Finite Group Databasebeta

← #14533952

Subgroups — #14533952

Order 2176.

564 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 151 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×#10587630C211yesmaximal, characteristic, complemented
2C2×#10587567C211yesmaximal, characteristic
3#11762722C211yesmaximal, characteristic, complemented
4#11762721C211yesmaximal, characteristic, complemented
5#11762720C211yesmaximal, characteristic, complemented
6#11762709C212yesmaximal
7#11762673C212yesmaximal, complemented
8#11762668C212yesmaximal, complemented
9#11762667C212yesmaximal, complemented
10#11762660C211yesmaximal, characteristic, complemented
11C17×#393C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C22×C22⋊D17C2211yescharacteristic
14C23×Dic17C2211yescharacteristic
15C22×C4×D17C2211yescharacteristic
16C22×Dic34C2211yescharacteristic
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊C4×C17C2212yes—
19C2×C22⋊Dic17C2212yes—
20#10587630—24no—
21C2×#63227C2211yescharacteristic
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×#63225C2212yes—
25C2×#63225C2211yescharacteristic
26C2×#63225C2212yes—
27C2×#63225C2212yes—
28C2×C4×Dic17C2212yes—
29C2×C4×Dic17C2211yescharacteristic
30#10587567—24no—
31C17×#95C2211yescharacteristic
32C17×#95C2212yes—
33#10587502C2211yescharacteristic
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C23×D17C2311yescharacteristic
39C2×Q8×C17—22no—
40C2×Q8×C17—42no—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2312yes—
43C22×C4×C17C2312yes—
44C22×C4×C17C2311yescharacteristic
45C2×C22⋊D17—22no—
46C2×C22⋊D17—22no—
47C2×C22⋊D17Q814yescomplemented
48C22×Dic17—22no—
49C22×Dic17—24no—
50C22×Dic17C2312yes—
51C22×Dic17C2311yescharacteristic
52C22×Dic17C2312yes—
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2311yescharacteristic
55C2×C4×D17—22no—
56C2×C4×D17—24no—
57C2×Dic34—22no—
58C2×Dic34—42no—
59C22⋊C4×C17—22no—
60C22⋊C4×C17—42no—
61C22⋊Dic17—42no—
62#63227—22no—
63#63227—42no—
64#63227—42no—
65#63225—22no—
66#63225—24no—
67#63225—24no—
68#63225—24no—
69C4×Dic17—24no—
70C4×Dic17D414yes—
71C23×C17—21no—
72C23×C17—22no—
73C23×C17—21no—
74C23×C17—22no—
75C23×C17C2×Q812yes—
76C23×C17C2411yescharacteristic
77C22×D17—21no—
78C22×D17—22no—
79C22×D17—21no—
80C22×D17—22no—
81C22×D17C2×Q812yes—
82Q8×C17—44no—
83C2×C4×C17—22no—
84C2×C4×C17—21no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17C2×D412yes—
93C22⋊D17—28no—
94C2×Dic17—22no—
95C2×Dic17—21no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—44no—
103C2×Dic17C2×D412yes—
104C2×Dic17C2×D412yes—
105C2×Dic17C2×Q812yes—
106C2×Dic17#4112yes—
107C2×Dic17#4114yes—
108C4×D17—44no—
109Dic34—44no—
110#2563—171nomaximal, Sylow
111C22×C17—22no—
112C22×C17—24no—
113C22×C17—41no—
114C22×C17—41no—
115C22×C17—42no—
116C22×C17C22×D411yescharacteristic
117C22×C17C22×Q811yescharacteristic
118C22×C17C2×#4111yescharacteristic
119C22×C17C2×#4112yes—
120C22×C17H(5,𝔽2)12yes—
121C2×D17—22no—
122C2×D17—24no—
123C2×D17—41no—
124C2×D17—41no—
125C2×D17—42no—
126C4×C17—22no—
127C4×C17—41no—
128C4×C17—42no—
129C4×C17—42no—
130Dic17—41no—
131Dic17—42no—
132Dic17—42no—
133Dic17C22⋊Q814yes—
134Dic17#12414yes—
135C2×C4⋊Q8—171no—
136C2×C4×D4—171no—
137#393—171no—
138#393—171no—
139#393—171no—
