Finite Group Databasebeta

← #14534130

Subgroups — #14534130

Order 2176.

584 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 161 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×D4×Dic17C211yesmaximal, characteristic
2C22×#63225C211yesmaximal, characteristic, complemented
3#11762743C212yesmaximal, complemented
4#11762732C212yesmaximal, complemented
5#11762692C212yesmaximal, complemented
6#11762689C211yesmaximal, characteristic
7#11762684C211yesmaximal, characteristic, complemented
8#11762682C211yesmaximal, characteristic, complemented
9#11762681C212yesmaximal, complemented
10#11762667C211yesmaximal, characteristic, complemented
11C17×#386C211yesmaximal, characteristic
12C22×D4×C17C2211yescharacteristic
13C23×C4×C17C2211yescharacteristic
14C23×Dic17C2211yescharacteristic
15C23×Dic17C2211yescharacteristic
16C23×Dic17C2211yescharacteristic
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊C4×C17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2211yescharacteristic
21C2×C22⋊Dic17C2211yescharacteristic
22C2×C22⋊Dic17C2211yescharacteristic
23D4×Dic17—24no—
24C2×C4⋊Dic17C2212yes—
25C2×C4⋊Dic17C2211yescharacteristic
26C2×#63225—24no—
27C2×#63225C2212yes—
28C2×#63225C2212yes—
29C2×#63225C2212yes—
30C2×#63225C2212yes—
31C2×C4×Dic17C2211yescharacteristic
32C17×#95C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C24×C17C2311yescharacteristic
39C2×D4×C17—22no—
40C2×D4×C17—22no—
41C2×D4×C17Q814yes—
42C22×C4×C17—22no—
43C22×C4×C17—42no—
44C22×C4×C17C2312yes—
45C22×C4×C17C2312yes—
46C22×C4×C17C2311yescharacteristic
47C22×Dic17—22no—
48C22×Dic17—24no—
49C22×Dic17—22no—
50C22×Dic17—42no—
51C22×Dic17—42no—
52C22×Dic17D414yes—
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2312yes—
55C22×Dic17C2311yescharacteristic
56C22×Dic17C2311yescharacteristic
57C22×Dic17C2311yescharacteristic
58C22×Dic17C2312yes—
59C22⋊C4×C17—22no—
60C22⋊C4×C17—42no—
61C22⋊Dic17—22no—
62C22⋊Dic17—22no—
63C22⋊Dic17—22no—
64C22⋊Dic17—24no—
65C4⋊Dic17—22no—
66C4⋊Dic17—24no—
67#63225—24no—
68#63225—24no—
69#63225—42no—
70#63225—42no—
71C4×Dic17—22no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—21no—
75C23×C17—22no—
76C23×C17—21no—
77C23×C17—22no—
78C23×C17C2×D412yes—
79C23×C17C2×Q812yes—
80C23×C17C2×Q812yes—
81C23×C17#4112yes—
82C23×C17C2411yescharacteristic
83D4×C17—28no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
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—44no—
93C2×C4×C17C2×Q812yes—
94C2×Dic17—28no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—21no—
100C2×Dic17—21no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—44no—
107C2×Dic17—44no—
108C2×Dic17C2×D412yes—
109C2×Dic17C2×D412yes—
110C2×Dic17#4114yes—
111#2579—171nomaximal, Sylow
112C22×C17—22no—
113C22×C17—22no—
114C22×C17—24no—
115C22×C17—24no—
116C22×C17—22no—
117C22×C17—24no—
118C22×C17—41no—
119C22×C17—41no—
120C22×C17—42no—
121C22×C17C22⋊Q814yes—
122C22×C17C22×D411yescharacteristic
123C22×C17C22×Q811yescharacteristic
124C22×C17C2×#4111yescharacteristic
125C22×C17C2×#4112yes—
126C22×C17H(5,𝔽2)12yes—
127C4×C17—22no—
128C4×C17—42no—
129C4×C17—42no—
130Dic17—24no—
131Dic17—22no—
132Dic17—22no—
133Dic17—41no—
