Finite Group Databasebeta

← D17⋊#352 #14545033

Subgroups — D17⋊#352 #14545033

Order 2176.

278 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 54 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
1D17⋊#100C211yesmaximal, characteristic
2D17⋊#99C211yesmaximal, characteristic
3#11763712C212yesmaximal, complemented
4#11763711C212yesmaximal, complemented
5#123×D17C211yesmaximal, characteristic
6C2×D4×D17C2211yescharacteristic
7C22×C4×D17C411yescharacteristic, complemented
8C2×C22⋊#595—21no—
9D17⋊M16—21no—
10C17×#123C411yescharacteristic, complemented
11#10587621C411yescharacteristic, complemented
12#10587615C2211yescharacteristic
13#10587580—21no—
14C4⋊C4×D17—21no—
15#10587572C2211yescharacteristic
16#10587569—21no—
17C22⋊C4×D17—21no—
18C22⋊C4×D17C411yescharacteristic, complemented
19#10587553—21no—
20#10587553C2212yes—
21#10587551C2212yes—
22#10587539—21no—
23C23×D17C2×C411yescharacteristic
24C23×D17D411yescharacteristic
25C22×#595—41no—
26C2×D4×C17C2×C411yescharacteristic
27C22×C4×C17C2×C411yescharacteristic, complemented
28C2×C22⋊D17D411yescharacteristic
29C2×C22⋊D17C2311yescharacteristic
30C22×Dic17C2×C411yescharacteristic, complemented
31D4×D17—41no—
32D4×D17—41no—
33C2×D68D411yescharacteristic
34C2×C4×D17—21no—
35C2×C4×D17—21no—
36C2×C4×D17—21no—
37C2×C4×D17—41no—
38C2×C4×D17—41no—
39C2×C4×D17D411yescharacteristic
40C22⋊#595—22no—
41C22⋊#595—41no—
42#63247—22no—
43#63243—41no—
44#63242—41no—
45C4⋊C4×C17—21no—
46C22⋊C4×C17—21no—
47C22⋊C4×C17C2×C411yescharacteristic, complemented
48C22⋊Dic17—21no—
49C22⋊Dic17C2×C411yescharacteristic, complemented
50#63227—21no—
51#63227—21no—
52#63227—21no—
53C4⋊Dic17—21no—
54#63225—21no—
55#63225—21no—
56C23×C17C22⋊C411yescharacteristic
57C23×C17C22×C411yescharacteristic
58C22×D17—21no—
59C22×D17—21no—
60C22×D17—21no—
61C22×D17—21no—
62C22×D17—42no—
63C22×D17—42no—
64C22×D17C22⋊C411yescharacteristic
65C22×D17C22⋊C411yescharacteristic
66C22×D17C2×D411yescharacteristic
67C2×#595—42no—
68C2×#595—82no—
69D4×C17—41no—
70C2×C4×C17—21no—
71C2×C4×C17—21no—
72C2×C4×C17—41no—
73C2×C4×C17C22⋊C411yescharacteristic
74C22⋊D17—41no—
75C22⋊D17—41no—
76C2×Dic17—21no—
77C2×Dic17—21no—
78C2×Dic17—41no—
79C2×Dic17C2×D411yescharacteristic
80D68—41no—
81C4×D17—21no—
82C4×D17—21no—
83C4×D17—42no—
84C4×D17—42no—
85C4×D17—41no—
86C4×D17—41no—
87#3624—42no—
88C2×#352—171nomaximal, Sylow
89C22×C17—21no—
90C22×C17—21no—
91C22×C17—41no—
92C22×C17—41no—
93C22×C17C2×C22⋊C411yescharacteristic
94C2×D17—22no—
95C2×D17—21no—
96C2×D17—21no—
97C2×D17—42no—
98C2×D17—42no—
99C2×D17—42no—
100C2×D17—41no—
101C2×D17—41no—
102C2×D17—81no—
103C2×D17#9911yescharacteristic
104C2×D17#9911yescharacteristic
105#595—82no—
106C4×C17—21no—
107C4×C17—41no—
108C4×C17—41no—
109Dic17—21no—
110Dic17—41no—
111Dic17—41no—
112C2×#123—171no—
113C2×#100—171no—
114C2×#99—171no—
115#352—172no—
116#352—172no—
117C2×C17—21no—
118C2×C17—41no—
