Finite Group Databasebeta

← #14534643

Subgroups — #14534643

Order 2176.

500 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#10587571C211yesmaximal, characteristic
2C2×Q8×Dic17C211yesmaximal, characteristic, complemented
3#11762746C211yesmaximal, characteristic, complemented
4#11762745C211yesmaximal, characteristic, complemented
5#11762730C212yesmaximal, complemented
6#11762725C212yesmaximal
7#11762720C211yesmaximal, characteristic, complemented
8#11762702C212yesmaximal, complemented
9#11762698C212yesmaximal, complemented
10#11762676C211yesmaximal, characteristic, complemented
11C17×#393C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C22×C22⋊D17C2211yescharacteristic
14C22×Dic34C2211yescharacteristic
15C2×C22⋊C4×C17C2212yes—
16C2×C22⋊C4×C17C2211yescharacteristic
17C2×C22⋊Dic17C2211yescharacteristic
18C2×C22⋊Dic17C2212yes—
19Q8×Dic17—24no—
20C2×#63227C2211yescharacteristic
21C2×#63227C2212yes—
22C2×#63227C2211yescharacteristic
23C2×#63227C2212yes—
24C2×C4⋊Dic17C2212yes—
25C2×C4⋊Dic17C2211yescharacteristic
26C2×C4⋊Dic17C2212yes—
27C2×#63225C2212yes—
28C2×C4×Dic17C2212yes—
29C2×C4×Dic17C2211yescharacteristic
30#10587571—24no—
31C17×#95C2212yes—
32C17×#95C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2211yescharacteristic
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C23×D17C2311yescharacteristic
39C2×Q8×C17—22no—
40C2×Q8×C17—24no—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2312yes—
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2312yes—
45C2×C22⋊D17—22no—
46C2×C22⋊D17—42no—
47C22×Dic17C2311yescharacteristic
48C22×Dic17C2312yes—
49C22×Dic17C2312yes—
50C22×Dic17C2311yescharacteristic
51C22×Dic17C2311yescharacteristic
52C2×Dic34—22no—
53C2×Dic34—42no—
54C22⋊C4×C17—22no—
55C22⋊C4×C17—24no—
56C22⋊Dic17—22no—
57C22⋊Dic17—42no—
58#63227—22no—
59#63227—24no—
60#63227—22no—
61#63227—42no—
62C4⋊Dic17—22no—
63C4⋊Dic17—24no—
64C4⋊Dic17—42no—
65#63225—42no—
66C4×Dic17—24no—
67C4×Dic17D414yes—
68C23×C17—21no—
69C23×C17—22no—
70C23×C17—22no—
71C23×C17—21no—
72C23×C17—21no—
73C23×C17C2411yescharacteristic
74C22×D17—21no—
75C22×D17—22no—
76C22×D17—22no—
77C22×D17—21no—
78C22×D17—21no—
79Q8×C17—28no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
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×C4×C17C2×D412yes—
89C2×C4×C17#4114yes—
90C22⋊D17—44no—
91C2×Dic17—22no—
92C2×Dic17—21no—
93C2×Dic17—21no—
94C2×Dic17—21no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17C2×D412yes—
103C2×Dic17C2×D412yes—
104Dic34—44no—
105#2632—171nomaximal, Sylow
106C22×C17—41no—
107C22×C17—41no—
108C22×C17—41no—
109C22×C17—42no—
110C22×C17—42no—
111C22×C17C22×D411yescharacteristic
112C22×C17C2×#4112yes—
113C22×C17H(5,𝔽2)12yes—
114C22×C172−1+411yescharacteristic
115C22×C172−1+411yescharacteristic
116C2×D17—41no—
117C2×D17—41no—
118C2×D17—41no—
119C2×D17—42no—
120C2×D17—42no—
121C4×C17—22no—
122C4×C17—24no—
123C4×C17—41no—
124C4×C17—42no—
125Dic17—22no—
126Dic17—22no—
127Dic17—41no—
128Dic17—42no—
129Dic17—42no—
130C2×#124—171no—
131C2×C4×Q8—171no—
132#399—171no—
133#399—171no—
134#395—172no—
135#394—172no—
136#393—171no—
