Finite Group Databasebeta

← #14534726

Subgroups — #14534726

Order 2176.

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

normal subgroups only

This group has 584 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×#10587626C211yesmaximal, characteristic
2C2×#10587575C211yesmaximal, characteristic, complemented
3#11762727C211yesmaximal, characteristic, complemented
4#11762726C211yesmaximal, characteristic, complemented
5#11762725C212yesmaximal, complemented
6#11762724C212yesmaximal, complemented
7#11762723C211yesmaximal, characteristic, complemented
8#11762709C212yesmaximal, complemented
9#11762709C212yesmaximal, complemented
10#11762654C211yesmaximal, characteristic, complemented
11C17×#394C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C22×C22⋊D17C2212yes—
14C22×C22⋊D17C2212yes—
15C22×D68C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2212yes—
18C2×C22⋊C4×C17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2212yes—
21#10587626—24no—
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×C4⋊Dic17C2211yescharacteristic
27C2×#63225C2211yescharacteristic
28C2×#63225C2211yescharacteristic
29C2×#63225C2211yescharacteristic
30C2×#63225C2211yescharacteristic
31C2×C4×Dic17C2212yes—
32C2×C4×Dic17C2211yescharacteristic
33#10587575—24no—
34C17×#95C2211yescharacteristic
35#10587502C2211yescharacteristic
36#10587502C2212yes—
37C24×C17C2312yes—
38C23×D17C2312yes—
39C2×D4×C17—22no—
40C2×D4×C17—42no—
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C2×C22⋊D17—24no—
46C2×C22⋊D17—24no—
47C2×C22⋊D17—24no—
48C2×C22⋊D17—24no—
49C2×C22⋊D17—42no—
50C2×C22⋊D17—42no—
51C22×Dic17C2312yes—
52C22×Dic17C2311yescharacteristic
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2311yescharacteristic
55C22×Dic17C2311yescharacteristic
56C2×D68—22no—
57C2×D68—42no—
58C4⋊C4×C17—22no—
59C22⋊C4×C17—42no—
60C22⋊C4×C17—42no—
61C22⋊Dic17—42no—
62C22⋊Dic17—42no—
63#63227—42no—
64#63227—42no—
65#63227—42no—
66#63227—42no—
67C4⋊Dic17—22no—
68#63225—22no—
69#63225—22no—
70#63225—22no—
71#63225—22no—
72C4×Dic17—24no—
73C4×Dic17D414yes—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17—22no—
78C23×C17—22no—
79C23×C17—22no—
80C23×C17—22no—
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—44no—
90C2×C4×C17—22no—
91C2×C4×C17—21no—
92C2×C4×C17—21no—
93C2×C4×C17—22no—
94C2×C4×C17—21no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—21no—
99C2×C4×C17C2×D412yes—
100C22⋊D17—48no—
101C22⋊D17—48no—
102C2×Dic17—21no—
103C2×Dic17—21no—
104C2×Dic17—21no—
105C2×Dic17—21no—
106C2×Dic17—22no—
107C2×Dic17—21no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—21no—
112C2×Dic17C2×D412yes—
113C2×Dic17C2×D412yes—
114C2×Dic17#4114yes—
115D68—44no—
116#2641—171nomaximal, Sylow
117C22×C17—42no—
118C22×C17—42no—
119C22×C17—42no—
120C22×C17—42no—
121C22×C17—42no—
122C22×C17—42no—
123C22×C17—42no—
124C22×C17C22×D411yescharacteristic
125C22×C17C2×#4112yes—
126C22×C17H(5,𝔽2)12yes—
127C22×C17H(5,𝔽2)11yescharacteristic
128C22×C17H(5,𝔽2)11yescharacteristic
129C2×D17—42no—
130C2×D17—42no—
131C2×D17—42no—
132C2×D17—42no—
133C2×D17—42no—
134C2×D17—42no—
135C2×D17—42no—
136C4×C17—22no—
137C4×C17—41no—
138C4×C17—41no—
139C4×C17—42no—
140Dic17—22no—
141Dic17—41no—
142Dic17—41no—
