Finite Group Databasebeta

← #14534413

Subgroups — #14534413

Order 2176.

582 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 141 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×C4×D68C211yesmaximal, characteristic, complemented
2#11762739C211yesmaximal, characteristic, complemented
3#11762734C211yesmaximal, characteristic, complemented
4#11762726C211yesmaximal, characteristic, complemented
5#11762718C211yesmaximal, characteristic
6#11762709C212yesmaximal, complemented
7#11762699C212yesmaximal, complemented
8#11762685C212yesmaximal, complemented
9#11762680C212yesmaximal, complemented
10#11762655C211yesmaximal, characteristic, complemented
11C2×C17×#126C211yesmaximal, characteristic, complemented
12C22×C22⋊D17C2212yes—
13C22×D68C2211yescharacteristic
14C22×C4×D17C2212yes—
15C17×#126—24no—
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C42×C17C2211yescharacteristic
23C2×C22⋊Dic17C2212yes—
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×#63227C2212yes—
27C2×#63227C2212yes—
28C2×C4⋊Dic17C2211yescharacteristic
29C2×#63225C2212yes—
30C2×C4×Dic17C2212yes—
31C4×D68—24no—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2211yescharacteristic
35#10587502C2211yescharacteristic
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C23×D17C2312yes—
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C2×C22⋊D17—24no—
46C2×C22⋊D17—24no—
47C2×C22⋊D17—42no—
48C22×Dic17C2312yes—
49C22×Dic17C2312yes—
50C22×Dic17C2312yes—
51C2×D68—24no—
52C2×D68—22no—
53C2×C4×D17—24no—
54C2×C4×D17—42no—
55C2×C4×D17—42no—
56C4⋊C4×C17—22no—
57C4⋊C4×C17—22no—
58C4⋊C4×C17—22no—
59C22⋊C4×C17—22no—
60C22⋊C4×C17—22no—
61C22⋊C4×C17—22no—
62C42×C17D414yescomplemented
63C22⋊Dic17—42no—
64#63227—24no—
65#63227—24no—
66#63227—24no—
67#63227—42no—
68C4⋊Dic17—22no—
69#63225—42no—
70C4×Dic17—24no—
71C23×C17—21no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—21no—
75C23×C17—21no—
76C23×C17C2411yescharacteristic
77C22×D17—22no—
78C22×D17—22no—
79C22×D17—22no—
80C22×D17—22no—
81C22×D17—22no—
82C22×D17—22no—
83C22×D17#4114yes—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—21no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—21no—
94C2×C4×C17—21no—
95C2×C4×C17C2×D412yes—
96C2×C4×C17C2×D412yes—
97C2×C4×C17C2×D412yes—
98C22⋊D17—48no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17#4114yes—
108D68—28no—
109C4×D17—48no—
110#2603—171nomaximal, Sylow
111C22×C17—41no—
112C22×C17—41no—
113C22×C17—41no—
114C22×C17—42no—
115C22×C17—42no—
116C22×C17C22×D411yescharacteristic
117C22×C17C2×#4112yes—
118C22×C17C2×#4112yes—
119C22×C17H(5,𝔽2)11yescharacteristic
120C22×C17H(5,𝔽2)11yescharacteristic
121C2×D17—24no—
122C2×D17—24no—
123C2×D17—24no—
124C2×D17—42no—
125C2×D17—42no—
126C2×D17—42no—
127C2×D17—42no—
128C4×C17—22no—
129C4×C17—22no—
130C4×C17—22no—
131C4×C17—41no—
132C4×C17—41no—
133C4×C17—41no—
134Dic17—24no—
135Dic17—42no—
136Dic17—42no—
137C2×#126D1711yescharacteristic, complemented
138C2×C4×D4—171no—
139#398—171no—
140#397—171no—
141#394—171no—
142#392—171no—
143#390—172no—
144#388—172no—
145#386—172no—
146#385—172no—
147#376—171no—
148C2×C17—81no—
149C2×C17#52511yescharacteristic
150C2×C17#53211yescharacteristic