137#393D1711yescharacteristic, complemented
138#388—172no—
139#388—172no—
140#383—171no—
141C2×C17—81no—
142C2×C17#52511yescharacteristic
143C2×C17#53511yescharacteristic
144C2×C17#53611yescharacteristic
145C2×C17#54212yes—
146C2×C17#54212yes—
147D17—81no—
148C22×Q8—171no—
149C22×Q8C2×D1711yescharacteristic
150C22×D4—171no—
151#124—344no—
152C4×Q8—344no—
153C2×C4⋊C4—172no—
154C2×C4⋊C4—171no—
155C2×C4⋊C4—172no—
156C2×C4⋊C4—172no—
157C2×C22⋊C4—171no—
158C2×C22⋊C4—171no—
159C2×C22⋊C4—171no—
160C2×C22⋊C4—172no—
161C2×C22⋊C4—172no—
162C2×C22⋊C4—172no—
163C2×C22⋊C4C2×D1711yescharacteristic
164C2×C22⋊C4C2×D1712yes—
165C2×C42—171no—
166C2×C42—172no—
167#95—172no—
168#95—171no—
169#95—172no—
170#95—172no—
171#95C2×D1711yescharacteristic
172#95C2×D1712yes—
173C17#263211yescharacteristic, complemented, Sylow
174C24—171no—
175C24C22×D1711yescharacteristic
176C2×Q8—22no—
177C2×Q8—24no—
178C2×Q8—342no—
179C2×Q8—682no—
180C2×D4—342no—
181C2×D4—682no—
182C22×C4—171no—
183C22×C4—171no—
184C22×C4—172no—
185C22×C4—171no—
186C22×C4—172no—
187C22×C4C22×D1712yes—
188C22×C4C22×D1711yescharacteristic
189C22×C4C22×D1711yescharacteristic
190C22×C4C22×D1712yes—
191C4⋊C4—344no—
192C4⋊C4—342no—
193C4⋊C4—682no—
194C4⋊C4—682no—
195C22⋊C4—22no—
196C22⋊C4—24no—
197C22⋊C4—342no—
198C22⋊C4—342no—
199C22⋊C4—342no—
200C22⋊C4—344no—
201C22⋊C4—682no—
202C22⋊C4—682no—
203C42—174no—
204C42—344no—
205C23—21no—
206C23—22no—
207C23—21no—
208C23—22no—
209C23—21no—
210C23—341no—
211C23—342no—
212C23—341no—
213C23—342no—
214C23—341no—
215C23C23×D1711yescharacteristic
216Q8—28no—
217Q8—684no—
218D4—684no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—172no—
228C2×C4—172no—
229C2×C4—341no—
230C2×C4—342no—
231C2×C4—341no—
232C2×C4—342no—
233C2×C4—341no—
234C2×C4—342no—
235C2×C4—342no—
236C2×C4—342no—
237C2×C4—342no—
238C2×C4—342no—
239C2×C4—342no—
240C2×C4D4×D1712yes—
241C2×C4#6325414yes—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—42no—
246C22—42no—
247C22—681no—
248C22—681no—
249C22—681no—
250C22—682no—
251C22—682no—
252C22C2×D4×D1711yescharacteristic
253C22C2×#6325412yes—
254C22#1058769411yescharacteristic
255C22#1058769612yes—
256C22#1058769711yescharacteristic
257C4—22no—
258C4—24no—
259C4—41no—
260C4—42no—
261C4—342no—
262C4—342no—
263C4—681no—
264C4—682no—
265C4—682no—
266C2—81no—
267C2—1361no—
268C2#1176331511yescharacteristic
269C2#1176337711yescharacteristic
270C2#1176338211yescharacteristic
271C2#1176342812yes—
272C2#1176343212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#393C2
prime radicalC22×C22⋊D17C22
hypercenter#393D17
hypocenterC17#2632
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic17C2semidirect
3. #11762746C2semidirect
4. #11762745C2semidirect
5. #11762730C2semidirect
7. #11762720C2semidirect
8. #11762702C2semidirect
9. #11762698C2semidirect
10. #11762676C2semidirect
11. C17×#393C2semidirect
137. #393D17semidirect
173. C17#2632semidirect

Maximal subgroups

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

subgroupindex
C2×#105875712
C2×Q8×Dic172
#117627462
#117627452
#117627302
#117627252
#117627202
#117627022
#117626982
#117626762
C17×#3932
#263217