134Dic17—41no—
135Dic17—42no—
136C2×C4×D4—171no—
137C22×C4⋊C4—171no—
138#398—172no—
139#396—172no—
140#387—172no—
141#386—171no—
142#386D1711yescharacteristic
143#385—171no—
144#385—172no—
145#385—171no—
146#380—171no—
147C2×C17—24no—
148C2×C17—42no—
149C2×C17C2×C22⋊Q811yescharacteristic
150C2×C17#52511yescharacteristic
151C2×C17#53411yescharacteristic
152C2×C17#55312yes—
153C2×C17#55412yes—
154C22×D4C2×D1711yescharacteristic
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—172no—
166C2×C4⋊C4—344no—
167C2×C22⋊C4—171no—
168C2×C22⋊C4—171no—
169C2×C22⋊C4—171no—
170C2×C22⋊C4—172no—
171C2×C22⋊C4C2×D1711yescharacteristic
172C2×C22⋊C4C2×D1712yes—
173C2×C42—171no—
174#95—172no—
175#95—172no—
176#95—172no—
177#95—172no—
178#95C2×D1712yes—
179C17#257911yescharacteristic, complemented, Sylow
180C24C22×D1711yescharacteristic
181C24C22×D1711yescharacteristic
182C2×D4—22no—
183C2×D4—22no—
184C2×D4Dic3414yes—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—171no—
188C22×C4—172no—
189C22×C4—171no—
190C22×C4—174no—
191C22×C4—172no—
192C22×C4—171no—
193C22×C4—171no—
194C22×C4—344no—
195C22×C4—342no—
196C22×C4—342no—
197C22×C4—682no—
198C22×C4—682no—
199C22×C4C22×D1712yes—
200C22×C4C22×D1712yes—
201C22×C4C22×D1711yescharacteristic
202C4⋊C4—344no—
203C4⋊C4—344no—
204C4⋊C4—342no—
205C4⋊C4—344no—
206C4⋊C4—682no—
207C4⋊C4—682no—
208C22⋊C4—22no—
209C22⋊C4—42no—
210C22⋊C4—344no—
211C22⋊C4—342no—
212C22⋊C4—342no—
213C22⋊C4—342no—
214C42—342no—
215C23—22no—
216C23—22no—
217C23—21no—
218C23—21no—
219C23—22no—
220C23—22no—
221C23C2×Dic3412yes—
222C23C2×Dic3412yes—
223C23D4×D1712yes—
224C23#6325412yes—
225C23C23×D1711yescharacteristic
226D4—28no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—44no—
236C2×C4—172no—
237C2×C4—172no—
238C2×C4—174no—
239C2×C4—348no—
240C2×C4—342no—
241C2×C4—342no—
242C2×C4—342no—
243C2×C4—342no—
244C2×C4—341no—
245C2×C4—341no—
246C2×C4—342no—
247C2×C4—342no—
248C2×C4—342no—
249C2×C4—342no—
250C2×C4—342no—
251C2×C4—684no—
252C2×C4—684no—
253C2×C4C2×Dic3412yes—
254C22—24no—
255C22—24no—
256C22—22no—
257C22—22no—
258C22—24no—
259C22—22no—
260C22—42no—
261C22—41no—
262C22—41no—
263C22#1058756414yes—
264C22C22×Dic3411yescharacteristic
265C22C2×D4×D1711yescharacteristic
266C22C2×#6325411yescharacteristic
267C22#1058769112yes—
268C22#41×D1712yes—
269C4—22no—
270C4—42no—
271C4—42no—
272C4—344no—
273C4—342no—
274C4—342no—
275C4—682no—
276C4—681no—
277C4—681no—
278C2—24no—
279C2—42no—
280C2C2×#1058756411yescharacteristic
281C2#1176331711yescharacteristic
282C2#1176336911yescharacteristic
283C2#1176350212yes—
284C2#1176351812yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#386C2
prime radicalC22×D4×C17C22
hypercenter#386D17
hypocenterC17#2579
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×#63225C2semidirect
3. #11762743C2semidirect
4. #11762732C2semidirect
5. #11762692C2semidirect
7. #11762684C2semidirect
8. #11762682C2semidirect
9. #11762681C2semidirect
10. #11762667C2semidirect
179. C17#2579semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic172
C22×#632252
#117627432
#117627322
#117626922
#117626892
#117626842
#117626822
#117626812
#117626672
C17×#3862
#257917