151C2×C17#53511yescharacteristic
152C2×C17#55112yes—
153C2×C17#55312yes—
154D17—44no—
155C22×D4—172no—
156C22×D4—171no—
157C23×C4—172no—
158#126—24no—
159C4×D4—344no—
160C2×C4⋊C4—171no—
161C2×C4⋊C4—172no—
162C2×C4⋊C4C2×D1711yescharacteristic
163C2×C4⋊C4C2×D1711yescharacteristic
164C2×C4⋊C4C2×D1711yescharacteristic
165C2×C22⋊C4—172no—
166C2×C22⋊C4—172no—
167C2×C22⋊C4—172no—
168C2×C22⋊C4—172no—
169C2×C22⋊C4—172no—
170C2×C22⋊C4C2×D1711yescharacteristic
171C2×C22⋊C4C2×D1711yescharacteristic
172C2×C22⋊C4C2×D1711yescharacteristic
173C2×C42—172no—
174C2×C42C2×D1711yescharacteristic
175#95—172no—
176#95—171no—
177#95—171no—
178#95—172no—
179#95—172no—
180C17#260311yescharacteristic, complemented, Sylow
181C24—172no—
182C24C22×D1711yescharacteristic
183C2×D4—344no—
184C2×D4—344no—
185C2×D4—344no—
186C2×D4—342no—
187C2×D4—682no—
188C22×C4—172no—
189C22×C4—172no—
190C22×C4—172no—
191C22×C4—344no—
192C22×C4—682no—
193C22×C4—682no—
194C22×C4C22×D1711yescharacteristic
195C22×C4C22×D1711yescharacteristic
196C22×C4C22×D1711yescharacteristic
197C22×C4C22×D1711yescharacteristic
198C22×C4C22×D1711yescharacteristic
199C22×C4C22×D1711yescharacteristic
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—342no—
204C4⋊C4—682no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—344no—
209C22⋊C4—344no—
210C22⋊C4—344no—
211C22⋊C4—682no—
212C22⋊C4—682no—
213C42—344no—
214C42C22⋊D1714yescomplemented
215C23—22no—
216C23—22no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—174no—
221C23—342no—
222C23—342no—
223C23—342no—
224C23—342no—
225C23—342no—
226C23—342no—
227C23C23×D1711yescharacteristic
228D4—348no—
229D4—688no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—174no—
242C2×C4—342no—
243C2×C4—342no—
244C2×C4—342no—
245C2×C4—342no—
246C2×C4—342no—
247C2×C4—342no—
248C2×C4—342no—
249C2×C4—342no—
250C2×C4—688no—
251C2×C4C2×C22⋊D1712yes—
252C2×C4C2×C22⋊D1712yes—
253C2×C4C2×C22⋊D1712yes—
254C22—42no—
255C22—42no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—344no—
260C22—344no—
261C22—344no—
262C22—682no—
263C22—682no—
264C22—682no—
265C22—682no—
266C22#41×D1712yes—
267C22#41×D1712yes—
268C22#1058769611yescharacteristic
269C22#1058769611yescharacteristic
270C22C22×C22⋊D1711yescharacteristic
271C4—22no—
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—344no—
278C4—682no—
279C4—682no—
280C2—81no—
281C2—684no—
282C2#1176331611yescharacteristic
283C2#1176335511yescharacteristic
284C2#1176337311yescharacteristic
285C2#1176348412yes—
286C2#1176351012yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#126C2
prime radical#11762718C2
hypercenterC2×#126D17
hypocenterC17#2603
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×C4×D68C2semidirect
2. #11762739C2semidirect
3. #11762734C2semidirect
4. #11762726C2semidirect
6. #11762709C2semidirect
7. #11762699C2semidirect
8. #11762685C2semidirect
9. #11762680C2semidirect
10. #11762655C2semidirect
11. C2×C17×#126C2semidirect
62. C42×C17D4semidirect
137. C2×#126D17semidirect
180. C17#2603semidirect
214. C42C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D682
#117627392
#117627342
#117627262
#117627182
#117627092
#117626992
#117626852
#117626802
#117626552
C2×C17×#1262
#260317