143Dic17—42no—
144Dic17#12414yes—
145C2×#127—171no—
146C2×#125—171no—
147#394—171no—
148#394—171no—
149#394—172no—
150#394—172no—
151#394—171no—
152#394D1711yescharacteristic, complemented
153#390—172no—
154#390—172no—
155#376—171no—
156C2×C17—82no—
157C2×C17C2×#12411yescharacteristic
158C2×C17#53511yescharacteristic
159C2×C17#53511yescharacteristic
160C2×C17#54012yes—
161C2×C17#54012yes—
162D17—82no—
163C22×D4—172no—
164C22×D4—172no—
165C22×D4—171no—
166C22×D4C2×D1711yescharacteristic
167#127—344no—
168#125—344no—
169C2×C4⋊C4—171no—
170C2×C4⋊C4—171no—
171C2×C4⋊C4—171no—
172C2×C4⋊C4—171no—
173C2×C4⋊C4—171no—
174C2×C4⋊C4C2×D1711yescharacteristic
175C2×C22⋊C4—172no—
176C2×C22⋊C4—172no—
177C2×C22⋊C4—172no—
178C2×C22⋊C4—172no—
179C2×C22⋊C4—172no—
180C2×C22⋊C4—172no—
181C2×C22⋊C4C2×D1712yes—
182C2×C22⋊C4C2×D1712yes—
183C2×C42—171no—
184C2×C42—172no—
185#95—172no—
186#95—171no—
187#95C2×D1711yescharacteristic
188C17#264111yescharacteristic, complemented, Sylow
189C24—172no—
190C24C22×D1712yes—
191C2×D4—22no—
192C2×D4—42no—
193C2×D4—344no—
194C2×D4—344no—
195C2×D4—344no—
196C2×D4—344no—
197C2×D4—342no—
198C2×D4—682no—
199C2×D4—682no—
200C2×D4—682no—
201C22×C4—171no—
202C22×C4—171no—
203C22×C4—172no—
204C22×C4—171no—
205C22×C4—171no—
206C22×C4C22×D1712yes—
207C22×C4C22×D1711yescharacteristic
208C22×C4C22×D1711yescharacteristic
209C22×C4C22×D1711yescharacteristic
210C4⋊C4—22no—
211C4⋊C4—342no—
212C4⋊C4—342no—
213C4⋊C4—342no—
214C4⋊C4—342no—
215C4⋊C4—342no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—682no—
219C22⋊C4—682no—
220C22⋊C4—682no—
221C22⋊C4—682no—
222C22⋊C4—682no—
223C22⋊C4—682no—
224C42—174no—
225C42—344no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—342no—
234C23—342no—
235C23—342no—
236C23—342no—
237C23—342no—
238C23—342no—
239C23—342no—
240C23C23×D1711yescharacteristic
241D4—44no—
242D4—684no—
243D4—688no—
244D4—688no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—172no—
255C2×C4—174no—
256C2×C4—172no—
257C2×C4—341no—
258C2×C4—341no—
259C2×C4—341no—
260C2×C4—341no—
261C2×C4—342no—
262C2×C4—341no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—341no—
266C2×C4—342no—
267C2×C4D4×D1712yes—
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#41×D1712yes—
286C22#1058769611yescharacteristic
287C4—22no—
288C4—41no—
289C4—41no—
290C4—42no—
291C4—174no—
292C4—342no—
293C4—681no—
294C4—681no—
295C4—682no—
296C2—82no—
297C2—1362no—
298C2#124×D1711yescharacteristic
299C2#1176337411yescharacteristic
300C2#1176337511yescharacteristic
301C2#1176340912yes—
302C2#1176341112yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
prime radicalC2×#10587626C2
hypercenter#394D17
hypocenterC17#2641
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#10587575C2semidirect
3. #11762727C2semidirect
4. #11762726C2semidirect
5. #11762725C2semidirect
6. #11762724C2semidirect
7. #11762723C2semidirect
8. #11762709C2semidirect
9. #11762709C2semidirect
10. #11762654C2semidirect
11. C17×#394C2semidirect
152. #394D17semidirect
188. C17#2641semidirect

Maximal subgroups

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

subgroupindex
C2×#105876262
C2×#105875752
#117627272
#117627262
#117627252
#117627242
#117627232
#117627092
#117626542
C17×#3942
#264117