Finite Group Databasebeta

← #14534136

Subgroups — #14534136

Order 2176.

584 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 153 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×#10587567C211yesmaximal, characteristic
2C22×#63225C211yesmaximal, characteristic, complemented
3#11762740C212yesmaximal, complemented
4#11762733C212yesmaximal, complemented
5#11762691C212yesmaximal, complemented
6#11762688C211yesmaximal, characteristic
7#11762686C211yesmaximal, characteristic
8#11762681C211yesmaximal, characteristic, complemented
9#11762680C212yesmaximal, complemented
10#11762667C211yesmaximal, characteristic, complemented
11C17×#385C211yesmaximal, characteristic, complemented
12C23×C4×C17C2211yescharacteristic
13C22×C22⋊D17C2211yescharacteristic
14C23×Dic17C2211yescharacteristic
15C23×Dic17C2211yescharacteristic
16C22×C4×D17C2211yescharacteristic
17C2×C4⋊C4×C17C2212yes—
18C2×C22⋊C4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C22⋊Dic17C2211yescharacteristic
21C2×#63227C2211yescharacteristic
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×#63225—24no—
25C2×#63225C2212yes—
26C2×#63225C2212yes—
27C2×#63225C2212yes—
28C2×#63225C2211yescharacteristic
29C2×#63225C2212yes—
30C2×C4×Dic17C2211yescharacteristic
31#10587567—24no—
32C17×#95C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C23×D17C2311yescharacteristic
39C22×C4×C17—22no—
40C22×C4×C17—42no—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2312yes—
43C22×C4×C17C2312yes—
44C22×C4×C17C2311yescharacteristic
45C2×C22⋊D17—22no—
46C2×C22⋊D17—22no—
47C2×C22⋊D17Q814yes—
48C22×Dic17—22no—
49C22×Dic17—24no—
50C22×Dic17—42no—
51C22×Dic17D414yes—
52C22×Dic17C2312yes—
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2311yescharacteristic
55C22×Dic17C2311yescharacteristic
56C22×Dic17C2312yes—
57C2×C4×D17—22no—
58C2×C4×D17—42no—
59C4⋊C4×C17—42no—
60C22⋊C4×C17—22no—
61C22⋊C4×C17—22no—
62C22⋊Dic17—22no—
63#63227—22no—
64#63227—24no—
65#63227—42no—
66#63225—22no—
67#63225—24no—
68#63225—24no—
69#63225—24no—
70#63225—42no—
71C4×Dic17—22no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17C2×D412yes—
75C23×C17C2×Q812yes—
76C23×C17#4112yes—
77C23×C17C2411yescharacteristic
78C22×D17—21no—
79C22×D17—21no—
80C22×D17—22no—
81C22×D17—22no—
82C22×D17C2×Q812yes—
83C2×C4×C17—21no—
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×C17—22no—
93C2×C4×C17—44no—
94C22⋊D17—28no—
95C2×Dic17—28no—
96C2×Dic17—22no—
97C2×Dic17—21no—
98C2×Dic17—22no—
99C2×Dic17—21no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—44no—
106C2×Dic17C2×D412yes—
107C2×Dic17C2×D412yes—
108C2×Dic17C2×Q812yes—
109C2×Dic17#4114yes—
110C4×D17—44no—
111#2579—171nomaximal, Sylow
112C22×C17—22no—
113C22×C17—22no—
114C22×C17—24no—
115C22×C17—24no—
116C22×C17C22⋊Q814yes—
117C22×C17C22×D411yescharacteristic
118C22×C17C22×Q811yescharacteristic
119C22×C17C2×#4111yescharacteristic
120C22×C17C2×#4112yes—
121C22×C17H(5,𝔽2)12yes—
122C2×D17—22no—
123C2×D17—24no—
124C2×D17—41no—
125C2×D17—41no—
126C2×D17—42no—
127C4×C17—41no—
128C4×C17—41no—
129C4×C17—42no—
130C4×C17—42no—
131Dic17—22no—
132Dic17—24no—
133Dic17—22no—
134Dic17—22no—
135Dic17—42no—
136C2×C4×D4—171no—
137C22×C4⋊C4—171no—
138#398—172no—
139#396—172no—
140#387—172no—
141#386—171no—
