Finite Group Databasebeta

← #14533690

Subgroups — #14533690

Order 2176.

604 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 147 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×#10587582C212yesmaximal, complemented
2C2×#10587573C211yesmaximal, characteristic, complemented
3C2×#10587579C211yesmaximal, characteristic
4#11762740C211yesmaximal, characteristic, complemented
5#11762727C211yesmaximal, characteristic
6#11762726C211yesmaximal, characteristic
7#11762699C212yesmaximal, complemented
8#11762698C212yesmaximal, complemented
9#11762673C211yesmaximal, characteristic, complemented
10#11762668C212yesmaximal, complemented
11C17×#398C211yesmaximal, characteristic, complemented
12C22×D68C2211yescharacteristic
13C22×C4×D17C2212yes—
14C22×Dic34C2211yescharacteristic
15C2×C4⋊C4×C17C2212yes—
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×#63225C2212yes—
24C2×#63225C2211yescharacteristic
25C2×#63225C2211yescharacteristic
26C2×#63225C2212yes—
27C2×C4×Dic17C2211yescharacteristic
28C2×C4×Dic17C2211yescharacteristic
29C2×C4×Dic17C2211yescharacteristic
30#10587582—28no—
31#10587579—24no—
32#10587573—24no—
33C17×#95C2212yes—
34C17×#95C2211yescharacteristic
35#10587502C2212yes—
36#10587502C2212yes—
37#10587502C2211yescharacteristic
38C23×D17C2312yes—
39C22×C4×C17C2312yes—
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×Dic17C2312yes—
45C22×Dic17C2311yescharacteristic
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2311yescharacteristic
48C22×Dic17C2311yescharacteristic
49C2×D68—24no—
50C2×D68—22no—
51C2×C4×D17—24no—
52C2×C4×D17—24no—
53C2×C4×D17—24no—
54C2×Dic34—22no—
55C2×Dic34—22no—
56C2×Dic34D414yescomplemented
57C4⋊C4×C17—22no—
58C4⋊C4×C17—22no—
59C4⋊C4×C17—24no—
60#63227—24no—
61#63227—24no—
62#63227—24no—
63#63227—24no—
64#63227—24no—
65#63225—22no—
66#63225—22no—
67#63225—24no—
68#63225—24no—
69C4×Dic17—22no—
70C4×Dic17—22no—
71C4×Dic17D414yes—
72C23×C17C2411yescharacteristic
73C22×D17—22no—
74C22×D17—22no—
75C22×D17—22no—
76C22×D17—22no—
77C22×D17—22no—
78C22×D17—22no—
79C22×D17#4114yes—
80C2×C4×C17—21no—
81C2×C4×C17—21no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—21no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17C2×D412yes—
92C2×C4×C17C2×D412yes—
93C2×Dic17—21no—
94C2×Dic17—21no—
95C2×Dic17—22no—
96C2×Dic17—21no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17C2×D412yes—
102C2×Dic17C2×D412yes—
103C2×Dic17C2×D412yes—
104C2×Dic17C2×D412yes—
105C2×Dic17#4112yes—
106D68—28no—
107C4×D17—48no—
108Dic34—28no—
109#2536—171nomaximal, Sylow
110C22×C17C22×D411yescharacteristic
111C22×C17C22×D411yescharacteristic
112C22×C17C2×#4111yescharacteristic
113C22×C17C2×#4112yes—
114C22×C172−1+412yes—
115C2×D17—24no—
116C2×D17—24no—
117C2×D17—24no—
118C2×D17—42no—
119C2×D17—42no—
120C2×D17—42no—
121C2×D17—42no—
122C4×C17—22no—
123C4×C17—22no—
124C4×C17—42no—
125C4×C17—41no—
126C4×C17—42no—
127Dic17—22no—
128Dic17—22no—
129Dic17—41no—
130Dic17—42no—
131Dic17#12114yes—
132C2×C22⋊Q8—172no—
133C2×C4×Q8—171no—
134C2×C4×D4—171no—
135#398—171no—
136#398D1711yescharacteristic, complemented
137#394—171no—
