Finite Group Databasebeta

← #14534137

Subgroups — #14534137

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×Q8×Dic17C211yesmaximal, characteristic, complemented
2C2×#10587567C211yesmaximal, characteristic
3#11762746C211yesmaximal, characteristic, complemented
4#11762745C211yesmaximal, characteristic, complemented
5#11762720C211yesmaximal, characteristic, complemented
6#11762709C212yesmaximal
7#11762706C212yesmaximal, complemented
8#11762684C212yesmaximal, complemented
9#11762683C212yesmaximal, complemented
10#11762673C211yesmaximal, characteristic, complemented
11C17×#393C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C22×C22⋊D17C2211yescharacteristic
14C23×Dic17C2211yescharacteristic
15C22×C4×D17C2211yescharacteristic
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊Dic17C2212yes—
19Q8×Dic17—24no—
20C2×#63227C2212yes—
21C2×#63227C2211yescharacteristic
22C2×#63227C2212yes—
23C2×C4⋊Dic17C2211yescharacteristic
24C2×C4⋊Dic17C2212yes—
25C2×C4⋊Dic17C2212yes—
26C2×#63225C2211yescharacteristic
27C2×#63225C2212yes—
28C2×#63225C2212yes—
29C2×C4×Dic17C2212yes—
30C2×C4×Dic17C2211yescharacteristic
31#10587567—24no—
32C17×#95C2212yes—
33C17×#95C2211yescharacteristic
34#10587502C2212yes—
35#10587502C2211yescharacteristic
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C23×D17C2311yescharacteristic
39C2×Q8×C17—22no—
40C2×Q8×C17—24no—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2312yes—
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2312yes—
45C2×C22⋊D17—22no—
46C2×C22⋊D17—22no—
47C2×C22⋊D17Q814yescomplemented
48C22×Dic17—22no—
49C22×Dic17—42no—
50C22×Dic17C2311yescharacteristic
51C22×Dic17C2312yes—
52C22×Dic17C2312yes—
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2311yescharacteristic
55C2×C4×D17—22no—
56C2×C4×D17—42no—
57C22⋊C4×C17—22no—
58C22⋊C4×C17—24no—
59C22⋊Dic17—42no—
60#63227—22no—
61#63227—24no—
62#63227—42no—
63C4⋊Dic17—22no—
64C4⋊Dic17—24no—
65C4⋊Dic17—42no—
66#63225—22no—
67#63225—24no—
68#63225—42no—
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—21no—
79C22×D17—22no—
80C22×D17—22no—
81C22×D17C2×Q812yes—
82Q8×C17—28no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—21no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17C2×D412yes—
92C2×C4×C17#4114yes—
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—22no—
103C2×Dic17—44no—
104C2×Dic17C2×D412yes—
105C2×Dic17C2×D412yes—
106C2×Dic17C2×Q812yes—
107C2×Dic17#4112yes—
108C4×D17—44no—
109#2580—171nomaximal, Sylow
110C22×C17—22no—
111C22×C17—24no—
112C22×C17—41no—
113C22×C17—41no—
114C22×C17—42no—
115C22×C17C22×D411yescharacteristic
116C22×C17C22×Q811yescharacteristic
117C22×C17C2×#4112yes—
118C22×C17C2×#4111yescharacteristic
119C22×C17H(5,𝔽2)12yes—
120C2×D17—22no—
121C2×D17—24no—
122C2×D17—41no—
123C2×D17—41no—
124C2×D17—42no—
125C4×C17—22no—
126C4×C17—24no—
127C4×C17—41no—
128C4×C17—42no—
129Dic17—22no—
130Dic17—42no—
131Dic17—41no—
132Dic17—42no—
133Dic17C22⋊Q814yes—
134C2×C4×Q8—171no—
135C2×C4×D4—171no—
136#399—171no—
137#399—171no—
