Finite Group Databasebeta

← #14536161

Subgroups — #14536161

Order 2176.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C17⋊#401C212yesmaximal, complemented
2#11762722C212yesmaximal, complemented
3#11762721C211yesmaximal, characteristic, complemented
4#11762718C211yesmaximal, characteristic
5#11762688C212yesmaximal, complemented
6#11762687C212yesmaximal, complemented
7#11762685C212yesmaximal, complemented
8#11762671C212yesmaximal, complemented
9C17×#381C211yesmaximal, characteristic, complemented
10C23×C4×C17C2211yescharacteristic
11C22×C22⋊D17C2212yes—
12C22×D68C2211yescharacteristic
13C22×C4×D17C2212yes—
14C22×Dic34C2211yescharacteristic
15C2×C22⋊C4×C17C2211yescharacteristic
16C2×C22⋊C4×C17C2211yescharacteristic
17C2×C22⋊Dic17C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C17×#95C2212yes—
23C17×#95C2212yes—
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2211yescharacteristic
27#10587502C2212yes—
28#10587502C2211yescharacteristic
29#10587502C2212yes—
30#10587502C2212yes—
31C24×C17C2311yescharacteristic
32C23×D17C2312yes—
33C22×C4×C17—22no—
34C22×C4×C17—41no—
35C22×C4×C17—41no—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2312yes—
40C2×C22⋊D17—24no—
41C2×C22⋊D17—24no—
42C2×C22⋊D17—42no—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C2×D68—24no—
47C2×D68—41no—
48C2×C4×D17—24no—
49C2×C4×D17—42no—
50C2×C4×D17—42no—
51C2×Dic34—42no—
52C2×Dic34—41no—
53C22⋊C4×C17—41no—
54C22⋊C4×C17—41no—
55C22⋊Dic17—42no—
56#63227—42no—
57#63227—42no—
58#63227—42no—
59#63227—42no—
60C23×C17—22no—
61C23×C17—22no—
62C23×C17—21no—
63C23×C17—21no—
64C23×C17#4112yes—
65C23×C17C2411yescharacteristic
66C22×D17—22no—
67C22×D17—22no—
68C22×D17—22no—
69C22×D17—22no—
70C22×D17—22no—
71C22×D17—22no—
72C22×D17#4114yes—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17—21no—
77C2×C4×C17—22no—
78C2×C4×C17—21no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—44no—
84C22⋊D17—48no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94D68—44no—
95C4×D17—48no—
96Dic34—44no—
97#2760—171nomaximal, Sylow
98C22×C17—22no—
99C22×C17—24no—
100C22×C17—42no—
101C22×C17—41no—
102C22×C17—41no—
103C22×C17C2×#4111yescharacteristic
104C22×C17C2×#4112yes—
105C22×C17H(5,𝔽2)11yescharacteristic
106C22×C17H(5,𝔽2)11yescharacteristic
107C22×C17H(5,𝔽2)12yes—
108C2×D17—24no—
109C2×D17—24no—
110C2×D17—24no—
111C2×D17—42no—
112C2×D17—42no—
113C2×D17—42no—
114C2×D17—42no—
115C4×C17—42no—
116C4×C17—42no—
117C4×C17—41no—
118C4×C17—41no—
119Dic17—42no—
120Dic17—42no—
121Dic17—42no—
122#401—172no—
123#393—171no—
124#393—172no—
125#392—171no—
126#386—172no—
127#386—172no—
128#386—172no—
129#381—172no—
130#381D1711yescharacteristic, complemented
131C2×C17—42no—
132C2×C17#53812yes—
133C2×C17#53811yescharacteristic
134C2×C17#55112yes—
135C2×C17#55111yescharacteristic
136C2×C17#56011yescharacteristic
137D17—44no—
138C22×Q8—171no—
139C22×D4—172no—
140C22×D4—171no—
141C23×C4—172no—
142C23×C4C2×D1711yescharacteristic
143C2×C22⋊C4—172no—
144C2×C22⋊C4—172no—
145C2×C22⋊C4—172no—
146C2×C22⋊C4—172no—
147C2×C22⋊C4—172no—
148C2×C22⋊C4C2×D1711yescharacteristic
149C2×C22⋊C4C2×D1711yescharacteristic
150#95—171no—
151#95—172no—
152#95—172no—
153#95—172no—
154#95—172no—
155#95—172no—
156#95—171no—
157#95C2×D1712yes—
158#95C2×D1712yes—
159C17#276011yescharacteristic, complemented, Sylow
160C24—172no—
161C24C22×D1711yescharacteristic
162C2×Q8—681no—
163C2×Q8—682no—
164C2×D4—344no—
165C2×D4—344no—
166C2×D4—344no—
167C2×D4—681no—
168C2×D4—682no—
169C22×C4—22no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—172no—
173C22×C4—172no—
174C22×C4—172no—
175C22×C4—344no—
176C22×C4—682no—
177C22×C4—682no—
178C22×C4C22×D1711yescharacteristic
179C22×C4C22×D1712yes—
180C22×C4C22×D1712yes—
181C22×C4C22×D1711yescharacteristic
182C22⋊C4—41no—
183C22⋊C4—41no—
184C22⋊C4—682no—
185C22⋊C4—682no—
186C22⋊C4—682no—
187C22⋊C4—682no—
188C22⋊C4—682no—
189C23—21no—
190C23—21no—
191C23—22no—
192C23—22no—
193C23—174no—
194C23—342no—
195C23—342no—
196C23—342no—
197C23—342no—
198C23—342no—
199C23—342no—
200C23#6325212yes—
201C23C23×D1711yescharacteristic
202Q8—684no—
203D4—684no—
204D4—688no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—44no—
216C2×C4—342no—
217C2×C4—342no—
218C2×C4—342no—
219C2×C4—342no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4—342no—
224C2×C4—342no—
225C2×C4—688no—
226C22—24no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—344no—
232C22—344no—
233C22—344no—
234C22—682no—
235C22—682no—
236C22—682no—
237C22—682no—
238C22C2×#6325211yescharacteristic
239C22#1058769111yescharacteristic
240C22#1058769111yescharacteristic
241C22#41×D1712yes—
242C22#1058769612yes—
243C4—41no—
244C4—41no—
245C4—42no—
246C4—42no—
247C4—682no—
248C4—682no—
249C4—682no—
250C2—42no—
251C2—684no—
252C2#1176339512yes—
253C2#1176339711yescharacteristic
254C2#1176348312yes—
255C2#1176348711yescharacteristic
256C2#1176354911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#381C2
prime radical#11762718C2
hypercenter#381D17
hypocenterC17#2760
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C17⋊#401C2semidirect
2. #11762722C2semidirect
3. #11762721C2semidirect
5. #11762688C2semidirect
6. #11762687C2semidirect
7. #11762685C2semidirect
8. #11762671C2semidirect
9. C17×#381C2semidirect
130. #381D17semidirect
159. C17#2760semidirect

Maximal subgroups

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

subgroupindex
C17⋊#4012
#117627222
#117627212
#117627182
#117626882
#117626872
#117626852
#117626712
C17×#3812
#276017