Finite Group Databasebeta

← #14536809

Subgroups — #14536809

Order 2176.

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

normal subgroups only

This group has 644 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
1C2×#10587625C212yesmaximal, complemented
2#11762738C212yesmaximal, complemented
3#11762737C212yesmaximal, complemented
4#11762718C211yesmaximal, characteristic, complemented
5#11762718C212yesmaximal, complemented
6#11762717C212yesmaximal, complemented
7#11762717C212yesmaximal, complemented
8#11762670C211yesmaximal, characteristic, complemented
9C17×#392C211yesmaximal, characteristic, complemented
10C22×D4×C17C2212yes—
11C22×D4×C17C2211yescharacteristic
12C22×C22⋊D17C2212yes—
13C22×C22⋊D17C2212yes—
14C22×C22⋊D17C2212yes—
15C23×Dic17C2211yescharacteristic
16C22×D68C2211yescharacteristic
17C2×C22⋊C4×C17C2212yes—
18C2×C22⋊C4×C17C2211yescharacteristic
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2212yes—
21C2×C22⋊Dic17C2211yescharacteristic
22#10587625—28no—
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×#63225C2212yes—
27C17×#95C2211yescharacteristic
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2211yescharacteristic
31#10587502C2212yes—
32C24×C17C2312yes—
33C24×C17C2311yescharacteristic
34C23×D17C2312yes—
35C2×D4×C17—24no—
36C2×D4×C17—42no—
37C2×D4×C17—41no—
38C2×D4×C17—42no—
39C2×D4×C17—42no—
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2312yes—
43C2×C22⋊D17—24no—
44C2×C22⋊D17—24no—
45C2×C22⋊D17—42no—
46C2×C22⋊D17—42no—
47C2×C22⋊D17—42no—
48C2×C22⋊D17—42no—
49C2×C22⋊D17—42no—
50C2×C22⋊D17—42no—
51C2×C22⋊D17—42no—
52C22×Dic17—41no—
53C22×Dic17—41no—
54C22×Dic17D414yescomplemented
55C22×Dic17C2312yes—
56C22×Dic17C2312yes—
57C22×Dic17C2312yes—
58C2×D68—41no—
59C2×D68—42no—
60C22⋊C4×C17—41no—
61C22⋊C4×C17—42no—
62C22⋊Dic17—24no—
63C22⋊Dic17—41no—
64C22⋊Dic17—42no—
65#63227—24no—
66#63227—42no—
67#63227—42no—
68#63225—24no—
69C23×C17—21no—
70C23×C17—21no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17—22no—
78C23×C17—22no—
79C23×C17—22no—
80C23×C17C2×D412yes—
81C23×C17C2411yescharacteristic
82C22×D17—22no—
83C22×D17—22no—
84C22×D17—22no—
85C22×D17—22no—
86C22×D17—22no—
87C22×D17—22no—
88C22×D17—22no—
89D4×C17—48no—
90D4×C17—44no—
91C2×C4×C17—22no—
92C2×C4×C17—21no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—21no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C22⋊D17—48no—
99C22⋊D17—48no—
100C22⋊D17—48no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—44no—
110C2×Dic17C2×D414yes—
111D68—44no—
112#2815—171nomaximal, Sylow
113C22×C17—22no—
114C22×C17—24no—
115C22×C17—41no—
116C22×C17—42no—
117C22×C17—41no—
118C22×C17—42no—
119C22×C17—42no—
120C22×C17—42no—
121C22×C17—42no—
122C22×C17—42no—
123C22×C17—42no—
124C22×C17—42no—
125C22×C17C22×D411yescharacteristic
126C22×C17H(5,𝔽2)11yescharacteristic
127C22×C17H(5,𝔽2)12yes—
128C22×C17H(5,𝔽2)12yes—
129C22×C17H(5,𝔽2)11yescharacteristic
130C2×D17—42no—
131C2×D17—42no—
132C2×D17—42no—
133C2×D17—42no—
134C2×D17—42no—
135C2×D17—42no—
136C2×D17—42no—
137C4×C17—42no—
138C4×C17—41no—
139C4×C17—41no—
140Dic17—24no—
141Dic17—42no—
142Dic17—42no—
143C2×#121—172no—
144#397—172no—
145#397—172no—
146#392—172no—
