Finite Group Databasebeta

← #14535830

Subgroups — #14535830

Order 2176.

560 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 125 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#11762737C212yesmaximal, complemented
2#11762734C211yesmaximal, characteristic, complemented
3#11762721C212yesmaximal, complemented
4#11762721C211yesmaximal, characteristic, complemented
5#11762703C212yesmaximal, complemented
6#11762702C212yesmaximal, complemented
7#11762697C212yesmaximal, complemented
8#11762665C212yesmaximal
9C17×#379C211yesmaximal, characteristic
10C25×C17C2211yescharacteristic
11C22×Dic34C2211yescharacteristic, complemented
12C2×C22⋊C4×C17C2212yes—
13C2×C22⋊C4×C17C2211yescharacteristic
14C2×C22⋊C4×C17C2212yes—
15C2×C22⋊C4×C17C2211yescharacteristic
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2212yes—
21C2×C22⋊Dic17C2212yes—
22C2×C4⋊Dic17C2211yescharacteristic, complemented
23C2×#63225C2212yescomplemented
24C2×C4×Dic17C2212yescomplemented
25#10587502C2212yescomplemented
26#10587502C2212yescomplemented
27#10587502C2212yescomplemented
28#10587502C2211yescharacteristic, complemented
29#10587502C2212yescomplemented
30#10587502C2211yescharacteristic, complemented
31C24×C17—22no—
32C24×C17—24no—
33C24×C17—42no—
34C24×C17—41no—
35C24×C17—41no—
36C24×C17C2311yescharacteristic
37C24×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C2×Dic34—42no—
46C2×Dic34—41no—
47C22⋊C4×C17—24no—
48C22⋊C4×C17—42no—
49C22⋊C4×C17—41no—
50C22⋊C4×C17—41no—
51C22⋊Dic17—24no—
52C22⋊Dic17—24no—
53C22⋊Dic17—42no—
54C22⋊Dic17—42no—
55C22⋊Dic17—42no—
56C22⋊Dic17—42no—
57C4⋊Dic17—41no—
58#63225—42no—
59C4×Dic17—42no—
60C23×C17—22no—
61C23×C17—22no—
62C23×C17—22no—
63C23×C17—22no—
64C23×C17—22no—
65C23×C17—22no—
66C23×C17—22no—
67C23×C17—21no—
68C23×C17—21no—
69C23×C17—22no—
70C23×C17—48no—
71C23×C17—44no—
72C23×C17—48no—
73C23×C17—44no—
74C23×C17—44no—
75C23×C17#4112yes—
76C23×C17#4114yes—
77C23×C17C2411yescharacteristic
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—21no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—21no—
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—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97Dic34—44no—
98#2736—171nomaximal, Sylow
99C22×C17—24no—
100C22×C17—22no—
101C22×C17—24no—
102C22×C17—24no—
103C22×C17—24no—
104C22×C17—416no—
105C22×C17—42no—
106C22×C17—42no—
107C22×C17—42no—
108C22×C17—42no—
109C22×C17—41no—
110C22×C17—42no—
111C22×C17—41no—
112C22×C17C2×#4111yescharacteristic
113C22×C17C2×#4112yes—
114C22×C17H(5,𝔽2)11yescharacteristic
115C22×C17H(5,𝔽2)12yes—
116C22×C17H(5,𝔽2)11yescharacteristic
117C4×C17—42no—
118C4×C17—41no—
119C4×C17—41no—
120Dic17—42no—
121Dic17—42no—
122Dic17—42no—
123Dic17—42no—
124#397—171no—
125#397—172no—
126#393—171no—
127#393—172no—
128#388—172no—
129#388—172no—
130#388—172no—
131#379—172no—
132#379D1711yescharacteristic
133C2×C17—42no—
134C2×C17—44no—
135C2×C17#53812yes—
136C2×C17#53811yescharacteristic
137C2×C17#55112yes—
138C2×C17#55111yescharacteristic
139C2×C17H(3,𝔽4)11yescharacteristic
140C25C2×D1711yescharacteristic
141C22×Q8—171no—
142C2×C4⋊C4—172no—
143C2×C4⋊C4—171no—
144C2×C22⋊C4—172no—
145C2×C22⋊C4—172no—
146C2×C22⋊C4—172no—
147C2×C22⋊C4—172no—
148C2×C22⋊C4—172no—
149C2×C22⋊C4—172no—
150C2×C22⋊C4C2×D1712yes—
151C2×C22⋊C4C2×D1711yescharacteristic
152C2×C22⋊C4C2×D1712yes—
153C2×C22⋊C4C2×D1711yescharacteristic
154C2×C42—172no—
155#95—172no—
156#95—171no—
157#95—172no—
158#95—172no—
159#95—172no—
160#95—171no—
161C17#273611yescharacteristic, complemented, Sylow
162C24—22no—
163C24—24no—
164C24—42no—
165C24—41no—
166C24—41no—
167C24C22×D1711yescharacteristic
168C24C22×D1712yes—
169C2×Q8—682no—
170C2×Q8—681no—
171C22×C4—172no—
172C22×C4—172no—
173C22×C4—172no—
174C22×C4—172no—
175C22×C4C22×D1711yescharacteristic
176C22×C4C22×D1711yescharacteristic
177C22×C4C22×D1712yes—
178C4⋊C4—682no—
179C4⋊C4—681no—
180C22⋊C4—24no—
181C22⋊C4—41no—
182C22⋊C4—42no—
183C22⋊C4—41no—
184C22⋊C4—344no—
185C22⋊C4—344no—
186C22⋊C4—682no—
187C22⋊C4—682no—
188C22⋊C4—682no—
189C22⋊C4—682no—
190C42—682no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—21no—
196C23—21no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—48no—
202C23—44no—
203C23—48no—
204C23—44no—
205C23—44no—
206C23#6325212yes—
207C23#6325414yes—
208C23C23×D1711yescharacteristic
209Q8—684no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
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—342no—
226C2×C4—342no—
227C2×C4—342no—
228C2×C4—342no—
229C22—24no—
230C22—24no—
231C22—24no—
232C22—24no—
233C22—22no—
234C22—416no—
235C22—42no—
236C22—41no—
237C22—41no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22C2×#6325211yescharacteristic
243C22C2×#6325412yes—
244C22#1058769112yes—
245C22#1058769111yescharacteristic
246C22#1058769111yescharacteristic
247C4—41no—
248C4—41no—
249C4—42no—
250C4—682no—
251C4—682no—
252C4—682no—
253C4—682no—
254C2—42no—
255C2—44no—
256C2#1176339412yes—
257C2#1176339711yescharacteristic
258C2#1176348512yes—
259C2#1176349011yescharacteristic
260C2#1176355211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#379C2
prime radicalC25×C17C22
hypercenter#379D17
hypocenterC17#2736
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. #11762737C2semidirect
2. #11762734C2semidirect
3. #11762721C2semidirect
4. #11762721C2semidirect
5. #11762703C2semidirect
6. #11762702C2semidirect
7. #11762697C2semidirect
11. C22×Dic34C22semidirect
22. C2×C4⋊Dic17C22semidirect
23. C2×#63225C22semidirect
24. C2×C4×Dic17C22semidirect
25. #10587502C22semidirect
26. #10587502C22semidirect
27. #10587502C22semidirect
28. #10587502C22semidirect
29. #10587502C22semidirect
30. #10587502C22semidirect
161. C17#2736semidirect

Maximal subgroups

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

subgroupindex
#117627372
#117627342
#117627212
#117627032
#117627022
#117626972
#117626652
C17×#3792
#273617