119C2×C17—41no—
120C2×C17C2×#9911yescharacteristic
121D17—21no—
122D17—41no—
123D17—41no—
124D17#35212yescomplemented
125C22×D4—171no—
126C23×C4—171no—
127C2×M16—341no—
128#123—171no—
129#123—171no—
130#123—171no—
131#123—341no—
132#123—341no—
133#123#59511yescharacteristic, complemented
134C2×C4⋊C4—341no—
135C2×C22⋊C4—171no—
136C2×C22⋊C4—341no—
137C2×C22⋊C4—341no—
138#100—172no—
139#100—341no—
140#99—172no—
141#99—341no—
142C17C2×#35211yescharacteristic, complemented, Sylow
143C24—171no—
144C24—171no—
145C2×D4—171no—
146C2×D4—171no—
147C2×D4—171no—
148C2×D4—681no—
149C2×D4—681no—
150C2×D4C2×#59511yescharacteristic
151C22×C4—171no—
152C22×C4—171no—
153C22×C4—341no—
154C22×C4—341no—
155C22×C4—341no—
156C22×C4—681no—
157C22×C4—681no—
158C22×C4—681no—
159C22×C4C2×#59511yescharacteristic, complemented
160M16—342no—
161M16—681no—
162C2×C8—681no—
163C4⋊C4—21no—
164C4⋊C4—341no—
165C4⋊C4—341no—
166C4⋊C4—341no—
167C22⋊C4—21no—
168C22⋊C4—171no—
169C22⋊C4—341no—
170C22⋊C4—341no—
171C22⋊C4—341no—
172C22⋊C4—341no—
173C22⋊C4—342no—
174C22⋊C4—681no—
175C22⋊C4C2×#59511yescharacteristic, complemented
176C23—171no—
177C23—171no—
178C23—171no—
179C23—341no—
180C23—341no—
181C23—341no—
182C23—341no—
183C23—682no—
184C23—682no—
185C23C22⋊#59511yescharacteristic
186C23C22×#59511yescharacteristic
187D4—41no—
188D4—681no—
189D4—681no—
190D4—681no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—171no—
195C2×C4—341no—
196C2×C4—341no—
197C2×C4—341no—
198C2×C4—341no—
199C2×C4—681no—
200C2×C4—681no—
201C2×C4—682no—
202C2×C4—682no—
203C2×C4—681no—
204C2×C4—682no—
205C2×C4—1362no—
206C2×C4C22⋊#59511yescharacteristic
207C8—682no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—171no—
213C22—171no—
214C22—342no—
215C22—341no—
216C22—341no—
217C22—681no—
218C22—681no—
219C22—682no—
220C22—682no—
221C22—682no—
222C22—1361no—
223C22C2×C22⋊#59511yescharacteristic
224C4—21no—
225C4—41no—
226C4—41no—
227C4—341no—
228C4—681no—
229C4—681no—
230C4—1362no—
231C2—21no—
232C2—41no—
233C2—41no—
234C2—172no—
235C2—341no—
236C2—681no—
237C2—681no—
238C2D17⋊#9911yescharacteristic
subgroupG/H
centerC2D17⋊#99
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC17×#123C4
socleC2×C17C2×#99
prime radicalC2×D4×D17C22
cosocleC2×C22⋊D17C23
hypercenter#123#595
hypocenterC17C2×#352
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
3. #11763712C2semidirect
4. #11763711C2semidirect
7. C22×C4×D17C4semidirect
10. C17×#123C4semidirect
11. #10587621C4semidirect
18. C22⋊C4×D17C4semidirect
27. C22×C4×C17C2×C4semidirect
30. C22×Dic17C2×C4semidirect
47. C22⋊C4×C17C2×C4semidirect
49. C22⋊Dic17C2×C4semidirect
124. D17#352semidirect
133. #123#595semidirect
142. C17C2×#352semidirect
159. C22×C4C2×#595semidirect
175. C22⋊C4C2×#595semidirect

Maximal subgroups

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

subgroupindex
D17⋊#1002
D17⋊#992
#117637122
#117637112
#123×D172
C2×#35217