147#392—172no—
148#392—172no—
149#392—171no—
150#392D1711yescharacteristic, complemented
151#381—171no—
152C2×C17—42no—
153C2×C17—82no—
154C2×C17#53411yescharacteristic
155C2×C17#53512yes—
156C2×C17#56011yescharacteristic
157C2×C17#56012yes—
158C2×C17#56011yescharacteristic
159D17—82no—
160C22×D4—172no—
161C22×D4—172no—
162C22×D4—172no—
163C22×D4—171no—
164C22×D4C2×D1712yes—
165C22×D4C2×D1711yescharacteristic
166C23×C4—171no—
167#121—348no—
168C2×C4⋊C4—172no—
169C2×C22⋊C4—172no—
170C2×C22⋊C4—172no—
171C2×C22⋊C4—171no—
172C2×C22⋊C4—172no—
173C2×C22⋊C4—172no—
174C2×C22⋊C4—172no—
175C2×C22⋊C4C2×D1711yescharacteristic
176C2×C22⋊C4C2×D1712yes—
177#95—172no—
178#95—172no—
179#95—172no—
180#95—171no—
181#95C2×D1711yescharacteristic
182C17#281511yescharacteristic, complemented, Sylow
183C24—172no—
184C24C22×D1712yes—
185C24C22×D1711yescharacteristic
186C2×D4—24no—
187C2×D4—42no—
188C2×D4—41no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—344no—
192C2×D4—344no—
193C2×D4—682no—
194C2×D4—681no—
195C2×D4—682no—
196C2×D4—682no—
197C2×D4—682no—
198C2×D4—682no—
199C2×D4—682no—
200C2×D4—682no—
201C2×D4—682no—
202C22×C4—172no—
203C22×C4—174no—
204C22×C4—172no—
205C22×C4—172no—
206C22×C4—681no—
207C22×C4—681no—
208C22×C4C22×D1711yescharacteristic
209C22×C4C22×D1711yescharacteristic
210C22×C4C22×D1712yes—
211C4⋊C4—344no—
212C22⋊C4—41no—
213C22⋊C4—42no—
214C22⋊C4—344no—
215C22⋊C4—344no—
216C22⋊C4—682no—
217C22⋊C4—682no—
218C22⋊C4—682no—
219C22⋊C4—681no—
220C23—21no—
221C23—22no—
222C23—22no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—342no—
232C23—342no—
233C23—342no—
234C23—342no—
235C23—342no—
236C23—342no—
237C23—342no—
238C23D4×D1712yes—
239C23C23×D1711yescharacteristic
240D4—48no—
241D4—44no—
242D4—684no—
243D4—688no—
244D4—688no—
245D4—688no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—174no—
254C2×C4—342no—
255C2×C4—342no—
256C2×C4—342no—
257C2×C4—342no—
258C2×C4—342no—
259C2×C4—342no—
260C2×C4—342no—
261C2×C4—342no—
262C2×C4—684no—
263C22—22no—
264C22—24no—
265C22—41no—
266C22—42no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—682no—
276C22—682no—
277C22—682no—
278C22—682no—
279C22—682no—
280C22—682no—
281C22—682no—
282C22C2×D4×D1711yescharacteristic
283C22#1058769112yes—
284C22#1058769111yescharacteristic
285C22#1058769112yes—
286C22#1058769611yescharacteristic
287C4—41no—
288C4—41no—
289C4—42no—
290C4—344no—
291C4—682no—
292C4—682no—
293C2—42no—
294C2—82no—
295C2—1362no—
296C2#1176336711yescharacteristic
297C2#1176337412yes—
298C2#1176354611yescharacteristic
299C2#1176354712yes—
300C2#1176354811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#392C2
hypercenter#392D17
hypocenterC17#2815
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#10587625C2semidirect
2. #11762738C2semidirect
3. #11762737C2semidirect
4. #11762718C2semidirect
5. #11762718C2semidirect
6. #11762717C2semidirect
7. #11762717C2semidirect
8. #11762670C2semidirect
9. C17×#392C2semidirect
54. C22×Dic17D4semidirect
150. #392D17semidirect
182. C17#2815semidirect

Maximal subgroups

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

subgroupindex
C2×#105876252
#117627382
#117627372
#117627182
#117627172
#117626702
C17×#3922
#281517