Finite Group Databasebeta

← #14539827

Subgroups — #14539827

Order 2176.

420 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 109 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
1#11763413C211yesmaximal, characteristic, complemented
2#11763372C211yesmaximal, characteristic, complemented
3#11763196C211yesmaximal, characteristic, complemented
4#11763173C211yesmaximal, characteristic, complemented
5#11763164C211yesmaximal, characteristic, complemented
6#11763022C212yesmaximal, complemented
7#11763002C212yesmaximal, complemented
8#11762951C212yesmaximal, complemented
9#11762941C212yesmaximal, complemented
10#11762868C211yesmaximal, characteristic, complemented
11C17×#493C211yesmaximal, characteristic, complemented
12C2×#63254C2211yescharacteristic
13C2×D4×D17C2211yescharacteristic
14C2×D8×C17C2212yes—
15C2×D8×C17C2212yescomplemented
16C17×#127C2211yescharacteristic
17C17×#127C2211yescharacteristic, complemented
18#10587626C2211yescharacteristic
19#10587625—22no—
20#10587625—22no—
21#10587624C2212yes—
22#10587623—22no—
23#10587622C2211yescharacteristic
24#10587621—22no—
25D4×Dic17C2212yescomplemented
26C2×#63229C2212yescomplemented
27C2×#63228C2212yes—
28#10587561C2211yescharacteristic
29#10587558C2211yescharacteristic
30#10587556C2211yescharacteristic, complemented
31C4×C8×C17C2211yescharacteristic, complemented
32#10587503C2212yes—
33#10587503C2212yescomplemented
34#10587492C2212yes—
35#10587491C2212yes—
36#10587487C2212yescomplemented
37#10587485C2212yescomplemented
38#10587474C2211yescharacteristic, complemented
39C23×D17—21no—
40C2×D4×C17—22no—
41C2×D4×C17—22no—
42C2×D4×C17C2311yescharacteristic
43C2×D4×C17C2311yescharacteristic
44C2×D4×C17C2312yes—
45C2×C22⋊D17—21no—
46C2×C22⋊D17—22no—
47C2×C22⋊D17—21no—
48C2×C22⋊D17—22no—
49C22×Dic17—21no—
50C22×Dic17—22no—
51#63254—42no—
52D4×D17—42no—
53C2×D68C2311yescharacteristic
54C2×C4×D17C2311yescharacteristic
55C2×Dic34C2311yescharacteristic
56D8×C17—42no—
57D8×C17—42no—
58C2×C8×C17C2312yes—
59C42×C17C2311yescharacteristic
60C22⋊Dic17—22no—
61C22⋊Dic17—22no—
62C22⋊Dic17—22no—
63#63229—42no—
64#63228—42no—
65#63227—22no—
66#63227D412yes—
67C4⋊Dic17C2312yes—
68#63225—22no—
69#63225D412yes—
70C4×Dic17C2311yescharacteristic
71C2×#3622C2312yes—
72C23×C17—22no—
73C23×C17—21no—
74C23×C17—21no—
75C22×D17—21no—
76C22×D17—42no—
77C22×D17—82no—
78C22×D17C2×D411yescharacteristic
79D4×C17—22no—
80D4×C17—22no—
81D4×C17—22no—
82D4×C17—22no—
83D4×C17—82no—
84D4×C17—82no—
85C2×C4×C17C2×D412yes—
86C2×C4×C17C2411yescharacteristic
87C22⋊D17—42no—
88C22⋊D17—42no—
89C22⋊D17—82no—
90C22⋊D17—82no—
91C2×Dic17—21no—
92C2×Dic17—22no—
93C2×Dic17—42no—
94C2×Dic17—82no—
95C2×Dic17C2×D411yescharacteristic
96C2×Dic17C2×D412yes—
97D68—22no—
98C4×D17—22no—
99Dic34—22no—
100C8×C17—22no—
101#3622—22no—
102#3220—171nomaximal, Sylow
103C22×C17—41no—
104C22×C17—41no—
105C22×C17—42no—
106C22×C17—42no—
107C22×C17—42no—
108C22×C17—42no—
109C22×C17—42no—
110C22×C17C22×D411yescharacteristic
111C2×D17—21no—
112C2×D17—22no—
113C2×D17—41no—
114C2×D17—42no—
115C2×D17—84no—
116C4×C17—22no—
117C4×C17H(5,𝔽2)12yes—
118Dic17—21no—
119Dic17—22no—
120Dic17—41no—
121Dic17—42no—
122#540—171no—
123#535—171no—
124#496—171no—
125#493D1711yescharacteristic, complemented
126#490—171no—
127#488—171no—
