Finite Group Databasebeta

← #14534158

Subgroups — #14534158

Order 2176.

544 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 159 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×#10587583C211yesmaximal, characteristic
2C2×C4⋊C4×D17C211yesmaximal, characteristic
3#11762740C211yesmaximal, characteristic, complemented
4#11762730C211yesmaximal, characteristic, complemented
5#11762729C211yesmaximal, characteristic, complemented
6#11762708C212yesmaximal, complemented
7#11762706C212yesmaximal, complemented
8#11762683C212yesmaximal
9#11762680C212yesmaximal
10#11762656C211yesmaximal, characteristic
11C17×#398C211yesmaximal, characteristic, complemented
12C22×C4×D17C2211yescharacteristic
13C22×C4×D17C2211yescharacteristic
14C22×C4×D17C2211yescharacteristic
15C22×Dic34C2211yescharacteristic
16C2×C4⋊C4×C17C2212yes—
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×C4⋊Dic17C2212yes—
22C2×C4⋊Dic17C2211yescharacteristic
23C2×C4⋊Dic17C2212yes—
24C2×#63225C2211yescharacteristic
25C2×#63225C2212yes—
26C2×#63225C2211yescharacteristic
27C2×#63225C2212yes—
28C2×C4×Dic17C2212yes—
29#10587583—24no—
30C4⋊C4×D17—24no—
31C17×#95C2212yes—
32C17×#95C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C23×D17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2312yes—
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C22×Dic17C2312yes—
44C22×Dic17C2311yescharacteristic
45C22×Dic17C2311yescharacteristic
46C22×Dic17C2312yes—
47C22×Dic17C2311yescharacteristic
48C2×C4×D17—22no—
49C2×C4×D17—22no—
50C2×C4×D17—22no—
51C2×C4×D17—22no—
52C2×C4×D17—42no—
53C2×C4×D17—42no—
54C2×C4×D17D414yes—
55C2×Dic34—22no—
56C2×Dic34—42no—
57C4⋊C4×C17—22no—
58C4⋊C4×C17—22no—
59C4⋊C4×C17—24no—
60#63227—24no—
61#63227Q818yescomplemented
62C4⋊Dic17—22no—
63C4⋊Dic17—24no—
64C4⋊Dic17—42no—
65#63225—22no—
66#63225—22no—
67#63225—24no—
68#63225—42no—
69C4×Dic17—42no—
70C23×C17C2411yescharacteristic
71C22×D17—21no—
72C22×D17—21no—
73C22×D17—22no—
74C22×D17C2×D412yes—
75C22×D17C2×Q814yes—
76C2×C4×C17—21no—
77C2×C4×C17—22no—
78C2×C4×C17—21no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—21no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17C2×D412yes—
87C2×C4×C17C2×Q814yes—
88C2×C4×C17#4112yes—
89C2×Dic17—21no—
90C2×Dic17—21no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—21no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17C2×D412yes—
100C2×Dic17C2×Q814yes—
101C2×Dic17#4112yes—
102C4×D17—28no—
103C4×D17—44no—
104C4×D17—44no—
105Dic34—44no—
106#2583—171nomaximal, Sylow
107C22×C17C22×D411yescharacteristic
108C22×C17C22×Q812yes—
109C22×C17C2×#4111yescharacteristic
110C22×C17C2×#4111yescharacteristic
111C22×C17H(5,𝔽2)12yes—
112C2×D17—22no—
113C2×D17—22no—
114C2×D17—24no—
115C2×D17—24no—
116C2×D17C4⋊Q814yes—
117C4×C17—22no—
118C4×C17—22no—
119C4×C17—24no—
120C4×C17—42no—
121C4×C17—41no—
122Dic17—22no—
123Dic17—22no—
124Dic17—24no—
125Dic17—42no—
126Dic17—41no—
127C2×C22⋊Q8—171no—
128C22×C4⋊C4—171no—
129#398—171no—
130#398D1711yescharacteristic, complemented
131#395—171no—