140#393D1711yescharacteristic, complemented
141#390—172no—
142#382—172no—
143#380—172no—
144#380—172no—
145C4×C4⋊C4—171no—
146C2×C17—42no—
147C2×C17C2×C22⋊Q811yescharacteristic
148C2×C17C2×#12411yescharacteristic
149C2×C17#53511yescharacteristic
150C2×C17#55412yes—
151C2×C17#55512yes—
152D17—42no—
153C22×Q8—171no—
154C22×Q8C2×D1711yescharacteristic
155C22×D4—171no—
156C23×C4—171no—
157C23×C4—171no—
158C4⋊Q8—344no—
159C4×D4—344no—
160C2×C4⋊C4—171no—
161C2×C4⋊C4—172no—
162C2×C4⋊C4—172no—
163C2×C4⋊C4—172no—
164C2×C22⋊C4—171no—
165C2×C22⋊C4—172no—
166C2×C22⋊C4—172no—
167C2×C22⋊C4—172no—
168C2×C22⋊C4C2×D1711yescharacteristic
169C2×C22⋊C4C2×D1712yes—
170C2×C42—171no—
171C2×C42—172no—
172#95—171no—
173#95—172no—
174#95—172no—
175#95—172no—
176#95C2×D1711yescharacteristic
177#95C2×D1712yes—
178C17#256311yescharacteristic, complemented, Sylow
179C24—171no—
180C24C22×D1711yescharacteristic
181C2×Q8—22no—
182C2×Q8—42no—
183C2×Q8—342no—
184C2×Q8—682no—
185C2×D4—174no—
186C2×D4—342no—
187C2×D4—342no—
188C22×C4—171no—
189C22×C4—171no—
190C22×C4—171no—
191C22×C4—172no—
192C22×C4—172no—
193C22×C4—342no—
194C22×C4—344no—
195C22×C4—342no—
196C22×C4—344no—
197C22×C4C22×D1711yescharacteristic
198C22×C4C22×D1712yes—
199C22×C4C22×D1712yes—
200C22×C4C22×D1711yescharacteristic
201C4⋊C4—342no—
202C4⋊C4—344no—
203C4⋊C4—344no—
204C4⋊C4—344no—
205C22⋊C4—22no—
206C22⋊C4—42no—
207C22⋊C4—342no—
208C22⋊C4—682no—
209C22⋊C4—682no—
210C22⋊C4—682no—
211C42—174no—
212C42—344no—
213C23—21no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—172no—
218C23—341no—
219C23—342no—
220C23—342no—
221C23—341no—
222C23Q8×D1712yes—
223C23C23×D1711yescharacteristic
224Q8—44no—
225Q8—684no—
226D4—348no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—172no—
237C2×C4—172no—
238C2×C4—172no—
239C2×C4—172no—
240C2×C4—174no—
241C2×C4—341no—
242C2×C4—342no—
243C2×C4—342no—
244C2×C4—342no—
245C2×C4—342no—
246C2×C4—342no—
247C2×C4—342no—
248C2×C4—342no—
249C2×C4—684no—
250C2×C4—684no—
251C2×C4D4×D1712yes—
252C22—24no—
253C22—22no—
254C22—42no—
255C22—41no—
256C22—41no—
257C22—344no—
258C22—342no—
259C22—682no—
260C22—681no—
261C22—681no—
262C22C2×D4×D1711yescharacteristic
263C22#1058769112yes—
264C22C2×Q8×D1711yescharacteristic
265C22#41×D1712yes—
266C22#41×D1711yescharacteristic
267C4—22no—
268C4—42no—
269C4—41no—
270C4—42no—
271C4—174no—
272C4—174no—
273C4—682no—
274C4—681no—
275C4—682no—
276C2—42no—
277C2—682no—
278C2C22⋊Q8×D1711yescharacteristic
279C2#124×D1711yescharacteristic
280C2#1176337411yescharacteristic
281C2#1176351312yes—
282C2#1176352112yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#393C2
prime radicalC22×C22⋊D17C22
hypercenter#393D17
hypocenterC17#2563
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#10587630C2semidirect
3. #11762722C2semidirect
4. #11762721C2semidirect
5. #11762720C2semidirect
7. #11762673C2semidirect
8. #11762668C2semidirect
9. #11762667C2semidirect
10. #11762660C2semidirect
11. C17×#393C2semidirect
47. C2×C22⋊D17Q8semidirect
140. #393D17semidirect
178. C17#2563semidirect

Maximal subgroups

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

subgroupindex
C2×#105876302
C2×#105875672
#117627222
#117627212
#117627202
#117627092
#117626732
#117626682
#117626672
#117626602
C17×#3932
#256317