138#393—171no—
139#393D1711yescharacteristic, complemented
140#390—172no—
141#389—172no—
142#385—172no—
143#385—172no—
144#382—171no—
145C2×C17—42no—
146C2×C17C2×C22⋊Q811yescharacteristic
147C2×C17#52511yescharacteristic
148C2×C17#53511yescharacteristic
149C2×C17#55412yes—
150C2×C17#55912yes—
151D17—42no—
152C22×Q8C2×D1711yescharacteristic
153C22×D4—171no—
154C23×C4—171no—
155C23×C4—171no—
156C4×Q8—344no—
157C4×D4—344no—
158C2×C4⋊C4—171no—
159C2×C4⋊C4—172no—
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—172no—
173#95—171no—
174#95—172no—
175#95C2×D1711yescharacteristic
176#95C2×D1712yes—
177C17#258011yescharacteristic, complemented, Sylow
178C24—171no—
179C24C22×D1711yescharacteristic
180C2×Q8—22no—
181C2×Q8—24no—
182C2×D4—174no—
183C2×D4—342no—
184C2×D4—342no—
185C22×C4—171no—
186C22×C4—171no—
187C22×C4—172no—
188C22×C4—171no—
189C22×C4—172no—
190C22×C4—342no—
191C22×C4—342no—
192C22×C4—682no—
193C22×C4—682no—
194C22×C4C22×D1712yes—
195C22×C4C22×D1711yescharacteristic
196C22×C4C22×D1711yescharacteristic
197C22×C4C22×D1712yes—
198C4⋊C4—342no—
199C4⋊C4—344no—
200C4⋊C4—344no—
201C4⋊C4—342no—
202C4⋊C4—682no—
203C4⋊C4—682no—
204C22⋊C4—22no—
205C22⋊C4—24no—
206C22⋊C4—342no—
207C22⋊C4—344no—
208C22⋊C4—682no—
209C22⋊C4—682no—
210C42—174no—
211C42—344no—
212C23—21no—
213C23—22no—
214C23—21no—
215C23—22no—
216C23—172no—
217C23—341no—
218C23—342no—
219C23—341no—
220C23—342no—
221C23Q8×D1712yes—
222C23C23×D1711yescharacteristic
223Q8—28no—
224D4—348no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—172no—
234C2×C4—172no—
235C2×C4—172no—
236C2×C4—172no—
237C2×C4—342no—
238C2×C4—342no—
239C2×C4—341no—
240C2×C4—342no—
241C2×C4—342no—
242C2×C4—342no—
243C2×C4—342no—
244C2×C4—342no—
245C2×C4—342no—
246C2×C4—684no—
247C2×C4—684no—
248C2×C4D4×D1712yes—
249C2×C4#6325414yes—
250C22—24no—
251C22—22no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—344no—
256C22—342no—
257C22—682no—
258C22—681no—
259C22—681no—
260C22C2×D4×D1711yescharacteristic
261C22C2×#6325412yes—
262C22C2×Q8×D1711yescharacteristic
263C22#41×D1711yescharacteristic
264C22#1058769612yes—
265C4—22no—
266C4—24no—
267C4—42no—
268C4—41no—
269C4—174no—
270C4—342no—
271C4—682no—
272C4—681no—
273C4—682no—
274C2—42no—
275C2—682no—
276C2C22⋊Q8×D1711yescharacteristic
277C2#1176331511yescharacteristic
278C2#1176337711yescharacteristic
279C2#1176351712yes—
280C2#1176354112yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#393C2
prime radicalC22×C22⋊D17C22
hypercenter#393D17
hypocenterC17#2580
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic17C2semidirect
3. #11762746C2semidirect
4. #11762745C2semidirect
5. #11762720C2semidirect
7. #11762706C2semidirect
8. #11762684C2semidirect
9. #11762683C2semidirect
10. #11762673C2semidirect
11. C17×#393C2semidirect
47. C2×C22⋊D17Q8semidirect
139. #393D17semidirect
177. C17#2580semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic172
C2×#105875672
#117627462
#117627452
#117627202
#117627092
#117627062
#117626842
#117626832
#117626732
C17×#3932
#258017