
201C22—21no—
202C22—172no—
203C22—172no—
204C22—172no—
205C22—171no—
206C22—171no—
207C22—344no—
208C22—344no—
209C22—341no—
210C22—341no—
211C22#1058754912yes—
212C22C2×C22⋊#59511yescharacteristic
213C4—21no—
214C4—21no—
215C4—41no—
216C4—41no—
217C4—341no—
218C4—341no—
219C4—684no—
220C4—681no—
221C4—681no—
222C2—22no—
223C2—174no—
224C2—341no—
225C2—341no—
226C2D17⋊#9911yescharacteristic
227C2C2×#1058754911yescharacteristic
228C2D17⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×C22⋊Q8C4
socleC22×C17C2×C22⋊C4
prime radicalC23×D17D4
cosocleC2×C4×D17C23
hypercenterC22⋊Q8#595
hypocenterC17C2×#328
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #11763663C2semidirect
4. #11763662C2semidirect
7. C2×Q8×D17C4semidirect
10. C17×C22⋊Q8C4semidirect
12. #10587610C4semidirect
16. C4⋊C4×D17C4semidirect
28. C2×Q8×C17C2×C4semidirect
40. C2×Dic34C2×C4semidirect
44. C4⋊C4×C17C2×C4semidirect
50. C4⋊Dic17C2×C4semidirect
119. D17#328semidirect
129. C22⋊Q8#595semidirect
137. C17C2×#328semidirect
142. C2×Q8C2×#595semidirect
164. 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
#117636632
#117636622
C22⋊Q8×D172
C2×#32817