Finite Group Databasebeta

← #14535825

Subgroups — #14535825

Order 2176.

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

normal subgroups only

This group has 552 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#11762745C211yesmaximal, characteristic, complemented
2#11762737C212yesmaximal, complemented
3#11762734C211yesmaximal, characteristic, complemented
4#11762732C212yesmaximal, complemented
5#11762703C212yesmaximal, complemented
6#11762702C212yesmaximal, complemented
7#11762697C212yesmaximal, complemented
8#11762665C212yesmaximal
9C17×#379C211yesmaximal, characteristic
10C25×C17C2211yescharacteristic
11C2×C22⋊C4×C17C2212yes—
12C2×C22⋊C4×C17C2212yes—
13C2×C22⋊C4×C17C2211yescharacteristic
14C2×C22⋊C4×C17C2211yescharacteristic
15C2×C22⋊Dic17C2212yes—
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2212yes—
21C2×C4⋊Dic17C2211yescharacteristic, complemented
22C2×C4⋊Dic17C2211yescharacteristic, complemented
23C2×#63225C2212yescomplemented
24C2×#63225C2212yescomplemented
25C2×C4×Dic17C2212yescomplemented
26#10587502C2211yescharacteristic, complemented
27#10587502C2212yescomplemented
28#10587502C2212yescomplemented
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×C17C2312yes—
40C22×C4×C17C2311yescharacteristic
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C22⋊C4×C17—24no—
46C22⋊C4×C17—42no—
47C22⋊C4×C17—41no—
48C22⋊C4×C17—41no—
49C22⋊Dic17—24no—
50C22⋊Dic17—24no—
51C22⋊Dic17—42no—
52C22⋊Dic17—42no—
53C22⋊Dic17—42no—
54C22⋊Dic17—42no—
55C4⋊Dic17—41no—
56C4⋊Dic17—41no—
57#63225—42no—
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—22no—
82C2×C4×C17—22no—
83C2×C4×C17—21no—
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—
97#2735—171nomaximal, Sylow
98C22×C17—24no—
99C22×C17—22no—
100C22×C17—24no—
101C22×C17—24no—
102C22×C17—24no—
103C22×C17—416no—
104C22×C17—42no—
105C22×C17—42no—
106C22×C17—42no—
107C22×C17—42no—
108C22×C17—41no—
109C22×C17—42no—
110C22×C17—41no—
111C22×C17C2×#4111yescharacteristic
112C22×C17C2×#4112yes—
113C22×C17H(5,𝔽2)11yescharacteristic
114C22×C17H(5,𝔽2)12yes—
115C22×C17H(5,𝔽2)11yescharacteristic
116C4×C17—42no—
117C4×C17—41no—
118C4×C17—41no—
119Dic17—42no—
120Dic17—42no—
121Dic17—42no—
122Dic17—42no—
123#399—171no—
124#397—171no—
125#397—172no—
126#396—172no—
127#388—172no—
128#388—172no—
129#388—172no—
130#379—172no—
131#379D1711yescharacteristic
132C2×C17—42no—
133C2×C17—44no—
134C2×C17#53812yes—
135C2×C17#53811yescharacteristic
136C2×C17#55112yes—
137C2×C17#55111yescharacteristic
138C2×C17#56011yescharacteristic
139C25C2×D1711yescharacteristic
140C2×C4⋊C4—171no—
141C2×C4⋊C4—172no—
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—171no—
160C17#273511yescharacteristic, complemented, Sylow
161C24—22no—
162C24—24no—
163C24—42no—
164C24—41no—
165C24—41no—
166C24C22×D1711yescharacteristic
167C24C22×D1712yes—
168C22×C4—172no—
169C22×C4—172no—
170C22×C4—172no—
171C22×C4—172no—
172C22×C4C22×D1711yescharacteristic
173C22×C4C22×D1712yes—
174C22×C4C22×D1711yescharacteristic
175C4⋊C4—682no—
176C4⋊C4—682no—
177C4⋊C4—681no—
178C4⋊C4—681no—
179C22⋊C4—24no—
180C22⋊C4—42no—
181C22⋊C4—41no—
182C22⋊C4—41no—
183C22⋊C4—344no—
184C22⋊C4—344no—
185C22⋊C4—682no—
186C22⋊C4—682no—
187C22⋊C4—682no—
188C22⋊C4—682no—
189C42—682no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—21no—
195C23—21no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—48no—
201C23—44no—
202C23—48no—
203C23—44no—
204C23—44no—
205C23#6325212yes—
206C23#6325414yes—
207C23C23×D1711yescharacteristic
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—342no—
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—342no—
226C2×C4—342no—
227C22—24no—
228C22—24no—
229C22—24no—
230C22—24no—
231C22—22no—
232C22—416no—
233C22—42no—
234C22—41no—
235C22—41no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22C2×#6325211yescharacteristic
241C22C2×#6325412yes—
242C22#1058769112yes—
243C22#1058769111yescharacteristic
244C22#1058769111yescharacteristic
245C4—41no—
246C4—41no—
247C4—42no—
248C4—682no—
249C4—682no—
250C4—682no—
251C4—682no—
252C2—42no—
253C2—44no—
254C2#1176339412yes—
255C2#1176339711yescharacteristic
256C2#1176348512yes—
257C2#1176349011yescharacteristic
258C2#1176354811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#379C2
prime radicalC25×C17C22
hypercenter#379D17
hypocenterC17#2735
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. #11762745C2semidirect
2. #11762737C2semidirect
3. #11762734C2semidirect
4. #11762732C2semidirect
5. #11762703C2semidirect
6. #11762702C2semidirect
7. #11762697C2semidirect
21. C2×C4⋊Dic17C22semidirect
22. C2×C4⋊Dic17C22semidirect
23. C2×#63225C22semidirect
24. C2×#63225C22semidirect
25. C2×C4×Dic17C22semidirect
26. #10587502C22semidirect
27. #10587502C22semidirect
28. #10587502C22semidirect
29. #10587502C22semidirect
30. #10587502C22semidirect
160. C17#2735semidirect

Maximal subgroups

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

subgroupindex
#117627452
#117627372
#117627342
#117627322
#117627032
#117627022
#117626972
#117626652
C17×#3792
#273517