132#395—171no—
133#389—172no—
134#389—172no—
135#385—172no—
136#385—172no—
137C4×C22⋊C4—171no—
138C2×C17#52511yescharacteristic
139C2×C17C2×C4⋊Q811yescharacteristic
140C2×C17#53411yescharacteristic
141C2×C17#55412yes—
142C2×C17#55412yes—
143D17—24no—
144C22×Q8—171no—
145C23×C4—171no—
146C23×C4—171no—
147C23×C4—171no—
148C22⋊Q8—344no—
149C2×C4⋊C4—171no—
150C2×C4⋊C4—171no—
151C2×C4⋊C4—171no—
152C2×C4⋊C4—172no—
153C2×C4⋊C4—172no—
154C2×C4⋊C4—172no—
155C2×C4⋊C4—172no—
156C2×C4⋊C4—344no—
157C2×C4⋊C4C2×D1711yescharacteristic
158C2×C4⋊C4C2×D1711yescharacteristic
159C2×C4⋊C4C2×D1712yes—
160C2×C22⋊C4—172no—
161C2×C22⋊C4—172no—
162C2×C42—172no—
163#95—171no—
164#95—172no—
165#95—172no—
166#95—172no—
167#95C2×D1711yescharacteristic
168#95C2×D1712yes—
169C17#258311yescharacteristic, complemented, Sylow
170C24—171no—
171C2×Q8—342no—
172C2×Q8—682no—
173C22×C4—172no—
174C22×C4—171no—
175C22×C4—171no—
176C22×C4—174no—
177C22×C4—172no—
178C22×C4—171no—
179C22×C4—342no—
180C22×C4—342no—
181C22×C4—342no—
182C22×C4—342no—
183C22×C4—682no—
184C22×C4—682no—
185C22×C4C22×D1712yes—
186C22×C4C22×D1711yescharacteristic
187C22×C4C22×D1711yescharacteristic
188C22×C4C22×D1711yescharacteristic
189C22×C4C22×D1712yes—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—24no—
193C4⋊C4—342no—
194C4⋊C4—342no—
195C4⋊C4—344no—
196C4⋊C4—344no—
197C4⋊C4—342no—
198C4⋊C4—682no—
199C4⋊C4—682no—
200C22⋊C4—178no—
201C22⋊C4—344no—
202C42—682no—
203C23—172no—
204C23—174no—
205C23—341no—
206C23—341no—
207C23—342no—
208C23C23×D1711yescharacteristic
209Q8—684no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—172no—
221C2×C4—174no—
222C2×C4—172no—
223C2×C4—348no—
224C2×C4—341no—
225C2×C4—341no—
226C2×C4—342no—
227C2×C4—342no—
228C2×C4—342no—
229C2×C4—342no—
230C2×C4—341no—
231C2×C4—342no—
232C2×C4—342no—
233C2×C4—342no—
234C2×C4—684no—
235C2×C4—684no—
236C2×C4D4×D1712yes—
237C2×C4#6325412yes—
238C2×C4Q8×D1714yes—
239C22—174no—
240C22—344no—
241C22—344no—
242C22—342no—
243C22—342no—
244C22C2×D4×D1711yescharacteristic
245C22C2×#6325411yescharacteristic
246C22C2×Q8×D1712yes—
247C22#41×D1711yescharacteristic
248C22#1058769612yes—
249C4—24no—
250C4—22no—
251C4—22no—
252C4—42no—
253C4—41no—
254C4—344no—
255C4—342no—
256C4—342no—
257C4—681no—
258C4—682no—
259C2—344no—
260C2#1176331911yescharacteristic
261C2C4⋊Q8×D1711yescharacteristic
262C2#1176337011yescharacteristic
263C2#1176351512yes—
264C2#1176351712yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#398C2
prime radicalC23×D17C23
hypercenter#398D17
hypocenterC17#2583
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. #11762740C2semidirect
4. #11762730C2semidirect
5. #11762729C2semidirect
6. #11762708C2semidirect
7. #11762706C2semidirect
11. C17×#398C2semidirect
61. #63227Q8semidirect
130. #398D17semidirect
169. C17#2583semidirect

Maximal subgroups

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

subgroupindex
C2×#105875832
C2×C4⋊C4×D172
#117627402
#117627302
#117627292
#117627082
#117627062
#117626832
#117626802
#117626562
C17×#3982
#258317