Finite Group Databasebeta

← #14535826

Subgroups — #14535826

Order 2176.

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

normal subgroups only

This group has 552 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
1#11762738C212yesmaximal, complemented
2#11762734C211yesmaximal, characteristic, complemented
3#11762731C212yesmaximal, complemented
4#11762699C212yesmaximal, complemented
5#11762699C212yesmaximal, complemented
6#11762697C212yesmaximal, complemented
7#11762664C211yesmaximal, characteristic
8#11762664C212yesmaximal
9C17×#399C211yesmaximal, characteristic, complemented
10C24×D17C2211yescharacteristic
11C2×C4⋊C4×C17C2211yescharacteristic, complemented
12C2×C4⋊C4×C17C2212yescomplemented
13C2×C22⋊C4×C17C2212yes—
14C2×C22⋊C4×C17C2211yescharacteristic
15C2×C22⋊Dic17C2212yes—
16C2×C22⋊Dic17C2211yescharacteristic
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×C4⋊Dic17C2211yescharacteristic, complemented
24C2×#63225C2212yescomplemented
25C2×C4×Dic17C2212yescomplemented
26C17×#95C2211yescharacteristic, complemented
27#10587502C2212yescomplemented
28#10587502C2212yescomplemented
29#10587502C2211yescharacteristic, complemented
30#10587502C2212yescomplemented
31C24×C17C2311yescharacteristic
32C23×D17—22no—
33C23×D17—24no—
34C23×D17—41no—
35C23×D17—42no—
36C23×D17—41no—
37C23×D17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2312yes—
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C4⋊C4×C17—41no—
46C4⋊C4×C17—42no—
47C22⋊C4×C17—42no—
48C22⋊C4×C17—41no—
49C22⋊Dic17—24no—
50C22⋊Dic17—41no—
51#63227—24no—
52#63227—24no—
53#63227—42no—
54#63227—42no—
55#63227—42no—
56#63227—42no—
57C4⋊Dic17—41no—
58#63225—42no—
59C4×Dic17—42no—
60C23×C17—22no—
61C23×C17—21no—
62C23×C17—21no—
63C23×C17—22no—
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—44no—
73C22×D17—48no—
74C22×D17—48no—
75C22×D17—44no—
76C22×D17—44no—
77C22×D17#4114yes—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—21no—
82C2×C4×C17—22no—
83C2×C4×C17—21no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97#2735—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)12yes—
106C22×C17H(5,𝔽2)11yescharacteristic
107C22×C17H(5,𝔽2)11yescharacteristic
108C2×D17—24no—
109C2×D17—24no—
110C2×D17—24no—
111C2×D17—416no—
112C2×D17—42no—
113C2×D17—42no—
114C2×D17—42no—
115C2×D17—42no—
116C4×C17—42no—
117C4×C17—41no—
118C4×C17—42no—
119C4×C17—41no—
120Dic17—42no—
121Dic17—42no—
122Dic17—42no—
123#399D1711yescharacteristic, complemented
124#397—171no—
125#397—172no—
126#396—172no—
127#388—172no—
128#388—172no—
129#388—172no—
130#379—172no—
131#379—171no—
132C2×C17—42no—
133C2×C17#53812yes—
134C2×C17#53811yescharacteristic
135C2×C17#55112yes—
136C2×C17#55111yescharacteristic
137C2×C17#56011yescharacteristic
138D17—44no—
139C25—171no—
140C2×C4⋊C4—172no—
141C2×C4⋊C4—171no—
142C2×C4⋊C4C2×D1711yescharacteristic, complemented
143C2×C4⋊C4C2×D1712yescomplemented
144C2×C22⋊C4—172no—
145C2×C22⋊C4—172no—
146C2×C22⋊C4—172no—
147C2×C22⋊C4—171no—
148C2×C22⋊C4—172no—
149C2×C22⋊C4—172no—
150C2×C22⋊C4—172no—
151C2×C22⋊C4—172no—
152C2×C22⋊C4C2×D1712yes—
153C2×C22⋊C4C2×D1711yescharacteristic
154C2×C42—172no—
155#95—172no—
156#95—171no—
157#95—172no—
158#95—172no—
159#95C2×D1711yescharacteristic, complemented
160C17#273511yescharacteristic, complemented, Sylow
161C24—172no—
162C24—344no—
163C24—342no—
164C24—682no—
165C24—681no—
166C24—681no—
167C24C22×D1711yescharacteristic
168C22×C4—172no—
169C22×C4—172no—
170C22×C4—172no—
171C22×C4C22×D1711yescharacteristic
172C22×C4C22×D1712yes—
173C22×C4C22×D1712yes—
174C22×C4C22×D1711yescharacteristic
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—682no—
178C4⋊C4—681no—
179C22⋊C4—42no—
180C22⋊C4—41no—
181C22⋊C4—344no—
182C22⋊C4—344no—
183C22⋊C4—344no—
184C22⋊C4—682no—
185C22⋊C4—682no—
186C22⋊C4—682no—
187C22⋊C4—682no—
188C22⋊C4—681no—
189C42—682no—
190C23—22no—
191C23—21no—
192C23—22no—
193C23—21no—
194C23—174no—
195C23—342no—
196C23—342no—
197C23—342no—
198C23—342no—
199C23—342no—
200C23—342no—
201C23—688no—
202C23—684no—
203C23—684no—
204C23—688no—
205C23—684no—
206C23#6325612yes—
207C23C23×D1711yescharacteristic
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—342no—
219C2×C4—342no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4—342no—
224C2×C4—342no—
225C2×C4—342no—
226C2×C4—342no—
227C22—24no—
228C22—22no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—344no—
233C22—344no—
234C22—344no—
235C22—6816no—
236C22—682no—
237C22—682no—
238C22—682no—
239C22—682no—
240C22#1058769112yes—
241C22#1058769111yescharacteristic
242C22C2×#6325611yescharacteristic
243C22#41×D1712yes—
244C22#1058769611yescharacteristic
245C4—41no—
246C4—42no—
247C4—41no—
248C4—42no—
249C4—682no—
250C4—682no—
251C4—682no—
252C2—42no—
253C2—684no—
254C2#1176339512yes—
255C2#1176339911yescharacteristic
256C2#1176348612yes—
257C2#1176348711yescharacteristic
258C2#1176354611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#399C2
prime radicalC24×D17C22
hypercenter#399D17
hypocenterC17#2735
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. #11762738C2semidirect
2. #11762734C2semidirect
3. #11762731C2semidirect
4. #11762699C2semidirect
5. #11762699C2semidirect
6. #11762697C2semidirect
9. C17×#399C2semidirect
11. C2×C4⋊C4×C17C22semidirect
12. C2×C4⋊C4×C17C22semidirect
23. C2×C4⋊Dic17C22semidirect
24. C2×#63225C22semidirect
25. C2×C4×Dic17C22semidirect
26. C17×#95C22semidirect
27. #10587502C22semidirect
28. #10587502C22semidirect
29. #10587502C22semidirect
30. #10587502C22semidirect
123. #399D17semidirect
142. C2×C4⋊C4C2×D17semidirect
143. C2×C4⋊C4C2×D17semidirect
159. #95C2×D17semidirect
160. C17#2735semidirect

Maximal subgroups

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

subgroupindex
#117627382
#117627342
#117627312
#117626992
#117626972
#117626642
C17×#3992
#273517