128#463—172no—
129#459—172no—
130#449—172no—
131#447—172no—
132#433—171no—
133C2×C17—81no—
134C2×C17—81no—
135C2×C17—82no—
136C2×C17#53511yescharacteristic
137C2×C17#57612yes—
138D17—41no—
139D17—81no—
140C2×#41—171no—
141C22×D4—171no—
142C2×SD16—172no—
143C2×D8—172no—
144C2×D8C2×D1712yes—
145C2×D8C2×D1712yescomplemented
146#127—171no—
147#127C2×D1711yescharacteristic
148#127C2×D1711yescharacteristic, complemented
149#125—171no—
150#124—171no—
151#124—171no—
152#123—342no—
153#121—172no—
154#121—342no—
155#121—342no—
156#120—342no—
157C4×D4—172no—
158#117—171no—
159C4⋊C8—172no—
160#103—172no—
161#102—172no—
162#102—172no—
163#102—172no—
164C22⋊C8—172no—
165#97—171no—
166C4×C8C2×D1711yescharacteristic, complemented
167C17#322011yescharacteristic, complemented, Sylow
168C24—341no—
169#41—682no—
170C2×Q8—171no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—171no—
174C2×D4—341no—
175C2×D4—342no—
176C2×D4—341no—
177C2×D4—342no—
178C2×D4—682no—
179C2×D4C22×D1711yescharacteristic
180C2×D4C22×D1711yescharacteristic
181C2×D4C22×D1712yes—
182C22×C4—171no—
183C22×C4—341no—
184C22×C4—342no—
185SD16—682no—
186D8—42no—
187D8—42no—
188D8—682no—
189C2×C8—172no—
190C2×C8C22×D1712yes—
191C4⋊C4—172no—
192C4⋊C4—172no—
193C4⋊C4—342no—
194C22⋊C4—172no—
195C22⋊C4—342no—
196C22⋊C4—342no—
197C22⋊C4—342no—
198C22⋊C4—342no—
199C42—171no—
200C42C22×D1711yescharacteristic
201C23—22no—
202C23—21no—
203C23—21no—
204C23—171no—
205C23—341no—
206C23—682no—
207C23—1362no—
208Q8—342no—
209D4—22no—
210D4—22no—
211D4—22no—
212D4—22no—
213D4—82no—
214D4—82no—
215D4—342no—
216D4—682no—
217D4—682no—
218D4—1362no—
219D4—1362no—
220C2×C4—172no—
221C2×C4—171no—
222C2×C4—342no—
223C2×C4—341no—
224C2×C4—342no—
225C2×C4—682no—
226C2×C4—1362no—
227C2×C4D4×D1712yes—
228C2×C4C23×D1711yescharacteristic
229C8—22no—
230C8—342no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—41no—
236C22—41no—
237C22—42no—
238C22—342no—
239C22—341no—
240C22—682no—
241C22—681no—
242C22—1364no—
243C22C2×D4×D1711yescharacteristic
244C4—22no—
245C4—342no—
246C4—341no—
247C4—682no—
248C4—681no—
249C4#1058769112yes—
250C2—81no—
251C2—82no—
252C2—81no—
253C2—681no—
254C2—1361no—
255C2#1176337211yescharacteristic
256C2#1176361012yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#493C2
socleC22×C17C22×D4
hypercenter#493D17
hypocenterC17#3220
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #11763413C2semidirect
2. #11763372C2semidirect
3. #11763196C2semidirect
4. #11763173C2semidirect
5. #11763164C2semidirect
6. #11763022C2semidirect
7. #11763002C2semidirect
8. #11762951C2semidirect
9. #11762941C2semidirect
10. #11762868C2semidirect
11. C17×#493C2semidirect
15. C2×D8×C17C22semidirect
17. C17×#127C22semidirect
25. D4×Dic17C22semidirect
26. C2×#63229C22semidirect
30. #10587556C22semidirect
31. C4×C8×C17C22semidirect
33. #10587503C22semidirect
36. #10587487C22semidirect
37. #10587485C22semidirect
38. #10587474C22semidirect
125. #493D17semidirect
145. C2×D8C2×D17semidirect
148. #127C2×D17semidirect
166. C4×C8C2×D17semidirect
167. C17#3220semidirect

Maximal subgroups

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

subgroupindex
#117634132
#117633722
#117631962
#117631732
#117631642
#117630222
#117630022
#117629512
#117629412
#117628682
C17×#4932
#322017