Finite Group Databasebeta

← #14536868

Subgroups — #14536868

Order 2176.

472 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 117 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#11762751C212yesmaximal, complemented
2#11762736C212yesmaximal, complemented
3#11762727C212yesmaximal, complemented
4#11762725C212yesmaximal, complemented
5#11762720C212yesmaximal, complemented
6#11762718C211yesmaximal, characteristic
7#11762686C212yesmaximal, complemented
8#11762681C211yesmaximal, characteristic, complemented
9C17×#393C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C22×C22⋊D17C2212yes—
12C23×Dic17C2211yescharacteristic
13C22×D68C2211yescharacteristic
14C2×C22⋊C4×C17C2211yescharacteristic
15C2×C22⋊C4×C17C2212yes—
16C2×C22⋊Dic17C2211yescharacteristic
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×C4⋊Dic17C2212yes—
23C2×#63225C2212yes—
24C17×#95C2212yes—
25C17×#95C2211yescharacteristic
26#10587502C2211yescharacteristic
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31C24×C17C2311yescharacteristic
32C23×D17C2312yes—
33C2×Q8×C17—42no—
34C2×Q8×C17—41no—
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2312yes—
39C2×C22⋊D17—24no—
40C2×C22⋊D17—42no—
41C2×C22⋊D17—42no—
42C22×Dic17—42no—
43C22×Dic17—41no—
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2312yes—
47C2×D68—41no—
48C2×D68—42no—
49C22⋊C4×C17—41no—
50C22⋊C4×C17—42no—
51C22⋊Dic17—22no—
52#63227—42no—
53#63227—42no—
54#63227—42no—
55#63227—42no—
56#63227—42no—
57C4⋊Dic17—42no—
58#63225—42no—
59C23×C17—21no—
60C23×C17—22no—
61C23×C17—22no—
62C23×C17—21no—
63C23×C17#4112yes—
64C23×C17C2411yescharacteristic
65C22×D17—22no—
66C22×D17—22no—
67C22×D17—22no—
68C22×D17—22no—
69C22×D17—22no—
70C22×D17—22no—
71C22×D17—22no—
72Q8×C17—44no—
73C2×C4×C17—21no—
74C2×C4×C17—22no—
75C2×C4×C17—21no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C22⋊D17—48no—
84C2×Dic17—22no—
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—44no—
94D68—44no—
95#2824—171nomaximal, Sylow
96C22×C17—24no—
97C22×C17—22no—
98C22×C17—41no—
99C22×C17—41no—
100C22×C17—42no—
101C22×C17C2×#4111yescharacteristic
102C22×C17H(5,𝔽2)12yes—
103C22×C17H(5,𝔽2)11yescharacteristic
104C22×C17H(5,𝔽2)12yes—
105C22×C172−1+411yescharacteristic
106C2×D17—42no—
107C2×D17—42no—
108C2×D17—42no—
109C2×D17—42no—
110C2×D17—42no—
111C2×D17—42no—
112C2×D17—42no—
113C4×C17—42no—
114C4×C17—41no—
115C4×C17—41no—
116C4×C17—42no—
117Dic17—42no—
118Dic17—42no—
119Dic17—42no—
120#400—172no—
121#397—172no—
122#394—172no—
123#394—172no—
124#393—172no—
125#393D1711yescharacteristic, complemented
126#392—171no—
127#386—172no—
128#385—171no—
129C2×C17—42no—
130C2×C17#53812yes—
131C2×C17#53911yescharacteristic
132C2×C17#56011yescharacteristic
133C2×C17H(3,𝔽4)11yescharacteristic
134C2×C17#56212yes—
135D17—82no—
136C22×Q8C2×D1711yescharacteristic
137C22×D4—172no—
138C22×D4—171no—
139C23×C4—171no—
140C2×C4⋊C4—172no—
141C2×C4⋊C4—172no—
142C2×C22⋊C4—171no—
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×D1712yes—
150#95—172no—
151#95—172no—
152#95—172no—
153#95—171no—
154#95—172no—
155#95C2×D1712yes—
156#95C2×D1711yescharacteristic
157C17#282411yescharacteristic, complemented, Sylow
158C24—172no—
159C24C22×D1711yescharacteristic
160C2×Q8—41no—
161C2×Q8—42no—
162C2×D4—344no—
163C2×D4—681no—
164C2×D4—682no—
165C2×D4—682no—
166C2×D4—682no—
167C22×C4—172no—
168C22×C4—172no—
169C22×C4—172no—
170C22×C4—681no—
171C22×C4—682no—
172C22×C4C22×D1711yescharacteristic
173C22×C4C22×D1712yes—
174C22×C4C22×D1712yes—
175C22×C4C22×D1711yescharacteristic
176C4⋊C4—682no—
177C4⋊C4—682no—
178C22⋊C4—42no—
179C22⋊C4—41no—
180C22⋊C4—342no—
181C22⋊C4—682no—
182C22⋊C4—682no—
183C22⋊C4—682no—
184C22⋊C4—682no—
185C22⋊C4—682no—
186C23—21no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—342no—
191C23—342no—
192C23—342no—
193C23—342no—
194C23—342no—
195C23—342no—
196C23—342no—
197C23#6325612yes—
198C23C23×D1711yescharacteristic
199Q8—44no—
200D4—684no—
201D4—688no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—342no—
213C2×C4—342no—
214C2×C4—342no—
215C2×C4—342no—
216C2×C4—342no—
217C2×C4—342no—
218C2×C4—342no—
219C2×C4—342no—
220C2×C4—342no—
221C2×C4—684no—
222C22—24no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—682no—
228C22—682no—
229C22—682no—
230C22—682no—
231C22—682no—
232C22—682no—
233C22—682no—
234C22#1058769111yescharacteristic
235C22#1058769112yes—
236C22C2×#6325611yescharacteristic
237C22#1058769612yes—
238C22#1058769711yescharacteristic
239C4—41no—
240C4—42no—
241C4—41no—
242C4—42no—
243C4—682no—
244C4—682no—
245C4—682no—
246C2—42no—
247C2—1362no—
248C2#1176339912yes—
249C2#1176340811yescharacteristic
250C2#1176354811yescharacteristic
251C2#1176355111yescharacteristic
252C2#1176355312yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#393C2
prime radical#11762718C2
hypercenter#393D17
hypocenterC17#2824
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. #11762751C2semidirect
2. #11762736C2semidirect
3. #11762727C2semidirect
4. #11762725C2semidirect
5. #11762720C2semidirect
7. #11762686C2semidirect
8. #11762681C2semidirect
9. C17×#393C2semidirect
125. #393D17semidirect
157. C17#2824semidirect

Maximal subgroups

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

subgroupindex
#117627512
#117627362
#117627272
#117627252
#117627202
#117627182
#117626862
#117626812
C17×#3932
#282417