138#394—171no—
139#388—172no—
140#388—172no—
141#382—171no—
142#380—172no—
143C2×C17C2×#12111yescharacteristic
144C2×C17#52511yescharacteristic
145C2×C17#53611yescharacteristic
146C2×C17#54812yes—
147C2×C17#55212yes—
148D17—44no—
149C22×Q8—171no—
150C22×D4—171no—
151C23×C4—172no—
152C22⋊Q8—348no—
153C4×Q8—344no—
154C4×D4—344no—
155C2×C4⋊C4—171no—
156C2×C4⋊C4—171no—
157C2×C4⋊C4—172no—
158C2×C4⋊C4—172no—
159C2×C4⋊C4C2×D1711yescharacteristic
160C2×C4⋊C4C2×D1711yescharacteristic
161C2×C4⋊C4C2×D1712yes—
162C2×C22⋊C4—172no—
163C2×C22⋊C4—172no—
164C2×C22⋊C4—172no—
165C2×C22⋊C4—172no—
166C2×C22⋊C4—172no—
167C2×C42—171no—
168C2×C42—171no—
169C2×C42—171no—
170#95—172no—
171#95—172no—
172#95—171no—
173#95C2×D1712yes—
174#95C2×D1711yescharacteristic
175C17#253611yescharacteristic, complemented, Sylow
176C24—172no—
177C2×Q8—174no—
178C2×Q8—342no—
179C2×Q8—342no—
180C2×D4—344no—
181C2×D4—342no—
182C22×C4—171no—
183C22×C4—171no—
184C22×C4—171no—
185C22×C4—172no—
186C22×C4—171no—
187C22×C4—344no—
188C22×C4—344no—
189C22×C4—344no—
190C22×C4C22×D1712yes—
191C22×C4C22×D1712yes—
192C22×C4C22×D1711yescharacteristic
193C22×C4C22×D1711yescharacteristic
194C22×C4C22×D1711yescharacteristic
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—24no—
198C4⋊C4—342no—
199C4⋊C4—344no—
200C4⋊C4—344no—
201C4⋊C4—342no—
202C22⋊C4—344no—
203C22⋊C4—344no—
204C22⋊C4—344no—
205C22⋊C4—344no—
206C22⋊C4—344no—
207C42—174no—
208C42—342no—
209C42—342no—
210C23—174no—
211C23—342no—
212C23—342no—
213C23—342no—
214C23—342no—
215C23—342no—
216C23—342no—
217C23C23×D1711yescharacteristic
218Q8—348no—
219D4—348no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—172no—
232C2×C4—172no—
233C2×C4—172no—
234C2×C4—172no—
235C2×C4—172no—
236C2×C4—341no—
237C2×C4—342no—
238C2×C4—341no—
239C2×C4—341no—
240C2×C4—342no—
241C2×C4—342no—
242C2×C4—342no—
243C2×C4—342no—
244C2×C4—688no—
245C2×C4D4×D1712yes—
246C2×C4D4×D1712yes—
247C22—344no—
248C22—344no—
249C22—344no—
250C22—682no—
251C22—682no—
252C22—682no—
253C22—682no—
254C22C2×D4×D1711yescharacteristic
255C22C2×D4×D1711yescharacteristic
256C22#1058769412yes—
257C22#41×D1712yes—
258C22#41×D1711yescharacteristic
259C4—22no—
260C4—22no—
261C4—42no—
262C4—42no—
263C4—41no—
264C4—174no—
265C4—342no—
266C4—342no—
267C4—682no—
268C4—681no—
269C2—684no—
270C2#121×D1711yescharacteristic
271C2#1176331711yescharacteristic
272C2#1176338411yescharacteristic
273C2#1176346912yes—
274C2#1176349112yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#398C2
prime radicalC22×D68C22
hypercenter#398D17
hypocenterC17#2536
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#10587582C2semidirect
2. C2×#10587573C2semidirect
4. #11762740C2semidirect
7. #11762699C2semidirect
8. #11762698C2semidirect
9. #11762673C2semidirect
10. #11762668C2semidirect
11. C17×#398C2semidirect
56. C2×Dic34D4semidirect
136. #398D17semidirect
175. C17#2536semidirect

Maximal subgroups

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

subgroupindex
C2×#105875822
C2×#105875732
C2×#105875792
#117627402
#117627272
#117627262
#117626992
#117626982
#117626732
#117626682
C17×#3982
#253617