142#386—171no—
143#385—172no—
144#385—171no—
145#385D1711yescharacteristic, complemented
146#380—171no—
147C2×C17—24no—
148C2×C17C2×C22⋊Q811yescharacteristic
149C2×C17#52511yescharacteristic
150C2×C17#53411yescharacteristic
151C2×C17#55312yes—
152C2×C17#55412yes—
153D17—42no—
154C22×D4—171no—
155C23×C4—171no—
156C23×C4—171no—
157C23×C4—171no—
158C23×C4C2×D1711yescharacteristic
159C4×D4—344no—
160C2×C4⋊C4—172no—
161C2×C4⋊C4—172no—
162C2×C4⋊C4—171no—
163C2×C4⋊C4—172no—
164C2×C4⋊C4—172no—
165C2×C4⋊C4—344no—
166C2×C4⋊C4C2×D1712yes—
167C2×C22⋊C4—171no—
168C2×C22⋊C4—171no—
169C2×C22⋊C4—172no—
170C2×C22⋊C4—172no—
171C2×C22⋊C4C2×D1711yescharacteristic
172C2×C22⋊C4C2×D1711yescharacteristic
173C2×C42—171no—
174#95—172no—
175#95—172no—
176#95—172no—
177#95—172no—
178#95C2×D1712yes—
179C17#257911yescharacteristic, complemented, Sylow
180C24—171no—
181C24C22×D1711yescharacteristic
182C2×D4—174no—
183C2×D4—342no—
184C2×D4—342no—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—171no—
188C22×C4—171no—
189C22×C4—172no—
190C22×C4—171no—
191C22×C4—174no—
192C22×C4—172no—
193C22×C4—342no—
194C22×C4—344no—
195C22×C4—342no—
196C22×C4—682no—
197C22×C4—682no—
198C22×C4C22×D1712yes—
199C22×C4C22×D1711yescharacteristic
200C22×C4C22×D1712yes—
201C22×C4C22×D1711yescharacteristic
202C4⋊C4—42no—
203C4⋊C4—344no—
204C4⋊C4—344no—
205C4⋊C4—342no—
206C4⋊C4—344no—
207C4⋊C4—682no—
208C22⋊C4—22no—
209C22⋊C4—22no—
210C22⋊C4—342no—
211C22⋊C4—344no—
212C22⋊C4—342no—
213C22⋊C4—682no—
214C42—342no—
215C23—22no—
216C23—22no—
217C23—172no—
218C23—341no—
219C23—341no—
220C23—342no—
221C23—342no—
222C23#6325212yes—
223C23D4×D1712yes—
224C23Q8×D1712yes—
225C23C23×D1711yescharacteristic
226D4—348no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—44no—
238C2×C4—172no—
239C2×C4—172no—
240C2×C4—172no—
241C2×C4—174no—
242C2×C4—348no—
243C2×C4—341no—
244C2×C4—341no—
245C2×C4—342no—
246C2×C4—342no—
247C2×C4—342no—
248C2×C4—342no—
249C2×C4—342no—
250C2×C4—342no—
251C2×C4—342no—
252C2×C4—684no—
253C2×C4—684no—
254C22—24no—
255C22—24no—
256C22—22no—
257C22—22no—
258C22—344no—
259C22—342no—
260C22—682no—
261C22—681no—
262C22—681no—
263C22#1058758214yes—
264C22C2×#6325211yescharacteristic
265C22C2×D4×D1711yescharacteristic
266C22#1058769112yes—
267C22C2×Q8×D1711yescharacteristic
268C22#41×D1712yes—
269C4—42no—
270C4—41no—
271C4—42no—
272C4—41no—
273C4—342no—
274C4—344no—
275C4—342no—
276C4—342no—
277C4—682no—
278C2—24no—
279C2—682no—
280C2C2×#1058758211yescharacteristic
281C2#1176331711yescharacteristic
282C2#1176336911yescharacteristic
283C2#1176350812yes—
284C2#1176351312yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#385C2
prime radicalC22×C22⋊D17C22
hypercenter#385D17
hypocenterC17#2579
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C22×#63225C2semidirect
3. #11762740C2semidirect
4. #11762733C2semidirect
5. #11762691C2semidirect
8. #11762681C2semidirect
9. #11762680C2semidirect
10. #11762667C2semidirect
11. C17×#385C2semidirect
145. #385D17semidirect
179. C17#2579semidirect

Maximal subgroups

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

subgroupindex
C2×#105875672
C22×#632252
#117627402
#117627332
#117626912
#117626882
#117626862
#117626812
#117626802
#117626672
C17×#3852
#257917